EPA-650/4-74-038
October 1974 Environmental Monitoring Series
PROCEEDINGS
OF THE SYMPOSIUM
ON STATISTICAL ASPECTS
OF AIR QUALITY DATA
Meteorology Laboratory
National Environmental Research Center
Office of Research and Development
U.S. Environmental Protection Agency
Research Triune, It Park, N.C. 27711
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Research reports of the Office of Research and Development, Environmental
Protection Agency, have been grouped into five series. These five broad
categories were established to facilitate further development and application
of environmental technology. Elimination of traditional grouping was
consciously planned to foster technology transfer and a maximum interface
in related fields. The five series are:
1. Environmental Health Effects Research
2 . Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Soci©economic Environmental Studies
This report has been assigned to the ENVIRONMENTAL MONITORING series.
This series describes research conducted to develop new or improved
methods and instrumentation for the identification and quantification of
environmental pollutants at the lowest conceivably significant concentrations.
It also includes studies to determine the ambient concentrations of pollutants
in the environment and/or the variance of pollutants as a function of time
or meteorological factors.
Copies of this report are available free of charge to Federal employees,
current contractors and grantees, and nonprofit organizations - as supplies
permit - from the Air Pollution Technical Information Center, Environmental
Protection Agency, Research Triangle Park, North Carolina 27711. This
document is also available to the public for sale through the Superintendent
of Documents, U. S. Government Printing Office, Washington, D. C. 20402.
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EPA-650/4-74-038
PROCEEDINGS
OF THE SYMPOSIUM
ON STATISTICAL ASPECTS
OF AIR QUALITY DATA
Editor:
Dr. Lawrence D. Kornreich
Executive Director, Triangle Universities
Consortium on Air Pollution
Symposium Sponsors:
Meteorology Laboratory, National Environmental Research Center
and
Triangle Universities Consortium on Air Pollution
Contract No. 68-02-0994
(with University of North Carolina, Chapel Hill)
ROAPNo. 21ADO
Program Element No. 1AA009
EPA Project Officer: Dr. Ralph Larsen
Prepared for
U.S. ENVIRONMENTAL PROTECTION AGENCY
Office of Research and Development
National Environmental Research Center
Research Triangle Park, N.C. 27711
October 1974
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This report has been reviewed by the Environmental
Protection Agency and approved for publication. Ap-
proval does not signify that the contents necessarily
reflect the views and policies of the Agency, nor
does mention of trade names or commercial products
constitute endorsement or recommendation for use.
Publication No. EPA-650/4-74-038
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PREFACE
The symposium on Statistical Aspects of Air Quality was held on November
9 and 10, 1972, at the Carolina Inn, in Chapel Hill, N. C., in accordance with the
terms of a contract between the Division of Meteorology* of the U. S. Environ-
mental Protection Agency (EPA) and the University of North Carolina at Chapel
Hill (UNC).
Although UNC was the contractor, it was agreed that the symposium would
be sponsored by the Triangle Universities Consortium on Air Pollution
(TUCAP), an association of Duke University, North Carolina State University
and tne University of North Carolina at Chapel Hill. The project officer for
EPA was Mr. Charles R. Hosier; the responsible officer for TUCAP was Dr.
Lawrence D. Kornreich. The detailed planning for the symposium was done by a
steering committee representing both TUCAP and EPA.
All papers that were presented at the symposium are included in this vol-
ume. Most of the technical papers were reprinted and distributed to the partici-
pants prior to the meeting. An open discussion followed the presentation of each
technical paper, and the questions and answers were recorded and transcribed.
Each discussant was given the opportunity to review and edit his comments.
Only those comments which were reviewed and approved by the discussants
appear in this volume.
For their outstanding performances at the banquet, special thanks are due
Mr. Donald Pack of the National Oceanographic and Atmospheric Administra-
tion, who acted as Toastmaster, and Dr. Roy Kuebler of the UNC Department of
Biostatistics who was the featured speaker.
The registration of participants and preparation of information packets was
ably handled by Continuing Education and Field Service of the U,NC School Of
Public Health. The audio-visual arrangements throughout the meeting were ca-
pably managed by Mr. Lewis Kontnick and Mr. E. James Dale, graduate students
in the air pollution curriculum at UNC.
I am especially grateful to Professor Arthur C. Stern of the UNC
Department of Environmental Sciences for his help and guidance in preparing
this volume. I also want to thank my secretaries for their outstanding
service—Mrs. Jean Lang during the planning and holding of the symposium, and
Mrs. Ann Harrell during the preparation of this volume.
Lawrence D. Kornreich
Chapel Hill, N. C.
February, 1974
*Now the Meteorology Laboratory of the National Environmental Research Center, Re-
search Triangle Park, N. C.
Ill
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STEERING COMMITTEE
Kenneth L. Calder, Chief Scientist, Division of Meteorology, Environmental Pro-
tection Agency.
Kenneth R. Knoerr, Associate Professor, Biometeorology, Duke University.
Lawrence D. Kornreich, Executive Director, Triangle Universities Consortium on
Air Pollution
Ralph I. Larsen, Environmental Research Engineer, Division of Meteorology,
Environmental Protection Agency.
Arthur C. Stern, Professor of Air Hygiene, University of North Carolina at
Chapel Hill
Allen H. Weber, Associate Professor, Geosciences, North Carolina State
University
SESSION CHAIRMEN
Irving Singer, Smith-Singer Meteorologists
Donald Rote, Argonne National Laboratory
James Arvesen, Purdue University
Arnold Court, California State University
George Tiao, University of Wisconsin
Ray Wanta, Consulting Meteorologist
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WORKSHOP LEADERS
Ralph Larsen, Division of Meteorology, U. S. Environmental Protection Agency
Donald McNeil, Princeton University
Warren Johnson, Division of Meteorology, U. S. Environmental Protection
Agency
Victor Hasselblad, Human Studies Laboratory, U. S. Environmental Protection
Agency
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CONTENTS
/. Robert A. McCormick 1-1
Welcoming Remarks
2. Arthur C. Stern 2-1
Keynote Address: Statistical Analysis of Air Quality Data
3. Frank A. G if ford. Jr. 3-1
The Form of the Frequency Distribution of Air Pollution
Concentrations
4. F. K. Wipperman 4-1
Meteorological Parameters Relevant in a Statistical Analysis
of Air Quality Data
5. Michael Benarie 5-1
The Use of the Relationship between Wind Velocity and
Ambient Pollutant Concentration Distributions for the
Estimation of Average Concentrations from Gross
Meteorological Data
6. Richard E. Barlow and Nozer D. Singpurwalla 6-1
Averaging Time and Maxima for Dependent Observations
7. Allan H. Marcus 7-1
A Stochastic Model for Estimating Pollutant Exposure by
Means of Air Quality Data
8. Harold E. Neustadter, Steven M. Sidik and John C. Burr, Jr. 8-1
Evaluating Conformity with Two-Point Air Quality Standards,
Polludex
VI
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9. Joseph B. Knox and Richard I. Pollack 9-1
An Investigation of the Frequency Distributions of Surface
Air-Pollutant Concentrations
10. D. Bruce Turner 10-1
Air Quality Frequency Distributions from Dispersion Models
Compared with Measurements
/ 7. Bernard E. Saltzman 11-1
Fourier Analysis of Air Monitoring Data
12. F. Barry Smith and G. H. Jeffrey 12-1
The Prediction of High Concentrations of Sulfur Dioxide in
London and Manchester Air
13. David A. Lynn 13-1
Fitting Curves to Urban Suspended Particulate Data
14. Joseph R. Visa/li, David L. Brenchley and Howard Reiquam 14-1
A Proposed Ambient Air Quality Sampling Strategy and
Methodology for the Design of Surveillance Networks
15. Yuji Horie and John H. Overton 15-1
The Effect on Rollback Models Due to Distribution of Pollutant
Concentrations
16. Symposium Participants 16-1
17. Bibliographic Datasheet 17-1
VII
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1. WELCOMING REMARKS
ROBERT A. McCORMICK*
Division of Meteorology
National Environmental Research Center
Environmental Protection Agency
Research Triangle Park, North Carolina
On behalf of the National Environmental Research Center in the Research
Triangle Park, North Carolina, and in particular the Division of Meteorology who
have actually sponsored the Symposium, I would like to warmly welcome you
all and say how pleased we are that so many distinguished investigators have
found time to attend, especially those from overseas.
Before Professor Stern's opening remarks, I would like to say a few words as
to why we in the Division of Meteorology (DMT)** wereianxious to support this
Symposium. Following the pioneering efforts of Fran Pooler and Bruce Turner,
our primary efforts have been in the development of source-oriented
diffusion-type models. Because of their source-oriented structure, they have
potentially wide application in the consideration of the effects on air quality of
hypothetical and arbitrary emission control strategies. They thus provide a
rational basis for air quality management through control of selected sources of
pollution. At the present time the development and improvement of such air
quality simulation models has the highest priority of all items in the research
program of the DMT.
In contrast are those models that primarily involve some form of statistical
regression analysis and that depend entirely on the availability of extensive
meteorological and air quality data for a particular urban location. Over the
years these statistical approaches have become increasingly sophisticated and
now include such things as "multiple-discriminant analysis", "empirical
orthogonal functions", "factor analysis", and most recently "computerized
adaptive pattern classification". Although these developments have had useful
applications to specific problems, the fact that they are receptor- rather than
source-oriented and do not involve any explicit input of information concerning
pollution emissions, so far makes them not applicable to comparative studies of
control strategies. This is the reason for their lower priority in our research
program as compared with the source-oriented dispersion-type models. I have
intentionally said "so far" in the preceding as it seems to us that a question
*On assignment from the National Oceanic and Atmospheric Administration, U. S. Depart"
ment of Commerce at the time of the Symposium. Mr. McCormick has since retired from
Federal service.
**Now the Meteorology Laboratory.
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which ought to be resolved is whether some form of statistical predictive scheme
could be evolved that would incorporate hypothetical changes in the pollution
emissions distribution without any direct use of meterological diffusion theory.
Such a model would, of course, then possess the desirable source-oriented
property.
By analogy with the striking developments of the last decade or so in
turbulent fluid mechanics, we can hope that advances and improved
understanding of air quality simulation will result from a marriage of statistical
techniques with more precise physical formulations of the problems. It was this
strong feeling that suggested the need for the present Symposium. Perhaps this
will stimulate a stronger interaction between some of the more meteorologically
inclined of our air quality modelers and non-meteorological statistical experts.
No less important is the hope that the workshop sessions will more clearly define
for our- air quality modeling fraternity, those areas and approaches where
improved cooperation between meteorologist and statistician might be most
helpful and fruitful in the immediate future, in much the same manner as the
clarification now being achieved between the meteorologists and atmospheric
chemists.
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Keynote Address
2. STATISTICAL ANALYSIS OF AIR QUALITY DATA
ARTHUR C. STERN
Department of Environmental Sciences and Engineering
University of North Carolina
Chapel Hill, North Carolina
Introduction
The need for a Symposium on Statistical Analysis of Air Quality Data arises
from a quite diverse assortment of challenges to our understanding of the
meaning of air quality data now coming into sharper focus. A closer look at
these challenges will give a better understanding of some of the answers we are
seeking at this Symposium.
Much of the research effort in the field of air pollution is and has been
directed to the development of air quality criteria, and through them to the
establishment of air quality standards. Since air quality criteria are explorations
of the relationship between levels of air quality and the adverse effects found in
receptors exposed to these levels, it is essential that there be precise description
of both these levels and their associated adverse effects. It is not the function of
this Symposium to discuss the precise description of the adverse effects, but it is
our function to discuss the precise description of the levels of exposure that
cause them.
Simple Chamber Atmospheres
Our simplest task is to describe the exposure in a chamber in which one or
more receptors are being experimentally exposed. These receptors may variously
be materials specimens, such as textiles, paper or leather; vegetation, such as
plants, lichens, or bacteria; animals, such as monkeys, guinea pigs, or mice; or
human volunteers. However, even in this most simple situation an adequate
description of the quality of the air in the chamber must reflect the variance in
the system generating the chamber atmosphere; the decay in contaminant level
in the chamber due to wall effects and absorption by chambers contents, such as
the fur of exposed animals, their urine and feces, and cyclic response associated
with the effect of activity on animal uptake and light-intensity on plant uptake.
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Complex Chamber Atmospheres
Description of the level of exposure becomes more complicated when
long-term exposure is adjusted to the working hours of the experimenters, as
when exposures are for 8 hours a day, 5 days a week, with occasional hours or
days lost due to staff holidays, equipment maintenance or malfunction, or other
causes.
A still higher level of skill is needed to describe the level that results when
the experimenter, instead of attempting to maintain a constant level,
deliberately attempts to simulate in the chamber some of the changes in air
quality level observed in the ambient air. This is a problem with which some of
my colleagues at UNC-Chapel Hill are now having to cope in connection with the
introduction of reactants into a 12,000 cubic foot, naturally irradiated "Teflon"
chamber under construction a few miles from here. The intent is to introduce
the reactants at a rate that will simulate what happens in the ambient air of a
community such as Los Angeles on a typical weekday morning. Precise
description of the quality of the contents of the chamber will be as complex as
describing the quality of ambient air.
In effects research, be it of materials, plants, animals, or humans, one of the
things we most need to know is the relative influence on the adverse effects
observed of short-duration, high-concentration spikes superimposed on long
sustained average levels. Very few, if any, chamber experiments attempt this
type of superimposition and raise the problem of air quality description which it
would impose. The importance of this kind of information arises because we
need to know in which classes of receptors adverse effects are proportional to
integrated dose, and in which classes protective or defense mechanisms are
inhibited by short-duration, high-concentration spikes so that adverse effect is
more than proportional to integrated dose.
Ambient Air Quality Data for Specific Effects
As great as is the challenge of providing precise description of exposure
chamber atmospheres, ev^n more challenging is the task of precisely describing
the ambient air. Air quality criteria development also requires that the exposure
of materials, livestock, forests, crops, and human populations to the ambient air
be described during both short-term episodic conditions and for long periods of
time, up to the lifetimes of viable receptors. Here we have three situations of
increasing complexity. The first, and least complex, is providing a description of
air quality at a fixed location in the field where material specimens are being
exposed; test crops are being grown; or animals are being maintained under
observation and air quality is being monitored.
Next in level of complexity is providing a description of the quality of the
air associated with a specific observed adverse effect occurring at other than a
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fixed experimental location, as was the former case. Typical situations are:
(a) Gradual or sudden increase in clinic or hospital admissions for asthmatic
attack, respiratory or cardiac disease.
(b) Gradual or sudden awareness of specific damage to trees, crops, or
livestock.
(c) Gradual or sudden awareness of specif ic damage to materials.
In such situations, it is rare that air was being monitored at the precise
location where the person who was hospitalized lived, the damaged crop was
being grown, or the damaged material was in use. However, part of the reason
why we are here today is to better define the kinds of air quality measurements
that must be made to provide air quality data to relate to these types of
situations. A closely related reason for our Symposium is the as-yet-unresolved
role of air quality measurement in the prediction and management of short-term
air pollution episodes. There is no question but that we need good description of
air quality to understand what has happened and is happening during air
episodes. However, the extent that air quality data can be used to forecast the
occurrence or course" of an episode is still not clear and is an area into which a
group such as this should be able to provide some insight.
Ambient Air Quality Data for Epidemiological Studies
Our most difficult and imperative task in the statistical analysis of air
quality data is to provide descriptors of air quality that are meaningful for
understanding epidemiological data on human mortality and morbidity, since
this is the description of air safe tp breathe. We need to know where to measure,
what to measure, how frequently to measure, and how to analyze and interpret
the data we measure.
Once air quality criteria are endowed with meaningful descriptions of air
quality, we are in a position to select some of these descriptors as air quality
standards. To date, we have chosen such descriptors rather sparingly, using only
averaging time and simple statements of frequency of occurrence. Can you
supply better ones?
Federal Air Quality Standards for CO, HC, and NO2
So much for the generalities. Now let's get down to a few examples of
specifics—to some real problems created by the way air quality data and air
quality standards are presently described. Among these problems have been the
descriptors used for the Federal Air Quality Standards for CO, HC and IMO2.
First letfs look at the CO standard of 9 ppm (8-hour average), not to be
exceeded more than once a year. This is ambiguous since it does not specify
which 8-hour period to use. There are an infinite number of possible 8-hour
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running average values if the running averages may start randomly, not
necessarily on the hour, at any instant of time during the year. In practice, to
determine compliance with this national standard, data must be organized. Since
there is also a 1-hour average national standard, the unit of time apparently
intended was the clock hour rather than any randomly started period of 60
minutes. For clock hour data, the 8-hour average possibilities start with the use
of one specific 8-hour period (e.g. 8 a.m. -4 p.m.) per calendar day. Since there
are 365 such periods possible per year, if one period exceeds the standard, the
standard represents the 99.7 percentile value. There are 1095 possible
values. There are 1095 possible non-overlapping 8-hour periods per calendar year,
and between 8752 and 8760 possible running 8-hour periods, which if similarly
used would set the standard at the 99.9 and 99.99 percentiles, respectively. It is
thus unclear whether the 99.7, 99.9 or 99.99 percentile values were intended as
the standard. The problem for the person who must establish compliance with
the National standard is then to determine which of these possibilities was
intended and is acceptable.
Next let's look at the N02 standard of 0.05 ppm—annual average. In order
to relate this value meaningfully to the National standard for hydrocarbons
(non-methane) of 0.24 ppm (3-hour average—6-9 a.m.), and to the NOX
reduction required in automobiles, it is necessary to convert the Federal NO2
standard to its equivalent 3-hour average NOX value. This requires a double
conversion—first of annual average NO2 to equivalent 3-hour average NO2then
to equivalent 3-hour average NOX. The hydrocarbon standard with which this
latter value must be considered is for non-methane hydrocarbons, while the
National automobile emission standard which is intended to achieve it, is for
hydrocarbons including methane. Finally the National oxidant standard that is
intended to be achieved by the control of HC (3-hour average) and NO2 (annual
average) is expressed as a 1-hour average. We hope that this Symposium will help
provide more rational bases for understanding and expressing air quality data
and air quality standards.
Conclusion
In October, 1969, we ran a predecessor to this Symposium, one on Multiple-
Source Urban Diffusion Models. There are several important ties between these
two symposia. Both were under the same joint sponsorship of the Division of
Meteorology (EPA) and the Triangle Universities Consortium on Air Pollution.
Some of the areas opened up at the 1969 symposium form the basis for research
papers at this one. Finally as diffusion modeling comes of age, its requirements
for real air quality data for model calibration and validation become increasingly
important. Some of the air quality data analysis techniques discussed here
should help improve the quality of diffusion models just as diffusion modeling
should help improve the analysis, tabulation and presentation of air quality data.
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One can envision a combined air quality monitoring-air quality modeling effort
in which more accurate air quality data for a community can be obtamed at
lower cost in monitoring equipment and manpower, by using the model to fill in
the data where there are no monitoring stations, and using the monitoring"
stations to calibrate and validate the model. In the past, air quality monitoring
and modeling have been considered quite separate and disparate operations.
What is proposed is that they can be operated as a joint enterprise with benefit
to both aspects.
If this Symposium can point the way to these kinds of interactions, we may
well be planting the seed from which another symposium on such interactions
may arise a few years hence, just as this one arose from the seeds planted in 1969.
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3. THE FORM OF THE FREQUENCY DISTRIBUTION
OF AIR POLLUTION CONCENTRATIONS
FRANK A. GIFFORD
Air Resources Atmospheric Turbulence & Diffusion Laboratory
National Oceanic & Atmospheric Administration
Oak Ridge, Tennessee
Introduction
The practical need to be able to estimate the frequency distribution of air
pollution concentrations doesn't have to be pointed out to the participants in
this symposium. It will I'm sure be emphasized in many of the papers that we
will hear. Instead, I'm going to try to bring out some of the possible reasons for
the concentration frequency distributions that we observe. Along the way I hope
to emphasize the basic difference between the frequency distribution of urban
air pollution, which results from the combined effects of many sources, and that
of concentrations from a single, isolated source of pollution, like an electric
power plant. I'll also mention several different proposals that have been given in
the literature for the mathematical form of these distributions, and will try to
bring out some relationships among several of these.
The Lognormal Distribution of Urban Air Pollution
Larsen (1970; 1971) has established, by means of a large number of data
comparisons, that observed air pollution concentration distributions are closely
approximated by the lognormal function. This is an interesting fact about air
pollution, which calls for some kind of explanation.
The concentration X due to an urban air pollutant, the ambient air quality
in other words, is ordinarily observed over successive, short, time intervals. A
record of X at an air pollution sampling point consists of a series of observed
values, Xj, i = 0,1,2, . . . n. The irregular change in these numbers from one
sampling interval to the next reflects all the obviously complex variability of
source, meteorological, and other factors. But at any time it will most strongly
depend on the existing air pollution concentration level. This is true for at least
two reasons. First, relevant meteorological factors, principally the wind speed
and direction, tend to be strongly self-correlated. Second, urban air pollution is
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more uniform than that from isolated rural sources because the urban source is
distributed more uniformly, over a very large area.
This suggests that the X; are generated by the following simple process:
(1)
X| = X0 + y,X0| X2 = X,+y2X|; , Xn=Xn_, + ynXn_,
The quantities y-t are specified to be irregular, stochastic variables, known only in
terms of their means and standard deviations. Nothing more definite than this
needs to be said about the y-t, but the implication is that yj results from a large
number of small, irregular effects acting on Xj.1f such as brief shifts in the wind,
changes in traffic, and so on. The lognormal distribution of X follows directly.
From Equation 1 it is seen by rearranging that
2 y; = zr=2 (Xj-Xi-|)/Xj_, (2)
i=l i=l
According to the central limit theorem, zn, the sum of n independent stochastic
variates, will be normally distributed for large n; but Equation 2 is equivalent to
Xn
zn= f X"' dX = ln(Xn/X0) (3)
X0
That is, Xn has a lognormal distribution. The conditions of validity of the central
limit theorem are very general. The y-t do not have to be normally distributed, or
even to have the same distribution.
This derivation of the lognormal distribution is just a particularization of
the standard explanation of how skew distributions are generated; see for
example Hald (1952). Since the cause of the irregular changes in yj did not have
to be specified exactly, the derivation only gives the* form and not the pa-
rameters of the distribution. Nevertheless it seems to be an adequate explanation
of the observed strong tendency towards Ibgnormality of air pollution concen-
tration distributions.
The Distribution of Concentration from an Isolated Point Source of
Pollution
In some contrast to the above simple, but rather general result, I proposed a
specific model of the frequency distribution of concentrations from an isolated
point source (Gifford (1959)). See also Scriven (1965). In this model, the
concentration Xp at a point (x, y, z) due to a fluctuating plume of contaminant
originating at (0, 0, 0) is given bv a randomly positioned, spreading-disk,
Gaussian plume mpdel:
1 txp-[(y-Dy)2/
(z-Dz)2/2crz2]
(4)
/ - »•---«, j
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Q is the source strength, U is the (constant) mean wind speed, and D and D2
are the fluctuating distances of the center of the instantaneous plume from the
mean plume axis (x, 0, 0). D and Dz are assumed to be normally distributed,
and ay and az are the standard deviations of the instantaneous plume spreading.
If the new variables
Y=(y-Dy)/(2o-y2)l/2, and Z = (z-Dz)/( 2<7Z2)"2
are defined, it follows from Equation 4 that
L = Y2 + Z2= -ln(cXp/Q) (5)
where _ ,,
C = 27T CTy CTZ U
For suitably standardized values of the variables, and for concentrations
measured on or near the mean plume axis, it was demonstrated that the
distribution of L is p (|_) = e-L/2/2 (6)
Thus the logarithm of concentration from a single source is distributed
exponentially, which is the same as chi-square with two degrees of freedom. The
degrees of freedom correspond to the two directions, y and z, into which the
plume fluctuations are resolved by the model. The mathematical form of the
dis'tribution for sampling points off the mean axis turns out to be considerably
more complicated. However the parameters of the distribution, the mean and
standard deviation, are given explicity in terms of the plume parameters. See the
references for details.
Gartrell (1966) pointed out that the distribution of concentrations from an
isolated source is qualitatively different from that due to urban air pollution.
The most probable concentration from an isolated source is clearly zero, and the
distribution is strongly skewed. Gartrell found that a semi logarithmic diagram, in
which concentration is plotted as the logarithm of frequency, yielded a good
linear correlation of large amounts of TVA SC>2, i.e., essentially isolated
point-source, data. Urban air pollution distributions are, on the other hand,
flatter, and richer in the low concentration range with a higher modal value; high
concentrations are less frequent.
More recently, Barry (1971) has also plotted extensive amounts of argon-41
data semilogarithmically and shows an excellent fit to the semi logarithmic
distribution
bXn
P(Xp) = oe P (?)
The sources of these concentrations are isolated, tall stacks. Since there is little
or no background contamination, these give the isolated point source in a
particularly clear-cut example. He and Scriven (1971) discussed this result,
concluding that because of the simplicity of Equation 7 and the high quality of
the agreement with data, this empirical, semilogarithmic distribution is to be
preferred to Equation 6. Actually the two distributions refer to different
quantities.
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The mean wind is assumed to be constant in Equation 6, over a time period of
the order of an hour, whereas in Equation 7 all wind fluctuations are included.
For this reason Equation 7 is a more practically useful result. There must be
some fairly direct relationship between the two results, but so far it has not been
found.
The Distribution from n Point Sources
If there are a large number, n, of sources of a pollutant whose plumes affect
a particular point, the i-th one will make a contribution to the concentration at
the point given by
Xpj/Qj = UTTcrjyCrjzUr1 exp{-[(yj-Dyj)2/ 2cryj2 +
(8)
(Zj-DZJ)2/2crZi]}
from Equation 4. Following the same procedure,
-ln(CjXpj/Qj) = Yj2+zf (9)
Summing over n such sources gives
n p
-L=ln n CjXpj/Qj = 2 (Yj2 + Zj2) (10)
If the quantities Dyi, Dzj are independently normally distributed with mean
= 0 and a = (D2)1/2, then the quantities (yj-Dyj)2 and (Zj-Dzj)2 are also
normally distributed, with mean = yi/(2ayj2)1/2 and a = (D2j/2ayj2)1/2 and
similarly for the z-term. By the central limit theorem, the quantity L is therefore
asymptotically normally distributed, with mean and standard deviation obtained
by summing those for the individual sources. Equation 10 can be written
r A i
ln[n CjXpj/QjJ
l/n
= -L/n
and the quantity in parentheses is just the geometric mean of the concentration,
weighted by c-r If the arithmetic mean is simply related to the geometric mean
(for instance if they are proportional), then Equation 11 says that the logarithm
of the concentration due to a large number of point sources is normally
distributed.
A distribution function for point sources based on the Poisson distribution
was proposed by Wipperman (1966). His model essentially assumed a uniform,
"top-hat" distribution of the instantaneous plume, which lends itself very neatly
to the Poisson representation. Similarly Prinz and Stratmann (1966) developed a
model using the negative binomial distribution. These have also been used to
describe multiple-source, urban pollution data. Probably any of these general,
skewed, frequency distributions could be used successfully to describe urban air
pollution distributions. Most of the empirical comparisons, as a result of
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Larsen's extensive studies, have been made with the lognormal distribution. The
fit of urban pollution data to the lognormal curve, while good, is not perfect.
Systematic departures occur, particularly for low concentration values. For this
reason extrapolation of the Ibgnormal, or any other distribution function out of
the usual range of observed frequencies, should be made cautiously.
A final point is that many observed air pollution frequency distributions
must be composites, reflecting the presence of both the multiple, distributed,
urban pollution sources and nearby, strong, isolated point sources as well, in
varying degrees. Study of all the resulting distribution types should be
rewarding, not only for their theoretical interest but also as clues to the nature
of urban air pollution sources. Concentration distributions are, so to speak, the
"fingerprints" of air pollution, and their characteristics may help us detect and
analyze urban air pollution patterns.
Acknowledgement
This research was performed under an agreement between the Atomic
Energy Commission and the National Oceanic and Atmospheric Administration.
References
Barry, P. J., 1971: Use of argon-41 to study the dispersion of stack effluents.
Proc. of Symposium on Nuclear Techniques in Environmental Pollution,
Int. Atomic Energy Agency, pp. 241-253.
Cramer, H., 1946: Mathematical methods of statistics. Princeton Univ. Press,
Princeton, N. J.
Gartrell, F. E., 1966: Control of air pollution from large thermal power stations.
Revue Mensuelle 1966 de la Soc. Beige des Ingenieurs et des Industriels
Bruxelles, pp. 1-12.
Gifford, F. A., 1959: Statistical properties of a fluctuating plume model.
Proceedings of Symposium on Atmospheric Diffusion and Air Pollution,
Advances in Geophysics. 6: 117-137.
Hald, A., 1952: Statistical theory with engineering applications. Wiley, New
York, N. Y.
Larsen, R., 1970: Relating air pollutant effects to concentration and control../.
Air Pollution Control Association. 20: 214-225
Larsen, R., 1971: A mathematical model for relating air quality measurements
to air quality standards. Office of Air Programs, USEPA Pub. No. AP-89.
Prinz, B. and Stratmann, H., 1966: The statistics of propagation conditions in
the light of continuous concentration measurements of gaseous air
pollutants. Staub. 26: 4-12 (English translation edition).
Scriven, R. A., 1965: Some properties of ground level pollution patterns based
upon a fluctuating plume model. CEGB Lab. Note No. RD/L/N 60/65.
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Scriven, R. A., 1971: Use of argon-41 to study the dispersion of stack effluents.
Proc. of Symposium on Nuclear Techniques in Environmental Pollution,
Int. Atomic Energy Agency, pp.^254-255.
Wipperman, F., 1966: On the distribution of concentration fluctuations of a
harmful gas propogating in the atmosphere. (Translation of unpublished
MSS.) 17 p.
DISCUSSION
Arnold Court: Your closing comment that the lognormal does not fit very well
at low concentrations is obvious in the derivation. That derivation is not valid
when x can approach 0, because you would be dividing by 0 in your derivation.
The lognormal will fit only when the fluctuations are small compared to the
concentration value.
Gifford: I have no comment; that seems reasonable.
F. B. Smith: Referring back to one of your very first equations. I don't think
this equation which relates the concentration at time (tf) to the concentration at
time (tj..j) can be unique. In fact the relationship normally used, for example
with wind velocities, is that the velocity, or the concentration in this case, is
related to the velocity or concentration of the previous time through a
correlation coefficient. In other words, looking at this first equation, x1 would
equal the value XQ times some sort of decay function, which might be
exponential but related to the correlation between the concentrations at those
two times, plus some random element, which would have zero mean and some
specified standard deviation. I think this would probably give-quite a different
distribution, a different answer. I'm not quite sure whether it would give a
lognormal.
Gifford: Yes, I would be interested to know the answer to this. I think that
certainly the important problem is to try to say something more about why,
other than just that it's an irregular function. I would certafnly think it would be
a good idea to use different kinds of generating functions and see what the
resulting distributions are.
M. M. Benarie: I really am not here to disprove or attack lognormal
distributions, which I use in the next paper to a great extent. All this discussion
about the genesis of describing functions reminds me very much of the
discussion in aerosol physics which began about 25 years ago, but was luckily
ended about 10 years ago, about the exponential, Risen-Rammler, lognormal
and other descriptive functions for the distribution of aerosols. For me, any
function which is mathematically easy to handle and is a good approximation is
good. So why not take the lognormal?
James Arveson: I have two questions - one is for information, the other might
open a Pandora's Box. The first one is for your equation 4. Why is there circular
3-6
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symmetry? Why not allow the possibility of some kind of correlation in the
model? The second question is why is there a need to have a parametric
formulation for these distributions? Perhaps we should be content to deal with
just quantiles of generally non-parametric distributions. Why is there the need to
fit a lognormal, a Weibull, etc.?
Gifford: This equation 4 doesn't have cross-correlation terms in it simply
because, in the usual form, the distributions with respect to y of the
concentration is assumed not to involve a cross-product term in ayand az. In
conditions of strong stability this is undoubtedly a poor assumption and I don't
think there is any particularly good reason for not setting up the model on the
basis that you suggest; I just haven't done it here. As to the second question, I
don't see offhand—it goes back to what the introductory speakers were
saying—you want to be able to rationalize what you observe in terms of physical
variables. Now the exact details of the model for doing this could be debated,
but it seems to me that whether you use...I am trying to see rather desperately,
not being a statistician, how what you would call a non-parametric approach
would apply here, but it seems to me that it is something like, "Look Ma, no
hands." What vou want is a wav of relating the atmosphere, which is the.
physical medium, to the observed concentration values, because you need to
specify the transfer function in the atmosphere, and I certainly don't have any
strong feeling about how that should be done. Here I intended to show only a
couple of possibilities that occurred to me.
Ron Snee: I agree that if you had to pick a single distribution function, the
lognormal would be the one to pick. I would point out, however, that the
Pearson system of distribution functions includes a variety of distribution curves
and has been used by statisticians for many years in the characterization of
empirical distributions. You did not mention the Pearson system. I wonder if it
hasn't been used or if perhaps there is some reason why it shouldn't be used?
Gifford: I don't know of any reason why any rational approximation to what is
observed couldn't be used and I certainly didn't intend to imply that it shouldn't
be. They're all equivalent. If you, for instance, look at a table of the parameters
of distributions, you will find that all of the skew frequency distributions are
related. The main difference among the different families of distributions has to
do with whether they are discrete or continuous variables. For the rest of it they
are mostly more or less all inter-related.
Snee: I would encourage the group to investigate the Pearson system. I believe
the Pearson system will help get us out of the problem of deciding whether the
lognormal distribution is appropriate in a given instance.
Gifford: The problem isn't the form of the distribution, in my opinion. It's how
you go about specifying the physical mechanisms involved, and that's the reason
that I don't really care too much about this little explanation here without some
rational way of characterizing the y's, which is what Dr. Smith was getting at.
The problem is to be able to express the parameters in some suitable distribution
in terms of atmospheric physical variables.
3-7
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4. METEOROLOGICAL PARAMETERS RELEVANT IN A
STATISTICAL ANALYSIS OF
AIR QUALITY DATA
F.WIPPERMANN
Department of Meteorology
Technical University, Darmstadt, Germany
Introduction
Today many measurements (continuous or at discrete times) of air quality
are carried out at many places in industrial countries. Statistical evaluations of
these measurements are as different (and therefore incomparable) as the places
are. Mostly concentration of a gaseous component or of particulate matter is
considered as depending on surface wind (speed and direction), on surface
temperature, and sometimes on humidity. However these three or four
parameters are not able to describe completely the turbulent state in the
atmospheric boundary layer and, therefore, not the diffusion condition in it.
This paper intends to show which meteorological parameters can be
considered as relevant in a statistical analysis of air quality data. However the
conclusions drawn are valid only if the atmospheric boundary layer satisfies the
conditions of a planetary boundary layer (PBL), i.e., stationarity and horizontal
homogeneity. In general, since the actual boundary layer differs from the PBL,
the conclusions are therefore only approximately correct.
This basic concept has been developed together with Dr. Yordanov (Sofia,
Bulgaria) and is the subject of a recent joint paper (Wippermann and Yordanov
(1972)).
Planetary Boundary Layer (PBL) and Rossby Number Similarity
The PBL is defined as a steady state horizontally homogeneous boundary
layer. In a PBL, there exists, for z » ZQ (ZQ = roughness length), a so-called
Rossby number similarity, which means that the vertical profiles of certain
variables (non-dimensionalized correctly by internal parameters) are
independent of the given external parameters. They depend only on an internal
parameter ju for thermal stratification, and on two internal parameters Xx and Xy
4-1
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for baroclinicity in the PBL. Of course, height above ground as an independent
variable has also to be made nondimensional bv a scale height H of the PBL.
•
Z* z/H H = /cu/f (1)
where K is the von Karman constant; u» = (rn//o)1/2 the friction velocity; r the
Reynolds stress; p the density; and f the Coriolis parameter. Variables for which
Rossby number similarity is valid are for instance
Q=AC(v-Vg)/U* T=T/(j5ll|) (2)
= km/(H2f)
Ex£x(n)
Ez£z(n) T= (T -)/^# S=(sT-s)/s
P and Q are non-dimensional velocity defects; u and v are velocity
components in a coordinate system, the x-axis of which coincides with the
direction of the surface stress trn (an internal parameter); and E is the rate of
dissipation of turbulent energy; u", v" and w" are the fluctuations, and Ex, Ey
and Ez are the three parts of turbulent kinetic energy; fx, f and fz are spectral
density functions with a frequency n. F is the non-dimensional difference of
temperature to the temperature at the top of the PBL (index T), and S is the
non-dimensional difference of water vapor to the water vapor content at the top
of the PBL. #* = -q0/(/cpcpu*) is a characteristic temperature fluctuation, with
cp the specific heat and q0 the turbulent heat flux at the ground, s* = j0/(/cpu*)
is a characteristic moisture fluctuation, with J0 the turbulent moisture flux at
the ground, i.e., the rate of evaporation.
All the variables listed in Equation 2 form universal vertical profiles,
depending only on the three internal parameters, ]U, Xx, Xy. For instance
T= T(Z,/i, Xx, Xy) for Z»Z0 (3)
where
(4)
is the internal parameter for thermal stratification with L* = — cp/5U*3/(Kj3qg), the
Monin-Obukhov stability length; |S = g/# with & a reference temperature.
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(5)
are the two internal parameters for baroclinicity^of the PBL.They are internal
ones because they contain the components of dV /dz in a coordinate system
oriented with the x-axis in the direction of the internal parameter tr0-
For all the variables listed in Equation 2 the same as in Equation 3 is valid.
This means that the vertical profiles depend only on u, Xv, A.,. This means
x y
furthermore that the state of turbulence in the PBL, and, therefore also the
turbulent diffusion, is completely described by these three parameters.
The Vertical Profile of Concentration Caused by a Horizontal Sur-
face Source
The condition of horizontal homogeneity of the PBL is satisfied as long as
the source of a gas or of particulate matter is a horizontal and infinite surface
source. This means that the concentration rs [g cm"3], made dimensionless by
the characteristic fluctuation of concentration r, = in/(Kpu.), with iQ [g cm"2
sec"1] the source strength, must have a universal vertical profile
Rs= -- (2,fJi, Xx, Xy) for Z»Z0 (6)
Actually the Rossby number similarity is valid for the non-dimensional
difference (Rs)j — Rs. Here (Rs)T/-the concentration at the .top of the PBL, is
made zero (vanishing background concentration). Examples of natural sources of
this kind are an evaporating sea surface or a very large sand desert with sustained
strong winds. Water vapor or sand is added to the air. The vertical profile of
concentration of these admixtures depends only on the three parameters ju, Xx,
X .
The Concentration in the Case of a Continuous Point Source
If one considers a continuous point source, the condition of horizontal
homogeneity is violated and Rossby similarity can no longer be used to conclude
on which meteorological parameters the concentration pattern depends. (In the
case of .an instantaneous point source, the condition of stationarity is also
violated).
One can try to make a statement concerning the relevant meteorological
parameters by assuming that the diffusion process is described by the
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steady-state Fickian diffusion equation
Z r d Tn 1
(7)
and by replacing the velocities of u and v and the turbulent diffusion coefficients
kx, ky, kz by variables, for which universal vertical profiles exist. The horizontal
coordinates should be made dimensionless by the internal scale height H given in
Equation 1b, X = x/H and Y = y/H, and the diffusion coefficients by H2f.
Since
,/\
flu ,N
Xxz
and
"go3 '^go' cos(a0)
-^go= '^go1 sin(ia0D
there results
u/(Hf) = [P+KCOS (a0)/Cg]//c2 (8)
v/(Hf)= [o-Ksin(la0l)/Cg]//c2
a0 is the cross-isobar angle and Cg = u*/|vgo| the geostrophic drag
coefficient. Both an and Cg can be eliminated in Equation 8 by making use of
the resistance law for the PBL
K cos (a0)/Cg= -Mm(/t,XXIXy) + ln(Ro0Cg) (9)
K sin (Ia0l)/Cg= N(ft, Xx, Xy)
where Ro0 = |\y |/fzn is the surface Rossby number, a non-dimensional
4-4
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combination of given external parameters. The functions N and Mm appearing in
this law are universal functions, i.e. they are independent of external parameters
and depend only on ju, Xx, Xy. The resistance law was first derived by Kazanskii
and Monin (1961) for the barotropic and neutral case. It was extended to the
diabatic case by Blackadar (1967) and by Monin and Zilitinkevich (1967),
Recently it was extended to baroclinic cases by Yordanov and Wippermann
(1972). By using the resistance law Equation 9a and 9b and the definition
Z0= (K Ro0Cg)"'
one obtains for the diffusion Equation 7, the following form
dR0 % dRp
( P-Mm+ Xx Z)
The boundary conditions are
X = 0 , Y = 0 , Z = Zb: Rp=I
Z = Z0: dRp/dZ = 0
R = fp/(b f5 g"3) is the non-dimensional concentration caused by a
continuous point source; b [g sec'1 ] is the strength of the source; and Zb = zb/H
is the non-dimensional effective height of the source. If one assumes that the
vertical profiles of the non-dimensional diffusion coefficients for matter Kv, K..,
A y
Kz are universal ones like the vertical profile of the diffusion coefficient Km for
momentum, all coefficients in Equation 11—except ZQ—depend only on p., Xx
and Xy and, of course, some of them on the independent variable Z. They all are
universal functions. However the non-dimensional roughness length ZQ depends
on Ro0 and Cg as seen in Equation 10, and Cg itself depends on Ro0,/z, Xx, Xy,
as seen in the resistance law, Equations 9a and 9b. Therefore since ZQ depends
on Ro0/ JJL, Xx, Xy the non-dimensional concentration Rp must also depend on
Ro0.
R* D / V V 7 "7 ^)A • • \ \\ / "1 O V
p - npiAtit£,£bl nO0, ^i, AX» Ay J (\o)
Universality is now lost; dependence from Ro0 (i.e., from external
parameters) enters because of violation of the condition of horizontal
4-5
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homegeneity by having a continuous point source. A comparison of R in
Equation 13 with R. in Equation 6 shows the difference.
The Concentration (at a Fixed Point) Caused by Multiple Sources
A statement can be made only if one assumes that the sources do not
change their coordinates and their strengths during the period of measurement.
Furthermore one has to assume that the period of measurement is long enough
to cover all possible cases (wind direction, Rossby number, thermal stratification
and, possibly, the baroclinicity) in almost equal parts. This later assumption has
to be made in almost all statistical analyses of measured concentrations.
However, the first assumption will be only incompletely fulfilled and therefore
causes errors.
For each_ direction (of the geostrophic surface wind) the mean
concentration rd at the point under consideration should be evaluated and a
non-dimensional concentration
Rd = r/7d (14)
should be formed therefrom. Fluctuations of RH should be independent of the
source positions (xb)j, (yb)j, (zb)j and of the source strengths bj. They should
depend on the remaining parameters in Equation 13.
Rd = Rd (R0OI fJL, Xx> Xy) (15)
The surface Rossby number RoQ = IWgol/(fzn) can be determined directly
from the given geostrophic surface wind, when zn in the denominator is known.
This can be obtained by conventional measurements of the wind profile near the
ground, but should be representative of the whole area in which diffusion takes
place. It may vary with the wind direction (and is therefore a "meteorological"
parameter) and, possibly, with the vegetation period.
The internal parameter p. for thermal stratification has to be found by
converting the external parameter a for thermal stratification
A= *3
with 6£ = #T -#0- When the temperature difference 5# from top to bottom and
^oo the geostropic wind at the ground are given, parameter a then can be
4-6
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formed. This parameter must be converted to parameter/H- Zilitinkevich (1970)
gives diagrams for the conversion of given external parameter a into the wanted
internal parameter ju- For this conversion RoQ is again needed.
Of course difficulties^ will arise in forming external parameter a from the
given parameters #T, #Q'^go' because the first two are difficult to find. A PBL
can only have a monotonically decreasing or increasing temperature profile #(z).
It does not know temperature inversions or layers with unstable stratification or
similar things very frequently observed in the real atmospheric boundary layer.
Therefore an observed temperature profile has to be smoothed in order to obtain
the corresponding profile belonging to the PBL. The difference #T — #0 should
be taken from such a profile.
The baroclinicity seems to be less important. It has to be considered only
for muchly elevated sources, e.g., very tall stacks. An example has been given by
Wippermann and Yordanov (1972) showing a case with a pronounced minimum
of eddy diffusivity in 320 m caused by baroclinicity. When baroclinicity must be
considered, one has first to form the two external parameters (in a coordinate
system x*, y* oriented with the x-axis in the direction of the geostropic wind
at the surface)
2
K f ^ \o ^ i IT
** fzT i g i u
r\ (|
where ZT is the height of the top of the PBL. For a conversion into the internal
parameters Xx, Xy of baroclinicity, the cross-isobar angle a0 is needed:
a0) + -n »sin(|a0l) (19)
y
sin
The angle ao can be obtained from the resistance law Equations 9a and 9b.
Concluding Remarks
The baroclinicity of the boundary layer flow has an effect on the
concentration pattern only when these are caused by much elevated sources; the
effect can be neglected in most cases. The remaining meteorological parameters
are the internal parameter ju for the thermal stratification of the boundary layer
and the surface Rossby-number ROQ. Both these parameters determine the
concentration uniquely (as long as the assumptions of a PBL are satisfied).
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It seems that the parameter /i is just the one which has been sought as a
measure of diffusion characteristics which depend mainly on thermal
stratification. The empirical "dispersion categories" can possibly be replaced by
this parameter, if we succeed in determining the PBL which is equivalent to the
(measured) actual boundary layer.
References
Blackadar, A. K., 1967: External parameters of the wind flow in the barotropic
boundary layer. GARP Study Conference Report, Stockholm, Sweden,
Appendix IV, 11 p.
Kazanskii, A. B., and Monin, A. S., 1961: On the dynamic interaction between
the atmosphere and the earth surface. Bull, (tzv.) Acad. Sci. USSR,
Geophys. Ser. Nr. 5, 786-788, English translation. 514-515.
Monin, A. S.,^nd Zilitinkevich, S. S., 1967: Planetary boundary layer and large
scale atmospheric,dynamics. GARP Study Conference Report. Stockholm,
Sweden, Appendix V, 37 p.
Wippermann, F., and Yordanov, D., 1972: A perspective of a routine prediction
of concentration patterns. Atmospheric Environment. 6: 877-888.
Yordanov, D., and Wippermann, F., 1972: The parameterization of the
turbulent fluxes of momentum, heat and moisture at the ground in a
baroclinic planetary boundary layer. Beitr. Phys. Atm. 45: 58-65.
Zilitinkevich, S. S., 1970: The dynamics of the atmospheric boundary layer.
Gidrometeor., Leningrad, (in Russian) pp. 291.
DISCUSSION
Smith: Could I ask you. Dr. Wipperman, if you consider that the depth of the
boundary layer could be adequately given by the Rossby similarity theory. My
experience with this is that one can use the theory very adequately to give you
estimates of the surface stress and the turning of the wind at the surface, but in
unstable conditions it doesn't normally give very good estimates of the depth of
the boundary layer which you've used in your scaling. Usually the depth of the
boundary layer depends much more on the historical development of the
boundary layer due to the input of heat over the daytime period.
Wipperman: This is a question of how one defines the depth of the boundary
layer. So if you have, for instance, an inversion on the top and you consider the
height of this inversion as depth of the boundary layer, this could not be
considered in a planetary boundary layer in which an inversion is not possible.
I'm just choosing this "H" as a scale height for the boundary layer, which does
not mean somewhere is the top of this boundary layer.
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5. THE USE OF THE RELATIONSHIP BETWEEN WIND
VELOCITY AND AMBIENT POLLUTANT CONCENTRATION
DISTRIBUTIONS FOR THE ESTIMATION OF AVERAGE
CONCENTRATIONS FROM GROSS METEOROLOGICAL DATA
M. M. BENARIE
Institut National de Recherche Chimique Appliquee
Vert-le-Petit, France
Introduction
A synonym for "computation of pollutant concentrations from
meteorological data" is atmospheric modeling. In this matter, one has on the ojie
hand mechanistic (or explaining) models, which seek the breakdown as well as
the comprehension of the elementary physical processes of dispersion. On the
other hand, one has formal (or phenomenological) models, which look for the
necessary and sufficient coefficients for the computation of some mean or
probability of given concentrations. After explaining the reasons why I have not
chosen a mechanistic model for concentration frequency computation, I will
deal further with one specific model.
This distinction is very near to the one defined by Stern (1970) at this place
just two years ago, in his Symposium Summary on Multiple-Source Urban
Diffusion Models: mechanistical models are source-oriented and the
phenomenological ones are receptor-oriented. It should nevertheless be stressed
that phenomenological and statistical models do not necessarily mean the same
thing. Mechanistical (source-oriented) models are constrained by considerations
of material balance, as opposed to statistical (empirical) ones, which are not
(Calder (1970)). The receptor-oriented phenomenological model proposed
herewith does not implicitly avoid the use of the law of the conservation of
matter (even if in the present paper we fail yet to attain its ultimate
consequences) and it has the pretension to give more insight into physical
processes than just a correlation between measured pollutant concentrations and
simultaneously observed meteorological parameters.
Concentration frequency distributions are generally obtained by calculating
the concentration values for all possible combinations of meteorological
parameters: wind direction, wind speed and stability category. Afterwards, we
take the sum of the joint frequencies of all combinations of classes that rise to a
given concentration. The first objection to this way of proceeding is economical.
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Since the chosen dispersion equation has to be evaluated numerically at least a
few hundred times for each receptor location of any interest, such extensive
computation can quickly involve prohibitive time, even with a high-speed
computer.
Secondly, the relevant meteorological data or statistics have to be
extensively known for the given location. While these data may be available in
sonie cases, in others not less important, the next meteorological
station—perhaps a hundred miles away—may not be at all representative.
Thirdly, the result of individual computations of the dispersion equation has
the character of a differential. In the case of important point sources, the result
justifies careful consideration of the constants, which are often based on
extensive surveys of emission and meteorological parameters. As maximum
concentrations are mostly sought, the results are acceptable even when off the
target by a factor of two or even more. But if concentration frequency
distributions, or one of their derivatives as a mean, are sought, the input errors in
the dispersion equations (including lack of basic meteorological information) can
easily be amplified by the effect of the summation. As experience shows, the
final result is, as often as not, off by ±50%. This error is unacceptable since any
air pollution engineer, worthy of this name, should be able to estimate in ten
minutes on a slide rule from very few data (such as population density, space
heating habits, industrial context and some general knowledge about
climatology), a mean with the specified accuracy.
The fourth objection that can be made is that the basic concept of plume
computations is a short-time process, say typically of 30 minutes. The passage to
long-time averages is especially awkward at the level of transition from gaussian
plume of constant direction to the normally meandering plume. Next follows
another transition to changing wind directions, as described by a specific wind
rose. At least two different physical processes are involved, one of which is
definitely not gaussian.
The fifth and last, but not least, objection is that a model should require a
minimum of basic assumptions and discard everything not absolutely needed.
This should apply to stability classes, as far as averages are concerned. Not that a
frequency table of the stability classes for towns and most populated areas
should be so difficult to obtain. However such tables are usually unavailable
when and where they are most needed, e.g., for the location of a new plant (see
also the second argument above).
And thus arises the question as to whether frequency tables of stability
classes are necessary. Looking at Table I, we are inclined to answer that, at least
for the limited purpose of computation of averages, they are not absolutely
indispensable.
It would be highly interesting to supplement the data of Table I (the only
ones we have been able to locate), with non-European statistics and with figures
concerning other than temperate climates. From what is available for the time
being, we can see that the near-neutral classes represent 76% ±5% of all
5-2
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situations*. We effectuated a few trial computations which show that the mean
is but slightly—and the median not at all—sensitive to the frequency changes in
the stable and unstable classes. If this is the case, and the overall class frequency
distribution is an approximately constant property of the temperate climates,
then why bother to split up first into categories and afterwards totalize them
during expensive computer hours?
The purpose of the present paper is to provide a simplified method of
estimation of pollutant concentrations, in cases where detailed meteorological
data are not at hand. As far as possible, the method is an empirical one and has
no pretension to give insight into the physical processes of atmospheric
dispersion. The purpose is to provide an easy and rapid means for atmospheric
modeling around point sources.
The Experimental Data
The experimental basis for the present work is due to Prof. P. Bourbon and
his staff who obligingly permitted us to make the following statistical analysis of
the data of their survey around the natural gas sweetening plant at Lacq
(southwest France). Gratitude is expressed at this point for this important
contribution.
For several years, 24-hour mean concentrations of SO2, NO2, H2S and
other pollutants were measured at 37 points. Figure 1 shows their repartition
around the plant. For the time being, we will use only the data for the two.
years, 1968-1969, which are complete and homogeneous insofar as SO2 and
NO2 concentrations are concerned. The former was analysed by the very specific
nitroprussate-pyridine method (Bourbon et al. (1971)); the latter by the
Jacobs-Hochheiser method.
The concentrations were represented in the form of cumulative frequency
diagrams, one for each survey point and pollutant. Of the 74 diagrams, 54 are
very nearly straight lines on the lognormal coordinates which were used. In
Figure 1 the survey points with two straight-line distributions are marked-^;
those with one by-O-; Figures 2, 3, 4 and 5 give samples of these distributions.
Discussion
It has been found that the cumulative frequency distributions of suspended
particulates at CAMP (urban) sites have a tendency toward lognormality
(U.S.D.H.E.W. (1958)). Earlier applications, referring also to other pollutants
*) With the restrictive condition that classification criteria should be the same. This meant
that we had to leave the Szepesi (1964) data out of Table I because they are based on a
perhaps better but nevertheless different criterion, i.e., the measurement of the
temperature gradient.
5-3
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but always to receptors located in or at area sources, are to be found in Zimmer
et al. (1959) and Gould (1961). As stated by Gifford (1969), the lognormal
concentration distribution can be mathematically derived by the
particularization of the general explanation of skew distributions.
It was shown by Benarie (1969) (1971): (a) that the lognormal distribution
is strictly valid for concentration frequency distributions in any given direction
around a point source. This is a consequence of the facts that wind velocity
distributions in any given direction may be approximated by a lognormal and of
some very general mathematical properties of this function.* (b) that in the
special case of the point source without thermal plume rise, the geometrical
standard deviations of the wind velocity distribution and that of the
concentration distribution are numerically equal. From this equality it follows as
a corollary: the concentration frequency distributions for receptors, situated at
various distances along the same radius and from the point source, should have
the same geometrical standard deviation. The general case of the source with
plume rise, will be discussed further when speaking about S02 (c) that the
observed lognormal distributions for area sources follow directly by summation
of the effect of a large number of likely distributed point sources.
At this point, we should distinguish between NO2 emitted at nearly ambient
temperature and the SO2 contained in plumes of higher temperature. The
former contributes evidence to points (a) and (b) above. As these general
affirmations were obtained from relatively few data, this further evidence is
useful. The discussion of the S02 results below will add a new contribution.
Figures 2, 3, 4 and 5 illustrate the above affirmations (a) and (b). Geometric
standard deviations for (unperturbed) NO2 receptors in the same radial
orientation are identical and nearly equal to the geometrical standard deviation
of corresponding wind velocity just as required by the corollary to the theorem
cited in the Appendix.
Figures 2, 3, 4 and 5 are only a fractional sample of the evidence on hand.
Although they are quite convenient for interpolation, as will be shown below,
space does not permit displaying similar figures for all 37 survey points. Instead,
the principal information from them have been summarized in Figure 6,
which displays values of kNQ2 = o NO^ /aw (ONO^ and aw are the respective
geometrical standard deviations for N02 concentrations and wind velocity).
Values of kNO deviating from unity, are found in the western half of the
pattern, where topographical accidents are more pronounced. In the eastern half,
in the first approximation level the behaviour of k|\io2 is as theoretically
expected.
As for the S02 which is definitely associated with a thermal plume rise and
is emitted by 60 to 80 m high stacks, the hypothesis of (concentration) prop w"1
cannot be assumed and a kSO2 = °S°2 value different from unity should be
Ow
expected. The numerical value of ksoo mav readily be computed by applying a
*See Appendix, for discussion of the frequency distribution of wind velocity and the
properties of the lognormal distribution which are of interest here.
5-4
-------
dispersion formula to a plume rise expression. As one has a rather wide choice
from both sort of equations and naturally an even larger one of combinations, it
is easy to find one or more to "prove" that the kso2 values reported in Figure
7 are correct. For us, as far as they were empirically observed, they are indeed.
It has already been mentioned that cumulative frequency diagrams such as
Figures 2, 3, 4 and 5 are convenient for interpolation purposes. At least for level
terrain, the sequence of (almost) parallel, straight lognormal representations is
related to distance. This is easy to understand, as concentrations diminish with
distance, or, what is equivalent, a given concentration occurs with decreasing
frequency and with increasing distance.
This relation between concentration at constant frequency and distance, is
illustrated by Figure 8. As most survey points show some (topographical)
singularity, it is not easy to align enough points, in order to judge the form of
the regression and the exactitude of fit of some function. Therefore Figure 8
should be considered as an empirical data collection and will be used in the
following as such, as means for interpolation.
We may now investigate whether there is a correlation for a given distance
between the frequency of exceeding some concentration and the frequency of
wind blowing from the source in the direction of the receptor.
Figure 9 is the wind rose observed between 1961 and 1965 near the plant
site. Frequencies corresponding to the opposite wind directions are the abscissae
of Figure 10, concerning only receptors at approximately the same distance,
between 5 and 7 km, in this case. The ordinates are the frequencies by which
some given concentration—here 50jug NO2/m3—are exceeded. It might be
expected that a correlation should exist between these frequencies. At first
approximation, this assumption seems to be verified.
The observed scatter is due, among other causes, presumably to the lateral
wind turbulence and its directional change during a 24-hour sampling period.
Probably, with shorter sampling times, this scatter would diminish.
Example of Application
Up to this point, we have presented these experimental data somewhat
differently from the usual tabular or isoconcentration-map form. How far
reaching is this special presentation? Should it be called a model; or, more
modestly, a relationship between wind and ambient concentration.
Suppose we ask for a cumulative concentration frequency diagram for the
point marked X on Figure 1, a location at which no receptor was operated.
From Figure 9 it can be seen that the frequency of wind blowing from the stacks
in the direction of the receptor is 5.5%. Entering Figure 10 at this abscissa value,
the frequency of 25% is read at the ordinate. This would be the frequency of
5-5
-------
exceeding 50 jug NO2/m3/ if the receptor were 6±1 km distant from the source.
Actually it is «* 3.5 km from the source. The concentration corresponding to.this
distance (and direction) is interpolated as shown in Figure 8. Thus 62 jug
N02/m3 is found. This pair of values (62 /ig N02/m3, 25%) is one point of the
cumulative frequency diagram. As its geometric standard deviation should be
equal to that of the wind, the concentration distribution at this hypothetical
receptor is defined.
For a thermal source, the same procedure should be followed except that an
experimental k-value should also be determined and stack height, effluent
temperature and velocity also taken into account. In our case this is defined only
by Figure 7 and therefore cannot be considered of general validity. Nearer
details will be supplied in a subsequent paper.
Outlook
These interpolations and perhaps slight extrapolations have limited uses in
a dense survey network, as the one just discussed. However the empirical
relations (a) wind direction frequency versus frequency of exceeding an arbitrary
concentration (Fig. 9) and (b) concentration versus distance (Fig. 8) are
generally established. Then a few points per diagram, perhaps three, will be
sufficient to obtain concentration versus frequency diagrams for a multiplicity
of geographically scattered points. The only additional information needed, are
wind roses and frequency distributions of wind velocities. This seems true for
plane, undisturbed topography. The evidence under consideration is just enough
to say that topographic relief does something to the constants. But for the time
being, we are unable to express this effect in a general and quantitative way.
If, with more evidence at hand, the functional form and the general
constants of both these relations can be found, we shall have a modeling method
which will need very little meteorological input. In this way, computer time can
be replaced by equivalent graph reading time, and, what is even more important,
results will be of irrefutable empirical character. Its advantage, above purely
statistical models, is a greater generality, as cause and effect are more evidently
related in the present model. We hope to continue working in this direction.
Acknowledgement
I wish to express my gratitude to Mr. P. Bessemoulin and Mrs. T. Menard for
all their help, computations, computer programs, tedious graph drawing, etc.
involved in the present work.
5-6
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Table 1. FREQUENCIES OF STABILITY CLASSES
Frequency %
Un- Near-
stable neutral Stable
A,B C,D,E F,G
4.2 76.8 19.0
10.1 69.6 20.3
5.0 85.7 6.7
14.3 70.6 15.1
10.2 82.2 7.6
7.8 81.4 10.7
Principle of
classification
Pasquill-Turner
Pasquill
t measured
Pasquill
Pasquill-Turner
Pasquill
Year Reference
1961-62 Nester (1966)
? Bryant (1964)
1958-63 Szepesi (1964)
1964 Polster-Vogt (1965)
1965-70 Hodin
1967-71 Bessemoulin (1972)
Site
Frankfort, G.
? ,Br.
Budapest, Hung.
Julich, G.
Trappes, F.
Rouen, F.
Figure 5-1. Location of sampling stations around gas sweetening plant at Lacq,
France.
5-7
-------
0.01
0.09
ai
az
as
10
"
X
40
90
60
TO
•O
•0
• Station 10
+ • II
a - 12
o - is
1 1 I I I 1 I I I
i i i i i i
8 S
8 3 £888
CONCENTRATION
§ § §§!
Figure 5-2. Cumulative frequency diagram for S02 concentrations at stations
located in the 289° -308° sector from the source, at Lacq, France.
0.01
0.09
O.I
0.2
0.9
I
2
I
X 9
(J
2 .„
> 40
; 90
60
70
80
90
»9
96
o Stolkm 10
+ « II
Q - 12
O - 19
8 ? 8 8!? 888 § 8888 8888
— N 10 * K 3 F?A3)O
ODNCENTRATION t/ig/m1)
Figure 5-3. Cumulative frequency diagram for NO2 concentrations at stations
located in the 289°-308° sector from the source, at Lacq, France.
5-8
-------
aoi
aos
0.1
0.2
as
40
90
60
TO
•0
90
I I
II i III
2 8 8 ? 8 82888 g 8 8 8 8&S88
CONCENTRATION I
Figure 5-4. Cumulative frequency diagram for SO2 concentrations at stations
located in the 108°-121° sector from the source, at Lacq, France.
I I i I i I I i I I I I I I I I I I
8O Q Q Q O OO Q O O OQ9QOfiC
8 ? s gessg i 1 ; § sesS|
CONCENTRATION
Figure 5-5. Cumulative frequency diagram for N02 concentrations and wind
speed at stations located in the 108°-121° sector from the source at Lacq,
France.
5-9
-------
Figure 5-6. Values of kNO = pN02 for sampling stations at Lacq, France.
2 awind
Figure 5-7. Values of kso^ = ° . 2 for sampling stations at Lacq, France.
^ /T\A/irirl
awm
5-10
-------
too -
16%
10
DISTANCE (km)
Figure 5-8. The concentration at three frequencies as a function of distance from
the source.
NW
NE
SE
• 1%
Figure 5-9. Wind rose for 1961 - 1965 at plant site in Lacq, France.
5-11
-------
70
6O
50
30
20
10
2
I
as
0.1
7
or
as
2 5 10 20 30 40
WIND FREQUENCY (%)
Figure 5-10. Frequency of exceeding 50|ug NO2/m3 as a function of wind
frequency from the source to the receptor for eight sampling stations.
O.OS
O.I
o.t
OS
2
2 •
n 1—i—r i i i i i 1 1—i—i i i i i
Mil™.'1"*
RESULTANT WIND SPEED (rtl/sec)
Figure 5-11. Cumulative frequency of resultant wind speeds for three averaging
times.
5-12
-------
References
Aitchinson J. and Brown, J.A.C., (1969): The lognormal distribution. Cambridge
University Press, p. 9 ff.
Benarie, M., (1969): Le calcul de la dose et de la nuisance du polluant e'mis par
une source ponctuelle. Atmospheric Environment. 3: 467.
Benarie, M., (1971): Sur la validite de la distribution logarithmico-normale des
concentrations de polluant. Proceedings of the 2nd Internat. Clean Air
Congress, 1970, Washington, D. C., Academic Press, New York, N. Y., pp.
68-70.
Bessemoulin, P., (1972): Contribution a I'etude de la diffusion des polluants
gazeux dans /'atmosphere. Thesis, Paris, France.
Bourbon, P., Malbosc, R., Bel, M. J., Roufiol, F. and Rouzaud, J. F., (1971),:
Contribution a la determination specif ique dans I'atmosphere du dioxyde
de soufre. Poll. Aim. 52: 271-275.
Brooks, C.E.P. and Carruthers, N., (1953): Handbook of statistical methods in
meteorology, Her Majesty's Stationery Office, London, England, chapters
3 and 11.
Bryant, P. M., (1964): Methods of estimation of the dispersion of windborne
material and data to assist in their application. AHSB (RP) R42.
Calder, K. L., (1970): Some miscellaneous aspects of current urban pollution
models. Proc. Symposium on Multiple-Source Urban Diffusion Models,
Research Triangle Park, N. C., APCO Pub. No. AP-86.
Gifford, F. A. Jr., (1969): The lognormal distribution of air pollution
concentrations. Air Resources Atmospheric Turbulence and Diffusion
Laboratory, ESSA, Oak Ridge Tennessee (preprint, 3p.).
Gould, G., (1961): The statistical analysis and interpretation of dustfall data.
Proc. 54th Annual Meeting Air Pollution Control Association, New York,
N.Y.
Hodin, M.: Personal Communication.
Nester, K., (1966): Hauf igkeitsstatistische Aussagen uber
Maximalkonzentrationen von Schornsteinabgasen auf Grund synoptischer
Wetterbeobachtungen.Sfat/6. 26: 521.
Polster, G. and Vogt, K. J., (1965): Grundsatze and Untersuchungen zur
Beurteilung der Ausbreitung radioaktiver Abluft. Protokoll zur
Informations- und Arbeitstagung im Kernforschungszentrum Karlsruhe,
Germany, p. 15.
Stern, A. C., (1970): Utilization of air pollution models. Proc. of the
Symposium on Multiple-Source Urban Diffusion Models, Research
Triangle Park, N. C., APCO Pub. No. AP-86.
Szepesi, P., (1964): Computations of concentrations around a single source.
Idojaras. 68: 257.
5-13
-------
U.S.D.H.E.W., (1958): Air Pollution Measurements of the National Air Sampling
Network - Analyses of Suspended Particulates, 1953-57. PHS Publication
No. 637. p. 245.
Zimmer, C. E., Tabor, E. C. and Stern, A. C., (1959): Particulate pollutants in
the air of the United States. J. Air Pollution Control Association. 9: 136.
Appendix
It is fairly well known in meteorology (Brooks, et al. (1953)) that the
distribution of wind velocities is skew in a given direction, with high frequencies
at low velocities. Several two or more parameter laws present a fair
approximation of the experimentally observed distributions.
It has been observed that among two-parameter skew distributions the
logarithmic normal function is an experimentally convenient representation of
the wind velocity (Benarie (1969)),
At first, it seems singular to use a mathematical function which will not
accomodate the zero value of the variable for wind. Measured with the usual cup
anemometer, wind velocity values almost everywhere show a high percentage of
"calm" periods. Closer scrutiny of sensitive thermoanemometric data seems to
suggest that this abundance of calms is purely instrumental. In reality, very low
velocities occur with finite frequencies, and a true zero does not physically exist.
As our present purpose is not to get into meteorological arguments, we avoid
this difficulty by defining wind velocity classes as "less than 1 m/sec." (the
starting point of the anemometer) and by including in this class all observations
between 0 and 1 m/sec. The fraction of observed "calms" is proportionately
attributed to each directional frequency.
A second difficulty arises from the fact that (except for specially conceived
survey networks which we do not possess) wind data are from meteorological
stations, following international meteorological conventions (i.e., one
observation every three hours.); while pollutant concentration data are
integrated for shorter or longer periods (24 hours in our case). Figure 11 which
presents the cumulative frequency distributions of: (a) 3-hour, (b) 24-hour and
(c) 1-week wind vectors from the same station, shows that the error committed
by using (a) instead of (b) is slight. Anyway, this is a minor point, as the 24-hour
wind vector, which should be physically better justified, can be obtained easily
from the original data by a minor computational program.
This rather lengthy argument: about the approximation of the observed
distribution of wind velocity frequency by a lognormal function was necessary
because of the interesting reproductive properties of this two-parameter
distribution (Aitchinson and Brown (1969)), which are the immediate
consequences of those forjthe normal distribution.
Theorem: if w is A (w,a) i.e. a lognormal function with the geometric mean
w and the geometric standard deviation a, and k and c are constants, where c >
o (say c = ea), then cw is A (a + bw, ka).
This theorem implies the corollary result: if w is A (w,a) then w1 is A (-w,a).
5-14
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DISCUSSION
Donald Rote: I'm not completely sure but it seems to me that your approach
depends very heavily upon having a uniform wind field. If you have topographic
features that in some way influence the wind field, you will have differences at
the same time between wind directions at the source, and at a given receptor. As
a consequence, this will greatly distort your capability of generating curves of
constant percentile. Could you comment on that please.
Benarie: I fully agree with you. The fact, in Figure 7 I think, of having ratios
different from the expected value of 1 in the western part of the pattern, and
about 1 in the eastern part, which is level, illustrate your point very well. But
survey data are very expensive; I had to do with the data I had and these were
the many meteorological data I had. The correct experiment to verify would be
to have had wind vanes at at least 8 stations. Then the conclusion would be
immediate or almost immediate. I agree fully with your point.
Harold Neustadter: Have you had an opportunity yet to attempt any internal
check op the validity of your conclusion. Namely taking three or four of your
receptors and seeing if you can generate the results of your dense network
within the set you already have?
Benarie: Sure, I did. That was the first check, and it was as reliable as the
receptors and measurement results. You know, of course, that manual chemical
and analytical methods are good to, say, plus or minus 20 percent. I can't
pretend more.
Singer: Work like this is being done by Brookhaven National Lab where they are
studying a network outside of New York City. A paper was just presented by Gil
Raynor at the Philadelphia meeting where he had concentration vs. distance
from New York City out to a hundred kilometers and it is very similar to yours.
Predictions were done very similar to yours, and it is related to Frank Gifford's
paper this morning. Using the lognormal distribution, the predictions worked
very well as long as you stayed near the mean. But when you went to the
extremes, if you tried to predict the extremes near ninety-nine percent, which is
needed for many problems, the whole system fell apart. While near the means it
worked verv well.
Benarie: It's quite a general statistical property that if you don't have any
infinite samples, then at the ends of the sample distribution you go wrong.That's
sure. One more point, I stressed that I am interested here in long-time means,
and as you have seen in the first table, I neglected the stable and the unstable
situations, saying that the mean is mainly influenced by the 75% of neutral
situations. I know that I can't do anything with the extremes.
Singer: It's true, but I know the normal situation. People will then take your
curve and extrapolate it to 99 percent.
5-15
-------
Benarie: No, they shouldn't do that.
Arnold Court: The apparent relation between the distributions of the
concentrations and the wind speeds may be valid, but this does not mean that
either is necessarily lognormal. For one thing, we as meteorologists and
climatologists cannot accept lognormality for wind speeds. By the argument
which the speaker made earlier in the discussion, we are looking for the most
simple relation. Winds are basically three-dimensional vectors. The third
dimension, up, is generally one or two orders of magnitude less than the two
horizontal vectors, so we tend to ignore it. However our general attitude toward
wind is that we have two orthogonal components, and we represent it by a
bivariate distribution. We generally accept the bivariate normal largely in default
of any other bivariate distribution that we can handle. If winds and components
are bivariate normal, then the wind speed itself, independent of direction, has a
Chi distribution of two degrees of freedom, also called the Rayleigh distribution.
Now this is quite similar in appearance to a lognormal, but is a different
distribution. On the other hand if you accept lognormality for wind speed, you
have a very difficult time deriving the distribution for winds by components.
Therefore I think that if the speaker's argument holds that the distributions of
concentrations and wind speeds must be similar, this indicates that
concentrations also may have a Rayleigh rather than a lognormal distribution.
Benarie: Thank you very much and mostly I agree with you, Mr. Court. Firstly, I
stressed one of your points in my appendix which I didn't read here. Normally
it's known that wind speed having lots of zeros is not a function to be
represented by a lognormal. I am asking the meteorologists present here if they
can provide me any data. I have made some experiments with a sensitive
thermistor anemometer in a wind field. Because it's not a cup anemometer, it
registers lots of values down near zero. It seems there are values everywhere. As I
told already at the end of Mr. Gifford's paper, I am looking for a convenient
engineering fit and an easy mathematical manipulation and not a theoretical
explanation. Lognormal is good for me, but the argument is open as to how far
it's physically good, and I leave it open.
Joseph Knox: I would like to ask you a question if I'may about two of the
figures pertaining to direction 108° to 121° in regard to pollutants SO2 and
N02. These figures have different slopes for the pollutants on lognormal paper,
and the wind is shown as being approximately a lognormal function paralleling
the NO2 distribution. Since the slopes for these two pollutants are different, it
doesn't parallel the S02 distribution.
Benarie: It should not be. I stressed in the paper that for the NO2, which is a
non-thermal emission, parallelity is requested. For a thermal emission, if we put
concentration against distance with the parameter of wind speed, by a
combination of the effective stack height with a formula like Brigg's, and a
diffusion formula, you get .... (writing on board) things like that. In this case
the concentrations have a slope in logarithmical representation. The exponent is
-1 only when there is no thermal elevation. With thermal elevation it is different
5-16
-------
from 1. But I stress the point that you can choose a chimney thermal elevation
formula which fits just the numerical value which gives you a good slope.
Knox: My point is this, I would also be afraid of chemical reactivity or
photochemical reactivity affecting these distributions. As noted by Larsen
several years ago, the pollutants with the steepest slopes for the largest standard
geometrical deviations on lognormal paper are the most reactive pollutants. And
so, I really want to comment that I see some cause for caution about dealing
with photochemical reactive pollutants in this manner.
Benarie: It has to be remembered; your argument is quite valid. I asserted up to
now that only the thermal rise is a cause of the variation of this ratio from 1.
Another could be a sink, a reaction. Because I cannot yet give a quantitative
evaluation or a theoretical explanation of the differing values, I note them only,
so any tenative explanation is good.
5-17
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-------
6. AVERAGING TIME AND MAXIMA
FOR DEPENDENT OBSERVATIONS
RICHARD E. BARLOW
Department of Industrial Engineering and Operations Research
and Department of Statistics
University of California, Berkeley, California
and
NOZER D. SINGPURWALLA
Department of Operations Research
The George Washington University, Washington, District of Columbia.
Introduction
Monitoring Air Pollutant Concentrations
Under the Continuous Air Monitoring Program of the Environmental
Protection Agency, pollutant concentrations are punched into a computer tape
every five minutes. Let tvt2, . . ., tk, . . ., tn denote the instants of time, spaced
five minutes apart, at which concentrations of a certain pollutant, say xt ,xt , . .
., xt . . ., xt are recorded on the tape (Larsen (1969)).
K n
We assume, for now, that the observations represent a time series in which
the successive observations are highly correlated. Consider averages of length k
where k« n.
For purposes of evaluating air quality, it is important to know the
6-1
-------
probability of maximum pollutant concentrations exceeding state standards
which are stated for various averaging times. Let
,-l
+ X
tn
We are interested in obtaining the distribution of 77k n for k moderate and n
large.
A Survey of Results Assuming Independence
If the sequence of observations xt., i = 1,2, . . ., n were independent, as
was assumed by Barlow (1972) and Singpurwalla (1972), we could use extreme
value theory to determine the limiting distribution of Tjk n as a function of the
averaging time k. Under the hypothesis of independence, it is easy to verify that
when the distribution of pollutant concentration, F, is assumed to be either a
normal, a lognormal, a gamma or a Weibull
LimP
n—CO
*
k,n
= exp(-e~*) = A(x)
- CD < x < CO
exists and is nondegenerate, where ak n> 0 and |3k n are a sequence of norming
constants.
LetG(x) = 1-e'*forx>0and
Rk(x) = G 'Fk(x)
where Fk is the k-fold convolution of F with itself. Gnedenko (1943) (cf.
Marcus and Pinsky (1969)) showed that the norming constants could be
expressed as
Rfc'Uogn)
(2)
and
log n) - R^1 (log n)
(3)
6-2
-------
Hence for large n,
k.n
(4)
j3k n is the location parameter and also approximately the 37th percentile of
X -
'k,n
°k,n
and thus provides a convenient way of summarizing 17 k n.
The main difficulty in using j3. occurs in computing the convolution F. .
K f n K
In the case where F is the gamma (normal) distribution, then of course F. is
again a gamma (normal) distribution and there is no problem in computing
Rk(x).
For large n, and k« n, Gurland (1955) has approximated j3k n and ak n
when F is the gamma distribution, i.e.,
x X-l -u/0
For this case
n)' and
~-
(5)
If we let
where
F(x) = O [* ^ ] -00 < x < GO
A
-00
k-u2/2
A/2TT
du
that is, if F is a normal distribution with mean ji and variance a2, then we can
immediately verify (cf. Cramer (1951) pp. 374-375) that for large n
£M~o-k-|/2(2logn)l/2 +/i
and a^n~crk~l/2(2logn)-|/2
(6)
6-3
-------
Barlow (1972) Corollary 4.3 has obtained bounds on ]3kn when F is
continuous, F(0) = 0, and R(x) is convex (concave). He has shown that
(7)
sk)^^) FT1 [ -i- f"1 G(log n)]
where
if i *
f
TU) = l-e-x[ Z Tfl forx>0
K L j=0 J! J
is the gamma distribution and G = F-j .
For example, if we let
F(x)= >-
that is, if F is a Weibull distribution with scale parameter 5 and shape parameter
1/b, then for large n Equation 7 gives
kH8(logn)b< £kn < k~bS(logn)b (8)
when 0 < b < 1 .
Motivation and Summary
Since air pollutant data are often correlated, as will be illustrated in the next
section, the assumption of independence for the sequence xt. , i = 1,2, . . ., n
is clearly incorrect. We can overcome this difficulty if it is reasonable to assume
that the sequence of observations [xtj] is associated. Association is a
strengthening of the concept of positive correlation and is defined and discussed
in the next section. In that section we show that certain air pollutant data can be
modelled by an autoregressive process of suitable order. In the next section and
the one following it, we show that the extreme value approximation function
given by Equation 4 is a lower bound on the distribution function of the
maxima of averages of associated observations. Based on this result, j3k n, (or its
upper bound), is an upper bound on the 37th percentile of the distribution of
the maxima of averages of associated observations.
6-4
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Time Series Models for Air Pollutant Concentrations
Preliminaries
Suppose than n observations
*t,» *t2>-- • . xtk.---.Mn
which are generated sequentially in time represent a discrete time series. We
regard these observations as a particular realization of a stochastic process.
We focus attention on those processes which are strictly stationary. For
such processes, the joint distribution of any set of observations is unaffected by
shifting all the times of observation forward or backward by any integer
amount k. The mean n of the process can be estimated as
I n
"*,?,""
and the variance a^of the process can be estimated as
.2 i r -2
•
The covariance between Xtj and Xti+k is called the autocovariance at lag k,
and is defined as (capital Xtj's are random variables)
yk=Cov[xt.,X,i+k]=E[(X,.-M)(Xti+k-/i)]
For a stationary process, the autocorrelation at lag k is defined as
The most satisfactory estimate of pk is given as (cf. Box and Jenkins
(1969)).
ck
rk = •£"•" (9)
<"o
where
Ck=7TnZ [xtj-x][xti+k-x],k = 0,l,2,...,K.
i=l u
In practice, to obtain a useful estimate of the autocorrelation function, we
would need at least 50 observations and the estimated autocorrelations rk would
be calculated for k = 0,1,2, . . ., K, where K is not larger than n/4. (cf. Box and
Jenkins (1969), p. 33.)
6-5
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Associated Processes and Air Pollutant Measurements
Random variables X (, X2,. . ., Xn are said to be associated if
CovtfCXD, Ar.)U) >0
for all pairs of binary, increasing functions F and A where
£ = ( Xj, X2,..., Xn)
(Binary functions are 0 and 1 valued functions.) Essentially, this is a
strengthening of the concept of nonnegative correlation. The definition is due to
Esary, Proschan and Walkup (1967), who also prove many important properties
of associated random variables. For example, two binary random variables X and
Y, are associated if and only if
Cov (X,Y) >0
This is not true for arbitrary random variables. They also show that independent
random variables are associated.
It follows easily from the definition that increasing functions (not
necessarily binary) of associated random variables are associated. Hence if air
pollutant measurements
Xf,i Xf2, . . . , Xtn
are associated then so are their averages
Now let [XT ; T e D ] be a stochastic process, where D = [1,2,3, . . .] or D =
[0,°°], for example. The process is said to be associated if, for all [T1fr2, • • -,
rn] e D (the TJ'S need not be equally spaced) and all n>1, the random variables
are associated. The definition can be found in Esary and Proschan (1970). They
study special performance processes of interest in reliability theory. It follows
from the definition of an associated process that the autocorrelation function,
p(t) > 0. However, p(t) > 0, does not of course imply in general that the process
is associated. Additional restrictions on p(t) are required, in general, to assure
association.
Air pollutant concentrations follow a diurnal cycle which results in an
autocorrelation which may assume both positive and negative values. Hence we
cannot expect hourly averages to be associated. If we record only the high-hour
daily average the association concept is more reasonable if we also confine
observation to a single season. Figure 1 is a plot of oxidant data for Livermore,
California covering the period June-August 1970. The sample autocorrelation
shown in Figure 2 shows the existence of a 6 - 8 day weather cycle phenomenon.
6-6
-------
Since the autocorrelation shows negative values, it is unreasonable to assume
oxidant values are associated in time according to our definition. However,
oxidant is a secondary pollutant and highly dependent on meteorological
conditions. Figures 3 and 4 are plots of carbon monoxide data for Livermore,
California covering the period June-August, 1970. The autocorrelation seems to
remain positive within the range of sampling error. The assumption of
association may be reasonable for primary pollutants over a time period not
exceeding a season. Also, the less dependent the pollutant is on the weather
cycle, the more likely the assumption of association will be valid. As we shall
see, the association assumption, when valid, will enable us to obtain useful
bounds on quantities of interest.
Ash, Bloomfield and McNeil (1972) have used a fourth root transformation
on S02 data. The resulting data was modelled using a Brownian motion process.
Such processes have independent increments and are always associated, since
independent random variables are associated.
The Autoregressive Process
Most of the time series occuring in practice can be reasonably well explained
by an autoregressive process. In this section, abstracted from Box and Jenkins
(1969), we review some well-known properties of such processes.
The models that are usually employed in time series analysis are based on
the idea that a time series in which the successive values Xt. , Xt2, . . . are highly
dependent, can be regarded as generated from a series of independent shocks
[atj] i = 1,2, .... These shocks are random drawings from a fixed distribution,
usually assumed normal, and having a mean zero and a variance af . The [atj]
process is transformed to the [Xt.] process by what is known as a linear filter.
A [Xtj] process which is extremely useful for representing certain
practically occuring situations is called the autoregressive process. Let Xt.-/u =
*-^/ '
Xt. = 1,2 ..... Then the process
is called an autoregressive process of order p. In the next section we establish
conditions under which an autoregressive process is associated.
If we define a backward shift operator B as
BXtn= Xtn_,
then, the above autoregressive process can be written as
where 0(B) = (1 - 0., B - 02B2 - ... - 0pBP).
The equation 0(B) = 0 is called the characteristic equation of the process.
Several properties of the autoregressive process have been given by Box and
Jenkins (1969). We summarize below a few pertinent ones.
6-7
-------
(a) An autoregressive process is stationary if the roots of its characteristic
equation lie outside the unit circle.
(b) The autocorrelation function pk of an autoregressive process satisfies a
difference equation whose general solution is
where Gj1 are the roots of the
characteristic equation. Thus, the autocorrelation function of an autoregressive
process tails off either exponentially, or as a mixture of exponentials and
damped sine waves, depending on the nature of the roots Gj1 (or equivalently,
the parameters <£j).
(c) If we let
Rp=
and
P\
P2" ' Pp-\
P\ " ' Pp-2
P>-< Pp-Z VV ' '
then 0 = TTp Rp can be used to obtain what are known as the Yule Walker
estimates of the parameters 0j, by replacing the p-{ by their estimates r-t.
(d) The partial autocorrelation function of an autoregressive process is
defined as
I
P\ " •
-3
(10)
Pi
I
.. p
" P
k-l
k-2 rk-3
6-8
-------
For an autoregressive process of order p , the partial autocorrelation function
0kk will be non-zero for k less than or equal to p, and will be zero for k greater
than p.
Estimates for the partial autocorrelation function can be obtained by using
rk in place of pk. If $kk is an estimator of kk then
Vor ( kk) * k > p + I ,
and this can be used to test if the partial autocorrelation function has a cut-off
at lag (p + 1).
Examples
As an example, we consider an autoregressive process of order two. Most of
the time series commonly occuring in practice can be described by this process.
The process can be written as
For stationarity, the roots of 1 — (^B-02 B2 = 0 must be outside the unit
circle. This implies that the parameters $^ and 02 must lie in the triangular
region given by
2 ~
-I «f>2< I
If G~.|1 and G21 are the roots of the characteristic equation, the autocorrelation
function is
When the roots are real (i.e., 0 + 402=** 0). the autocorrelation function
consists of a mixture of damped exponentials. Additionally, if <^>1 and $2 are
both positive, the process is associated and the autocorrelation function remains
positive as it damps out. If the roots are complex the autocorrelation function
6-9
-------
damps out sinusoidally. A necessary condition for the association of an autore-
gressive process of order two with positive coefficients, is that its autocorrelation
function remain positive as it damps out. The coefficients ^ and 02 can De
estimated using the relationships
Application to Carbon Monoxide Data
We estimate the autocorrelation function pk of the carbon monoxide data
given in Figure 3 using Equation 9. The estimates are r^ = .736, r2 = .676, r3 =
.560, r4 = .461, .... The partial autocorrelation function 0k k is estimated using
Equation 10 and replacing the p-t's by the TJ'S, for k = 2 and k = 3. These
estimates are 022 = -294 and 033 = -.018. Since $33 « 0, it is reasonable to
conclude that the carbon monoxide data can be reasonably well described by an
autoregressive process of order 2.
Estimators of the parameters of the autoregressive process 0
6-10
-------
which implies 0 > 0.
Lemma 1:
If 0j > 0 (i = 1,2, . . ., p), then the autoregressive process of order p is
associated. (See also Theorem 2.)
Proof:
Esary, Proschan and Walkup (1967) prove that independent random
variables are associated and also that increasing functions of associated random
variables are associated. Hence
Xf=ot
Xt2s*lflt,*flts
and
are associated. The lemma follows by induction. ||
Clearly, it follows from the lemma and the previous remarks that an
autoregressive process of order 1 is associated if and only if 01 > 0.
Bounds on the Distribution of the Maxima of Averages for Stationary
Associated Processes
Let Xt1 ,Xt2, • • •, X tk,.. ., Xtn be an associated process and
Lemma 2:
Let [Xt1, Xt2, . . .] be a stationary associated process with marginal
distribution F, /3k nand ak n as defined in the introductory section. If Fk is such
that Equation 1 holds, then
•n - < x
'k,n k,n
ak,n
> exp (-
6-11
-------
for sufficiently large n.
Proof:
Let Xv X2, . . ., Xn be associated random variables. Esary, Proschan and
Walkup (1967, pp. 1472-73) prove that
r 1 £
P[Mox(X,,X2,.. , Xn)< xj > II P(Xj
-------
standard will be violated. The standard deviation of this data, a, was estimated as
2.8. Basing our total sample size as hourly observations for 90 days, we have n =
2160, and considering averages of length 8 (because of the 8-hour averaging
time), we take k = 8. Thus
20-8'l/2(2.8)(2loq2l60)"2-5.l6
8-"*<2.8)<2lo, 2I60)-'2
= A (43.39)
n
-(43.39)
< 20) > e~e w I
and this bears out the fact there were no violations of the specified standard.
In the light of the observed data it appears that the specified standard is
unreasonably high.
In general, if F is difficult to convolute and if R(x)= -log [1 - F(x)] is
convex, then
as noted in Equation 7. Let £ 37 be the 37th percentile of P[i?k n< x] . Thus,
even in the presence of association
Additional Associated Processes
It is difficult, in general, to verify that a process is associated from the
definition of association. Another useful concept which implies association is
that of conditionally increasing in sequence.
Definition:
Random variables X1f X2, . . ., Xn are conditionally increasing in sequence if
P(Xj>x| X| = x,,X2= X2,... , Xj_| = Xj_,)
is increasing in x1, x2, • . ., Xj.., for i = 1,2, . . ., n.
A stochastic process is conditionally increasing in sequence if any subset of
random variables based on the process is conditionally increasing in sequence.
6-13
-------
This concept is due to Esary and Proschan (1968) who also proved the
following theorem.
Theorem 1: (Esary and Proschan)
If X1 ,X2, . . -, Xn are conditionally increasing in sequence, then X1 ,X2, - - •,
Xn are associated.
The concept has an immediate application to autoregressive processes of
order p, which, according to Theorem 2, are associated if 0; > 0, for i = 1,2, . . .,
P-
Theorem 2:
Autoregressive processes of order p are conditionally increasing in sequence
if and only if 0; >0, i = 1.2,. . ., p.
Proof:
For an autoregressive process of order p, it is easy to verify that
P[Xtn>xl Xtn-| = xtn-.|'"-»Xtn-p-l = xtn_p.,]
is increasing in xt ^ , . . ., xtn_D_i if ar|d on'y if 0X- ^ 0 for i = 1,2, . . ., p.||
Lemma 3:
If [Xt; t e D] is a Markov process and if
P(xt>x| xs = y)
is increasing in y for s< t, then the process is associated.
Proof:
It is sufficient to prove that the process is conditionally increasing in
sequence, i.e.,
P[xtn>x| X,|rxt|l...,X,n_| = xtn.
But the Markov property implies that this equals
which completes the proof.
6-14
-------
Theorem 3:
A stationary, Gaussian process with autocorrelation function
00
/D(t)=Je'XtdH(X)
0
for some distribution, H, on [0,°°] is associated. (Note that time may be either
continuous or discrete.)
Proof:
It is well known that a stationary Gaussian process is completely determined
by its autocorrelation function together with the marginal mean and variance.
By Lemma 3, the stationary Gaussian Markov process with autocorrelation
is associated pdl s 6
(Note P[Xt > x|Xs = y) = /x.py exp[-u2/[2(1-p2)]] du/(27r(1-p2)) 1/2. )
K
To complete the the proof, let PJ > 0, i = 1,2, . . ., k and 2 PJ = 1 . Also
specify Xj > 0, i = 1,2, . . ., k. Let [Xj(t);t ^ Q] be a stationary Gaussian process
with autocorrelation
for i = 1,2, . . ., k. Assume that the k processes are mutually independent. Since
each process is associated, it follows that the process
k
Yt= I -/Pi Xj(t)
i=l
is associated. (Recall that increasing functions of associated random variables are
associated.) Also
= Cov[Y(t), Y(t+s')]
r k k
= Cov £ yPj Xj (t), £ -/Pi Xj
L in i=l
.-X"
= Z P.,e
By a limiting argument we can show that if
00
e'XtdH(X)
6-15
-------
and the process is a stationary Gaussian process then the process is associated. II
The previous theorem has useful applications to data which is believed to be
generated by a stationary Gaussian process. If we can approximate the sample
autocorrelation function by a convex combination of exponentials then this is
evidence that the process is associated.
Discrete State Markov Processes
An example of a discrete state Markov process is the birth and death process
assuming states [0,1,2, . . .]. Such processes, it turns out, are always associated if
the time variable t > 0, can assume any non-negative value. When such
processes are restricted to integer values they of course remain associated. On
the other hand, a random walk process in discrete time with transition matrix
- b c 0 0 ...
a b c 0 0
0 a b c 0 ...
is associated if and only if b2> a c.
The above remarks follow from
Theorem 4:
If [Xt; t e D] is a Markov process with transition probability matrix (Pjj(t))
which is totally positive in i and j for all t > 0; i.e..
2,j2
(t)
>0
then the process is associated. (The time variable may be either continuous or
discrete.) We assume here that i 1 < i2 and j1 < j2.
Theorem 4 was proved by D. J. Daley (1968).
Karlin (1968) showed that birth and death processes with state space [0,1,2,
. . .] always satisfy the conditions of Theorem 4 and hence are associated. Esary
and Proschan (1970) showed that two-state birth and death processes are asso-
ciated.
6-16
-------
Conclusion
Our objective in this paper has been to present a new and different approach
to the analysis of air pollution data, which can be, and perhaps should be
modelled as a time series. The results presented here are based on more realistic
considerations than those of a similar nature presented before, and should be
useful in setting and monitoring air pollution standards.
Though the primary motivation in this paper has been the analysis of air
pollution data, the results obtained here should be of a more general interest.
The results on associated stochastic processes presented in the section headed
"Associated Stochastic Processes" should have applications in time series
analysis, queueing theory, and reliability theory.
By showing that the extreme value distribution is a lower bound on the
distribution function of the maxima of observations generated by an associated
stochastic process, we have expanded the scope of applications of extreme value
theory. However, the extreme value approximation may be too conservative in
many applications.
Theorem 3 asserts that if [Xt; t > 0] is a stationary Gaussian process and
p(t) can be represented as a mixture of exponentials, then
p[Mox(Xf|lX,2f ... , Xtn) 0
S. M. Berman [Annals of Mathematical Statistics, Vol. 35, pp. 502-516, (1964)]
has shown that, in general, if EXt = 0, EXt = 1 and EXtXt = rn, then
,.. . , Xfn) < x] - ft P[xt.2 's a two dimensional normal density with mean vector Q and
correlation | r j |. Berman further shows that if either
Mm rn logn = 0
or 00
I r*n and a^n given by Equation 6 where // = 0 and
0=1.
6-17
-------
30
28
26
24
r
* 20
§,6
14
12
10
| 08
§06
04
O2
01
.00
June ID
1970
21
30 July I
II
DAYS
21
31 Aug. I 10
Figure 6-1. Oxidant Concentrations in ppm for Livermore, California, June
August, 1970.
.5
.4
.3
.2
.1
-ai
-0.2
12 4 6 6 10 12 14 16 18 20 22 24 26 28
LAG (days)
Figure 6-2. Sample Autocorrelation Function for Oxidant Data From Livermore,
California, June - August, 1970.
6-18
-------
19
18
17
16
E 16
13
Z 12
9 M
5 10
June I
15
July I 15 August I
TIME (day»)
Figure 6-3. Carbon Monoxide Concentrations in ppm for Livermore, California,
June-August, 1970.
Figure 6-4. Sample Autocorrelation Function for Carbon Monoxide Data from
Livermore, California, June - August, 1970.
6-19
-------
Acknowledgement
This research has been partially supported by the Office of Naval Research
under Contract N00014-69-A-0200-1036 and the National Science Foundation
under Grants GP-29123 and GK-23153 with the University of California.
Research supported in part by the Office of Naval Research under
Contract N00014-67-A-0214 Task 001, Project NR 347 020 and the National
Science Foundation Institutional Grant GU3287 with the George Washington
University, D. C. 20006. This work was begun while the author (N.D.S.) was a
visitor at the Operations Research Center, University of California, Berkeley.
Reproduction in whole or in part is permitted for any purpose of the United
States Government.
References
Ash, D., Bloomfield, P., and McNeil, D. R., 1972: On the Statistical Analysis of
Air Pollution Data. Department of Statistics, Princeton University,
Princeton, N. J., Technical Report 19, Series 2.
Barlow, R. E., 1972: Averaging Time and Maxima for Air Pollution
Concentrations. Proceedings of the Sixth Berkeley Symposium on
Mathematical Statistics and Probability, Vol. VI, pp. 433-442.
Berman. S. M., 1969: Limit Theorems for the Maximum Term in Stationary
Sequences. The Annals of Mathematical Statistics. 35: 512-516.
Box,G. E. P., and Jenkins, G. M., 1969: Time Series Analysis Forecasting and
Control. Holden-Dav, Inc., San Francisco, California.
Cramer, H., 1951: Mathematical Methods of Statistics. Princeton University
Press, Princeton, New Jersey.
Daley, D. J., 1968: Stochastically Monotone Markov Chains. Z.
Wahrscheinlichkeitsth. 10: 305-317.
Daley, D. J., 1969: Integral Representations of Transition Probabilities and
Serial Covariances of Certain Markov Chains. J. App: Prob. 6: 648-659.
Esary, J. D., and Proschan, F., 1972: Relationships Among Some Concepts of
Bivariate Dependence. The Annals of Mathematical Statistics. 43:
651-655.
Esary, J. D., and Proschan, F., 1970: A Reliability Bound for Systems of
Maintained, Interdependent Components. Journal of the American
Statistical Association. 65: 329-338.
Esary, J. D., and Proschan F., 1968: Generating Associated Random Variables.
Boeing Scientific Research Laboratories, Doc. D1-82-0696.
Esary, J. D., Proschan F., and Walkup, D. W., 1967: Association of Random
Variables, With Applications. The Annals of Mathematical Statistics. 38:
1466-1474.
Gurland, J., 1955: Distribution of the Maximum of the Arithmetic Mean of
Correlated Random Variables. The Annals of Mathematical Statistics. 26:
294-300.
6-20
-------
Karlin, S., 1968: Total Positivity, Volume I, Stanford University Press, Stanford,
California.
Larsen, R. I., 1969: A New Mathematical Model of Air Pollutant Concentration
Averaging Time and Frequency. J. Air Pollution Control Association.
75/24-30.
.j^
Marcus, M. and Pinsky, M., 1969: On the Domain of Attraction of e"e . J.
Math. Anal. Appl. 28: 440-449.
Singpurwalla, N. D., 1972: Extreme Values from a Lognormal Law with
Applications to Air Pollution Problems, Technometrics. 14: 703-711.
DISCUSSION
Don Pack: The question is as follows—in working with a time series of data you
know that it is contaminated in various ways either by the position of the
sampler or otherwise but I'd like to particularly direct the question toward the
instrumental contamination. We have remarked that the tails of the distribution
are qf particular interest. Yet, if I understand instrumentation, if the dynamic
range of the instrument is just about that of the range of the pollution
concentrations you can expect, the maximum error will occur at the threshold
and at the very high values—such things as poisoning bubblers by a spike of
concentration. Can the statistician clean up the time series distribution through
examination of the instrument characteristics so you don't have to examine each
individual observation for its validity?
Singpurwalla: If I understand your question correctly, I would say possibly yes.
Pack: Do you know how long it takes to go through each one of these? And yet,
you know that there are errors in here. Can you establish the probability that
the observation is real and not instrumental?
Singpurwalla: There are techniques, outlier techniques, or there are probably
techniques of some kind of pattern recognition, discriminant analysis, which
could be used to put a particular piece of data in category A or category B,
where category A might be something that is real, and Category B might be
something that is phony for somebody else. I would imagine yes.
Pack: I simply haven't seen it done. That is why I asked.
Singpurwalla: I think it could be done.
6-21
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-------
7. A STOCHASTIC MODEL FOR ESTIMATING POLLUTANT
EXPOSURE BY MEANS OF AIR QUALITY DATA
ALLAN H. MARCUS
Department of Mathematical Sciences
University of Maryland-Baltimore County
Baltimore, Maryland
Introduction
Air quality data can and should be made more useful in determining the
public health implications of air pollution control strategies. The problem is that
the performance of pollution control strategies is tied to time-averaged pollutant
concentrations at spatially fixed sampling or monitoring stations. Different
individuals in the population receive vastly different exposures from the same
polluted environment. For example, an executive who drives from a nearly rural
suburb to a polluted urban center business district in an enclosed air-conditioned
car, and works in the upper stories of an air-conditioned office building, receives
a much smaller pollutant exposure than does, say, a traffic policeman working in
the same urban center. The executive may actually receive most of his dosage
while waiting in a poorly ventilated parking garage for his car. An individual
exposure thus depends significantly on the personal "trajectory" or movement
of the individual in the urban area through space and time, and on other
hard-to-predict factors. There are also important differences in individual
response to exposure, such as the age and state of health of the person, history
of smoking, and the time required for intake and elimination of pollutants by
various body organs. For this reason, the urban poor (who have to live with a
variety of environmental stresses, and who have a high proportion of children,
elderly, ill and other susceptible types) are particularly vulnerable (U.S.E.P.A.
(1971)). Age, health, income, travel patterns and other elements of "lifestyle"
are interdependent, and make the prediction of exposure and response more
difficult.
The purpose of this paper is to show that many of these questions can be
formulated mathematically in terms of the excursions of filtered stochastic
integrals of pollutant concentration. When pollutant concentrations are
functions of a Gaussian random field, some of the questions raised can be
answered analytically, and most can be studied numerically.
7-1
-------
The data base required to actually use the model for predictive purposes is
enormous. These include: meteorological variables such as wind speed, height of
the mixing layer, and ground-level turbulence; an inventory of the major point,
line and area emission sources; demographic data for the estimation of personal
trajectories for various population types; and reliable dosage-response data for
various pollutants. The advantage of the analytical approach is that it may prove
possible to combine much of the above data into a relatively small number of
parameters which determine the level-crossing properties of the stochastic
integrals. In this way it may prove possible to study simultaneously the
performance of air quality standards and health effects on various segments of
the population of alternative pollution control strategies, without resorting to
extensive (and expensive) computer simulations.
Performance of Air Quality Standards
Air quality standards are defined in terms of average pollutant
concentrations with respect to a specified averaging time T, which are not to
exceed a threshold level I_T more than nT times in a period of length ST. We can
define the air quality standards problem in the following rather formal way. Let
C(R,t) be the pollutant concentration at a point R at a time t. The time-averaged
concentration is
CT(R,t) = ~ f* C(R,u)du
(D
Define NT(R,t) as the number of excursions of CT(R,t) above I_T during the
interval of time (t-ST,t] . That is, for u in the interval (t-ST,t], there will be a
random number NT(R,t) of episodes in which CyfFKu) exceeds LT
continuously. Let D: be the duration of the j^tl such excursion, which starts at
time tj(t-ST LT for tjnT, !0< t< tf]
(3)
7-2
-------
Now, if there are k monitoring stations at points RI, . . ., R|< in the
region, the standards are harder to interpret. What might be meant is either that
the standards are satisfied for all sites, so that the measure of severity of the
pollution problem is
l-p[NT(R|,t) < nT,...,NT(Rk,t)0,
(4)
not exceed Lj more than nj times. Thus, letting Nj(t) be the number of times
Cj(u) exceeds Lj for t-S < unT] (6)
The quantities may differ substantially.
One approach to these problems is by modeling. We could start by assuming
that C(R,t) is a stationary stochastic process, and develop the needed results in
terms of familiar level-crossing probabilities, but this is a very difficult problem
(Marcus (1972)). Even here, what we would need is the matrix of
cross-correlations Pjj(t) between the transformed concentration at R( and
transformed concentration t units of time later at Rj.
For purposes of evaluation of performance probabilities, it may be
sufficient to consider only a two-state stochastic process
IT(R,O = I '* CT(R,t) > LT (7)
= 0 if CT(R,t) < LT
we could then enquire whether the times between successive crossings of level
LT constitute a realization of an alternating renewal process. If so, the intensity
function (i.e., renewal density) and distributions of duration and frequency of
exceedances are easily estimated. We could then answer the usual "quality
control" question of whether or not a certain adversely high concentration
proves that the underlying process is out of bounds.
The analysis of pollutant concentrations as a (multi-variate) time series is
essential with regard to the stationarity of the underlying processes (i.e.,
statistical homogeneity with respect to time). We should examine concentrations
at each site for:
(a) secular variations, such as trends (increase due to increasing regional
population; decrease due to movement of industry or conversion to
low-polluting fuels).
7-3
-------
(b) cyclical variations, including seasonal, weekly, daily, and other regular
periodic climatic or human movements.
(c) other persistent but irregular events.
Some useful first steps in the time series analysis of air pollution data have
been made by Merz, etal. (1972) and by Ash, et. at. (1972). We will discuss these
in more detail in a later section on "Stochastic Models for Pollutant
Concentrations."
Individual Dosage Histories
What we are really interested in are the public health implications of
pollutant control policies. Policies are often tied to physical performance
characteristics of spatially fixed monitoring and sampling stations. These only
indirectly characterize individual physiological response to pollutants. It would
be more directly meaningful to measure cumulative dosage or dosage-response
on an individual history basis. Let P j be the space-time "trajectory" of person i,
i.e., the entire history of his or her movements in the metro region during some
interval of time. Most people travel extensively during the day and may be
exposed to quite different concentrations at various times and places.
There are several possible indicators of total dosage in the interval (tn,tf).
Total dosage is given by
»f
Q(Pj) = / / C(R,u)du dR (8)
»o Pi
If the threshold level L is important, then in terms of an indicator function
IL(C) = I if C>L (9)
= 0 if C
-------
If there is a possible non-linear physiological response at time t due to after-effect
of a pollutant contentration C(R,u) at some point T at an earlier time u, then we
define an after-effect or response function f(C, t-u) so that response at t is
t
r(Pj,t) = f f f [c(R,u),t-u]dudR (11)
-CO PJ
which is a variable, stochastic response to a stochastically variable exposure for
each trajectory. Because of the highly non-stationary nature of the exposures, an
analytical study of the distribution of the indicators Q, QL, DL or r seems less
promising than a simulation study. The required individual trajectories can be
estimated from demographic data.
These simulated histories could be compared with personal pollutant
monitoring devices analogous to individual total radiation dosimeters. It should
then be possible to more adequately evaluate epidemiological studies, e.g., data
collection by the CHESS or CHAMP networks.
Stochastic Models for Pollutant Concentrations
Much of what is called "random" variation in a system is merely due to
ignorance—we often do not know which factors affect the evolution of the
system, or else we know (or suspect) that certain factors are significant, but
cannot relate them precisely to system performance, and so choose incorrect
functional relationships. If important variable factors are not included in
predictions of system performance, they may contribute greatly to the
unexplained "random" variation, and their exclusion could greatly modify the
structure of the statistical data analysis. This is, in fact, the most serious problem
in finding a stochastic model for statistical interpretation of pollutant
concentration data.
The state of the art in predicting urban air pollution by multiple-source
diffusion models was thoroughly explored in a symposium held here in 1969
(Stern (1970)) and the field has continued to develop rapidly. We assume the
usual continuous point- and line-source Gaussian plume dispersion model. Let
the ground-level monitor be affected by np point sources and by nL line sources,
both emitting continuously but with a possibly slowly varying emission rate. Let
the mean wind speed be U m/sec. The ith point source emits Qpj micrograms/sec
of a given pollutant, and is located at elevation Hj and distance Sj meters from
the monitor. Let 4>-t be the angle between the mean wind direction and the line
from the source to the monitor. The downward distance is then Xj = Sj cos fy
and the crosswind distance is y{ = Sj sin 0;. Similarly, let the shortest distance
from the monitor to the ith infinite continuous line source be Rjf and let QLj be
its emission rate in micrograms/sec/m. if 6-t is the angle between the line source
7-5
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and the direction of the mean wind, then Xj = Rj/(sin 6-t) is the downwind
distance of the monitor. Then, defining the crosswind dispersion variance 02 by
CTy2 = Oy2 X2"" d2)
and the vertical dispersion variance a\ by
-------
large number of roughly equal sources, individual source strength and wind
direction may contribute less to the distributional variation of C(t) than does
wind speed and turbulence. The latter situation may be approximated in urban
centers.
Some studies are available which give the distribution of wind speeds and
turbulence in urban environments (Holzwofth (1967); Brook (1972); Luna and
Church (1972)). The wind speed distribution is significantly non-Gaussian and
positively skewed (Brook (1972)), and may possibly be of lognormal form.
However, wind speed and turbulence parameters are strongly dependent. Let aA
be the elevation standard deviation in radians (note that if n = 0, then a = aA
and az = aE). The product oAoE\J (and equivalently, its reciprocal) does not
have a lognormal distribution except for stability classes E and F, corresponding
to low winds and night or overcast conditions (Luna and Church (1972)). These
are, however, conditions of very high air pollution potential. One might suspect
then, that urban air pollutant concentration distributions are a mixture of
distributions, but with an approximately lognormal upper tail. This is also
suggested by Fig. 7 of Holzworth (1967).
Larsen (1971) has shown that the lognormal concentration distribution is
applicable to many sets of observations, such as hourly averages of S02 at the
CAMP monitor in Washington, D. C. from 1961 to 1968. However, examination
of smaller data sets shows that some are well described as lognormal, but others
are not (see e.g., U.S.P.H.S. (1966)). A recent, very thorough study by Ash,
Bloomfield and McNeil (1972) of CO and S02 concentrations in Camden and
Bayonne, N. J. in 1970-1971, shows that the lognormal distribution is not
particularly satisfactory, especially at low concentrations. They suggest that a
"fourth-root Gaussian" distribution may be better (although suggesting a
lognormal distribution for NO20; the fourth-root transformed concentrations
also have the property of approximating a stationary Gauss-Markov
(Ornstein-Uhlenbeck) random process.
What is needed is a combination of the deterministic diffusion modeling and
the purely empirical statistical data analysis. The deterministic model provides a
structural framework for predicting pollutant concentrations, with important
variables such as wind speed and direction, and turbulence parameters, as
predictors rather than unexplained sources of variation. The residual
unexplained variation is the true "random" variation, and understanding its
structure should suggest the most useful statistical analysis methods.
Some Useful Results on Level Crossings and Exceedances
It would be convenient to study the frequency, duration, and intervals
between pollution episodes using well-known results about curve-crossings of
random processes (see Cramer and Leadbetter (1967) for a rather complete
exposition of known results). However, we can use the extensive body of results
7-7
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about Gaussian random processes only if we can first find a monotone
transformation of pollutant concentrations such that the transformed
concentrations approximate a realization of a Gaussian process. Explicitly, let
g(C) be a monotone increasing function of C. We are looking for a representation
g[cT(t)] = o-(T)Z{t)
(15)
where Z(t) is a zero-mean, unit-variance, Gaussian random process with
auto-correlation function Py(t); the parameters may depend on the averaging
time T and implicitly on the location of the monitor. Hence, if LT is the T-hour
standard, a pollution episode occurs if
CT(t) > LT (16)
i.e.,
Z(t)>h= [g(LT)-fL(T)]/T(t) = I - X2(T) t2/2j + X4(T) t4/4| + o(t4) (18)
We then have, e.g.:
(a) The expected number of episodes in (tQ,tf),
E[NT(R,tf-t0)] r X(tf-to) (19)
where
X = [X2(T)/2tr] "2 <£(h) (20)
(b) The expected duration of an episode,
E[D|]= [i-(h)]/X (2D
where
h
x <22a)
-CD
and
) = (2TT)-"2 exp(-x2/2)
Higher moments of these and related random variables have more complicated
formulae. Simple asymptotic results are available if h is very large.
Integral functionals of CT(t) are particularly important for estimating
cumulative dosage and dosage-response. The mean and variance of the so-called
"Zn-exceedance measures" are relatively easy to compute (Cramer and
7-8
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I n
2n(z,T) = / [Z(t)-z] Iz[z(t)] dt/T
Leadbetter (1967)). These are random variables defined by
r i" _ r •}
(23)
L J ' L J
0
where IZ(Z) is the Heaviside step function defined in the previous section on
Individual Dosage Histories. Thus
GO
. «>
(24)
~Z
and a more complicated formula for the variance. In particular, for:
(a) Lognormal distribution, g(C) = logeC
(25)
(26)
uu
E[zn(z,T)] = f (x-z)n <£(x) dx
CT(t) = exp [erZ(t) +
Zn(z,T)= ] dt/T
(28)
Note that if (as is often the case) physiological response is proportional to the
logarithm of the stimulus above a certain theshold level ;u + zo, then Equation 26
is most appropriate for health impact predictions.
These results are readily extended to curve crossings by non-stationary
Gaussian processes. This generalization is needed to discuss the effects of
pollution episode control strategies. An episode is declared on the basis of values
of CT(t) and its time derivative Cy(t) over some interval, or equivalently, on the
basis of the values of the jointly Gaussian process Z(t), Z'(t) during some
interval. Once the episode is declared, the effect of the controls is to decrease
7-9
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the mean value with time. The correlation matrix of the controlled process is the
matrix of partial correlations conditioned on the history of the process up to the
time the control strategy is initiated. The results are rather complicated and
details will be presented elsewhere.
The correlation structure of the process Z(t) is crucial in predicting episode
frequency and duration, but is not well known. Merz, et at. (1972) find that in
Los Angeles, there is a weak, yearly trend in weekly averages of hourly maxima
for oxidants, CO, NO, and HC; superimposed on this are strong semi-annual
(seasonal) and weekly regressions, and a possible bi-weekly regression for CO and
NO. Ash, et al. (1972) find that the fourth-root transformation reduces the
process C.,(t) to a stationary Gaussian random process with approximately
Markov dependence,
/> (t)=exp(-Wltl) (29)
for T = 1 hour, CO and S02 in Camden and Bayonne in 1970-1971.
Unfortunately, Equation 29 cannot apply for small times, since it implies that
the process Z(t) is not differentiate. One possible solution is that Z(t) is doubly
stochastic, and that w is itself a random variable. Larsen has observed (1971)
that the standard deviation and the maxima of log C-r(t) are very slowly
decreasing power functions of T. This suggests (Marcus (1972)) an average
correlation function for large times t of the form
p (t) = a|t|~b + 0(|t|~b) 0 I hour
' o
7-10
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The predictions are:
Standard
Primary
Secondary
Secondary
Averaging
Time T, hr
24
3
24
Standard
LT, ppmSO2
0.14
0.50
0.10
Avg. No. Yearly
Exceedances
1.95
0.20
6.74
Avg. Episode
Duration, hr
210
7.84
176
These predictions could, in principle, be compared with observations, e.g.,
Highway Research Board (1966). Unfortunately, my research has not been
funded or supported and I do not personally have the time or computing
resources needed to carry out the data analysis. The predicted values are typical
and plausible.
The importance of the correlation structure of the process, and the
formulation of health impact problems in terms of stochastic integrals, suggests
that it may not be useful to study only the maxima of a series of independent
random variables (e.g. Barlow (1971) and Singpurwalla (1972)). Maximum
concentrations, and maxima of integral functionals, are of considerable interest,
but it would be more informative to study them for non-stationary correlated
random processes.
Applications to Human Populations: Some Problems
One of the first problems is that of defining an appropriate physiological
response function for exposure to pollutant concentrations which vary with
time. This depends significantly on the pollutant, the most sensitive organ, the
time scale and method of elimination as well as other factors. These are
discussed in a review by Saltzman (1970). See also Rossin and Roberts (1971).
The second problem is the personal trajectory estimation for different types
of individuals. We first need to classify individuals by potential health effects
(the preceding problem), and then to relate these to demographic characteristics
of the individual—age, sex, income, state of health, occupation, etc. These
demographic factors, and an inventory of land uses in the metropolitan region,
largely determine the personal trajectory. The techniques for doing this are an
essential part of travel demand forecasting (e.g., Hanafani (1972); Wooten and
Pick (1967); Highway Research Board (1972)). A recent, very useful approach
involves the estimation of personal trip patterns as a Markov chain (Sasaki
(1972)).
7-11
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Finally, meteorology and human activities interact in complicated ways not
readily accessible to modeling. For example, on a hot, calm, overcast summer
day (light winds and stable turbulence conditions being conducive to high
pollutant concentrations), an unusually large number of people might absent
themselves from downtown offices, reducing motor vehicles emissions
downtown but possibly increasing them near roads leading to parks and beaches.
Power plant emissions might also change as a result of the redistribution of air
conditioning demands, etc. A severe winter storm would introduce a different
set of interactions.
The interpolation and extrapolation of pollution in space and time from air
quality data at fixed monitors is not an extremely difficult problem. The
prediction of multivariate time series is well known (Cramer and Leadbetter
(1967)). The extrapolation of spatial random fields can be conveniently done by
the use of empirical eigenvectors Peterson (1970) (1972), thus also exposure
along trajectories.
References
Highway Research Board, 1972: Transportation Demand and Analysis
Techniques. 18 reports. Highway Research Record No. 392.
Ash, D., Bloomfield, P., and McNeil, D. R., 1972: On the statistical analysis of
air pollution data. Statistics Dept., Princeton Univ., Princeton, N. J., Tech.
Rept. 19, Ser. 2.
Barlow, R. E., 1971: Averaging time and maxima for air pollution
concentrations. Univ. of Calif., Berkeley, Calif., Operations Research
Center Rept. ORC-71-17.
Brook, R. R., 1972: The measurement of turbulence in a city environment. J.
Applied Meteorology. 11: 443-450.
Cramer, H., and Leadbetter, M., 1967: Stationary and Related Stochastic
Processes. John Wiley, New York, N. Y.
Hanafani, A. K., 1972: An aggregative model of trip making. Transportation
Research. 6: 119-124.
Holzworth, G. C., 1967: Mixing depths, wind speeds and air pollution potential
for selected locations in the United States. J. Applied Meteorology. 6:
1039-1044.
Larsen, R. I., 1971: A Mathematical Model for Relating Air Quality
Measurements to Air Quality Standards. U. S. Env. Prot. Agency Publ.
AP-89.
Luna, R. E., and Church, H. W., 1972: A comparison of turbulence intensity and
stability ratio measurements to Pasquill stability classes. J. Applied
Meteorology. II: 663-669.
Marcus, A. H., 1972: Air pollutant averaging times: Notes on a statistical model,
and predicting the frequency and duration of air pollution emergencies: A
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statistical model. Johns Hopkins Univ., Baltimore, Md., Statistics Dept.
Tech. Repts.
Merz, P, H., Painter, L. J., and Ryason, R. R., 1972: Aerometric data
analysis-time series analysis and forecast and an atmospheric smog
diagram. Atmospheric Environment. 6: 319-342.
Peterson, J. R., 1970: Distribution of sulfur dioxide over metropolitan St. Louis
as described by empirical eigenvectors and its relation to meteorological
parameters. Atmospheric Environment. 4: 501-518.
Peterson, J. T., 1972: Calculations of sulfur dioxide concentrations over
metropolitan St. Louis. Atmospheric Environment. 6: 433-442.
Rossin, A. D., and Roberts, J. J., 1971: Episode control criteria and strategy for
carbon monoxide. Paper 71-55, presented at 64th Annual Meeting of the
Air Pollution Control Association.
Saltzman, B., 1970: Significance of sampling time in air monitoring. J. Air
Pollution Control Association. 20: 660-665.
Sasaki, T., 1972: Estimation of person trip pattern through Markov chains. Proc.
Fifth International Symposium on Traffic flow Theory and
Transportation, G. W. Newell (ed.), American Elsevier, New York, N. Y.
Singpurwalla, N. D., 1972: Extreme values from a lognormal law with
applications to air pollution problems. Technometrics. 14: 703-712.
Stern, A. C. (ed.), 1970: Proceedings of Symposium on Multiple-Source Urban
Diffusion Models, U. S. Env. Prot. Agency Publ. AP-86.
U.S.E.P.A., 1971: Our Urban Environment and Our Most Endangered People.
Report, U. S. Environmental Protection Agency.
U.S.P.H.S., 1966: Continuous Air Monitoring Program, Washington, D. C.,
1962-1963. U. S. Public Health Service Publ. AP-23.
Wooten, H. J., and Pick, G. W., 1967: A model for trips generated by
households.^. Trans. Econ. Policy. 1: 137-153.
DISCUSSION
Joseph Vasalli: I noticed that in a great many of the purely statistical papers that
have been presented there's been a concern with the time variance of air
pollution concentration. If you think of it in a slightly different fashion there is
a spatial variation that is superimposed on the time variance such that if you are
looking at a time-series average concentration over an area with time, you get a
band instead of a line. I am wondering what is the effect of superimposing the
spatial variance on the time variance?
Marcus: You mean a spatial variance in the individual movements or what? I
don't quite follow that.
Visalli: If you attempt to sample over an area, instead of at one point . . .
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Marcus: Yes. You are right. If you attempted to sample over an area instead of
at a single point you would have a problem. That would be exactly analogous in
a formal sense to the moving average problem that we've had in here. That is, if
instead of an instantaneous spatial point concentration, you were somehow
able to simultaneously accumulate measurements over a very large area, then
you would have the two dimensional analog of the moving average sort of
formulation we have here. You would have a moving average with respect not
only to the time axis, but also to three dimensions. This sort of thing could be
dealt with if you went over to, say, Gaussian random processes with
multi-dimensional index sets, say, time and three space dimensions. In
laboratory studies of turbulence, this kind of representation is necessary.
Unfortunately it becomes very difficult to do anything in the spatially
completely isotropic and homogeneous case, and this doesn't describe very
many cities I know—Los Angeles, maybe. Even in Los Angeles there are places
that are distinctive from other places.
Ralph Larsen: Your observation that if 1-hour concentrations of a pollutant are
lognormal then the 24-hour observations by theory cannot be
lognormal, but that the fit may be still good to lognormal, is confirmed in
another field by^R. L. Mitchell in the September, 1968, issue of the Journal of
the Optical Society of America, in an article titled "Permanence of the
lognormal distribution." His abstract states that the distribution of the sum of
lognormal variates is shown for most cases of interest to be accurately
represented by a lognormal distribution instead of a normal or Rayleigh
distribution that might be expected from the Central Limit Theorem. Then he
goes on to show in his analysis that he does tend to get summations which look
lognormal, but they're not perfect, they're just quite close.
Marcus: I wasn't aware of that paper, I would be interested in seeing it. The
lognormal distribution has a number of strange properties which haven't come
out yet, which I think should be mentioned. It doesn't have a moment
generating function that has a unique inverse, which is rather embarassing in
some applications. As far as heavy tailed, I wasn't aware that the sums of
lognormal variates don't coverge to normality very quickly. I suppose it
shouldn't be too surprising. I'd like to believe in the Central Limit Theorem. But
there is another family of heavy tailed distributions which hasn't been
mentioned yet, the so-called stable distribution laws, which have a number of
extremely awkward properties like having infinite variance and, in many cases,
infinite mean values. On the other hand, besides being a very heavily skewed sort
of distribution, they do have one good property and that is that sums, or more
generaHy, moving averages, of stably distributed variates have a stable
distribution law. This gives us a useful kind of reproductive property. The
problem about dealing with distributions that don't have a theoretical finite
variance I find rather horrifying and would prefer not to look into. The question
of the underlying distribution structure is one I absolutely haven't discussed. I
did it in the paper a little bit. I even tried to go into how, starting out from a
7-14
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fundamental diffusion model, you could try to derive a lognormal distribution
either by assuming that some of the components of the concentration like
reciprocal wind speed, and reciprocal product of the azimuthal times the
elevation standard deviation times wind speed might be approximately
lognormally distributed. But when you have a large number of point, line and
atea sources, as you do in an urban region, you have a combination of both
multiplicative and additive factors which will give you a distribution which is not
evidently lognormal or anything else for that matter, and I don't know how to
handle that. I'm afraid you have a mixture of distributions with a heavy tail, and
that's about all I can say.
Benarie: Being an engineer and not at all a mathematician I would have checked
the theory by available monitors. In radiation contamination protection personal
monitors that are movable with the person are very extensively used, which
could be used as a check of your theory first of all. In industrial hygiene there
are several types of portable paniculate monitors and it should be checked on
them.
Marcus: I thoroughly agree with that, and the analogy with the radiation
dosimeters is perfect.
7-15
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8. EVALUATING CONFORMITY WITH TWO-POINT AIR
QUALITY STANDARDS, POLLUDEX*
HAROLD E. NEUSTADTER AND STEVEN M. SIDIK
Lewis Research Center
National Aeronautics and Space Administration
Cleveland, Ohio
and
JOHN C. BURR, JR.
Air Pollution Control Division
City of Cleveland, Ohio
Introduction
This report presents the results of various statistical analyses of data
obtained by the Air Pollution Control Division (APCD) of Cleveland, Ohio. It
contains a tabulation of averages, statistics relevant to lognormal distributions,
and good ness-of-fit statistics. In addition, a pollution-level index is introduced
which relates the measured pollution levels over a year to the existing air quality
standards.
The air sampling program of APCD is currently in its sixth year.
Twenty-four-hour samplings have been made of total suspended particulate
(TSP) since January 1967, and of nitrogen dioxide (NO2) and sulfur dioxide
(SO2) since January 1968. The sampling methods used are high-volume air
sampling, Jacobs-Hochheiser, and West-Gaeke, respectively. The geographic de-
ployment of sampling sites is shown in Figure 1. The meandering heavy line in
the center of the city is the Cuyahoga River, about which is centered most of the
region's heavy industry.
At present, there are 21 stations monitoring the air. Fifteen of these stations
monitor all three pollutants, while the remaining six (stations O to T in Figure 1)
"This paper has also been released as a LeRC publication, NASA TN D-6935 entitled
"Statistical Summary and Trend Evaluation of Air Quality Data for Cleveland, Ohio, in
1967 to 1971: Total Suspended Particulate, Nitrogen Dioxide, and Sulfur Dioxide."
8-1
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measure TSP only. Seventeen of these sites have been in operation for more than
5 years. Stations B, D, K, and N have undergone relocation since their initial
installation. However, because of the proximity of their present sites to their
former sites, we have assumed that essentially the same environment has been
measured throughout the 5-year period. Currently, the air is sampled every third
day, although the sampling frequency has varied over the 5 years and has been as
low as once-a-week. Some of these data have been presented elsewhere in a more
preliminary manner (Neustadter, et al. (1972)). The data analysis reported
herein was performed by the Environmental Research Office of the NASA Lewis
Research Center (LeRC) as part of the preliminary phase of a joint APCD-LeRC
program to study trace elements and compounds in airborne paniculate matter.
Cleveland Aerometric Data
Pertinent results are presented in Tables I, II, and III for TSP, N02, and
SO2, respectively. In each table, the first column gives an alphabetic designation
of the monitoring site corresponding to the code shown in Figure 1. The second
column lists the various parameters of interest for each of the pollutants. These
parameters are (a) number of days observed (readings); (b) geometric (TSP) or
arithmetic (SO2 and IMO2) averages; (c) standard geometric deviation; (d)
estimated value of the second largest pollution level for the year; and (e) an
adjusted Kolmogorov-Smirnov goodness-of-fit statistic for lognormality, denoted
as(N)1/2D.
Air quality standards are set nationally by the Environmental Protection
Agency (EPA) of the Federal Government (Anon. (1971)) and statewide by the
Air Pollution Control Board of the Department of Health (DoH) of the State of
Ohio (Ohio, (1972)). Whenever these two standards differ, we have chosen to
work with the DoH (more stringent) standard, which is listed in the third
column. In the remaining five columns are the various statistics for each of the
years 1967 to 1971.
Number of Readings
For each pollutant, both EPA and DoH require a minimum of one sampling
every sixth day, or an equivalent set of at least 61 random samples per year.
Thus, we designate this standard as > 60 in the tables. Even though early in the
program some stations did not achieve 60 samples per year for each pollutant,
we have included the analyses of these data sets in this report. At present, the
nominal schedule of APCD calls for monitoring the environmental air every
third day. In practice, this procedure generally allows sufficient margin for
unanticipated disruptions (e.g., equipment failure) while still exceeding 60
readings per year.
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Geometric and Arithmetic Averages
The geometric average is used in Table I, and the arithmetic average is used
in Tables II and III. This corresponds to the particular averaging method
stipulated by EPA and DoH standards. Calculations were performed whenever
the number of readings exceeded 10. The values listed as standards are the DoH
primary standards, which correspond to the EPA secondary standards.
Standard Geometric Deviation (SGD)
It has been noted that, irrespective of sampling duration or location, air
sampling data are generally distributed lognormally (Larsen (1971)). When such
is actually the case, the entire data set is sufficiently described by its geometric
average and SGD. The higher the SGD, the greater the spread between the lower
and higher values. As with the averages, SGD was calculated for data sets of
more than 10 readings.
Second Largest Value
Both EPA and DoH standards for TSP and SO2 specify that a certain level
of pollution is ". . . not to be exceeded more than one time per year." This
implies that for the 365 daily pollution levels per year (366 for leap years), there
is no upper bound on the largest single level. However, the next largest value
(i.e., the second most polluted day of the year) is required to be at or below the
standard. Thus, Tables I, II, and'lll include estimates of the second highest
pollution level for each year. As with the averages, the Values listed here are the
DoH primary standards, which correspond to EPA secondary standards. While
N02 has only a standard for the annual average, we believe the estimated second
largest level for a year is useful information and we have included it in Table II.
An approximation to the second largest pollution le"vel estimate, for a year
of n days, and a sample of N observations, is obtained by the following
procedure. (The transformation to the logarithms of the data values is made
because the expected values of normal order statistics are well developed in the
literature, whereas we are not aware of any comparable development for
lognormal distributions.) The logarithms yf = In(Xj) of the pollution levels Xj are
computed. According to the assumption of lognormality, the y; values follow a
normal distribution. The sample mean y and sample standard deviation s of the
set of logarithms are computed. From Harter (1961), the expected value of the
second largest observation in a sample of 365 (366 in a leap year) independent
values from a normal distribution is 2.63 (to three significant digits) standard
deviations from the mean. This value, along with the average y and the standard
8-3
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deviation sy of the set of logarithms, is used in the following equation to obtain
the estimate of the second largest pollution level of the year:
y2nd= y + 2-63sy (D
The values of x2nd listed in Tables I, II, and III are obtained by exponentiation,
as
*2nd =
Because of the decreased precision which occurs when extrapolating to the
tail of a distribution and because the sample mean and standard deviation are
used, the minimum number of readings for this calculation was increased to 30
as opposed to 10 used for the averages. Implicit in using Equation 1 is the
assumption of lognormality of the data, which leads us to the final entry in these
tables.
Kolmogorov-Smirnov Statistic
The Kolmogorov-Smirnov statistic is a goodness-of-fit statistic which can be
applied to any distribution (Noether (1967)). In testing for a lognormal
distribution, it is easier for calculation purposes to take the logarithms of the
values and test for goodness-of-fit to a normal distribution. This statistic was
originally intended for use when the distribution which the data is suspected of
following is completely specified. For the normal distribution, this is equivalent
to knowing the mean ju and the standard deviation a. In this case, the
Kolmogorov-Smirnov statistic is denoted D and is calculated as
max
i=l,
(3)
where the function 3>(z) denotes the cumulative standard normal distribution
function.
The statistic D measures the maximum deviation of the observed cumulative
distribution function from the theoretical cumulative distribution function.
Thus, D is always a value between 0 and 1. A value of 0 would indicate a perfect
fit of the sampled data to a lognormal distribution, and larger values indicate an
increasing deviation from lognormality.
When the mean and the standard deviation are unknown, it is common to
use the estimates y and sy = [2j(yj - y)2 /(N - 1)]1/2 in place of /i and a.
Lilliefors (1967) has studied the use of the Kolmogorov-Smirnov statistic in this
situation. Table IV of this report presents the significance levels of (N)1/2D
from Lilliefors (1967) for samples of N > 30. Thus, the statistics in Tables I, II,
and III are presented as (N)1/2D.
8-4
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It should be recognized that the observed pollution levels are but a sample
of levels from some distribution. Thus, even if the distribution of the complete
set of pollution levels is indeed lognormal, some of the samples will lead to large
values of (N)1/2D. The interpretation of the tabulated significance levels a is
that if the distribution is indeed lognormal, then about 100a percent of the
samples tested will lead to a value of (N)1/2D which exceeds ((N)1/2D)a,
whereas about 100(1 - a) percent will lead to a value of (N)1/2D lower than
((N)1/2D)a. Because subsequent calculations in this report depend heavily on
the assumption of lognormality, the value of a = 0.20 was chosen. Choosing this
large value for a has the drawback of rejecting the assumption of lognormality a
substantial proportion of the times that the distribution is lognormal. However,
it has the compensating advantage of being more discriminating against
distributions which are not lognormal.
Lognormality
Lognormal Plots
As a graphical means of assessing the goodness-of-fit of the data to a
lognormal distribution, we can enter the observed data on lognormal probability
graphs. Figures 2 and 3 show two plots for TSP. The solid line indicates the plot
of the cumulative sample distribution of all measurements over the 5-year
period. The data points present the separate sample distributions for the 5 years
(1967 to 1971). Any steady increase or decrease in the pollutant concentration
would be discernible as a vertical sequence of the data points representing those
years. In the two cases shown, there is no overall trend. Figure 2 is for station I
in the industrial valley. The overprinting of the data points shows the TSP levels
to be fairly uniform at a rather high average level for the 5-year period. Figure 3
represents station K, in a residential neighborhood, predominantly upwind from
the industrial region.
A full set of lognormal curves for all 21 stations for the 3 pollutants is
available on microfiche from the authors upon request.
Goodness of Fit
To indicate the decreasing likelihood of lognormality as (N)1/2D increases,
all values calculated on the assumption of lognormality for which the
goodness-of-fit statistic is outside the 20-percent confidence level (i.e., the data
having (N)1/2D > 0.736) are footnoted in the Tables. For a further indication of
lognormality, as well as for a check on the consistency of our data, we examined
the distribution of sets for which (N)1/2D > 0.736.
8-5
-------
Table V summarizes the results of the goodness-of-fit tests in which the a -
0.20 significance level was used. The first column lists the station identification.
The remaining columns list for each of the pollutants the number of yearly tests
which were performed, and the number of these tests which rejected the
assumption of lognormality. For TSP, there are 85 tests, of which 20 were
rejections. This is very close to the expected number of rejections and implies
that the distribution of TSP may very safely be considered to be lognormal. For
NO2 and S02, however, there are more than twice as many rejections as would
be expected, and hence their close:ness to a lognormal distribution is somewhat
suspect. On the basis of an examination of the lognormal plots of SO2 and the
fact that the SO2 departure from lognormality, as indicated by (N)1/2D, is not
severe, we will proceed on the assumption that the lognormal is still a useful
approximation to the distribution of SO2.
Further examination of Table V shows that the lognormality of TSP, SO2
and NO2 is most questionable at stations E, F, and I. Benarie (1970) and
Mitchell (1968) have each considered the additivity of lognormal distributions.
Mitchell has shown that under certain conditions the sum of independent and
identically distributed lognormal variates also follows a lognormal distribution.
Benarie has considered a more general situation, where the lognormal variates
have differing geometric means and standard geometric deviations. His
conclusions are that when a large number (>10) of lognormal variates with
slightly differing geometric means are superimposed, the resulting distribution is
still well approximated by a lognormal distribution. However, when a small
number (<10) of lognormal variates with differing means are superimposed, the
resulting distribution generally is not a lognormal. Thus, it is possible to assume
that pollution levels at stations E, F, and I are dominated by a small number of
major sources, whereas the remaining stations reflect the influence of either a
single large source or a superposition of many sources.
Air Quality
Among the goals of APCD are monitoring of the environmental air,
determination of its quality, and initiation of action to improve the local air
quality, where indicated. There are well established techniques for analyzing
lognormal plots to extract information pertinent to determining compliance
with air quality standards and/or the existence of long-term trends (Larsen
(1971)). However, it is often desirable to have available some single number, or
index, which presents as simply as possible a maximum of information. To this
end we have developed an index, which we call Polludex, which gages the
conformity of the measured environment to the established standards.
8-6
-------
Polludex, An Air Pollution Index
Many indices have been proposed and a number are in use by various
agencies (Babcock (1970)). Polludex is a variation of an index proposed by Pikul
(1971). The rationale for constructing this modified index is as follows. The
standards for TSP and S02 specify values for the annual mean which may not be
exceeded and also values which may not be exceeded more than once per year.
In relation to a lognormal plot of the underlying population, these standard
values specify the coordinates of two points on a straight line. If the data
obtained during a 1-year period conform to lognormality and conform to the
required standards, the plot of the data will closely approximate a straight line
falling entirely below (or on) the line segment joining the standard points.
For each of the three pollutants, define
Sample average
Standard for average
Estimate of second largest level
s =
Standard not to be exceeded more than once yearly
Then Polludex, P (pollutant), is defined for TSP and SO2 by
P(TSP, S02) = 50 x [max(0,r-l)+ max(0,s-l)l (4)
and for NO2 by
P(N02) = IOO x mox(0,r,-l) (5)
where max(a,b) means that the larger of the two values, a or b, is to be used. The
geometric average is to be used in calculating r for TSP and the arithmetic
average is to be used in calculating r for SO2 and NO2. For the estimate of the
second largest level to be used for s, we used the approximate value listed in
Table I for TSP and in Table III for S02.
With this definition, the same weight is given to the long-term (chronic)
effects of pollution as is given to the severe short-term (episode) incident. The
standards for these pollutants have presumably been set with regard to
maximum acceptable levels for reasons of public health and/or welfare. Thus, we
assume that normalization of the estimated mean and second highest values by
the standards will, in a sense, put each P on an equal basis with respect to the
potential harm caused by excesses. If the air quality is equal to or better than
the standards, Polludex = 0. A value of Polludex =100 can be understood to
mean that the air is, in a sense, 100 percent polluted, in that a value of 100 is
obtained when the average and the second highest values are each 100 percent
8-7
-------
higher than their respective permissible levels. Of course, Polludex = 100 would
also result from a continuum of other combinations, as, for example, when the
second highest value is three times its standard, provided the average was at or
below its standard. Figure 4 graphically illustrates several of these possibilities.
Figure 4(a) shows three possible examples which have P = 0. Figure 4(b) shows a
line having P = 100, where both the mean and second largest standards are
exceeded. Figure 4(c) shows a line where again P = 100, but the standard for the
mean has been met. Finally, Figure 4(d) shows a line with P = 50, where the
standard for the mean is not met but the other standard is.
Four-Year Trends
Polludex was evaluated for the APCD data and is listed for all three
pollutants in Table VI. The State of Ohio standards were used in these
calculations.
Where there are adequate data, the 1968 and 1971 values are also presented
as bar graphs overprinted on the Cleveland map. The Polludex values for TSP,
NO2, and S02 are shown in Figures 5(a), (b), and (c), respectively. If there are
two bars, the left bar represents 1968 and the right bar 1971. With the exception
of site M of Figure 5(c), a single bar represents 1971. It is clear that, in general,
TSP levels have increased to the west of the Cuyahoga River and decreased to
the east. The most pronounced improvements are downwind of the valley (in
Cleveland, the winds are predominantly out of the southwest) at sites A, I, and
E. The levels of N02 show much less variation, except for the increased levels at
sites H and C. With one exception, there has been a significant reduction in the
levels of S02 throughout the city, with the most pronounced improvements
occurring, as with TSP, at sites A, I, and E. Since space heating is fueled
primarily by natural gas, this implies a reduction in SO2 contamination by
industrial and power-producing sources. At this time we do not have sufficient
information to determine whether the improvements in the valley are due to the
general decline in business activity in recent years, the abatement efforts by the
industrial community, both of these reasons, or, possibly, neither of these
reasons.
Concluding Remarks
Air quality data (total suspended particulate, nitrogen dioxide, and sulfur
dioxide) for Cleveland, Ohio, for the period of 1968 to 1971 have been collated
and subjected to statistical analysis. It is apparent that the data for total
suspended particulate and, to a lesser degree, the data for sulfur dioxide and
nitrogen dioxide are lognormally distributed. The air quality standards of the
8-8
-------
State of Ohio are met only sporadically by sulfur dioxide in isolated residential
neighborhoods. The available data indicate that definite improvement in air
quality has taken place in the industrial region. Overall, there appears to be a net
improvement in air quality, which would be a reflection primarily of the striking
reduction in sulfur dioxide levels.
A pollution index has been introduced which directly displays information
regarding the degree to which the environmental air conforms to the mandated
standards for the environment. As such, it is a useful tool in air quality
monitoring programs.
8-9
-------
Table I. Total Suspended Particulate Data Summary for 1967 to 1971
Monitoring
station (m
fig. 1)
A
B
C
D
E
F
G
H
|
1
Statistic
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-f it statistic, (NP^D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-f it statistic, (N)1'2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-f it statistic, (N)1'2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-f it statistic, (N)1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-f it statistic, (N)1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-f it statistic, (N)1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-f it statistic, (N)1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
Standard
>60
60
150
>60
60
150
>60
60
150
>60
60
150
>60
60
150
>60
60
150
>60
60
150
>60
60
150
>60
60
150
1967
19
190
1.4
36
112
1.5
351
0.76
64
124
1.6
343
0.55
44
134
1.5
371
0.37
61
139
1.4
352
0.59
64
101
1.5
303
1.0
8
55
210
1.4
a543
1.08
1968
70
242
1.7
919
0.53
64
104
1.6
349
0.72
79
121
1.6
a429
0.76
72
126
1.5
390
0.42
75
147
1.5
a410
0.83
75
103
1.6
357
0.67
75
99
1.6
317
0.56
65
83
1.6
280
0.53
75
232
1.5
694
0.60
1969
73
199
1.6
"711
0.84
66
94
1.4
226
0.63
72
107
1.6
346
0.50
74
123
1.5
378
0.50
75
119
1.4
276
0.61
75
88
1.6
297
0.64
73
82
1.6
a292
0.79
68
84
1.6
299
0.59
75
223
1.5
8639
0.97
'970
76
188
1.6
a682
0.81
b?2
113
1.6
370
0.48
97
124
1.6
420
0.39
b62
154
1.6
487
0.40
93
136
1.5
a395
0.80
82
109
1.5
307
0.07
103
94
1.7
358
0.59
96
94
1.7
384
0.48
101
225
1.5
701
0.51
1971
69
183
1.7
730
0.73
63
92
1.6
319
0.53
89
121
1.7
502
0.65
C30
163
1.8
80
120
1.5
a328
0.80
74
105
1.5
304
0.72
83
91
1.6
337
0.57
70
89
1.7
352
0.68
93
196
1.6
a658
0.83
8-10
-------
Table I (cont'd). Total Suspended Particulate Data Summary for 1967 to 1971
Monitoring
station (see
Fig. 1)
j
K
L
M
N
O
P
Q
Ft
S
Statistic
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (NI1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (Nt1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Standard
>60
60
150
>60
60
260
>60
60
260
>60
60
260
>60
60
260
>60
60
150
>60
60
150
>60
6O
150
>60
60
150
>60
60
150
1967
63
174
1.5
474
0.62
74
53
2.5
399
0.55
53
50
1.9
220
0.72
69
92
1.5
265
0.62
62
135
1.4
343
0.71
63
105
1.5
310
0.62
57
81
1.6
265
0.44
1968
76
161
1.6
"538
0.78
75
58
2.1
320
0.57
73
55
1.9
235
0.67
75
86
1.6
298
0.39
74
139
1.5
390
0.40
69
95
1.5
277
0.42
72
80
1.7
304
0.69
1969
74
151
1.7
"613
0.76
b105
&
1.9
258
0.64
42
157
1.7
569
0.62
98
58
2.3
309
0.67
35
68
2.6
a548
0.76
72
79
1.6
"270
0.83
•72
?27
1.6
407
0.64
70
96
1.4
241
0.67
65
81
1.6
285
0.52
1970
103
156
1.6
"530
0.98
81
49
2.4
8 359
0.83
79
1«I6
2.6
a1013
0.98
58
41
2.6
"372
0.74
81
72
2.9
"755
0.90
90
89
1.7
333
0.71
93
137
1.5
412
0.55
88
106
1.8
8495
0.97
90
89
1.6
309
0.49
1971
90
163
1.7
645
0.73
78
92
1.6
312
0.52
73
212
1.6
637
0.64
72
82
1.6
284
0.59
86
138
2.0
905
0.71
76
90
1.8
422
0.55
74
146
1.4
371
0.60
79
101
1.4
256
0.65
66
89
1.7
384
0.60
51
92
1.5
290
0.71
8-11
-------
Table I (cont'd). Total Suspended Paniculate Data Summary for 1967 to 1971
Monitoring
station (see
Fig. 1) Statistic Standard 1967 1968 1969 1970 1971
T
U
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2b
Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
>60
60
150
>60
60
150
41
170
2.0
1014
0.48
d34
114
2.3
137
0.55
aThe calculation used to obtain this estimate assumed lognormality despite 60
100
>60
100
>60
100
>60
100
>60
100
>60
100
1968
71
211
1.4
517
0.60
76
177
1.5
a495
0.87
55
207
1.4
497
1.65
69
203
1.4
497
0.70
47
212
1.4
a511
0.78
1969
73
220
1.4
470
0.57
75
248
1.3
a454
0.88
70
219
1.3
424
0.70
74
237
1.3
a437
0.90
74
197
1.3
a370
0.76
1970
84
214
1.4
464
0.61
9
115
234
1.4
a576
0.88
b83
217
1.5
576
1.03
108
217
1.4
a504
1.39
96
215
1.3
444
0.70
1971
86
202
1.5
538
0.59
81
190
1.5
a539
0.77
96
255
1.6
835
0.64
C47
205
1.6
686
0.62
9b
205
1.6
a686
1.69
86
203
1.5
a518
0.93
8-12
-------
Table II (cont'd). Nitrogen Dioxide Data Summary for 1968 to 1971
Monitoring
station (see
Fig. 1) Statistic
G Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
H Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
I Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
J Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
K Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
L Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
M Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
N Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
U Number of readings
Geometric average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1|/2D
Standard 1968
>60 72
100 201
1.5
571
0.56
>60 66
100 166
1.5
a471
1.03
>60 67
100 247
1.4
535
0.45
>60
100
>60 74
100 162
1.5
433
0.53
>60
>60 55
157
a1'4
342
0.80
>60
>6U
100
1969
72
221
1.3
a432
0.91
71
225
1.3
a443
0.75
76
253
1.3
495
0.71
52
225
1.4
488
0.65
74
192
1.4
417
0.67
74
168
1.3
335
0.60
1970
104
224
1.3
453
0.43
114
213
1.4
464
0.70
111
238
1.3
a495
1.1
113
255
1.4
a548
0.82
b104
209
1.4
a486
0.76
41
220
1.4
513
0.68
96
176
1.3
341
0.65
39
208
1.6
647
0.65
1971
89
203
1.5
516
0.65
78
202
1.6
a633
1.1
88
217
1.5
a615
0.93
93
240
1.5
600
0.58
88
183
1.6
565
0.67
80
219
1.5
572
0.71
73
159
1.6
507
0.54
88
223
1.6
a712
0.95
d36
230
1.9
a1030
1.34
^he calculation used to obtain this estimate assumed lognormality despite (N) ' D > 0.736.
bSampling site was relocated within same general neighborhood in midyear. It is assumed that for sampling
purposes the environmental air was the same at both locations.
cTemporarily discontinued because of construction at sampling site.
^Sampling was initiated in the latter part of the year.
8-13
-------
Table III. Sulfur Dioxide Data Summary for 1968 to 1971
Monitoring
station
Fig.
A
B
C
D
E
f
Q
H
I
J
(see
1 ) Statistic
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N|1/2D
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fi-. statistic, (N)1/2D
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)"2D
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1'2D
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Number of readings
Arithmetic average
Standard geometric deviation
Second highest reading
Goodness-of-fit statistic, (N)1/2D
Standard
>60
60
260
>60
60
260
>60
60
260
>60
60
260
>60
60
260
>60
60
260
>60
6O
260
>6Cr
60
260
>60
60
260
>60
60
260
1968
71
137
2.4
8972
0.75
72
95
2.4
644
0.61
53
106
1.8
413
0.52
/I
112
1.9
476
0.68
47
84
1.9
"364
0.80
69
77
2.1
414
0.57
62
64
2.3
a416
0.85
64
129
1.8
"522
1.04
1969
74
135
2.0
a674
0.96
76
85
2.3
546
0.48
72
103
1.7
278
0.47
75
107
1.6
314
0.42
75
76
2.1
8 409
1.04
71
58
2.0
294
0.70
71
63
2.3
390
0.69
77
110
1.8
467
0.64
32
113
1.9
543
0.53
1970
82
116
1.9
"518
0.88
9
105
74
2.3
476
0.54
b79
109
2.0
a538
0.91
107
96
1.8
8 397
0.88
97
90
1.8
373
0.68
105
63
1.9
295
0.70
113
66
2.2
408
0.47
108
101
1,9
8449
0.87
113
124
1.8
504
0.70
1971
88
84
2.2
523
0.66
86
50
2.1
284
0.70
93
67
2.4
485
0.73
C45
89
2.0
8469
0.76
U4
65
2.1
375
0.71
86
59
2.3
8401
0.83
86
5O
2.4
8 363
0.75
72
48
2.4
336
0.72
83
67
2.1
8358
0.90
93
79
2.0
8410
1.23
8-14
-------
Table III (cont'd). Sulfur Dioxide Data Summary for 1968 to 1971
Monitoring
station (see
Fig. 1) Statistic
K Total suspended paniculate
Nitrogen dioxide
Sulfur dioxide
L Total suspended paniculate
Nitrogen dioxide
Sulfur dioxide
M Total suspended paniculate
Nitrogen dioxide
Sulfur dioxide
N Total suspended paniculate
Nitrogen dioxide/
Sulfur dioxide
O Total suspended paniculate
Standard 1968
*SS a59
62
27
61 62
57
0
205 293
65 71
1969
43
92
11
37
68
0
268
a56
1970
b59
b109
bo
222
120
141
70
76
9
b436
108
a62
85
1971
81
83
819
280
119
a192
63
59
a 22
317
a127
a105
116
P Total suspended paniculate 127 146 142 151 145
Q Total suspended paniculate 91 71 60 a153 69
R Total suspended paniculate 56 68 62 77 102
S Total suspended paniculate 73
T Total suspended paniculate 380
U Nitrogen dioxide d129
Sulfur dioxide d138
aThe calculation used to obtain this estimate assumed lognormality despite (N) ' D ^ 0.736.
'-'Sampling site was relocated within same general neighborhood in midyear. It is assumed that for sampling
purposes the environmental air was the same at both locations.
cTemporarily discontinued because of construction at sampling site.
Sampling was initiated in the latter part of the year.
8-15
-------
Table IV. - Significance Levels for the
Kolmogorov-Smirnov Goodness-of-Fit Statistic
[From Lilliefors( 1967)]
Significance level, a 0.20 0.15 0.10 0.05 0.01
Statistic, (N)1/2Da 0.736 0.768 0.805 0.886 1.031
Table V. - Summary of Results of Goodness-of-Fit Tests
Monitoring
station (see
fig. 1)
A
B
C
D
E
F
G
H
I
J
K
L
M
N
0
P
Q
R
S
T
U
Total suspended
Number
of tests
4
5
5
4
5
5
4
4
5
5
5
2
5
5
5
5
5
5
1
1
part icu late
Rejected
2
0
1
0
3
2
1
0
3
3
2
0
0
1
1
0
1
0
0
0
N itrogen
Number
of tests
4
1
4
4
4
4
4
4
4
3
4
2
4
2
1
dioxide
Rejected
0
1
3
2
3
3
1
3
2
1
1
0
1
1
1
Sulfur
Number
of tests
4
1
4
4
4
4
4
4
4
3
4
2
4
2
1
dioxide
Rejected
3
0
0
2
1
3
1
1
3
1
1
1
1
2
0
Total
85
20
49
23 49 20
Percentage of
tests rejected
24
47
41
Expected number
of rejections
17
9.8
9.8
8-16
-------
Table VI. Polludex Values for 1967 to 1971.
Monitoring
station (see
fig. 1)
A
B
C
D
E
F
G
H
1
J
Pollutant
Total suspended paniculate
Nitrogen dioxide
Sulfur dioxide
Total suspended participate
Nitrogen dioxide
Sulfur dioxide
Total suspended paniculate
Nitrogen dioxide
Sulfur dioxide
Total suspended paniculate
Nitrogen dioxide
Sulfur dioxide
Total suspended paniculate
Nitrogen dioxide
Sulfur dioxide
Total suspended paniculate
Nitrogen dioxide
Sulfur dioxide
Total suspended particulate
Nitrogen dioxide
Sulfur dioxide
Total suspended particulate
Nitrogen dioxide
Sulfur dioxide
Total suspended particulate
Nitrogen dioxide
Sulfur dioxide
Total suspended particulate
Nitrogen dioxide
Sulfur dioxide
1967 1968
408
111
a201
111 103
117 a144
77
103
135 135
107
68
133 a159
103
85
a85 104
112
a40
89
101
44
62
66
a34
"255 324
147
a108
203 a213
1969
a303
120
a142
54
105
148
75
129
119
58
91
137
50
72
97
a42
a66
121
7
70
125
27
8299
153
82
a230
125
99
1970
a284
114
a97
b117
144
134
55
b191
b117
a,b94
a145
117
a56
a93
115
47
98
124
10
106
113
34
321
138
8 70
"207
155
100
1971
296
102
70
82
90
5
167
155
49
(c)
C99
a.c64
a109
105
26
89
103
27
89
103
a20
91
102
15
a283
117
25
251
140
a45
*The calculation used to obtain this estimate assumed lognormality despite (N)1^D ^0.736.
Sampling site was.relocated within same general neighborhood in midyear. It is assumed that for sampling
purposes the environmental air was the same at both locations.
"^Temporarily discontinued because of construction at sampling site.
Sampling was initiated in the latter part of the year
8-17
-------
Air Pollution Control Oltict' ?7h5 Broddttdy
Audubon Junior High School 3055 East Boulevard
Brooklyn > MCA We st 25 St and Demson
Cleu'ldnrt Health Museum 8111 tuclid
Clrieland Pneumatic Tool 3701 East 71 (near Broadway)
Collirmood Hiqli School East 15? and St Clair
Cudell Recreation Center West Boulevard and Detroit
Estalirook Recreation Center Fulton and Memphis
Fire Station H 474" Broadway
I ire Station 11 East 5' and St Clair
G \VashinqtonElementarySchool 16210 Lorain
Harvard Nards 4150 East 49 St
J F Kennedy High School. 17100 Harvard
P I Ounbar Elementary School 22(10 West 28 St
Alnira Elementary School West 98 St and Almira
Fire Station ?9 East 105 St dnd Superior
John Adams High School 3817 East 116 St
J f Rhodes High School 5100 Biddulph
St Joseph High School 18491 Lake Shore Blvd
Supplementary Education Center 1365 E
St Vincent Charity Hospital, E. 22 St
Lakewood
Rockv River
Figure 8-1. Air pollution monitoring sites for Cleveland, Ohio. The heavy line
down the center is the Cuyahoga River. The municipal boundaries have
been straightened somewhat but are accurate in their essential features.
600-
400-
200-
I 100
§ 80
5 60
40
20
J
.1
50
Frequency
90
99 99.9
Figure 8-2. Lognormal plot of distribution by weight of total suspended
particulate (24-hr, sampling) for monitoring station I (see Fig. 1)
downwind of the industrial region.
8-18
-------
600
400
tf\
200
g
1 1Qo
§ 80
o
0 60
40
20
Year
o 1967
o
o
Ci
o
1968
1969
1970
1971
I
i i i
1
I
1
1
10
90
99 99.9
50
Frequency
Figure 8-3. Lognormal plot of distribution by weight of total suspended
paniculate (24-hr, sampling) for monitoring station K (see Fig. 1) upwind
of the industrial region.
'P-100
^Standard
P-0
= (a) Standards for air quality (b) Pollution levels twice the
g are met. allowed standards.
/Standard
P-50
Frequency
Figure 8-4. Examples of Polludex levels.
8-19
-------
(a) Total suspended paniculate
Figure 8-5. Bar graph presentations of Polludex values for the three pollutants at
the various monitoring stations. Left bar represents 1968 level of
pollution; right bar or a single bar represents 1971 level. Alphabetic coding
of monitoring sites corresponds to that of Figure 1.
8-20
-------
References
Anon., 1971: Federal Register. 36: 8186.
Babcock, L. R., 1970: A Combined Pollution Index for Measurement of Total
Air Pollution,./. Air Pollution Control Association. 10: 653.
Benarie, M., 1971: Sur la Validitie de la Distribution Logarithmical-normalie des
Concentrations de Polluant. Proc. Second Intern. Clean Air Congress, pp.
68-70, eds., H.M. Englund and W. T. Beery, Academic Press, New York.
Harter, H. L., 1961: Expected Values of Normal Order Statistics. Biometrika.
48: 151-165.
Larsen, R. I., 1971: A Mathematical Model for Relating Air Quality
Measurements to Air Quality Standards. Environmental Protection
Agency, Office of Air Programs, U. S. Rep. AP-89.
Lilliefors, H. W., 1967: On the Kolmogorov-Smirnov Test for Normality With
Mean and Variance Unknown./ Am. Stat. Assoc. 62: 399-402.
Mitchell, R. I., 1968: Permanence of the Log-Normal Distribution./ Opt. Soc.
Am. 58: 1267-1272.
Neustadter, H. E.; King, R. B.; Fordyce, J. S.; and Burr, J. C., Jr., 1972: Air
Quality Aerometric Data for the City of Cleveland from 1967 to 1970 for
Sulfur Dioxide, Suspended Particulates, and Nitrogen Dioxide. NASA TM
X-2496.
Noether, G. E., 1967: Elements of Nonparametric Statistics. John Wiley & Sons,
Inc. New York, N. Y.
Ohio, 1972: State of Ohio, Department of Health, Air Pollution Unit.
Regulations AP-3-02, AP-7-01.
Pikul, R., 1972: Development of Environmental Indices. Mitre Corp., Rep.
M71-47.
DISCUSSION
Singpurwalla: When you tried these Kolmogorov-Smirnov goodness of fit tests
could you care to tell which tables you used for the levels of significance?
Neustadter: I believe they are referenced in the report. I don't have it in my
head. The statisticians did it, but I believe the tables are in the report and the
reference is there.
Singpurwalla: The reason for questioning this is because if you estimate the
parameters from the data and go ahead and use Kolmogorov-Smirnov tables
which are generally available, then you are apt to make sdme kind of an error.
But there are modified tables.
Neustadter: We are aware of that. We used the modified table.
8-21
-------
Singpurwalla: O.K., and I was wondering if that might be any reason why it
might change the answer.
Neustadter: No. We are aware of the problem of using estimated parameters and
we did use modified tables. [See Lilliefors (1967).]
Rustagi: This morning I have heard quite a bit of glorification of lognormal
distribution. I want to add one more reference to that. In 1964, Archives of
Environmental Health I have the paper, titled "Stochastic behavior of trace
substances," in which many of these substances have been studied including air
pollutants. The amazing thing was that many substances were in liquids. For
example, amino acids in urine also followed lognormal distribution. The second
point I want to ask is for Mr. Marcus, who used the concept of a very interesting
cumulative dose. I think most of the trace metals such as lead, about which I am
familiar with, in the human body or biological systems are excreted in also
certain random fashion. Could the deposition of a substance like lead or other
gases be put into the model? A simple model in that connection was also
mentioned by me in Archives of Environmental Health giving a model of body
burden where intake and output were used in the model, however, not as the
formal stochastic processes, rather as probability distributions without any
assumptions for parametric form such as lognormal. I would like to mention the
physiological experiments connected with air pollutants. There are very few
studies but I think the audience should be aware of two famous studies—one is
on human subjects over the past thirty years on lead by Dr. R. A. Kehoe and I
think it is given in a series of lectures by Dr. Kehoe, "Metabolism of lead in man
in health and disease," where he studied whatever metabolism could be studied
in man. In animals Professor Herman Cember of Northwestern has studied the
metabolism of mercury in rats over a period of time and I think these two
studies should be noted. I'm not aware of others.
8-22
-------
9. AN INVESTIGATION OF THE FREQUENCY DISTRIBUTIONS
OF SURFACE AIR-POLLUTANT CONCENTRATIONS
J. B. KNOX
R. I. POLLACK
Lawrence Livermore Laboratory
University of California, Livermore, California
In several papers, Larsen and co-workers (1965; 1967; 1969) have discussed
the frequency distributions of various pollutant concentrations calculated from
data taken at CAMP (Continuous Air Monitoring Program) sites for 3 years in
various cities. The data consist of instantaneous measurements taken at 5-minute
intervals. When used in this fashion, or averaged over any period of time, the
resulting frequency distribution is in all cases approximately lognormal. It was
also noted that median concentration is proportional to averaging time to an
exponent.
This result allowed these investigators to relate the geometric mean,
standard geometric deviation and averaging time to the probable number of
times during a year that a given level of pollution would be exceeded. This type
of data presentation is useful because ambient air quality standards are set in the
form of a maximum allowable average over a given period of time e.g., 0.03 ppm
for 8-hr-duration samples would be allowed once a year for oxidant.
The CAMP data indicates that reactive pollutants may have a larger SGD
(standard geometrical deviation) because of the. additional variability
introduced by the nature of chemical or photochemical reactions. It was noted
by Larsen and verified by Knox and Lange (1972) that continuous point sources
give concentration distributions with larger SGD's than area sources. This is
attributed to the dependence of the pollutant concentration on the lateral and
vertical standard deviation of the plume.
Barlow (1971) suggested that the lognormal distribution may not be
appropriate because the averaging process implies that the sum of lognormally
distributed random variables is itself lognormal. This is contrary to statistical
theory. He suggests that a Weibull distribution would be more appropriate. This
suggestion is supported by Milokaj (1972) who has argued the validity of the
Weibull distribution for a variety of situations involving pollutant
concentrations. He emphasizes the importance of the threshold parameter (7).
9-1
-------
The probability of occurrence of a value of the random variable smaller than 7 is
essentially zero. The lognormal distribution also has a formulation including a
threshold parameter, however it is usually ignored due to the fact that low
concentrations are ordinarily beneath the sensitivity of the measuring
instruments. Also, the most interesting cases are the higher concentration levels
where the lognormal fits well, and the significance of a threshold parameter is
minimal. There is some indication, justifying further investigation, that the
lognormal fits well at both high and low concentration levels, but with slightly
different parameters. This suggests that two adjacent lognormal distributions
may be present, perhaps caused by different types of meteorology.
The motivation for using the Weibull distribution is largely empirical. This
distribution, with density function
f(x) = kxme-kxm+l/
-------
until their increased weight causes them to fall out. This growth process dictates
that the size of a particle is a function of its previous size multiplied by the rate
of coagulation—a multiplicative process. This can be shown to yield a lognormal
distribution. The rate of coagulation is a function of a variety of other variables.
Another important variable which has been found to be lognormally
distributed is the rate of dissipation of energy in turbulence (e). This distribution
was postulated by Kolmogorov (1941) based on the assumption of a cascade
process by which energy is transferred from a large-scale turbulent motion to
progressively smaller-scale motions. The transfer stages are assumed similar and
independent. Thus, the amount of energy dissipated at any stage is a function of
the amount dissipated at the previous stage. The process may be viewed as
€i = (€j_,)Y; (1)
where ej is the energy dissipated at stage i and Yj is a characteristic of the
transfer stage causing the change in e j_i.
Due to the reproductive properties of the lognormal distribution, the
distribution of e implies the lognormality of several related variables including
the dissipation of temperature by thermal eddy conduction, the squared-space
differences of temperature and velocity, which imply that the differences them-
selves are lognormally distributed on either side of the origin, and the
horizontal eddy diffusivity. These imply that the diffusive transfer between
adjacent volumes of air is lognormally distributed. Furthermore, the
lognormality of wind speeds, which has been demonstrated empirically, implies
that the advective transfer is also lognormally distributed. These distributions
have all been verified experimentally (Knox and Lange (1972); Gibson, et al.
(1970) and (1970a)).
The Lognormal Process
The fundamental question has yet to be answered: Why are all these
variables lognormally distributed? What underlying physical phenomena cause
the lognormality of these variables? The answer to these questions requires a
basic understanding of the theory behind the generation of the lognormal
distribution.
Consider a stochastic process of the form
xi = xi-| + *i-l Yj (2)
where Yj is an independent stochastic variable, arbitrarily distributed.
If we solve Equation 2 for Yj.
X - X_
9-3
-------
and sum both sides.
i-l 0
We can approximate*the left side by
n=0
N
N
by the Central Limit Theorem ZYj is normally distributed, hence Xn is
lognormally distributed. 0
This is known as the law of proportional effect; the percentage change in a
variable is equal to a constant plus an error. If the absolute change had been
equal to this same constant + error term, the normal distribution would have
resulted. Hence, the lognormal distribution is the result of a multiplicative
process whereas the normal distribution results from an additive process.
The basic properties of the lognormal are all multiplicative analogies to the
normal distribution. This includes the reproductive properties. In particular the
product of two lognormal distributions is lognormal, the sum, however, is not.
Aitchison and Brown (1957) discuss these matters more fully, but the above is
sufficient for the purposes herein.
We recognize at this point that the processes leading to the particle size
distribution, and the lognormality of e, are similar to Equation 2. In the latter
case we need merely to replace Y1 by (1+Y.,). This adds a constant to the error
term, but makes no fundamental change in the process.
A Simple Model of Pollutant: Concentrations
It has been found that
= KQ/uU) (3)
where i// is pollutant surface concentration
Q is source strength
u is wind speed
v is frequency
\&
Approximation error small as At>0.
9-4
-------
is an appropriate model for predicting concentration of S02 and particulates in a
well mixed urban environment (Gifford and Hanna (1972)). The constant K has
been measured rather extensively and a range constructed for K for each
pollutant.
Based upon the lognormality of 1/u, which has been verified empirically
(Knox and Lange (1972)), and the relative invariance of K, which has been said
to be a weak function of city size, we conclude that i// is lognormally
distributed.
Knox and Lange (1972) have estimated K' = KQ by experiment and by
using a box model to predict concentrations. Their findings indicate that a
suitable value of K can be found either by using the box model or comparing i//
and 1/u by visual superposition, and adjusting K' so that i// and K'/u have
approximately equivalent geometric means. In addition, with this value the
variances of i// and K'/u are approximately equal. (See Figs. 2-6 of Knox and
Lange (1972)).
For continuous point sources Knox and Lange (1972) fitted the model
Q 2
-------
^_ ik -TU> d . y ~T I "•" T~ V >>7 ^ I
0y ' oy oy dz
where t//a is the concentration of pollutant a; u, w, v are the velocity
components; Kx, Ky are the lateral vertical eddy diffusitives, which are
lognormally distributed based upon the lognormality of e and the reproductive
properties; Kz is the vertical eddy diffusivity; Sa is the source term for pollutant
a; P is the term representing changes in concentration due to photochemistry; V
is the volume of air for which S and P act.
This equation can be manipulated to represent a box model formulation
(MacCracken, et al. (1972)) wnere we are concerned with the concentration
averaged over a box which is surrounded by M other boxes.
«[*k(mft)] M
- Tt - = -£ TA(m,j) + TD(m,j) hK(m,t) (6)
at L j *
Z [TA(J,m) + TD( j,m)] \//k ( j ,t) + Sk(m,t)
j=0
where TA (m,j) and TD (m,j) are the advective and eddy diffusive transfer
coefficients from box m to box j. The lognormal distribution can bearguedfor
these latter variables in a similar manner as for Kx and K .
This equation is also consistent with the generating process. Equation 2,
when certain reasonable restrictions hold:
(a) The contribution of advection and diffusion terms are larger than the
contribution of the source term. It has been found empirically that if this is not
the case, lognormality does not result (Hopper (1972)).
(b) The concentrations in the surrounding boxes are, on the average over
long periods of time, close to that of the box we are interested in, due to the
fact that they are subjected to similar stimuli.
9-6
-------
These restrictions transform Equation 4 to
M) M
dt .=Q
M
(7)
r T
I TA(j,m) + TD(j,m)|
i=o L J
Suppose we let
^(J.O = ^k(m,t) + Ek(j,t) (8)
the equation becomes
diMm.t) M r 1
— T: - = - I TA(m,j) + TD(m,j)J //(j,t) and i//(m,t) will be small.
Hence the term tends to zero. Conversely, in the case where the error term
Ek(jft) is large, the constant term will usually be small indicating light winds.
Furthermore, in either case, or any combination of cases occurring between T
and TR, we can expect that the sign of the term will vary, implying that the
positive and negative terms will cancel each other.
This argument implies that Equation 9 is consistent with the law of
proportional effect.
The solution will be source-dominated only when the magnitude of the
source terms is comparable to the magnitude of the current concentration. There
9-7
-------
is reason to believe (Hopper (1972)) that in such cases the concentrations will
not be lognormally distributed, as the model indicates. This result has been
noted in investigations of particle size distributions also (Blifford and Gillette
(1971)).
This reasoning is most easily justified for a well mixed urban region. It is not
clear that the lognormal distribution will fit as well for non-urban, poorly mixed
areas. We do feel, however, that the characteristics of an area's topography and
typical meteorology would have to be highly unusual tor (10) to be so large that
the lognormal distribution would fit poorly.
We have not yet discussed the At interval necessary for these results. We
recognize that it must be sufficiently small not to obscure the generating
process. If, for an extreme example, At was 6 months, we would not see the
process defined by Equation 2 because the effect of (//j^ on i//j would have long
since died out. Larsen's (1965) data is for 5-minute instantaneous readings. We
accept this as an appropriate time scale for our purposes, based on the fact that
meteorology certainly does not change enough in a 5-minute period to
obscure the relevant correlations. Of course, the distribution of pollutant
concentration remains unaltered.
When the data is averaged over other time periods within the realm of
atmospheric motion, the averaging time acts as a filter which smooths out
motions of a smaller time scale. This has the effect of allowing us to see only
motion of a time scale comparable to the averaging time in the averaged data.
Hence the process described by Equation 2 still holds for larger averaging times,
but the TA, TD terms now represent motion of a larger scale. This results in
lognormality over a large spectrum of averaging times.
We are presently investigating the magnitudes of the first order
multiplicative autocorrelations for all averaging times. Preliminary results
indicate that significant positive autocorrelation is present for averaging times up
to at least 2 weeks. This lends credence to the assumption that Equation 2
acts over a large spectrum of averaging times.
Applications
Ambient air quality standards (AAQS) are set in terms of the number of
times a concentration of a particular pollutant shall exceed a specified limit,
averaged over a specified number of hours. For example, an 8-hour average of
CO may not exceed 30 ppm more than once per year.
Concentration distributions must be calculated to compare ambient air
quality with these types of standards. An air quality prediction has meaning only
when the averaging time and level of confidence of the estimate are included.
This requires knowledge of the concentration distributions. Thus, whether we
are interested in prescribing standards, describing levels, real time monitoring or
land use planning, knowledge of concentration distributions is indispensable.
9-8
-------
The foregoing discussion indicates that there is an increasing amount of
evidence supporting the contention that surface air pollutant concentration
frequency distributions are lognormal. This evidence includes empirical
investigative results, arguments regarding the relationship of meteorological
variable distributions to pollutant frequency distributions from simple diffusion
models, and deductions of the nature of the pollutant frequency distributions
from considerations of the complete set of governing equations for a multiple
box model of photochemical pollutants. The possible exceptions to lognormality
of pollutant distributions have been indicated. However, it is now pertinent to
explore the practical and research implications of large portions of air quality
regions having pollutant distributions that are lognormal; significant implications
include:
(a) The application of air quality simulation models to land-use planning
assessments for consistency with AAQS or to the design of measures to achieve
consistency with AAQS in growing areas should be expedited in principle.
(b) The validation tests of air quality simulation models should include the
requirement that calculated pollutant frequency distributions, or key portions of
those distributions, correspond to reality.
(c) Knowledge that the pollutant concentration distributions are lognormal,
should eventually lead to simplifications in data acquisition by air monitoring
networks and to the feasibility of real time control mechanisms.
Land Use Plan Assessment
Consider the future when a verified and acceptable numerical simulation
model for air pollution exists. The question then is, how can such an acceptable
numerical simulation model be employed in land-use plan assessments? Given a
region of interest for planning purposes and a suite of pollutants of concern, one
could examine, for instance, the frequency distribution of hourly-average values
of surface air concentrations and identify the portion of the distribution which
•
is equal to or greater than the ambient air quality standard involved.
Conceptually, the days or episodes involved in that part of the distribution could
be composited into mesoscale or regional weather types. The meteorological
fields and air quality data from those days or episodes would constitute case
studies for model calculations. In Figure 3, 3 weather types are illustrated,
corresponding to high, moderate, and low levels of pollution.. The solutions of
such numerical modeling case studies would delineate a spatial distribution of
the excess over ambient air quality standards in the region which might not
necessarily be defined by the network of monitoring stations. From examining
those excesses and their spatial distributions, one could determine the degree of
control and a location of control necessary to remedy the excess. In principle,
the same set of analytical steps could be applied to forecast emission zonings
9-9
-------
associated with either growth or alternative land-use plans for the same set of
identifed days. Hence, in this matter, one could evaluate the degree of control
necessary for an existing situation in a region of interest to bring the air quality
of that region into line with ambient air quality standards, or else to assess the
excess of ambient air quality standards in need of control that correspond to
various land-use plans.
Kennedy, et al. (1971) developed such a program for Chicago utilizing a
sub-model to predict the effects of a certain type of emission zone in a particular
place. These "coupling coefficients" are essentially a linear model of dispersion.
They are used as coefficients in a linear program, the objective of which is to
minimize the social and financial burden of restrictions while satisfying air
quality constraints. Of course, non-linearities caused by interaction between
pollutants and such are overlooked, and extremes are calculated through the use
of coupling coefficients and extrapolation of the frequency distribution. But the
model seems appropriate for making a land-use plan assessment or corresponding
emission zoning.
Model Validation
Application of the model to such economically sensitive problems as land
use planning requires that the model predict the surface concentration
distributions quite accurately. In order to discuss validation of numerical
simulation models of regional air pollution we reference some recent results in
the development and initial verification of an air pollutant model for the San
Francisco Bay Area (MacCracken, et al. (1972); Gelinas (1972)). This model uses
historical meteorological data to predict the mean and surface air concentration
in each of the model cells, including transport and diffusion by the ambient
wind field between the irregular earth surface and the time and space variable
marine inversion layer. (See Figs. 3-4 of MacCracken, etal. (1972)). The verifica-
tion work was carried out on a 48-hour test period during July, 1968. Figure 4
displays the observed hourly-average concentrations of CO in parts per million
during the case study, as well as the computed vertical average and computed
surface hourly-average CO concentration. There is very reasonable
agreement between the observed and the computed surface concentrations. This
information of calculated versus observed concentrations can also be displayed
as a lognormal frequency distribution plot, Figure 5. The significant feature to
be noted here is that the frequency distribution of the predicted hourly-average
concentrations on lognormal paper parallels the observed (Knox and Lange
(1972)). In addition, it is parallel to that obtained by Larsen for the frequency
distribution of hourly averages of CO for a year. Frank Gifford
(ARATDL-NOAA) has recently noted that several of the numerical simulation
models under development at this time render numerical solutions which are
9-10
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"noisier" than the observed distributions. A numerical solution, that is
contaminated with noise will, in general, not be able to predict the frequency
distribution of the surface pollutant and, therefore, will have severe limitations
in regard to a comparison of predicted frequency distributions to ambient air
quality standards. Hence, one criterion for an acceptable model for numerical
simulation of air pollution is whether the model is able to reproduce the
frequency distribution characteristics of the pollutants involved and in the
region of interest.
Monitoring
Knowledge of the particular distribution and its parameters allows us to
make statistical comparisons between predicted air quality and air quality
standards. Alternatively, we may simplify the procedure by taking random
samples and manipulating only this reduced volume of data. The resulting
estimates would be measures of typical long-term concentrations and variability.
Figure 6 shows estimates of the distribution of hourly averages of CO in San
Francisco from 1968 through 1970. The good agreement between estimates
made from 100 random samples and 10 random samples, with the distribution
obtained from continuous monitoring suggests the possibility that an
appropriate spatial and temporal random sampling scheme would allow one
movable receptor to estimate annual averages in a number of locations. This
method has potential for use with land use models where long-term information
is desired. Methods of sampling local air quality, as contrasted to continuous
monitoring of local air quality, are not well suited to comparison of predicted
concentrations from a model to short term AAQS.
Nonparametric methods were also investigated, but they tend to be less
powerful than parametric methods in cases where the assumptions of parametric
statistics apply. The latter methods also have the advantage of ease of
manipulation and the simplicity of exact specification of the distribution.
A natural extension of the principles of air pollution monitoring is real-time
control. This is a potentially effective method for controlling air pollution
episodes. It requires a model with the ability to predict future pollutant
concentration distributions at all points in the region sufficiently far in advance
so that control actions can be taken to avoid an impending episode. These
actions may be quite selective, in that they need only be taken during
emergencies and then only in offending emission zones. We recognize that this is
not within present capabilities, but we look ahead to the construction of such
"feed forward" control schemes.
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Conclusions
There is increasing evidence to support the theory that air pollutant
concentrations are lognormally distributed in areas devoid of strong sources,
whether they be passive or photochemical, and in the absence of meteorological
or topographic effects resulting in sharp differences in concentrations between
adjacent volumes of air. This lognormal distribution is supported by (a)
empirical evidence, (b) the simple model of urban pollutant concentrations
proposed by Gifford when examined in the light of the lognormal distribution of
the reciprocal of wind speed verified by Knox and Lange, and (c) the theoretical
derivation from the full set of equations governing the time evolution of
pollutant concentrations presented herein.
Weibull wrote ". . . it is utterly hopeless to expect a theoretical basis for
distribution functions such as ... particle sizes," and yet one has been provided.
In fact, it seems reasonable to expect that the physics describing a process
should be consistent with a distribution function describing the results of that
process, indeed, anything else would be suspect. This is what has been provided
here, a consolidation of empirical evidence with physical theory.
Knowledge of pollutant concentration distributions is necessary for land-use
plan assessment to compare predicted air quality with ambient air quality
standards. It is useful as a method of verification of a numerical simulation
model of air pollutant evolution, and it is a potentially valuable tool for use in a
real-time model predicting short-term fluctuations in pollutant concentrations.
Acknowledgement
This work was performed under the auspices of the U. S. Atomic Energy
Commission.
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100
I
10
5
u
§
1.0
100
E
Q.
O.
o
I 10
LJ
8
10
40 60
FREQUENCY
80
Figure 9-2. Lognormal probability plot
of CO concentration vs frequency for
San Francisco.
6 8 10 20 40
FREQUENCY
60 80
Figure 9-1. Weibull probability plot
of CO concentration vs frequency for
San Francisco.
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lf
I'
Z 9
High Pollution
Annuol Average •
I I I I I I
Moderate Pollution
Low Pollution
001 QOSQIQ2 05 I 2 5 10 20 30 4O 50 60 70 80 90 95 98 99 99.5 99.8 | 99.99
FREQUENCY (%) 99.9
Figure 9-3. Carbon monoxide concentration vs frequency for San Francisco for
various categories of pollution-days.
i«
O 3
H
P 2
Z I
00
T I I I I I
-Computed vertical average
Observed
12 18 00
12 18 00
(July 10) T|ME (July II)
Figure 9-4. Carbon monoxide concentrations for San Francisco, July 10-11,
1968.
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e 10
0.
0.
Id
(J
O
O
0.03
I I I I I I I I I I I I I I T
LARSEN MODEL CONC.
BAY AREA MODEL
SURF CONC.
OBSERVED SURF CONC.
x^v
BAY AREA MODEL *^X
AV CONC.
I I I I I i I I I I I I I I
0.01
I
10
50
90
FREQUENCY-% of time
concentration is exceeded
Figure 9-5. Carbon monoxide concentration vs frequency for San Francisco.
~i—p—i—i—i—r
i—i—i—r
9O
eo
70
6O
5O
4O
30
2O
5
4
3 I-
2 -
O - 10 Random Samptn
A - 100 Random Samplti
Continuous Monitoring
'(20,100 data pomlt)
0.01 0050102 0.5 2 3 10 20 30 4O SO 60 70 80 90 95 969999.5 (99.9 99.99
FREQUENCY (%) 99.8
Figure 9-6. Carbon monoxide vs frequency for San Francisco—hourly averages.
9-15
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References
Aitchison and Brown, 1957: The Lognormal Distribution. Cambridge Univ.
Press, 176.
Barlow, R. E., 1971: Averaging time and maxima for air pollution concentra-
tion. NTIS AD-729 413, ORC 71-17.
Blifford, I. H., and Gillette, D. A., 1971: Applications of the lognormal fre-
quency distribution to the chemical composition and size distribution of
naturally occurring atmospheric aerosols. Water, Air & Soil Pollution. 1:
106-114.
Friedlander, S. K., 1960: On the particle size spectrum of atmospheric aerosols.
Journal of Meteorology. 17: 373-374.
Gelinas, R. J., 1972: Stiff systems of kinetic equation, a practioner's view. J. of
Computational Physics. 9: 222-236.
Gibson, Stegen, and Williams, 1970: Statistics of the fine structure of turbulent
velocity and temperature fields measured at high reynolds number. J. Fl.
Mech. 41: 153-167.
Gibson, Stegen, and McConnel, 1970: Physics of Fluids. 13: No. 10.
Gifford, F. A., and Hanna, S. R., 1972: Modeling urban air pollution. ARATDL
Contribution No. 63.
Hopper, C., 1972: personal communication.
Kennedy, A. S., Cohen, A. S., Croke, F. J., Croke, K. G., Stork, J., and Hurter,
A. P., 1971: Air pollution-land use planning project, phase I. Final Report,
ANL/ES-7.
Knox, J. B. and Lange, R., 1972: Surface air pollutant concentration-frequency
distribution: implications for urban air pollution modelling. University of
California, Lawrence Livermore Laboratory, Report UCRL-73887.
Kolmogorov, A. N., 1941: Dokl AN SSSR. 30: 301.
Larsen, R. I., Zimmer, C. E., Lynn, D. A., and Blemel, K. G., 1967: Analyzing
air pollutant concentration and dosage data. J. Air Pollution Control As-
sociation. 17: 85-93.
Larsen, R. I., 1969: A new mathematical model of air pollutant concentration,
averaging time, and frequency. J. Air Pollution Control Association. 19:
24-30.
MacCracken, M. C., Crawford, T. V., Peterson, K. R., and Knox, J. B., 1972:
Initial Application of a Multi-Box Air Pollution Model to the San
Francisco Bay Area. Univ. of California, Lawrence Livermore Laboratory,
Report UCRL-73348.
Milokaj, P. G., 1972: Environmental applications of the Weibull distribution
function: oil pollution. Science. 176: 1019-1021.
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Singpurwalla, N. D., 1972: Extreme values from a lognormal law with
applications to air pollution problems. Technometrics. T4: 3.
Weibull, W., 1951: A distribution function of wide applicability. J. Appl. Mech.
293-297.
Zimmer C. E., and Larsen, R. I., 1965: Calculating air quality and its control. J.
Air Pollution Control Association. 15: 565-572.
DISCUSSION
J. Arvesen: Regarding your model itself, the box model that you applied to the
San Francisco data, how do you go about estimating the parameters in that
model to fit that data? Is there a problem involved with that? It would seem to
be a problem to me. There seemed to be a lot of parameters in there and I was
wondering how you can estimate them reasonably well on 2 days data. Am I
missing something?
Knox: Let me see if I can answer the question. The predicted frequency
distribution for the 48-hour test period was generated from the predicted 48
1-hour average CO concentrations for San Francisco receptor from the model.
This distribution was compared to the actual data from San Francisco—the 48
average hourly values at the sampling station. And so the frequency associated
with the highest CO value corresponds to 1 in 48. There is an interesting aspect
of this: the obvious question is how do we know that the model has an averaging
time that is appropriate to be compared to the average hourly data. If one looks
at the boxes used, they are "T" shaped, "l_" shaped, or any arbitrary shape that
fits the area roughness or source strength. Their average*dimension divided by
wind velocity is about an hour, so that the travel time across the boxes is
comparable to sampling period. If we had used 5-minute integrations, then the
comparison to actual data should be performed with 5-minute average CO data.
9-17
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10. AIR QUALITY FREQUENCY DISTRIBUTIONS FROM
DISPERSION MODELS COMPARED WITH MEASUREMENTS
D.BRUCE TURNER*
Environmental Protection Agency
National Environmental Research Center
Division of Meteorology
Research Triangle Park, North Carolina
Introduction
Cumulative frequency distributions (hereinafter abbreviated CFD) of air
quality can be estimated by dispersion models. By comparison with CFD'sfrom
air quality measurements at the same location, some indication of the accuracy
of these estimates can be made. Extremes of the estimated CFD for specific
locations can be compared with air quality standards. Not only can estimates be
made for existing pollution sources, but projected estimates can be made for
expected degrees of control of existing sources and inclusion of additional
sources. These projected estimates can also be compared with air quality
standards.
It is the purpose of this paper to present CFD's estimated from short-term
dispersion models and determined from measurements for the same locations,
periods of record and averaging times, and to compare these, especially the
maximum value, to indicate the accuracy of the estimates.
Background
National ambient air quality standards have been set in response to the
Clean Air Act. In most cases the standards consist of a long-term average, usually
the annual average, and a short-term standard, such as a maximum 24-hour or
3-hour concentration not to be exceeded more than once per year. For existing
sources, it is possible to monitor ambient air quality at selected sites to
determine if air quality standards are met at these locations. Due to the small
*0n Assignment from the National Oceanic and Atmospheric Administration, Department
of Commerce
10-1
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number of monitoring stations, it is highly likely that maximum concentrations
occur that exceed those measured at these stations.
It is desirable to supplement present air quality measurements by estimating
concentrations at additional locations. It is also desirable to estimate projected
ambient air quality at a number of locations for proposed source configurations
including both dditional sources and various assumptions as to degree of control.
These estimates can also be compared with air quality standards. Air quality
dispersion models have been developed to meet this need.
Long-term or climatological models have been used to estimate mean annual
concentrations at specific locations. These models typically require mean annual
emission rates from point and area sources and joint frequency distributions of
wind direction, wind speed, and stability. The relative accuracy of these models
is discussed elsewhere (Turner, Zimmerman and Busse (1972)). Summarizing this
paper, comparison of model estimates with measurements at a number of
sampling locations indicates that the ratio of root mean square error to the
measured mean for all stations is typically from 0.3 to 0.5. This indicates that
annual means can be estimated quite well. These estimated means can be
compared with the standards for the annual mean.
Dispersion models that calculate concentrations for averaging times of 1 to
2 hours can be used to make estimates for comparison with short-term
standards. Calculations can be made for each hour of the period of record and,
in addition to determining the extreme'Concentration occuring once during the
period, a frequency distribution of concentrations can be obtained. Hourly
concentrations can also be averaged for any longer averaging time, such as
24 hours, and a frequency distribution determined for this longer averaging time.
These short-term dispersion models require both meteorological and emission
information. Meteorological information typically consists of;(a) wind speed and
direction or wind flow fields, and (b) atmospheric stability class and mixing
height or temperature variation with height. Emission information typically
consists of emission rates for both significant point sources and all other sources
considered collectively as area sources. To be realistic the variations in emissions
from season to season, weekday to weekend and for various times of the day
should be included. It has been the experience of the author that this
information is difficult to obtain and also difficult to organize into a convenient
form. Stack parameter data are usually included for the point sources in order to
calculate plume rise. Because of inclusion of most emissions near the ground into
area sources, the resulting concentration estimates represent concentrations
averaged over an area the size of the smallest area source, usually 1 km2. On the
other hand, air quality measurements represent the concentration at the specific
point of measurement and are therefore particularly sensitive to any nearby
sources. Validation of dispersion models in urban areas is therefore difficult,
since it is necessary to compare the point measurements with estimates that are
more representative of an area.
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Frequency Distributions From Dispersion Models
Fortak (1970) and Koch and Thayer (1972) have estimated CFD's for
locations in urban areas from short-term dispersion models. Both used Gaussian
plume models, making separate calculations for point and area sources.
Fortak had 30-minute measurements of sulfur dioxide for four locations in
the city of Bremen, Germany. He made estimates using short-term dispersion
models for the same locations and averaging times and determined the frequency
distributions over various periods. The following results are for the heating
period (20 September 1967 - 31 May 1968). At two stations estimates are higher
than measurements for corresponding percentiles throughout the distribution.
At another station, estimates are less than measurements over the entire
distribution. For the remaining station, estimates are higher than measurements
except beyond the 99.6 percentile of the CFD where estimates are too low. At
the extreme end of the CFD, at the 99.5 percentile, Fortak's estimates for all
four stations are well within a factor of 2 of the measurements. The worst
estimate is off by a factor of 1.7.
Koch and Thayer (1972) of Geomet, working on a contract for EPA, also
used a short-term dispersion model to estimate 1-hour concentrations for 8
locations in Chicago for a 1-month period, (January 1967), and to estimate
2-hour concentrations for 10 locations in St. Louis for a 3-month period
(December 1964 - February 1965). CFD's were determined from these estimates
and compared with CFD's from measurements at the same locations.
In Chicago, the model underestimates concentrations for the entire CFD at
one of the stations. Four stations have concentrations overestimated for the
entire CFD. At one of the stations, concentrations are overestimated at the low
end of the CFD with slight underestimates past the 55 percentile. The other two
stations have concentrations underestimated at the low end of the CFD and
overestimated beyond the 65 percentile for one station and beyond about the 90
percentile for the other. Only one station has an estimate at the 90 percentile off
by more than a factor of 2. The error at this station is a factor of 2.8. For these
CFD's the 90 percentile is the highest cumulative frequency for which data is
presented.
In St. Louis, the model underestimates concentrations for the entire CFD
for five of the stations. One station has concentrations overestimated for the
entire distribution. At the other four stations concentrations are generally
underestimated, but are overestimated at the top end of the CFD with the
cross-over ranging from the 55 percentile to the 90 percentile. Only one station
has an estimate at the 90 percentile off by more than a factor of 2 from the
measurement at the same point in the CFD. The error at this station is a factor
of 2.6.
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24-Hour Frequency Distributions
The author, using a short-term Gaussian plume model similar to that used
by Koch and Thayer (1972), calculated 2-hour concentrations for 40 locations
in St. Louis. Measurements of 24-hour sulfur dioxide concentrations were made
at these stations during 89 consecutive days in December, 1964 through
February, 1965. Estimates of 24-hour concentrations at these stations were
made by averaging 12 successive 2-hour estimates. Frequency distributions of
24-hour concentrations for the period were determined for both estimated
concentrations and for measured concentrations for all 40 stations.
Because of the interest in the extreme end of the CFD (at the frequency of
the air quality standards), the extreme estimated value and the extreme
measured value were compared. These are near the 99 percentile for these three
months of 24-hour concentrations. The ratio of calculated concentration to
observed concentration was determined for each station. These ratios for the
extreme, arranged in ascending order, are 0.63, 0.70, 0.78, 0.81, 0.82, 0.84,
0.84, 0.87, 0.88, 0.88,0.90, 0.97, 1.07, 1.07, 1.09, 1.10, 1.11, 1.20, 1.23, 1.24,
1.30, 1.32, 1.42, 1.42, 1.44, 1.45, 1.45, 1.46, 1.59, 1.63, 1.65, 1.67, 1.69, 1.79,
1.90, 2.05, 2.23, 2.34, 2.35, 2.37.
Note that 35 of the 40 stations have estimated extreme values within a
factor of 2 of the measured extreme (ratio between 0.5 and 2.0). Also 15
stations have errors of less than or equal to ±20%.
Examples of agreement of estimates from the model and measurements at
the extreme (around the 99 percentile) are shown in Figures 1 through 3. Station
4 (Fig. 1) has the best agreement (a ratio of 0.97). Station 23 (Fig. 2) has the
highest overestimate (off by a factor of 2.37). Station 27 (Fig. 3) has the
greatest underestimate (a ratio of 0.63).
The comparison of the CFD's for the 40 locations is characterized
subjectively as follows: At ten stations the CFD's for estimates and
measurements are close. At ten stations overestimates occur throughout the
entire CFD. At three stations underestimates occur throughout the entire
distribution. At four stations overestimates occur primarily, but underestimates
occur at the higher percentiles (beyond the 88, 93, 95, and 96 percentiles). At
10 stations both underestimates and overestimates occur, with overestimates
beyond the crossover points of 7, 10, 25, 25, 25, 25, 30, 40, 83, and 90
percentiles. At two stations, although both underestimates and overestimates
occured, the comparison could be described as mostly underestimates. At one
station underestimation occured except at each end of the distribution.
Other visual comparisons of the estimated and measured CFD's for 24-hour
concentrations are given in Figures 4 and 5. Station 8 (Fig. 4) has the best
agreement between estimates and measurements over the whole CFD and has a
ratio of 1.07 at the extreme. Station 23 (Fig. 2), discussed previously, has the
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worst overestimate irregardlessof place in the distribution, with the estimate four
times the measurement at the 7 percentile. Station 38 (Fig. 5) has the worst
underestimate with an estimated 2 and a measured 46 at the 4 percentile, off by
a factor of 23. This is probably because low levels of background concentration
exist due to emissions from distant sources that are not included in the
calculations made by the model.
It is also desirable to consider if the CFD's appear to be lognormal (straight
lines on log-probability plots), particularly in view of the frequent use of the
Larsen statistical model (Larsen (1971)) to estimate extremes of concentrations
in urban areas. It appears that there is some deviation from the lognormal
distribution in the figures previously discussed, especially Stations 27 (Fig. 3)
and 38 (Fig. 5). Stations 28 (Fig. 6) and 16 (Fig. 7) seem to have two different
slopes in their distributions of measured concentrations, with the transition
taking place in the vicinity of the 50 percentile. Stations 2 (Fig. 8) and 6 (Fig. 9)
have a sudden transition to higher measured concentrations around the 95 to 97
percentiles. Station 19 (Fig. 10) has two portions of the CFD of measured
concentrations with the same slope but with a displacement occurring near the
50 percentile. For the most part, measured concentration CFD's appear to be
near lognormal. Although many of the CFD's from estimated concentrations are
also nearly lognormal, some of them appear to deviate more than those of the
measurements and to have an "S" shape such as station 28 (Fig. 6).
Two-Hour Frequency Distributions
At 10 of the 40 measurement stations in St. Louis, 2-hour measurements of
sulfur dioxide were also made. At these 10 locations, estimates and
measurements were used to determine CFD's for 2-hour concentrations over the
89 day period (December 1964 - February 1965).
For each station, the extreme estimated value and the extreme measured
value were compared. Since the data period consisted of 12 periods per day for
89 days, the extreme represents a frequency near the 99.9 percentile. The ratio
of calculated concentration to observed concentration was determined for each
station. These ratios for the extreme, arranged in ascending order, are 0.52, 0.66,
0.74, 0.75, 1.12, 1.47, 1.51, 1.60, 1.76, 1.88. All 10 of the stations have the
estimated extreme value within a factor of 2 of the measured extreme (ratio
between 0.5 and 2.0).
A selected number of these 2-hour CFD's are shown in Figures 11 through
13. Station 17 (Fig. 11) has the best agreement at the 99.9 percentile. Station 36
(Fig. 12) has the highest overestimate at the 99.9 percentile (off by a factor of
1.88). Station 10 (Fig. 13) has the greatest underestimate at the 99.9 percentile
(a ratio of 0.52).
The comparison of the 2-hour CFD's for the ten locations is characterized
subjectively as follows: Two stations (4 and 12) have cumulative frequency
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distributions close to those of the measurements. At two stations (3 and 23)
overestimates of concentration occur for the entire CFD with the largest errors
less than a factor of 3. At two stations (10 and 33) underestimates of
concentration occur for the entire distribution with errors as large as a factor of
4. At three of the stations (17, 28 and 36) concentrations are overcalculated
beyond the following percentiles: 99, 56, and 63. At one station (15)
concentrations are undercalculated beyond the 45 percentile.
Other visual comparisons of the estimated and measured CFD's for 2-hour
concentrations are given in Figures 14 through 16. Station 4 (Fig. 14) has the
best agreement between estimates and measurements over the whole
distribution. Stations 23 and 28 (Fig. 15 and 16) have poor agreement between
estimates and measurements throughout most of the CFD. At station 23
concentrations are primarily overestimated. At station 28 concentrations are
underestimated at low percentiles and overestimated at high percentiles.
The two measured CFD's that are least lognormal occur at stations 33 and
36. Station 33 (Fig. 17) appears to have two slopes, and at the highest
concentration (greater than 99.8 percentile) there is a sudden increase in
concentration. Station 36 (Fig. 12) also seems to have two different slopes with
the transition occurring around the 70 percentile. For estimated CFD's, station
28, 33, and 36 (Figs. 16, 17, and 12) appear to be least lognormal.
Discussion
These CFD's from measured air quality data and from dispersion model
estimates have been determined for averaging times from 30 minutes to 24
hours, for periods of record from 1 month to a heating season. These are all
for locations within urban areas. These cannot be compared directly to present
U. S. air quality standards since the standards specify periods of record of 1
year. However, it is quite likely that during the heating season in Bremen, and
during December through February in St. Louis, the highest sulfur dioxide
concentration of the year occurs, due to the number of space heating sources
that produce sulfur dioxide. Concentrations with the extreme frequency of once
per year should be expected to vary considerably from year to year, due to the
high variability of occurrence of stagnant or other special meteorological
conditions that cause the extreme.
The number of stations with extreme estimates from the dispersion models
within a factor of 2 of the extreme measurements for the investigators
mentioned herein are summarized in Table I.
Dr. Frank Pasquill (1971) in his presidential address delivered before the
Royal Meteorological Society on April 21, 1971 stated, "The agreement as close
as 20 or 30 percent which may be achievable in the most favorable
circumstances for a long-term multi-station average, is obviously unattainable in
respect of an individual value even when this is averaged over an hour or so. In
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this case the only prospect of useful prediction lies in the statistics of the
cumulative frequency distribution of a large number of such values, and it would
appear. . . that prediction of the rather extreme high concentrations encountered
only occasionally may be achievable with an error factor of about two."
Since most of the extreme value estimates are within a factor of two of the
extreme measurements, these results are in agreement with Dr. Pasquill's
statement.
It should be pointed out that these model results mentioned here contain
both overestimates and underestimates so that no constant correction factor can
be used to bring the estimate of these extremes more in line with the measured
extremes. Errors in both directions with regard to emissions and small sources
near the receptor probably account for a large proportion of the differences.
Keep in mind that model estimates representative of areas a square kilometer or
larger are being compared with point measurements.
There are many other comparisons and statistical tests than can be
performed with these CFD's in addition to the consideration of the extremes
and the rather cursory examination of the lognormality of them. Some of the
possibilities for further examination of this data follow: Perform statistical tests
to determine how close the given CFD's are to lognormal. Determine standard
geometric deviations (slope of distribution) from two percentiles in the
distribution and see how these vary with location in the urban area. From the
40-station sampling network determine measured and estimated concentration
patterns at various percentile levels. Determine what meteorological conditions
cause the extreme value estimated concentrations and the extreme value
measured concentrations at each station.
Conclusions
Gaussian plume dispersion models for urban areas produce CFD's at
individual sampling locations similar to the distributions determined from
measurements. These distributions subjectively appear similar to lognormal
distributions. The maximum 24-hour concentration estimated during an 89-day
period was within a factor of 2 of the measured maximum at 35 of 40 sampling
stations in St. Louis, Missouri. The maximum 2-hour concentration estimated
during the same 89-day period was within a factor of 2 of the measured 2-hour
maximum at all 10 sampling stations, having 2-hour measurements
available. Estimates of air quality concentration at a downwind receptor for a
given hour from a point source are generally regarded as accurate only within a
factor of 2 because of uncertainties in estimates of emission rate, turbulence
structure, plume height, wind direction and wind speed. It is encouraging to find
similar accuracies for the extreme value (99 percentile for 24-hour, 99.9
percentile for 2-hour) estimates for urban locations influenced by multiple
sources. (Note that the maximum estimate may be calculated for a different
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2-hour period than the period that has the maximum measured concentration.)
This gives somewhat increased confidence to the air pollution
meteorologist asked to estimate urban air quality concentrations to be compared
with standards. One must keep in mind that good estimates of concentrations
from dispersion models can only result from good emission estimates and
reliable measurements of meteorological parameters.
Acknowledgements
The author wishes to thank Adrian D. Busse for his development some years
ago of a computer program to routinely produce a cumulative frequency
distribution from a time series of data. Dale H. Coventry for programming the
computer-plotter routine to produce log-probability plots, Ralph I. Larsen for
suggesting the preparation of this paper, and Lea Prince for her valuable
assistance.
Table I
Number of stations with extreme estimates from models within
a factor of two of extreme measurements, and worst error.
Investigator
City
Averaging Time
Extreme Percentile
Within a factor of 2
Worst error, a factor of:
Fortak
Bremen
30-min.
99.5
4 of 4
1.7
Koch and Thayer
Chicago
1-hour
90
7 of 8
2.8
St. Louis
2-hour
90
9 of 10
2.6
Turner
St. Louis
2-hour
99.9
10of 10
1.9
St. Louis
24-hour
99
35 of 40
2.4
FREQUENCY DISTRIBUTION ST. LOUIS 24-HOUR S02. DEC64-FEB6S STflTION 4
Figure 10-1. Best agreement at the 99th percentile.
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10-10
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10-11
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10-12
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10-13
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10-14
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10-15
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Figure 10-17. Example of sudden transition to higher measured concentrations.
10-16
-------
References
Fortak, G., 1970: Numerical simulation of temporal and spatial distributions of
urban air pollution concentration. Proceedings of Symposium on
Multiple-Source Urban Diffusion Models. EPA, Air Pollution Control
Office Publication No. AP-86.
Koch, R. C., and Thayer, S. D., 1972: Validation and sensitivity analysis of the
Gaussian plume multiple-source urban diffusion model. Final Report
prepared under Contract Number CPA 70-94, Geomet, Inc. EPA Office of
Air Programs Publication No. APTD-0935.
Larsen, R. I., 1971: A mathematical model for relating air quality measurements
to air quality standards. Environmental Protection Agency, Office of Air
Programs Publication No. AP-89.
Pasquill, F., 1971: Atmospheric dispersion of pollution. Quarterly J. Royal
Meteorol. Soc. 97: 369-395.
Turner, D. B., Zimmerman, J. R. and Busse, A. D., 1972: An evaluation of some
climatological dispersion models. Presented at 3rd meeting of Panel on
Modeling of NATO Committee on the Challenges of Modern Society,
Paris, France.
DISCUSSION
J. Visalli: It has been suggested in some papers that were presented earlier that,
particularly for S02, the body is susceptible to even shorter-term fluctuations
than 2 hours; I'm talking about fluctuations more on the order of 2 to 4
minutes. I was wondering how good you feel your model would be in predicting
variations for this short of a time interval.
Turner: I would like to have good representative 2- to 4-minute
meteorological information and good emission information that includes the
variability of all sources with the time interval of 2 to 4 minutes, in order to
attempt such short-term concentration estimates.
Singer: One comment touched on a bit before by Don Pack and also questioned
by Frank Gifford, and a slight comment which you made at the end of your
paper, pleased me. Everyone has been dealing with numbers and just using them
blindly without making any comment about the accuracy of the data and
bringing in the statistics. The source term, Q, many times is out by an order of
magnitude when you actually look at the data itself. The meteorology may be
out by a factor of 2. When you start verifying it and looking at the SO2 data,
which can also easily be out. I would like to see someone say what is the
accuracy of the data and try to bring that into the statistics, the meteorology
10-17
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and also the final verification. You said you were out by 20, this could easily
just simply be an instrument error. But I would like to see this aspect of
statistics. I mean, Don asked that question this morning—can you bring in the
error into the analysis or can you verify it. I have always heard the answer; yes
(it can be done) but I have never seen anyone do it.
Turner: With regard to the accuracy of the data Irv, I have to give the people
credit who did the sampling in St. Louis for their care to obtain good data. For
24-hour samples they did have replicate bubblers side by side. Air was drawn
through them by the same pump but with two different critical orifices, one for
each of the samplers. These duplicate samples were compared in the laboratory.
I forget the exact numbers, but on the order of 3 to 5 percent of the total
number of samples deviated by more than 10 percent. Most of these were
thrown out for the reason that they did not duplicate within 10%.
Singer: I knew your data was good, but it was just a general warning to the
statisticians who take the number that we provide and blindly use it. We know
better in that respect.
J. Rustagi: I also notice this kind of behavior is lognorrnal. Actually, there are
outlier-prone distributions. I don't know whether lognorrnal is one of them but
Professor Neyman has given a detailed analysis of outlier proneness of
distributions in a symposium which was conducted at Columbus in 1971.*The
gamma distribution is one of them. This is one approach which could be taken.
Secondly, as has been mentioned before, that if the concentration is too low or
too high we have different kinds of errors of measurement. Suppose that you use
the same model as lognorrnal and you have the variance dependent on the mean.
In my data it was noticed that at low levels the errors were proportional to that
of the mean. So if you put in the model the variance as a certain function of the
mean, the estimate of mean and so on can be correspondingly calculated.
Helmut Lieth: I can verify your statement for the low values from our analysis
of the national air pollution network data, but what do you do with the
variability in the high levels?
Rustagi: As you said if there are different kinds of behavior at lower end and
upper end one could put the variance as a quadratic function of the mean, cubic
function of the mean, some other function of the mean, or some other
complicated function. What I mean is that the variance should reflect the error
in measurement as noticed by instrumentalists.
Lieth: Yes, but there is a problem. There is a logical difference in the production
of the high values and subsequently their variability, and the low values. It is a
factual problem as well that you have more of some kind of pollutant at a
certain weather condition. So this is not plainly a statistical deviance. How can
you get this logical difference out with a straightforward statistical method?
D. McNeil: We attempted to look at that problem with some data in New Jersey
and the point of view that we adopted was to try and find a transformation of
the pollutant concentration which would make the variance constant. In fact,
that was how we arrived at the fourth root transformation. In doing so we rather
*J. S. Hustagi (Editor), Symposium on Optimizing Methods in Statistics, Academic Press,
New York, N. Y., 1971.
10-18
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fortuitously found it made the mean value of the increment in the level linear as
the function of time. That would be one way of solving that problem. You can
do that . . . you might find you need a different transformation depending on
the weather conditions, but we did it just by lumping all the values for one year
together.
Don Pack: I think we could belabor this to death, but I would like to point out
one thing that when one is dealing in trying to identify extreme values you want
the longest possible period of record. I did a little arithmetic back there. New
York is generating around three million two hundred thousand estimates of an
individual pollutant for about ten pollutants. Alright, you've got thirty million
values, very attractive to people with large computers. However, the error
function in a real measurement system is not stationary. It trends with time.
Initially, and I point specifically to Mr. Turner's data—research data carefully
controlled with operators who were dedicated to producing the best possible
information. On the other hand, the kind of data that is becoming available to us
in the many urban areas of order 10 to 20 cities are not of that kind at all. The
technicians may be v,ery devoted initially and the equipment will be new. But
with time everything deteriorates so that we would have error functions such
that as the length of your record increases, errors also increase. The only point
that I am trying to make is that deductions on the kinds of distributions can be
markedly affected by the character of data.
L. Crow: In studying extreme values'and measurements of particulates in a
natural background in Wyoming, some important meteorological influences have
been noted. Natural dust produces the very high extremes due to high winds, but
the high winds are not neatly distributed throughout this year or any other year.
The extremely low values are affected by precipitation. Wind blown dust can be
locked-in during winter by heavy snow cover. Is there ;a way that we as
meteorologists and statisticians can treat these extrerhe ends using real data
instead of some arbitrary formula? Can we bring about an adjustment for the
extremes that do occur if we add the meteorological parameters to actual
instances of extreme data?
Rustagi: There would be a way of mixing distributions if we know enough about
the distribution at the other extremes. There are procedures available for
estimating parameters with mixtures and you can put the distributions in two
different tails with corresponding probabilities—that would be one possible way.
Lieth: I think we have listed, in the paper that we handed out here a little while
ago, the new program NONREG, which probably solves that problem
mathematically for you. NONREG is in a package available here on the UNC
campus.
Court: \ can't help being impressed by the great similarity of the problems being
discussed today and those with which we have been dealing for many years in
the field of hydro-meteorology-such problems as the inaccuracies of rain gages,
the non-normality of rainfall, the various procedures such as cube root and
fourth root to obtain homoskedacicity for regressions and many other similar
relations.
10-19
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Neustadter: I would desire to talk with anyone who might be able to help me
with a problem that we have almost at hand in our program. I mentioned very
passingly at the beginning of my presentation that we have hundreds of samples
which we're subjecting to analysis. These are samples collected on high-vols on
high quality analysis paper, and we are doing a lot of analysis. Essentially what
we are going to end up with is a set of hundred's of items each characterized by
ten's of parameters. We have been looking for techniques and so far we don't
know that much about it, but pattern recognition seems to sound like the best
thing. The only thing we are aware of is one article from Livermore that seems
to indicate that pattern recognition is now coming into the field of chemistry
and handling multiple parameter chemical reactions and phase changes.
10-20
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11. FOURIER ANALYSIS OF Al R MONITORING DATA
BERNARD E. SALTZMAN
Department of Environmental Health
University of Cincinnati
Cincinnati, Ohio
The proliferation of pollutant monitoring activities is now providing massive
amounts of data. Effective utilization of this information requires proper
analysis. This may be as costly as the collection of the data. A major problem
has been to obtain the "signal" from the data in the presence of overwhelming
amounts of "noise" produced by environmental fluctuations. Computers have
been utilized to provide information on the statistical distributions of the
numbers. Tabulations also have been presented of data averaged by time of day
and/or by season (NAPCA (1969)). The purpose of this paper is to explore the
application of another technique, Fourier analysis of data, which offers the
promise of extracting significant new types of information.
In Figure 1, a plot is given of monitoring data for particulate lead in air
(Cholak, et al. (1968)). This was obtained by continuously sampling outside air
from the second floor window at the Kettering Laboratory, Cincinnati, Ohio,
through 4-inch membrane filters. The filters were changed on Mondays,
Wednesdays and Fridays; thus the lead analyses represented values averaged for
2- or 3-day-periods. Application of the usual statistical calculations provided the
following information: mean 1.07 jug/m3, standard deviation 0.55 jig/m3,
geometric mean 0.95 jug/m3, standard geometric deviation 1.60. A plot of
cumulative and differential frequency distributions is given in Figure 2, which
shows a tendency to a lognormal distribution. What other significant
information can be extracted from this data?
Examination of Figure 1 indicates irregular fluctuations with time. These
can be regarded as analogous to colored light, comprised of the sum of a mean
value and of a series of fluctuations of differing periods, amplitudes and phases.
In the case of a mixture of colors of visible light, resolution into a spectrum can
be obtained by the means of a spectroscope. In the case of sound or radio wave
mixtures, tuned circuits can be utilized to obtain the spectra. Recent
developments in computer science now make practical Fourier analysis of data,
11-1
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which is the equivalent of a spectroscope in providing the spectra of
fluctuations. A good explanation of this technique for chemists has been
presented by Horlick (1972). In order to explore these possibilities, computer
programs were prepared utilizing a Wang computer and plotter, which was
available and convenient for program development. Table I provides a summary
of the programs that were developed.
Explanation of Program
Data for this program should consist of a series of values at uniform time
intervals. The time units are usually hours or days. Provisions are made for
missing data. The fluctuations are resolved as the sum of a series of sine and
cosine waves of different amplitudes and periods. Thirty-six periods are used
covering 7 octaves (doublings) from 3 times to 384 times the time interval of the
data; each octave is divided into 5 equal, logarithmically-spaced steps.
Data Processing
For each data point, the time from the middle of its interval to a selected
initial reference time and date (e.g., midnight on a Sunday) is calculated. This
time is divided by the first period (3 time units), and converted to a time
phase angle; the date value is multiplied by the sine of the angle and stored in
one register, and by the cosine and stored in a second register. This calculation is
repeated for successively longer periods (up to 384 time units), and the data
stored in 70 other registers. About 10,000 computer steps are required for each
data point. Each of the 72 data registers accumulates sine or cosine products for
its assigned period. Mathematically, the calculations are as follows:
For each value of data Xj taken at time tj, a series of 36 calculations is made
by assigning to an index, i, consecutive integral values from 0 to 35:
Period, p. = 3x2i/5
360 t:
Sine term, SM = X-. sin L
IJ J PJ
360 t:
Cosine term, GJ: = Xj cos
The 36 pairs of registers are each assigned to a specific period, PJ. They
accumulate the corresponding sums 2:Sj: and 2:Cj: for all data points.
11-2
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Data Printout
In the data printout, the final time, t, from the reference time to the end of
the last data value is given. The number of time units of data, d, is tabulated.
The mean value is calculated as follows:
X = 2Xj/d
Rather than presenting the sine and cosine results separately, a clearer
picture is obtained by combining them into a vector sum and a phase angle. The
latter is combined with the period to calculate the first peak time after the
selected reference time.
For each period tabulated, the results are calculated as follows:
amplitude,
The peak time, tjf (past the reference time) is calculated as follows:
peak degrees, . ^
Zsjj
Q\ = arc tan
peak time, _ Q.
tj = ["seo"
If the fluctuations are in phase with the cosine wave (peak at reference
time), the resulting angle, and peak time are zero.
Data Plotting
In both types of data plots, the horizontal scale is a logarithmic scale of
periods. The initial point represents 3 time units, and each inch represents 1
octave (doubling of period). In the amplitude plot, the vertical scale above the
origin is a linear scale, on which 5 inches is equal to the amplitude range
selected. The plotted points present the spectrum of fluctuation intensities. In
the peak time plot, the vertical scale below the origin is a linear time scale
beginning at the time selected. Each 0.1 inch represents 6 time units. The scale
can be marked off in appropriate divisions, e.g., days of the week or months of
the year. The plotted points represent the first peak time and 6 consecutive
11-3
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subsequent ones. For periods exceeding 64 time units, fewer peaks are plotted
because the maximum ordinate is 384 time units, or 6.4 inches downward.
Results
Table II illustrates the data printout of the computer program. This program
required that the data values be entered as integers. The values, which were
expressed in micrograms per cubic meter to the nearest hundredth, were
therefore multiplied by 100 before entry. The first column gives the selected
values of period, which are the same for all data processing. Each is
approximately 1.149 times the previous one. Each fifth line represents exactly
1 octave, or a doubling of the period. It will be noted that due to slight
inaccuracies in the computer, the values for 48, 192, and 384 are listed
respectively as 47.999, 191.999, and 383.999. Amplitude values are given in the
second column. It can be seen that there are several peak values. The third
column presents the peak times as phase angles. Since these are not convenient
to visualize, the fourth column presents the times for the first peak past the
selected reference time. In this case, the units are days after January 1, 1967. -
A clearer visualization of these results can be seen from the plot of the data
in the first and second columns of Table II, shown in Figure 3. Surprisingly,
there is a dip in the amplitude line at a period of 7 days, although there are
peaks at 3 1/2, 6, and 8 days. There are also successive amplitude peaks at the
following multiples of 8 days: 2, 4, 6, 8, 12. Since the Fourier calculations are
not accurate unless the data cover a time interval of at least 4 periods, the
plotted values for periods exceeding 100 days cannot be considered as accurate.
These data represent 364 days of measurement.
The computer output also includes phase information. Figure 4 is a plot of
values in the first and fourth columns of Table II. The vertical scale downward
represents a linear peak time scale, which is marked off in months of the year.
The horizontal scale is a logarithmic representation of cycle periods, identical
with that of the horizontal scale in Figure 3. To understand the significance of
this plot, one may visualize a straight line vertically downward for the period 96
days. It can be seen from column 4, Table II that the first peak time occurs 51
days after January 1st (or February 21st). This is indicated in Figure 4 by a dot
towards the bottom of the box representing February. There are successive dots
vertically downward for each 96 days thereafter. Thus the dots and the
connecting lines represent the times during the year when each cycle maximum
occurs. If Figure 4 is viewed vertically below Figure 3, peak times are shown in
correspondence with each amplitude value plotted in Figure 3. To avoid
crowding on the left side of the figure, for each period no more than 7 peaks
are plotted. In this plot the vertical time scale begins at January 1, 1967. The
computer program also permits starting this vertical scale at any desired time
after the reference time. This is the equi"~!ent of shifting the plotted lines and
11-4
-------
the time scale vertically upward and viewing any selected lower portion. The
lower end can be understood to extend to infinite time.
In the preceding discussion it was indicated that data for many cycles of
period were required for accurate results. Figure 5 presents Fourier spectral
amplitude data for the 3-month period of October-December, 1968 for total
hydrocarbons in Cincinnati. The data for this and the two following figures were
hourly-averaged values reported (NAPCA (1969)) by the Continuous Air
Monitoring Program of the National Air Pollution Control Administration.
Surprisingly, again there is no peak at 7 days. Major amplitude peaks can be seen
at 12 hrs., 18 hrs., and 1,31/2,6, 8, and 12 days. Similar plots were made for
the hydrocarbon data for each individual month of October, November and
December. The patterns of amplitude peaks showed a similarity although their
proportions were altered for the different months. The combined data for the
3 months eliminated some of the erroneous high peaks that were obtained for
periods exceeding 1 week. As the amount of data increases, the sharpness
increases of the "tuning" of the calculations for each period, and some of the
peaks are reduced.
Figure 6 shows the Fourier amplitude spectrum for sulfur dioxide
concentrations in Cincinnati, for the month of October, 1968. Amplitude peaks
are evident at periods of 12 and 18 hours, and 1,4 1/2, 6, and 8 days. If this
figure is compared with the hydrocarbon results in Figure 5, it can be seen that
there is a remarkable similarity, even though these pollutants come from entirely
different sources. This suggests that the atmospheric dispersion processes, which
are similar for both pollutants, exert the major controlling role in determining
the atmospheric levels of these pollutants. Figure 3 also shows peaks at
corresponding periods. All of these figures show a dip at 7 days and peaks at 6
and 8 days. They all show evidences of peaks at 3 1/2 days.
Figure 7 shows the phase results for the sulfur dioxide data. The downward
time scale in this case is from 0 to 16 days. The days of the week are indicated
on it. The computer program can view any portion of these results, which can be
assumed to extend downwards to infinity.
Discussion
The significance of the Fourier spectra presented will become clearer after
more types of data from more locations are analyzed. Interesting possibilities are
opened up by this technique. Common factors operating to determine pollutant
levels should become evident by amplitude spectra peaks in alignment. Differing
periods indicate differing sources of variation. The Fourier analysis technique
also offers a means of correcting the data for the incomplete response
characteristics of the sampling methods or of the instrumentation. It has been
shown (Saltzman (1970); Schnelle and Neeley (1972); Horlich (1972)) that the
resultant data include distortions because of failure to respond to rapid changes.
11-5
-------
If the transient and frequency responses are known, they can be incorporated in
the Fourier computer program. This may permit recalculation of the data to
correct for the distortions and more closely approximate the actual levels in the
atmosphere.
The calculations described above were carried out on a small computing
system which was readily accessible and convenient to rapidly develop a
program. The Wang system requires approximately 1 millisecond for each
step. Approximately 10,000 steps were required for the calculations on each
data point. Thus the calculations for hourly data for 1-month period (720
data points) required 100 minutes of computer time. A program is being
developed for an IBM S/360/65 computer. Preliminary results are in agreement
with those already presented. The IBM computer, of course, has a much greater
capacity, and can calculate for more intervals of period, allowing finer detail or
greater range. Calculating time was found to be 500 times as fast as that of the
Wang system. Future work should show whether results in other cities are
parallel to those in Cincinnati.
Acknowledgement
This work was supported in part by the Center for the Study of the Human
Environment, under U. S. Public Health Service Grant ES00159, and in part by
the Environmental Protection Agency Grant R800869.
Table I
Summary of Fourier Programs for
Wang Model 700B Computer
With Model 702 Plotting Output Writer
Name of Program No. Blocks
Total No.
of Steps
Functions
3-Digit Data Recording
Fourier Analysis of Data
Recording and Retrieval 1
of Fourier Data
718 Records, edits, and retrieves
3-digit numbers on magnetic tape
cassette; 48 numbers in each
block, 100 blocks on each side of
cassette.
911 Retrieves 3-digit numbers from
tape cassette, performs Fourier
calculations, tabulates and plots
results.
352 Records on magnetic tape cassette
and retrieves contents of 74
Fourier calculation registers
before they'are altered by data
printout and plotting. Permits
adding and subtracting of blocks
of data.
11-6
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Table II
Data Printout of Program
Cincinnati, Lead in Air (/ug/m3 x 100}
2nd Floor, Kettering Laboratory
1/13/67 to 12/31/67
Reference Time: Sunday, January 1,1967
Mean
Period
3.000
3.446
3.958
4.547
5.223
6.000
6.892
7.917
9.094
10.446
12.000
13.784
15.834
18.188
20.893
24.000
27.568
31 .668
36.377
41.786
47.999
55.137
63.336
72.754
83.572
96.000
110.275
126.672
145.508
167.145
191.999
220.550
253.345
291.017
'334.291
383.999
106.812, Data Time
Amplitude
1.666
2.550
2.000
3.789
5.270
7.042
3.955
10.084
4.754
5.065
8.175
8.230
12.176
2.727
8.819
7.131
9.846
15.490
3.478
9.145
15.615
10.204
23.046
6.397
9.391
12.771
10.484
18.866
28.376
15.357
16.456
33.781
38.536
32.181
28.957
44.370
352.00, Final Time
Peak, Degrees
65.082
181.068
58.181
282.193
141.800
90.440
31 3.694
263.186
91 .834
239.137
242.803
152.877
132.093
341 .883
265.902
330.826
68.748
45.089
56.746
58.009
156.919
84.015
170.716
299.028
98.225
190.476
121.677
156.635
103.599
67.377
120.906
92.264
48.657
355.468
281.102
216.058
364.00
Peak Time
.542
1.733
.639
3.564
2.057
1.507
6.005
5.787
2.319
6.939
8.093
5.853
5.809
17.273
15.432
22.055
5.264
3.966
5.734
6.733
20.922
12.867
30.034
60.432
22.802
50.793
37.272
55.115
41.874
31 .282
64.483
56.524
34.241
287.354
261 .027
230.462
11-7
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2.5
^2.0
| 1.9
5 ..0
a
0.5
JAN.
FEB.
MAR.
APR. MAY
TIME
JUNE
JULY
,2.56
AUG.
Figure 11-1. Concentrations of lead in air sampled from the second floor
window at the Kettering Laboratory, Cincinnati, for the period January 1
to December 31, 1967.
100
80
UJ
20
X of Time
Equal or Greater
0.5
1.0
LEAD
1.5
2.0
2.5
Figure 11-2. Differential and cumulative frequency distributions of the
concentrations of lead in air shown in Figure 1.
11-8
-------
0.5 r
0.4
0.3
o
E!02
0.1
3 4 6 8 10 15 20 3O 40 60 8OIOO 150200 300400
PERIOD (Days)
Figure 11-3. Fourier amplitude spectrum of the concentrations of lead in air
shown in Figure 1.
PERIOD (Days)
34 68 10 20904000 80100 ISO 200 300400
ii t i i i i i i i i iii • i i i i I i i i i • i
Figure 11-4. Fourier peak time data for concentrations of lead in air shown in
Figure 1.
11-9
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0.8 r
0.6
I
gO.4
0.2
0 -
_L
I i
6 12
(Hours)
I 2
PERIOD
4
(Days)
8
16
Figure 11-5. Fourier amplitude spectrum of concentrations of total
hydrocarbons in Cincinnati for Oct.-Dec., 1968 (as reported by CAMP).
10 i-
6 12
(Hours)
PERIOD
Figure 11-6. Fourier amplitude spectrum for concentrations of sulfur dioxide in
air in Cincinnati for October 1968 (as reported by CAMP).
11-10
-------
PERIOD
M
T
W
T
F
S
Ul
T
F
S
S
M
T
(Hours)
6 12
(Days)
4
8
16
Figure 11-7. Fourier peak time data for concentrations of sulfur dioxide in
Cincinnati for October, 1968.
References
Cholak, J., Schafer, L. J., and Yeager, D., 1968: The air transport of lead
compounds present in automobile exhaust gases.'Amer. Industrial Hygiene
Assoc. J. 29: 562-8.
Horlick, G.,1971: Fourier transform approaches to spectroscopy. Anal. Chem.
43: 61A-66A, July.
Horlick, G., 1972: Digital data handling of spectra utilizing Fourier trans-
formations. Anal. Chem. 44:943-7.
NAPCA, 1969: 1968 Data Tabulations and Summaries, Cincinnati, Continuous
Air Monitoring Projects. National Air Pollution Control Administration,
Raleigh, N. C., Publication No. APTD 69-16.
Saltzman, B. E., 1970: Significance of sampling time in air monitoring. J. Air
Pollution Control Association. 20: 660-5.
Schnelle, K. B., Jr., and Neeley, R. D., 1972: Transient and frequency response
of air monitors. J. Air Pollution Control Association. 22: 551-5.
11-11
-------
DISCUSSION
Marcus: I think this approach of time series analysis of pollutant concentration
data is absolutely essential. One thing about focusing on periodicities, is that it is
another way of looking at or trying to look at some of the fundamental
mechanisms that produce these diurnal patterns. Have you tried a logarithmic
transformation of the concentrations? To use the concentrations as they are
gives you a Fourier analysis of the . . .
Saltzman: The virtue of using a linear transformation is that is simplifies adding
all the components. One can then add all the amplitudes and reconstruct the
original data. I don't see any advantage to converting to logarithms. It just
complicates everything. You can get an exact representation with the linear data
input.
Marcus: Have you done any statistical analysis to test the significance or the
reality of the existence of these peaks?
Saltzman: No. What you see are preliminary results. As a matter of fact I want
to mention how the calculations were made. You may laugh, but this was done
on a Wang computer with a plotter output. To run one month's data required 7
million steps. It took 100 minutes on the Wang to execute, but it was convenient
for me. We are now putting it on the IBM/65 which is about 500 times as fast.
We hope to get this program going by about March. What you see now are
preliminary results. I can say that we are getting spectra and that they do persist
for data time periods as long as 3 months.
Marcus: One advantage perhaps of going into a transformation of the data would
be to reduce the concentration observations to a somewhat more nearly
Gaussian-distributed form and then you could .. .
Saltzman: This procedure has nothing to do with statistics. This is an analytical
representation. I am not talking about probabilities here. This is an exact
representation. If you use linear terms, you can add and subtract everything.
Marcus: We have an exact representation of intrinsically noisy data and perhaps
just transforming that, and trying to again extract a signal, we can get rid of
some of the uncertainties that are built into the observations at the beginning.
Benarie: I am very impressed with this spectral representation of air pollution
data. It is a great idea. Some amplitudes can be caused by meteorological factors
or by human activity. For instance, the 24-hour peak amplitude is almost
certainly produced by the early morning inversions. It appears in the Fourier
spectrum of every pollutant. The human activity has a weekly cycle, so it can be
easily recognized in the amplitudes, even if the data are not for a long period.
Secondly, such analysis gives us immediately the answer, following Shannon's
communication theory on what should be the sampling frequency of the
apparatus. It should be 2N if the highest frequency is N. So we don't have to ask
any more if the best sampling time is 5 minutes or 30 minutes or 24 hours. We
get the answer out of these spectra.
11-12
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For the lead, you took 2- and 3-day data. As the concentrations are related
to sampling time, an artificial sampling component which is not real has been
introduced. All sampling times should be either uniform or random, but not 2
and 3.
Saltzman: In this particular case each data item was weighted for the length of
its sampling period for calculations. Now with regard to the proper sampling
time, in my paper published in the October, 1970 issue of the Journal of the Air
Pollution Control Association, the viewpoint was that if we are interested in the
effects on the body, then we are only interested in the frequencies that the body
can see. For contaminants with a long biological half-life such as lead (which
could be several years), high frequency fluctuations are attenuated and
determining the high frequency components is a waste of money. So we would
only sample to determine the significant frequencies remaining after attenuation
by the biological window through which the body views the data if this is the
purpose of sampling.
11-13
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-------
12. THE PREDICTION OF HIGH CONCENTRATIONS OF
SULFUR DIOXIDE IN LONDON AND MANCHESTER AIR
F. BARRY SMITH AND G. H. JEFFREY
Meteorological Office, Brae knell, United Kingdom
Introduction
High concentrations of sulfur dioxide (SO2) in the atmosphere can cause
considerable upset to people with bronchial troubles, particularly if the
concentrations are maintained over a period of days. One of the most unpleasant
results of the famous London smog of 1952 was the very high mortality rate
caused by bronchitis and other related ailments during the subsequent week;
overall it was estimated the smog caused between 3,500 and 4,000 deaths (see
"Air Pollution and Health", 1970). Hospital places were also in tremendous
demand by less seriously affected sufferers.
Quoting from the same source, absenteeism tends to rise rapidly among
London factory and office workers whenever the daily average S02
concentration exceeds 250 jug/m3 (not a particularly high value in London) and,
in Salford, absenteeism is twice that of the daily average amongst all workers
when the concentration reaches 1000 /ig/m3 (a rather more exceptional level).
Even the most cursory investigation of weather conditions on days of high
S02 concentration reveals that cold and relatively calm days in winter are
frequently the most dangerous. The Meteorological Office in London was
therefore asked over 8 years ago to provide a forecasting service of those
meteorological conditions which were likely to lead to significantly high
concentrations and a subsequent demand on hospital beds. The criterion chosen
was for a concentration of at least 1000 jug/m3. During the 1952 smog the
maximum daily average SO2 concentration over 10 sampling stations was
approximately 2000 M9/m3; however the effects of the Clean Air Act are such
that this is about double the greatest SO2 concentration experienced from 1968
to 1970 inclusive when averaged over 4 stations, with a typical mean
somewhat above the Inner London average. In the original scheme developed to
meet this demand, the meteorological conditions which were expected to lead to
critical concentrations were as follows:
(a) an expectation of less than 2/8th of cloud, or of sky obscured by fog at
18Z, OOZand06Z.
12-1
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(b) an expectation of a mean of surface wind speeds at 18Z, OOZ and 06Z of
less than 3 knots, the actual speed at each of these hours being less than 5 knots,
(c) an instability index S = (2TX - 3Tn - 12) > 0, where Tx is the highest
temperature expected at midnight at Urawley at any level up to and including
900 mb (but excluding the surface) and Tn is the forecast minimum temperature
at Heathrow (temperatures are in °C). If (a), (b) and (c) are satisfied, a forecast
of high pollution is issued, but if in (c) we only have —3 < S < 0 then a more
cautious forecast is made.
The London Weather Centre, which is responsible for making the forecasts,
has felt that a re-appraisal of the scheme is called for, partially because the
scheme did not appear to be highly successful and partially because the Clean
Air Act has reduced the overall SO2 low-level emission rates.
With growing concern all over the world over the state of urban
environments, many alternative forecasting schemes have been developed, and
several of these are 'reported in the literature. These generally fall into one of
three groups.
(a) Numerical models. Whenever source distributions are reasonably well
known both in time and space, the equations of diffusion may be used to
calculate spatial distributions of pollution, provided the wind and turbulence
characteristics can be adequately prescribed and predicted. Such calculations
require considerable computer facilities, and can only be meaningful on a scale
that is large compared with the typical distance separating those sources which
are not individually represented but are merged into area sources.
(b) Physical models. Detailed models of urban areas have been created in
large wind tunnels and the dispersion of pollution emitted in life-like manner
from one or more sources studied. The advantage of this system is that the
proposed addition of a new major source into an urban environment can be
studied fairly realistically, even when the local topography is quite complex.
Perhaps their chief disadvantage lies in the difficulty in simulating the wide range
of meteorological parameters that affect dispersion: low-level inversions, fogs,
solar radiation, wind direction and so on. Their use is therefore more in the urban
planning field than in routine day-by-day predictive work.
(c) Empirical models. The scheme outlined above is one such model. The
physics of the whole dispersion process only enters in at a comparatively low
level, but the scheme does have the advantage that it is based on real data taken
in real situations. Considering the very considerable complexity of the problem
in an urban environment, the empirical approach may be the only really
practical one on a day-by-day basis whenever a sufficient body of data is
available for post-facto analysis (say at least 2 years of measurements of SO2
and the weather). Since such measurements are readily available in London, our
revised scheme described in later sections is also of this type.
12-2
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The Measurements of Sulfur Dioxide in London
Inner (central) London as defined by Weatherley and Gooriah (1970)
comprises an area 30 km by 20 km encompassing Hendon in the NW, Dagenham
in the NE, Sidcup in the SE and Wimbledon in the SW. Within this area the
National Survey sampling network has nearly one hundred sites in operation (the
exact number varies between 90 and 100 from year to year.) The area contains
industry, scattered mainly around the River Thames and along the Lee Valley, as
well as housing and commerce regions with substantial fuel consumption. Parks
and comparatively low density housing areas (less than 5000 people per square
km) are also present, so that the source distribution and the actual concentration
distribution are far from simple (see Figs. 1, 2 and 8). Inspection of the Figures
shows that the correspondence between source and concentration, as
represented, is not particularly strong on a scale of 1 or 2 kilometers, but is
much better on a scale of 5 to 10 kilometers. This perhaps indicates that
individual sites may often be significantly influenced by one or two fairly
dominant local sources, and only when the concentrations are averaged over,
say, four or more sites do they begin to have an obvious meaning in relation to
broad area source-values. Figure 1 shows population density and the main
industrial areas and comes from Weatherley and Gooriah. Figure 2 shows values
of the mean SO2 winter-values derived from the ten yearly values for each Inner
London station in which the smoothed overall trend over the period is linearly
extrapolated one year to 1969-70. The mean for all stations is 231 jug/m3;
however the area-density of stations is not uniform and if isopleths of mean
concentration are drawn (ignoring all the possible pitfalls in doing this) the mean
concentration determined on an area basis is approximately 213 jug/m3. The
overall pattern appears to change little from year-to-year but on a shorter time
scale significant changes from day-to-day probably occur due to changes in
source strength and wind direction. If Figure 2 is representative, concentrations
within Inner London vary from at least half, to twice the area mean on any
occasion. The highest values are in Westminster, where, since industrial
undertakings are few, road traffic and office-block central heating systems may
be the most significant polluters of the urban environment.
Figure 3 shows two concentration-direction roses, one for Kensington (site
4), the other for Deptford (site 3). The radius in any direction represents the
smoothed mean concentration, relative to the mean for all conditions, when the
wind is coming from that direction. An almost 3 to 1 variation in mean
concentration with wind direction is implied at both stations, and this appears to
be fairly typical.
12-3
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Meteorological Parameters for London
The analysis of S02 concentrations at a rural site which preceded the
present London analysis, revealed that day-to-day values depended significantly
upon the following parameters:
(a) wind direction. Effective source strengths may vary sufficiently with
direction as exampled in the last section.
(b) temperature. Source strength in the UK tends to be greater at lower
temperatures. Temperature is also correlated with other meteorological factors
that influence the dispersion of the S02-
(c) wind speed. Wind speed affects the stability of the atmosphere and
hence the vertical dispersion of E>02. For a specified emission rate of SO2 the
concentration immediately downwind of the source tends to be inversely
proportional to the wind speed. It is probable that when ventilation by the
exterior wind significantly affects offices and homes, the production of S02
increases, following the increase in compensatory heating. Some of these trends
are clearly in opposing directions and, at the rural site investigated, were almost
self-cancel ing. In London itself wind speed appears to remain important,
particularly at light winds when accumulation of S02 within the same mass of
air leads to the highest concentrations recorded.
(d) mixing depth or stability. Dispersion through the vertical of S02
depends on the intensity of vertical turbulence. Quite frequently a layer near the
ground which is well mixed by turbulence is "capped" by a thermal inversion
which inhibits further spreading of the pollutant to greater heights. The
pollutant is thus trapped, and concentrations tend to a value inversely
proportional to the height of the inversion. At places well away from the major
source of pollution, the mixing depth is one of the most important parameters,
since the approach to uniform mixing below the inversion has time to take place.
Within London itself where the typical distance between source and receptor is
much less, the mixing depth ceases to have this importance, except when it is
very small. (See (d) below.)
The post-facto meteorological data have been obtained from Kew records in
Parts I and II, and from London Airport in Part III of the forecasting scheme.
After consideration and experiment it seemed that the most relevant
parameters could be defined as follows:
(a) wind direction. 10 meter wind directions, using the tabulated mean over
the preceding hour, averaged over 12 hours centered at 15Z during the day when
the concentration sample is started. (National Survey 1-day samples start in the
morning at an assumed time between 09Z and 10Z and finish 24 hours later). If
12-4
-------
the wind direction varied by more than 60° during the period, the direction is
described as "Variable" and treated as a separate category. Further if there are at
least 3 hours of calm (wind speed effectively zero) during the period direction is
described as "Calm" and treated as a further separate category.
(b) temperature. In Parts I and II of the forecasting scheme the minimum
hourly temperature, during the period 10Z to 24Z on the day when the sample
is started, is used. The reasons for this choice are:
(i) temperatures after midnight are not expected to be very relevant
since emission rates are then normally quite low.
(ii) the minimum temperature is likely to be well-correlated with the
overall coldness of the late afternoon and evening, and hence the domestic
heating output.
In Part III of the forecasting scheme the minimum temperature for the whole
24-hour period is used.
(c) wind speed. Two wind speed parameters are extracted. The first is the
number of hours when the hourly-mean wind speed (10-meter value) is 2 knots
or less (Parts I and II) or less than 5 knots (Part III). For simplicity we call this
the number of hours of calm. The second parameter is the mean wind speed for
the full day on which the sample is started. Locally a mean speed over the
precise period of the sample should have been taken but the sidereal-day mean
was already tabulated and thus saved quite an amount of laborious computation
at the expense of some accuracy.
(d) the mean reciprocal mixing depth (MRMD). The London analysis
indicates that only in situations with low mixing depths, did the MRMD become
significant as a predictor. During the winter months either of the following
criteria almost always are necessary and sufficient for a significant MRMD:
(i) Surface inversion sets in before 18Z, and during the day cloud height
at or below 500m, or
(ii) Surface inversion sets in between 18 and 21Z, and during the day in-
version or cloud height at or below 300m.
The rules for surface inversions during the winter are:
(i) At 18Z, assume a surface inversion unless wind speed > 8 kts or
cloud amount > 5/8ths.
(ii) At 21Z and 24Z, assume a surface inversion unless wind speed >
8 kts or cloud amount 8/8ths.
The SO2-Concentration Data
Ideally all sampling stations in the Inner London area should have been used
in the analysis. However certain factors weighed against this. For various reasons
not all stations maintain a regular day-by-day sampling routine. Further it was
decided in this exploratory analysis to limit the amount of data to that which
12-5
-------
could be handled and analyzed fairly easily using a desk electronic computer, the
Olivetti Programma 101.
Consequently 4 stations with a good record of completeness were selected,
and permission to use their data was kindly granted by the Councils concerned.
These stations are:
Kensington, Site 4
City of London, Site 17
Hackney, Site 2
Deptford, Site 3.
Mean concentrations for a particular day were evaluated whenever either
3 or 4 or tne stations gave readings. In the former case the mean was given the
appropriate weighting to balance the omission of one of the readings:
Expected mean concentration when C4is missing
(m,+ m2+m»+mA \
•M+Vm, )
where C1 , C2 and C3 are the day's readings at the 3 given sites; m1 , m2 , m3 and
m4 are the long-term mean concentrations. For the 2 winter periods that are
studied in detail in this analysis (the winter of 1968-69 and that of 1969-70)
they take the following values:
m (Kensington) = 364/Lig/m3
m (City of London) = 415Mg/m3
m (Hackney) = 376 jt/g/m 3
m (Deptford) = 253 /ug/m 3
Winter covers the months from October to March inclusive.
No readings were taken on Saturdays or Sundays, an
-------
at one site, and C is a 1-day concentration at the same site) has a statistical
day-by-day distribution which is virtually the same irrespective of site, then
-------
and
S2=(C2-C£)C2/C£ (2)
Cm is calculated by forming the geometric mean of all the concentration values
in the sample and is a theoretically better estimate of the parent population
median concentration than is C, the arithmetic mean, of the parent population
mean concentration. Similarly a is more reliable than s.
Applied to the 73 data values involved in Figure 5:
FROM THE DATA CALC. FROM EQ. (1) and (2)
Cm = 227.1 C =238.6 C= 238.0
a =0.313 s = 76.8 s = 74.6
The median evaluated by its fundamental definition, namely by the value which
equally divides the data points (50% having a higher concentration and 50% a
lower) is Cm = 231 jug/m3. However this is a less accurate method of estimating
the parent population Cm from a sample on the assumption of a lognormal
distribution.
The close agreement between the calculated and derived values of C and s
strongly support the lognormal hypothesis. The advantage of this hypothesis is
that it enables us to estimate the likely area in Inner London in which the
concentration of SO2 may exceed some defined critical level at any time.
However the hypothesis must remain of doubtful validity "out on its tails", i.e.,
when the area becomes smaller than about 10 sq km, and too much reliance
should not be placed on forecasts in these circumstances without a much more
detailed investigation than is given here.
One final point concerning these statistics may be made. The geographical
distributions of
(a) the mean concentrations for the winters under analysis, and
(b) the number of days with concentrations exceeding 500jU9/m3 shown in
Figures 7 and 8, are very similar. The following approximate correspondence
apply:
Number of days with Mean concentration
C > 500 jug/m3 over for the same two
two winters winters
0
10
25
50
100 ...
150
200
300
360
400
These relations should be roughly consistent with the lognormal time
distributions of concentration.
12-8
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The Data for Manchester
Sulfur dioxide data were obtained for the same 2 winters as for London
from 7 regular sampling stations in the National Survey Network in
Manchester: Numbers 11, 13, 15, 16, 17, 18 and 19.
Meteorological data came from Manchester Airport (Ringway) some 9 miles
to the south of the city.
Both sets of data were treated in the same way as in the analysis of the
London data and thus no further explanations will be given.
The Variation of Concentration with Meteorological Parameters
The previous section headed "Meteorological parameters for London"
described the meteorological parameters that appeared to be significant.
Table I gives the variation of mean concentration, averaged over the 4 sites,
with wind direction.
TABLE I. Variation of Mean Concentration With Wind Direction
Mean concentration. Mean concentration.
Wind Direction jug/m3 Wind Direction Aig/m3
001-030
031-060
061-090
091-120
121-150
151-180
Variable
243
271
351
395
302
268
307
181-210
211-240
241-270
271-300
301-330
331-360
Calm
235
204
223
232
323
279
306
Table II sets out in detail all the basic data, some of which has already been
defined in section "Meteorological parameters for London." The column headed
MRMD gives the mean reciprocal mixing depth described as H when it is
significantly important. The penultimate column represents the results of the
objective post-facto forecasting scheme (Part I).
The forecast scheme was developed empirically by considering the
concentration values and the appropriate meteorological parameters for the first
winter 1968-69. When applied to the second winter 1969-70, the scheme proved
to be equally successful without any further modification or elaboration of the
rules. The rules may be stated quite simply as follows.
The Forecasting Scheme: Part I
(a) A concentration averaged over the usual 24-hour period at the 4
stations: Kensington 4, City of London 17, Hackney 2, Deptford 3, will exceed
400 jug/m3 (or in the case of those wind directions which on average have low
12-9
-------
S02 concentrations, a normalized concentration exceeding 1.5), whenever at
least one of the following conditions is fulfilled:
(i) the number of hours with mean wind speed lessor equal to 2 knots
greater or equal to 8 hours (see column 5, Table II)
(ii) minimum temperature (col 7) < 0°C, and at least 1 hour light winds
(col 5)
(iii) the MRMD (col 8) = H and minimum temperature (col 7) < 6°C,
and mean wind (col 6) < 1 0 knots
(iv) (3 tca(m -2Tmin) is between 0 and 25 if c (previous day) > 600
/Ltg/m3, or between 1 0 and 35 if C (previous day) > 400 jug/m3
(b) The concentration defined in (a) above will exceed 600 /zg/m3 whenever
(i) the minimum temperature (col 7) is less than 5°C; ^nd light winds
(col 5) for 19 or more hours
(ii) (3tca|m - 2Tmin) exceeds 25 if C (previous day) > 600 jUQ/m3, or
exceeds 35 if C (previous day )> 400 jug/m3
The results displayed in Table II may be summarized in the following
tables:
(A) Contingency Table for success in forecasting A*
(i.e., London: either O 400 jug/m3, or normalised C > 1.5; Manchester O
270A*g/m3)
high concentration lower concentration
forecast A forecast not A Total
.London M/C London M/C London
forecasting 55 49 106 144 161
success ~ 28% 21% 54% 61% 82%
forecasting 15 20 17 22 32
failure x 8% 9% 9% 9% 18%
Total 70 123 193
London 36% 64%
M/C 69 166
30% 70%
M/C
193
82%
42
18%
235
Exactly equivalent information is included for the Manchester data, without
giving the basic data equivalent to Table II.
In both cases when a forecast of high pollution is made, a success rate of
about 80% is achieved.
Some important points must be made:
(a) The London threshold values 400 and 600 jug/m3 are not universal
values. They are only meaningful in so far as the source distribution and output
remains basically unaltered. While it is virtually impossible over a short period of
time to identify any such change, it is recommended that the two values be
12-10
-------
(B) Contingency Table for success in forecasting B*
(i.e., C > 600M9/m3 for London; C ^ 450jug/m3 for Manchester)
very
high concentration
lower concentration
forecasting
success
forecasting
failure x
Total
forecast
London
11
6%
0
0
11
6%
B
M/C
11
5%
8
3%
19
8%
forecast
London
182
93.5%
1
0.5%
182
94%
not B
M/C
212
90%
4
2%
216
92%
Total
London
192
99.5%
1
0.5%
193
M/C
223
95%
12
5%
235
* Percentages in general have been rounded to the nearest whole number.
suitably modified if necessary once every 5 years in the light of the overall
changes in mean winter concentration at the 4 sites over the preceding 5 years.
(b) No attempt has been made in this analysis to relate the concentration
values to the effect on people's health and the likely demands on the facilities at
the two hospitals concerned. This is largely a medical problem and lies outside
our capabilities.
(c) In the previous scheme a forecast had to be made before 1600Z of the
chances of high pollution during the evening and night that followed. We have
moved to a different problem, partially because our basic concentration data are
daily mean values (rather than hour-by-hour values) and also because we feet
that the problem is not solely a night-time problem. At night many people, and
particularly bronchitic sufferers, are likely to be in the shelter of their own
homes where they can to some extent regulate the condition of the air they
breathe, whereas during the day they are more likely to be out and about, being
affected by atmospheric concentrations of SO2 which are not necessarily a great
deal less than the evening concentration. Our aim has therefore been to forecast
the mean concentration for the whole 24-hour day. The forecast of the
meteorological conditions is therefore longer-range and to that extent more
liable to error.
The percentage of forecasts made by
Forecast the scheme that were correct
London Manchester
The percentage of actual cases
correctly forecast by the scheme
London Manchester
C>A 79
CB 100
C
-------
Ideally, then, a forecast should be made in the early morning, before 1000Z,
of the meteorological conditions for the next 24 hours and hence the likely
mean concentration. Since many of the criteria in the forecasting scheme relate
to evening conditions (and hence are not altogether different in intent from the
previous shceme), some revision of the concentration forecast could be made as
late as 1600Z whenever this seemed called for.
(d) The results set out above refer to a post-facto application of the scheme
using meteorological data as it actually occurred. In day-by-day application of
the scheme in the future these data will have to be forecast and this is bound to
introduce further significant errors.
Some of the parameters, such as the minimum temperature and the cloud
amounts, are already estimated on a routine basis for other purposes. The
criterion of the number of hours when the mean wind falls to 2 knots or below
is probably the hardest to estimate with any certainty, and for this reason Part
III explores the effect of a relaxation of this condition.
A scheme designed to forecast actual concentration values: Part 1 1
This part is concerned with predicting actual concentration, as distinct from
forecasting whether or not certain threshold values are exceeded. The variation
of concentration with the same meteorological parameters that were successfully
used in the "threshold" method was studied for the 2 winter periods for the
London data. The following fairly simple formula yielded reasonably
satisfactory estimates of the average daily concentrations at the 4 sites:
where C = long term mean concentration
Cest = estimated concentration (24-hour average)
Cp = concentration for the previous 24 hours
T = minimum temperature (°C) expected up to midnight
t = number of hours of mean wind less than 3 knots during the 24 hours
2 2
a = — , b = ± if the mean wind for the day exceeds 6 knots
3 9
•3 Q
a = _, b = _ otherwise
7 21
5d = 1, if the mean wind comes from the "dirty" sector, 060 to 120
0, otherwise
5m = 1, if the mixing depth is low (as defined at the end of section on
London)
0, otherwise
12-12
-------
Although the formula has been verified only for the 4 sites, it is probably
equally applicable to any group of sites in Inner London, and can be applied to
other cities pro'vided the appropriate value of C, the mean concentration, is
inserted. This has been done for the Manchester data.
The formula can either be expressed graphically as a nomogram (see Figure
9) or it can be programmed for a desk electronic calculator.
The root-mean-square errors have been evaluated for the 4 sites over the 2
winters. The significance of the errors have to be assessed in relation to the
inherent "error" due to local quasi-random variations in concentration at the 4
sites, which can be estimated from the inter-site correlation studies described in
the section about SO2 concentration data. The inherent error in the 4-site
average concentration was shown to be about 50 /ig/m3.
The root-mean-square error in the formula-estimates is only 66 jug/m3 (little
more than the inherent error) if the actual value of Cp, the previous day's
concentration, is known and used, but rises to nearly 80 jug/m3 when Cp is only
known by the application of the formula using the actual meteorological data at
the end of the previous 24-hour day (see Figure 10). This is still a satisfactorily
small margin of error when compared with the inherent error, and is certainly
considerably less than that obtained using a persistence forecast (142 jug/m3) i.e.
by using yesterday's concentration as an estimate for today's. If Et is the total
error, Ej is the inherent error, and Es is the basic error of the scheme, then
For the whole of Inner London (nearly 100 sites), Ej will fall from 50jug/m3 to
about 10 /Ltg/m3. The expected value of Et would then be
E2=(80)2-(50)*+(IO)S
i.e. a little over 60
The formula displays the relative importance of the basic parameters. It is
clear that an error of 4°C in T, the minimum temperature, would introduce an
error in Cest of about only 50 jug/m3. A similar error would follow from an error
of 3 hours in t, the hours of light winds, or of about 150 jUQ/m3 in Cp. The
method does not therefore demand impossible precision in evaluating the basic
meteorological parameters. Nevertheless if it has to be used at an operational
office, such as the London Weather Centre, evaluation of these parameters may
take rather more time than the fully occupied staff may wish to spend. The next
section (Part III) describes a simplified scheme designed to overcome this
difficulty.
12-13
-------
The simplified forecasting scheme: Part III
In order to help the operational forecaster by minimizing the analysis
required in making a pollution forecast, an investigation has been made into the
effect on the accuracy of the scheme when:
(a) London Airport meteorological data is used instead of Kew data for
London. Unlike Kew, London Airport data is received by London Weather
Centre on an hourly basis.
(b) The "hours of calm" criterion is relaxed to include all hours when the
mean wind falls below 5 knots. This criterion should be much easier to forecast.
(c) The minimum temperature up to midnight is replaced by the minimum
temperature over the 24 hour period of the forecast. The forecaster will already
have this temperature estimate for other reasons.
(d) The effects of the daily mean wind speed and direction are ignored.
The empirical formula for forecasting concentration in terms of the revised
parameters is
where 5m = 1 if MRMD = H, and is 0 otherwise
T = minimum temperature 09Z to 09Z
t = hours when mean wind falls below 5 kts
C = mean concentration
Cp = yesterday's concentration
The results using this scheme are summarized in Table III. As expected the
errors are somewhat bigger than in Part II, but not appreciably so. It seems that
this very simple scheme still gives a very satisfactory means of forecasting
pollution.
For many cities, including Manchester, the number of days with a low
mixing depth (MRMD = H) is very small (only a few days per year), and
experience may show that the factor (1 + 5m/6) can be then fairly safely
ignored.
12-14
-------
TABLE II. Basic Forecasting Data
Date
1968
Oct. 1
2
3
4
10
11
15
16
17
18
22
23
24
25
29
30
31
Nov. 1
5
6
7
8
12
13
Ca
127
104
121
198
172
123
182
174
161
314
493
548
480
331
253
197
204
202
418
378
311
283
546
506
ddb
260
240
240
260
220
240
260
230
260
250
140
360
050
070
190
230
180
200
030
070
070
060
150
110
C/c~c
0.57
0.51
0.59
0.89
0.84
0.60
0.82
0.85
0.72
1.41
1.63
1.96
1.77
0.94
1.08
0.96
0.76
0.86
1;72
1.08
0.89
1.04
1.81
1.28
'calm01
0
0
0
4
1
0
2
0
0
13
15
18
8
0
0
7
7
2
12
2
2
2
11
0
ve
13
13
14
9
8
13
8
12
9
7
5
3
6
10
12
12
8
12
11
9
14
11
6
10
W MRMD«
14 H
15
15
16
14
12
10
11
9
8
10
12
11
12
12
11 -
14
13
2
7 H
7
6
4
4
' F/Ch
=
=
=
=
=
5=
=
=
=
Ax
A=
A=
A=
=
as
=
=
=
A=
=
=
=
A=
x
Estimated C'
—
121
119
164
—
155
—
166
185
367
—
497
477
259
—
277
236
177
—
431
360
357
—
405
aCorrected mean S02
-------
TABLE II (continued). Basic Forecasting Data
Date
Dec.
14
15
19
20
21
22
26
27
28
29
3
4
5
6
10
11
12
13
17
18
24
31
Ca
431
394
284
425
382
317
225
250
585
529
892
443
303
481
618
374
469
864
419
300
278
570
ddb
100
090
340
320
150
160
190
190
120
030
080
160
150
140
030
050
340
060
220
180
270
330
C/cc
1.09
1.12
1.02
1.21
1.26
1.18
0.96
1.06
1.48
2.18
2.54
1.65
1.00
1.59
2.54
1.38
1.68
3.19
2.05
1.12
1.25
1.76
t d u e
Tcalm w
0
0
3
20
5
1
0
2
17
12
12
10
2
12
20
1
16
13
7
2
3
12
14
17
10
5
8
9
12
11
6
4
7
5
6
8
5
6
7
5
9
10
13
7
min
6
5
6
7
7
6
10
12
7
8
2
4
4
1
2
3
4
0
0
6
6
0
MRMD9 F/Ch
—
—
—
—
H
H
—
H
H
H
H
—
—
—
—
—
H
H
—
—
H
—
X
=
=
A=
=
Ax
=
=
A=
A=
B=
A=
=
A=
B=
=
A=
B=
A=
—
=
A=
Estimated C'
362
344
—
519
382
310
—
222
616
557
—
668
318
470
—
349
612
723
—
289
—
—
1969
Jan.
Feb
1
2
3
7
8
9
10
14
15
16
17
21
22
23
24
28
29
30
31
. 4
5
6
7
11
12
13
677
565
499
500
346
435
421
321
271
287
248
290
319
351
281
274
353
341
197
449
670
706
329
635
318
233
340
270
330
090
150
190
120
190
190
170
270
150
190
230
240
200
270
270
240
340
280
280
240
320
V
360
2.43
2.53
1.54
1.42
1.14
1.85
1.06
1.36
1.15
1.07
1.11
0.96
1.36
1.72
1.38
1.16
1.58
1.53
0.96
1.61
2.89
3.04
1.61
1.96
1.03
0.83
17
2
14
0
0
10
0
0
0
3
1
0
0
8
13
0
9
1
0
4
17
16
0
5
0
0
7
8
8
10
14
12
8
16
13
10
13
8
12
10
5
12
8
8
14
14
6
4
6
9
11
14
2
0
5
1
6
2
2
8
4
4
3
6
10
11
10
7
3
4
7
-1
-1
4
2
-3
1
0
H
H
—
—
—
—
H
—
—
—
—
—
—
H
H
—
H
H
—
—
H
H
—
—
—
—
B=
A=
A=
X
=
A=
A=
=
=
=
=
=
=:
A=
Ax
=
A=
A=
=
A=
B=
B=
X
B=
=
=
537
493
546
—
271
443
477
—
261
296
280
—
191
357
428
—
461
330
232
—
733
783
362
—
361
304
12-16
-------
TABLE II (continued). Basic Forecasting Data
Date
14
18
19
20
21
25
26
27
28
Mar. 4
5
6
7
11
12
13
14
18
19
20
21
25
26
27
28
Oct. 1
2
3
7
8
9
10
14
15
16
17
21
22
23
24
28
29
30
31
Nov. 4
5
6
7
11
Ca
503
379
409
346
475
591
381
276
289
387
396
396
581
552
399
525
216
472
498
369
372
280
251
171
280
175
235
165
195
149
255
380
207
177
197
255
328
429
301
126
333
223
263
205
135
224
417
296
248
ddb C/Cc tcg|md
360
360
050
050
180
060
040
030
350
360
040
050
150
080
V
070
220
070
070
020
010
360
030
030
050
280
310
250
220
210
200
130
120
240
190
160
170
040
250
250
270
200
340
280
230
300
270
160
230
1.80
1.36
1.51
1.28
1.77
2.18
1.40
1.13
1.03
1.39
1.46
1.46
2.14
1.57
1.30
1.50
1.06
1.34
1.42
1.52
1.53
1.00
1.03
0.70
1.03
0.75
0.73
0.74
0.95
0.63
1.08
1.26
0.52
0.97
0.84
0.95
1.22
1.58
1.35
0.56
1.50
0.95
0.94
0.88
0.66
0.96
1.87
1.10
1.21
15
5
0
0
1
9
0
0
0
0
0
0
12
0
0
0
1
0
0
2
4
0
0
0
4
4
9
0
2
2
14
19
1
2
1
11
19
17
12
0
15
4
7
0
0
2
13
4
0
ve T . f
v 'mm
11
9
10
21
13
10
6
9
11
11
14
14
8
13
11
10
12
10
10
6
2
10
10
12
12
3
4
7
3
7
5
1
5
7
7
6
1
1
2
7
2
3
7
6
17
11
4
3
9
-1
-1
0
-1
2
6
4
0
2
3
0
1
-3
4
3
3
8
6
6
5
5
1
2
1
1
7
3
12
15
15
10
7
14
9
10
8
9
12
11
13
5
10
5
9
14
5
-2
5
8
MRMD9 F/Ch
H
-
—
H
—
H
—
—
—
—
—
H
—
—
—
—
—
H
H
—
—
—
—
—
—
—
—
H
H
—
—
—
—
—
—
H
—
—
H
—
—
—
H
—
—
H
—
—
-
A=
Ax
X
=
X
A=
=
=
=
=
=
=
A=
X
=
X
=
A=
A=
X
X
=
=
=
=
=
Ax
=
=
=
Ax
Ax
=
=
=
Ax
Ax
A=
Ax
=
A=
=
Ax
=
=
=
A=
=
=
Estimated C1
594
-
318
392
304
—
313
318
274
—
319
394
549
—
318
383
312
—
413
317
297
—
274
279
246
—
291
190
—
186
278
420
—
217
189
326
—
391
441
161
—
233
387
198
—
283
405
315
-
12-17
-------
TABLE II (continued). Basic Forecasting Data
Date
Dec.
12
13
14
18
19
20
25
26
27
28
2
3
4
9
10
11
12
16
17
18
19
23
24
30
31
Ca
163
192
279
461
292
161
320
280
395
287
529
356
218
661
1161
632
357
304
421
411
463
320
318
428
265
ddb
180
220
210
290
210
230
320
310
320
240
230
230
280
330
V
170
180
260
340
060
020
290
230
090
040
C/cc
0.61
0.94
1.61
1.99
1.24
0.79
0.99
0.87
1.22
1.41
2.59
1.74
0.94
2.05
3.78
2.36
1.33
1.36
1.51
1.52
1.90
1.30
1.56
1.22
0.98
tcalm
2
0
16
9
0
0
0
0
1
0
1
0
0
6
24
13
2
0
7
0
10
0
0
0
0
ve
14
10
4
8
8
11
10
12
9
8
4
7
9
7
1
1
4
12
6
13
9
12
5
12
17
^min
12
6
0
-2
2
10
1
1
-1
5
2
5
4
1
3
0
6
3
-1
1
-2
6
4
0
0
MRMD9 F/Ch Estimated C'
—
—
H
—
—
—
—
H
—
H
—
—
—
H
H
—
—
H
—
—
H
H
—
-
=
=
A=
A=
=
=
=
=
Ax
=
A=
X
=
X
B=
B=
=
=
A=
X
A=
=
A=
X
=
194
212
402
_
312
192
—
294
374
264
—
288
268
—
981
1140
346
—
486
420
510
_
272
—
328
1970
Jan.
Feb.
1
2
6
7
8
9
13
14
15
16
20
21
22
23
27
29
30
3
4
5
6
10
11
215
387
453
556
659
438
284
249
232
283
178
286
250
295
423
399
257
173
278
277
317
197
421
030
V
270
270
C
090
170
140
140
130
130
160
160
190
230
120
100
230
250
230
350
250
270
0.88
1.26
2.03
2.49
2.15
1.25
1.06
0.82
0.77
0.94
0.59
1.07
0.93
1.25
2.07
1.01
0.65
0.85
1.25
1.36
1.14
0.88
1.89
0
0
1
7
4
0
5
0
2
8
0
7
2
3
11
0
0
0
0
5
1
0
9
16
9
4
9
4
13
9
7
10
4
8
5
9
5
3
6
11
13
12
9
7
11
8
-1
-1
-4
-2
-5
-2
5
8
8
8
7
7
8
3
4
3
5
7
4
3
2
2
-4
—
—
H
H
H
—
H
—
—
—
—
H
—
H
H
—
—
—
—
—
—
—
-
=
=
A=
A=
B=
A=
Ax
=
=
=
=
=
ss
Ax
A=
=
=
=
=
=
=
=
A=
303
291
—
554
582
543
—
212
235
266
—
280
242
298
—
—
353
—
233
337
314
—
441
12-18
-------
TABLE II (continued). Basic Forecasting Data
Date
Ca ddfc
C/c
calm
Tminf MRMD9
F/Ch Estimated C1
12
13
17
18
19
20
24
25
26
27
Mar. 3
4
5
6
10
11
12
13
17
18
19
20
24
26
505
318
366
253
243
158
247
269
307
239
262
354
340
447
683
495
444
429
231
240
189
212
282
314
V
030
300
270
230
220
250
290
330
350
320
300
V
320
V
210
200
070
280
240
270
270
V
010
1.67
1.31
1.58
1.13
1.19
0.77
1.11
1.16
0.95
0.86
0.81
1.52
1.11
1.38
2.22
2.11
1.89
1.22
0.99
1.18
0.85
0.95
0.92
1.29
3
2
6
0
0
0
0
2
5
0
0
0
0
0
12
1
3
0
0
0
0
0
3
8
4
16
8
13
9
12
8
9
5
9
10
8
11
7
3
4
8
8
7
11
13
11
5
7
-1 H
1
-3 H
3
4
8
6
3
1 -
2
-1
-1
0
-1
-1 H
1
3 H
2
7
10
5
7 H
6
-1
A=
=
A=
=
=
=
=
=
=
=
=
X
=
X
B=
A=
A=
X
=
=r
=
=
=
Ax
429
368
—
281
248
205
—
286
307
280
—
302
312
320
—
416
418
411
—
183
235
241
—
425
TABLE III
Results of applying the Forecasting Schemes to London and Manchester data
Correlation between actual
and forecast concentrations
Standard error,
M9/m3
Detailed Scheme: Part II
Inherent error from taking
only 4 sites:
All data: actual cone, for
previous day known:
As above, excluding 11.12.69:
Previous day's concentration
replaced by mean C :
Previous day's cone, replaced
by f/c value for that day:
Persistence f/c: previous day's
cone, used as f/c for today:
London
0.94
0.87
0.90
0.80
0.85
0.55
Manchester London
50
0.80 76
66
95
79
142
Manchester
-
65
—
—
-
—
Simplified Scheme: Part III
All data: actual cone, for previous 0.81
day known:
As above, excluding 11.12.69:
Standard deviation of all the
observations:
0.84
-
0.79 88
83
150
66
-
106
12-19
-------
Figure 12-1. Population density of Inner London in thousands per square
kilometer.
Figure 12-2. Mean winter concentrations for Inner London for 1969-70 based on
extrapolation from data of previous ten years. Sulfur dioxide
concentrations in jug/m3.
12-20
-------
.0-5 \l I -1-5
* / '•
Figure 12-3. Normalized concentration-direction roses for Kensington &
Deptford.
I0
~ 9
X
5 8
•5
r
I 6
4
li)
£3
Distribution
Best fit
log-normal
ICO 200 300
AVERAGE CONCENTRATION
40O
500
Figure 12-4. Histogram based on 290 values (mean winter concentrations) for all
Inner London sites for the winters 1965-66 to 1969-70. Data grouped into
20jug/m3 classes.
12-21
-------
1000
i loo
UJ
u
8
10
I
i I
O.I I
10 50 90
CUMULATIVE FREQUENCY
99 99.9
Figure 12-5. Spatial distribution for mean winter 1965-70, all Inner London
sites.
1000
6
9
100
8
10
111
I I I I l_
0.1
10 50 90
CUMULATIVE FREQUENCY
99 99.9
Figure 12-6. Daily-values average over the 4 sites, Winters 1968-69 & 1969-70.
12-22
-------
Figure 12-7. Number of days when C > 500 for the winters 1968-69 & 1969-70.
Figure 12-8. Mean winter concentration for Inner London 1968-69-70.
12-23
-------
CONCENTRATION ON MCVIOM MY
MM* «ll>< i • Ml.
-• -4 -t 0 t 4 • • 10 II 14 !• M
Figure 12-9. Nomogram for SO2 concentrations at the four sites in Inner
London.
900
800
700
I
600
500
400
300
200
100
// ./. /••' (Mfldord «rw
•/'/ /\ sj? Ming m*oiurtd Cp
'
• •// r* s •
V;**V
\ I
J I
0 100 200 300 400 500 600 700 800 900 1000
ESTIMATED CONCENTRATION (Mg/m»)
Figure 12-10. Results of the forecasting scheme of Part II.
12-24
-------
Ack nowledgements
The authors are grateful to the Warren Spring Laboratory for their helpful
cooperation in the supply of their National Survey data, and to the following
Councils who gave permission for the data from their areas to be used:
The Corporation of Manchester
The Royal Borough of Kensington and Chelsea
The Borough of Hackney
The London Borough of Lewisham
The Corporation of London
The Rural District Council of Epping and Ongar.
We also wish to acknowledge the very helpful work done by Mr. P. Bushby,
a vacation student, who analysed the Manchester data. This paper is published
by permission of the Director-General of the Meteorological Office.
References
Annual Reports of the National Survey, Warren Spring Laboratory.
The Royal College of Physicians, 1970: Air Pollution and Health. Pittman.
Weatherly, M. L. and Gooriah, B. D., 1970: National survey of smoke and
sulphur dioxide: The greater London area. Warren spring Laboratory.
DISCUSSION
Gifford: You mentioned in your written presentation that you tried to avoid a
numerical scheme because of the complexity that the computer required. Of
course we have a scheme that is perfectly numerical which doesn't require a
computer and since you had been good enough to include a whole rather long
series of data we thought it would be fun to compare our scheme with yours.
The results are summarized in the form of a comment to the preprint of your
paper, which is presented after the discussion of the paper. Our scheme is very
simple. In your notation it simply says that the concentration is proportional to
the source strength divided by the wind speed. This might look like a box model
to you, and in fact it is. This number here (C in Equation 1 of my comments on
the preprinted paper), which can be expanded, depends rather weakly on the
city size and on stability and if you are interested in seeing some other
comparisons these were mentioned in the references. This model really works
very well. For instance, it gives a correlation with the data that Joe Knox
showed yesterday for carbon monoxide in San Francisco that is just about as
good as the one from his model. Unfortunately, as you pointed out, you don't
12-25
-------
have the source strength to be entered into the model so we simply turn the
formula around and use yesterday's source strength, which I call QQ, to put into
today's formula. So if 1 represents today and 0 represents tomorrow, our
formula says that the concentration is given by the ratio of today's vs.
tomorrow's wind speed, times the existing concentration level. Your model gives
a 0.9 (rounded off) correlation; our model gives a 0.7 correlation. Your
simplified model is 0.81 or perhaps it is 0.84, I don't remember. Anyway,
persistence was low, 0.55. I put in confidence limits to indicate the fact that all
of these models beat persistence. I discovered in going through the data that our
model had a serious tendency to over predict. It being so simple, it's very
subject, for instance, to error for low wind speed. When V1 is 1 meter per
second, things got pretty bad. So I tried an arbitrary correction, namely, I put a
modified model which took the square root of the wind ratio and found this
gave a correlation of 0.76. I would expect that these correlations, whichever one
you choose to use, would repeat, since ours is essentially an a priori model. This
I admit was somewhat suggested by our experience with your data, but I would
expect that ours would repeat, that yours would come down a little bit, and that
probably Manchester was more representative of the sort of effect that you are
likely to get with it, and would guess that in all likelihood the best way to
improve predictions of this item would be if you went to work on the source
strength term. In fact I tried to incorporate some information and this is the
question: Do you not think some information on the source spatial and
temporal variability janitor functions and so on would improve the model?
I made the point in my prepared comment on your written paper that I
thought that you had used both of your winters to develop this model and
you've now told us that you only used one winter so I have to that extent more
confidence. Just the same, yours is a very high correlation. If it holds up
anywhere near that value it is going to be an awfully tough forecast to beat.
Smith: I am very pleased to see this and I would certainly agree that wind speed
is important in this sense as you saw in the nomograms I showed you. Wind
speed obviously was playing a very important role in that both in the number of
hours of calm and in the influence of the previous day's concentration. I also
agree that if one could somehow get a better understanding of the Q
distribution, the source distribution in London, one might very easily improve
the situation. The only hesitation I have in this is that both Dr. Gifford's and my
correlations depend on using meteorological data which has really been
measured, its real data. Whereas the forecaster's problem is he has to actually
forecast this data which is not an easy task and I am quite sure the correlations
will drop in practice quite significantly when the forecaster has to face the
music and try to forecast the wind speeds and temperatures for the following
period. And perhaps there comes the level in this when it's not worth going to
great effort and expense to get a Q distribution when you know fairly well that
even if you had the most perfect Q values the meteorology would let you down
and you'll still not get a very accurate forecast.
12-26
-------
Comment by F. A. Gifford on the Paper, "The Prediction of High
Concentrations of Sulphur Dioxide in London and Manchester Air,"
by F.B. Smith and G.H. Jeffrey.
I would like first to record my agreement with Dr. Smith's remarks on the
need for a simple approach, guided by the data, in air pollution forecasting. And
also to second his remarks concerning the limited effect that mixing depth has
on urban air concentrations, as a rule.
The authors approach this problem using the empirical techniques of
objective weather forecasting and one can only applaud both their methodology
and the workmanlike result that they obtain. They mention as alternatives to
their empirical approach, numerical models and physical models, rejecting the
former because they require a large computer facility. One seemingly valid
numerical urban air pollution model does not however require such a computer
facility, namely the simple ATDL model described by my colleague Dr. Hanna
and me in a series of papers. (See all the references cited.)
It is of some interest to compare our model with the present results,
particularly as Smith and Jeffrey have included a fairly long series of London
S02 air concentration data, together with related meteorological data, the
developmental data for their model. Application of our simple model to the
present data is similar to the applications discussed in the first four references
cited. Chemical S02 removal, which could be included using the scheme
suggested by Hanna (1972), will not specifically be taken into account. Then the
simple model gives, in the notation of Smith and Jeffrey,
C= cQ/v (1)
where Q is source strength and c is a dimensionless parameter. Unfortunately no
data on Q are included. To apply our model to forecasting London S02
concentration, we have to use today's value of C as a measure of Q, i.e., where
subscript zero means today. Then
Q0=V0C0/C (2)
is the prediction of our model.
C|=(V0/V,)C0 (3)
Table I displays the results of this comparison, in the form of correlation
coefficients (with 95% confidence limits) between predicted concentration
values from Equation 3, using the data in Smith and Jeffrey's Table II. In this
table, "ATDL-modified" refers to a second prediction using the following
modification of Equation 3:
C| = (v0/v,)"2 C0 (4)
12-27
-------
This entirely arbitrary modification attempts to account for the lack of data on
source strength variability. Variation of Q with several of the meteorological
parameters of Table 11 is probable. For instance, higher winds in winter are likely
to be correlated with higher domestic fuel consumption. The modification to
our simple model is an arbitrary attempt to take this into account.
Table I - Correlations of concentration predictions
by various models with 140 measurements of London 24-hour SO2 concentrations.
Model
Met. Office
ATDL
ATDL-Modified
Met. Office-Simplified
Persistence
Correlation Coefficient
.87
.70
.76
.81
.55
95% Limits
.91-.82
.79-.60
.82-.G8
.8G-.74
.6G-.42
Several conclusions can be drawn from Table I. None of the other
correlation coefficients fall within the confidence limits of persistence, so in this
sense all methods "beat" persistence. Of these methods "Met. Office" correlates
best with the data sample. But remember that this is an empirical method,
developed from the given data sample. The "ATDL" correlation can on the
other hand be considered a genuine test and should repeat on independent
London data or on data from other cities. "ATDL-Modified" incorporates an a
priori hypothesis and should also hold up. However in all honesty it has to be
pointed out that the data suggested the hypothesis.
More important, it seems safe to suggest that there would be no inherent
difficulty in proposing further empirical modifications to "ATDL" to bring it up
to the level of "Met. Office," based on the data sample. "ATDL-Modified" is
already quite competitive. But notice the following implication of that fact.
"ATDL" says that only Q, v, and c can vary. The last varies with stability. To
whatever extent Q, the source strength pattern, is a variable factor, "Met.
Office" takes this into account indirectly, through correlations of Q with
meteorological factors. If you believe "ATDL" on the other hand, it seems that
the most sensible way to improve SO2 forecasts would be to include the best
available estimates of Q(x,y,t).
References
Gifford, F. A., and Hanna, S. R., 1971: Modeling urban air pollution. Atmos.
Environ. 6.
Gifford, F. A., and Hanna, S. R., 1971: Urban air pollution modeling.
Proceedings 2nd Int. Clean Air Congress, Washington, D. C., Academic
Press, pp. 1146-1151.
12-28
-------
Gifford, F. A., 1972: Application of a simple urban pollution model.
Proceedings of Conference on Urban Environment and Second Conference
on Biometeorology, Philadelphia, Pa., American Meteorological Society,
pp. 62-63.
Hanna, S. R., 1971: Simple methods of calculating dispersion from urban area
sources../. Air Pollution Control Association. 21: 774-777.
Hanna, S. R., 1972: A simple model for the analysis of photochemical smog.
Proceedings of Conference on Urban Environment and Second Conference
on Biometeorology, Philadelphia, Pa., American Meteorological Society,
pp. 120-123.
12-29
-------
-------
13. FITTING CURVES TO URBAN
SUSPENDED PARTICULATE DATA
DAVID A. LYNN
Department of Statistics
Harvard University
Cambridge, Massachusetts
Introduction
The effort described here is a comparison of a number of theoretical
frequency distributions with respect to their ability to fit bodies of urban air
quality data. I have used almost exclusively some quite extensive suspended
particulate data from Philadelphia, and so the work, and of course the
conclusions, are to this extent limited and in a sense preliminary. It is presented
at this meeting in particular, principally in the hope that it might interest others
in actually trying out various distributions on their data.
The reaction of people in the air pollution field to the mention of this topic
is often one of surprise. The question is usually believed to have been long
settled by the use of the lognormal distribution and its parameters, the
geometric mean and geometric standard deviation. It is certainly true that the
lognormal is widely accepted in this role. I think we have all observed that air
quality data is typically positively skewed, so that the normal distribution
doesn't fit at all, and then taken the lognormal as the simplest positively-skewed
alternative, without ever really considering alternative distributions very deeply.
(Phinney and Newman (1972) is a recent example.) We need to remember that
the decision to use the lognormal distribution was made mainly by the staff of
the National Air Sampling Network back in the 1950'swhen computer-handling
of data was almost unheard of. At that time, it was a major accomplishment to
achieve enough computer processing to publish the NASN data summary
reports, let alone to do anything very lengthy or complicated.
Now, however, the situation is much different. Not only do most sizable
agencies and installations have computerized data processing, but we are
beginning to be asked for increasingly sophisticated statistical inferences from
our air quality data. This will likely become increasingly true as the typical
pollution levels in our cities decline to the general vicinity of the National
Ambient Air Quality Standards. It is my view that as we in the field of data
131
-------
analysis begin to provide more sophisticated methodologies to be used for these
inferences, we should re-examine one of our most basic assumptions. In a sense,
the increasing importance of gaseous pollutants during the 1960's has already
forced this re-examination on us somewhat, and the results are not completely
consistent—we have to use arithmetic means with sulfur dioxide data in order to
avoid the embarrassment of too many logs of zero.
We might also consider briefly, by ways of further motivation, just what we
have, and might in the future do, with a distributional form once we've chosen
it, and how we verify our choice. When the NASN first introduced the
lognormal, they used it in two ways. They included the geometric parameters in
their data summaries, and they used logarithmic spacing of tallying intervals in
the frequency distribution program that calculated the familiar summary line of
percentiles. More recently, however, statistical applications have included
significance tests, confidence intervals (Hunt (1972)), and extreme value
statistics (Singpurwalla (1972)). In the future we may need inferences about
spatial distributions, decision-theoretic-inference procedures, and so on, all of
which involve making distributional assumptions of some sort. We might note
for the sake of completeness that in some fields, notably actuarial science, some
of the jobs done years ago with fitted theoretical curves have more recently been
approached with smoothed empirical distributions. In fact at least one of these
smoothing techniques has been applied to air quality data, specifically NASN
paniculate data (Spirtas and Levin (1970)). The reasons we might prefer a
theoretical model to a set of smoothed data are the availability of probability
theory, or at least more probability theory, the ability to use somewhat smaller
data sets, and the ability to follow trends in the value of a parameter over time.
Distributions and Fitting Methods
Our purpose here is to compare the several theoretical distributions under
consideration by actually fitting each to a number of sets of data, seeing which
typically fits best, or which fits best in some overall sense, if indeed any do. The
data used are suspended particulate data gathered at three sites in Philadelphia
by the City control agency. The data have been gathered daily for many years.
Here I have used data from 1960 to 1968, comprising 25 annual data sets in
all—nine years from three stations with two sets missing.
There are a number of ways to determine the parameter values that will fit a
specified distribution to a set of observed data. The use of the word "determine"
rather than "estimate" the parameter values is deliberate. For most of the data
under consideration we have essentially complete data for the entire year, so we
consider the year's frequency distribution as "known", and view the problem as
fitting a distribution with known parameters rather than as estimating the
parameters from a sample. Of the various methods available, I have used the
"method of moments," which selects the specific distribution out of the given
13-2
-------
family that has the same moments as our observed distribution, in each case
equating as many moments as there are parameters in the algebraic formulation
of the distribution. The first two moments are the mean and the variance, so this
is of course just what we routinely do when we calculate the sample mean and
standard deviation and then use a normal distribution with that mean and
standard deviation, similar to what we would do with the two geometric
parameters. Used in this way, the mean and variance of a normal distribution are
effective as location and scale parameters. That is, changes in the mean are
equivalent to changing the location of the distribution back and forth along the
axis, while changes in scale, such as changes in the size of the units. Beyond
these simple additive and multiplicative variations, however, a two-parameter
distribution has no flexibility left to change its shape.
We can, however, have more than two parameters, and can determine their
values by equating more than two moments from the observed distribution to
the algebraic expressions for their theoretical counterparts. In common practice,
four moments are the most used, because the sensitivity of the higher-order
moments to small sampling errors is very great. The third and fourth moments,
when used are commonly used in modified forms rather than in their raw
numerical form. If we let ju2, (JL3, and ju4 be the variance and 3rd and 4th
moments respectively, we commonly deal instead with the coefficients
The division by the proper power of the variance makes jS1 and /?2 small
dimensionless numbers, while squaring ju3 makes j31 independent of its sign.
These two constants ()31 and ]32) are called the coefficients of skewness and
kurtosis, respectively. Figures 1 and 2 give some indication of how they operate
to measure the shape of a distribution. The point to note here is that every j31 ,j32
pair represents a different possible distribution shape, if the theoretical
distributional form we are using has enough parameters to make use of them.
This is of course not to say that any observed distribution with the same mean,
standard deviation, skewness, and kurtosis will be the same. We have reduced
some 300-plus data points down to four numbers, and have given up some
information in the process, but we've given up less than if we dealt with only
two parameters. The first few columns in Table I present the mean, standard
deviation, and coefficients of skewness and kurtosis for the various data sets.
To pass from general considerations to the specific distributions used here,
let's begin briefly with the lognormal. We have used not only the ordinary
lognormal, here called the two-parameter or 2-p lognormal, but a
three-parameter version which includes as the third parameter a location
parameter, additive in the p.g/m2 scale before the logs are taken. The density, y,
13-3
-------
written in terms of the parameters GM3 and GSD3, and 6, is given in Equation
2, with GM3 and GSD3 denoting the parameters analogous to the geometric
mean and standard deviation. The 2-p lognormal is obviously just a special case
of the more general form. It doesn't really have a true location parameter,
because it's tied to the fixed origin of the coordinate system by the nature of the
log function. As the geometric mean varies along the axis, it does take the bulk
of the density with it, but the shape also changes, as illustrated in Figure 3. The
third parameter, here 6 permits the distribution to slide along the axis
independent of changes in its general size and shape.
In(X-Q) - lnGM3 I2
i —uzi•
y =
The other three-parameter distribution was the Gamma distribution with
density
Zir (X-0) In GSD3
a >0
y= for B>0
As one can see in the term x'g. the parameters o and ]3 are location and scale
parameters, respectively; a is a shape parameter that is determined from the
skewness coefficient 0.,. If the shape parameter is an integral multiple of 1/2, say
n/2, and |3 - 2, a = 0, the Gamma becomes a chi-square distribution with n
degrees of freedom. As we see in Figure 4, the Gamma distribution can take on a
variety of shapes with changes in a. For a < 1, it becomes "J-shaped", that is,
has an infinite ordinate at x = a; as a gets very large, the shape approaches that
of a normal distribution. Although it often isn't apparent in the graphs, the
lower tail of the density curve does come down tangent to the axis.
The other distribution considered was the four-parameter Beta distribution
or, as it turned out, the Bete distribution and another four-parameter
distribution in different instances. The Beta is the closest there is to a common
four-parameter distribution, although it's most commonly used in a
two-parameter form. Its density,
(X-A)m» (B-X)m2 m,= p-l
" B-Ap + q-' m2=q-
13-4
-------
depends on two parameters, A and B, that are very clearly the minimum and
maximum, and two others, p and q, that are symmetrical, representing a sort of
relative contribution from the two ends. The shape of the distribution and its (3-|
and /32 coefficients are functions of the latter two parameters, and are quite
flexible. The various distributions in Figures 1 and 2 are all Betas. The bounded
nature of the Beta and the p-q symmetry make it a sort of continuous analogue
of a binomial random variable and it is most often used as a Bayesian prior
distribution for the binomial parameters p and q, with the max and min
parameters set at 0 and 1.
Actually, the Beta distribution is specifically a Type I member of the system
of theoretical density curves developed by Karl Pearson many years ago.
Pearson's system is an attempt to provide a theoretical density function for
every point in the j31f j32 P'ane- that is, for every possible combination of the
skewness and kurtosis coefficients. As we see in Figure 5, his system consists of
three "main types", I, IV, and VI, and a number of "transition types". The three
main types are four-parameter distributions, representing the areas in Figure 5,
while the transition curves are represented by lines and points at the boundaries
of the areas. The three-parameter curves, such as the Gamma distribution, which
is Pearson's Type III curve, appear as lines in the plane, while the two-parameter
Normal distribution is only a point. Thus if one is fitting Pearson curves to data,
the correct main type to be used depends on the values of ^ and j32—if we try
to fit the wrong one, we get imaginary numbers during the calculation process.
As it turned out, the Philadelphia data used here, as often as not, fell outside the
range of the Beta distribution, we also included Type VI when needed. There
developed no need for Type IV, though there very well might have. The Type VI
density function is most simply written in terms of an arbitrary origin. Equation
5. A more complicated formulation, with the data expressed in their normal
way, is more logical (Eq. 6). In either case, it's apparent why Type VI is not
often used.
We note in closing that the lognormal densities, though not a part of
Pearson's system, can still be represented by a line in the j8}-/?2 Plane,
TYPE VI DENSITIES
+l) (X*)qi
with X* and A* referred to an origin at
(q.-')A*
-
13-5
-------
with origin at Ojug/m3,
(q2 +
y = £ : & x (6)
X0(q,-l)q' r(q,-q2-l) r
-------
to be, however; the symmetric normal fits the skewed data so badly that the
effect of the poor fit dominated the effect of the roughness of the observed
data. This comparison did, however, provide an interesting way of viewing the
process of choosing a distribution. With the value of the criterion for the normal
being 100%, the choice of any of the skewed distributions would reduce this to
50-70%, while the distributions typically differed among themselves by less than
10%, and rarely by more than 30%.
In fact, when we look at the results in Table III, it's apparent that there is
frequently very little choice among the distributions. The figures in the table are
the values of the absolute difference criterion at the 20 jug/m3 level of
aggregation; to provide a simple summary, averages for each station and for the
whole group are provided. It's clear, first of all, that the normal distribution
doesn't fit, which is of course no surprise. It is also apparent that the two
lognormal distributions have noticeably lower values than the Gamma and the
four-parameter Pearson. It is not really clear that the differences between the
two distributions in each pair are real because position changes of 1 to 2 units or
so did sometimes occur just in changing from one to another of the various fit
criteria.
Because the data used here are quite a narrow sampling from the many
monitoring sites (with their possibly-different distributional shapes), we
probably want to attach relatively little weight to choosing an overall "winner"
among the distributions, and rather consider just why the various distributions
react the way they do to the different data sets. In Table I are tabulated the
fitted values of the parameters of the various distributions, and with this
information in conjunction with Table III and some figures we'll consider the
performance of the various distributions.
First we consider the Pearson curves, the Gamma distribution and the two
types of four-parameter curves. They are usually closer together than they are
close to the performance of the two lognormals. But except for a very few
instances, they aren't too far from the lognormals, at least in comparison with
the normal. The extremely bad instances are the cases where one or both of
them becomes J-shaped. Recalling that in the jS.,-/^ plane, the Gamma represents
the dividing line between the Type I and Type VI areas, we can view the Gamma
as an approximation to either. The equation of the Type III, or Gamma line, is
2j32 = 6+3j31f so we would expect the approximation to be better, the closer j31
and j32 for our distribution are to that line.
In fact, the Gamma and the Type I or VI are quite close for most of the
data sets (or station-years), and those that are farthest apart are in fact those
farthest from the line. Considering the data sets where the Gamma and the
four-parameter are the most different, we find that those where the
four-parameter is better (1-60, 1-65, 2-65, 3-65) are all in the Type VI area,
while those where the Gamma distribution does better (1-68, 2-62, 2-67, 2-68,
3-68) are all in the Type I area. In fact, with a single exception, closer
examination shows this to be true for all the data sets. Except for 3-66, the
13-7
-------
Gamma always does better than the Type I and worse than the Type VI, though
in some cases the differences are trivial. It's not immediately clear why this
should be, and given the fact that the years 1965 and 1968 occur so often in the
lists above, it is likely nothing more than the fact that a year's meteorology may
give a distinct shape to the distributions at the several stations.
Figure 6 presents histograms and the fitted curves for two examples of cases
where the Pearson Type VI does much better than the Gamma. Figure 7 presents
two cases where the Gamma does better than the Pearson Type I. In Figure 6 it's
clear why the Gamma does poorly - it makes a much larger peak than necessary,
and then underestimates on the downslope of the upper side of the distibution.
In contrast, in Figure 7, where sharper peaks are needed, the Pearson Type I
tends to exaggerate the peak or, as in the 2-62 case, even to go off to an
infinite ordinate. In these cases, the curves are almost identical on the upper
side of the peak. On the lower side, the Gamma does better with its moderate
peak, though neither does really well. Interestingly, the factor which controls
these cases is hardly evident in the resulting curves. Those where the Type VI
does better than the Gamma are typically those with high maxima, or long tails,
and those where the Gamma does better than the Pearson Type I are those with
quite short tails—those with few or no values over 400 )Ltg/m3. The 1-65 case in
Figure 6 is a good illustration. The Gamma comes more steeply down the upper
side of the curve in order to throw more probability mass out into the tail, and
in order to do so, builds itself a higher peak, losing good fit not only in the peak
but on the downslope as well.
Before considering the two lognormal distributions more thoroughly, it
might be of value to consider when and why the two pairs of distributions differ
more or less. Although the two Pearson distributions are poorer overall here,
they have a number of possibly-useful properties, and ought not be discarded
outright. Clearly, the situation where they differ most is in those high-skewness
cases where the Pearson curves peak wildly or become J-shaped. No matter what
the skewness, the lognormals retain their zero ordinate at the lower boundary,
and hence fit much better.
The two pairs, however, do sometimes differ fairly markedly even when the
Pearson curves don't have infinite ordinates. Data set 3-68 in Figure 8 is an
example where the two lognormals are better, and set 3-66 is Figure 9 one where
the two Pearson curves are better. These two sets of data illustrate fairly well the
typical result in those situations when there is a noticeable difference among the
distributions, typically the cases where the distributions'tails are moderate-well
out into the 300-400 jug/m3 range, but not into the 600 jug/m3 range. On the
upper side of the distribution, there is very little difference among the several
distributions. To accomplish this, the two Pearson curves typically have
somewhat sharper peaks than the two lognormals, and more abrupt slopes on the
lower tails. Thus if the observed distribution has a clean, sharp peak as does
3-66, the Pearson curves fit better, while if it has a broader peak as in 3-68 or
2-64, the lognormals fit better. Actually, in these data, the former is a rarity; the
13-8
-------
3-66 data set is quite unusual, being the only case where the Pearson Type I is
the best of the four, and the only exception to the pattern of the Gamma doing
better than the Type I, regardless of the rank.
If we consider the differences between Figures 8 and 9, we would rather
expect that those observed distributions with peaks, in some sense, midway
between would produce roughly the same fit with all the curves. This is precisely
what happens, higure ID shows two data sets tor which the results are quite
close, with only the best and worst of the four distributions plotted. Their peaks
do seem midway, being a little more blocky on the high than on the low side. In
addition, there is less noise than in some of the distributions, leaving less oppor-
tunity for one of the distributions to appear better or worse, fortuitously. In
fact, the 3-62 data set can be considered the best in this sense, because by a clear
margin, it had the lowest overall value of the fit criterion.
There are several such data sets where all four distributions fit about equally
well; in about half of all the sets, three of the four could be described as
essentially tied. The balance of cases on either side of this center is far from
even, though; there are several data sets with square peaks, while only one (3-66)
with a sharp peak. It is largely this predominance of relatively square, blocky
peaks (low kurtosis) that gives the lognormals their overall advantage, at least
among these data. And at this point it might be prudent to point out that
shafper peaks often seem to go with overall lower concentration levels. This is
the case here, and since this is relatively high data (especially for the early years),
r
we might expect somewhat different overall results with other data.
We now turn to a closer look at the differences and similarities between the
two lognormals. The first general observations are most readily seen in the list of
parameters in Table I, comparing the geometric mean and standard deviation
with their analogues in the three-parameter model, labelled GM3 and GSD3. The
location of the distributions remains strikingly constant, the sum of the
threshold location parameter 6 and GSD3 is rarely more than 3 or 4 jzg/m3 from
the two-parameter geometric mean. The parameter 6 takes on both positive and
negative values, many near zero but with a few sizeable cases in each direction.
We also note that when the value of 6 is negative, GSD3 is less than GSD, and
vice versa, in rough proportion to the magnitude of 6.
In terms of performance, we've already noted that the two-parameter
lognormal does overall slightly better than the three-parameter, and in fact does
the best of all four distributions tried. This is, of course, in conflict with what
might be expected, since with only two parameters, it should have the least
flexibility. We've seen why this happens, though. The three-parameter
lognormal, the Gamma, and the four-parameter Pearson are in order of
increasing flexibility, but they use this flexibility primarily to adjust to the
upper tail of the distribution. While the 3-p lognormal can't go off to a wild peak
or an infinite ordinate even remotely as easily as the Pearson curves, it can do so
more readily than the 2-p lognormal, as in Figure 11. This permits the 2-p
distribution to "win" by default, even though it fails badly to fit the blocky
13-9
-------
peak. Thus the success of the lognormals is a bit left-handed, as it were. They
succeed not because they are sensitive to the bulk of the data but simply because
they are insentitive to the upper tail, and are tied down at the lower tail.
Figure 12 also illustrates this peaking tendency, but here the result is the
opposite; the observed distribution (2-62) has a very sharp peak, and the
flexibility of the 3-p lognormal reaches it a little better, though neither does
really well. It is also possible to associate this tendency toward high peaks and
long tails with the value of 6. The larger positive values are associated with those
that have shorter peaks and shorter tails than their 2-p counterparts, such as 3-66
in Figure 9.
An attempt to summarize is not as difficult as it first appears. The overall
level of performance is the reverse of what would be expected on the basis of the
number of parameters involved in the distributions, largely because of the
sensitivity of the distributions to the extreme upper tail. As a consequence,
square, blocky distributions can't be fit well unless they have a very short tail.
And I should add as a closing footnote, though this isn't the place to discuss it,
that this sensitivity to the extreme upper tail is characteristic of fitting by the
method of moments. Thus the first place to look for improvements is likely in
less touchy, though possibly less simple, fitting methods.
13-10
-------
TABLE Ma)
Fitted Parameter Values
Sta/Yr
1 1960
1961
1962
1963
1964
1965
1966
1967
1968
2 1960
1961
1962
1963
1964
1965
1966
1967
1968
3 1960
1961
1962
1963
1964
1965
1966
1967
1968
9 1968
11 1968
Mean
169
167
156
167
174
166
174
152
136
142
137
136
137
141
147
127
118
151
153
138
152
134
138
131
117
123
152
MOMENTS
Std.
Dev. ]3i
71
74
58
80
69
68
69
69
55
58
54
55
56
56
61
57
49
67
70
53
65
65
67
64
56
59
74
1.75
2.84
1.35
5.81
1.65
2.88
1.54
1.99
1.43
2.19
2.08
2.96
1.71
2.56
2.52
1.85
1.61
2.18
2.14
1.33
1.70
3.87
1.61
1.71
1.78
1.42
1.58
02
5.85
7.28
4.85
12.69
5.47
8.77
5.45
6.08
4.83
6.61
6.07
6.88
5.36
7.90
7.18
5.38
5.05
6.60
5.93
4.99
5.37
11.76
4.92
5.44
5.37
4.57
4.87
Geo.
Mean
156
153
147
153
162
154
162
139
125
133
128
128
127
132
136
116
109
138
139
129
140
122
124
117
105
110
137
LOGNORMALS
GSD GM3 GSD3
1.50
1.51
1.43
1.51
1.46
1.46
1.46
1.53
1.49
1.46
1.45
1.43
1.47
1.44
1.46
1.53
1.49
1.52
1.53
1.45
1.50
1.55
1.59
1.60
1.60
1.62
1.59
158
128
146
95
157
117
163
142
136
113
109
93
125
102
111
123
114
133
140
135
147
95
155
143
123
147
171
1.49
1.63
1.43
1.87
1.48
1.63
1.46
1.53
1.44
1.55
1.54
1.64
1.49
1.60
1.59
1.51
1.47
1.55
1.54
1.43
1.48
1.72
1.47
1.49
1.50
1.44
1.47
0
-2
23
0
57
4
34
-1
-3
-10
18
17
31
2
27
23
-6
-5
4
-1
-6
-7
24
-30
-24
-17
-34
-32
13-11
-------
TABLE Kb)
Fitted Parameter Values
Sta/Yr
1
2
3
9
11
1960
1961
1962
1963
1964
1965
1966
1967
1968
1960
1961
1962
1963
1964
1965
1966
1967
1968
1960
1961
1962
1963
1964
1965
1966
1967
1968
1968
1968
a
2.29
1.41
2.96
0.69
2.42
1.39
2.59
2.01
2.79
1.82
1.92
1.35
2.33
1.56
1.58
2.15
2.48
1.83
1.87
3.00
2.36
1.03
2.48
2.34
2.24
2.82
2.53
GAMMA
&
47.1
62.7
33.6
96.7
44.4
58.0
42.7
48.6
33.1
42.7
38.9
47.5
36.4
45.0
48.1
39.0
31.4
49.7
51.3
30.5
42.4
63.4
42.6
41.7
37.6
35.4
46.2
7
61
79
57
101
67
85
63
55
43
65
62
72
52
71
70
43
40
60
57
46
52
69
32
33
33
23
35
Criterion
4.04
87.89
PEARSON
X. (VI) A
1553
30908
-3.82
5.46
1396
-104.
1.30
5
11
-2
4
-24
-3
-4
1
.74
.92
.31
.06
.70
.49
.48
.51
3.89
227
2346
4493
1134
272
1064
-2.49
-2
3
-4
-48
-5
1
-1
.32
.85
.31
.48
.24
.01
.71
1249
26
-6.99
-3
-1
-1
.19
.31
.65
50
78
64
93
67
43
56
52
55
56
63
80
58
44
61
54
50
49
65
47
58
-11
50
38
41
44
56
TYPES I
m1/q1
44.03
500.5
1.32
19.93
1.39
16.57
66.60
101.84
0.87
36.07
0.84
-0.06
0.86
17.79
30.87
0.43
0.64
34.73
0.44
1.94
0.94
47.32
0.43
1.01
0.63
6.86
9.73
& VI
n\2/c\2
2.06
0.43
38.83
-0.04
1001.
4.44
2.14
1.21
19.86
1.47
184.1
13.87
35.97
3.42
1.21
15.99
17.37
1.53
26.74
575.1
43.49
38.32
11.48
59.26
22.92
0.44
0.44
B(l)
1731
1731
44973
1034
7569
1040
1703
996
876
1839
17954
2302
903
2901
1228
922
1122
13-12
-------
TABLE II
Typical Summary Tables With Criterion At
5, 10, and 20jug/m3 Class Widths
STATION 2, 1964
5/zg/m3
10jUg/m3
20jUg/m3
STATION
5jUg/m3
10jUg/m3
20jUg/m3
K
SUM
DIFF
REL
DIFF
REL
DIFF
REL
3,1961
K
SUM
DIFF
REL
DIFF
REL
DIFF
REL
Normal
360.5632
146.4098
1.0000
125.9653
1 .0000
125.6084
1 .0000
Normal
320.3271
168.2665
1.0000
148.1114
1.0000
126.3808
1.0000
2-P Log
363.0000
105.9059
0.7234
66.1073
0.5248
38.8479
t
0.3093
2-P Log
324.9995
109.4671
0.6506
79.7891
0.5387
39.1026
0.3094
3-P Log
363.0000
106.0737
0.7245
67.0827
0.5325
39.1794
0.3119
3-P Log
324.9990
110.7158
0.6580
81.1142
0.5477
39.2929
0.3109
Pearson
363.0208
120.8522
0.8254
86.2695
0.6849
62.3205
0.4961
Pearson
325.7625
121.5336
0.7223
101.8797
0.6879
56.9197
0.4504
Gamma
362.8891
114.4267
0.7816
80.6849
0.6405
53.4492
0.4255
Gamma
324.5805
114.4060
0.6799
90.2807
0.6095
53.0919
0.4201
13-13
-------
TABLE III
Summary of Total Absolute Deviations (20/ig/m3 classes)
Normal
Dist.
Station
1
Station
2
Station
3
Station
9
Station
11
1960
1961
1962
1963
1964
1965
1966
1967
1968
Average
1960
1961
1962
1963
1964
19C5
1966
1967
1968
Average
1960
1961
1962
1963
1964
1965
1966
1967
1968
Average
1968
1968
Average
109.8
135.1
129.9
159.4
119.5
129.1
129.0
131.0
120.3
129.2
114.2
125.7
177.2
125.6
119.5
143.5
143.7
134.5
135.5
108.6
126.4
106.0
122.9
123.8
134.9
131.2
136.9
123.8
94.0
60.2
125.6
Lognormals
2-P 3-P
43.4
56.6*
45.0*
60.8*
59.3*
59.0
52.2
51.2*
60.2
54.2
46.6
41.6*
82.2
38.8*
43.5*
43.3
59.8*
60.4*
52.0
47.3*
39.1*
27.7*
56.4
56.0
58.6
53.7
61.1*
50.0
60.3
31.6*
51.7
43.2*
62.6
46.7
93.1
60.3
55.9
52.4
52.5
59.4*
58.5
39.5
46.1
64.4*
39.2
46.5
35.8*
63.6
61.3
49.6
47.8
39.3
28.1
57.8
54.6*
66.8
53.8
65.3
51.7
58.9*
36.0
53.0
Pearson
4-P Gamma
45.6
109.3
49.7
131.6
70.9
55.1*
48.3*
60.6
86.6
73.1
36.4*
69.3
105.7
62.3
48.3
52.1
75.6
80.9
66.3
57.2
56.9
29.6
48.0
57.4
50.0*
52.2
82.5
54.2
72.8
36.9
64.1
54.0
110.1
49.4
210.6
70.5
84.8
48.4
63.9
69.3
84.6
39.1
68.1
68.2
53.4
63.5
67.0
62.9
68.9
61.4
66.7
53.1
29.1
45.1*
89.4
51.7
49.7*
73.1
57.2
60.7
32.0
66.8
13-14
-------
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13-15
-------
0.004 -
3.0 NORMAL
100 200 300 400
x = 150.0
VAR - 3000.0
a, = i.o
o.ooo
100
CONCENTRATION
300
Figure 13.2. Effect of kurtosisj32 on distribution shape.
13-16
-------
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15
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OJ
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a
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13-17
-------
0.016
0-014
0.002
0.000
£•40.0
y »0
100 200
CONCENTRATION (/tg/m»)
300
Figure 13-4. Variation of Gamma distribution with shape parameter a.
13-18
-------
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en
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UJ
Ul
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Q.
CD
c
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13-19
-------
STATION 1-1965
Peorson 21
Gamma
100
200 300 400
CONCENTRATION (/tg/m»)
500
600
STATION 3-1965
Pearson 3ZI
Gamma
200 300 400
CONCENTRATION (ua/m8)
500
600
Figure 13-6. Pearson VI does better than gamma.
13-20
-------
80
60
£
40
20
STATION 1-1968
Gamma
Beta (Type I)
200 300
CONCENTRATION
400
500
600
STATION 2-1962*
Gamma
Beta (Type I)
100 200 300 400
CONCENTRATION (Mg/m8)
500
600
Figure 13-7. Gamma does better than Pearson I.
13-21
-------
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13-23
-------
STATION 3 - 1962
2-P Uog
Pearson I
200 300 400
CONCENTRATION (/ig/m8)
500
STATION 3 - 1967
~ Gommo
3-P Log
100
200 300
CONCENTRATION
400
500
Figure 13-10. Cases where all distributions fit.
13-24
-------
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13-26
-------
Ack nowledgement
My appreciation is gratefully extended to the Harvard Department of
Statistics for their support of the computing necessary for this effort, and to the
Philadelphia Department of Public Health for permission to use their data.
References
Elderton, W. P., and Johnson, N. L., 1969: Systems of Frequency Curves,
Cambridge University Press.
Hunt, W. F., Jr., 1972: The Precision Associated with the Sampling Frequency
of Log-Normally Distributed Air Pollution Measurements. J. Air Pollution
Control Association. 22: 687.
Johnson, N. L., and Kotz, S., 1970: Continuous Univariate Distributions, Vol. I
and II. Houghton-Mifflin.
Phinney, D. E., and Newman, J. E., 1972: The precision associated with the
sampling frequencies of total paniculate at Indianapolis, Indiana. J. Air
Pollution Control Association. 22: 692.
Singpurwalla, N. D., 1972: Extreme Values from a Lognormal Law with
Applications to Air Pollution Problems, Technometrics. 14: 703.
Spirtas, R. and Levin, H. J., 1970: Characteristics of Particulate Patterns
1957-1966. PHS, NAPCA, Publication AP-61.
DISCUSSION
Larsen: As you mentioned in the beginning, Dave, there are several distributions
that could be used for expressing skewed data. Now that we are considering
controlling air pollution, there is one simple characteristic where the lognormal
is handy for the man in the regional office. For instance, if we have a lognormal
distribution of SO2 concentration and we use a control plan that reduces all
sources by 90%, then we expect the new distribution to be parallel and just
down one log cycle on the plot. So the lognormal provides a very simple way to
predict what is going to happen to a future distribution based on the present
distribution. Also it gives a physical feeling of what might happen if, for instance
only tall sources were reduced, maybe all the power plants in the region, and all
the low, area sources were not controlled. You might tend to chop off the
highest observed concentrations to give a new distribution with a shallow slope.
Or if only all the low sources were controlled, a steeper slope would be
expected.
13-27
-------
Lynn: There's not much I would say; I certainly agree that nothing much fancier
than the lognormal is ever going to be what we call handy for that type of field
calculation. I think that the place of more complicated distributions is in fitting
large bodies of data as entire bodies of data, rather than making assumptions
about control strategies. And of course the case when one wants to do some
mathematics which the lognormal may not be amenable to, or just that nobody
has done that particular derivation yet. I hesitated there in saying whether
another distribution would ever be of use in this field because I think the
extreme value distribution can be handy and do well if one is interested only in
predictions about the maximum levels.
Hershfield: We have been talking a great deal about the 2-parameter lognormal
distribution but we haven't mentioned some important characteristics of the
distribution. We have a distribution which is skewed to the right. It is outlier
prone. If you plot the coefficient of skew (Cs) on the vertical axis versus the
coefficient of variation (Cv) on the horizontal axis, the lognormal distribution
has the relationship given by the equation, Cs = 3CV + Cv3. Some mention has
been made of the extreme-value or Fisher-Tippett type I (Gumbel) distribution.
Gumbel, in his work postulates a coefficient of skew 1.139 and at Cv equal to
37%; estimates from both the lognormal and the Gumbel distributions are equal.
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14. A PROPOSED AMBIENT AIR QUALITY SAMPLING
STRATEGY AND METHODOLOGY FOR THE DESIGN
OF SURVEILLANCE NETWORKS
JOSEPH R. VISALLI AND DAVID L. BRENCHLEY
Environmental Engineering Laboratory
School of Civil Engineering
Purdue University
West Lafayette, Indiana
and
HOWARD REIQUAM
Battelle Memorial Institute
Columbus, Ohio
Introduction
Determination of the concentration of a specific pollutant in the ambient
air over a defined area is a complex problem because of spatial and temporal
variations in air quality. Many factors such as local topographical differences,
atmospheric chemistry phenomena, fluctuating emission rates, non-stationary
sources, and varying meteorological conditions contribute to these variations.
This space-time dependence complicates immensely the acquisition of statisti-
cally accurate estimates of true air quality. This in turn confounds the inter-
pretation and utility of the data.
The Federal Clean Air Act of 1970 requires the monitoring of ambient air
for various pollutants. Well defined and vigorous federal directives have been
issued indicating which pollutants are to be monitored, and the preferred
reference testing methods and procedures for measuring these pollutants (anon.
(1971)). In contrast, the guidelines for the establishment of air quality
surveillance networks are essentially of a subjective nature as implied by the
following quotation: "Experience and technical judgment are essential for
determining the number and location of sampling sites because adequate
mathematical models or other models have not been formulated" (E.P.A.
(1971)). The importance and high costs associated with ambient air monitoring
14-1
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demand that mathematically rigorous methods for determining the numbers and
placement of sampling instrumentation be developed, such that the precision
associated with the test methods and procedures is of equivalent magnitude to
that of the actual data collection. It is of little value to take precise
measurements in a subjective manner. The result is high cost data, having poor
reliability and statistical validity, and of little utilitarian value in determining
compliance with Federal standards.
The Problem
Federal air quality regulations essentially limit the concentration of specific
pollutants to certain levels at all locations within defined areas (Anon. (1971)).
Basically then, the problem is reduced to one of finding the point or points
within the defined areas at which the concentration of the specific pollutant is
the greatest. Noting however, that an infinite number of points exist in the
defined space, and that the concentration levels vary both spatially and
temporally, the deterministic approach of finding the points where the
concentration is the highest is a practical impossibility. Instead, a probabilistic
approach is indicated. A random sample of pollutant concentration within the
space over the defined area should be taken. This sample must take into account
both space and time factors. From the resultant distribution, an estimate of the
probability that a certain level of pollutant concentration has been exceeded can
be made. Two problems arise as a result of choosing this approach. One is
concerned with the mechanics of designing and conducting a simple random
sample with respect to an infinite population that is a function of time and
space. The other pertains to a strategy and methodology that will enable an
unbiased estimate of the mean and variance of pollutant concentration to be
calculated, with pre-determined accuracy and confidence in the statistical sense.
An approach to the first problem can be devised by taking into account the
nature of the turbulence regime at the microscale level. To accomplish the latter
requires an advance estimate of the sample size needed for assurance that the
desired degree of precision is attained (Cochran (1963)). This in turn requires an
advance estimate of the variance of the random variable under consideration.
With this advance information, a random sample of sufficient size can be
conducted, such that an accurate and precise estimate of the air pollutant
concentration distribution over the defined area can be made. The probabilistic
question of whether air quality standards have been exceeded for the sampled
area can then be asked. More important is that with such a procedure, the
question can be answered with a mathematically determined degree of
confidence.
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The Strategy and Methodology
Strictly speaking, federal standards can be interpreted to include not just
ground level concentrations, but concentrations as a function of the vertical
coordinates up to the depth of the mixing layer. To accomplish such a task
would require, as will soon be evident, a virtual armada of airplanes or
helicopters, or a remote sensing method. The primary purpose of establishing
federal standards is to protect public health and welfare. Thus we feel it can be
reasonably argued that the interests of public health and welfare will be
adequately served if pollutant concentration is monitored with respect to both
space and time at people (ground) level only. Dispensing of the vertical aspect
brings the problem into the realm of practicality as far as economics, and the
satisfaction of sampling criteria (to be discussed later) are concerned. The scope
of the problem, even with this significant deletion, remains however, beyond
practical consideration at this point. The results of the Nashville study (Stalker
et al. (1962)) revealed that at least 245 sampling stations, approximately four
per square mile, would be required to estimate the daily mean concentrations of
sulfur dioxide for the whole city with 95% confidence of ±20% accuracy.
Establishment of this many monitoring stations is probably beyond the financial
capabilities of most communities. The only alternative then, if statistical
accuracy and precision are to be maintained, is to consider that only certain
sub-areas within the whole bounded area require air monitoring. These sub-areas
would be specific sectors of the whole bounded area, that by some subjective or
objective criteria are judged to contain the highest levels of pollutant
concentration.1 Effectively then, we would be utilizing the statistical technique
of stratification, but departing from the usual method of analysis. Ordinarily,
sampling would be conducted in randomly selected stratified sub-areas, and an
estimate of the whole area mean value for some variable calculated from these
samples. In this approach, we purposely select the stratified sub-areas judged to
contain the highest values of the variable under consideration, and attempt to
1A certain amount of subjectivity is inherent in most engineering approaches or solutions to
problems. This is due to the two classical compromises between mathematical
requirements and technical capabilities, and between this resolution and available financial
resources. It will become evident later, that the approach taken in the proposed
methodology would theoretically eliminate all subjectivity if the imposed economic
demands could be met. In any case, deciding what constitutes a critical area is a complex
question that can be approached in any one of, or a combination of the following ways. It
could be based on existing air quality isopleth data, simulation dispersion model
predictions, attitudinal-behavioral studies of the citizenry, demographical-medical-specific
area data, wind rose data in conjunction with industrial location distribution, or prior
complaints. The point is that some criteria can be developed.
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establish from the sampling of these sub-areas an upper bound to the variable.
We can then conclude that the remaining sub-areas, based on the developed
criteria, will be enjoying better air quality (the variable) than that indicated by
the established upper bound.2 With this approach, the scope of the problem is
within the limits of practical economic consideration, the legal aspect is satisfied,
and the mathematical accuracy and precision that is desired can be attained.
Consider one of the sub-areas discussed earlier. It is a defined area of
constant (with time) topographical features. Assume a hypothetical situation
where the emissions polluting this sub-area are of constant strength and flow
rate. If the meteorological factors that influence the transport and dispersion of
these pollutants are also assumed to remain constant with time, then the mean
and variance of pollutant concentration across the sub-area will remain constant
with time. Larsen (1971) indicates such a situation for a single point. One can
extrapolate the concept to cover an area simply by taking the average of many
single points over the area.
Consider now the hypothetical case where emission factors polluting this
sub-area remain invariant with time, but the meteorological conditions are
changing with time. During time periods when the meteorological conditions are
similar, our previous argument of a constant mean and variance of pollutant
concentration is valid. That is to say, varying meteorological conditions can be
classified according to some finite scheme that attempts to categorize these
varying conditions into specific meteorological regimes (e.g. Pasquill's scheme).
If these regimes are described by the factors that influence transport and
dispersion phenomena, then during periods when a specific regime is dominating
the weather, a specific mean and variance of pollutant concentration will
accompany it.
In reality, however, emission characteristics will vary with time in a manner
which is very difficult to predict. Thus even during periods of "similar
meteorology", the mean pollution concentrations will differ. However, since the
variance is not directly proportional to the mean it is reasonable to hypothesize
that the variance of pollutant concentration will remain constant with time
during periods of "similar meteorology". Effectively we are hypothesizing that
since the variance of a group of numbers depends only on their relative
magnitudes, and not their absolute values, the variance of pollutant
concentration is not dependent upon the mean pollution level. Hence the
variance is independent of emission factors, and the original argument for the
variance of pollutant concentration holds.3 Thus, differing concentrations of
^Considering the extreme complexity of the problem, it is entirely possible that points in
the sub-areas not sampled could violate the standards. Nothing can be done about this
problem using this methodology. Thus it is important to consider this aspect when
determining the sampling criteria.
3The strategy and methodology developed in this paper does not consider the effect of
non-stationary emission sources such as automobiles. Hence this approach is applicable
only to those pollutants (such as SC^) emitted primarily from stationary sources.
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pollutants will be dispersed over a given area in a similar manner during periods
of "similar meteorology".
This hypothesis parallels closely the statistical concepts of homogeneity
and stationarity as applied to horizontal turbulence levels over an area.
Stationarity essentially means that the statistical characteristics which describe
the frequency distribution of the horizontal turbulent eddies remain constant
with time. Thus the mean, variance, and the rest of the statistical moments that
characterize the frequency distribution remain constant with changes in time.
Homogeneity implies that the frequency distributions of these turbulent
eddies are the same throughout the area under consideration. Dispersion models
derived from both gradient transport and statistical theory must employ these
concepts in order to solve their respective equations. It is well known that the
statistical properties of turbulence vary in the vertical direction. However,
horizontal turbulence characteristics at constant height do fulfill to a certain
extent, the statistical concepts of homogeneity and stationarity provided that
major changes in the meteorological regime do not occur (U.S. Atomic Energy
Commission (1968)).4 The first hypothesis contained in our methodology states
that a stationary process will occur, provided that no changes in the
meteorological characteristics that govern the transport and dispersion
phenomena occur. This is a less restrictive stationary process than the one
postulated for the turbulence regime. We have proposed the concept that the
variance of pollutant concentration, not the mean, will remain constant with
time under specific conditions. In many stationary processes, assuming the mean
of a random variable to remain constant with time is a questionable assumption.
The stationary process we propose does not require the mean value to remain
constant over time, and thus is less restrictive than the typical case.
Consider an infinitesimal section of a defined area. The variance of pollutant
concentration will be very small, and in the limit will equal zero as the area
approaches zero. As the section is increased in size, the variance will exhibit a
tendency to increase according to some function toward the variance of the
entire defined area. A decrease appears unlikely because the variance will
decrease in magnitude only if observations tend to cluster about the actual
mean. This indicates that the concentration gradient is tending toward zero.
Experience demonstrates the reverse. There is a tendency for pollutants to
accumulate in certain areas, creating large gradients when the whole area is
considered. Thus it is hypothesized that an increasing area-variance relationship
will exist. This is also in accordance with turbulent theory, which indicates that
the statistical properties of turbulence in the horizontal will tend toward greater
dissimilarity as larger areas are considered. Thus the variance between turbulent
eddies will in all likelihood increase as larger areas are considered (U. S. Atomic
4lt must be noted that these concepts are generally assumed to hold for flat plains and
rolling countryside. Little is known about the effects of irregular terrain.
14-5
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Energy Commission (1968)). As shown in Figure 1, knowledge of the
variance-area relationship could be combined with cost data and available
financial resources to maximize the area covered by the sampling strategy.
Knowing what financial resources are available for sampling purposes and the
cost per sample, we can easily determine the sample size we must work with. If
we select a confidence interval (precision) and a margin of error (accuracy) that
are deemed adequate for the purpose, then from the equation:
^ /t
n = (-r)2 (variance) = k; (variance)
d '
where:
n = sample size, if the population size is very large
t = the abscissa of the normal curve that cuts off an area a at the tails
d = chosen margin of error
1 — % = confidence probability
kj = (t/d)2, where the value of t depends upon the sample size
we can determine an estimate of the variance that would be required to
accomplish our sampling task. Knowing this and the variance-area size
relationship, we can determine the size of the area we should cover in our
random sample in order to achieve the prescribed precision and accuracy with
the financial resources available. We obviously would like to maximize the area
covered since this reduces the amount of subjectivity that has entered our
strategy and methodology.
The Experiment
The correct method of actually obtaining the samples to be used in
estimating the mean and variance of pollutant concentration is not at all
obvious. To apply probability theory to the data with any degree of confidence,
it is imperative that random sampling techniques be employed (Cochran (1963)).
Keeping this in mind, one might propose that to overcome the spatial-temporal
dependence noted earlier, continuous air monitors should be randomly scattered
throughout the defined sub-area. This approach would be correct if the
continuous monitors were not fixed at every randomly chosen location. Two
reasons dictate that continuous monitors should be continually moved in
random fashion, if accurate estimates of pollutant concentration and variance
are to be obtained. First, the population (molecules of S02) is continually
changing its spatial distribution with time, and its size (number of molecules) is
also continually changing with time (molecules are added, deposited,
transformed to other forms, remain with the area, etc.). Essentially then, a
14-6
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new random sample is needed for each new population. Secondly, and perhaps
more important, the response of any monitoring instrument placed in an
irregular topographical setting will be biased. This is due to specific effects of
building geometry on diffusion parameters. If a monitor is fixed at one
point—even if randomly located—the response of the instrument will include this
bias on a continual basis with time. Effectively this states that no single point
can be representative of a large area (Corn (1970)). This is especially true over
relatively short periods of time such as a day. The Nashville study (Stalker et al.
(1962)) demonstrated that relatively few instruments were needed to accurately
estimate seasonal and yearly concentration levels over an area. This is a
predictable result, as long term sampling tends to average out fluctuations and
converge on the true mean. The study also revealed how inaccurate their results
were when a few instruments were used to estimate 24 hour averages. This was
due, in part, to the bias encountered from fixed point sampling. Thus, to
continually "remove" or reduce the inaccuracy due to this time factor bias, one
must continually relocate the monitors in a random manner with time.
Randomized mobile monitoring is thus essential to proper sampling procedure in
this case.
In accordance with this type of reasoning, we should also note that a
restriction to the interpretation of data collected from continuous fixed point
monitoring is indicated. If the purpose of the monitoring'is merely to indicate
trends at that specific location with time, continuous fixed station monitoring is
certainly reasonable. Caution must be exercised however, if such data is to be
used to determine why the trends occur over time. Changes in the topography,
both local and in the vicinity of the continuous fixed point monitor (urban
renewal, road construction, etc.), the addition of new sources, etc. can confound
meaningful interpretation. It would be impossible to discern whether changes in
air quality are due to an effective (or ineffective) air conservation management
program, or merely due to charges jn topography that alter the bias suggested
previously. Problems of this nature would not arise if a mobilized monitoring
approach was used. This is because the mobilized monitoring approach
eliminates (or at least reduces) this bias.
Data analysis and prediction models
The utility of this whole approach is dependent upon an ability to predict
accurate advance estimates of the variance. The choice of a suitable prediction
model depends to a great extent upon the nature of variance data presented as a
time series. Figure 2 depicts some of the possible results of graphically
representing such data. All of these possibilities have distinct features that would
influence the choice of a suitable stochastic prediction model. Figure 2A, for
example, illustrates a process that incorporates features of both time stationarity
and non-stationarity. This time series depicts discrete periods of stationarity that
14-7
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obviously change mean values in an almost instantaneous jump fashion. Figure
2B exhibits a similar tendency, but has a smoother transition between time
stationarity intervals. Figure 2C shows a random fluctuation process. Figure 2D
depicts a time series that is stationary in the wide sense.5 Certainly other
possibilities exist. Subsequent data analysis would reveal if the random variables
are dependent or independent, and if long period, seasonal, or cyclic trends exist
along with inherent random fluctuations.6 A procedure outlining a general
method of determining the nature of a particular set of data can be found in
many texts on time series analysis (e.g. Box and Jenkins (1971)).
At this point we can only speculate (since data is not available) as to the
nature of the time series. Since many possibilities exist, it is not feasible here to
outline a data analysis for each situation. Instead, a single case where a time
series that is in accordance with the hypotheses stated earlier will be considered.
One possible approach to the data analysis will be outlined. The stress will be on
simplicity.
Table I represents the type of data required for the approach outlined in
this paper. The variance estimates and concomitant meteorological parameters
are arranged in the sequence in which the data were taken. Each variance
estimate is calculated from the average of simultaneous measurements taken over
equal averaging times if discrete sampling techniques are employed, or the
average of the measurements over equal time intervals if continuous monitors are
utilized.7 As shown in Figure 3, the variance can be plotted as a continuous
function of time. Since the sequence does not appear to fluctuate in a totally
random manner, and many atmospheric phenomena are known to be dependent
processes (U. S. Atomic Energy Commission (1968)) it will be assumed that the
variance measurements are dependent random variables. The data analysis should
concentrate on achieving five objectives:
The concept of stationarity in the wide sense essentially provides for a process where only
up to a certain statistical moment is constant with time. A good possibility exists that
many atmospheric processes exhibiting stationary features are actually stationary in the
wide sense.
first hypothesis described in this paper provides for transient variations due to short
term meteorological conditions. Cyclic trends due to diurnal effects and seasonal weather
trends may also affect the time series. Long term trends, due perhaps to changing topology
(urban renewal, etc.) or new emission sources, are also a possibility that should be
considered.
'Averaging time is defined as some time interval over which the continuously varying trace
of the variable may be represented by a constant (average) value. We note also that this
averaging time (or the time interval if continuous monitors are used) should be as small as
is technically possible to avoid the time bias discussed previously.
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Table I. Data Sequence Required
V ...................... V
n
Xn1
Xlm Xim Xnm
where:
V = spatial variance of pollutant concentration at time i
xij = recorded values of meteorological variables assumed to have a
significant relationship to the variance
i = time period number
j = representative of a specific meteorological parameter
1. Determine periods of time stationarity over which the variance is
homogeneous or constant.
2. Determine which meteorological parameters influence the changes in
variance levels.
3. Characterize the meteorology during each period of stationarity
according to some finite classification scheme based on the parameters found to
influence variance changes. Test the hypothesis of "similar meteorology" based
on this classification scheme.
4. Choose an algorithm that will predict advance estimates of the variance
across the sub-area, given that the variance of the time period immediately
preceding it is known, and the estimated meteorological, parameters of the time
period to the predicted are known.
5. Develop estimates of the length of the stationary time period for the
predicted variance.
Keeping these objectives in mind, the data analysis can proceed as follows:
(a) Find the longest possible continuous time intervals over which the
variance is constant in the statistical sense, i.e., no significant difference exists,
or limit the standard deviation of the data to a certain percentage, etc. This will
have the effect of breaking down the data into discrete step functions of time,
where the variance will be constant over the interval. These resultant time
intervals may not, as a consequence of the analysis, be of equal length, but
should be as long as statistically possible. The overall result could look
something like Figure 4.
14-9
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(b) Run an analysis of variance or a regression analysis on all of the data to
determine the meteorological variables that significantly influence the variance
change.
(c) Based on the results of the analysis of variance (or the regression
analysis), devise a classification scheme of a finite number of divisions or classes.
The scheme should probably be both qualitative and quantitative in
nature—perhaps similar to that proposed by Pasquill (1962). After each time
interval is classified, we can proceed to test the hypothesis of "similar
meteorology" by simple inference tests on the variance values of each class.
(d) Assuming that the hypothesis is not rejected, we are now in a position to
propose a simple algorithm. This will predict an estimate of the variance in
advance, given that we know what the variance was at the previous time interval,
and have available an estimate of the values of the pertinent meteorological
variables for the time interval to be predicted. This can be accomplished in the
following manner:
(1) Breaking down the data into a finite number of classifications, say six
(A,B,C,D,E,F), will result in six frequency distributions being developed (see
Figure 5). The variance values for each class stem directly from re-arranging and
grouping the data after classification. We thus have a mean value and a variance
associated with each variance class. These distributions need not be similar. The
only requirement for further analysis is that the variance of the variance
distributions be finite.
(2) Choose a simple linear least-squares algorithm such as:
cov
where:
(vj. ) = variance estimate for time period Tj+1
E[V-p J = expected value of the distribution of the variance for
time period Tj+1
cov[VT. VT. ] = covariance of the random variables VT.VT.
var[V-|-j] = variance of the distribution of the variance for time period Tj
(Vj.) = variance for time period i
E [ VT. ] = expected value of the distribution of the variance for time
period T;
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We possess all the inputs to such an equation from the six frequency
distributions. The
The estimate for E [VT.VT ] would be obtained from the bivariate
distributions resulting from a Markov chain analysis of the sequence of order
that the classification followed.
Time interval 12345 etc.
Class A C D A B etc.
We can, as a result of this sequence determine the bivariate frequency
distributions of AC, CD, DA, AB, etc. for all possible combinations of
classifications to determine the cross-correlation function above.8
This whole analysis thus far depends upon the availability of advance
weather forecasts. These forecasts must include estimates of the parameters
found by the analysis of variance to be pertinent to variance changes. It must be
noted that in the event such forecasted parameters are not available, we can
determine from the analysis just outlined, the k-step transitional probabilities
needed to estimate the probability that a certain predicted state will follow the
present state. We thus have meteorological class prediction capabilities inherent
within the normal data analysis routine.
The remaining problem is to predict the length of time interval that will be
associated with the predicted meteorological class (and hence the variance level).
The approach to this problem lies in determining whether the length of the time
interval of successive meteorological classes is described by a dependent or
independent random process. If the process is independent, then the best
estimate for the length of the time interval of the predicted meteorological class
is merely the mean value of the time interval frequency distribution for that
class. If the process is found to be dependent on the preceding event, an
algorithm similar to the one proposed for the variance estimates can be used. If
the process is found to be dependent in a more complex manner, such as being
dependent upon the length of the time interval for the previous 2 or 3 or n
classes, then a probability model such as
i = f|LTj + f2LJi_, + ........... + fnLTj_n + ru,
lf the time series is found to be an independent random process, the
would be zero.
The best advance estimate for the variance would then be the mean value of the
distribution for the predicted meteorological class.
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can be used,
where:
LTj+1 = length of the time interval to be predicted
LTj = length of the time interval of the present meteorological class9
f = weighted factor of previous values of LT
ri+1 = the random element that must be taken into account
There are some distinct advantages to a program and an analysis such as the
one proposed in this paper. The method is obviously extremely flexible, as
mobile monitors can be moved wherever desired. The method can be adopted
by almost any community with limited financial resources due to the
variance-area relationship.10 The analysis is basically simple and is Bayesian in
nature. That is, new data are continually put into the system so that better
and better estimates of predicted values can be realized. Most important, a
mathematical statement of accuracy and confidence can be associated with each
estimate of the upper bound concentration level for a specific pollutant.
Summary
A sampling strategy and methodology were proposed that enable an
optimum size air monitoring surveillance network to be developed at a given
cost. The estimates of pollutant concentrations that result from such a network
are statistically accurate and valid to a predetermined level. The model develops
two hypotheses into a prediction algorithm that enables advance estimates of the
variance of pollutant concentration to be made. Knowing this, one is able to
allocate the correct number of samples to best estimate pollution concentration
levels with predetermined accuracy and confidence.
The mobilized monitoring approach taken by this strategy is probably the
method that first should have been implemented. Unfortunately, the fast
response time and portability features that were necessary for such an approach
were not available when ambient air monitoring was first started. These
requirements are being met by the new instrumentation available now, and
planned for the future. If the mobilized approach by any methodology is
deemed the proper way to go, we should not let tradition stand in the way.
that LTj must be estimated also, as we are dealing with a real time situation. That is,
we don't really know how long LTj will actually last until it ends. We can then get a
revised estimate for LTj+^.
"It should be noted that as a result of the analysis, the variance-area relationship becomes a
family of curves—one curve for each classification. Thus the final graph depicted in Figure
1 will contain a family of curves.
14-12
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UJ
N
tf)
UJ
SAMPLE SIZE
VARIANCE
LIMITING VALUE FOR TOTAL BOUNDED AREA
SUB AREA SIZE
Figure 14-1. Use of variance-area relationship irf sampling strategies.
14-13
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CE
-------
o: t
UJ <
> UJ
o rr
LU
O CD
2 =>
< C7)
o
TIME PERIOD NUMBER-
Figure 14-3. Spatial variance as a function of time.
1
Ul
u<
a:
=J<3
-------
CLASS A
o
z
UJ
z>
a
UJ
rr
SPATIAL VARIANCE FOR A
SPECIFIC SUB AREA
>-
o
o
UJ
or
u_
CLASS B
SPATIAL VARIANCE FOR A
SPECIFIC SUB AREA
Figure 14-5. Frequency distributions for various meteorological classes (similar
plots for all classes).
14-16
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REFERENCES
Aitchison, J. and Brown, J.A.C., 1957: The Lognormal Distribution. Cambridge
University Press.
Anon., 1971: Requirements for Preparation, Adoption, and Submittal of
Implementation Plans. Federal Register. 36: No. 158.
Box, G.E.P., and Jenkins, G.M., 1971: Time Series Analysis Forecasting and
Control. Holden-Day.
Cochran, W. G., 1963: Sampling Techniques. John Wiley.
Corn, M., 1970: Measurement of Air Pollution Dosage to Human Receptors in
the Community. Environmental Research. 3'. 218-233.
Davenport, W. B., 1970: Probability and Random Processes. McGraw-Hill.
E.P.A., 1971: Guidelines: Air Quality Surveillance Networks. United States
Environmental Protection Agency, O.A.P., AP-98.
Larsen, R. I.,1971: A Mathematical Model for Relating Air Quality
Measurements to Air Quality Standards. Environmental Protection
Agency, O.A.P. Publication No. AP-89.
Pasquill, F., 1962: Atmospheric Diffusion. D. Van Nostrand Co. Ltd.
Stalker, W. W., Dickerson, R. C., and Kramer, G. D., 1962: Sampling Station and
Time Requirements for Urban Air Pollution Surveys. J. Air Pollution
Control Association. 12: 361-375..
United States Atomic Energy Commission, 1968: Meteorology and Atomic
Energy, 1968. A.E.G./Division of Technical Information.
DISCUSSION
Court: A third hypothesis that you should consider is that the variance, your
spatial variance, will decrease as the sampling time increases. If you take samples
for an entire year over your area, the difference between the sampling sites will
be much less than if you read them minute by minute. This all comes back then
to the question of what you are trying to measure. If all that is wanted is the
total concentration over an area, then your procedure may be valid. If we want
to know where the hot spots are so we can track them down to a certain emitter,
then we want to keep the stationary sampling sites. Furthermore, if we have
stationary sites we can use the correlation between stations (the covariance) to
look at the measure of the variability of the concentrations over area. A
covariance between mobile stations would be meaningless whereas the
covariance between fixed stations will indicate just how spotty the pattern is.
Visalli: Yes, I think you're absolutely correct. I would like to comment on one
other thing, that is the concept of the biological half-life. If indeed the primary
purpose of monitoring air is to protect public health, then the time period of
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analysis should be something that is going to be effective, that is, comparable to
the biological half-life. In the case of SO2 the biological half-life happens to be
on the order of several minutes—I think 2-4 minutes. Well, if we're trying to get
an estimate of what the air quality is in an area, I don't think that there is a way
that we can do this by placing a monitor at a fixed point. We've got to have at
least several monitors, in one area, whether fixed or not, to get an idea of what
the probable concentration is for the entire area; and then see if we get
meaningful cause and effect relationships. However I do agree with your point.
Marcus: One thing I would just like to point out is that as far as getting
correlations between moving monitors, this seems to be a little bit analogous to
the association between the Eulerian and Lagrangian approaches to correlation
turbulence theory.
Dave Spiegler: One of the problems that National Weather Service has (I'm not
with the National Weather Service) is to determine the optimum spacing of
stations for an analysis which is similar to the problem you're talking about.
What they do is have an objective analysis of an area and the spacing depends on
the scale and how big the grid is. Here again it depends on whether, as Dr. Court
said, you want to analyze over a period of an hour, or 6 hours, or a day, or
what. The space scale is related to the time scale. The longer the period of time
that you want to analyze, the fewer the number of stations necessary to describe
the analysis over the area; i.e., the shorter wavelength features are less important
as the time period increases. An objective analysis procedure has initial guess
values on a grid of points and uses the station observations near those points to
adjust the initial guess. I think that something like this might be useful in the
work you are doing.
/. Enger: Just a small comment about the role of sampling, based upon just a
very preliminary result that we came across a few weeks ago. Apparently a very
small number of samples can approximate the distribution of an annual average
very closely. What we did was for each 10 points we took a random sample of
100 and then a random sample of 10; and the means for two samples came very
close. Not that I'm trying to say something statistical about it, just that it is an
indication that a very small number of samples can estimate a distribution quite
effectively. I think that's a point favorable to mobile sampling, provided you
have the sampler returned to the same place. On a randomized scheme, 100 or
200 times a year you can get an annual average for a whole variety of places with
only one instrument.
Wanta: I suppose that with the passage of time one will become acquainted with
the characteristics of each of the sampling sites in the sense that one does with a
single site. He learns what the near and more distant sources are.
14-18
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15.THE EFFECT ON ROLLBACK MODELS DUE TO
DISTRIBUTION OF POLLUTANT CONCENTRATIONS
YUJI HORIE AND JOHN OVERTON
Department of Environmental Sciences and Engineering
School of Public Health
University of North Carolina
Chapel Hill, North Carolina
Introduction
One of the important uses of air quality data is determination of the
emission reduction required to achieve desired air quality. So-called "rollback
equations" are often employed to compute a reduction in emissions. Basic
Questions have been raised on the application of rollback equations to emission
standards for automobiles, and the effect of the distributions of present, future,
and desired concentrations on the applicability of these equations.
The rollback equations that have been proposed are expressed as
L gLxp-B
gj(Xp-B)-(Xd-B)
where
RL = reduction ratio (according to Larsen (1969))
Rj = reduction ratio (according to Jensen (1971))
X = present air quality
Xd = desired air quality
B = background concentration
9l_' 9j = 9rowtri factors over the period from the present to the goal year in
which the desired air quality will be achieved.
The reason for the differences between these two equations is not being
explored. The differences, however, are negligible when the background
15-1
-------
concentration, B, is much smaller than the present air quality, Xp. Both
equations, for most practical purposes, can be reduced to
D~D ~D ~ 9xP-xd
R ~*~ Ri ~"~ Kj ~~" lo)
Q Xn
where g = gL = gj.
Once the reduction ratio, R, is determined, the permissible emissions can be
calculated by using
where
ed = future desired emission per unit source
ef = future emission without controls per unit source.
If the future emission without controls, ef, can be assumed to be the same as for
the present (this is the case for automotive emissions), then Equation 4 can be
written as
ed = ep(l-R) (5)
where
ep = present emission per unit source.
The major question in using a rollback equation is how to compute the
growth factor, g (gL or gj). According to Equation 1 the growth factor appears
to be defined by
— ' IR}
"" XP
where
Xf = future air quality without controls.
According to Equation 2 the growth factor appears to be defined by
Values of pollutant concentration are described by both their averaging
times and percentiles. The national air quality standards, for example, are stated
as "D milligrams per cubic meter -maximum t-hour concentrations not to be
exceeded more than once per year" (Anon. (1971)). Therefore the values of Xp,
Xd, and B in the rollback equations must have the same averaging times, t-hours,
etc. In addition to this, the growth factor defined either by Equations 6 or 7
should be calculated using the values of Xf, Xp, and B that correspond to the
same percentile at which the value of Xd is designated, i.e., "once per year."
The usual method for determining the growth factor (Ott et al. (1967);
Larsen (1961)) is: First, future emissions are estimated by some method such as
15-2
-------
projection of past car registrations to the goal year. Second, future
concentrations are estimated by applying an air pollution display model to the
future emissions using presently available meteorological data. Third, growth
factor is determined from the ratio of the estimated future concentration to the
present air quality that is either estimated by the same air pollution model or
actually measured.
A question arises as to whether the ratio of future concentration to present
concentration remains the same at every percentile value. This question may be
restated—does the linear relation between emissions and concentrations, which
leads to the rollback equations, extend to the percentile values of present, future
and desired concentrations. In order to explore the assumption of linearity,
which is implicitly employed in the usual method, a simple proportional model
is assumed. As will be seen, a proportional model does not in general imply
linearity between emissions and concentrations.
The discussion and consequence of this model are developed in the next
section. Using the assumption of linearity over percentile values, the effect of
concentration distributions on the rollback equations is investigated in the
section titled "Rollback equations for percentile values." Several numerical
examples are given in the succeeding section in which imaginary cities have been
constructed and empirical distributions of the "city's" present and future
concentrations have been calculated; growth factor for each percentile is
graphically displayed showing the dependence of the growth factor on percentile
and emission growth pattern. The implications of these calculations are discussed
in the last section.
Simple Proportional Model
The air quality concentrations at any location in a city is assumed to be
given by (Appendix I)
X=B+eF (8)
where
e = emission per unit source
F = function of all relevant variables such as weather factors and source
distribution.
Even though B has a distribution, its value (even at the relevant high percentiles)
is assumed much smaller than e F and may be neglected. We assume it to be
constant and independent of city growth.
15-3
-------
In Equation 8, F is assumed not to be a function of e; thus the distribution
of F determines the distribution of X. Future concentration at the a-th
percentile, without emission controls, is
Xfa=B + efFfa (9)
Xfa depends, in part, on the future source distribution. This source distribution
includes the effect of any city planning and regulatory practices. The desired
future concentrations due to an emission reduction at the same percentile is
Since F is independent of emission per unit source, we have
Fda=Ffa
Using Equations 5 and 1 1 , Equations 9 and 1 0 can be transformed to
(12)
Xda-B = (I-R) epFfa (13)
Thus we have
X,a-B
The reduction ratio given by the above equation is independent of percentile, a.
There is, however, no established means to estimate the percentile value of
future concentration, Xfa.
A gcowth factor, Ga, can be defined as
r = Xfot"B (15)
Ga — T nr
*pa °
This growth factor is a measure of the growth of concentration value at different
percentiles due only to the sources that can be controlled and thus excludes the
background which is assumed to be independent of any growth. Substitution of
the growth factor into Equation 14 yields
Ga(Xpa-B)
Now the reduction ratio is expressed in terms of known variables except the
growth factor.
Since (Xpa- B) = epFpa and (xfa- B) = epFfa, the growth factor can be
written as (Appendix I)
Ga = l!2 (17)
""pa
15-4
-------
There is no reason to assume that the ratio, (Ffa/Fpa), is independent of
percentile since both Fpa and Ffa are dependent on the emission inventories.
The spatial distribution of emission sources changes as the city grows.
Consequently, the distributions of Fpa and Ffa may be significantly different.
In this sense, the proportional model does not necessarily imply a linear
relationship between emissions and concentrations.
Rollback Equations for Percentile Values
This section is concerned with the dependence of the rollback equations on
percentile and in particular the relation of these equations to the proportional
model of the previous section.
For simplicity the following assumptions are made: (a) The growth factor is
independent of percentile, i.e., G = (Ffa/F a) = constant, (b) The rollback
equations are valid for some percentile, say the 50-th. Then, the 50-th percentile
reduction ratios in terms of any other percentile concentration values can be
expressed as (Appendix II)
qLXPa" Xda + fiqCG-gJ , 9Lxpg-xda
jj — , <+
gLXpa-B
and
RJ =
9 I v A pa - o i T f?a v u ~9j j
gj(Xpa~B) — (Xda~^)
where
£a= (Xpa-Xp50)/ep (20)
One can see that the form of Equations 18 and 19 is not invariant with a change
in percentile a nor are RL and Rj independent of a unless the growth factors,
gL and gj are properly defined. A "natural" definition of gL and gj is
Fpa
In this case Equations 18 and 19 reduce to the same form as the original rollback
equations, Equations 1 and 2.
15-5
-------
In a more general case the growth factor, G, will be a function of percentile.
In this situation the reduction ratio, R:, defined in Equation 2 is the same as R
in Equation 16 when the growth factor, gj( is defined as in Equation 21, while
the reduction ratio, RL, is different from R even when the definition. Equation
21, is used.
Numerical Examples
A simple diffusion model (Hanna (1971)) is used to simulate the
distribution of concentrations due to several source density configurations. In
this model, as used, concentrations are a function only of the source density and
wind velocity; i.e.,
(22)
where
u = wind speed
PJ = density of sources in the i^ grid upwind from the central block "o"
c = constant
N = the number of upwind grid blocks included in the sum.
The source density, p, has been calculated using the sum of several Gaussian
functions with the same parameters, each centered at different positions. This
gives some structure to the "city." Future "cities" differ from present "cities"
by the addition of a Gaussian function to the present city's source density
configuration. The statistical distributions of concentrations are determined for
each city by using Equation 22 with 2000 "wind" velocity vectors taken from a
normal population; i.e., the X and Y components of wind velocity were
computed independently by the use of a normal random number generator.
From this pair of values the wind speed and direction were calculated.
Concentrations were determined, using Equation 22, from 2000 pairs of wind
speed and direction. Using the 2000 values of concentrations, empirical
distributions were formed. Then the growth factors, Ga, were determined by the
use of Equation 15, with zero background concentration, and the empirical
distributions.
Three different source density configurations as shown in Figure 1 are used
in the numerical examples. The present city's source densities (first source
density configuration) are given by the sum of two Gaussian functions. The
center of the second function is located at 5.0 kilometers east of that of the
first. The receptor is at the center of the first function, the origin. The second
configuration is the present city's configuration plus another Gaussian centered
at 7.1 kilometers northeast of the origin. In the third configuration, the growth
15-6
-------
is represented by another Gaussian centered at 5.0 kilometers north of the
origin.
The growth factors, Ga, have been calculated and plotted, in Figure 2
through 6, as a function of percentile for several normal velocity distributions.
Of the 2000 values of growth factor, only seventy-five evenly spaced (along
percentile axis) values have been plotted. On each figure are two sets of growth
factors corresponding to the two different emission growth patterns. The
solid circles give the growth factor when the future city is the second configura-
tion, while the closed circles give the growth factor when the future city
is the third configuration as described above. The wind velocity distribution is
indicated on each diagram by Ux ~ N (mean wind in x-direction, variance of
x-wind component) and U ~ N (mean wind in y-direction, variance of y-wind
component). The growth factor diagram for the case of no preference in wind
direction rs shown in Figure 2. Beginning with a southerly dominant wind in
Figure 3, the dominant wind direction keeps rotating clockwise by 90° for each
of the remaining growth factor diagrams shown in Figures 4 through 6. Thus the
effect of differing wind patterns relative to the cities'source configurations can
be observed. The variances have been held constant, 25 and 100, respectively,
for the wind components in East-West and North-South directions.
The growth factors appear to be more dependent on emission growth
pattern than on concentration distribution. Although the two future cities have
the same amount of increase in emissions, the growth factors of future city 2
(third configuration) are appreciably greater in every wind pattern and at every
percentile than those of future city 1 (second configuration). The reason
probably is that the third configuration (p (r_= 0) = 1.90) yields a higher source
density around the receptor point than the second configuration (p (x= 0) =
1.65) does, and that pollutant concentrations are influenced in a greater extent
by nearby sources than remote sources.
The dependence of the growth factor on percentile is fairly sensitive on the
source configuration relative to the wind pattern. As the dominant wind
direction rotates, the slope of the trends changes. Strong dependence on
percentile occurs when the dominant wind blows from the North or South (see
Figures 3 and 5 respectively). Weak or no dependence on percentile occurs when
the dominant wind blows from the East or West. When wind does not have a
preferred direction the growth factors show a mild dependence on percentile.
It is difficult to determine how realistic the model simulations are. The wind
roses were constructed from the wind speeds and wind directions generated by
the method mentioned above. The simulated wind frequency distributions are
not much different in nature from those observed at CAMP cities.
15-7
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Discussion and Conclusions
The effect of statistical distribution of pollutant concentrations on rollback
equations has been investigated by using the concept of the proportional model
as well as by using numerical examples. The form of the rollback equations, in
general, changes with percentiles of pollutant concentration when incorrect
growth factors are used. This conclusion derives from the percentile forms of the
rollback equations that has been obtained for the case of a constant growth
factor by using the proportional model.
Most cities may not be expected to grow in any simple fashion. To see the
dependency of the growth factor on percentiles and emission growth patterns,
the growth factors at different percentiles of an imaginary city were calculated
for several emission growth patterns and for several different wind velocity
distributions. The percentiles were constructed from 2000 concentration values
that were calculated using a simple model and normal random wind velocity
vectors as meteorological input. The results are shown in Figures 2 through 6.
In all cases the figures show that the growth factor of future city 2 (open
circles) are greater than those of future city 1 (solid circles) although the amount
of emission growth is the same for the two future cities. The reason is that
emission growth around the receptor point is larger in the 2nd future city than
in the first, and that the receptor concentrations are affected in a greater extent
by nearby sources than remote sources. Here, the source densities at the origin
where the receptor is located are 1.45, 1.65 and 1.90, respectively, for present
city, future city 1, and future city 2 (Fig. 1). The "emission growth factors" at
the receptor point, therefore, are 1.14 and 1.31, respectively, for future cities 1
and 2. This indicates that not only the amount of emission growth but also the
emission growth pattern is important to correctly estimate the growth factor.;
This in turn suggests that redistribution of emission sources through city
planning can be an effective measure to improve the air quality at dirty spots in
a city.
The growth factor may increase or decrease with percentile depending upon
the emission growth pattern and wind pattern. When there is no preference in
wind direction (Fig. 2), the growth factors gradually increase with percentile.
The reason is that a weak wind, which results in higher concentrations, tends to
magnify the effect of emission growth on concentrations. When a dominant
wind blows from the East or West (Figs. 4 and 6), the growth factors become
less dependent on percentile. Strong dependence on percentile occurs when the
dominant wind blows from the North or South. When the dominant wind blows
from the South (Fig. 3), the growth factors increase with percentile. For a
northerly dominant wind the growth factors decrease with percentile (Fig. 5).
This can be explained as follows:
15-8
-------
(a) Lower percentile concentrations in Figure 3 result from a strong
southerly wind that blows over the southern part of the city where the emission
growth is lower than the other part of a city. Thus, the concentration growth
factors at lower percent!les are smaller than the "emission growth factors" at the
receptor.
(b) Higher percentile concentrations in Figure 3 result from a weak
northerly wind that carries pollutant from the northern part of the city where
the emission growth is higher than the other part. Thus, the concentration
growth factors at higher percentiles are greater than the "emission growth
factor" at the receptor.
(c) As a result of (a) and (b), the growth factors in Figure 3 increase rapidly
with percentile.
(d) Lower percentile concentrations of future city 2 result from a strong
northerly wind that carries pollutant from the high emission growth northern
part to the receptor. Thus, the concentration growth factors at lower percentiles
are much greater than the "emission growth factor" at the receptor (=1.31). On
the other hand, higher percentile concentrations of future city 2 result from a
weak southerly wind that blows over the low emission growth southern part of a
city. Since a weak wind tends to magnify the effect of emission growth on
concentrations as mentioned before, the concentration growth factors at higher
percentiles are about the same or a little higher than the "emission growth
factor" at the receptor. As a result of this, the growth factors of future city 2
decrease sharply with percentile as seen from Figure 5.
(e) The growth factors of future city 1 decrease with percentile by a similar
reason to the above. However, the downward trend is much milder than that of
future city 2 because the center of the emission growth is located at the
northeast of the receptor instead of the North, and is more distant from the
receptor point than that of future city 2.
From the preceding discussion the following qualitative statement can be
made as to percentile dependence of growth factors. When there is no preference
in wind direction and speed, growth factors tend to increase with percentile
unless source density configuration is point symmetric and the receptor is
located at the symmetric point. When there is a dominant wind direction from
which wind blows strong and frequently, growth factors at downwind receptors
from the emission growth center decrease with percentile and those at upwind
receptors increase with percentile.
The effects of concentration distribution and emission growth pattern on
growth factors have been discussed mainly because the growth factors are used
extensively in the literature (Larsen (1961) (1969); Jensen (1971); Ott et al.
(1967)). The real concern, however, is the effects of concentration distribution
and growth pattern on the reduction ratio that is given either by Equations 1 or
2 or 16. The sensitivity of the reduction ratio, R, on the growth factor, G, can
15-9
-------
be checked by expanding R in a Taylor series about the correct value of G, G0,
and in terms of the deviation of G from the correct value, 5 = G — G0-
'o
r j 11 i r_
'o
*R(G0) + 8^- (23)
Setting B = 0 in Equations 1, 2, and 16 we can obtain
dR i = ' ~ R (Qo) (24)
dGlGo G0
Thus we can write the reduction ratio as
8(I-R CG03)
R(G) ^
(25)
Suppose that one made a mistake in estimating future growth pattern and
ignored the effect of concentration distribution on growth factors. Thus, from
Figure 3 the difference between the growth factors at the 50-th percentile of
tuture city 1 and the 99-th percentile of future city 2 is about —0.25. Taking R
(G0) = 0.90, then from Equation 25, R(G) s 0.87. This appears to be a negligible
effect. Real importance, however, is as to how much the total amount of
pollutants will remain when emissions are reduced according to the reduction
ratio. The relative difference in the amounts of remaining pollutants according
to the correct and incorrect reduction ratios is given by
{(l-RrG03)-{l-RCG])}ep/£/of(dr')
(26)
(l-RCG0])ep /r, />f (dr')
_ R(G)-R(G0)
I-R(G0)
Substituting the values into Equation 26, the example yields about 30%
difference in the amounts of remaining pollutants. This can not be a negligible
effect.
The results of this work are not conclusive as to the extent of the effects of
concentration distribution and emission growth pattern on calculating reduction
ratios. It is, however, obvious that a correct reduction ratio can not be obtained
from emission growth only. Consideration on emission growth patterns and
concentration distributions should be included in rollback calculations.
15-10
-------
/»(£)- 2 />n
n=l
Present City:
N=2
R, =
R2=5000i
Future City I
Future City 2'-
<' R3S 5000 i
+5000]
R| = 0 R2=5QOOi
R3=500of
-• A
R,=0 R^SOOOi
Figure 15-1. Source configuration of imaginary city
15-11
-------
1.4
O I 3
J2
o
oc
O 00 00
I
_r
0.0
25.0 50.0
FREQUENCY (%)
75.0
9995
Figure 15-2. Cumulative frequency diagram for the growth factor for case I: U,
N (0,100), Uy ~N (0,100)
1.4
"
u> 1.2
I.I
I
I
I
J
0.0
25.0 50.0 75.0
FREQUENCY (%)
99.95
Figure 15-3. Cumulative frequency diagram for the growth factor for case II: U,
~N (0,25), Uy ~N (5,100)
15-12
-------
1.4
o
2
1.3
o
o:
o
1.2
I.I
ooo
0000 n on
o
ooo o o o
-o—oo-o—-oo o o-o-
^_^____—^MI^^fcB^^^^to^—-•••^"^^••"•^•••"^^^—^— •"' W W—V WW ^f » *r
U UUUUO 000 000000000 000 00 O 000 000 000 O
o o o ooo o
oo
I
I
I
I
0.0
25.0 50.0 75.0
FREQUENCY (%)
99.95
Figure 15-4. Cumulative frequency diagram for the growth factor for case
Ux ~N (5,25), Uy ~N (0,100)
1.5
KI.4
o
<
1.3
o
cr
o
1.2
o o
o o
000
oo
b
I
I
•
J_
25.0 50.0 75.0
FREQUENCY (%)
99.95
Figure 15-5. Cumulative frequency diagram for the growth factor for case IV:
Ux ~N (0,25), Uy~N (-5,100)
15-13
-------
Q CO O OO I
g UUU -QUO oo-oo oo o coo..
O 00 000 O OO O O OO O OO
13
*
o
250 50.0 75.0
FREQUENCY (%)
9995
Figure 15-6. Cumulative frequency diagram for the growth factor for case V: Ux
~N (-5,25), Uy~N (0,100)
Acknowledgments
The authors wish to thank Mr. Richard Kamens for fruitful discussions and
Professor Arthur C. Stern for his encouragement and helpful suggestions. This
work was supported by the Environmental Protection Agency research project
R-800901.
References
Anon, 1971: National Primary and Secondary Ambient Air Quality Standards.
Federal Register. 36: 8187-8188.
Hanna, S. R., 1971: A Simple Method of Calculating Dispersion from Urban
Area Sources. J. Air Pollution Control Association. 21: 774-777.
Jensen, D., 1971: From Air Quality Criteria to Control Regulations. Ford Motor
Co. Publication No. 710303, pp. 67-74.
Larsen, R. I., 1961: A Method for Determining Source Reduction Required to
Meet Air Quality Standards. J. Air Pollution Control Association. 11:
71-76.
Larsen, R. I., 1969: A New Mathematical Model of Air Pollutant Concentration
Averaging Time and Frequency. J. Air Pollution Control Association. 19:
24-30.
Ott, W., Clarke, J. F., and Ozolins, G., 1967: Calculating Future Carbon
Monoxide Emissions and Concentrations from Urban Traffic Data. U. S.
Dept. of HEW, Public Health Service, Bureau of Disease Prevention and
Environmental Control, Public Health Service Publication No. 999-AP-41.
15-14
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APPENDIX I
Simple Multiple-Source Model
The concentrations at position £and time t due to all relevant physical
factors w can be expressed for a single type of sources as
2/
l',t',w) A(r-r', t-t', w ) (dr/)2dt
where
(1-1)
Xs = concentration due to one type of source
e = emission per unit source
p = density of emission sources
A = transfer function that relates sources at£f, t' to concentration atj, t.
The total concentration is given by the sum of the background
concentration, B, and the concentration due to the one type of source, Xc.
o
Defining a function F d't'.w) A(r-r', t-t', ffi)(dr/)2dt/
the total concentration can be written as
X = B + e F (8)
Equation 8 indicates that the distribution of the total concentration, X, is
determined by the distribution of F through the physical factors w, position £,
and time t. Thus the a-th percentile value of X, Xa, is determined by the a-th
percentile of F, Fa/ where the background concentration, B, is assumed to be
constant.
The growth factor defined by the ratio of (Xfa — B)/(Xpa — B) can be
expressed as
-Ffa
/2/
L//° (I7, t/wf) A(r-r',tf-r', tf-t! gf )(dr/)
dt
dt7
where the integrals in Equation I-3 can be computed with variables, t and w, that
would give the a-th percentiles of Fp and Ff .
15-15
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APPENDIX 11
Derivation of the Percentile Form of the Rollback Equations
Assume that G = (Ffa/F a) is independent of percentile a and that the
rollback equations are valid for the 50-th percentile, i.e.,
D - gLXP50~ Xd50 (,|.-,)
RL ~3 J
(I I-2)
9j (Xp50- B)
where
Xp50 = the 50-th percentile value of Xp
Xd50 = the 50-th percentile value of Xd.
The proportional model, in general, can be written as
Xj = B + ejFj (11-3)
where subscript i is used to generalize quantities for future, present and desired.
From the preceding discussions we have
Ffa= Fda = GFPa O1'4)
ef = ep
ed = (l-R)ep
R = RLor Rj
The a-th and 50-th percentiles oi the present and desired air qualities can be
obtained, using RL, from Equations 11-3 and 11-4 as
X— Q -U A C* /i i c'\
pa ~ ° eprpot v""^
= B + (|-RL) epG Fpa (ii-6)
6PFP50
epGFp50 (u-8)
15-16
-------
Any percentile value of F can be related to the 50-th percentile value of
FP' Fp50. by a constant, )3a.
pa
0"-9)
Substitutions of Equations 11-5 and 11-9 into Equation 11-7 and Equations 11-6
and 11-9 into Equation 11-8 yield, respectively,
^P50 = xpa~" epPa
and
-{l-RL)epG/3a
Substitution of Equations 11-10 and 11-11 into Equation 11-1 yields
gL(xpa-ep)3a)-{xda-Cl-RLDepG pY
R, =
gL(Xpa-ep/3a)-B
Solving for RL we obtain
R _
- B
By similar steps Equation 19 can be derived from Equation II-2.
(11-10)
(n-11)
(H-12)
15-17
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DISCUSSION
Larsen: Thank you, Dr. Horie, for a thorough look at this problem. We might
consider one factor. Three rollback equations could be used according to the
behavior of background concentrations. We have heard of two rollback
equations, Rj which refers to Jensen's article and RL which refers to Larsen's
equation. Jensen's equation is the correct equation if background now remains
the same as background later; a second rollback equation would be one in which
the concentration might be doubling and the background would also be
doubling. In other words, a second rollback equation would be one in which the
background growth factor equaled the urban growth factor. The Larsen rollback
equation is between these two, between a background growth factor of one and
a background growth factor equal to the urban growth factor. For a situation
involving no background growth, the Jensen equation should be used. If growth
is intermediate the Larsen equation should be used. This intermediate situation
might be experienced, for instance, with a northeast wind blowing from Boston
to New York to Philadelphia to Baltimore to Washington. As these places grow
together, the background grows together and background increases. The Larsen
equation could be used for places growing together. In the middle of the great
plains, not affected by background from other cities, the Jensen equation could
be used.
Horie: Thank you very much. This is exactly so. We noticed this difference when
we consider the growth for the background concentration.
Smith: I think one fact which ought be taken into account in this sort of
calculation is the variation due to the modification of the weather by the
pollutants themselves. If you reduce the concentration of such things as smoke
or photochemical components, then you may very well change the statistics of
the weather and hence, get a change in your reduction factor.
15-18
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16. SYMPOSIUM PARTICIPANTS
Pat Adomitis
Dept. of ESE
School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
Gerald G. Akland
EPA — Div. of Atmospheric Surveillance
Research Triangle Park, N. C. 27711
James N. Arvesen
Dept. of Statistics
Math Science Bldg.
Purdue University
Lafayette, Indiana 47907
C. W. Ash
Dept. of Biostatistics
School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
R. E. Barlow
University of California
Berkeley, Ca. 94700
Joel Barnett
Div. of Air Pollution Control
C2-212Cordell Hull Building
Nashville, Tennessee 37219
John J. Beauchamp
Oak Ridge National Laboratory
Statistics Department
P. O. Box Y
Bldg. 9704-1
Oak Ridge, Tennessee 37830
Michael M. Benarie
IRCHA Research Center
Boite Postale 1
91 Vert-le-Petit
France
Bernard Bloom
Allegheny Cty. Air Poll. Control Board
301 39th St.
Pittsburg, Penn. 15201
John M. Bowman
501 N. 9th St.
Room 130
Safety, Health & Welfare Building
Richmond, Virginia 23219
Franklin Briese
Div. of Biometrics
Mail Container 2355
Univ. of Colorado Med. Center
Denver, Colorado 80220
Kenneth Calder
EPA — Div. of Meteorology
Research Triangle Park, N. C. 27711
Norman L. Canfield
19 Bayberry Ave.
Garden City, Ntew York 11530
Charles R. Case
TRC
125 Silas Deane Highway
Wethersfield, Conn. 06109
C. Andrew Clayton
EPA
Research Triangle Park, N. C. 27711
David C. Collins
Technology Service Corp.
225 S. Monica Blvd.
Santa Monica, Ca. 90401
R. E. Cooper
Environmental Analysis and Planning
Savannah River Laboratory
Aiken, S. C. 29801
16-1
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Arnold Court
California State University
Northridge, California
J. M. Craig
Southern Services Inc.
P. O. Box 2625
Birmingham, Alabama 35202
Alexander R. Craw
National Bureau of Standards
Washington, D. C. 20234
T. V. Crawford
Environmental Analysis and Planning
Savannah River Laboratory
Aiken, S. C. 29801
Loren W. Crow
2422 South Downing St.
Denver, Colorado 80210
E. James Dale
Dept. of ESE
School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
Eugene M. Darling, Jr.
DOT Transportation Systems Center
55 Broadway
Cambridge, Massachusetts 02147
William Delaware
Dept. of Envir. Conservation
Div. of Air Resources
50 Wolf Road
Albany, New York
John B. Edwards
Chrysler Corp.
CIMS 418-18-22
P. O. Box 1118
Detroit, Michigan 48231
I sod ore Enger
Geomet, Inc,
50 Monroe St.
Rockville, Maryland 20850
16-2
James A. Fay
Room 3246
MIT
Cambridge, Massachusetts 02139
Robert Faoro
EPA
Research Triangle Park, N. C. 27711
Martin A. Ferman
Research Laboratories
Department E.V.
General Motors Technical Center
Warren, Michigan 48090
Doug Fox
EPA/NERC
Research Triangle Park, N. C. 27711
Neil H. Frank
EPA
Research Triangle Park, N. C. 27711
Joel Frockt
Department of Biostatistics
School of Public Health
University of North Carolina
A. S. Galbraith
EPA
Research Triangle Park, N. C. 27711
Frank A. Gifford
NOAA-ATDL
P. 0. Box E
Oak Ridge, Tennessee 37830
Steve Goranson
Office of Statistical Services
EPA/NERC
Research Triangle Park, N. C. 27711
Rebecca A. Gray
EPA/NERC
Div. of Chemistry and Physics
Research Triangle Park, N. C. 27711
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George W. Griff ing
EPA/Div. of Meteorology
Research Triangle Park, N. C. 27711
Nathaniel Guttman
National Climatic Center
Federal Building
Asheville, North Carolina 28801
Howard R. Hammond
Baltimore Gas and Electric
Room 231
531 E. Madison St.
Baltimore, Maryland
David M. Hershfield
ARS-W Soils Building
Beltsville, Maryland 20705
C. Doyce Hester
Reynolds Metals Co.
P.O. Box 9177
Corpus Christ!, Texas 78408
M. Eugene Hoffman
North Carolina State University
Raleigh, North Carolina 27607
George C. Holzworth
EPA/ Div. of Meteorology
Research Triangle Park, N. C. 27711
Yuji Horie
ESE, School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
William F. Hunt
EPA
Research Triangle Park, N. C. 27711
Peter Imrey
Department of Biostatistics
School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
John S. Irwin
University of North Carolina
Chapel Hill, N. C. 27514
Warren G. Johnson
EPA
Research Triangle Park, N. C. 27711
Howard C. Jones
Dept. of Environmental Conservation
Div. of Air Resources
50 Wolf Road
Albany, New York
James Jones
Dept. of Chemical Engineering
University of Kentucky
Lexington, Kentucky
Surendra Joshi
ESE, School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
R. H. Ketterer
General Electric
Schenectady, New York
K. R. Knoerr
308 Biological Sciences
Duke University
Durham, North Carolina 27706
Joseph B. Knox
Lawrence Livermore Laboratory
University of California
P. O. Box 808
Livermore, California 94550
Lewis Kontnik
ESE, School of Public Health
University of North Carolina
Chapel Hill, North Carolina
Stanley L. Kopczynski
EPA/NERC
Research Triangle Park, N. C. 27711
16-3
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Lawrence L. Kupper
Dept. of Biostatistics,
School of Public Health
University of North Carolina
Chapel Hill, N. C.
Ralph Larsen
EPA/Met. Laboratory
Research Triangle Park, N. C. 27711
Russell F. Lee
EPA/OAP
Research Triangle Park, N. C. 27711
Stephen A. Lesh
ESE, School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
Helmut Lieth
Department of Botany
University of North Carolina
Chapel Hill, N. C. 27514
Jiumn W. Lin
Room 402,
Dept. of Environ. Control
320 N.Clark St.
Chicago, Illinois 60610
James W. Lingle
Industrial Bio-Test Labs Inc.
1810 Frontage Rd.
Northbrook, Illinois 60002
Gene R. Lowrimore
EPA
Research Triangle Park, N. C. 27711
David A. Lynn
33 Cogswell Ave.
Cambridge, Massachusetts 02140
George Manire
EPA
Research Triangle Park, N. C. 27711
16-4
Allan H. Marcus
Dept. of Mathematical Sciences
Johns Hopkins University
Baltimore, Maryland 21218
David McLeod
EPA
Research Triangle Park, N. C. 27711
Thomas E. McMullen
EPA
Research Triangle Park, N. C. 27711
Donald McNeil
Dept. of Statistics
Princeton University
Princeton, New Jersey 08540
David McNeils
Dept. of ESE
University of North Carolina
Chapel Hill, N. C. 27514
W. S. Meisel
Technology Service Corp.
225 Santa Monica Blvd.
Santa Monica, Ca. 90401
Edwin L. Meyer
EPA/OMP
Research Triangle Park, N. C. 27711
Mark Murray
Dept. of Biostatistics
School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
Carl Nelson
Research Triangle Institute
Box 12194
Research Triangle Park, N. C. 27711
Harold E. Neustadter
NASA-LERC
21000 Brookpack Road
Cleveland, Ohio 44135
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Everett C. Nickerson
EPA
Research Triangle Park, N. C. 27711
Lawrence Niemeyer
EPA/Div. ot Meteorology
Research Triangle Park, N. C. 27711
John Overton
ESE, School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
Don Pack
NOAA
8060 13th Street
Silver Springs, Maryland 20910
John J. Paulus
Westinghouse Electric Corp.
P. O. Box 9533
Raleigh, North Carolina
Robert Papetti
EPA
Waterside Mall
4th and M St. S.W.
Washington, D. C.
M. M. Pendergrast
Envir. Analysis and Planning
Savannah River Laboratory
Aiken, S. C. 29801
Bier Peters
North Carolina State University
Raleigh, N. C. 27607
James Peterson
EPA/Div. of Meteorology
Research Triangle Park, N. C. 27711
John M. Pierrard
Petroleum Laboratory
E. I. DuPont de Nemours
Wilmington, Delaware 19898
Richard I. Pollack
University of California
Lawrence Livermore Laboratory
Livermore, California
Charles Proctor
Statistics Department
North Carolina State University
Raleigh, North Carolina 27607
William J. Raynor
Dept. of Biostatistics
School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
John H. Reynolds
2615Selwyn Ave.
Charlotte, N. C. 28209
Donald M. Rote
Argonne National Laboratory
9700 S. Cass Ave.
Argonne, Illinois 60439
R. Ruff
EPA/Div. of Meteorology
Research Triangle Park, N. C. 27711
J. S. Rustagi
Division of Statistics
The Ohio State University
Columbus, Ohio 43210
Bernard E. Saltzman
Kettering Laboratory
Eden and Bethesda Ave.
Cincinnati, Ohio 45219
Don Thomas
Air Management Branch
880 Bay St. 4th Floor
Toronto, Ontario
Irving A. Singer
Smith-Singer Meteorologists, Inc.
189 Brooklyn Ave.
Massapequa, N. Y. 11758
16-5
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Nozer Singpurwalla
School of Engineering
George Washington University
Washington, D, C. 20006
Bjarne Sivertsen
135 Clinton
Apt. 1U
Hemptead, New York 11550
Ralph C. Sklarew
EPA
Research Triangle Park, N. C. 277
F. Barry Smith
Meteorological Office
Met-0-14
Bracknell, Berkshire, ENGLAND
Vernon M. Smith
Box 2723
Geography Department
East Carolina University
Greenville, N.C. 27834
Ronald D. Snee
Engineering Dept.
E. I. DuPont de Nemours
Wilmington, Delaware 19898
David B. Spiegler
22 Fiske Rd.
Lexington, Mass. 02173
Robert Spirtas
Department of Biostatistics
School of Public Health
University of North Carolina
Chapel Hill, N.C. 27514
William Stasick
Dept. of Environmental Conservation
Div. of Air Resources
50 Wolf Road
Albany, New York
16-6
David J. Svendsgaard
Dept. of Biostatistics
School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
Arthur C. Stern
ESE, School of Public Health
University of North Carolina
Chapel Hill, N. C. 27514
Philip Sticksel
Battelle Columbus Laboratories
505 King Ave.
Columbus, Ohio 43201
George C. Tiao
Dept. of Statistics
University of Wisconsin 53706
George Touchton
ESE, School of Public Health
University of North Carolina
Chapel Hill, N.C. 27514
D. Bruce Turner
EPA/Div. of Meteorology
Research Triangle Park, N. C. 27711
Joseph R. Visalli
School of Civil Engineering
Purdue University
Lafayette, Indiana 47907
Raymond C. Wanta
Consulting Meteorologist
28 Hayden Lane
Bedford, Massachusetts
Lowell Wayne
Pacific Environmental Services
2932 Wilshire Blvd.
Santa Monica, California 90403
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Robert Wevodau
Air Pollution Consulting Group
E. I. Dupont De Nemours
Wilmington, Delaware
J.G.Williams, Jr.
501 N. 9th St.
Room 130
Safety, Health and Welfare Bldg.
Richmond, Va. 23219
Peggy Wingard
VEPCO
P. O. Box 26666
Richmond, Virginia 23261
F. K. Wippermann
Technische Hochschule Darmstadt
Sektion Meterologie
6100 Darmstadt, GERMANY
Donald F. Worley
EPA
Research Triangle Park, N. C. 27711
Charles E. Zimmer
8160Trabant Dr.
Cincinnati, Ohio 45242
16-7
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1 REPORT NO.
EPA-650/4-74-038
3. RECIPIENT'S ACCESSION-NO.
4 TITLE AND SUBTITLE
Proceedings of the Symposium on Statistical Aspects
of Air Quality Data
5. REPORT DATE
October 1974
6. PERFORMING ORGANIZATION CODE
7 AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT NO
Lawrence D. Kornreich, Editor, Executive Director,
Triangle Universities Consortium on Air Pollution
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Triangle Universities Consortium on Air Pollution
Post Office Box 2284
Chapel Hill, North Carolina 27514
10. PROGRAM ELEMENT NO.
1AA009
11. CONTRACT/GRANT NO.
68-02-0994
12 SPONSORING AGENCY NAME AND ADDRESS
Meteorology Laboratory
National Environmental Research Center
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
13. TYPE OF REPORT AND PERIOD COVERED
Symposium Proceedings
14. SPONSORING AGENCY CODE
15 SUPPLEMENTARY NOTES
16. ABSTRACT
The 15 papers in these proceedings analyze air quality data as a function of
frequency, maxima, the form of the frequency distribution, and averaging time.
Concentrations and frequency distributions calculated with meteorologic diffusion
models are compared with observed values. Discussions that followed the paper
presentations are included.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
Air quality data
Statistical analyses
Frequency distribution
Lognormal
Averaging time
Meteorology
Diffusion
Sulfur dioxide
Suspended
particulate
c. COSATI Field/Group
3 DISTRIBUTION STATEMENT
Unlimited
19 SECURITY CLASS (This Report)
Unclassified
21. NO OF PAGES
266
20 SECURITY CLASS (Thispage)
Unclassified
22 PRICE
EPA Form 2220-1 (9-73)
17-1
G P O 1975— 64O-878 J 639, REGION NO. 4
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