Environmental Monitoring Series
NUMERICAL OPTIMIZATION TECHNIQUES IN
AIR QUALITY MODELING
Objective Interpolation Formulae for
the Spatial Distribution of
Pollutant Concentration
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina 27711
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. 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. Socioeconomic 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.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/4-76-058
December 1976
NUMERICAL OPTIMIZATION TECHNIQUES IN AIR QUALITY MODELING
Objective Interpolation Formulae for the
Spatial Distribution of Pollutant Concentration
by
S-A Gustafson
The Royal institute of Technology
Stockholm
K. 0. Kortanek
Carnegie-Melion University
Pittsburgh, Pennsylvania 15213
J. R. Sweigart
University of South Carolina
Columbia, South Carolina 29208
Grant No. R 8O3632
Project Officer
Kenneth L. Calder
Meteorology and Assessment Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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DISCLAIMER
This report has been reviewed by the Environmental Sciences
Research Laboratory, U.S. Environmental Protection Agency, and
approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the U.S.
Environmental Protection Agency, nor does mention of trade names
or commercial products constitute endorsement or recommendation
for use.
11
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ABSTRACT
A technique is proposed for objective interpolation of the
air quality distribution over a region in terms of sparse
measurement data. Empirical information provided by the latter
is effectively combined with knowledge of atmospheric dispersion
functions of the type commonly used in source-oriented air qual-
ity models, to provide improved estimates of the concentration
distribution over an extended region. However, the technique is
not primarily source-oriented since, in contrast to the real
source distribution of a source-oriented model, it utilizes fic-
titious or pseudo-sources that are estimated in terms of the
measured air quality data. This involves the use of interpola-
tion functions that are computed using numerical optimization
techniques based on the method of least squares. Due to the
large number of different "weather" states that affect the atmos-
pheric dispersion of pollution, considerable computation is
required, although the bulk of this can be done in advance, so
that the final interpolation from the measured values only re-
quires very simple calculation. Thus the proposed method has the
potential for application on a real-time basis.
In addition to the mathematical formulation of the problem,
this preliminary study includes some numerical experiments, using
a current multiple-source EPA air quality model, to illustrate
the technique that is proposed.
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CONTENTS
Abstract . . . ill
Acknowledgement ........... vi
1. introduction . . . . . ....,,... 1
2. Multiple-Source Plume Models and the Concept
of Pseudo Sources ....... . 4
3. Further Development of the Pseudo-Source Concept . . 8
4. The interpolation Formula for Concentration 13
5. interpolation Formula for Multiple Weather States . . 22
6. Least Squares Computation for interpolants 26
7. A Numerical Experiment in Concentration
interpolation ... ...... 3o
8. The Same Example with Aggregation of Pseudo-Sources . 37
9. Recommendations 41
References 42
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ACKNOWLEDGEMENTS
The authors gratefully acknowledge several extended discus-
sions with the Project officer, which led to identification of
the air quality interpolation problem as one particularly appro-
priate for analysis-by numerical optimization techniques.
Mr. K. L. Calder also had a major part in clarifying some of the
interpretations and developing the final report on the research,
based on an early draft.
The authors are also grateful to Professor E. L. Rubin,
Carnegie-Me lion University for communications regarding a hypo-
thetical 25-point source inventory used in numerical experiments.
J. R. Sweigart received financial support during the year as a
Richard King Mellon Fellow in Environmental Studies, Carnegie-
Mellon University. Finally, we are indebted to R. Edahl and
D. N. Lee (also a Richard King Mellon Fellow) for their assist-
ance in the computational experiments.
VI
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SECTION 1
INTRODUCTION
Even when attention can be restricted to near-ground values
of the pollutant concentration, the distribution of air quality
over an urban area represents a two-dimensional time-dependent
scalar field of great complexity. Any proper understanding of
its complex structure must involve theoretical considerations,
like those in mathematical source-oriented air quality models
which estimate the air quality in terms of specified pollutant
emissions and appropriate atmospheric dispersion functions, or
alternatively the use of measured air quality data based on the
operation of an extensive sampling network. If a mathematical
air quality model is available, it will generally be possible to
estimate the pollutant concentration at an arbitrary receptor-
location, and hence with sufficient computational effort to
determine the air quality distribution over the entire area.
However, in contrast, when only empirical air quality data from a
sampling network of fixed stations is available, it will normally
be necessary to interpolate concentration values for the inter-
mediate locations, and in this case it is desirable that an
objective procedure be used, i.e., one that does not involve per-
sonal or subjective judgment.
The preliminary analysis of the present paper shows that
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these two approaches can be complementary and that real air
quality measurements at typical locations can be combined with
appropriate atmospheric dispersion functions to provide an air
quality interpolation function for an entire region. The latter
does not require complete information on the real source
strengths of the pollutant emissions such as would be required
for application of a conventional source-oriented air quality
model. The interpolation function, however, actually involves
the concept of a hypothetical or fictitious "pseudo-source" dis-
tribution, that is computed for a given atmospheric dispersion
situation or weather state, by optimized fitting techniques from
the measured air quality values at selected stations and the
specified atmospheric dispersion functions. Similar ideas have
recently been exploited by Heimbach and Sasaki (1975) in fitting
an analytical dispersion model to sparse data on air quality.
Their work was motivated by the fact that detailed emission in-
ventories are frequently difficult to obtain, and the predic-
tions of source-oriented air quality models should be capable of
improvement when air quality measurements are available.
Since the interpolation approach is self-correcting, the
air quality estimates are less affected by inaccuracies in the
real emissions data. In a sense, it provides a technique for
adjusting or "calibrating" the air quality estimates provided by
a conventional source-oriented model in order to correct for er-
rors in the estimates of the emissions. The calibration
technique is, however, not subject to the fundamental objection
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that can be raised in connection with some others that have been
used with source-oriented models (Brier, 1973). it is, in prin-
ciple, the complement of another approach using empirical data
recently suggested by Calder et.al. (1975). in the latter, mul-
tiple-station observations of air quality are coupled with
estimates of the emissions for a multiple-source distribution,
to deduce the effective atmospheric dispersion functions. The
latter can then be used as a basis for a conventional source-
oriented air quality model. The pseudo-source interpolation
technique, on the other hand, estimates "effective" emissions in
terms of the observed air quality and prescribed atmospheric
dispersion functions. In principle, the end result is a "cali-
brated" interpolatory air quality model. Since the interpolation
functions can be computed in advance, the technique has potential
for use in air quality prediction on a real-time basis.
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SECTION 2
MULTIPLE- SOURCE PLUME MODELS AND THE CONCEPT OF PSEUDO- SOURCES
The general structure of multiple- source plume models for
urban air pollution has recently been discussed by Calder (1976) .
The starting point for most models is the working assumption
that the long-term variability of emissions, air quality and at-
mospheric conditions can be treated as though it resulted from a
time- sequence of different steady- state situations. The sequence
interval is normally taken to be relatively short, and perhaps
only of the order of an hour. For pollutants that can be regard-
ed as chemically inert, it is further assumed that the concen-
tration contributions produced at any receptor location from
several sources combine additively, so that the individual plumes
can be simply superimposed. Then the total concentration C(x)
at the point x can be written
N
C(x) = L q.v (x) , (1)
where
q. is the strength of source j
v . is the concentration produced by a unit source at the
location of j .
The function v. is, of course, independent of source strength
and is assumed to be a computable function of the meteorological
4
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conditions, such as wind direction, wind speed, atmospheric sta-
bility, mixing depth, etc. The particular combination of these
conditions existing during any one steady-state interval of the
time sequence is regarded as constituting one of an ensemble of
possible meteorological or "weather" states. The expression of
the total concentration C(x) as a linear superposition extend-
ing over the atmospheric dispersion functions v.(x) is mathe-
matically analogous to the use of polynominal or trigonometric
functions for finite expansion approximation of continuous func-
tions.
Before considering the general pseudo-source concept, we
first consider a very simple example to illustrate the basic con-
cept and its use for interpolating air quality over a region in
terms of a limited number of observed values. We consider a
single source of strength q at x = x , and a hypothetical one-
dimensional concentration distribution C(x) produced by it.
(Note that here x is not to be confused with that in equation
(1) above, where it denotes a general position vector). Then,
corresponding to just a single term of the sum in (1), we have
C(x) = qv(x-xQ) (2)
so that C(x) can be determined once q is specified. However,
in the absence of direct knowledge of q, it may be estimated in
terms of a "measured" concentration value, say C, at location
x = x, , and the corresponding value v(x.,-x ) of the dispersion
functions, since q = C,/v(x,-x ). This value constitutes a
5
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"pseudo-source" strength which when substituted into the equa-
tion for C(x) then provides the general interpolation formula
£(x) = C1v(x-xo)/v(x1-xo) . (3)
This might be written C(x) = m(x)C, , where
m(x) = v(x-x )/v(x..-x ) vis a simple interpolation function that
"converts" the measured value c, into a field of values cor-
responding to the different values of x. The function m(x) is
completely independent of the actual strength of the source and
depends only on the positional parameters of the source and the
meteorological conditions that affect the dispersion function.
Thus the interpolation function could be calculated once and for
all without regard to the value of the pseudo-source strength
This simple idea can be immediately generalized for equa-
tion (1) . Thus if concentrations are measured at p sampling
stations x^,x2,...,x yielding values C., i = l,2,...,p, then
we have the following linear system to determine the N pseudo-
source strengths q. (j = 1,2,...,N)
N
(4)
2 q^v. (x.) = C., i = 1,2,...,p.
j=l 3 .3 i i
Under appropriate conditions this system of equations will deter-
mine a set of pseudo-source values, which we will denote q.. On
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substituting these back into equation (1) we obtain the desired
air quality interpolation formula for the region, namely
. N
C(x) = S q,v (x) . (5)
j=l : D
The "hat" notation is used here, since as will be seen later, it
will sometimes be necessary to consider inconsistent sets of
equations having no exact mathematical solutions, but only ap-
proximate solutions as, for example, in a least squares sense.
In this case, of course, the values of C(x.) for the sampling
locations x. will not exactly duplicate the measured values
C., i = 1,2,...,p. Finally, we note that in principle the dis-
persion functions v.(x) could be precalculated, once and for
all, for a grid of points. .The interpolated concentration dis-
tribution could then be very simply calculated from (5) in terms
of the q., which from (4) are functions only of the observed
values C. and v.(x.)(i = l,2,...,p). This offers the hope for
interpolation on a real-time basis. The remainder of the paper
in effect only develops these simple concepts further. The com-
plications arise from the need to consider many sources and the
use of approximate solutions in a least-squares sense, together
with the need for consideration of an ensemble of possible
weather states.
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SECTION 3
FURTHER DEVELOPMENT OF THE PSEUDO-SOURCE CONCEPT
The number of real identified sources, N in equation (1)
may be very large for an urban area. However, an aggregated
pseudo-source formulation in terms of a much smaller number of
sources n < N can be developed. Formally, we proceed as fol-
lows: Let j , j.,...,j be integers such that
O = j < J-, < . .< j = N.
Jo Jl Jn
Then we combine sources with indices between j + 1 (= 1) and
j into the first pseudo-source, those with indices from j.. + 1
to j- into the second pseudo-source, and so on. Generally,
pseudo-source number r consists of sources with indices from
j , + 1 to j . Let the concentration contribution from the
r-th pseudo-source be U . Thus (1) becomes
n
C(x) = S U (x) , (6)
r=l r
where
U_(xj = L q.v. (x) .
We now write U (x) in the following manner
8
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Ur(x) = Qrur(x) ,
where
q , ur(x) = Ur(x)/Qr. (7)
Q is the combined strength of all the sources in source class
r. The functions u (x) are still dependent on source-strength
considerations since they depend on the ratio of the individual
source strengths in the class, to the class total. This is in
contrast to the v.(x) functions of equation (1) which are true
dispersion functions that are entirely independent of source
strength considerations. The pseudo-source concept is only en-
tirely independent of knowledge of the real source distribution
in the obvious case n = N, j = r, when Q = q , so that
u (x) = v (x). However the determination of the large number of
values of q. (j = 1,2,... ,13) in (5) by the pseudo-source method
could be computationally prohibitive, and this is the primary
reason for the aggregated source concept.
Another case is very important as an approximation. For it
is obvious that the real sources may be aggregated into classes
in such a manner that within each class the strengths are
"approximately" equal, i.e., to within some prescribed variation.
This motivates an important specialization of the aggregated
source concept, for which every source in a class is replaced by
the average value, say q(r), for the class. This approximation
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can be made as close as we please up to the limit of the com-
pletely unaggregated situation. It will be designated as the
aggregated-averaged approximation (A.A.A.). For this case we
see from (6) that
U (x) = q(r)V_(x); V (x) = £ v (x) , (8)
r r r r class :
and the function V (x) is now completely independent of source-
strength considerations. Like the original functions v.(x) it
depends only on positional parameters for the sources constitu-
ting the class, and on the meteorological conditions as affecting
dispersion. Evidently from (7) the same will be true for the
functions
u (x) = V (x)/No. in r-class.
in terms of the aggregated pseudo-source classes equation
(1) is now replaced by a sum that only involves n terms, namely
n
C(x) = S Q u (x) . (9)
r=l r r
Now, if concentrations are measured at p sampling stations
x..,x2,...,x yielding values C^, i = l,...,p, we have the fol-
lowing linear system of equations to determine the n pseudo-
source strengths Q (r = 1,2, . .-,n)-
n
2 Q u (x.) = C. , i = 1,2,...,p. (10)
r=1 r r i i
10
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If p < n, so that there are fewer equations than unknowns, then
the system is underdetermined and will not suffice for a unique
determination of the n unknown pseudo-source strengths
Q (r = l,2,...,n). If p = n, so that there are as many equa-
tions as unknowns, the system is even determined and, except in
a singular case (i.e., vanishing of the determinant of the co-
efficients), will determine the Q 's uniquely. if p > n, so
that there are more equations than unknowns, the system is over-
determined and there will generally be no exact mathematical
solution for the Q 's. However, in the latter case, there will
still be a best approximate solution in a least squares sense
(cf., Lanczos Applied Analysis, Prentice-Hall, 1965) for which
p n 2
A = min Z { X Q u (x ) - C. } , (11)
Q i=l r=l r r x 1
and the vector Q = (Q, ,Q2* « ,»Qn) is such as to minimize A.
Thus in the even determined case we have a mathematically exact
solution for the Q 's and for the overdetermined case a least
squares approximation. If Q denotes the appropriate solution
for these two cases, then on substituting back in (9), we obtain
the corresponding air quality interpolation formula for the en-
tire concentration field based on the use of pseudo-sources, viz.,
£(x) = S du (x). (12)
r=l r r
This is a solution in the least-squares sense of equation (11) ,
11
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namely
p A 2
£ (c(x.) - C.j = Tninimura.
1
It should be noted that equation (12) is analogous to equation
(5) for the example that was considered earlier. The case of a
uniform background concentration B extending over the whole
region is realized by setting Q. = B and u (x) = 1 for all x.
12
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SECTION 4
THE INTERPOLATION FORMULA FOR CONCENTRATION
Still restricting attention to the interpolation problem
for a single meteorological or weather state, we next consider
the form of the interpolation formula, for the two cases just
considered of an even- determined and over- determined system.
The discussion in this Section is based on well-known results in
linear algebra. See, e.g., Dahlquist et.al., 1974, Chapter 5.
EVEN- DETERMINED SYSTEM
We consider the set of equations (10) for the case p = n.
This set has a unique solution for the Q ' s provided the deter-
minant of its matrix of coefficients is not zero (a matrix having
determinant zero corresponds to the singular case) . We treat
ur(x.) [r = l,2,...,n; i = l,2,...,p] as a (nxp) matrix, say Uri.
Let U . = U. = say A. (so that A = U is a (pxn) matrix) .
ri ir * ir ^
Then (1O) becomes
n
C. = T, &irQr> * = l,2,...,p, (13)
1 r=l ir r
or in matrix notation i
C = AQ, (14)
where generally C and Q are column vectors of p and n
13
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elements respectively. We solve this equation for Q (say Q)
and then set
= S 6 u (x) = uT(x)6 (15)
r=l
to obtain the required interpolation formula for the entire con-
centration field C(x) . [For given x, u(x) is a column vector
T
of n elements, so u (x) is a row vector of n elements , while
Q is a column vector of n elements. For given x, C(x) is a
one-element matrix or scalar, in contrast to the vector C.]
Equation (15) is, of course, equivalent to (12) . We now show
that (15) can be written in a different, although exactly equiva-
lent form as follows :
£(x) = uT(x)£ = uT(x)A~1C = [CT(A~1)Tu(x) ]T
T T T T
since for 3 matrices L, M, N, (LMN) = N M L where the super-
script T denotes a transpose. Hence
) = [CTrn(x)]T = mT(x)C, (16)
where
m(x) = (A~1)Tu(x) . (17)
Here for given x, m(x) is a column vector of p(= n) elements
m. (x) . Put B & (A""1)7 = (A1")'1 so that m(x) = Bu (x) . Hence
n
m. (x) = S B. u (x) , i = 1,2,..., p.
1 r=l ir r
14
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T
Also, .since A m (x) = u(x) ,
T P T P
u (x) = [Am(x)] = S A .TO. (x) = 2 A. m. (x)
r ri i ir i
= £ m. (x)u (x.) ,, r = l,2,...,n. (18)
X x
Equation (16) in expanded form is
A p
6(x) = L m. (x)C. , (19)
which is thus the required interpolation formula (15) . Since for
the even- determined case under consideration the solution of (14)
for Q is exact, in this case we have C(x.) = C. (i =
l,2,...,p), at the sampling locations where the c. are the
measured values. We note that equation (18) can be computed
without any direct determination of the pseudo- source strengths.
Two very important features should be noted from the pre-
ceding analysis:
(a) Since the functions u (x) become "true" dispersion
functions when the sources are unaggregated (i.e., n = N) ,
and also when the " aggregated- average approximation" is used,
the interpolation functions m. (x) satisfying the system (18)
will be completely independent of source- s treng th considera-
tions under these conditions, and will only depend on the
source positional parameters and meteorological factors.
Thus the interpolation formula should be valid for arbitrary
15
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values of the source strengths at the specified locations
of the real source distribution. Likewise, the concentra-
tion interpolation formula (19) should be valid for
arbitrary values of the source strengths.
(b) From equation (18) it follows that the m. (x) do not
depend on the source class r, i.e., they serve as general
interpolation functions for all the u (x) functions. Since
the individual functions u (x) may be widely different for
different values of r, it follows that the interpolation
functions m (x) must be insensitive to the form of the
i
functions u (x) . in this case it may be conjectured that a_
similar simple relation might also be applicable even when
several different weather states are involved, as when these
are appropriately aggregated into a. single weather class.
As we shall see later, the aggregation of different weather
states into weather classes for which an equation like (18)
for the interpolation m.(x) may be an acceptable approxima-
tion must be the subject of empirical study, although this
is the basis for analysis of the multiple-weather state
interpolation problem.
OVER-DETERMINED SYSTEM
For p > n, the system becomes over-determined and in this
case we seek for a solution Q that is the best solution in a
least squares sense. Then an argument similar to that in the
preceding section may be used. Thus as before we take
16
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where Q is the best solution in the least squares sense of
C = AQ,
where the matrix A has p rows and n columns. This least
squares solution is the solution of (cf. Lanczos, loc cit)
T T
A AQ = AC.
The remarkable fact about this equation is that it gives an even-
determined system having just as many equations as unknowns, no
matter how strongly over-determined was the original system.
*-
Its solution is...
_ 1 T
LA1C. (20)
Then
uT(x) (ATA)~1ATC
m m -
[CXA(A A) -
[CTm(x)]T = mT(x)C, (21)
where
m(x) = A(ATA) ^(x) (22)
Note that now A$ = A(ATA) ~1ATC £ C.
17
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and, for given x, is a column vector with p elements
T
(m., i = l,2,...,p). Multiplying by A we obtain
ATm(x) = u(x) . (23)
The system (23) always has the exact solution given by (22), but
is generally not unique since (23) has p unknowns
(m. , i = l,2,...,p), but only n equations. However, if m(x)
is a solution of (23), then we set
A T P
C(x) = m (x)C= E m.(x)C. (24)
i=l
as before in equation (19). A particular solution obtained from
least squares is, of course, (22) and gives the^ least squares
estimate C(x) . From the above analysis, it is seen that even in
the over-determined case, provided a least squares solution is
adopted, then the same form of concentration interpolation formu-
la is available. However, this is now an approximation in a
least squares sense, so that C(x-) ^ C..
Finally, we may note the underdetermined case mentioned in
Section 3, where for equation (1O) (or equivalently (14)) p < n,
so that in general the system has many solutions. This implies
that the adjoint system to (14), namely, the system (23) is over-
determined and has no exact solution. It then follows from the
preceding analysis for the over-determined case, that the least-
squares solution of (23) (compare with (2O) ) is
m(x) = (AAT)~1Au(x)
18
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under the assumption that the rank of the (pxn) matrix A is p.
A corresponding exact but non-unique solution for the unknown
pseudo-source strengths Q from the underdetermined system
(14) is then
6 = AT(AAT)-1C.
Now as before on substituting for Q, the formula C(x) = u (x) Q
becomes
= uT(x)AT(AAT)~1C = mT(x)C,
i.e. ,
P
m. (x) C.
Thus the underdetermined case can be analyzed by the same least-
squares procedure. A classification of the solutions of the
systems (14) and (23) in terms of the above discussion can be
summarized as below.
19
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Case
Rank
Assumption
for A
Equation (14)
AQ = C
or
n
Ci
A = (pxn) ; Q = (nxl)
c = (pxl)
Equation (23)
_T
A m = u
or
S u (x. )m. (x) = u (x)
i=1 r i i r
A = (nxp) ; m = (pxl)
u = (nxl)
to
O
(1)
Overdetermined
p > n
n
No solution (exact)
Define (least-squares)
Has solutions
Choose
m(x) = A(ATA)~1u(x)
(2)
Underdetermined
p < n
Has solutions
Choose
= AT(AAT)-1C
No solution (exact)
Define (least-squares)
m(x) = (AAT)~1Au(x)
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As for the even-determined situation previously considered,
the interpolation function m.(x) (as given by equation (23) ,
which in expanded form is equation (18)) should be entirely in-
dependent of source-strength considerations when there are
either unaggregated or aggregated-averaged pseudo sources. Simi-
larly, the concentration interpolation formula for C(x) should
be valid independent of source-strength considerations. However,
as we are now dealing with equations that are only satisfied in a
least-squares sense of approximation, it is necessary to check
empirically to determine exactly how good such "approximate" con-
clusions may be.
21
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SECTION 5
INTERPOLATION FORMULA FOR MULTIPLE WEATHER STATES
in the preceding discussion we have been concerned with a
single (steady-state) meteorological dispersion condition, that
involves only a single set of source-receptor functions u (x),
r = l,...,n. Normally, however, multiple-source air quality
modeling will involve consideration of many different meteoro-
logical conditions or "weather states" (Calder, 1976).
Let Q be the set of all weather states w, each say as
specified by wind direction, wind speed, stability class and
mixing depth. For example, one might specify 16 wind directions,
1O wind speed classes, 6 stability classes and 4 different mix-
ing depths (cf. Calder, loc.cit.). Then 0 contains
16-1O-6-4 = 384O weather states. For each of these we shall
have an equation of the form (9) , namely
Cw(x) = S Q u^(x). (25)
r=l r r
in principle we could now proceed exactly as before and for
each weather state determine the appropriate interpolation func-
tion m(x) as a basis for the general interpolation formula
CW(x) for the concentration at an arbitrary location x for the
weather state w occurring. However, in view of the large num-
ber of weather states this could involve a prohibitive amount of
22
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computation. We thus consider the conjecture mentioned in Sec-
tion 4, and based upon the observation that for a given weather
state the single interpolation function m(x) applies for all
the (possibly widely) different source-receptor functions u (x)
in a class r. We therefore consider a partitioning of the set
of all weather states Q into a smaller number of subsets or
weather classes
1 * 2 '*"* -i'
and examine the question of whether an adequate general concen-
tration interpolation formula can be derived for CW(x) in terms
of interpolation functions m(x) calculated for each of the
smaller number of classes, rather than the large number of indi-
vidual states.
As partitioning is a combinatorial problem it may be accom-
plished in many ways, and a fuller study would be required to
compare the various possibilities. Only a very limited initial
study is presented in this report. However, for each weather
class 0, of a partitioning we shall need to examine whether a
single interpolation function m(x) can be employed (to avoid
confusion we refrain from using the index symbol k on m(x))
such that (see equations (23) and (24))
_T . . w, ,
A m (x) = u (x)
or
23
-------
E ur(xi)mi(x) ur(x; , r , ,...,n
all w in n, ,
and
AW T W " W
C (x) = in (x)C = 2 m. (x)C., all w in O.. (27)
1 1 *
Here c. denotes the measured concentration value at the samp-
ling station i (i = l,2,...,p) for the weather state w that is
occurring. it may be noted that (27) follows immediately from
(26) on using the pseudo-source strengths Q defined by the
^.
solution of (25) . For
1-L. . *
r=l
p n
= L m. (x){ E Q
i=l 1 r=l r
w
= E m. (x)CV.
The basic concept of the pseudo- source technique for the multi-
ple-weather state situation is thus to determine the interpola-
tion functions m. (x) by least- squares solution (if necessary) of
^f
To reduce to the very simple example of Section 2, we take
n = 1 = p, so u^(x) = v(x-x ) and m(x) = v(x-x )/v(x -x ).
Then (26) corresponds to v(x-x ) = m(x)v(x,-x ) and (27) to
= m(x) C-j^.
24
-------
the system (26), and then substitute into (27) to obtain the re-
quired interpolation formula. As we have noted the method can be
applied irrespective of whether the system is over-, even- or
underdetermined. In a fuller study it would be very desirable to
examine in detail the adequacy of the tentative conclusion that
the interpolation functions should be entirely independent of
source-strength considerations and only dependent on the source
positional parameters and the weather class. Because of the ap-
proximations that have been noted this conclusion may only be
valid in an approximate sense, and the degree of approximation
must be tested experimentally.
25
-------
SECTION 6
LEAST SQUARES COMPUTATION OF INTERPOLANTS
Calculation of the interpolation functions m.(x) from (26)
will first require calculation of the functions uw(x)
[r = l,2,...,n, all w in CI] . As was shown in Section 3,
these functions reduce to "true" dispersion functions when the
sources are unaggregated, i.e., n = N, so that
u^(x) = v(x) , r = 1,2,. . . ,N,
and- in the special case of the aggregated-averaged approxima-
tion, when
uw(x) = £ vw(x)/No. in class, r = l,2,...,n« N) .
r r class -*
The least squares solution of (26) will then require the consider-
ation of a set of n x s, equations for the p unknown inter-
polants m. (x) , where s, denotes the number of weather states
X K.
in the weather class Q, . in contrast, the system (27) , which
as we have just seen is actually the direct result of summing
.(26) over all the n source classes after first weighting by the
corresponding pseudo-source strengths Q , is a system of only
s, equations for the interpolants, which will therefore be much
K.
simpler for least-squares computation. in the present explora-
tory study, as discussed in more detail in Section 7,
26
-------
concentration estimates were generated using an existing EPA
source-oriented plume model of the form
N
CW(x) = E q,v(x) (28)
with an arbitrary source- strength distribution q.
(j = 1,2,...,N). For this reason also it was convenient to use
the direct output of the model to determine the interpolants
m. (x) by least squares applied directly to equations (27) and
using the model to provide estimates of both c(x) and cY. As
we have seen, in some approximate sense that has yet to be stud-
ied in detail, the interpolants should be independent of the
assumed source distribution and thus generally applicable to any
distribution of source strengths at the specified locations.
Before considering the numerical experiments, we first con-
sider some general aspects of the least- squares problem for the
w
system (27) and where the c (x) values are provided by a model
as in (28) . We first select a finite set of points x, like the
points of a rectangular grid R, at which we eventually desire to
Aw
interpolate values of the concentration C (x) for any weather
state w, using only the values C., i = l,2,...,p at the p
sampling stations x.. The number of weather states in the
weather class CI is s, . Thus for any fixed point x of R
we consider the set of linear equations
Z z.CW(x.) = CW(x), w 6 Cl . (29)
27
-------
This is a set of s, equations for the p unknowns z.,
JC X
i = r,2,...,p and we will consider the over-determined case
s, > p. Such equations may be solved in a least squares sense.
Jc
The solution z. defines a point in the interpolation function
table m.(x)(where the dependence of the functions on the weather
class Q, is not explicit in the notation).
The amount of computational effort is illustrated by return-
ing to the Q above, containing 384O weather states. We let £1
contain all states with a fixed wind direction. Then Q is
split into 16 subgroups, each containing 24O states. Assume now
that we have 1O sampling stations and that the interpolation grid
contains 4OO points. Hence for each point x of the grid we
must solve 16 over-determined systems of equations (29) with 10
variables z. and 24O equations. That is, for each x of R
we need to store 1O x 16 = 160 coefficients m. (x) , or a total
of 64,OOO numbers for the entire grid. This computational effort
is formidable although it only needs to be done once. Then when
concentration values are available from actual measurements C.
at a limited number of sampling stations during weather state w,
the concentration field over the entire grid may be very simply
calculated in terms of just these p sampled values from (27).
The important feature of the method is that it would permit in-
terpolation on a real-time basis, since the functions m.(x)
could be computed in advance.
Finally, we may note in the above example that if there had
been N = 2OO unaggregated sources in the region and the system
28
-------
(26) has been used for the least-squares determination of the
interpolants, then this would still have involved the lo vari-
ables z., but now 24O x 2OO = 48,OOO equations. If the 200
real sources had been aggregated into 2O pseudo-source classes
(so n = 20) with dispersion functions u (x) corresponding to
the aggregated-averaged approximation as above, then only
24o x 2O = 4800 equations would be involved for the 24o weather
states in the weather class. The storage requirements for the
interpolants m.(x) are, of course, the same as previously in
both cases, i.e., for each x of R we need 16o coefficients,
or a total of 64,OOO for the entire grid for the one weather
class £1 .
in the least-squares computations performed in Sections 7
and 8, a linear regression package, IAREG, of the Graduate
School of Industrial Administration Program Library at Carnegie-
Mellon University was used. These statistical programs are
available both for the IBM 36o/Time Sharing System and the
UNIVAC 11O8 at the Carnegie-Mellon Computation Center. IAREG
is a FORTRAN program developed in 1974 by Bouwman and Prescott
at Carnegie-Mellon. In addition to providing the standard good-
ness of fit measures such as the means and standard deviations
of the estimated coefficients, it prints the correlation matrix
of the estimated coefficients and also plots the normal proba-
bilities of the residuals. The package also performs adaptive
regression and multiple-regression.
29
-------
SECTION 7
A NUMERICAL EXPERIMENT IN CONCENTRATION INTERPOLATION
in lieu of real measurement data a preliminary numerical
experiment was performed using synthetic or simulated air quality
data generated by a current EPA multiple-source model (EPA code-
DBT51). An interactive version of this is available on the EPA
UNAMAP (Users' Network for Applied Modeling of Air Pollution)
computer system. This model is a point-source model that pro-
duces hourly concentration values for up to 3O receptors whose
locations are specified, and for up to 25 sources. It is based
on a Gaussian plume dispersion formulation. inputs to the pro-
gram consist of the number of sources to be considered, and for
each the emission rate, physical height of emission, stack gas
temperature, volume flow or stack gas velocity and diameter, and
the coordinates of the source location. The numbers of receptors,
their coordinates and heights above ground level, are also re-
quired. For each hour the meteorological or weather state is
specified in terms of wind direction, wind speed, stability class,
mixing height (and ambient air temperature which is required in
the estimation of plume rise) .
The above multiple-source code was applied to generate syn-
thetic concentration data for a hypothetical air quality region
containing 25 major point sources, and under the assumption of
30
-------
zero background concentration. Although hypothetical, the source
inventory reflects a realistic spatial distribution of sources
and stack characteristics. Likewise, the locations of 3 hypo-
thetical measurement stations were selected in an ad hoc fashion.
The source inventory is given in Table 1, and the DBT51 code then
provided synthetic concentration estimates corresponding to this
inventory by the superposition of 25 "Gaussian" plumes in the
form of equation (28) , namely
25
CW(x) = L q.v(x).
j=l D D
Figure 1 shows:
(a) the approximate locations of the individual sources on
a 15 x 15 grid, where the grid size is 1 mile x 1 mile.
(b) a sample synthetic concentration distribution corres-
ponding to a single weather state (say as below, w,,) for
which the wind direction was SW, wind speed 5 m/sec, sta-
bility was Pasquill category D, mixing height 1219m, and
the ambient air temperature was taken as 284°K.
(c) location of 3 hypothetical sampling stations x^ = [84],
x2 = [127] and x3 = [224].
(d) 1O other typical locations at which concentration esti-
mates will be compared, viz x = [111], [113], [13o], [143],
[161], [164], [175], [178], [2O6], [209].
Synthetic concentration data were computed by the EPA code
for all the weather states w in a hypothetical weather class
31
-------
TABLE 1. HYPOTHETICAL SOURCE INVENTORY
SOURCES
NO
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Q
(G/SEC)
191.
47.
21.
14.
7.
59.
87.
25.
101.
41.
222.
20.
20.
20.
20.
24.
67.
66.
63.
6.
36.
28.
8.
172.
171.
1
7
1
2
O
2
2
3
0
6
7
1
1
1
0
7
5
7
7
3
2
8
4
4
3
HP
(M)
61.0
63.6
30.5
38.1
38.1
21.9
61.0
36.6
36.6
18.0
35.7
45.7
50.3
35.1
34.7
3O.O
76.3
82.0
113.0
31.0
5O.O
5O.O
31.0
42.6
42.6
TS VS
(DEC K) (H/SEC)
6OO. O 6.1
6OO. O 4.8
811.0 29.2
727. O 9.2
727.0 7.0
616.0 4.3
616.0 5.2
477.0 11.9
477.0 16.0
727.0 9.0
477.0 5.7
727. O 2.4
'727.0 1.6
727.0 1.5
727.0 1.6
727.0 9.0
473.0 10.7
603.0 12.9
546.0 9.3
46O.O 5.0
46O.O 7.0
46O.O 7.O
46O.O 5.O
616.0 13.4
616.0 16.1
D
(M)
2.6
2.9
0.9
1.7
2.1
2.0
2.1
2.7
2.0
2.6
2.4
1.9
1.5
1.6
1.5
2.2
3.0
4.4
5.2
1.6
2.2
2.5
1.6
4.6
3.7
VF
(M**3/SEC)
32.
31.
18.
20.
25.
12.
18.
66.
49.
47.
26.
6.
2.
2.
2.
34.
78.
196.
196.
10.
26.
34.
10.
219.
169.
2
9
6
3
9
9
7
6
3
5
6
9
8
9
8
2
2
1
7
1
6
4
1
8
4
X
(KM)
9
9
9
9
9
9
9
8
8
8
8
8
8
8
8
9
5
5
4
8
8
11
6
14
14
.190
.190
.190
.190
.190
.190
.190
.520
.520
.520
.050
.O5O
.050
.050
.050
.190
.770
o620
.6OO
.230
.750
.240
.140
.330
.330
Y
(KM)
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
6
10
9
9
8
5
4
8
6
6
.300
.300
. 3OO
.300
.3OO
.300
.300
.840
.840
.840
.680
.680
.680
.680
.680
.3OO
.810
.820
.500
.870
.880
.560
.780
.200
.2OO
Q
HP
TS
VS
D
VF
X
Y
emission rate
physical stack
height
stack gas temperature
stack gas exit
velocity
inside stack diameter
stack gas volumetric
flow rate
x- coordinate of stack
y- coordinate of stack
32
-------
15
30 45 60 75 90 105
105
.
.
.
.
1
16
7
1
189
m
120
.
.
1
3
23
7
3
(34p
2
(20l)
135
*
1
6
25
7
CD
319
46
[343J
.
150
1
9
26
9
16
269
(Hi?
^-.ii_ --- ^
315
1
24
165
11
(26;
12
27
(226)
181
263
5
22
1
180
25
15
(&)
193
201
(229)
17
18
6
0
195
18
45
169
202
189
31
15
19
1
^
210
51
(152}
194
168
(Zj)
16
31
2
.
^
225
139
fl83|
153
50
19
38
6
.
.
^
U)
U)
16 31 46 61
76
91
106 121 136 151 166 181 196 211
= approximate location of individual sources
s sampling stations (3) x^ = [84] , x2 = [127]
= [224]
s 10 additional typical points x = [111], [113], [130], [143], [161], [164],
[175], [178], [206], [209]
Figure 1: Location of sources, sampling stations and 10 additional interpolation
positions. Entries are concentration values (/ig/m ) estimated by EPA
code, for one weather state w.
-------
fl defined to comprise the same wind direction, mixing height
(and ambient air temperature) as for Figure 1, but with
wind speeds of 3, 5 and 7 m/sec
stability categories B, D, and E.
With these synthetic data, the 3 interpolation functions m.(x),
(i = 1,2,3), were then computed through least squares solution of
the over-determined system of equations
W
C(x) = S m. (x)C we,
1 X k
where the "measurement" or "sampling" locations were as in Fig.
l(c), and the locations x were the 1O typical values selected
in Fig. 1(d). The values of m.(x) are given in Table 2.
TABLE 2.. COMPUTED INTERPOLATION FUNCTIONS m. (x)
FOR THE WEATHER CLASS 0^. AS DEFINED IN THE TEXT
AND FOR THE 3 SELECTED SAMPLING LOCATIONS, AND 10
HYPOTHETICAL INTERPOLATION LOCATIONS
X
m1(x)
m2(x)
m3 (x)
[111]
[113]
[130]
[143]
[161]
[164]
[175]
[178]
[206]
[209]
3.8520
.0637
.3159
- .4881
- .1281
1.2610
.3 6OO
.3355
.3563
.1367
.8170
.8866
.0119
.5729
-.O1O4
.O39O
.0557
.0837
.1097
-.1397
- .5307
.1718
.O04O
.6641
1.2720
.0756
- .03O3
.0265
.0033
1 . 1OOO
34
-------
As a numerical test of the suggested interpolation proce-
dure, the computed interpolation function values were then used
to estimate concentration values for certain selected weather
states w in the weather class fl , at the 10 selected loca-
tions, using the interpolation formula
m.
where cV here denotes the "measured" values at the 3_ sampling
i
locations. The results are shown in Table 3 for two weather
states w.. , Wp differentiated as follows:
w1 - wind speed 3 m/sec, stability category 5
w,, - wind, speed 5 m/sec, stability category 4,
but with all other weather conditions (wind direction, mixing
height and air temperature being the same) . in Table 3 the in-
terpolated values C (x) are compared with the synthetic values
C (x) generated by the EPA code. For both weather states a very
*
close agreement is obtained, covering a wide range of concentra-
tion values. This is a very encouraging result and particularly
so in relation to the rather complex structure of the typical
concentration field (shown in Figure 1 for weather state w_) and
in an expanded numerical investigation the generally accepted
statistical measures for the performance of the regressions
would be reported, see for example Draper and Smith (1966)8
Such measures are among the output of any of the standard com-
puter statistical packages, such as the IAREG Statistical
Package, loc. cit.
35
-------
the fact that the interpolation estimates only utilize 3 widely
separated measurement locations. From examination of Figure 1,
it is fairly obvious that in the absence of the interpolation
scheme that has been developed and with only measurements at the
3 sampling stations, it would be virtually impossible to predict
with any accuracy the fine structure of the concentration field.
TABLE 3_. COMPARISON FOR THE SELECTED CONCENTRATION INTERPOLATION
LOCATIONS, BETWEEN INTERPOLATED VALUES CW(x) BASED ON ONLY 3
SAMPLING STATIONS AND THE SYNTHETIC VALUES CW(x) GENERATED BY
THE EPA MODEL. COMPARISON IS FOR WEATHER STATES W-j^ AND W2,
BOTH ASSUMED IN THE SAME WEATHER CLASS C IN UNITS
X
w
C X(x)
w.
C^x)
w
C 2(x)
w^
62(x)
[111]
[113]
[130]
[143]
[161]
[164]
[175]
[178]
[206]
[209]
63.78
339.20
4.24
447.37
508.65
44.32
3.86
33.59
32.25
401.70
54.08
356.15
5.64
451.59
506.53
43.02
6.10
37.93
37.06
395.83
200.77
340.69
7.00
314.47
225.27
25.63
16.74
36.85
42.11
151.40
196.09
335.43
5.84
315.71
227.61
27.55
14.76
34.62
39.38
152.83
36
-------
SECTION 8
THE SAME EXAMPLE WITH AGGREGATION OF PSEUDO-SOURCES
As a simple example the 25 "real" sources of Table 1 in the
previous section were subdivided rather crudely into only 4 clas-
ses based on the magnitudes of the individual strengths, as shown
in Table 4. A more detailed analysis would produce more pseudo-
source classes with smaller within-class variation.
TABLE 4. SELECTION OF PSEUDO-SOURCE
CLASSES BY LEVEL OF EMISSIONS
Range of
Individual Source
Pseudo- Source Sources in Class Strengths
1
2
3
4
4,5,20
,23
3,8,12-16,21,22
2,6,7,
1,9,11
10,17-19
,24,25
6-
20-
41-
101-
in Class
14
36
68
222
Number of
Sources
4
9
7
5
t, . -;.;
1^ v^-A .. '. .-.
'-* *"--1i-, ' '.. - .- -.
:v '=₯^- "-For,-the experiment the aggregated^ aver aged approximation was
employed, corresponding to the "pure" dispersion function
u^(x) = 2 vW(x)/No. in class; r = 1,2,3,4.
r class ^
These were computed for r = 1,2,3,4 and each of the 9 weather
37
-------
states w in the weather class QL defined in Section 7. As
before, the EPA model was used to translate the source-positional
and meteorological information into values of u (x), for the
same three sampling locations x , x_ and x., and the same ten
receptor locations as in Figure 1. However, in the present case
the system of equations (26) (in contrast to the system (27) for
Section 7) were solved by least-squares for the three interpolant
functions m.(x), i = 1,2,3. These values are given in Table 5.
TABLE 5. COMPUTED INTERPOLATION FUNCTIONS m.(x) FOR
THE WEATHER CLASS 0^., AND FOR THE 3 SELECTED SAMPLING
LOCATIONS, 4 SELECTED PSEUDO-SOURCE CLASSES, AND 1O
HYPOTHETICAL INTERPOLATION LOCATIONS
X
[111]
[113]
[130]
[143]
[161]
[164]
[175]
[178]
[206]
[209]
m^x)
1.7800
.1086
- .1512
- .3752
- .02555
.1736
.1321
- .1658
.08797
.0311
m2(x)
.6172
-.4374
.2402
.9025
-.3024
.1O8O
.0340
.3158
.08334
-.1820
m3(x)
- .3O40
1.9360
- .1580
.2945
1.7920
- .05258
.003239
- .07981
.03493
1.2740
Then, as in Section 7 (see Table 3), the computed interpola-
tion functions of Table 5 were used to estimate the concentration
values for certain selected weather states in the weather class
fl at the 1O selected locations using the formula
38
-------
m.(x)CW,
1
where c here denotes the "measured" values at the 3 sampling
locations. Analogous to Table 3, the interpolated values 6w(x)
are compared with the synthetic values C (x) generated by the
EPA model in Table 6.
TABLE 6_. COMPARISON FOR THE SELECTED CONCENTRATION
INTERPOLATION LOCATIONS, BETWEEN INTERPOLATED VALUES
6W(x) (BASED ON ONLY 3 SAMPLING STATIONS AND 4 PSEUDO-
SOURCES) AND THE SYNTHETIC VALUES CW(x) GENERATED BY
THE EPA MODEL. COMPARISON IS FOR WEATHER STATES w.
AND w_ BELONGING TO WEATHER CLASS ii, (WITH UNITS
3
/igm/m ) . -
X
[111]
[113]
[130]
[143]
[161]
[164]
[175]
[178]
[206]
[209]
w
C (x)
63.78
339.20
4.24
447.37
508.65
44.32
3.86
33.59
32.35
401.70
w
G (x)
79.08
634.53
14.39
410.25
620.46
14.01
9.79
70.22
41.05
451.82
w
C (x)
200.77
340.69
7.OO
314.47
225.27
25.63
16.74
36.85
42.11
151.40
AW9
^ (X)
162.08
202.89
53.08
361.78
222.51
28.02
11.50
93.14
35.22
169.67
Some of the comparisons of Table 6, that is based on the
aggregated-averaged approximation, are good and some are poor,
although overall the agreement may be considered very encouraging
in view of the crude initial aggregation of the sources into 4
classes with unduly large variations of source strengths within
39
-------
each class. It is perhaps even surprising that without any know-
ledge of the real source-strengths (since for the aggregated-
averaged approximation the functions u (x) are "pure" dispersion
functions), except a very crude grouping by general level of
emission, the numerical features of the concentration field are
so well produced by the interpolation procedure. of course, the
errors arising in the aggregation approximation that was used can
be avoided if an unaggregated pseudo-source distribution is em-
ployed. As we have seen, this will involve more computation to
produce the required interpolation functions for the real world,
where the number of sources may be quite large, although still of
a magnitude well within the capabilities of modern computers. in
any case the interpolation functions may be precalculated for
subsequent use in a real-time air quality modeling situation.
40
-------
SECTION 9
RECOMMENDATIONS
The results reported herein relate to a small-scale and
limited exploratory study. They point to the possibility for
innovative and optimal interpolation procedures for real-time
estimation of the fine structure of the air quality distribution
over a region, in terms of sparse measurement data. The first
results are very encouraging and are suggestive of considerable
operational value in air quality management. Further effort
should be directed towards analysis of several theoretical ques-
tions that it was not possible to resolve completely in a limited
study. These include an analysis of the degree of independence
of the interpolation functions from the spatial distribution of
pollutant source-strengths (in contrast to purely source-position-
al parameters) together with an analysis of the errors of inter-
polation and the development of optimal operational formulae. At
the same time the techniques developed should be tested against
real air pollution data that is becoming available from appropri-
ate field programs, such as the EPA Regional Air Pollution Study.
41
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REFERENCES
Brier, G. W. , 1973. Validity of the Air Quality Display Model
Calibration Procedure, Environmental Protection Agency, EPA-R4-
73-017.
Calder, K. L. , et.al., 1975. Empirical Techniques for Analyzing
Air Quality and Meteorological Data: Part II. Feasibility of a
Source-oriented Empirical Air Quality Simulation Model, Environ-
mental Protection Agency, ESRL-RTP-O54, Series No. 4.
Calder, K. L. , 1976. Multiple-Source Plume Models for Urban Air
Pollution - Their General Structure, NATO/CCMS Panel on Modeling,
Documentation for Practical Demonstration of Urban Air Quality
Simulation Models.
ffahlquist, G. and Bjorck, A., 1974. Numerical Methods (Transl.
by N. Anderson), Prentice-Hall, Englewood Cliffs, N. J.
Draper, N. R. and Smith, H., 1966. Applied Regression Analysis,
J. Wiley and Sons, New York.
Heimback, J. A. and Sasaki, Y., 1975. A Variational Technique
for Mesoscale Objective Analysis of Air Pollution, J. Appl.
Meteorology 14, 194-2O3.
instructions for Using the IA Regression Package, 1974. Graduate
School of industrial Administration, Carnegie-Mellon University,
Pittsburgh, Pa. 15213.
Hrenko, J. M. and Turner, D. B., 1975. An Efficient Gaussian-
Plume Multiple-Source Air Quality Algorithm, #75-o4.3, 68th An-
nual Meeting of APCA, Boston, Massachusetts, June 15-2O.
Lanczos, C., 1965. Applied Analysis, Prentice-Hall, Englewood
Cliffs, New jersey.
42
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/4-76-058
2.
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
NUMERICAL OPTIMIZATION TECHNIQUES IN AIR QUALITY
MODELING. Objective Interpolation Formulae for the
Spatial Distribution of Pollutant Concentration
5. REPORT DATE
December 1976
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT NO.
S-A Gustafson, K. 0. Kortanek and J. R. Sweigart
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Carnegie-Mellon University
Pittsburgh, Pennsylvania 15213
10. PROG'RAM ELEMENT NO.
1AA603 (1AA009)
11. CONTRACT/GRANT NO.
R 803632
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park. North Carolina 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final 4/75-4/76
14. SPONSORING AGENCY CODE
EPA-ORD
15. SUPPLEMENTARY NOTES
16. ABSTRACT
A technique is proposed for objective.interpolation of the, air quality distri-
bution over a region in terms of sparse measurement data. Empirical information
provided by the latter is effectively combined with knowledge of atmospheric
dispersion functions of the type commonly used in source-oriented air quality models,
to provide improved estimates of the concentration distribution over an extended
region. However, the technique is not primarily source-oriented since, in contrast
to the real source distribution of a source-oriented model, it utilizes fictitious
or pseudo-sources that are estimated in terms of the measured air quality data. This
involves the use of interpolation functions that are computed using numerical
optimization techniques based on the method of least squares. Due to the large number
of different "weather" states that affect the atmospheric dispersion of pollution,
considerable computation is required, although the bulk of this can be done in
advance, so that the final interpolation from the measured values only requires very
simple calculation. Thus the proposed method has the potential for application on a
real-time basis.
In addition to the mathematical formulation of the problem, this preliminary
study includes some numerical experiments, using a current multiple-source EPA air
quality model, to illustrate the technique5
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
COS AT I Field/Group
Air pollution
Atmospheric composition
Meteorological data
Numerical analysis
Interpolation
Least squares method
Empirical equations
* Mathematical
models
13B
04A
04B
I2A
18. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (This Report)
UNCLASSIFIED
21. NO. OF PAGES
49
20. SECURITY CLASS (Thispage)
UNCLASSIFIED
22. PRICE
EPA Form 2220-1 (9-73)
43
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