HUD
EPA
United States
Department of Housing and
Urban Department
Office of Policy Development and Research
Office of Community Planning and Development
Washington DC 20410
United States
Environmental Protection
Agency
Environmental Monitoring and Support EPA-600/7-78-106
Laboratory June 1978
Research Triangle Park NC 27711
Research and Development
TheGEOMET
Indoor-Outdoor
Air Pollution
Model
Interagency
Energy/Environment
R&D Program
Report
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THE GEOMET INDOOR-OUTDOOR AIR POLLUTION MODEL
by
Demetn'os J. Moschandreas, Ph.D.
John W.C. Stark, M.S.
GEOMET, Incorporated
15 Firstfield Road
Gaithersburg, Maryland 20760
EPA Contract No. 68-02-2294
Project Officer
Steven M. Bromberg
Environmental Monitoring and Support Laboratory
Environmental Research Center
U.S. Environmental Protection Agency
Research Triangle Park, N.C. 27711
Prepared for
U.S. Department of Housing and Urban Development
Office of Policy Development and Research
Office of Community Planning and Development
Washington, D.C. 20410
ENVIRONMENTAL MONITORING AND SUPPORT LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, N.C. 27711
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DISCLAIMER
This report has been reviewed by the Environmental Monitoring and
Support Laboratory, U.S. Environmental Protection Agency, the U.S.
Department of Housing and Urban Development, and non-governmental personnel,
and approved for publication. Approval does not signify the contents
necessarily reflect the views and policies of the U.S. Environmental
Protection Agency, or the U.S. Department of Housing and Urban Development,
nor does the mention of trade names or commercial products constitute
endorsement or recommendation for use. The views, conclusions and
recommendations in this report are those of the contractor, who is solely
responsible for the accuracy and completeness of all information-and data
presented herein.
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ABSTRACT
This'report documents the formulation of the GEOMET Indoor-Outdoor
Air Pollution (GIOAP) model. The model estimates indoor air pollutant
concentrations as a function of outdoor pollutant levels, indoor pollutant
generation sources rates, pollutant chemical decay rates, and air exchange
rates. Topics discussed include basic principles, model formulation,
parameter estimation, model statistical validation, and model sensitivity
to perturbations of the input parameters.
The numerical estimates obtained from the GIOAP simulations have been
validated with observed hourly pollutant concentrations obtained from an
18-mo residential air quality sampling program. Statistically, the model
values of carbon monoxide are within 10% of the observed values in the
concentration range interval which includes 85% of all hourly measurements;
similarly, for nitric oxide the predicted values are within 15% of the
observed; for nitrogen dioxide the difference between the predicted value
and the ideal condition of exactly estimating the corresponding observed
value is 16% for 85% of the observed hourly concentrations. Also, the
GIOAP model predicts within 25% for nonmethane hydrocarbons and within
8% for carbon dioxide. Thus statistically the GIOAP model estimations
are within 25% of the ideal condition (estimated and observed values coin-
cide) for 85% of the observed values for CO, NO, N02, NMHC, and C02- The
model has not been validated against sulfur dioxide owing to the very low
values measured both indoors and outdoors. The predetermined validation
criteria were not satisfied by the ozone model estimations; however, the
calculated values were judged adequate because for about 85% of the
observed values the predicted concentrations were within 2 ppb.
Sensitivity studies on the GIOAP model parameters indicate that errors
in the estimation of the initial condition and the volume of the structure
dissipate with time. Errors in estimating the air exchange rate, the indoor
generation strength, and the indoor chemical decay rate are more significant.
Sensitivity coefficients have been formulated for all input parameters.
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The transient term is a unique feature of the GIOAP model; the impact
of this term is substantial for stable pollutants but insignificant for
ozone, which is a high reactive pollutant. The GIOAP Model, validated
against a variety of measured data, satisfactorily predicts indoor pollu-
tant concentrations, and it can be used to determine residential pollution
levels under any conditions.
-IV-
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CONTENTS
Abstract iii
Figures vi
Tables vii
1. Introduction. 1
2. Mathematical Formulation 3
General Principles 3
The GEOMET Indoor-Outdoor Air Pollution (GIOAP) Model. . . 4
3. Model Validation 9
Introduction 9
Parameter Estimation Procedure 10
Statistical Studies 18
4. Conclusions 55
Bibliography 58
Appendices
A. Derivation of the GIOAP Model Sensitivity Coefficients 59
B. Numerical Sensitivity Analysis Examples for the GIOAP Model . . 65
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FIGURES
Number Page
1 Graphical illustration for the three cases of constraints
on interval source rate 16
2 Estimated vs. observed pollutant concentrations for
7 consecutive days 20
3 Estimated vs. observed pollutant concentrations for
7 consecutive days 21
4 Scatter diagram with r = 0.96 26
5 Scatter diagram with r = 0.82 27
6 Scatter diagram with r = 0.72 28
7 Scatter diagram with r = 0.62. 29
8 Nominal values 51
9 Comparison of nominal values obtained by perturbing C-jn. . . 52
10 Comparison of nominal values with values obtained by
perturbing S 53
11 Comparison of nominal values with values obtained by
perturbing v 54
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TABLES
Number . , Page
1 Decay Factors Used in the GEOMET Indoor Air Pollution
Study 7
2 Air Exchange Rates . 18
3 Statistical Data Summary 24
4 Statistical Data Summary for Carbon Monoxide 31
5 Statistical Data Summary for Nitric Oxide 33
6 Statistical Data Summary for Nitrogen Dioxide 35
7 Statistical Data Summary for Sulfur Dioxide 36
8 S0£ Frequency Distribution 37
9 Negative C02 Interference on S02 Levels 38
10 Statistical Data Summary for Nonmethane Hydrocarbons .... 40
11 Statistical Data Summary for Methane 41
12 Statistical Data Summary for Carbon Dioxide 43
13 Nominal Conditions Used in the Sensitivity Study Examples. . 51
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SECTION 1
INTRODUCTION
Numerical simulation is considered among the most practical tools in
estimating indoor air pollutant levels as a function of the outdoor pollutant
levels plus other parameters. While its potential use has been realized by
many scientists in the field, its application has been rather restricted,
because the indoor-outdoor numerical models require inputs not readily avail-
able to the researcher. The extensive field program of the EPA-GEOMET project
on the "Indoor Air Pollution Assessment Control and Health Effects" has pro-
vided a large data base. Using this information we have formulated the GEOMET
Indoor-Outdoor Air Pollution (GIOAP) model.
The objective of the GIOAP model is to predict the indoor air pollu-
tion levels by simulating a series of complex interactions involving out-
door pollutant levels, structural characteristics of the examined dwellings,
and behavioral patterns of the inhabitants.
The motivation for the formulation and application of the numerical model
arises from the scientific recognition that measures to conserve energy within
buildings, through the introduction of new energy transfer systems and the
reduction of building ventilation rates, will result in changes in the indoor
air quality characteristics. These changes may affect air quality either
adversely or beneficially. Simulation of a large variety of indoor conditions
will quantify these effects. In addition, the validated GIOAP model can be
coordinated with an epidemiological study to determine the health effects of
indoor air pollution.
The approach followed in the generation of the GIOAP model is a two-
step procedure: (a) mathematical formulation, and (b) model validation. Each
of these steps constitutes a section of this report and will be discussed in
detail in the balance of this document.
The impact of a validated indoor-outdoor air pollution model will be
substantial in identifying the optimum scenario that meets the national
policy toward energy conservation measures in residential buildings without
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endangering the health of the segment of the population that spends a large
portion of its time in the indoor residential environment. The last section
of the document discusses the conclusions and impact of the 6IOAP model.
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SECTION 2
MATHEMATICAL FORMULATION
GENFRAL PRINCIPLES
The numerical simulation model formulated by GEOMET for assessing air
quality in the indoor residential environment follows the general principles
of a mass balance equation. In the sense that the GIOAP model specifically
addresses residential environments (detached dwellings, row buildings, mobile
homes, and apartments), it is different from the general models which include
terms that are not applicable in the nonworkplace environment.
Air pollution in an enclosure may be of either outdoor or indoor origin,
or both. If of outdoor origin, it enters through infiltration and ventila-
tion. If of indoor origin, it is generated from pollutant sources within the
enclosure. Regardless of source, air pollutants diffuse in the enclosure.
They are removed over varying periods of time by exfiltration and ventilation
to the outdoors and/or through indoor decay processes.
Air infiltration is defined as the change of air within a structure
without the interference of the inhabitants. Thus the ambient air entering
an enclosure through cracks in its walls is infiltrated air which, whether
clean or contaminated, influences the indoor pollution levels. Ambient pol-
lutants may also be introduced indoors through the ventilation process, which
may be defined as air changes induced by the occupants of an enclosure; this
can be natural ventilation through closing and opening of doors and windows,
or can be forced ventilation through the operation of attic fans and air
conditioning/heating systems. Air exfiltration is the opposite of air
infiltration; indoor air leaves an enclosure through structural cracks.
Exhaust ventilation moves air from indoors to outdoors through the vents of
a forced circulation system as well as through door and window openings.
Pollutants are introduced into the indoor environment by means of sources
such as fireplaces, stoves, smoking, and cleaning devices. Finally, pol-
lutants may be removed from the indoor air environment through indoor decay
processes such as chemical transformation, settling, and absorption and adsorp-
tion by walls and furnishings (collectively termed pollutant sinks), and by
filtering procedures in the makeup air or in the recirculated air.
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To characterize time-dependent aspects of pollutant behavior, it is useful
to deal with rates of change of air pollution within enclosures, rather than
simply with absolute amounts of pollution. Mass balance principles require
that the rate of change of an air pollutant quantity within an enclosure
equals the sum of the rates of all pollutant introduction and removal processes
that operate upon the enclosure.
THE GEOMET INDOOR-OUTDOOR AIR POLLUTION (GIOAP) MODEL
The GIOAP model illustrates the above general principles by specifically
simulating procedures present in residential environments and is based upon
the following mass balance equation:
dC.
V- = VvC + S - VvCin - VDCin W
where
C. = the indoor pollutant concentration, mass/volume
C t = the outdoor pollutant concentration, mass/volume
V = the volume of the building, volume
v = the air exchange rate of the building, air exchange/time
S = the indoor source strength rate (rate of indoor pollutant
emission), mass/time
D = the decay factor, time" .
The term on the left-hand of Equation (1) denotes the rate of change of
the indoor pollutant mass. The first two terms on the right-hand side of the
equation represent the rate by which the pollutant is introduced indoors, by
infiltration of air (VvC t) from outdoors, and by indoor pollutant generation
due to indoor sources (S). The last two terms represent the pollutant removal
rate due to exfiltration of indoor air (VvC.), and due to indoor sinks, such
as decay processes (VDCin). The factor v, the air exchange rate, appearing
in the first and third terms of the right-hand side in Equation (1) is a total
rate; it is the sum of the infiltration, exhaust, and ventilation rates.
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A series of approximations are necessary before the GIOAP model can be
applied to estimate the indoor air pollution levels. Over short time inter
vals simulated, the outdoor pollutant concentration is approximated by a
straight line (an approach used by Shair and Heitner(1) ). The line is
given by:
Cout = "out1 + bout
where mQUt is the slope, t is the time, and bQut is the y-intercept. It is
further assumed that the parameters v, S, and D are constant during the time
interval that is being modeled.
Thus the model equation becomes:
dCin
u — = Vu(m t + b } - VvC + S - VDC
v H+ Vv^mn,,tl * Dout' VvLin ^ vuuin
5 Cin
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The subject of relating the outdoor and indoor levels is relatively
recent, and research emphasis has been placed on field measurements of con-
taminant levels rather than on the development of numerical models. Several
scientists have attempted to formulate models employing relationships similar
to the one expressed by Equation (3); Milly,(2) Calder,(3) Turk,(4) Hunt
et al.,(5) Shair and Heitner,*1) and others have used more or less complex
versions of Equation (3). The GIOAP model is specifically designed to simu-
late residential conditions. The assumptions concerning elements of the
right-hand side of Equation (3) (S, v, and D are constant over the time
period being modeled, and C t can be approximated by a straight line over
the time period being modeled) are consistent with those made for other
models. Previous studies have empahsized and simulated steady-state condi-
tions; however, the GIOAP approach includes the transient portion of the
solution to Equation (3). As a result, it is possible to model both short-
and long-time intervals.
When Equation (4) is used in this study to model indoor air pollutant
concentration levels, several additional assumptions are required. These
assumptions are as follows:
1. The air exchange rate (v) remains constant for at least 1 h.
2. The internal pollutant source rate (S) remains constant for
at least 1 h.
3. The indoor pollutant removal procedure is modeled as a first-
order decay term with the decay factor D = InZ/t^, where \>2
is the half-life of the pollutant considered. For stable
pollutants with long half-lives, D is approximated by zero.
A list of the decay factors used in this study is given in
Table 1.
The GIOAP model is capable of simulating any time interval, because
the principles involved do not constrain this aspect. However, the time
unit of the generally available ambient pollution data has led us to specify
1 h as the time resolution of the model. Finally, 1 h is the interval
that is most appropriate for our purposes.
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TABLE 1. DECAY FACTORS USED IN THE GEOMET
INDOOR AIR POLLUTION STUDY
Pollutant Decay factor (per hour)
CO 0.00
SO2 1.04
NO 0.00
NO2 1.39
03 34.66
CH4 0.00
THC 0.00
CO 2 0.00
THC-CH4 0.00
In order to model the indoor air pollutant behavior over the time inter-
val [tn, tf], the interval must be decomposed into the set of subintervals
(t0, t,], ... (V,, tn] where tn = tf, and t,- - t^-, = 1 h. As a result,
Equation (3) becomes
• [C1-l - -1 (ujy '1-1 - (0^7) (^ * T - Bjft
S1 "Vl
"I B *1 (5)
where
C. = the indoor pollutant concentration level at time t^,
1 i = 1, ..., n
C t(t) = the outdoor pollutant concentration level at time t
mi = Ccn,,t^i) - cnnt^i i)]/(*! - t, ,), i = 1, .... n
i out i out i-i i ii
bi=
S. = internal source rate over the interval (t- ,, t.],
T ^ i « I ~ I I
i - 1, ..., n
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v. = air exchange rate over the interval (t,. n, t-L
1 n-] 1
n
V = volume of the building
D- = decay factor over the interval (t.. -, , t.],
1 1 = 1, ..., n. 1~1' 1
-8-
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SECTION 3
MODEL VALIDATION
INTRODUCTION
The problem of estimating indoor air pollution levels involves both
physical and behavioral parameters: the outdoor levels vary as a function
of the local meteorology and other factors, while the indoor source strengths
(indoor pollutant generation rates) and air exchange rates depend on the
meteorology and the activity of the occupants. The combination of all inputs
results in complex conditions that are rarely repeated or are very expensive
to duplicate in the laboratory. Numerical models enable scientists to simu-
late these complex conditions, to stage specific incidents, and most
importantly, to estimate values for the indoor pollutant concentrations.
Two essential stages determine the predictive capability of the GIOAP
model:
1. Initial model validity. Do the predicted concentrations
reflect observed data?
2. Model sensitivity. How do the predicted concentrations
change in relation to changes in input parameter values?
An intrinsic element of these two stages is the ability to demonstrate the
validity of the model using "best" estimates of the input model parameters.
We have formulated a parameter estimation procedure that enables us to esti-
mate the values of indoor source strengths and air exchange rates from the
raw outdoor and indoor pollutant data. Recurring modes of pollution behavior,
episodes, are of extreme importance in the Indoor Air Pollution EPA-GEOMET
project. Numerically, a new episode is defined each time a new initial con-
dition is introduced. It is our objective to associate each episode with
stratified levels of indoor activity so that in the future we can estimate
the indoor pollutant concentrations from outdoor pollution concentrations
and indoor activity levels. The validity of these estimates will be studied
in the balance of this section.
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A three-step evaluation design will be followed in the assessment of
the GIOAP model: we begin by estimating "best" values for input parameters,
continue with statistical validation studies which include tables and input/
output graphs, and conclude with a section on parameter sensitivity. This
design takes advantage of the extensive data base available to this project.
PARAMETER ESTIMATION PROCEDURE
In order to estimate the indoor pollution levels using the GIOAP numeri-
cal model, all parameters associated with the model must be given numerical
values. The monitoring data from the field studies of the indoor air pollu-
tion project constitutes a unique source of information for assigning numerical
values to the relevant parameters. Some parameter values, such as initial
indoor concentration and the volume of the structure, are easily determined;
others, such as the air exchange rate (v) of the building investigated and the
internal pollutant generation source strength (S), are more difficult to obtain.
The validity of the GIOAP model obviously depends on the values given to
these difficult-to-quantify parameters. Since the model is to be validated
under "best" conditions, we must obtain the best possible values for v and S;
the methodology used to obtain these values is the Parameter Estimation Pro-
cedure described in this section.
Theoretical Approach
Recall Equation (4):
C1n0 - "out
w
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Also, consider the following function of S and v:
n „
f(S,v) = £ [C.(S,v) - CM.r (6)
i=l
where
n = number of points
CM. = i measured value; i = 1, ..., n
th
C-(S,v) = i computed value via Equation (4) correspondinq
to CM..
The fundamental problem in parameter estimation is to find value(s) of
v and S that minimize Equation (6). Two points must be made: (a) in the
case of indoor air pollution studies, the parameters v and S are constrained
to lie within certain intervals specified by the nature of the investigated
dwelling and the particular pollutant and source examined; and'(b) Equation
(4) is not linear in v. These points combine to make the problem of esti-
mating values for v and S difficult.
A parameter estimation technique appropriate for the present problem is
the grid search method. Although this process is easily constrained, it is
lengthy, and its accuracy strongly depends on the number of grid points.
Because f(S,v) is a function of two variables, it is not possible to obtain
the required minimum by the technique used for finding extreme values of
functions of one variable that is taught to all students of beginning calculus,
This technique is easily constrained over a given interval, but it cannot be
applied to f(S,v) because it minimizes only functions of one variable. How-
ever, the Parameter Estimation Procedure used in this study is a cross
between the grid search and the optimization technique for a function of one
variable. A sequence of five steps must be followed in the parameter esti-
mation procedure.
1. Define the constraints on v and S, i.e., v-j <_ v <_ vu and
$1 <_ S <_ Su, where subscripts 1 and u define the lower and
upper values of the parameter intervals.
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2. Determine AV = (vu - v-|)/k; where k is the number of
intervals.
3. For a given v-j = vj + (i - 1) AV, i = 1 k + 1, find
the points SQ^ for which
df(S.v)
dS
= 0 .
(S,v) = (S,v.)
4. Determine Sm. as the value of S that gives
min (ffSpVj), f(SQ ,v.j), f(Su,v.)} .
5. The required estimates of v and S are those values
that give
min (f(Sm tvj):1 = 1 , ... , k} .
The theoretical approach used in deriving this parameter estimation
procedure is provided below. Owing to step 3 of the above sequence, all
equations below refer to a given fixed value of the parameter v. Equation
(4) can be rewritten as follows:
-(v+D)(t-t0)
= - e
-(v+D)(t-t0)
+ 1 CQ - m (^) t0 - (id-lM) - ^-}\e
where
c = cin(t)
C0 = CinQ
m = mout
b ' bout
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or as
C = aS + 3 (8)
where
-(v+D)(t-t0)
1 - e
and
(
*-&*
From Equation (6) we have
f(S.v) = £ («,-S + B, - CM,)2 (9)
1=1 1 n 1
where
a. is computed @ t = t^ and
3, is computed @ t = t.. , tg = t._-j, and C« = CM. ,
Equation (9) becomes
2
f(S,v) = S £ a + 2S E a.(gi - CM.) + £ ( B . - CM.)
f(S,v) = AS2 + 2BS + C (10)
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where
A =
B = E a. (B. - CM.)
i=l 1 1 1
C = £ (Bi - CM.)2 .
Equation (10) indicates that f(S,v) is parabolic in S. The critical point
is found by taking the derivative of f with respect to S:
= 2AS + 2B .
(11)
Setting df/dS = 0, it is seen that -(B/A) is a critical point; in order to
determine whether -(B/A) is a maximum or a minimum, the second derivative
of f with respect to S is taken:
dS
= 2A .
(12)
Since A is a sum of squares, A > 0, which means that
dS
_ B
" "A"
> 0
thus S = -(B/A) minimizes f(S,v)
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Completion of the square in Equation (10) leads to the same conclusion:
f(S,v) s As + + c - . (13)
Again, since A > 0, the vertex S = -(B/A) is the desired minimum.
The parameter S, representing the indoor pollutant generation rate, must
be constrained to lie in a physically meaningful interval [S], Su]. Thus,
while S = -(B/A) is an absolute minimum, -(B/A) may not be within the
interval [S^, Su]. Hence, in addition to computing -(B/A), a test must be
made to determine whether -(B/A) lies in the interval (i.e., whether
-(B/A) e [S-j, Su]); if not, the end points of the interval must be examined.
The three cases to be considered are illustrated in Figure 1. They are:
Case 1: Sm < S-j
Case 2: S-, < S^ < Sy
Case 3: Su < Sm3 .
In Case 1, the constrained minimum occurs at S]; in.Case.2, the constrained
minimum occurs at Sm ; and in Case 3, the constrained minimum occurs at Su.
Application of the Parameter Estimation Procedure to the Indoor Air Pollution
Data Base
In order to apply the Parameter Estimation Procedure to the Indoor Air
Pollution Data Base, the problem of degrees of freedom must be considered;
i.e., the fact that the number of observations used to estimate a set of
parameters must be greater than the number of parameters being estimated.
Thus as a means of increasing the amount of data available for use in the
Parameter Estimation Procedure, the instantaneous outdoor values and instan-
taneous indoor averages for each 20-min segment of each hour were used
because at this time the entire house is being modeled instead of individual
zones.
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CTi
I
Figure 1. Graphical illustration for the three cases of constraints on interval source rate.
-------
Hourly values of v and S are desired; however, there is a problem in
applying the Parameter Estimation Procedure to obtain hourly estimates of
v and S. The problem involves degrees of freedom. Even though three values
are available for each hour to be used in the estimation, one of these is
the initial value, which means that only two values can actually be used in
the estimation procedure. As a result, the number of parameters to be
estimated equals the number of points to be used, which means that no degrees
of freedom are left for the estimation procedure.
The problem mentioned in the previous paragraph is resolved as follows:
2-h estimates of v are calculated from the data of a "nonreactive pollutant,"
then, using these estimates of v, hourly estimates of S are found for all
pollutants. The "nonreactive" pollutant chosen was CO. If any of the 12 CO
values (6 indoor and 6 outdoor over a 2-h period) are missing, v is estimated
using NO data; however, if both CO and NO data are missing for a given 2-h
period, neither v nor any of the pollutant source rates for that period is
computed. Finally, if, for a given pollutant for a given hour, any values
are missing, the corresponding source rate is not computed.
Essentially, we are using CO as a tracer to estimate theoretical values
of air exchange rates for each of the investigated dwellings. As part of
the field monitoring program of the Indoor Air Pollution project, the air
exchange rate of each residence is determined experimentally. The tracer
used for the experimental determination of the air exchange rate is sulfur
hexafluroide (SFg); the monitoring protocol calls for three or four different
4-h experiments per residence. The theoretical and experimental values for
the air exchange rate agree in most of the investigated periods, and, in
those cases of disagreement, the difference between the two values is not
appreciable. Table 2 shows a comparison between the estimated and experi-
mental values for the air exchange rates.
The estimated indoor source strength value S in mg/h is an "effective"
pollutant production rate; i.e., the internal source is treated as if it
operates for the entire 1-h period. The estimated indoor source is also
comprehensive; that is, if two indoor sources are generating a pollutant
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simultaneously, the calculated theoretical value will be the sum of the
individual source strengths. Using the daily logs kept by the occupants of
the residences we monitored, we were often able to isolate a single source;
the estimated theoretical values due to isolated indoor sources compare
favorably with the available literature values.
TABLE 2. AIR EXCHANGE RATES
Residence
Chicago Experimental I
Pittsburgh Low-Rise Apt. I
Pittsburgh Mobile Home I
Denver Conventional
Washington Experimental I
Washington Conventional I
Baltimore Conventional I
Baltimore Experimental I
Air
Estimated
0.40
0 30
0. 20
0.64
0.58
0.60
0.98
0.44
1. 10
0. 10
0.6
0.4
0. 4
1.2
0. 64
exchange rates
Experimental
0.23
0.22
0.26
0.60
0.84
0.63
1.05
0. 52
1.02
0.60
0.24
0.2
0.43
0. 78
0. 72
STATISTICAL STUDIES
The objective of the statistical studies performed on the GIOAP model
is to define its ability to estimate indoor air pollution levels. In this
document we evaluate the results from a sample of eight residences; this
is a comprehensive review of the predictive power of the model.
In the first section we describe the procedure followed and outline the
motivation and the objectives of each step. The section on the statistical
assessment includes response graphs, statistical tables, and scatter diagrams.
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In addition, it contains comments and conclusions on the model for each gaseous
pollutant monitored. The final section provides theoretical details on sen-
sitivity coefficients and includes a series of simulations that illustrate
the errors introduced by not using the "best" estimated value for any given
input parameter.
Statistical Procedure and Methodology
The strength of a theoretical or numerical method to predict air pollu-
tion levels has often been demonstrated with graphical illustrations. This
has been the case in many stu'dies with a small data base. Figures 2 and 3
present a sample of indoor values estimated by the GIOAP model against the
observed indoor values for a number of pollutants during a 2-week monitor-
ing period. While this randomly chosen set of illustrations indicates the
predictive power of the model, it does not allow for general conclusions.
In order to validate the GIOAP model a statistical approach is required.
A statistical analysis is performed on data sets consisting of pairs of
hourly estimated and observed indoor air pollutant concentrations. A sta-
tistical approach is preferred to judgments made on the basis of comparing
corresponding estimated and observed indoor air pollutant concentrations for
two reasons. First, the GIOAP model simulates a variety of conditions for
a number of pollutants and generates a large number of data sets, each of
which contains many points (over 300). Second, direct comparison of estimated
and observed values involves difficult judgments in deciding when the estimated
value falls within an acceptable range of the observed value. Under such con-
ditions statistical techniques provide fast, efficient, and reliable methods
for assessing the data, thus making the judgments much less subjective.
The statistical analysis used to validate the GIOAP model is based on
the principle that if the model simulates realistically the involved complex
conditions, a plot of the estimated versus observed values would fall on or
near a line with a slope of one and an intercept of zero.
-19-
-------
1000 .
900 .
. Obs«rv«d
*• Eitlm»t«d
x Coincide
Slca : Waihlngton, 0. C. Exp«rlm«nal - Vlilt Number I
Hourly Plod
24-Hour Epitt>d«
Figure 2. Estimated vs. observed pollutant concentrations for 7 consecutive days.
-20-
-------
F\
i- L -7-5
n
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Site : V. .ifrincTon, ^*. C. Exp«nt;
-------
The problem of measuring the association between the observed and esti-
mated values is divided in the following three sequential steps:
1. The degree of linearity in the relationship between the
pairs of estimated and observed values is established.
Linearity is required to proceed to step 2.
2. The slope and intercept of the line expressing the linear
relationship between the observed and estimated values
are determined. Criteria on the proximity of the slope
and intercept to one and zero, respectively, are estab-
lished and must be met before proceeding to step 3.
3. The degree of dispersion about the line defined by the
slope and intercept in step 2 is calculated and must meet
certain acceptability criteria.
The statistical methods used in each of the three steps will be discussed in
the remainder of this subsection.
The Pearson product-moment correlation coefficient is calculated in
order to determine whether or not a linear relationship exists between the
pairs of estimated and observed values. The calculated correlation coeffi-
cient must be close to +1; if that is not the case, it should be concluded
that the numerical model does not realistically simulate the processes
involved.
The second step requires that the relationship be linear. A line is
characterized by calculating the regression parameters (slope and intercept)
of the plot of observed versus estimated values. In addition, the calculated
slope and intercept values are tested for statistical significance. Each of
the following hypotheses is statistically tested by a two-tailed t-test:
(a) the slope is +1, and (b) the intercept is 0. In the following section
it will become apparent that this statistical procedure will reject numerical
estimations well within the accuracy limits of the input values. Thus, it
is necessary to ease the limits that strict adherence requires by establish-
ing a set of less restrictive criteria.
-22-
-------
The final step is necessary only if a linear relationship exists between
the estimated and observed values, and if the regression line meets the estab-
lished criteria. This step determines how well the line fits the data points.
In order to estimate the dispersion of the data points from the regression
line, the Standard Error of Estimate (SEE) is calculated. If the SEE is small,
the model data set is acceptable; if the SEE is large, it indicates that the
points are widely scattered about the regression line and the modeled set
should be rejected.
If a data set meets all these criteria, then it is concluded that the
GIOAP model adequately represents the simulated event.
Statistical Assessment of the GIOAP Model
The statistical procedure outlined in the last section will be applied
to eight sets of data corresponding to continuous monitoring from eight
dwellings, each set consisting of seven gaseous pollutants. The model pre-
dicts the average indoor pollutant concentration for three different time ,
periods (episodes), 3, 8, and 24 h. The model performance is evaluated for
all days of the monitoring period.
The statistical information generated for each pollutant, each episode,
and each residence is presented in tabular form (for example, see Table 3).
The first column identifies the residence investigated, and the column labeled
tep-js indicates the duration of each episode. The column labeled r contains
the correlation coefficient; b is the intercept of the regression line cor-
responding to the plot observed versus estimated values, and m is the slope
of this line. The null hypothesis—that the intercept is zero—is tested
against the statistic tb, while the null hypothesis—that the slope equals
one—is tested against the statistic tm> The column to the right of tm con-
tains the range of the average pollutant concentration observed indoors;
this range provides a value against which the calculated standard error of
estimate can be judged. The next to the last column presents the number of
observations used for estimating the various statistics; it must be noted
that this number is not always the total number of possible pairs because
-23-
-------
TABIE 3. STATISTICAL DATA SUMMARY
Residence
epis
3-h
8-h
24 -h
3 h
8-h
24-h
3-h
8-h
24-h
3-h
8 h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
r
b
m
*b
t
m
Range .
of indoor
observed value
SEE
No.
of
observ.
Comments
I
ro
-------
of either missing observed values and/or missing calculated values due to
the lack of the initial value, the air exchange rate, and/or the effective
source strength.
Based on the information included in these columns, a conclusion is
reached on how well the model predicts the indoor observed values. Three
classes of acceptance or rejection comments are generated: Class I describes
the simulations that satisfy all predetermined tests; Class II refers to
numerical estimation of the indoor pollutant concentrations that, although
not statistically acceptable, are judged to meet predetermined criteria which
will be described below; and Class III refers to model estimations that do
not meet any of the above requirements and must be rejected because they do
not realistically simulate the observed indoor values.
The investigation of the model performance for each pollutant will be
presented in the balance of this section. It is however essential to begin
by stating a set of rules for each of the classes outlined in the previous
paragraph. Since there are varying degrees of linearity, a decision must be
made on the cutoff level of the correlation coefficient. Figures 4 through 7
are scatter diagrams with four different correlation coefficients; their
values are r = 0.96, r = 0.82, r = 0.72, and r = 0.66, respectively. The
cutoff value chosen for this study for the correlation coefficient is r =
0.7; thus if r is below 0.7 the relationship between the observed and esti-
mated value is considered nonlinear for the purposes of this study.
For Class racceptance the criteria on b and m are set by the two-tailed
statistical t-test. For a significance level of 0.01 the t value tQ 005,°° =
2.576 (the number of degrees of freedom is considered to be infinite because
in this project it is almost always larger than the maximum finite degrees of
freedom specified in the statistical tables); thus if either tb or tm is
outside the interval -2.576 <_ t <_ 2.576, the estimated indoor pollutant con-
centrations do not fall into Class I.
For Class II, predetermined acceptability limits can be placed on the
slope m for all pollutants. However, limits on the intercept must be
determined on a case basis since the proximity to zero of the intercept
-25-
-------
MODEL VALIDATION -
8-H. CO
Ofl/12/77
PAGE
FILE P6135
SCATTERGRAM
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-------
M erne LI VALIDATION
24-H THC-CH4
OS/1o/77
FILE
P&l 35
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-------
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-------
crucially depends on the range of the observed values which varies from pol-
lutant to pollutant. In Class II, slope values within the closed interval
[0.7, 1.3] are acceptable. In addition, values of the intercept that have
magnitude greater than 15% of the maximum observed pollutant concentration
are rejected. The last criterion for acceptance in Class II requires that
the Standard Error of Estimate (SEE) is <_ 10% of the maximum observed
value.
When examining the tables containing the statistical validation data for
each pollutant (i.e., Tables 4 through 7 and 10 through 12), it will be seen
that the following phenomenon occurs several times: for a given set of con-
ditions the 3- and/or 8-h episodes will be Class II, but the corresponding
24-h episode will be Class I. Further examination will reveal that the
24-h episode statistical data is based on fewer observations than either
the 3- or 8-h episode data. The reason for this is that, when missing data
are encountered, the model calculations cease regardless of whether the end
of the episode has been reached. Thus, some episodes span fewer hours than
are indicated by the headings, a condition which results in less overall
variation and gives better statistical data.
In the balance of this section each pollutant will be examined individ-
ually.
Carbon Monoxide--
Table 4 provides all the statistical information obtained by the pre-
viously outlined steps for CO. In all cases investigated an acceptable
degree of linearity exists (r >^ 0.821), the intercept is uniformly close
to zero, and the slope is outside the prescribed interval only once (m =
1.336 for the 24-h episode in the Pittsburgh Low-Rise Apt. I); this is
the only Class III case simulated. Let us investigate this Class III case
in more detail. The following points can be made:
1. Frequency distribution tables show that 91% of the sampled
values fall in the half-open interval [0, 2.8).
-30-
-------
TABLE 4. STATISTICAL DATA SUMMARY FOR CARBON MONOXIDE (CO ppm)
Residence
Pittsburgh
Mobile
Home I
Denver
Conventional
Chicago
Experimental I
Pittsburgh
Low -R ise
Apt. I
Baltimore
Experimental I
W ashington
Conventional I
Baltimore
Conventional I
Washington
Experimental I
epis
3-h
8-h
24 -h
3 h
8-h
24-h
3-h.
8-h
24-h
3-h
8 h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
r
0.943
0.900
0.912
0.970
0.959
0.949
0.947
0.888
0.920
0.884
0.821
0.878
0.967
0.969
0.960
0.961
0. 938
0.912
0.851
0. 876
0.916
0.930
0. 889
0.895
b
0. 174
0.242
0.173
0.041
0.018
-0. 024
0.289
0.683
0.70
-0. 029
0.023
-0. 105
0.001
0.004
-0. 003
0.040
-0. 003
0.019
0.107
0.036
0.034
0.080
0.117
0.087
m
0.911
0.874
0.919
0.980
1.006
0.993
0.943
0.862
0.796
1.128
1.218
1.336
1.005
1.096
1.034
0.972
0.949
1.004
0.778
0.882
0.894
0.942
0.883
0. 948
lb
4. 779
4.810
3.119
0. 744
0.259
-0. 286
5.768
9.350
9.293
-5.90
0.348
-1.378
0.264
0.709
0.525
1.222
-0. 068
0.390
3.683
1, 178
1.235
3.224
3/594
2.50
t
m
-5. 039
-5.096
-2. 553
-1.471
0. 333
-0. 285
-3.004
-4. 616
-6. 890
3.40
3.778
5. 143
0.308
5.062
1.419
-1.913
-2.543
0.130
-8.038
-3.734
-3.363
-2. 740
-4. 175
-1.427
Range
of indoor
observed value.
0. 0-6. 2
0.89-15. 9
0.33-6.33
0. 0-7. 0
0. 0-2. 0
0. 0-5. 2
0. 0-4. 56
0. 0-4. 22
SEE
0. 310
0. 389
0.268
0. 570
0.683
0. 549
0.352
0. 520
0.462
0.519
0.652
0. 571
0.077
0.075
0. 067
0.396
0.487
0.475
0.389
0. 349
0. 271
0.314
0. 391
0. 358
No.
of
observ.
337
293
174
320
291
168
282
224
133
253
217
126
235
213
162
360
305
200
305
240
154
313
265
169
Comments
II
II
II
I
I
I
II
II
II
II
II
III
I
II
I
I
I
I
II
II
II
II
II
II
-------
2. Straightforward calculations using the following relation-
ship
Observed Value = Intercept + (Slope) (Estimated Value)
indicate that the approximate maximum difference between
observed and estimated values is 22% of the observed
value.
A similar analysis applied to a randomly chosen Class II case gives an approx-
imate maximum difference between observed and estimated values of 2% of the
observed CO value. Analyses of this nature for a Class I case provide
similar or better results. It is concluded that the model predicts CO indoor
values acceptably.
Nitric Oxide--
Table 5 illustrates a strong linear correlation between observed
and estimated values, r >^ 0.875. Similarly, the slopes of the calculated
regression lines lie within the predetermined interval. As a first observa-
tion, the magnitude estimated for the intercept and the standard estimates of
error may seem large; however, when compared with the indicated ranges of the
monitored values, they are put in proper perspective and are judged acceptable.
Thus, 21 cases are accepted as Class II, while the remaining 3 cases are
accepted as Class I.
In order to provide a perspective on the model performance, an approxi-
mate estimate of the percent difference between observed and estimated values
will be calculated. The Denver Conventional Residence is chosen because it
was one of the extreme cases considered. Let us investigate the 8-h episode
simulations. The relevant statistics are r = 0.875, b = 10.135, and m =
0.793. Frequency distributions of observed indoor averages generated for the
data interpretation task of this study show that 94% of the hourly values
fall within the half-open interval [0, 120). Following the thinking expressed
for CO, we conclude that for this case, within the specified interval, the
approximate maximum difference between observed and estimated values is 15%
of the observed value; or within this interval the statistical model value
is at most 1.15 times the observed value.
-32-
-------
TABLE 5. STATISTICAL DATA SUMMARY FOR NITRIC OXIDE (NO ppb)
Residence
Pittsburgh
Mobile
Home I
Denver
Conventional
Chicago
Experimental I
Pittsburgh
Low -R ise
Apt. I
Baltimore
Experimental I
Washington
Conventional I
Baltimore
Conventional I
Washington
Experimental I
epis
3-h
8-h
24 -h
3 h
8-h
24 -h
3-h
8-h
24 -h
3-h
8 h
24 -h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
r
0.970
0.941
0.917
0.909
0.875
0.911
0.987
0.970
0.976
0.982
0.982
0.993
0.939
0.948
0.946
0.986
0.966
0.953
0.906
0.954
0.961
0.935
0.918
0.906
b
6.873
5.192
12.541
8.307
10.135
1.236
1.907
3.479
4.733
-1.135
-2.355
-2.235
1.231
1.740
2.402
5.542
15. 105
25. 878
2.314
0.706
0.207
1.120
2.052
1.841
m
0.962
0.990
0.931
0.798
0.793
0.912
0.941
0.913
0.836
1.034
1.066
1.048
1.026
1.042
1.039
0.975
0.919
0.809
0.841
0.927
0.931
1.058
1.052
1.066
-------
This specific analysis is of course an example; however, we think
that the model realistically simulates indoor average concentration for
nitric oxide.
Nitrogen Dioxide--
Numerical simulations of indoor NCL concentrations require the use of
a first-order decay term. The half-life used for these simulations is
30 min; this value is suggested by observations of the indoor instantaneous
N02 values of this project as well as by Wade et al.(6) and by Craig Hollowell
in a private communication. The use of first-order chemical decay terms,
instead of the zero order rate used originally, has substantially improved
the predictive power of the model. Six cases are rejected, 3 cases are
accepted as Class I, and 15 cases are accepted as Class II. The cases that
have been rejected are in residences without indoor N02 sources; the indoor
concentrations are persistently low with very little variation. In cases
like this the model may overestimate the indoor N02 concentrations by as
much as 50%. However, the model performs well for a total of 18 cases
(out of 25); see Table 6. The maximum difference between the statistical
model value and the observed value is 16%. This conclusion is reached by
the process described for CO, and it refers to a specific example; however,
the general assessment is that the GIOAP is realistically simulating indoor
average concentrations for N02.
Sulfur Dioxide--
The nature of the S02 data is a source of the apparent inability of the
model to estimate the observed indoor values (see Table 7). Table 8 shows
portions of the indoor average concentration frequency distribution for S02.
Note that the instrument used, a Meloy OA-185-2A commercial detector, has a
o
limit of detection of 13.1 ug/m (0.005 ppm). It is apparent that almost
all SOp values observed indoors are at or below the instrument's lower limit
of detection.
Three factors influence these low levels of 502= (a) the observed S02
outdoor levels are generally low; (b) the pollutant is a moderately reactive
gas; therefore, the indoor levels are lower than the outdoor levels; and
-34-
-------
TABLE 6. STATISTICAL DATA SUMMARY FOR NITROGEN DIOXIDE (NO2 ppb)
Residence
Pittsburgh
Mobile
Home I
Denver
Conventional
Chicago
Experimental I
Pittsburgh
Low -Rise
Apt. I
Baltimore
Experimental I
Washington
Conventional I
Baltimore
Conventional I
Washington
Experimental I
t .
epis
3-h
8-h
24 -h
3 h
8-h
24-h
3-h
8-h
24-h
3-h
8 h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
r
0.834
0.773
0.708
0.875
0.744
0.835
0.651
0.487
0.475
0.918
0.871
0.839
0.750
0.679
0.570
0.847
0.823
0.773
0.758
0.836
0.925
0.854
0.728
0.758
b
1.093
1.265
2.630
8.045
12.349
6.012
6.106
7.558
7.684
-0.400
-0.317
-0.255
2.779
3.448
4.176
0.192
0.567
0.856
7.398
5.574
1.879
3.492
5.981
4.372
m
0.838
0.832
0.718
0.803
0.711
0.837
0.725
0.677
0.868
0.946
0.873
0.837
0.661
0.571
0.492
0.925
0.809
0.739
0.707
0.764
0.934
0.968
0.888
0.939
-------
TABLE 7. STATISTICAL DATA SUMMARY FOR SULFUR DIOXIDE (SO2 ppb)
Residence
Pittsburgh
Mobile
Home I
Denver
Conventional
Chicago
Experimental I
Pittsburgh
Low -Rise
Apt. I
Baltimore
Experimental I
Washington
Conventional I
Baltimore
Conventional I
Washington
Experimental I
t .
epis
3-h
8-h
24 -h
3 h
8-h
24 -h
3-h
8-h
24 -h
3-h
8 h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
r
0.491
0.488
0.517
0.692
0.619
0.713
0.603
0.464
0.574
0.511
0.505
0.514
0.0
0.948
0.946
0.945
0.919
0.906
0.704
0.627
0. 825
0. 140
0.051
0.050
b
4.848
4.346
3.963
0.725
0.778
0.696
1.342
1.789
1.350
0.414
0.398
0.488
2.0
1.740
2.40
-0. 169
-0. 474
-0.800
1. 127
1.384
1.006
1.920
1.985
1.993
m
0. 142
0. 130
0.152
0.361
0.277
0.259
0.489
0.341
0.449
0.226
0.166
0.159
0.0
1.042
1.039
1.113
1.231
1.326
0.448
0.329
0.472
0.041
0.008
0.007
-------
(c) the flame photometric principle of detection, employed in the commercial
instrument used in the field operations, is subject to negative interference
from C02- The instrument, although "approved" by EPA for sampling in the out-
door ambient environment, is subject to quenching by COo, which is present in
high levels in the indoor environment. The extent of the negative CC^ inter-
ference on the S02 levels is illustrated in Table 9, which shows the results
of four tests performed by the field team of this project. In each case the
same correction factor is calculated; however, we have not undertaken such
corrections because the observed levels are almost always close to very low,
unreliable levels.
TABLE 8. SO2 FREQUENCY DISTRIBUTION
Residence Percentage SC>2 Range in ppb
Pittsburgh Mobile Home I 44. 2 3. 00 - 5. 60
35. 2 5.60 - 8.20
Denver Conventional 90. 5 1. 00 - 2. 50
Chicago Experimental I 80. 0 2. 00 - 3. 80
15.6' 3.80-5.60
Pittsburgh Low-Rise Apt. I 60.9 0.00-1.00
20.7 1.00-2.00
Baltimore Experimental I 99. 7 2. 00 - 2. 20
Washington Conventional I 60. 5 3. 70 - 4. 40
28. 8 7. 90 - 8. 60
Baltimore Conventional I 94. 9 2. 00 - 2. 90
Washington Experimental I 98. 8 2. 00 - 2.12
-37-
-------
TABLE 9. NEGATIVE CO2 INTERFERENCE ON SO2 LEVELS*
Test No.
[C02]
Introduced into the
SO2 Monitor
[so2]
Output
by the Instrument
[S02]
Introduced
into the Instrument
1
2
3
4
300
813
1460
1975
300
833
1450
1975
308
850
1500
2037
312
872
1525
2088
0 33
0.265
0.205
0. 165
0.23
0. 18
0. 14
0. 115
0.095
0.075
0.055
0.045
0.050
0.040
0.030
0.023
0.33
0.33
0. 33
0.33
0.23
0. 23
0. 23
0.23
0.095
0.095
0.095
0.095
0. 050
0 050
0.050
0. 050
* All concentration levels in ppm.
The point here is that while the model does not simulate the observed
SCX, concentrations, both the estimated values and the observed values are too
low and too close to zero to justify employment of correction factors. It is
concluded that the model's ability to correctly estimate SOg values has not
been tested by the available data. SC^ levels have been found to be low in
the indoor environment not only by the present study but in all similar studies.
Sulfur dioxide concentrations decay at rates similar to NOg; thus it is
expected that the GIOAP model would realistically simulate higher and more
variable indoor levels.
Ozone--
An ozone table similar to the statistical summary tables for the other
pollutants would indicate that the model does not satisfy the predetermined
criteria. However, it is misleading to consider the model performance as
-38-
-------
unsatisfactory because the majority of the estimated values are within
o
3.92 pg/m (2 ppb) of the observed values; this difference is smaller than
the monitor's precision. The observed ozone levels in the indoor environ-
ment are low and often constant for long periods. The small variations in
the predicted values weigh heavily in the estimation of correlation coeffi-
cients and other statistics that assess the power of the model to predict.
The capability of the GIOAP model to predict indoor ozone levels is judged
adequate because it gives a realistic picture of the ozone variation indoors.
Finally, the linear dynamic model(1) has been utilized to predict higher
indoor levels, and its use in conjunction with the GIOAP model is recommended.
Nonmethane Hydrocarbons--
The model simulates the majority of the investigated cases well. The
model requires knowledge of the molecular weight of the pollutant examined;
in the case of hydrocarbons we had to use an average molecular weight repre-
senting the hydrocarbons most often sampled. Thus the uncertainty intro-
duced may have caused some of the Class III judgments. In spite of this
uncertainty, Table 10 indicates that the GIOAP model estimates the indoor
nonmethane hydrocarbon levels satisfactorily in the majority of the cases
examined.
Methane--
Table 11 illustrates that the model estimates the indoor methane values
realistically. As always, 24 cases are run. For this pollutant, 2 are
judged Class III, 17 are accepted as Class II, and 5 are classified as
Class I. Following the techniques used in previous pollutant analysis, we
estimate a maximum difference of approximately 30% between the statistically
estimated CH^ concentration and its corresponding observed value. This is
one of the largest percent differences found in Table 11.
-39-
-------
TABLE 10. STATISTICAL DATA SUMMARY FOR NONMETHANE HYDROCARBONS (THC-CH4 ppm)
Residence
Pittsburgh
Mobile
Home I
Denver
Conventional
Chicago
Experimental I
Pittsburgh
Low -R ise
Apt. I
Baltimore
Experimental I
Washington
Conventional I
Baltimore
Conventional I
Washington
Experimental I
epis
3-h
8-h
24-h
3 h
8-h
24-h
3-h
8-h
24-h
3-h
8 h
24-h
3-h.
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
T
0.819
0.745
0.721
0.817
0.792
0.662
0.951
0.873
0.949
0.855
0.853
0.929
0.809
0. 821
0.832
0.879
0.863
0.747
0.727
0.677
0.771
0.961
0.942
0.984
b
1.036
0.884
0.765
0.030
0.120
0. 186
0.305
0.533
0.286
0.117
-0. 630
0.376
0. 119 .
0.246
0. 196
0.043
0.096
0.299
0.227
0. 198
0. 121
-0. 006
0.006
-0. 051
m
0.585
0.665
0.718
0.851
0.714
0. 542
0.899
0.788
0.845
1.046
1.29
1.167
0.840
0.621
0.603
0.915
0.798
0.602
0. 590
0. 533
0.633
0.936
0.886
0.921
-------
TABLE 11. STATISTICAL DATA SUMMARY FOR METHANE (CH4 ppm)
Residence
Pittsburgh
Mobile
Home I
Denver
Conventional
Chicago
Experimental I
Pittsburgh
Low -Rise
Apt. I
Baltimore
Experimental I
Washington ,
Conventional I
Baltimore
Conventional I
Washington
Experimental I
epis
3-h
8-h
24 -h
3 h
8-h
24-h
3-h
8-h
24-h
3-h
8 h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
3-h
8-h
24-h
r
0.914
0.895
0.880
0.812
0.689
0.708
0.921
0.847
0.844
0.832
0.816
0.870
0.871
0.859
0.850
0.943
0.886
0.918
0.940
0.940
0.951
0.928
0.919
0.904
b
0.352
0.251
0.307
0.579
0.930
0.813
0.086
0.068
0.120
0.226
0.268
0.215
0.441
0.483
0.474
0.127
0.210
0.108
0.212
-0. 332
-0.554
0.147
0.179
0.197
m
0.769
0.861
0.823
0.716
0. 556
0.576
0.931
0.917
0.840
0.779
0.753
0.799
0,763
0.731
0.723
0.907
0.836
0.913
1.008
1. 141
1.200
0.995
0.986
0.994
-------
Carbon Dioxide--
The last pollutant investigated in this section is C02; Table 12
illustrates that the GIOAP model estimates the indoor C02 concentrations
very well. Nine cases are judged Class II, and 15 are judged Class I.
Following the procedure used in the other pollutant analysis, the difference
between the estimated CO^ concentrations and the corresponding observed
indoor values is at no time greater than 8% of the observed value.
Model Sensitivity
An integral part of the model validation is a theoretical analysis of
the model sensitivity. A study of this nature shows how errors in the
estimation of a model parameter affect the model output(s). One of the
unique features of the GIOAP model is the transient term. Previous
numerical studies simulating the relative balance between the indoor and
outdoor environments have included only steady-state conditions. An assess-
ment of the transient term indicates that this term is most important for
stable pollutants, but its contribution is minimal for reactive pollutants.
Exclusion of the transient term reduces the correlation between observed
and estimated values by about 50% of its value with the transient term
included. Exclusion of the transient term does not have any effect on the
estimations of indoor concentrations of the chemically reactive ozone.
This behavior is expected from theoretical considerations, since for ozone
the decay term in the exponent of the transient term is very large; there-
fore, the transient term approaches zero, which is not the case for stable
pollutants. Thus the GIOAP model becomes a steady-state model for reactive
pollutants; the model, however, is very sensitive to the transient term for
the stable pollutants. A sensitivity analysis will further indicate which
parameters are highly sensitive to errors (i.e., a small error in such a
parameter would result in a significant error in the output(s)) and which
parameters are relatively insensitive to errors. Knowledge of the sensitivity
of the various parameters will assist us in determining priorities in the
utility of the model and in estimating parameter values when the model becomes
-42-
-------
TABLE 12. STATISTICAL DATA SUMMARY FOR CARBON DIOXIDE (CO2 ppm)
Residence
Pittsburgh
Mobile
Home I
Denver
Conventional
Chicago
Experimental I
Pittsburgh
Low -R ise
Apt. I
Baltimore
Experimental I
Washington
Conventional I
Baltimore
Conventional I
W ashington
Experimental I
t .
epis
3-h
8-h
24 -h
3 h
8-h
24 -h
3-h
8-h
24-h
3-h
8 h
24-h
3-h
8-h
24-h
3-h
8-h.
24-h
3-h
8-h
24-h
3-h
8-h
24-h
r
0.973
0. 929
0.943
0.837
0.730
0.656
0.948
0.919
0.981
0.967
0.953
0.929
0.921
0.900
0.910
0.979
0.954
0.979
0.932
0.950
0.943
0.893
0.896
0.905
b
42.175
-10. 878
12. 243
- 5. 166
-35.016
-93. 259
30.654
21.308
15.496
43. 778
21.220
43. 220
35. 860
5.939
-23. 461
12. 643
4.027
18.378
21. 506
20. 668
7.237
75. 750
41.831
53. 971
m
0.948
1.033
0.999
1.061
1.174
1.275
0.953
0.948
0.964
0.941
0.986
0.946
0.943
0.992
1.051
0.984
1.003
0.960
0.955
0.949
0.986
0.878
0.948
0.908
-------
an application tool. Model validation studies are often undertaken under
the best possible conditions; thus, it is necessary to study the model's
sensitivity in order to define its limitations and capabilities.
For a given general model y = f()T,F), model sensitivity is defined as:
af(X.P)
ap.
(X,P) = (X0,P0) (14)
where
f = the function defining the mathematical model
X = {x-|, X2, ..., xn> = the vector of independent variables
» X2n> •••' xnn^ = fixed value of )f
= (p-|, P2> ...» P|<} = the vector of parameters
I P> -••> P} = f1xed value of P".
For some insight as to why this formula is used to measure model sensi
tivity, one should recall the following equation:
Equation (15) indicates that the approximate error (df) of f is a linear com-
bination of the errors in the individual parameters (dpj, i=l, ..., k) where
the coefficient of each dpn-, 1=1, ..., k is the corresponding sensitivity
coefficient. This approach is used for error or sensitivity analysis when
Af, the actual change in the function, can be approximated by df.
The GIAOP model is a first-order initial value problem given by
Equation (3). Thus, for this case, the function f referred to in the
definition of model sensitivity is replaced by C^, Equation (4). The
GIOAP model sensitivity coefficients are as follows:
9C. -(D+v)(t-t0)
^7=e toltltf (16)
-44-
-------
3C.
(18)
3C. ,
IT g e t
-------
where
tQ = initial time
tf = final time.
In the balance of this document the subscripts referring to indoors and
outdoors will be eliminated; thus, we will denote Cjn by C0, C-jn by C, mou^
0
by m, and bout by b.
The study documented in this report requires several additional, though
nonrestrictive, assumpsions to be made (see Section 2) in order to implement
the model, Equation (4). These assumptions resulted in Equation (5). The
sensitivity coefficients for Equation (5) are as follows (see Appendix A
for the derivations of the sensitivity coefficients):
m
3C,
(23)
n- fJU\[/LJ_ * \
am. - ai Dj+v^ [lDi+vi - S-lj
(24)
3C
_
- a
(25)
8C
n,
1 - e
(26)
-46-
-------
9C
m
3Di
S.. 2m, v.
v,b, + - -
-
D.+V.
S 2m
(27)
m
s -(V.+D.
—' (e
D1+v) \
(28)
8C
3v-
(29)
where
3C
m 8C
( ) =
-------
Two major advantages become apparent when the closed form partial deri-
vatives are available. The first is the ease with which the sensitivity
coefficients can be computed. The second is that the sensitivity coeffici-
ents can be given a thorough analytical treatment which is difficult, if not
impossible, when the partial derivatives are not available in closed form.
Each of the seven sensitivity coefficients (Equations (23) through (29)) will
be discussed below.
The first sensitivity coefficient to be considered is 3Cm/3CQ. Referring
back to Equation (23), it is seen that 0 < 3Cm/3C0 < 1. Moreover, as time
increases, 3Cm/3C0 decreases, which means that the effect of an error in CQ on
C diminishes with time. Finally, since 3C /3Cg'1s positive, an increase in
CQ will cause an increase in Cm, and a decrease in CQ will result in a decrease
1n Cm'
Next, the sensitivity coefficients involving m, the slope of the line used
to approximate the outdoor pollutant concentrations, will be dealt with. By
rearranging and deleting terms from Equation (24), it is seen that
0 <
m
3m.
i, m <_ n . (30)
Equation (30) shows that the effects on C of an error in m at t •= t-j will
dissipate with time. On the other hand, it should be noted that if there is
a recurrent error in m (e.g., every value of m-,- , i = 1 , . . . , n is in error
by 20%, the effects will be additive (see Appendix A), i.e.,
m aC
dC = L .-•dm., * = 1, ..., n (31)
with the elements having the lowest indices contributing less and less to
the error in C.
-48-
-------
The sensitivity coefficients involving b, the intercept of the line used
to approximate the outdoor pollutant concentrations, will be examined in this
paragraph. From Equation (25), it is clear that
i i
(32)
Thus, the effects of an error in b at t^ on C diminish with time. Also,
8Cm/8bi 1 1 » which means that errors in b are not magnified when they are
transmitted to C. In addition, the fact that 3Cm/3b-j is positive means that
an increase in bn- causes an increase in Cm, and, similarly, a decrease in b..
results in a decrease in Cm. As in the case of m, recurrent errors in b are
additive.
The sensitivity coefficients involving S, the internal pollutant source
rate, will be discussed next. It is seen from Equation (26) that
9C a,
< ' 1 lL min . (33)
As before, due to the action of a.,-, the effects of an error in S at t-j on C
will decrease with time. In addition, considering the fact that V is a large
number and that the product V times v + D is large (even though v is usually in
the interval (0.1, 2.0)), it is seen that aCm/9S-j will be small. Thus, C is
relatively insensitive to errors in S. Moreover, since 9Cm/9Si is positive,
it is seen that an increase (decrease) in S^ will result in an increase
(decrease) in Cm. Finally, as in the case of m, recurrent errors in S are
additive.
The next sensitivity coefficients to be studied are those that deal
with chemical decay rate, D. From Equation (27) it is seen that the expres-
sion for 9Cm/9Dn- is a very complicated one. As a result, it is hard to make
any meaningful analysis or calculate any useful bounds. As before, the
effects of an error in vi at t = t-j will diminish with time, and the effects
of recurrent errors are additive.
-49-
-------
The next sensitivity coefficient to be examined is 9Cm/3V. Since V
does not change from hour to hour, any error associated with V will occur
initially and will dissipate with time as in the case of CQ. Also, it is
easily seen from Equation (28) that 3Cm/3V is less than zero. Thus, an
increase (decrease) in V would cause a decrease (increase) in Cm. Values
of V are usually obtainable from plans or blueprints, thus minimizing any
error connected with V; thus V is, for all practical purposes, a known con-
stant, and 9Cm/3V was presented here as a point of interest.
Finally, the sensitivity coefficients associated with the air exchange
rate v will be discussed. As can be seen from Equation (29), aC_/3V- is
a complex expression, and, due to the interactions of the various elements
of the equation, it is difficult to make any meaningful analysis or to com-
pute any useful bounds. As in previous cases, the effects of an error in
v-j at t = t-j on C will diminish with time, and the effects of recurrent
errors are additive.
The balance of this section will consist of graphical illustrations
representing the effects induced on the indoor pollutant concentrations by
errors imposed on different input parameters. Figure 8 represents the nominal
conditions, i.e., a carbon monoxide (CO) 8-h episode calculated by the
GIOAP model; this episode is extracted from the data set of the Baltimore
Conventional Residence, Visit Number 1. The baseline conditions are also
shown in Table 13.
Figure 9 shows how a 50% error on the initial condition affects C^
over the 8-h episode. The figure illustrates that for this example the
effects of the initial condition error on the indoor concentration are
essentially eliminated after 2 h. This case is an example of a nonrecur-
rent error; i.e., a single parameter is perturbed only once in the episode,
and the effects of the introduced perturbation are traced for the duration
of the episode. The same type of error behavior in C-jn seen in this example
will occur for any other parameter under similar conditions.
-50-
-------
3.0
2.7
2.4
2.1
O i c
•a 1'b
a
fi
-------
3.0
2.7
2.4
2.1
5,1.8
a.
a
El. 5
a
fi
§1.2
o
U
.9
.6
.3
BALTIMORE CONVENTIONAL RESIDENCE
CO ppm. 9/7/76
Key:
— Nominal
O O Perturbed
I I
J I
J I
I
12 123456789
10 11 12 1
Noon
23456
8 9 10 11 12
Time (hours)
Figure 9. Comparison of nominal values obtained by perturbing Cin.
Figure 10 illustrates the errors in the indoor pollutant levels caused
by recurrent time-dependent errors in the internal source (S) rate term. The
magnitude of the error in S at a given time is 30% of the corresponding inter-
nal source rate.
Figure 11 illustrates the errors in the indoor pollutant level caused by
a recurrent constant error in the air exchange rate (v). The nominal input
value for v is 1.2 air exchanges per hour, the error used is 0.2 air exchanges
per hour.
Numerical investigations of the above illustrated cases appear in
Appendix B, where the sensitivity coefficients, the parameter errors (AS,
Av, ACg), the actual output errors (AC-jn), and approximate output errors
(dC-jn) are tabulated.
-52-
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3.0
2.7
2.4
2.1
I
1.5
-
§ 1.2
o
U
.9
BALTIMORE CONVENTIONAL RESIDENCE
CO ppm. 9/7/76
Key:
0—0
Nominal
Perturbed
J I
I l I I I
I 1 I I I I I I I I I I I
12 1 2 3 4 5 6 7
9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
Noon
Time (hours)
Figure 10. Comparison of nominal values with values obtained by perturbing S.
-53-
-------
3.0
2.7
2.4
2.1
g-1.8
e
o
'°
a
ti
c
u
BALTIMORE CONVENTIONAL RESIDENCE
CO ppm . 9/7/76
U
.9
.6
Key:
Nominal
O—-O Perturbed
I I I
I I I I I I I
12 1 234 5678 9 10 11 12 1 2 3 45 6 7 8 9 10 11 12
Noon
Time( hours)
Figure 11. Comparison of nominal values with values obtained by perturbing u.
-54-
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SECTION 4
CONCLUSIONS
This study indicates that the GIOAP model performs well. The funda-
mental principles and assumptions used in the formulation of the GIOAP model
are similar to the concepts used for other indoor numerical models; however,
the GIOAP model is different from previous numerical efforts because it is
applicable to, and has been applied to, a large variety of pollutants and
because it simulates short periods of time since the transient term is
included. Most importantly, the GIOAP model stands alone because it is the
only indoor-outdoor numerical model that has been validated against a large
set of observed data.
The GIOAP model has been tested under a wide variety of meteorological
and behavioral conditions. Weather conditions encountered ranged from late
autumn in Denver to summer in Baltimore to winter in Pittsburgh. Behavioral
patterns varied widely, e.g., families with children versus families with-
out children, and families with smoking members versus nonsmoking families.
In addition the GIOAP has simulated conditions in residences of different
structural characteristics, e.g., detached dwellings, row houses, apartments,
and mobile homes. Thus, not only does the GIOAP model provide good estimates
of indoor air pollutant concentration levels for different pollutants, but also
it does so for various types of residential structures under diverse meteorologv
cal and behavioral conditions.
The transient term included in the GIOAP model does not appear in other
numerical models that estimate indoor air pollution levels. Examination of
the GIOAP model and simulations with and without the transient term have led
to the following conclusions: (a) the transient term contributes substantially
when the variation of indoor concentrations for stable pollutants is simulated;
(b) the transient term becomes less significant for moderately reactive pollu-
tants; and (c) the transient term is unimportant for ozone which is a highly
reactive pollutant.
-55-
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The model validation phase was undertaken under "best" conditions, i.e.,
the parameter values used for simulation were the best estimates available.
For CQ, m, b, and V the actual values were obtained from the monitoring data.
The large data base available for this study was utilized to compute con-
strained best least-squares estimates for S and v, the two parameters most
difficult to obtain. The estimates for these two parameters were computed
with the Parameter Estimation Procedure (PEP) (see Section 3). Finally,
values for D for various pollutants were obtained from the available litera-
ture.
The motivation for obtaining the "best" estimates for all the parameters
underscores our desire to validate the GIOAP model under ideal conditions, to
characterize its performance without considering the problems of obtaining
realistic values for the input parameters, and subsequently, to determine its
sensitivity resulting from errors in the values of the input parameters.
The GIOAP model was statistically tested using two set of rules: (a)
strict statistical tests, and (b) predetermined empirical criteria. Under
"best" conditions, the model performance is divided into three categories:
1. Not Validated—Due to low outdoor levels and a negative inter-
ference of C02 on the S02 monitor, a large number of hourly
sulfur dioxide concentrations are measured close to the thres-
hold value of the monitor. In the indoor environment S02 and
N02 have an approximately equal half-life; it is thus judged
that the GIOAP model would validate satisfactorily against
higher indoor S02 concentrations.
2. Adequate—Ei'ghty-five percent of the observed indoor ozone
values are in the low range of 0-6 ppb. The model estimated
values are mostly within 2 ppb (less than the instrument pre-
cision) of the observed values. The numerical output does not
satisfy the predetermined model validation criteria, but it
provides realistic estimations of the indoor ozone concentrations.
3. Satisfactory—Model estimated values are within 25% of the
observed indoor values for the following pollutants:
CO, NO, NMHC, CH4, C02, and N02.
-56-
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Under "best" conditions the GIOAP model provides realistic estimates
of the observed indoor pollutant concentrations. However, due to the nature
of the model usage, it is necessary to investigate the performance of the
model under less than ideal conditions. As a result, a sensitivity study
of the GIOAP model is necessary. The set of parameters associated with
the GIOAP model has been divided into two groups: (a) those parameters
that remain constant throughout an episode (CQ, V), and (b) those that must
be estimated for every hour of the episode (m, b, S, \>, D).
The model is comparatively insensitive with respect to parameters in the
first group, i.e., errors in CQ and V have relatively little effect on the
model output. In addition, the effects of errors in CQ and V dissipate with
time (see Section 3). Errors in the second group have more impact on the
model output. This is due to two factors: (a) estimates of parameter values
in this group are more susceptible to error than are the parameters in the
first group, and (b) errors can be introduced at each hour of the episode
because these parameters must be estimated every hour. In an effort to stratify
the parameters of this group on a relative basis, they are ranked as follows
from the least to the most sensitive: (1) S, (2) b, and (3) D, m and v. S
is the least sensitive due to the fact that the magnitude of the numerator
is less than one, and the denominator, which contains a factor of V, is
relatively large, b is considered to be somewhat more sensitive than S due
to the fact that 9C/3S = (l/vV)(aC/3b) and that, within the range of values
being used, 1/vV is less than 1. Finally, D, m, and v are considered to be
the most sensitive, even though the complexity of their respective sensi-
tivity coefficients does not allow any general conclusions to be drawn.
Intuition and the examples studied suggest that these are the three most
sensitive parameters.
-57-
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REFERENCES
1. Shair, F.H., and K.L. Heitner. Theoretical Model for Relating Indoor
Pollutant Concentrations to those Outside. Environ. Sci. Techno!.,
8:444-51, 1974.
2. Milly, G.H. A Theory of Chemical Attack of Tanks and Enclosed Forti-
fications. Report for the Chemical Corps, Chemical and Radiological
Laboratories, Army Chemical Center, Maryland, 1953.
3. Calder, K.L. A Numerical Analysis of the Protection Afforded by
Buildings Against BW Aerosol Attack. BWL Technical Study No. 2,
Office of the Deputy Commander for Scientific Activities, Fort
Detrick, Maryland, 1957.
4. Turk, A. Measurements of Odorous Vapors in Test Chambers: Theoreti-
cal. ASHRAE 9(5):55-8, 1963.
5. Hunt, C.M., B.C. Cadoff, and F.J. Powell. Indoor Air Pollution Status
Report. NBS Report 10-591, National Bureau of Standards Project
4214101. Gaithersburg, Maryland, 1971.
6. Wade, W.A., W.A. Cote, and J.E. Yocom. A Study of Indoor Air Quality.
J. Air Poll. Control Assoc. 25:933-9, 1975.
BIBLIOGRAPHY
Bevington, P.R. Data Reduction and Error Analysis for the Physical
Sciences. McGraw-Hill Book Company, New York, N.Y., 1969.
-58-
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APPENDIX A
DERIVATION OF THE GIOAP MODEL SENSITIVITY COEFFICIENTS
As stated in the text, a sensitivity study involves analyzing the change
in the output(s) of a model resulting from a change in the parameter(s). When
a function such as
i = f(tr P) (A-l)
where
t- = time, i = 1 , . . . , n
P" = {p-|, ..., pk> = vector of parameters
P"n = {p, , . . . , p. } = fixed value of P"
u IQ KQ
is to undergo a sensitivity analysis, the procedure is straightforward:
evaluate the first partial derivatives of f with respect to the parameter(s)
at a specified condition (i.e., 3f/8p,- /t- F \ » J = 1 » > • • » k) in order to
J v i > 0
determine how variations in the parameter(s) affect the output. However,
when the function is of the form
Ps y-) (A-2)
where
t.j = time, i = 1 , . . . , n
yg = initial value of the output variable
PI' = P"(t-j) = {p-|j, ... , Pk-j}, i = 1 , ... , n = vector of time-
dependent parameters
P"i = (Pii , .. . , Pki } = fixed value of F.:, i = 1 , ... , n
'0 M0 0
the analysis becomes considerably more involved. This 'is due to the fact
that at t = t-j , f is not only a function of P^ but also of all the previous
parameter vectors, Pj, j = 1, ..., i-1, and yg as a result of the dependence
-59-
-------
on y-j_]. Thus, the sensitivity coefficients that must be determined are
3f
j = 1, ..., n
(A-3)
3f
ap,
(tJ*
i = 1, .... k, j = 1, ..., n
(A-4)
The sensitivity coefficients will be derived below by first obtaining a
general expression for the total differential, then taking the sensitivity
coefficients to be the coefficients of the parameter differentials. Before
the derivation of the sensitivity coefficients for Equation (A-2) can be
developed, the following additional notation is required.
dP.
urvu^; -^up-,., ..., dpk-j)i i = 1, ..., n = vector of
time-dependent parameter variations
= (dp-,- , ..., dp. . ) = fixed value of dF., i = 1 n
llQ KlQ 1
= the change in the dependent variable at time, tj,
i = 1, ..., n, due to a change in the parameter(s)
(t,, P, , y, ,
1 nO n"'
= the vector of partial derivatives of the function
with respect to the parameter(s) evaluated at
3f
dP. =
* *
= the dot product of the two vectors,
and dP
.
-60-
-------
The general expression for dy , 1 ^m^n, is derived in the following set of
equations:
i i
dy = dy - ' dp (A~5)
3y2 3y2 _ 3y2 9y-i
dy2 = w~ dyi + ~ ' dP2 = "3^" w dyo
f- OY-I I ,-n f-n OVi OV/N U
ay-,
dP, + • dP? (A-6)
'0 3P ^0
_
dy3= ady?+ - ' dP3 = ^r ^r ar- dy
3 8y2 Z 3P 30 8y2 3yl 9yO
2 l
_
dpl
'
ay. m-\
pr . (A-8)
m
in 0
-61-
-------
In order to obtain the coefficients of the parameter differentials, we
expand the dot products in Equation (A-8)
/
=
m ay
»
k m- 1 m
i' m ' Viy
dp,
"o •
p"
'i' i '
1 ^
(A-9)
Now the desired sensitivity coefficients may be easily obtained from equa-
tion (A-9). For a given value of m, 1 <^m <^n, the sensitivity coefficients
are as follows:
3f
m ay
= n
(A-10)
_9f_
ap,
— I , . . . , K
j = 1, ..., n
(A-ll)
where
1, j = m
m
n
1 < j < m .
As mentioned previously, at t = tm, f is a function of yQ and P^, i =
1, ..., m. Equation (A-10) illustrates this dependence on y0 and shows how
an error in yg is propagated throughout the time period being modeled.
-62-
-------
Similarly, Equation (A-ll) for j = 1, ..., m-1 illustrates the dependence of
f on past parameter vectors and shows how an error in a parameter is propa-
gated: at the time the error appears, the change in the dependent variable
is due to the change in the parameter; however, at subsequent points in time,
this error is manifested as a change in the dependent variable resulting
from a change in the y-j_i term. Finally, Equation (A-ll) at j = m shows that,
when an initial error occurs in a parameter, the change in the dependent
variable is due only to the parameter error.
In order to apply Equations (A-10) and (A-ll) to the GIOAP model, let
tn- = tf, t^-i = tQ, and evaluate the expressions defined by Equations (4)
and (16) through (22) of Section 2 at t = tf. Equation (A-10), when applied
to the GIOAP model at t = tm, 1 ^ m <^ n, becomes
m
3f
(V V V V V v' V
= e
i=l i i
(A-12)
Similarly, the following sensitivity coefficients are obtained by applying
Equation (A-ll) at t = tm to Equations (17) through (22):
3f
3m.
( b' V' C-}
m
(A-13)
af
(tm'V bm' V V V'
\-14)
-63-
-------
3f
9Si
V V V
V> V Cm-l>
1 - e
-(V.J+D.)
(A-15)
3f
9Di
V V V Dm« V'
S 2m
S. m.v, \i -(V.+D.)
l n
^i+T-DTT^ViS-
(A-16)
9f
9V
(tm'
Si
(A-17)
3f
bm« V Dm«
-M! _-(VDT}
(A-18)
-64-
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APPENDIX B
NUMERICAL SENSITIVITY ANALYSIS EXAMPLES
FOR THE GIOAP MODEL
In this appendix, three examples are presented which will illustrate
the use of the sensitivity coefficients given in Section 3 of this report.
The data used in these examples are taken from actual model calculations (CO
data for the Baltimore Conventional House, first visit, hours 8-15). Table
B-l gives the nominal values for all parameters used in the examples.
TABLE B-l. NOMINAL CONDITIONS USED IN THE SENSITIVITY STUDY EXAMPLES
House: Baltimore Conventional (Visit #1)
Pollutant: CO
Volume: 13.575 ft3 ;
Hour
8
9
10
11
12
13
14
15
Cin
(ppm)
1.33
2.23
1.04
0.31
0.68
1.02
1.01
0.30
cout
(ppm)
1.33
1.33
0.00
0.00
0.00
0.00
0.00
0.00
S
(mg/h)
_.
677.77
0.00
0.00
440. 14
619.13
528.17
0.00
V
(air exchanges/h)
_
1.20
1.20
1.20
1.20
1.20
1.20
1.20
The first example deals with the case in which an error is made when,a
parameter value is estimated initially, but, after that initial error, no
other errors are introduced. This situation is most likely to arise in the
estimation of the initial indoor air pollutant concentration, CQ, because
it is estimated only once unlike some other parameters (e.g., S and v) which
must be estimated for each hour. Using Equation (23), Table B-2 gives the
approximate error (dCin) and actual error (AC-jn) for each hour due to an error
in C0. Here it should be noted that, since the GIOAP model is a linear func-
tion of CQ, dCin = AC-jn; however, this does not show up ,in some of the entries
in the table due to the fact that some values were rounded off. As mentioned
-65-
-------
in Section 3, since the sign of 3Cin/aCin is positive, a decrease (increase)
in C0 will cause a decrease (increase) in C which can be seen in Table B-2.
Finally, for this particular case, it is seen that by the llth hour the effects
of the error in CQ on C are minimal.
TABLE B-2. ERRORS IN C DUE TO AN ERROR IN C
in in
House: Baltimore Conventional (Visit #1)
Pollutant:
Hour
8
9
10
11
12
13
14
15
CO
ACinQ
(ppm)
-0.665
0.000
0.000
0.000
0.000
0.000
0.000
0.000
3Cin
5q^
_
0.3012
0.0907
0. 0273
0.0082
0.0025
0.0007
0.0002
dCin
(ppm)
_
-0. 2003
-0.0603
-0.0182
-0. 0055
-0.0016
-0. 0005
-0. 0001
Acin
(ppm)
_
-0.20
-0.06
-0.01
-0.01
0.00
0.00
0.00
The next example deals with the case in which the parameter errors vary
with time and are recurrent. This situation can occur in any number of the
GIOAP parameters (e.g., m, b, and v) which must be estimated. For this example
the internal source rate parameter (S) was chosen, and AS^ = - 0.3 Si, i = 9,
.... 15 (i.e., a negative 30% error in S). The data for this example
are presented in Table B-3. Here, as in the previous example, dCin = ACin
because the GIOAP model is linear with respect to S. As pointed out in
Section 3, the sign of aC^/aS is positive; thus a decrease (increase) in S
results in a decrease (increase) in Cin. This is illustrated in Table B-3.
Also, the error term consists not only of the error due to the current per-
turbation of S but also of the error due to the past perturbations of S which
are transmitted via the Ci_i term (see Equation (5)). Notice that during
hours 9 through 11, the situation is similar to that of the previous example
(i.e., an initial error with no errors introduced subsequently) and that the
-66-
-------
effect of the error in S has diminished by the. llth hour. However, during
hours 12 through 14, errors in S occur each hour so, even though the effects
of previous errors in S begin to dissipate, the overall error in Cin is not
reduced appreciably.
TAB1EB-3. ERRORS IN C. DUE TO ERRORS IN S
in
House: Baltimore Conventional (Visit #1)
Pollutant: CO
Hour
8
9
10
11
12
13
14
15
AS
(mg/h)
^
-203.331
0.000
0.000
-132.042
-185.739
-158.451
0.000
3Cin
9S
1 (ppm/mg/h)
_.
0.0015
0.0015
0.0015
0.0015
0.0015
0.0015
0.0015
ac.n
9Ci*0
_
-
0.3012
0.3012
0.3012
0.3012
0.3012
0.3012
dCin*
(ppm)
_
-0.2690
-0.0810
-0. 0244
-0. 1820
-0. 3005
-0. 3001
-0. 0904
ACin
(ppm)
_
-0.27
-0.08
-0.02
-0.18
-0.30
-0.30
-0.09
* In these tables, dCjn (Col. 5) was computed using the following
formula:
. . , 15
i-l .1
1 Col. 5 1
,
Col. 2
Col. 5
Col. 4 Col. 3
This formula is an equivalent form of equation (A-8) (see Appendix
A). Also, it should be noted that any attempt to calculate dC-
from its components appears to result in an error. Actually, some
of the values were obtained via computer and were rounded off when
entered in the table.
The final example illustrates two cases: (a) that of a recurrent con-
stant error, and (b) that in which the GIOAP model 'is not a linear function
of the parameter being considered. The parameter being dealt with here is
the air exchange rate (v). Table B-4 presents the data associated with this
example. In this example, as opposed to the previous one, the error (AV =
0.20) occurs every hour, thus the error induced in C^n does not have a chance
-67-
-------
to dissipate. Also, this case gives an idea of how well dCin approximates
AC- when the model is not linear in the parameter being studied. Here, the
approximation is fair since dC- , in most cases, agrees with AC- , in the
first decimal place. Ideally, Av should be small enough so that the GIOAP
model is approximately linear in the Av-neighborhood about the nominal point.
This is illustrated by Figure B-l which shows how, for y = f(x), Ay =
f(x+Ax) - f(x) differs from the approximation of Ay, Aay, found using the
formula Aay = f'(x)Ax. In this case, if AV were much larger, the validity
of using dC^ to approximate AC^n would be questionable. Finally, in this
case, 3Cin/9v is negative, which means that a decrease (increase) in v
results in an .increase (decrease) in C^. This is intuitively plausible;
however, due to the complexity of 3C1-n/8v, it is not possible to say that
this is always the case.
TABLE B-4. ERRORS IN Cin DUE TO ERRORS IN v
House: Baltimore Conventional (Visit ttl)
Pollutant: CO
Hour
Ac
(air exchanges/h)
dCin*
(ppm/air exchanges/h)
(ppm) (ppm)
. 8
9
10
11
12
13
14
15
.
-0. 20
-0.20
-0.20
-0.20
-0.20
-0.20
-0..20
.
-0. 3608
-0.5827
-0.3132
-0. 3277
-0.5344
-0.5884
-0.3042
_
-
0.3012
0.3012
0. 3012
0.3012
0. 3012
0.3012
—
0. 0722
0.1382
0.1043
0.0969
0.1361
0. 1587
0. 1086
_
0.07
0.16
0.13
0.11
0.16
0.18
0.14
* In these tables, dCjn (Col. 5) was computed using the following formula
vv i = 9, . . . , 15
| Col. 5 1 Col. 2
Col. 4 Col. 3
This formula is an equivalent form of equation (A-8) (see Appendix A). Also, it should
be noted that any attempt to calculate dQn from its components appears to result in an
error. Actually, some of the values were obtained via computer and were rounded off
when entered in the table.
-68-
-------
Figure B-l. Graph showing how Ay differs from Ay (the approximation to Ay computed as Afly = f (x_)Ax).
-69-
-------
TECHNICAL REPORT DATA
•(Please read Inunctions on the reverse before completing)
2.
3. RECIPIENT'S ACCESSION-NO.
1. 7ITLS AMC? SUBTITLE
The. CKOMT.T In. loor -Outdoor Air Pollution Model
6. PERFORMING ORGANIZATION CODE
7. AUTHORISI
Demi-trios j. Moschandreas
John W.C. SMrl;
8. PERFORMING ORGANIZATION REPORT NO.
CEOMET Report. Number EF-628
i. REPORT DATE (Issue date)
February 10, 1978
9. PEHPORMINC- ORGANIZATION NAME AND ADDRESS
GF,OMET, Incorporated
15 Firslfiold Ro;ul
Giii'tlif.rsburs, Maryland 20760
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
68-02-2294
12. SPONSORING AGENCY NAME AND ADDRESS
U.S. EPA
Environmental Research Center
Research Triangle Park, North Carolina
and
U.S. Department of Housing and
Urban Development
Office of Policy Development and
Research
13. TYPE OF REPORT AND PERIOD COVERED
Scientific Report
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
16. A
•TliiJ re-pert documents the formulation of the GEOMET Indoor-Outdoor Air Pollution (GIOAP) model. The model esti-
mates indoor ;iir pollutant concentrations as a function of outdoor pollutant levels, indoor pollutant generation sources rates,
pollui.in!. clii-mical decay rat<:s, and air exchange rates. Topics discussed include basic principles, model formulation,
paraiiu-ic.r '-Mini ilion, model statistical validation, and model sensitivity to perturbations of the input parameters.
'Mi.' 'iiimciiral estimates obtained from the GIOAP simulations have been validated with observed hourly pollutant con-
cenlr itions obtained from an J8-mo residential air quality sampling program. Statistically, the model values of carbon
monoxide .ire within 10% of the observed values in the concentration range interval which includes 85%.of all hourly measure-
ments; similarly, for nitric oxide the predicted values are within 1596 of the observed; for nitrogen dioxide the difference
httlween I lie predicted value and the ideal condition of exactly estimating the corresponding observed value is 16% for 85% of
the- ol-'sorved hourly concentrations. Also, the GIOAP model predicts within 25% for nonmethane hydrocarbons and within
S.y, for rarhon dioxide. Thus statistically the GIOAP model estimations are within 25% of the ideal condition (estimated and
observed values coincide) for S5-?S of the observed values for CO, NO, NO2, NMHC, and CO2- The model has not been vali-
dated .ip.ainst sulfur dioxide owing to the very low values measured both indoors and outdoors. The predetermined validation
criteria were not satisfied by the ozone model estimations; however, the calculated values were judged adequate because for
nlwnt W!f- of- the observed values the predicted concentrations were within 2 ppb.
Scnsit Ivi'.y studies on the GIOAP model parameters indicate that errors in the estimation of the initial condition and the
volume of i.!n: structure dissipate with time. Errors in estimating the air exchange rate, the indoor generation strength, and
the indoor chemical decay rate :tre more significant. Sensitivity coefficients have been formulated for all input parameters.
The transient t.erm is a unique feature of the GIOAP model; the impact of this term is substantial for stable pollutants but
17.
i.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
Air Quality
Nuiii'.'ric.al Mi,1 dels
M'-drl Validation
SriiMi.Mly Shulles
Si.'-jul;- Slate
Term
b.lOENTIFIERS/OPEN ENDED TERMS
Prediction of Indoor Pollutant
Concentrations
c. COSATl Field/Group
19. OlS'fl'OUTION STATEMENT
Kele.ise Unlimited
19. SECURITY CLASS (This Re pan)
Unclassified
21. NO. OF PAGES
75
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Unclassified
22. PRICE
SPA Form ZJ20-I (9-73)
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