EPA-600/4-76-016a
May 1976
Environmental Monitoring Series
CONTINUED RESEARCH IN MESOSCALE AIR
POLLUTION SIMULATION MODELING:
Volume I - Assessment of Prior Model
Evaluation Studies and Analysis
of Model Validity and Sensitivity
Environmental Sciences Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
lesearch Triangle Park, North Carolin
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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.
the Nalional Teohnical
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EPA 600/4-76-016 A
Hay 1976
CONTINUED RESEARCH IN MESOSCALE AIR
POLLUTION SIMULATION MODELING:
VOLUME I - ASSESSMENT OF PRIOR MODEL EVALUATION STUDIES
AND ANALYSIS OF MODEL VALIDITY AND SENSITIVITY
by
M. K. Liu
D. C. Whitney
J. H. Seinfeld
P. M. Roth
Systems Applications, Incorporated
950 Northgate Drive
San Rafael, California 94903
68-02-1237
Project Officer
Kenneth L. Demerjian
Meteorology and Assessment Divison
Environmental Sciences and Research Laboratory
Research Triangle Park, North Carolina 27711
U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND DEVELOPMENT
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
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11
DISCLAIMER
This report has been reviewed by the Office of Research and
Development, U.S. Environmental Protection Agency, and approved
for publication. Mention of trade names or commercial products
does not constitute endorsement or recommendation for use.
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CONTENTS
LIST OF ILLUSTRATIONS v
LIST OF TABLES xiii
ACKNOWLEDGMENTS xiv
I OVERVIEW 1
II ANALYSIS OF THE RESULTS OF PAST MODEL VERIFICATION STUDIES ... 5
A. Introduction 5
B. The Data Base 6
C. Selection of Analyses of the Data 12
D. Results 21
1. Data Set Selection 21
2. Statistical Analysis 22
3. Scatter Plots 30
4. Residuals Analyses 42
E. Conclusions 66
III ASSESSMENT OF THE VALIDITY OF AIRSHED MODELS 68
A. Introduction 68
B. A Theoretical Analysis of the Validity of the Airshed Models 72
1. The Trajectory Model 73
2. The Grid Model 77
C. Assessing the Validity of Airshed Models Through
Numerical Experiments 79
D. The Validity of the Trajectory Model ..... 81
1. The Effect of Horizontal Diffusion 84
2. The Effect of Vertical Winds 103
3. The Effect of Wind Shear 116
E. The Validity of the Grid Model—The Effect of
Numerical Errors ..... 133
F. Conclusions and Recommendations 143
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IV SENSITIVITY STUDY OF THE SAI URBAN AIRSHED MODEL 146
A. Introduction 146
B. Design of the Sensitivity Study 148
1. Plans for Carrying Out the Sensitivity Study 148
2. Criteria for Assessing the Sensitivity of
the SAI Model 151
C. Analysis of the Sensitivity of the SAI Model 156
1. The Effect of Random Perturbations in the Wind Field ... 156
2. The Effect of Variations in Wind Speed 164
3. The Effect of Variations in Turbulent Diffusivity .... 177
4. The Effect of Variations in Mixing Depth 181
5. The Effect of Variations in Radiation Intensity 202
6. The Effect of Variations in Emissions Rate 211
D. Discussion and Conclusions 213
1. Justification for a Complex Model 222
2. The Sensitivity of the SAI Model 222
APPENDICES
A The Nonuniqueness of Lagrangian Velocities 229
B Wind and Diffusivity Profiles in ''"he Lower Atmosphere .... 233
C A Theoretical Analysis of the Effect of Random
Perturbations of the Measured Wind 238
REFERENCES 242
FORM 2220-1 246
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ILLUSTRATIONS
II-l Locations of Monitoring Stations Relative to Major
Contaminant Sources in the Los Angeles Basin 8
II-2 Scatter Plots for the CO Results 31
II-3 Scatter Plots for the NO Results 33
II-4 Scatter Plots for the N02 Results 36
II-5 Scatter Plots for the 03 Results 38
II-6 Residuals (PESC Minus PESM) Analyses of the PES Results for CO . 43
II-7 Residuals (SAIC Minus SAIM) Analyses of the SAI Station
Results for CO 44
II-8 Residuals (CORC Minus CORM) Analyses 'of the Correlated
Station Results for CO 45
II-9 Residuals (GRCT Minus GRCI) Analyses of the GRC Results for CO . 46
11-10 Residuals (SAIT Minus SAII) Analyses of the SAI Trajectory
Results for CO 47
'11-11 Residuals (PESC Minus PESM) Analyses of the PES Results for NO . 48
U-12 Residuals (SAIC Minus SAIM) Analyses of the SAI Station
Results for NO „ 49
11-13 Residuals (CORC Minus CORM) Analyses of the Correlated
Station Results for NO 50
11-14 Residuals (GRCT Minus GRCI) Analyses of the GRC Results for NO . 51
11-15 Residuals (SAIT Minus SAII) Analyses of the SAI Trajectory
Results for NO • 52
11-16 Residuals (PESC Minus PESM) Analyses of the PES Results for N02- 53
11-17 Residuals (SAIC Minus SAIM) Analyses of the SAI Station
Results for N02 54
11-18 Residuals (CORC Minus CORM) Analyses of the Correlated -
Station Results for N00 55
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VI
11-19 Residuals (GRCT Minus GRCI) Analyses of the GRC Results for N02> 56
11-20 Residuals (SAIT Minus SAII) Analyses of the SAI Trajectory
Results for N02 57
11-21 Residuals (PESO Minus PESM) Analyses of the PES Results for 03 . 58
11-22 Residuals (SAIC Minus SAIM) Analyses of the SAI Station
Results for 03 59
11-23 Residuals (CORC Minus CORM) Analyses of the Correlated
Station Results for CL 60
11-24 Residuals (GRCT Minus GRCI) Analyses of the GRC Results for 03 . 61
11-25 Residuals (SAIT Minus SAII) Analyses of the SAI Trajectory
Results for 03 62
III-l Diagram of the Basic Relationships in the Validity Study .... 71
III-2 The Effect of Neglecting Horizontal Diffusion on the Trajectory
Model Predictions (for Instantaneous Line Sources) 91
III-3 Spatial Distribution of Carbon Monoxide Emissions
(10:00 A.M. PST) 97
III-4 Temporal Distribution of Carbon Monoxide Emissions 98
III-5 The Effect of Neglecting Horizontal Diffusion on the Trajectory
Model Predictions (for Urban-Type Sources) 102
III-6 The Effect of Vertical Wind on the Trajectory Model Predictions 108
III-7 Assessing the Effect of Wind Shear 117
II1-8 The Effect of Wind Shear on Trajectory Model Predictions
(for Line Sources) 120
III-9 The Effect of Wind Shear on Trajectory Model Predictions
(for Area! Sources) 127
III-l0 The Effect of Numerical Errors on Grid Model Predictions:
Results Using the First-Order Finite Difference Scheme
(Wind Speed = 4 MPH) 136
III-ll The Effect of Numerical Errors on Grid Model Predictions:
Results Using the Second-Order Finite Difference Scheme
and Realistic Spatial and Temporal Emission Patterns 138
111-12 A Smooth Pattern of Pollutant Emissions 139
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vn
111-13 The Effect of Numerical Errors on Grid Model Predictions:
Results Using the Second Order Finite Difference Scheme
and Smooth Spatial and Temporal Emission Patterns 140
111-14 The Effect of Numerical Errors on Grid Model Predictions:
Results Under the Same Conditions as Those of Figure 111-10,
Except for an Increase in Horizontal Diffusion 141
111-15 The Effect of Numerical Errors on Grid Model Predictions:
Results Under the Same Conditions as Those of Figure 111-10,
Except for a Reduction in Wind Speed 142
IV-1 The Effect—Expressed as Average Deviations—of Random
Perturbations in Wind Direction 158
IV-2 The Effect—Expressed as Standard Deviations—of Random
Perturbations in Wind Direction 159
PV-S The Effect—Expressed as Average Deviations—of Random
Perturbations in Wind Speed 162
IV-4 The Effect—Expressed as Standard Deviations—of Random
Perturbations in Wind Speed 163
IV-5 Relative Changes in Wind Speed and Direction at the Locations
of Maxima for the Base Case 166
IY-6 The Effect—Expressed as Average Deviations—of Variations
in Wind Speed for CO 167
IV-7 The Effect—Expressed as Average Deviations—of Variations
in Wind Speed for NO 167
IV'-S The Effect—Expressed as Average Deviations—of Variations
in Wind Speed for 03 168
IV-9 The Effect—Expressed as Average Deviations—of Variations
in Wind Speed for N02 168
I'V-10 The Effect—Expressed as Percentage Deviations—of Variations
in Wind Speed for CO 169
IV-11 The Effect—Expressed as Percentage Deviations—of Variations
in Wind Speed for NO 169
IV-12 The Effect—Expressed as Percentage Deviations—of Variations
in Wind Speed for 03 170
IV-13 The Effect—Expressed as Percentage Deviations—of Variations
in Wind Speed for N02 170
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IV-14 The Effect--Expressed as Maximum Deviations — of Variations
in Wind Speed for CO ..................... 171
IV-15 The Effect—Expressed as Maximum Deviations—of Variations
in Wind Speed for NO ..................... 171
IV-16 The Effect—Expressed as Maximum Deviations— of Variations
in Wind Speed for 0, ..................... 172
IV-17 The Effect— Expressed as Maximum Deviations—of Variations
in Wind Speed for N02 ..................... 172
IV-18 The Effect—Expressed as Maximum Percentage Deviations —
of Variations in Wind Speed for CO .............. 173
IV"-19 The Effect — Expressed as Maximum Percentage Deviations —
of Variations in Wind Speed for NO .............. 173
rV-20 The Effect — Expressed as Maximum Percentage Deviations—
of Variations in Wind Speed for 03 .............. 174
IV-21 The Effect — Expressed as Maximum Percentage Deviations —
of Variations in Wind Speed for N02 .............. 174
rV-22 The Effect — Expressed as Average Deviations — of Variations
in Vertical Diffusivity for CO ................ 182
IV-23 The Effect — Expressed as Average Deviations — of Variations
in Vertical Diffusivity for NO ................ 183
IV-24 The Effect— Expressed as Average Deviations— of Variations
in Vertical Diffusivity for 03 ................ 184
IV-25 The Effect— Expressed as Average Deviations— of Variations
in Vertical Diffusivity for N02 ................ 185
IV-26 The Effect— Expressed as Percentage Deviations— of
Variations in Vertical Diffusivity for CO ........... 186
IV-27 The Effect— Expressed as Percentage Deviations— of
Variations in Vertical Diffusivity for NO ........... 187
IV-28 The Effect— Expressed as Percentage Deviations— of
Variations in Vertical Diffusivity for 03 ........... 188
IV-29 The Effect— Expressed as Percentage Deviations— of
Variations in Vertical Diffusivity for N02 .......... 189
IV-30 The Effect— Expressed as Maximum Deviations— of Variations
in Vertical Diffusivity for CO ................ .190
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IV-31 The Effect—Expressed as Maximum Deviations—of Variations
in Vertical Diffusivity for NO 191
IV-32 The Effect—Expressed as Maximum Deviations—of Variations
1n Vertical Diffusivity for 03 192
IV-33 The Effect—Expressed as Maximum Deviations—of Variations
in Vertical Diffusivity for NCL 193
IV-34 The Effect—Expressed as Average Deviations—of Variations
in Mixing Depth for CO 194
IV-35 The Effect—Expressed as Average Deviations—of Variations
tn Mixing Depth for NO 194
IV-36 The Effect—Expressed as Average Deviations—of Variations
in Mixing Depth for 0- 195
IV-37 The Effect—Expressed as Average Deviations—of Variations
in Mixing Depth for N02 195
IV-38 The Effect—Expressed as Percentage Deviations—of
Variations in Mixing Depth for CO 196
IV<-39 The Effect—Expressed as Percentage Deviations—of
Variations in Mixing Depth for NO 196
IV-40 The Effect—Expressed as Percentage Deviations—of
Variations in Mixing Depth for 0., 197
IV-41 The Effect—Expressed as Percentage Deviations—of
Variations in Mixing Depth for NOp 197
IV-42 The Effect—Expressed as Maximum Deviations—of Variations
tn Mixing Depth for CO 198
TV-43 The Effect—Expressed as Maximum Deviations—of Variations
in Mixing Depth for NO 198
IV-44 The Effect—Expressed as Maximum Deviations—of Variations
in Mixing Depth for Oo 199
IV-45 The Effect—Expressed as Maximum Deviations—of Variations
in Mixing Depth for N02 199
IV-46 The Effect—Expressed as Maximun Percentage Deviations—of
Variations in Mixing Depth for CO 200
IV-47 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in Mixing Depth for NO 200
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IV-48 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in Mixing Depth for 0_ 201
IV-49 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in Mixing Depth for NCL 201
IV-50 The Effect—Expressed as Average Deviations—of Variations
in Radiation Intensity for CO 203
IV-51 The Effect—Expressed as Average Deviations—of Variations
in Radiation Intensity for NO 203
IV-52 The Effect—Expressed as Average Deviations—of Variations
in Radiation Intensity for 0~ 204
IV-53 The Effect—Expressed as Average Deviations—of Variations
in Radiation Intensity for N0? 204
IY-54 The Effect—Expressed as Percentage Deviations—of
Variations in Radiation Intensity for CO 205
IV-55 The Effect—Expressed as Percentage Deviations—of
Variations in Radiation Intensity for NO 205
IV-56 The Effect—Expressed as Percentage Deviations—of
Variations in Radiation Intensity for 0, 206
IV-57 The Effect—Expressed as Percentage Deviations—of
Variations in Radiation Intensity for N02 '. 206
IV-58 The Effect—Expressed as Maximum Deviations—of Variations
in Radiation Intensity for CO 207
I'V-59 The Effect—Expressed as Maximum Deviations—of Variations
in Radiation Intensity for NO 207
IV-60 The Effect—Expressed as Maximum Deviations—of Variations
in Radiation Intensity for Og 208
IV-61 The Effect—Expressed as Maximum Deviations—of Variations
in Radiation Intensity for N02 • 208
IV-62 The Effect—Expressed as Maximum Percentage Deviations—of
' Variations in Radiation Intensity for CO 209
IV-63 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in Radiation Intensity for NO 209
IV-64 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in Radiation Intensity for 0- 210
IV-65 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in Radiation Intensity for N02 210
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XI
IV-66 Sketch Illustrating the Effect of Changes in Radiation
Intensity 212
IV-67 The Effect—Expressed as Average Deviations—of Variations
in Emissions Rate for CO 214
IV-68 The Effect—Expressed as Average Deviations—of Variations
in Emissions Rate for NO 214
IV-69 The Effect—Expressed in Average Deviations—of Variations
in Emissions Rate for 0, 215
IV-70 The Effect—Expressed in Average Deviations—of Variations
in Emissions Rate for N0~ 215
IV-71 The Effect—Expressed as Percentage Deviations—of
Variations in Emissions Rate for CO 216
IV-72 The Effect—Expressed as Percentage Deviations—of
Variations in Emissions Rate for NO 216
IV-73 The Effect—Expressed as Percentage Deviations—of
Variations in Emissions Rate for 03 217
IV-74 The Effect—Expressed as Percentage Deviations—of
Variations in Emissions Rate for N0? 217
IV-75 The Effect—Expressed as Maximum Deviations—of Variations
in Emissions Rate for CO 218
IV-76 The Effect—Expressed as Maximum Deviations—of Variations
in Emissions Rate for NO 218
IV-77 The Effect—Expressed as Maximum Deviations—of Variations
in Emissions Rate for 0- 219
IV-78 The Effect—Expressed as Maximum Deviations—of Variations
in Emissions Rate for N02 219
IV-79 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in Emissions Rate for CO 220
IV-80 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in Emissions Rate for NO 220
IV-81 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in Emissions Rate for 03 221
IV-82 The Effect—Expressed as Maximum Percentage Deviations—of
Variations in'Emissions Rate for N00 221
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IV-83 The Average Effect of Changes in Input Parameters
on CO Concentration 224
IV-84 The Average Effect of Changes in Input Parameters
on NO Concentration 225
IV-85 The Average Effect of Changes in Input Parameters
on Oo Concentration 226
IV-86 The Average Effect of Changes in Input Parameters
on N0? Concentration 227
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TABLES
II-l Location of Contaminant Monitoring Stations of the
II-2
III-l
in -2
IV-l
IV-2
IV-3
IY-4
IV-5
Los Angeles Air Pollution Control District
Statistical Analysis for All Locations
Exact Solutions to the Diffusion Equation
Summary of the Cases Considered in the Validity Study ....
Summary of the Cases Investigated in the Sensitivity Study . .
The Largest Deviations in the Grid Generated by Randomly
Varying the Wind Direction
The Largest Deviations in the Grid Generated by Randomly
Varying the Wind Speed
The Largest Deviations in the Grid Generated by Randomly
Varying the Horizontal Diffusion
Ranking of the Relative Importance of the Input Parameters . .
14
23
82
83
150
160
165
179
228
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XIV
ACKNOWLEDGMENT
We_wish to thank Dr. Richard I. Pollack for his comments on Chapter II
and his modifications of some of the computations.
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I OVERVIEW
This report presents a series of studies carried out for the
Environmental Protection Agency (EPA) to evaluate three "first generation"
photochemical air pollution models to determine what modifications or
extensions should be made in developing a "second generation" model.
The three models studied were developed during the period 1969 to
1973 under the sponsorship of the EPA:
Type of Model Developer EPA Contract
Trajectory General Research Corporation 68-02-0336
Pacific Environmental Services, Inc. 68-02-0345
Grid Systems Applications, Inc. 68-02-0339
None of these models as constituted in mid-1973 appeared to be capable of
adequately simulating the physical and chemical processes that occur in a
polluted urban atmosphere, hence the motivation for further model develop-
ment. However, before a second generation model could be developed, certain
issues required resolution:
> In view of their different structures, what is the quality of
performance of each first generation model when compared with
observational data?
> From a theoretical point of view, what is the degree of validity of
each modeling approach?
> From a practical point of view, what are the processes that most
significantly affect urban photochemical air pollution levels?
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Chapters II, III, and IV of this volume describe our efforts to answer
these questions. An assessment of past model performance provides some
insight into the sources of inaccuracy of the first generation models and,
ultimately, an indication of their relative merits. The validity studies
we carried out point out some of the fundamental shortcomings of these
models. And the sensitivity analyses indicate which physical and chemical
processes must be included in the second generation model and to what level
of accuracy they must be represented.
Chapter II presents a comparative analysis of the predicted and
observed pollutant concentrations in the Los Angeles basin reported by the
three developers of first generation models. Owing to the different modes
of data usage and model operation, we could not directly compare the results
of the three studies; consequently, our analysis was limited to an investi-
gation of some of the statistical properties of predicted and measured
values for each study.
The results of the analysis indicate that none of the models can
consistently reproduce measured pollutant concentrations such that the
residual differences between the predicted and measured values can be
ascribed with reasonable assurance to random errors alone. Our investigation
of the causes of these discrepancies revealed both inadequacies in the
models and inappropriateness of the data base. Specifically, overly
simplistic kinetic mechanisms, emissions distributions, and diffusion
algorithms all contributed to discrepancies between the predicted and
spatially averaged observed values. Even more significant were the results
of our comparison of site-specific station measurements with pollutant
concentrations calculated using emissions that were averaged over a four
square mile area and winds that were interpolated among meteorological
stations ten to twenty miles apart. The disparity in scales involved in
these comparisons, together with the nonrepresentative locations of the
measuring stations, presented major barriers to our evaluation of the
relative performance of the models. A true test of model capabilities
will require a more suitable data base.
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Chapter III discusses the validity of airshed models based on the
trajectory and grid approaches. Predictions of both models for realistic
but simplified situations were compared with the exact solutions to the
full atmospheric diffusion equation. The sources of errors in each of the
two approaches were first identified as follows:
Type of Model Source of Error
Trajectory Neglect of horizontal diffusion
Neglect of the vertical component of the wind
Neglect of wind shear in the vertical direction
Grid Introduction of numerical errors due to finite
differencing
Further calculations showed that for trajectory models errors involved in the
neglect of horizontal diffusion were always less than 10 percent, but neglect
of the vertical wind component can lead to errors in prediction as great as
a factor of 2, and neglect of vertical wii.d shear, to errors in excess of
50 percent. Numerical errors in the grid model can result in prediction
errors as high as 50 percent after more than nine hours of simulation if a
conventional second-order finite difference scheme is used.
Chapter IV describes the use of the SAI model as a vehicle to assess
the sensitivity of photochemical air pollution levels to relative changes
in wind speed, horizontal and vertical diffusivities, mixing depth, radiation
intensity, and emis-sions rate. The results indicate that, in general, the
effect of changes in variables on predictions is:
> Highly time dependent, indicating that proper inclusion of
the initial pollutant distributions and other time-dependent
features is essential in urban airshed modeling.
> Strongly spatially dependent, revealing the importance of
adequate spatial resolution in a model. (It should be at least
comparable to that of the emissions distribution.)
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> Different for different chemical species. The relative
sensitivity of predictions to changes in variables or
parameters is given in the following ranking (A = most
important and D = least important):
Parameter or Variable
Wind speed
Horizontal diffusivity
Vertical diffusivity
Mixing depth
Radiation intensity
Emissions rate
CO
A
D
C
B
D
B
NO
A
D
C
B
A
A
°3
A
D
C
B
A
B
N02
A
D
C
B
B
B
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II ANALYSIS OF THE RESULTS OF PAST MEL VERIFICATION STUDIES
A. INTRODUCTION
Each of the three firms that developed the first generation photochemical
air pollution models discussed in this chapter carried out extensive valida-
tion studies of its model. The description of each study, the measured and
calculated concentration values used, and the statistical analysis of the
results appear in the following final reports:
> "Further Development and Evaluation of a Simulation Model for
Estimating Ground Level Concentrations of Photochemical Pol-
lutants," R73-19, Systems Applications, Incorporated, Beverly
Hills (now in San Rafael), California (February 1973).
> "Evaluation of a Diffusion Model of Photochemical Smog Simula-
tion," EPA-R4-73-012, Volume A (CR-1-273), General Research
Corporation, Santa Barbara, California (October 1972).
> "Controlled Evaluation of the Reactive Environmental Simulation
Model (REM)," EPA-R4-73-013a, Volume I, Pacific Environmental
Services, Incorporated, Santa Monica, California (February 1973).
These reports also contain discussions of the possible origins of the devia-
tions between measured and calculated concentrations.
*
The objective of the study reported here was to reevaluate the results
of these analyses in light of the additional experience gained over the past
two years. Toward this end, we used a self-consistent data base that ties
together the three sets of results of these statistical analyses as a basis
*
To avoid confusion between SAI's role as one of the three model developers
discussed and its role as the evaluator of all three studies, we use the
third person reference "SAI" to denote the former and the first person "we"
and "our" for the latter.
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for discussing the physical, mathematical, and modeling sources of the dis-
crepancies between the measured and predicted concentration values obtained
by the three firms. Our discussion draws heavily upon the three final reports
cited above for both data and analyses. For convenience, we refer to these
reports using the initials of the company (SAI, GRC, and PES) and the corres-
ponding page number.
B. THE DATA BASE
The use of trajectory models by GRC and PES and a grid model by SAI
created a disparity in the set of available data points. Based on hourly
calculations of pollutant concentrations within 2x2 mile squares of a 50 x
50 mile grid, the SAI model produces more data points than do the GRC and PES
models. Moreover, the SAI model can essentially reproduce any trajectory
reported by PES and GRC simply by giving the concentration values for each
square through which the trajectory passes. Given this disparity, we decided
to let the size of the data base depend or; the nature of the trajectories
reported by PES and GRC; the SAI contribution was thus simply a one-for-one
match of each of the PES and GRC points.
Unfortunately (or perhaps fortunately, as shown later), PES chose not to
present the results of its hour-by-hour predictions along each trajectory,
but instead set up its model trajectories so that they would pass near cer-
tain measuring stations at preselected times; thus, PES could directly com-
pare the measured and predicted values (PES, p. IV.3). PES tabulated only
the values calculated at or near one of these stations (PES, pp. A.2-A.13),
though consistent, detailed maps of each trajectory for all hours were pre-
sented (PES, pp. A.14-A.19). In contrast, GRC reported predicted concentra-
tions only along its model trajectories. GRC data, presented only in graphics,
(GRC, pp. 101-147) may contain small errors from interpolation due to digiti-
zation. More significantly, the GRC trajectories were depicted on somewhat
free-form maps of various sizes (GRC, pp. 100-146), thus introducing substan-
tial uncertainty in selecting the appropriate SAI grid square to match a
given point on a GRC trajectory.
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Another type of data used in our evaluation was a set of results produced
by the SAI automated input program for the airshed model (SAI, Volume III).
These results compare the measured pollutant concentration at a station with
the calculated concentration at that station obtained by interpolation from
measured values at a set of neighboring stations. Although these results were
originally intended to validate the SAI interpolation scheme, the disparities
between the interpolated and measured values reveal a general problem in the
use of station measurements in such a study, as explained in detail below.
In the discussion that follows, we refer to these values as "interpolated
station points."
As part of its validation effort, each of the three firms was required to
use the available measurement data on pollutant concentrations in the Los
Angeles basin for six smoggy days in late summer and early fall of 1969 as the
basis of comparison between observed and predicted pollutant concentrations.
These data were reported as hourly averages by 10 monitoring stations of the
Los Angeles Air Pollution Control District (LAAPCD) scattered throughout the
Los Angeles basin. In addition, measurements taken by Scott Research Labora-
tories were used by GRC (p. 86) and SAI (p. 67) in carrying out interpolation
calculations for measured values at nonstation locations; SAI (p. 67) also
used data from three stations of the Orange County Air Pollution Control
District (OCAPCD) for such calculations. The locations of all of these mea-
suring stations are shown in Figure II-l.
On the basis of these available data, we selected the following types of
values for inclusion in the "data base":
> All measured (PESM) and predicted (PESC) concentration values
reported by Pacific Environmental Services (PES, pp. A.2-A.13)
at each of six stations--Burbank (BURK), downtown Los Angeles
(CAP), Pasadena (PASA), Whittier (WHTR), Azusa (AZU), and
Pomona (POMA).
> All measured (SAIM) and predicted (SAIC) concentration values
reported by Systems Applications (SAI, pp. 102-143) at the six
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8
(2 /.? /<• /:>'
:T.:r;
CONTAMINANT MONITORING STATIONS
POI-'A
FIGURE II-l. LOCATIONS OF MONITORING STATIONS RELATIVE TO
MAJOR CONTAMINANT SOURCES IN THE LOS ANGELES BASIN
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stations listed above. SAI reported values for other stations
as well, but these six were selected to provide representative
coverage for comparison.
> All measured (CORM) and interpolated (CORC) concentration
values computed by SAI (unpublished data using programs described
in SAI, Volume III) at the six stations listed above (COR is
shorthand for correlated station.)
> All interpolated (GRCI) and predicted (GRCT) concentration
values reported by General Research Corporation (GRC, pp. 101-147)
for each of four trajectories on each day. These trajectory points
did not, in general, correspond to station locations.
> All interpolated (SAII) and predicted (SAIT) concentration
values computed by Systems Applications (SAI, unpublished data
generated using programs described in Volume II) in grid squares
corresponding to the four GRC trajectories mentioned above.
In every case, data were collected for all six validation days and for
four species--CO, NO, NCL, and (L. Hydrocarbon data were not compiled, since
neither PES nor GRC reported them; moreover, SAI (p. 40) had serious doubts
about the reliability of certain of the measurement data reported by the APCDs.
Direct comparison of these data (i.e., predicted and measured or interpo-
lated values) among the five sets of results listed above was difficult because
of discrepancies in the measurement data, differences in the methods used to
interpolate for missing data, and wide variations in the reporting of calcu-
lated pollutant concentrations among the three investigating firms. Since
these factors and the assumptions used strongly affect the significance of the
data analysis, they are discussed in detail below.
In addition to the basic set of LAAPCD measurement data used by all three
firms, SAI and GRC used data reported for the same days at Commerce and El
Monte by Scott Research Laboratories. SAI also used data for La Habra, Anaheim,
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10
and Santa Ana as reported by the OCAPCD. In its evaluation of these data,
SAI determined that the measurements carried out by both APCDs were subject
to a consistent bias; thus, for NO and 0.-,, these data were modified by SAI
(pp. 39-40) as follows:
Actual NO = Observed NO x 1.25
Actual 03 = Observed 03 + Observed S02 - 0.2 x Observed N02
The Scott data were obtained using different methods and did not need cor-
rection. In addition, the "correction" for ozone at some stations (notably
Long Beach, due to the high SOp concentrations in that area) was unreal is- :
tically large—sometimes resulting in negative concentrations—and thus could
not be applied. SAI (p. 39) also pointed out that PAN interferes to some
extent with NO^ measurement and that the pollutant concentration data in
general are of no better than ±10 percent precision. Numerous data items are
missing, and some of the reported values may be incorrect. An even more seri-
ous problem in using the APCD data is the disparity in scales involved in the
assumption that the readings at a single point can be compared with predictions
having a spatial resolution of four square miles, especially when many of the
measuring stations are located near strong emissions sources. This topic is
discussed extensively later.
Since the trajectories in the PES and GRC models do not, in general, pass
through the measuring stations, it was necessary for each firm to develop an
interpolation scheme to project station readings to nearby squares. The ones
used by GRC (p. 91) and SAI (Volume III) are similar in concept, but the dif-
ferent algorithms for station acceptance or rejection and the variations in
the measured values data base mentioned above introduced some discrepancies.
PES chose to accept any point within a five mile radius of a measuring station
as being "at" that station (PES, p. A.2); some of the PES outlying data points
were adjusted using the SAI interpolation scheme. Even when extrapolations of
30 miles or more were allowed, occasionally no observed data were available to
calculate values for some squares or stations; all three firms interpolated
these values temporally when needed.
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11
Selection and verification of the calculated data were troublesome.
The time of sampling of data varied among all three firms: PES (p. IV.3)
used an instantaneous reading (even though the resolution of emissions and
meteorological data was no finer than two miles), GRC (p. 82) used an hourly
average for the trajectory (which could have passed through as many as four
grid squares during the hour), and SAI (p. 59) used an hourly average for
each grid square. All PES readings began at 0830 and terminated at 1330,
SAI's ran from 0500 to 1400, and GRC used variable run times—with most runs
beginning between 0500 and 0800 and terminating in the early afternoon. With
the exception of 29 October, PES reported only one or two points for N02, NO,
and 0^; moreover, these were in the early afternoon, when the NO values had
decreased essentially to background levels. Assignment of the trajectories
to the rectangular grid was hampered by the lack of a scale on the GRC maps
and by the lack of times on the PES maps. Owing to the experimental data
adjustments mentioned above, SAI had different initial conditions from those
of PES and GRC. GRC reduced the NO emissions encountered along the trajectory
path by 75 percent (GRC, p. 96) and adjusted diffusion coefficients and rate
constants (GRC, p. 84).
Given all of these incongruences, it is amazing that any sort of consis-
tent data base could be constructed at- all. In fact, we did not attempt to
create a single set of measured station points or interpolated observations;
instead, each firm's sets of measured and predicted values were retained
intact for comparison.
This lack of uniformity should in no way be associated with incorrect
interpretation of EPA's contract requirements or laxness on the part of any
of the contractors in performing their validations studies. Each firm was
free to interpret the available data base in any appropriate fashion, and
each did indeed create a self-consistent validation study from these data and
the model used.
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12
Nevertheless, it is unfortunate that these disparities in the selection
of validation methods exist, since any efforts to match the results using the
different models immediately generates an "apples versus oranges" criticism.
Consequently, we did not attempt to compare across sets, and the ramifications
of this inability to make direct comparisons among the models are discussed in
the following section.
C. SELECTION OF ANALYSES OF THE DATA
Given the above defined sets of data points, one question summarizes the
initial focus of all three studies: How well do the pollutant concentration
values predicted by each model represent the actual pollutant concentrations?
Unfortunately, the term "actual pollutant concentrations" admits to a variety
of definitions, depending on the type of analysis and modeling being done. In
particular, the selection of definition is subject to the problem of disparity
in scales. This topic has been treated in depth elsewhere (SAI, p. 43); the
following is a brief summary of the SAI analysis.
Measurement of meteorological variables and pollutant concentrations is,
of necessity, instantaneous in both space and time, as are the chemical reac-
tions that may occur among the various pollutant species. Readings taken at
some arbitrary time will not necessarily reproduce those taken five minutes
earlier. Similarly, readings taken at some arbitrary location will not neces-
sarily agree with those taken a block away. In an urban area such as Los
Angeles, these observations are especially appropriate. Nevertheless, from a
purely practical standpoint, it is necessary to make compromises; the ideal of
continuous measurements at contiguous locations must give way to the reality
of fiscal responsibility in both modeling accuracy and station location and
operation. The question then becomes: Given the existence of known or estimated
variations in the spatial and temporal distributions of and statistical fluctu-
ations in ambient concentrations, what types of analyses of the data are meaning-
ful, and what sorts of explanations will account for the expected discrepancies?
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13
It is necessary from the start to'define the term "representative."
The measured value of pollutant concentration from a station is represen-
tative of the concentration at that site. The models, however, work from
an emissions base that is defined in terms of a 2 x 2 mile square and from
meteorological and initial conditions data that represent, on the average,
one station for every 25 to 40 square miles. If the smallest area for which
a meaningful average pollutant concentration can be calculated is the 2x2
mile square, then a station "representative" of the square in which it was
located would have to record consistently pollutant concentrations equal to
the average values in that square. Thus, we examine next the locations of
the stations and the likelihood of their being representative of the grid
squares in which they are located.
As previously mentioned, Figure II-l shows the locations of the moni-
toring stations relative to the major pollutant sources and the 50 x 50 mile
grid of the Los Angeles basin. Table II-l describes the physical site of
each station and the expected effect of that location on the representativeness
of the station readings. It can readily be seen that the majority of the
stations are exposed to relatively high emissions, with respect not only to
the basin as a whole, but also to the 2x2 mile grid square containing the
station* Since vehicular emissions are the major single source of pollutants
in the areas of most monitoring stations, one might expect that locally high
readings of the directly emitted pollutants, CO, NO, and hydrocarbons, would
be observed near major roadways and that the chemically formed pollutants,
N0,,,and.-.::0,., would be underrepresented.
Some stations are located so as to be representative of the grid squares
in which they are situated; in fact, some of them may even underrepresent
their areas (in terms of primary pollutant concentrations) under certain
meteorological conditions. Viewed basin-wide, the readings from these sta-
tions tend to "temper" the readings from those stations that are more heavily
influenced by vehicular traffic. This leads to the occurrence of an interest-
ing phenomenon whenever measurement data are prepared for grid squares that do
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Table II-l
LOCATION OF CONTAMINANT MONITORING STATIONS OF THE LOS ANGELES
AIR POLLUTION CONTROL DISTRICT
Station
Number
1
60
69
71
72
74
75
76
Code Name
CAP
AZU
BURK
WEST
LONB
RESD
POMA
LENX
Approximate Location
Surrounded by four freeways
1500 meters from each. Sampl-
ing probe is suspended outside
a sixth-floor window.
600 meters north of the Foothill
Freeway.
150 meters southwest of the
Burbank power plant and 300
meters southwest of the Golden
State Freeway.
400 meters northeast of the
San Diego Freeway and 400 meters
north of the Santa Monica Freeway.
200 meters north of the San Diego
Freeway.
3000 meters north of the Ventura
Freeway.
500 meters south of the San Bern-
ardino Freeway.
Immediately west of the San Diego
Freeway and immediately southeast
of Los Angeles International Air-
port.
Expectation of Local Effects
Strong.
Mild.
Strong.
Strong—predominant south-
westerly winds during the day.
Strong—predominant south
winds during the day.
None.
Mild.
Strong.
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Table II-l (Concluded)
Station
Number
78
79
80
91
92
93
98
99
Code Name
RB
PASA
WHTR
LAH
ANA
SNA
COM
'ELM
Approximate Location
300 meters northeast of the
Redondo Beach power plant. The
station measures only S02-
2000 meters east/northeast of
the Pasadena power plant.
On a main street—no other major
sources nearby.
Near Beach Boulevard and the
Imperial Highway, both of which
carry very light traffic in this
area.
Immediately northwest of the
Santa Ana Freeway.
Orange County Airport, 400 meters
south of the San Diego Freeway.
Immediately'west of the Long Beach
Freeway and 1500 meters south of
the Santa Ana Freeway.
El Monte Airport, 1500 meters north
of the San Bernardino Freeway.
Expectation of Local Effects
Moderate.
Mild for SOo measurements
during the day.
Mild.
None.
Strong—predominant south-
westerly winds during the day.
Moderate—from aircraft and
vehicular emissions.
Strong.
Mild.
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16
not contain a monitoring station: The pollutant concentration values
obtained by interstation interpolation for a grid square may be more repre-
sentative of that square than are the direct readings from a station within
the same square. In other words, a distribution of the (weighted) readings
from several stations of various degrees of representativeness over a large
area is more likely to approximate the results obtained from the determina-
tion of an "average" value for the pollutant concentration within a 2 x 2
grid square than would the reading from a single, and probably nonrepresen-
tative, monitoring station in the absence of complex topographical features.
Pollutant concentrations reported by the monitoring stations were used
as inputs to the models under discussion. All three models used station
readings (or, more often, interpolations among stations) to determine initial
conditions; in general, the significance of these values was attenuated by
the contribution from the emissions over the first few hours of the morning.
SAI also used station readings in the calculation of boundary conditions, but
the reported values were reduced (SAI, Volume III) to represent the presumably
lower pollutant concentrations at the border squares. The models relied
mainly on the emissions inventory (averaged over each grid square) and the
meteorology (smoothed into integral isotachs and streamlines), coupled with
chemical reactions, to calculate pollutant concentrations. No pretension was
made of presenting anything other than average values over the grid square
area. If a model is expected to reproduce accurately a reading from an admit-
tedly nonrepresentative monitoring station, it must carry out the calculations
at the subgrid-scale level; none of the models under consideration have this
capability.
The concept of subgrid-scale modeling is dealt with more fully in Volume
III of this report, but it is important here to recognize the implications of
dealing with nonrepresentative monitoring stations in attempting to analyze
the validation results presented in each final report. The question is not
how well the calculated values agree with the measured ones, but,rathei?v how
well the calculated values reflect the trends of the measurement data, and
whether it is possible to explain discrepancies between measured and calculated
values in terms of the general characteristics of those measurements. The
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17
chances are that calculations that agree too closely with measurements should
be looked upon with a certain amount of suspicion, lest the modeling process
might be degenerating into an exercise in glorified curve-fitting!
Despite the above warnings about the nonrepresentativeness of the measure-
ment data, several statistical analyses were worth performing on these valida-
tion results, if only to demonstrate that the conjectures about station sites
were correct and that significant data trends exist. They are described below.
> The correlation coefficient
3 =
I
E (Xi - x)(y. - y)
I \^ 9 T"^
LL (x, - x)2 L,
\l/2
(y, - y)2
is a measure of how well the values of x and y tend to
follow one another in their peregrinations about their
means; i.e., do the calculated and measured values show
the same trends?
> The deviation
" i
E
i=l
~
(^-y,)2
1 1
l-l
is a measure of the variation between the calculated and
observed concentrations; this measure provides an absolute
value for the discrepancy between the two sets of data.
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18
> The goodness-of-fit statistic can be expressed as
2 V K -*!> - fri.
where
f(x.j - y-) = number of occurrences of the observed
residuals x. - y. within a given interval,
fr. = number of occurrences of a similar residual
r- from a normal distribution with the same
mean and standard deviation as the observed
set of residuals,
I = number of intervals.
When compared with the expected chi-squared value for a given
probability level, the goodness-of-fit statistic indicates
whether the predicted values can be considered to have been
randomly drawn from the distribution defined by the measured
values.
> Scatter plots of predicted versus measured values graphically
illustrate the tendency of the model to under- or over-predict
the concentration values.
> Residual plots show trends in the deviations between predicted
and measured values; four are of particular interest:
- Histogram--the number of occurrences of each residual
value.
- Time series — variation of the residual values with time
of day.
- Prediction—variation of the residual values with the
size of the predicted value.
- Observation—variation of the residual values with the
size of the measured values.
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19
Each of the above types of statistics is useful in describing a partic-
ular facet of the behavior of the models. By measuring trends, the correla-
tion coefficient allows an assessment of the capability of the model to
respond to changes in pollutant concentrations, even if the calculated and
measured concentrations are not identical.because of some other defect in the
model. The deviation shows how well, in an absolute sense, the model is
matching its predicted values with the measured ones. The chi-squared
statistic permits determination of whether the differences between predicted
and measured values can be considered to-be attributable to chance or are
statistically significant and thus indicative of a flaw in the model. The
plots are particularly useful in assessing the accuracy of the models and
the nature of the errors. The results of statistical tests are relatively
insensitive indicators of model performance because of the limited quantity
of data, the varying conditionssancle-assumptions, the nondistributional char-
acter of the data, and the complexity of the potential sources of error. One
should not substitute statistical analysis results for an examination of the
plots.
Theoretically, the analyses described above could be applied to a com-
parison of any two sets of data from this study. Realistically, for reasons
delineated earlier, we felt that comparison of results from the three differ-
ent models, though feasible, would not be justified; the temptation to declare
one model "better" than another could become irresistible despite warnings
about incongruent data sets or incomparable assumptions used to obtain such
results. Therefore, the only comparisons we made involved, in every case, sets
of predicted and measured (or interpolated) values as reported by a single
firm using a particular set of assumptions.
However, one set of comparisons, although indirect, is highly pertinent
to the disparity in scale arguments presented above. Because PES, even though
it used a trajectory model, chose to present its results as station-based
(i.e., grid-point) data, it was necessary to create two sets of results for
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20
the SAI model--one set for PES station locations,and another for GRC trajec-
tories. Since the same SAI model, operating under the same set of assumptions,
was used to calculate these values, we could do a legitimate comparison of the
results of the statistical analyses of these two types of data. To the extent
that they differed, we could hypothesize relationships between these differences
and the disparity in scales between the station (point) measurements and the
trajectory (square-average) measurements. Again, we stress that, because of the
differing assumptions of the three contractors, no direct comparison of model
could be made; only the availability of both "station" and "trajectory" values
for the SAI model enabled the secondary effect of disparity in scales to be
addressed.
The availability of the interpolated station points offered additional
aid in the evaluation of the contributions of disparity-in-scales and nonrep-
resentativeness to discrepancies between predicted and measured values for all
three models. In the ideal case, there would be a smooth continuum of pollutant
concentrations throughout a region. Given this assumption, it should br pos-
sible, through distance-weighted interpolation, to calculate the concentration
at any point from the observations at a representative set of well-spaced
measurement stations within that region. Similarly, it should be possible to
calculate the expected concentration at the site of a particular .measuring
station by eliminating that station from the interpolation process. The extent
to which a value obtained by this interpolation process differs from that actu-
ally measured at the station reflects the questionability of the assumptions of
representativeness of the station measurements and the validity of the assump-
tions that underlie the interpolation scheme. More particularly, it should not
be expected that the statistical results for those models that depend on inter-
polation of station measurements for either input or comparative data would be
any better than that demonstrated by the station interpolation statistics
themselves.
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21
D. RESULTS
1. Data Set Selection
Using the five types of data listed earlier, we calculated the statistics
and plotted the deviations at three levels of detail. The finest level repre-
sents the comparison of predicted and measured (or interpolated, in the trajec-
tory model case) concentration values for.each of the four pollutants at each
of the six stations (or along each of the four trajectories) on each of the six
days represented by each of the five sets of results—a total of 720 runs. Al-
though we obtained significant findings at this level, we do not present them
in this report. The small number of data points per run, the repetitiveness of
the plots and -statistics, and the sheer bulk of information (3600 pages) do not
justify their reproduction. Instead, a single copy of the entire computer
printout will be forwarded to the EPA for researchers who may be interested in
the fine details of the analysis.
By combining the individual data sets described above, we reduced the
number of sets for analysis. Two such combinations were made: all stations
or trajectories for each day and all days for each station (since the trajec-
tories themselves were nonreproducible from day to day, they could not be
combined). These combinations gave rise to a total of 192 runs—again, too
many to include in this report (a single copy of the computer printout will
be sent to the EPA).
By combining the composite data sets described above, we reached the
third and most inclusive level. At this level, there was a single data set
for each pollutant from each of the five sets of results; each of these 20
data sets contained from 50 to 350 points. From a statistical point of
view, these were the most significant sets for interpretive work. However,
the large number of points tended to obscure some of the graphical results,
especially when multiple points with the same value overprinted one another.
These results are dealt with in detail below. The bulk of the discussion
centers on the statistical and graphical picture they portray.
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22
2. Statistical Analysis
Table II-2 presents the statistical analysis of the validation results
for all dates at all stations. In this table, the "standard chi-squared
value" represents the level at which there is a 90 percent probability that
the residuals could have been drawn from a normal distribution. The dis-
cussion begins with the emitted primary pollutants CO and NO, followed by
the secondary reaction products N0? and CL.
Carbon monoxide is an "inert" species in the sense that any chemical
reactions it undergoes are slow relative to its dispersion by winds and
diffusion. Thus, the CO results present a clearer picture than do those
of the other species of the effects of disparity in scales and nonrepresen-
tativeness of the measuring station locations on the comparison of predicted
and measured values. As shown in Table II-2, the correlation coefficients
for all sets of CO data are quite high, indicating that the models were able
to predict trends in the concentration values fairly well. It is especially
significant that the highest correlation coefficient was associated with the SAI
station results. As shown later, the SAI model did very well in relating
emissions and meteorology to CO concentrations, except during the morning
rush hours, when the model consistently underpredicted the high CO concen-
trations. If one assumes that most of the monitoring stations, owing to
their roadside locations, were measuring anomalously high CO concentrations
from engine emissions that had not yet dispersed evenly over the grid area,
this underprediction would be expected.
The results from the models were better, in all respects, than those
obtained using the input station correlation calculation for interpolation.
Since CO emissions arise almost solely from automobile traffic, measuring
stations generally tend to over- or under-represent their grid squares, de-
pending on whether they are downwind or upwind of the most heavily travelled
streets in their vicinity. If the location of a station with respect to the CO
dispersion were not similar to that of neighboring stations, we would expect
-------
Table II-2
Species
CO
NO
NO,
Data
Analyzed
PESC
SAIC
CORC
GRCT
SAIT
PESC
SAIC
CORC
GRCT
SAIT
PESC
SAIC
CORC
GRCT
SAIT
PESC
SAIC
CORC
GRCT
SAIT
Correlation
Coefficient
0.68
0.84
0.64
0.82
0.79
0.77
0.87
0.63
0.87
0.86
0.68
0.65
0.33
0.43
0.52
0.50
0.60
0.80
0.91
0.69
STATISTICAL ANALYSIS FOR ALL LOCATIONS
Deviation
4.27
3.52
4.61
3.39
3.18
3.18
9.13
13.90
8.70
8.13
8.36
6.82
8.14
10.05
12.62
8.26
8.56
5.32
3.82
8.75
Degrees of
Freedom
7
7
9
6
6
10
13
7
7
1
11
17
8
7
1
13
11
3
5
Measured
Chi -Squared
10.33
81.38
67.96
29.37
30.39
170.67
110.30
54.48
56.81
0.96
72.47
27.22
39.57
108.66
0.96
38.66
167.36
45.32
39.8/1
Standard
Chi-Squared
12.02
12.02
14.68
10.64
10.64
15.99
19.81
12.02
12.02
2.71
17.27
24.77
13.36
12.02
2.71
19.81
17.27
6.25
9.24
CO
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24
large discrepancies in predictions of CO concentrations at that station on
the basis of those measured at neighboring stations. However, the models
were able to use the variations in emissions and meteorology in the vicinity
of, though not necessarily adjacent to, the station. To the extent that the
winds and emissions are known within a 2 x 2 mile square, they are better
determinants of a point measurement within that square than are the point
measurements from stations 5 to 10 miles distant.
A similar effect appeared in the comparison of the PES and GRC results
for CO. One major difference between the PES and GRC models is the parti-
tioning by GRC of the moving column of air: The former used a single cell
containing the entire columns whereas the latter divided the column into
several superimposed layers. By keeping fresh emissions near the ground,
the GRC model was better able to follow significant rapid changes in CO
concentrations (e.g., during and after the morning rush hour) and thus to
simulate more closely the environment sensed by the network of CO monitoring
stations. Trie dispersion artifact intrinsic to the PES model, coupled with
the acceptance, for comparison purposes, of any trajectory that passed within
five miles of a station, considerably increased the effective volume over
which the CO concentration was being averaged and thus the disparity in
scales between this calculation and the point measurement.
The results of the chi-squared calculations clearly demonstrate the
effects of the nonrepresentativeness of the station measurements and the
problem of disparity in scales. As shown in Table II-2, the only set of
values that met the 90 percent chi-squared criterion is that of PES. Rela-
tively, the PES results met the chi-squared criterion better than either the
correlation coefficient or the deviation. In the.PES model, uniform mixing
over a five mile radius and the entire column height was assumed, and the
model is usually executed from 0830 onward, thus missing the major CO peak.
The greater averaging and the elimination of some of the high values appar-
ently enabled the differences between measured and calculated values to be
normally distributed while not greatly improving the ability of the model to
match trends in the data or to reduce the deviations significantly. Moreover,
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25
since the simulation began near the peak in CO concentrations, the model
had to reproduce only the slow descent from the morning peak to the midday
plateau in concentrations. In other words, no particular stress was placed
on the model's capability to follow rapidly changing emissions patterns and
the corresponding CO peak.
The contrast between the chi-squared values for the trajectory-type
(GRCT and SAIT) and the station-type (SAIC and CORC) calculations is striking.
Although neither was able to satisfy the 90 percent criterion, the former
values were significantly lower than the latter. The difference can most
readily be ascribed to the contrast between the attempt to match values
calculated for 2 x 2 mile grid square averages with (1) the point measure-
ments from the stations, which are likely to yield large and—since the
station measurements tend to be high rather than low with respect to the
predictions--nonrandomly distributed residuals, and (2) the interpolated,
and, therefore less disparate in scale, "measured" values used in the tra-
jectory comparisions. This conclusion is supported by the residual plots
and the high SAIC correlation coefficient, indicating that trends can be
reproduced well by both types of models and that the discrepancy lies in the
estimation of the overall magnitudes.
Turning next to the results for NO, the reader can see that the distri-
bution of correlation coefficients is quite similar to that for CO. Like CO,
NO is a primary emissions species; and for those monitoring sites that are
not representative of the grid squares, much of the discussion above for CO
pertains also to NO. However, major differences in behavior between NO and
CO affect this analysis. First, NO is emitted in significant amounts from
power plants, refineries, and other large point sources. Since most of these
sources are distant from any of the measuring stations, these emissions tend
to become reasonably well dispersed throughout the grid squares in which sta-
tions are located. Thus, the problem of disparity in scales pertains only to
that portion of NO emitted by automobiles, and the point readings can be
expected, by and large, to better reflect the average NO concentrations within
the corresponding grid square. The generally high correlation coefficients for
NO, compared with those for CO, may be partially ascribable to this "smoothing"
effect.
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26
The second, and more significant, difference between NO and CO is
the greater chemical reactivity of NO. Because NO is more reactive and
because its reaction times are short relative to the hourly averaging
process used in data reporting and modeling, the chemical kinetic mech-
anism and rate constants chosen for use in each model strongly affect
the calculated NO (and, correspondingly, N0? and CL) concentrations. As
C— O
demonstrated elsewhere in this report, the mechanisms used in all the
models are inadequate to represent the formation process of photochemical
smog in both temporal (time to reach peak concentrations) and quantitative
(concentrations of key species) senses. Thus, errors in the calculated
concentrations of NO (and the reaction products N0? and CL) can be antici-
pated, as borne out by the standard deviation and chi-squared calculations
for both the SAI and GRC models.
The low PES and high COR deviations (a.) deserve special comment.
Almost all the PES calculations were reported' only for midday conditions,
when NO concentrations had decreased to their background level of 1 pphm;
thus, the relatively low deviations represent essentially background con-
centration calculations. In contrast, the SAI and GRC calculations were
spread throughout the day, including the morning hours when NO concentra-
tions were on the order of 40 pphm; thus, they generated significantly
higher, but statistically more valid, concentrations and much more meaning-
ful predictions for evaluation.
The high value for station interpolation reflects to some extent the
high chemical reactivity of NO. NO, once released by an emissions source,
does not persist in an ozone-containing atmosphere but rather is rapidly
oxidized to N0?. Thus, during a major portion of the day, only insignifi-
cant amounts of NO generated near a given station are carried downwind and
can be detected at another station. As a result, if one station is in an
area of high emissions and some of its neighbors are in areas of low emis-
sions, any attempt to project the neighboring values into the area of high
-------
27
emissions is bound to produce a low result; similar problems arise when a
low-emissions area is adjacent to several areas with high emissions. The
models all include methods of taking the reactivity of NO into account and
of thereby "damping out" the downwind transport. The interpolation scheme,
which is much simpler than the models, projects the inappropriately high or
low numbers to nearby stations. Of the four pollutants studied, NO and its
reaction product N02 show this anomalous behavior most strongly; the inert
CO and the regenerative reaction product 03 are dispersed throughout the
basin, and less steep concentration gradients between stations are encoun-
tered for these two species.
Owing to the poor statistics for the distribution of NO residuals from
the PES model (almost all of the measured and calculated values were at the
background level of 1 pphm), the chi-squared value could not be calculated.
The station and trajectory values from the SAI and GRC models show behavior
similar to that demonstrated for the CO values, in that the chi-squared
statistic for the station values is higher. The reasoning given for CO
holds here: The smaller disparity in scales between calculated and inter-
polated values provides a more random distribution of residuals. The pro-
portionately higher numbers compared with those measured for CO are probably
due to biases introduced by the inadequacies in the chemical mechanism.
Since N0? is a secondary product of combustion, formed almost entirely
from emitted NO, a fair amount of mixing and dispersion of the NO will have
occurred by the time it is converted to NCL. Thus, station readings of N0?
concentration are likely to be more representative than the corresponding
readings for NO. Assuming that their chemical mechanism submodels are
reasonable approximations of the smog formation process, the models should
provide better estimates of the values that are both read by the stations
and obtained by interstation interpolation. The high chi-squared value
for the GRC model may indicate a systematic bias in the kinetics mechanism;
see the comments below on ozone. The PES values again are clustered around
midday, and the chi-squared statistic is probably unrepresentative.
-------
28
The correlation coefficients for NCL are uniformly lower than those
for NO. This result can be readily accounted for since the NO curves for
concentration as a function of time at most stations start high and fall
to background levels in a reasonably smooth manner, whereas the correspond-
ing N0? curves tend to pass through a maximum and then oscillate around a
fairly high afternoon concentration value. Since the correlation coeffi-
cient measures the extent to which the measured and predicted values follow
the same trends, the models all encounter extreme difficulty in attempting
to reproduce .the contours of these curves.
Turning finally to the ozone results in Table II-2, we note several
significant points. The exceptionally good GRC values are a direct result
of GRC's intent (p. 91) to achieve a good ozone fit, even at the expense of
fits involving other species. To reach this goal;,GRG (p. 84) used three of
the six validation days for "calibration" of the model, altering certain
parameters to improve the agreement between the measured and predicted con-
centration values. Such a "hands-on" process is useful and easily achieved
with a trajectory model, and the EPA specified in its statement of work that
up to three of the six days could be used for this purpose. As demonstrated
elsewhere in this report, the simple chemical kinetics mechanisms used in
these validation runs.:do not properly account for the behavior of concentra1,
tion as a function of time of both ozone and oxides of nitrogen. Since
oxidant concentration standards are the ones most often exceeded during pol-
lution episodes, GRC's choice to concentrate its "tuning" on ozone production
is reasonable. However, even GRC's efforts to better represent the ozone
concentration did not bring the chi-squared value within the 90 percent con-
fidence level. This failure again indicates the inadequacy of the chemical
mechanisms used to deal with the formation and distribution of a highly re-
active secondary pollutant such as ozone.
Equally striking are the results of the station interpolation (CORC).
For species other than ozone, the interstation correlation results ranked
near the bottom in terms of their correlation coefficients and deviations a ,.
-------
29
For ozone, the results were better than all but the finely tuned GRC results.
The reason for such exceptionally good results for ozone lies in the nature
of the "chemical stew" that produces this pollutant. The initial ingredients
of this "stew"--hydrocarbons and nitric oxide—are prepared during the
morning rush hour. The morning winds tend to disperse these ingredients
throughout the basin, but the winds are too weak to blow them out of the
basin; moreover, the previous day's pollutant load, which drifts out to sea
overnight, is returned to the basin by light onshore winds. Under the influ-
ence of the catalyzing effect of sunlight, ozone begins to form, but its rela-
tively slow reaction rate, combined with the continued presence of NO, keeps
the concentrations low. Only several hours after the emissions peak occurs
do appreciable amounts of ozone appear. Since, during this time, the ingre-
dients have become reasonably well dispersed, the ozone is also dispersed.
Moreover, in areas of abnormally high hydrocarbon emissions, such as those
near several monitoring stations, where one can presume that ozone would be
formed at a Caster rate, the higher concentration of NO acts as a scavenging
agent. Thus, station readings in high emissions areas are likely to slightly
underrepresent ozone concentrations.
In light of this dispersed and relatively constant ozone concentration,
it is not surprising that station correlation is so successful. Most of the
stations in a given area provide readings representative of the average grid
square concentrations, and the concentration gradients between areas are
small. Also, the nature of the ozone reaction kinetics tends to smooth out
any irregularities in pollutant concentrations. The three models, however,
must rely on the inadequately known kinetic mechanisms in their attempts to
calculate ozone concentrations, since ozone is a secondary product only ten-
uously related to the emissions pattern that enables the models to do so well
for CO.
-------
30
The high chi-squared statistic for COR is a consequence of the distri-
bution of the measured values. Most of these values are low and fairly
uniform throughout the basin, but occasionally in the afternoon one or two
stations show high values. When these values are used to calculate con-
centrations at other stations, the residuals (measured minus calculated
values) are always negative. It is this negative "hump," coupled with
the few large positive values when those anomalous station concentrations
are calculated, that produces the large chi-squared value.
3. Scatter Plots
Plots of measured versus calculated concentrations (i.e., "scatter
plots") for each of the five sets of CO results are presented in Figures
II-2(a) through II-2(e). Similar results for NO appear in Figures II-3(a)
through II-3(e); for N02, in Figures II-4(a) through II-4(e); and for 03,
in Figures II-5(a) through II-5(e).
At first glance, the PES, GRC, and SAI trajectory results shown ir,
Figure II-2 for CO generally appear to be randomly distributed about the
45° line. However, the SAI and correlated station results definitely indi-
cate a trend toward measured concentrations larger than the predicted values.
A more detailed examination of the PES, GRC,. and SAI trajectory results shows,
moreover, that a similar tendency is also present at high measured concentra-
tions (say, greater than 12 ppm). Similar but more highly skewed behavior is
shown in the NO scatter plots (Figure II-3).
Both CO and NO are primary emissions products that arise principally
from automobile exhaust. Possible reasons for the underprediction of their
concentrations, especially near the measurement stations, include measurement
inaccuracies, overestimation of vertical transport, and underestimation of
emissions rates. More likely, the consistent prediction of concentrations
lower than the measured values at high observed concentrations results from
the disparity in spatial scales between measurements (on the order of tens of
meters) and predictions (on the order of 3000 meters). These results provide
additional evidence that microscale models must be further developed to pro-
vide an adequate means of comparing airshed model predictions with point
measurements.
-------
40"
40- /
1 /
1
1 /
1 /
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Concentration
Concentration
(a) PES--Horizontal Axis = PESC,
Vertical Axis = PESM
(b) SAI —Horizontal Axis = SAIC,
Vertical Axis = SAIM
FIGURE II-2. SCATTER PLOTS FOR THE CO RESULTS
-------
40-
40-
°
£ 20.
c
Ol
(J
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Do.. !„., |.,|,
,.|-.|—|..|.,|,.|..|..|..|-.|..|..|,-|.
10 Z"0 30
Concentration
• i —I-
-i
40
(c) Correlated Station—Horizontal Axis = CORC,
Vertical Axis = CORN
I
I
t
I
0»,
o
0 0
ft 9 »<
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,|—I--I--I —I--I-, |,.|.,|..|.
10 ft
Concentration
.|..|..j..|..i —i..i
30 40
(d) GRC--Horizontal Axis = GRCT,
Vertical Axis = GRCI
FIGURE II-2. SCATTER PLOTS FOR THE CO RESULTS (Continued)
CO
ro
-------
» O 0 4
o « a
40-
30-
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0
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0 10 ZO 30 40
Concentration
(e) SAI--Horizontal Axis = SAIT,
Vertical Axis = SAII
• i--i — i — i —i--i —i.-i--i--i--i--i
10 zo
Concentration
• i — i — i — i--i —i — i — i
30 *Q
(a) PES--Horizontal Axis = PESC,
Vertical Axis = PESM
FIGURE 11-2. SCATTER PLOTS FOR THE
CO RESULTS (Concluded)
FIGURE II-3. SCATTER PLOTS FOR THE NO RESULTS
CO
CO
-------
c
o
4-1
E
J_t
onceni
w
80-
1
I
o
1
1
^
o
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1
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60 - oo
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| » « CO
| • 0 /
1 *
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40- » * * ' * »
, 0 O» * O
/ o
»« » 0 <• «
C » O 0 /
o e
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1 o o « /
1
1 -3
1 «« /
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20 - » o c / o
I « »« 0
| 0 J »* oo -00- »
^ O 4 C fi O O O O & O CO
. U^Oti/^ft l> 0
J .„.. . .. . .
80 0 '20 40 60 S
Concentration
(b) SAI--Horizontal Axis = SAIC, Vertical Axis = SAIM
Concentration
(c) Correlated Station—Horizontal Axis
= CORC, Vertical Axis - CORM
FIGURE II-3. SCATTER PLOTS FOR THE NO RESULTS (Continued)
CO
-------
60-
80 f
60
o
c
4)
u
c
o
40
O 44
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O 4 O C-O
o o o o
0 •..)..)..)..)..!«.(.. |— l ..I.. | —I —I —| — |..)..).. | — | —I — I
0 20 "0 60 80
Concentration
60
o
u
o
10
20
0- / 0
* o o
« o « /
* 04
20
16
60
Concentration
(d) GRC--Horizontal Axis - GRCT, Vertical Axis - GRCI (e) SAI--Horizontal Axis = SAIT, Vertical Axis = SAII
FIGURE II-3. SCATTER PLOTS FOR THE NO RESULTS (Concluded)
CO
en
-------
Q
•*•
c
c
u
c
o
W
80- 80*
1
1 '
1 '
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<*O*>OO* tf <>
0 20 40 60
80
Concentration
Concentration
(a) PES--Horizontal Axis = PESC, Vertical Axis = PESM
(b) SAI--Horizontal Axis = SAIC, Vertical Axis » SAIM
FIGURE II-4. SCATTER PLOTS FOR THE N02 RESULTS
CO
CTi
-------
80-
Concentration
i
i
1
i
1
I
60 -
' »
40. o „
1
i o
1
J 0 ^ 0
1 <><,»<, /
1 « •«•
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1
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0 20
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sol
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20- 0, * "(>'>ff *° *"
1
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1 * »*» / o o o o
| 0« O 0 «•«>
| *«• 0> O
1 *
10 60 80 n ,n ,. ' ' ' ' ' ' ' '
Concentration
(c) Correlated Station—Horizontal Axis = CORC,
Vertical Axis = CORM
80
Concentration
(d) GRC--Horizontal Axis = GRCT, Vertical Axis = GRCI
FIGURE II-4. SCATTER PLOTS FOR THE N02 RESULTS (Continued)
CO
-------
60
c
O)
u
c
o
/CO
«c ^
«3-*>>l««r>
30
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§
c
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1
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80 0 20 40, 60 t
Concentration
SO
Concentration
(d) GRC--Horizontal Axis = 6RCT, Vertical Axis = GRCI (e) SAI—Horizontal Axis = SAIT, Vertical Axis - SAII
FIGURE 11-5. SCATTER PLOTS FOR THE 03 RESULTS (Concluded)
-------
41
As in the case of CO, the PES, GRC, and SAT station results for N02
(Figure II-4) generally appear at first to be randomly distributed about
the 45° line. However, the SAI trajectory results for N(L definitely indi-
cate a trend toward measured concentrations smaller than the predicted
values. A more detailed examination of the PES, GRC, and SAI station
results shows, moreover, that a similar tendency is also present at high
observed concentrations (say, greater than 25 pphm). However, there is a
definite tendency to underestimate the highest NCL concentrations when per-
forming correlated station interpolations.
This last point can be readily explained. As noted in Table II-l, the
Burbank monitoring station is located very near a power plant, which pro-
vides a strong local source of nitrogen oxides. Peak hour NCL concentra-
tions at Burbank are double those of any other station. Since other stations
in the neighborhood of Burbank record much lower NCL concentrations, the
station correlation algorithm cannot calculate the high concentrations at
Burbank; this explains the anomalous points in Figure. II-4(c) and, to a
lesser extent, Figure II-3(c). This is an extreme,, but highly effective,
example of how nonrepresentative the location of a monitoring station can
be. In this instance, only microscale modeling could resolve the problem
of disparity in scales.
The trend toward overprediction of NCL concentrations by the models is
less easily explained. The problems associated with the station measure-
ments and the chemical reaction kinetics of the oxides of nitrogen have been
mentioned above. Both are likely contributors to the disparities between
measured and predicted nitrogen dioxide concentrations.
The ozone results (Figure II-5) demonstrate a rather striking anomaly.
The predictions at the station locations by PES, COR, and SAI are low rela-
tive to the measured values, whereas the predictions along the trajectories
for GRC and SAI are high relative to the interpolated values. In both cases,
the effect is more pronounced for the SAI data. This anomalous behavior
-------
42
probably reflects the smoothing aspects of the interpolation process, as
discussed in detail earlier, and indicates a need for further development
in the application of interpolation algorithms.
4. Residuals Analyses
Residuals analyses of the data are presented in Figures II-6 through
11-25. Each figure contains four residual plots:
> Histogram—residual value as a function of the number of
occurrences of that value.
> Time plot—residual value as a function of the time of day
of occurrence of that value.
> Calculated concentration—residual value as a function of
the calculated pollutant concentration that gave rise to
that value.
> Observed concentration—residual value as a function of
the measured pollutant concentration that gave rise to that
value.
Each of these types of plots is of potential value in attempting to uncover
deficiencies in either the model being evaluated or in the data with which
predictions are being compared. The results obtained using each model for
the pollutants CO, NO, N02, and 03 are analyzed separately below.
Residuals analyses of the PES data for CO are presented in Figures II-6(a)
thrugh II-6(d). A survey of the results given in the histogram of residuals
and the plot of residuals as a function of time and predicted concentrations
[Figures II-6(a), II-6(b), and II-6(c)] indicates no notable trends in the
residuals. However, a definite trend can readily be seen in Figure II-6(d):
At high measured concentrations, most of the residuals (calculated minus
measured values) are negative. This trend agrees with the discussion presented
above of undercalculation of some station values.
The residuals analyses of the SAI station results for CO are given in
Figures II-7(a) through II-7(d). Figure II-7(a) shows a definite trend toward
-------
20.
20-
10-.
a
I
to
-10
-20.
0500
0700
1300
0900 1100
Time (hours)
(b) Residuals as a Function of Time
-t
1500
20-
i
i
i
10-
i
r
-10--
-20.
»o G.
«0
0 10 20 30
Concentration
(c) Residuals as a Function of PESC Concentration
20-
10
re
3
HI
cs:
-10
a o o • •
-I—l-.l —I —I — I—l..l.-l.-l..l..l..l..,«.l..l
10 20 30 40
Concentration
(d) Residuals as a Function of PESM Concentration
FIGURE II-6. RESIDUALS (PESC MINUS PESM) ANALYSES OF THE PES RESULTS FOR CO
-------
20.
10
I
0)
c:
-10
HISTOGRAM LINE OVERFLOW FOR FOLLOWING VALUE:
HISTOGRAM LINE OVERFLOW FOR FOLLOWING VALUE:
0
-3
-20. —I —I--I--•! —I — I —I —I —I —I —I — I--1 —I--! —I —I —I —I —I
0 15 30 45 50
Number of Occurrences
(a) Histogram of Residuals
20-
-i
10-
v>
r—
10
-10-
20. — |
0500
0700 0900 110Q
Time (hours)
1300
1500
(b) Residuals as a Function of Time
20-
\
20-
10
r—•
id
-10
-20
•
*
e
»O * ft* * *
*
« «» « • « «
404* *
a o •
* *
1
0 TO 20
«
• « *
30 40
Concentration
(d) Residuals .as. a Function of SAIM Concentration
FIGURE II-7. RESIDUALS (SAIC MINUS SAIM) ANALYSES OF THE SAI STATION RESULTS FOR CO
-------
- 0
-20-.
I
1
I
I
.|-.!_!-. |..|..|..|..|..|..|..|..t..|..|.
15 30 45
Number of Occurrences
(a) Histogram of Residuals
60
40 -
I
zo-
J
I
I
I
o-.
I
I
I
I
,-40 ...| —I —f —f..|..|.
0500 0700
0900 1100
Time (hours)
I-. i — i— r«- 1
1300 1500
(b) Residuals as a Function of Time
40-
I
t
t
I
20-
I
r
i
.1
0-._.«
I 0
I
I •
I
-20-
I
I
I
I
0 *
10
20
Concentration
30
40
(c) Residuals as a Function of CORC Concentration
20-
o
-------
20 -
10-
tn
a
OJ
cc
HISTOGRAM LINE OVERFLOW FOR FOLLOWING VALUE:
-10
-20- — I — I — I — 1 — I— I — I — | —1.-|..| — ! — )..|..[._|..|..|..|..|
0 15 30 45 60
Number of Occurrences
(a) Histogram of Residuals
ra
20-
10
-o o-
u> I
-------
20.
-------
20-
10-
«o
I 0
-10
-20. — l — I —1..|..|..
0 15
30 45
Number of Occurrences
(a) Histogram of Residuals
60
20-
-4
10-
in
ro
0-
-10
0500 0700 0900 1100
Time (hours)
1300
1500
(b) Residuals as a Function of Time
20-
10
20-
!/>
a
X)
-10
• 9
1«| — I — I — I — I — 1««|
10 20
Concentration
30
(c) Residuals as a Function of PESC Concentration
OJ
az
1
10-
1
r
i
,f * *
1 00
1 0
1 0
1 »»
•10-
I
1
1
1
0 lo
•
20 30 40
Concentration
(d) Residuals as a Function of PESM Concentration
FIGURE 11-11. RESIDUALS (PESC MINUS PESM) ANALYSES OF THE PES RESULTS FOR NO
CO
-------
40-
I
20
VI
r—•
ra
I
•o o- <
"£ I .
£ '<
i .
( c
-20-
I
-40 ...
.|..|
I — I .-I — I — I — I — I — I — I — I
15 30 45
Number of Occurrences
(a) Histogram of Residuals
60
40-
I
t
I
I
20-
I
r
i
.t
0}
OL
-20-
I
0500 0700 0900 1100
Time (hours)
1300
• i—i
1500
(b) Residuals as a Function of Time
40-
1
t
I
I
20-
I
»# * •«
20
40
Concentration
eo
80
(c) Residuals as a Function of SAIC Concentration
40-
t
20-
VI
J o-'
I/I
OJ
IX
-20
-10 .V
0
• « *
4 a •
.. | .. | .. | .. | .. | .. | .. | .. | .„ | .
20 40
Concentration
60
60
(d) Residuals as a Function of SAIM Concentration
FIGURE 11-12. RESIDUALS (SAIC MINUS SAIM) ANALYSES OF THE SAI STATION RESULTS FOR NO
-------
40-
3
i2 o
t/l
<1J
IX
g
S
• I--I—I—I—l--l-.l.-l~l.-l—l"l--l--t--l-
15 30 45
Number of Occurrences
(a) Histogram of Residuals
40-
1
I
20-
I
r
9 f «» *
3 .t
*Tp Q ff
-20-
eo
fl- »
-<04..|..,..|..|M,..|..|..|..|..u.|..|..|.
0 20 40
Concentration
,.t..|..|..i~|..i
60 80
40 •
\fi
r—
n
3
•o
-------
40-
1
1
20-
OJ
o:
I •
I •
.2o::
i *
i
i
-40.-
0
15 30 45
Number of Occurrences
(a) Histogram of Residuals
60
40 -
4
V
20
-20
-4Q...).-!..I — |.. I-.| —| —| .-I —I-. |— I-.)..|.-I-. I-.|»-|..|».|
0500 0700 0900 1100 1300 1500
Time (hours)
("b) Residuals as a Function of Time
40-
20
i/i
r— •
ra
aj
cc
-20
-40
«. «« ««
.i..|..|..|..|..|..|..|..|M|v.i
40 60 80
(c) Residuals as a Function of GRCT Concentration
40-
i
20-
1
l
\fi i
1> ,
I
I
-20-
I
t
I
I
» «
• •
20
40
Concentration
•I — I"
80
(d) Residuals as a Function of GRCI Concentration
FIGURE 11-14. RESIDUALS (GRCT MINUS GRCI) ANALYSES OF THE GRC RESULTS FOR NO
c_n
-------
f
I
i
20.-
I •
*» I •«•
•5 '
o )••«»•«
^ 0. •>«•>•«
M ",.,...,
-20
40-
a
3
•o
-20
-40« —i —|.
0
IS 30 45
Number of Occurrences
(a) Histogram of Residuals
60
0500
0700
1300
0900 1100
T1ma (hours)
(_b) Residuals as a Function of Time
15CO
40-
20-
VI
ej
-20-
»»••« *
0 20 40 60 80
Concentration
(c) Residuals as a Function of SAIT Concentration
40-
i
i
i
i
20-
I
r
•o 0"
I
I
I
I
-20-
I
t
I
I
«« 4
« »
0 20 40 60 80
Concentration
(d) Residuals as a Function of SAII Concentration
FIGURE 11-15. RESIDUALS (SAIT MINUS SAII) ANALYSES OF THE SAI TRAJECTORY RESULTS FOR NO
en
IN3
-------
40 ».
f
20-
•s °
a)
Of,
-20
-40
I — I
|..|..|
l — l — i — l — |-
15 30 45
Number of Occurrences
(a) Histogram of Residuals
20-
a
r>
•a
-20
L
I
-40« — I — I — I — I — I — I — I — I — I "I — I— I — I — I — I — t —I — I — I — I
0500 0700 0900 1100 1300 1500
Time (hours)
(b) Residuals as a Function of Time
40-
«
»
c
i
20-
-20
ZQ
40
Concentration
•H —I —I — I —I — I
60 80
(c) Residuals as a Function of PESC Concentration
40-
20
-20
t
I
-40 ..
40
Concentration
60
(d) Residuals as a Function of PESM Concentration
FIGURE 11-16. RESIDUALS (PESC MINUS PESM) ANALYSES OF THE PES RESULTS FOR
-------
40
20-
f o
-20-
I
»
I
-40»~l —|~l«—l~l —
0 15
• I —|.
.)..(..)..)..)..)..)._)..)..(..I
30 45 60
Number of Occurrences
(a) Histogram of Residuals
40-
1
I
I
I
20-
I
r
i
^
•o
QJ
cn
— I —I —I —| — | — |..|..|..|..|..|..|..|..,..l..,..,..|fc,,
0500 0700 0900 1100 1300
Time (hours)
(b) Residuals as a Function of Time
40-
20-
o-<
I/I
m
a:
-20-
•a
3
•D
40 «
r
I
20-
fl
I
-40... i...I--I--I--I
0 20
40
Concentration
60
60
(c) Residuals as a Function of SAIC Concentration
20
•I — I —I —I--I-
40
Concentration
eo
so
(d) Residuals as a Function of SAIM Concentration
FIGURE 11-17. RESIDUALS (SAIC MINUS SAIM) ANALYSES OF THE SAI STATION RESULTS FOR
-------
40-
I
zo-
-------
40-
40'-
3
wi
IU
o:
-20
in
T-~
a
-20
-40. — I — I — I — I — I — I — I — 1 —I — |,-1 —|...|~|--I--I— I— I— I--I
0 15 30 45 60
Number of Occurrences
(a) Histogram of Residuals
-40
— I —I —I--I--I--I--I--I —I —I —I—1..|..|..(..|..|..|».|
0500 0700 0900 1100 1300 1500
Time (hours)
(b) Residuals as a Function of Time
40-
1
1
20-
0-
QJ
a:
-20-
O »
O 9
9tf »
40.
20°
v)
r—
IO
&
-20-
0 20 40 60 80
Concentration
(c) Residuals as a Function of GRCT Concentration
0 20 40 60 SO
Concentration
(d) Residuals as a Function of GRCI Concentration
FIGURE 11-19. RESIDUALS (GRCT MINUS GRCI) ANALYSES OF THE GRC RESULTS FOR N02
ir
en
-------
40
40-
-IW V „
9
9
• •
20-.
W 5 • • 9 »
r—
3 ...... ... .. ....
^ 9 0 «» « « «» 9 «««•««•.««*••.._„ ._
cs <>„»•
« « «
-20-.
1
1
I
1
0 IS 30 45 60
Number of Occurrences
(a) Histogram of Residuals
40- . .
<
'
i .
1 » 0
20 — •«•««•
| « • *0 9
f ff » * • *
« . ..*<,..... .. 0,
2 o --»»•«••••••«« »
- :• . •
-20
0 20 40 60 80
4
»
t
1
20-
1
1
£ i
S i *
_a 0 —o
w 1 «
1
OS '
1
-20-
1
I
1
1
osoo
40-
1
1
1
1
20-
1 •
^ !
§ i «»
n i
OJ 1
o: '
«
-20
0
«
0 •
* •*
e &
.v ** «
« 0
» <» *• •
9 « 4 « * *
» 0 * * * *>
4r * *. -O O O
0 «
e
o
0700 0500 1100 1JOO 1500
Time (hours)
•(b) Residuals as a Function of Time
« »
•
* 9
t>
9 » O • A
00 «
0 » « *
*o «
o *
9
20 40 60 80
fnnron+*»-ftt^nn
Concentration
(c) Residuals as a Function of SAIT Concentration
(d) Residuals as a Function of SAII Concentration
FIGURE 11-20. RESIDUALS (SAIT MINUS SAII) ANALYSES OF THE SAI TRAJECTORY RESULTS FOR NO,
-------
40 <
40-
20-.
a
20-
VI
i™»
«3
•20-
i — I — I — I-- I--I — I — 1--I — 1--I — I — I — I — I — I — I — I — I
15 30 45 60
Number of Occurrences
(a) Histogram of Residuals
»..,..,..,..,..,..,.., ..,..|._,.. | „_|__,__,__ |..{..| „. | __(... |
0500 0700 0900 1100 130Q 1500
Time (hours)
Cb) Residuals as a Function of Time
40'
20
-------
40-
20- .
40
20-
I
0-'
-20
§
-40. — I — I — I — I — I — I — ! — I — l«l — I — I — I — I — I — I — I — I — I — I
0 15 30 45 60
Number of Occurrences
(a) Histogram of Residuals
-40 .--!•
0500
0700
1300
0900 1100
Time (hours)
(b) Residuals as a Function nf Time
• i — i — i
1500
40 •
20
m
I
QJ
OC
-23
20
40
60
80
Concentration
(c) Residuals as a Function of SAIC Concentration
40-
20
3
•a o
OJ
a.
-20
-40._.,..|--| —I--I--I —I--I--I--I —I —I--I--I--I--I--I--I —l"l
0 20 40 60 80
Concentration
(d) Residuals as a Function of SAIM Concentration
FIGURE 11-22. RESIDUALS (SAIC MINUS SAIM) ANALYSES OF THE SAI STATION RESULTS FOR
en
i-O
-------
40-
20-
in
•0
•o
Si
-20
-40.
> « • tf
c • » » e
15 30 45
Number of Occurrences
(a) Histogram of Residuals
60
40
20-
«a
I «
-20
-40.— I — I — I — I — I — I — I — | — I — |.-|~|--l-.|~|~["l —
0500 0700 0900 1100 1300
Time (hours)
(b) Residuals as a Function of Time
I — I
40-
i
I
I
I
20-
I
r
i
II
o:
-20-
I
I
-40.,.,
20
40
Concentration
60
80
(c) Residuals as a Function of CORC Concentration
40-
20
"3
I
VI
» I — I — I — | — I •» | •• I »> |»| — i .> |». |.. |.»|.. |.. |.. |.. |.. I.. I
0 20 40 60 80
Concentration
(d) Residuals as a Function of CORM Concentration
CTl
O
FIGURE 11-23." RESIDUALS (CORC MINUS CORM) ANALYSES OF THE CORRELATED STATION RESULTS FOR
-------
20-
20-
10-
10-
V>
ro
3
"
-10
-10
-20. — I — I — I — I — I — I — I — 1 — I — I — I — I — l — I — l — I — l — I — I — i
0 15 30 45 60
Number of Occurrences
(a) Histogram of Residuals
-20. — |,
0500
0700 0900 1100
Time (hours)
1300
J 500
(b) Residuals as a Function of Time
20-
i
i
10-
i
r
I
•
2C
40
Concentration
60
SO
20-
t
I
I
I
10-
I
r
"% .(
2 o;.
lA I
Ol
-------
40 <
ZO-
I o
-20-
I
I
I
-40« — I —t — f — I —I —I — I —t—-I — I—I-
15 30 45
Number of Occurrences
(a) Histogram of Residuals
. I — 1 •• t — I — I — I — I —• I
60
40-
20
T3
V
I
I
-20-.
I
I
I
I --
-40. — I —I — | — l — I — I — |— |-.|~,|- -I —I- .] — ).. | _.[..|..|.,|..,
0500 0700 0900 1100 1300 7500
Time (hours)
(b) Residuals as a Function of Time
m
13
•o
40-
4
20-
I
I
I
I
O-'
aj
oc.
I
-20-
I
I
1
I
-10..-
o
.|..|..t..i..|..|.
40
Concentration
.|..i..|..|..|M|..i
60 SO
(c) Residuals as a Function of SAIT Concentration
40-
20-
VI
n
J 0
VI
(U
-20-
400*
-40...)..).. |..|M |..(.. |..(..).. |..|..|..|..|..|..|..|..|~fM|
0 20 40 60 60
Concentration
(d) Residuals as a Function of SAII Concentration
FIGURE 11-25. RESIDUALS (SAIT MINUS SAII) ANALYSES OF THE SAI TRAJECTORY RESULTS FOR
-------
63
underca1culation. Further insight can be obtained from Figure II-7(b), which
indicates a clear tendency to underpredict during the morning rush hours (0700
to 0900). In Figure II-7(d) the residuals are negative at high measured con-
centrations, a situation similar to that found for the PES results.
The station correlation results for CO are similar to those for PES,
except for a consistent tendency to underpredict [Figure II-8(c)] at almost
all concentration levels. This tendency arises from the attempt to calculate
particularly high, and probably nonrepresentative, CO concentrations using
lower, and probably more representative, concentrations from neighboring
stations. A similar effect is present in Figure II-8(d), where the negative
residuals at high concentrations also indicate a lack of high-concentration
neighbors.
In the residuals analyses of the GRC results [Figures II-9(.a) through
II-9 (d)], there is a trend toward underprediction during the rush period
and a trend toward overprediction in the afternoon [Figure II-9(b)]. This
particular behavior in the afternoon does not seem to be present in either
the PES or SAI station results [Figures II-6(b) and II-7(b)]. As with the
PES and SAI station results discussed previously, Figure II-9(d) points out
that the GRC results show underprediction at high measured concentrations.
Finally, an examination of the residuals for the SAI trajectory CO results
[Figures Il-lO(a) through Il-lO(d)] indicates much less of a tendency to
underpredict during the morning rush hours [see Figure Il-lO(b)], compared with
ithe GRC model and, in general, a weaker bias toward negative residuals.
The residual results for NO (Figures 11-11 through 11-15) do not reflect
the homogeneity that the CO results showed. In the PES results, the histogram
[Figure 11-11(a)] appears to indicate extremely strong agreement between cal-
culated and measured values. However, almost all of the data reported by PES
were for the early afternoon, when NO concentrations have reached their
background level of 1 pphm. For the one day when PES carried out calculations
for the morning period, the residuals were consistently negative (indicating
underprediction), as shown in Figures 11-11(b) and 11-11(d). Since NO is
a primary emissions product, these results probably reflect the problems of
disparity in scales and station nonrepresentativeness.
-------
64
The SAI residuals, for both the station and trajectory model results,
show a skewed distribution [Figures II-12(a) and II-15(a)], tending toward
either slightly high or markedly low values around zero. The greatest
(negative) deviations occurred at mid-morning [Figures II-12(b) and II-15(b)J,
which was also the time of the highest NO concentration [Figures II-12(d)
and II-15(d)]. Afternoon predictions by the SAI model were consistently
high, indicating its inadequate treatment of NO/NCL kinetics.
The GRC residuals, though they demonstrate a slight bias toward under-
prediction [Figure II-14(a)], do not exhibit any particular trends as a
function of time [Figure II-14(b)] or concentration [Figures II-14(c) and
II-14(d)]. The differences between the GRC and SAI trajectory results can
most likely be attributed to the differences between the chemical reaction
mechanisms used. The station interpolation values are reasonably well dis-
tributed, though highly scattered, except in Figure II-13(d), where the
interpolation scheme is unable to match the highest measured values. Closer
investigation of the data shows that almost all of these high values occur
at Burbank; the anomalous location of this station (and the correspondingly
high measured values) has already been discussed.
The N0? results (Figures 11-16 through 11-20) show a series of trends
opposite to those exhibited by NO. Again, the preponderance of PES data in
the early afternoon obscures some of the findings, but the tendency toward
overprediction [Figures II-16(a)] is evident.
The SAI and GRC results are similar in that they display a tendency toward
overprediction, most noticeably in the late morning hours [Figures II-17(b),
II-19(b), and II-20(b)]s with a predilection toward gross overprediction
at the highest calculated values [Figures II-17(c), II-19(C), and II-20(c)].
This latter trend results in part from a tendency of the GRC and SAI models
to predict occasionally the peak in the NOp concentration one or two hours
earlier than its actual occurrence. Similar behavior is observed in smog
chamber simulations and is presumably attributable to the inadequacies of the
reaction mechanism.
-------
65
In contrast, the station correlations do not show any particular trend
except underprediction of the largest measured values [Figure II 18(d)]; this
again is a result of the juxtaposition of the Burbank measuring station
and a neighboring power plant. The SAI station calculations also failed
\
to reflect these high measured values [Figure II-17(d)], though no such
trend is noticed for the SAI trajectory results [Figure II-20(d)]. This
is a particularly clear-cut example of the problem of disparity in scales.
Ozone is produced in the Los Angeles basin entirely as a result of
photochemical reactions. Its concentration rises sharply in the afternoon
after a morning "incubation period," because of the complex and relatively
slow reaction kinetics involved. Therefore, the trends observed in the 03
residuals are primary indicators of any systematic bias in the chemical
kinetics mechanism of a model.
For the PES results, the histogram [Figure 11-21(a)] shows a reasonably
even distribution, but the time sequence [Figure 11-21(b)] demonstrates that
this distribution is achieved through overprediction around midday and an
inability to calculate the steep CL concentration rise in the afternoon.
The concentration plots demonstrate the same behavior: positive residuals
at the highest predicted values [Figures 11-21(c)] and negative residuals
for the highest measured values [Figure 11-21(d)].
The SAI station residual plots somewhat parallel those of PES, though
the histogram [Figure. II-22(a)] is more negatively biased and the midday
overprediction [Figure II-22(b)] is less pronounced. The concentration
plots [Figures II-22(c) and II-22(d)] again show an inability to reproduce
the sharply rising measured CU concentrations.
The station correlations are dense and evenly distributed; only at the
highest values [Figures II-23(c) and II-23(d)] do the interstation discrepancies
become apparent.
-------
66
The GRC model produced a reasonably even distribution for both the
histogram [Figure II-24(a)] and concentration plots [Figures II-24(c) and
II-24(d)]. Only the time sequence [Figure II-24(b)] shows evidence of a
decided tendency to overpredict; this may be a consequence of the reduction
in NO emissions, since excess NO would tend to reduce the 0. concentration
O
chemically.
The SAI trajectory results are most interesting. Unlike the other three
data sets, this histogram [Figure II-25(a)] demonstrates a decided bias toward
overprediction, primarily at mid-morning [Figure II-25(c)]. The trend toward
underprediction of the highest measured values is not nearly as severe
[Figure II-25(d)].
Since the same SAI model was used both to underpredict the station
values (Figure 11-22) and to overpredict the trajectory values (Figure 11-25),
it is necessary to find some explanation beyond simple modeling error to
explain the discrepancy. A likely rationale for this behavior is the problem
of disparity in scales--a problem that would be expected to be especially
severe in the case of a slow-forming, fast-reacting pollutant such as ozone.
Owing to variations in pollutant concentrations and reaction conditions within
the grid squares, the "well-mixed chemical reactor" assumption used in modeling
is simply inadequate to represent the actual conditions in the "real world,"
and resolutions of such discrepancies as that noted in this case must await
more widespread application of subgrid models and better knowledge of the
chemical mechanisms of smog formation.
E. CONCLUSIONS
In general, it cannot be stated that any of the three models has or
has not been validated adequately. The intrinsic difficulties in attempting
to use a sparse and incomplete data base, with stations sited at what are,
from the modeler's point of view,'highly nonrepresentative'locations, leave
too many unknown factors to identify whether discrepancies are due to
modeling error or data inappropriateness. Better tests of the ability of
the models to simulate the formation and dispersion of photochemical smog
must await the availability of denser, more uniform, and more representative
measurements.
-------
67
Given the above caveat, it is sti-11 possible to draw a few general
conclusions about the performance of the three models under investigation.
All of the models were able to follow the changes in concentration of the
major pollutants as a function of time. The shortened time frame and
small number of data points offered by PES obviously did not stress the
model sufficiently, in the sense that it did not have the opportunity to
follow the rise and fall of pollutant concentrations during the early
morning hours; thus, its validity remains most in doubt. The GRC model
and especially the SAI model were placed in a "higher state of jeopardy"
by virtue of their earlier starting times and larger number of runs over
more varied conditions. Both performed well with regard to the primary
pollutants, CO and NO. The GRC model, "tuned" for ozone at the expense
of NOp, predicted the former well. The SAI model treated NO^ and ozone
equally successfully. None of the models exhibited a particular flair for
predicting the highest pollutant concentrations, which are of the greatest
interest from a pollution control standpoint. However, those highest con-
centrations are also the ones most suspect in terms of representativeness.
Again," satisfactory validation will ultimately depend on the availability
of more suitable data bases.
-------
68
III- ASSESSMENT OF THE VALIDITY OF AIRSHED MODELS
A. INTRODUCTION
Airshed models can be classified according to the type of coordinate
system used: fixed coordinate (grid) or moving coordinate (trajectory).
Grid models are based on a coordinate system that is fixed with respect to
the ground; hence, they are commonly referred to as Eulerian models. Tra-
jectory models "attach" their coordinate system to a hypothetical vertical
air column that moves horizontally with the advective wind; they are often
called Lagrangian models. Since the coordinate system used is one of the
basic differences among first generation photochemical air pollution models,
one of the first tasks that should be undertaken in the development of a
second generation model is a careful examination of the range and conditions
of validity of each of these modeling approaches. Such an assessment is
necessary to determine which type of model (or combination of models) pro-
vides a more suitable basis for the development of a second generation model.
In each of these two modeling approaches, we can phenomenologically
identify the sources of inaccuracies. First, we consider the trajectory
model. This formulation is based on the concept of a hypothetical vertical
air column that must maintain its integrity as it moves through the airshed.
For several reasons, this model may not be valid under certain conditions
in the turbulent atmospheric boundary layer. In the planetary boundary
layer, both the magnitude and the direction of the wind vary with height.
Therefore, strictly speaking, an air column cannot possibly remain vertical
as it is being advected by the wind over the time periods commonly of interest
Errors introduced by the assumption of a vertical air column are determined
by such factors as the wind profile in the vertical direction, the size of
the air column and the transverse distance it travels, and the concentration
gradients in the horizontal direction.
-------
69
Another possible source of errors in the trajectory model is the way in
which the trajectories are obtained. Conventionally, these trajectories are
computed from wind measurements made by a network of ground stations. The
question naturally arises as to whether a network of fixed wind stations can pro-
vide the trajectories of air colmns in a turbulent atmosphere. As shown in
Appendix A, two types of "Lagrangian" average velocities can be formed, and,
depending on the turbulent statistics, they can be quite different. Dyer (1973)
estimated values for many hypothetical cases and found that the two velocities
can differ under certain circumstances by more than 50 percent. If this is the
case, then which, if either, of these two velocities equals the corresponding
"Eulerian" velocity registered by a fixed wind station? Apparently, fundamental
difficulties exist in the construction of the trajectories used in the Lagrangian
model.
Second, we consider the grid model. The primary source of errors
associated with this type of model arises in the discretization of the
spatial coordinates. Consideration of both computational time and avail-
able core memory usually limits the amount of cells in each direction to
a number of the order of 50 or less. Unfortunately, in the advective
transport of material across the grid system, such a relatively small
number of grids produces the undesired effect of pseudo-diffusion. This
is evident in the following illustration, which shows that the concentration
distribution, as represented by a grid model, has been artificially smoothed.
I + 1
I + 1 1 + 2
t = 0
t = At
t = 2At
-------
70
Both the fixed-coordinate grid model and the trajectory model, as shown
in Figure III-l, are based on the diffusion equation,, which in turn can be
derived from a very general model, referred to here as the basic model. The
essential assumption upon which the diffusion equation for species that re-
act linearly--and, hence, the grid and trajectory models—are based, is
that the kernel Q in the basic model is Gaussian. This is true, however,
only for the case of homogeneous, stationary turbulence (Monin and Yaglom,
1971). Further assumptions must be made in deriving the grid and the tra-
jectory models. Solution of the grid model requires finite differencing,
whereas formulation of the trajectory model involves neglect of the spatial
derivatives. Therefore, a study that examines the validity of the airshed
models can be divided into the following two tasks:
> Evaluation of the validity of the diffusion equation.
> Determination of the magnitudes of the errors in the solution
of the diffusion equation under the assumptions made in the
grid and the trajectory models.
The evalution of the validity of the diffusion equation has been
examined by many investigators (e.g., Lamb and Seinfeld, 1973). However,
only qualitative results have been obtained with regard to the conditions
that must be met in applying the equation. Since these conditions involve
the statistics of atmospheric turbulence and linearity or nonlinearity
of photochemical reaction terms in the equations of continuity, quanti-
tatively stated conditions for the validity of the diffusion equation
under realistic situations are extremely difficult, if not impossible, to
obtain. Therefore, despite our suggestions of a scheme at the outset
of this task to assess the validity of the diffusion equation for certain
restricted cases, we later decided to omit this evaluation altogether in
the present study for two reasons:
> It would have required considerably more time and effort than
we could devote.
-------
ASSUMPTION:
FINITE DIFFERENCING
THE BASIC MODEL
r t
c^r.t) = ff Q^r.r'jt.f) S^r'.t1) dr'df
0 0
ASSUMPTIONS:
> GAUSSIAN KERNEL
> LINEAR REACTIONS*
THE ATMOSPHERIC DIFFUSION EQUATION
8 3\
x7
J
THE GRID MODEL
ASSUMPTIONS:
> NO HORIZONTAL DIFFUSION
> NO CONVERGENT OR DIVERGENT FLOWS
> NO WIND SHEAR
THE TRAJECTORY MODEL
*However, if one considers the atmospheric diffusion equation to be derived phenomenologically, the
reactions therein may not necessarily be linear.
FIGURE III-l. DIAGRAM OF THE BASIC RELATIONSHIPS IN THE VALIDITY STUDY
-------
72
> Cases for which analytic solutions can be obtained (such as
homogeneous, stationary turbulence, linear reactions) and,
thus, cases for which model evaluation can be undertaken,
may be too restrictive to be of practical interest.
Consequently, we chose to restrict the scope of the validity study
in this contract effort to the second task: assessment of the validity of
the grid and trajectory models when compared with the diffusion equation.
In the next section, we present a theoretical analysis of the errors in
the trajectory and grid models. This evaluation takes the form of a direct
examination of the basic mathematical formulation inherent in each of these two
approaches. In spite of the rigor of this theoretical analysis, it yielded only
qualitative, or order-of-magnitude, estimates. To provide a quantitative assess-
ment, we carried out a numerical experiment, which we describe in Section C.
Although they are still limited to two-dimensional cases, accurate comparisons
can be made for many realistic situations. Section D presents the results of
these comparisons for the trajectory model, and Section E, for the grid model.
Section F summarizes our conclusions regarding the relative merits of these
two models.
B. A THEORETICAL ANALYSIS OF THE VALIDITY OF THE AIRSHED MODELS
The most direct approach to investigating the difference between the
fixed-coordinate grid model and the moving-trajectory model is to compare
the mathematical equations upon which the two classes of models are based.
As we stated earlier, we chose the atmospheric diffusion equation as the
common basis of comparison. By invoking the assumption of eddy diffusivity
we can write the most general diffusion equation-describing the transport,
diffusion, and chemical reaction processes that, take place in the atmosphere
as follows:
3C. 3C. 3C. 3C. K 3C.\ L 3C.\ / 3c.
_L+ U_1+V__L+W__L= J_KH_L+1-KH —L + J- KV —""
3t 3x 3y 3z 3x \ 3x / 3y \ 3y / 3z \ 3;
3C.
W 3Z
3 ( K
9x y
:H 9Cl 1 + 3
3x / 3y
/
+ RI
\t Q C •
Kll ]
" ay
+ si
i = 1, 2, ..., N, (III-l)
-------
73
where c^ is the mean concentration of species i; R. is the average
reaction rate and S. is the emission rate for the species i; u, v, and
w are the average velocity components of the wind; and Ku and K are the
n v
horizontal and vertical diffusivities, respectively.
As we subsequently show, both the grid and the trajectory models
are derivatives of Equation (III-l) with the application of further--
and thus more restrictive—assumptions. In the following two subsections,
we attempt to evaluate the relative merits of each of these two classes of
modeling approaches, but first we derive the appropriate form of the model
equations for the two models. By comparing these model equations with the
atmospheric equation (III-l), we identify the deficiencies of each of the
modeling approaches.
1. The Trajectory Model
Despite the basically Eulerian nature of the atmospheric diffusion
equation, we can derive the modeling equation for trajectory models from
Eq. (III-l). As we stated earlier, the trajectory model attempts to des-
cribe, using a coordinate system that moves along a surface level wind
trajectory, physical processes that influence pollutant concentrations.
Toward this end, we can introduce the following general transformation
of variables in Eq. (III-l):
£ = ?(x,y,t)
n = n(x,y,t)
z = z ,
t = t , (HI-2)
-------
74
where the functional forms of £ and n are to be determined from the
trajectories. Invoking the chain rule to yield the derivatives, sub-
stituting these derivatives in Eq. (III-l), and rearranging some terms,
we obtain
12.
at
= K,
M.
3X
II
azc
+ 2
9X/\3X
9y/\9y
3n
9x
+
3x
!!)!£ flu + „ la + v la \ 1£
9y / 35 " 1 Bt u ax v ay I an
az
_
az
v az
(1II-3)
To the extent that the atmospheric diffusion equation is a valid
description of the physical processes under consideration, Eq. (III-3)
is still the "exact" equation describing the concentration changes rela-
tive to a moving coordinate system. We compare below Eq. (III-3) with
the most general form of the modeling equations that^have been adopted
in the trajectory approach,*
12. — IK 12.
at az I v az
+ R + S
(III-4)
Regardless of the size of its base area, we have considered the hypothetical
air column to be horizontally homogeneous. Thus, the pollutant concentrations
within the cell depend only on the distance above the ground, z, and the
time of travel, t, of the air column. If we assume further that K-theory
is valid, we can derive Eq. (III-4) phenomenologically for the variation
of the mean concentration in the air column.
-------
75
along a trajectory specified by
•i*
where u E u(x,y,zR5t) and v = v(x,y,zR,t) represent the surface wind1
used to construct the trajectory. Term-by-term comparisons of Eqs. (III-3),
(III-4), and (III-5) immediately disclose the following:
> The horizontal diffusion terms, i.e., the first group of terms
with the common multiplication factor K.., are neglected in the
trajectory models. As is clear from the terms in the braces
in Eq. (III-3), horizontal diffusion can be neglected theoreti-
cally, in general, only in the trivial case in which the concen-
tration field is constant throughout the entire region of interest.
However, strong localized sources, such as freeways and power
plants, are commonplace in any airshed. Consequently, the con-
centration field generally is far from constant. Thus, in the
validity study of trajectory models, the effect of the exclusion
of horizontal diffusion terms must be considered.
> The vertical component of the wind has been altogether neglected
in the conventional trajectory models. The occurrence of convergent
flows in an urban area due to many factors, such as the urban heat
island effect, certainly makes this assumption unrealistic.
> As shown in Eq. (II1-3), terms involving the first spatial
derivatives, 3C/3C and 8c/9n, vanish only if
* The surface wind is typically taken to be that at a height of 10 m; i,e,f
ZR = 10 m.
-------
76
"I,/ ^ »
a - o . (,,,-6)
where, in general, u = u(x,.y,z,t) and v = v(x,y,z,t). A
comparison of Eqs. (III-5) and (III-6) shows that in the tra-
jectory model, it is further assumed that
u = u ,
v = 7 . (III-7)
This assumption implies that only a constant horizontal wind
field at a reference height, ZR, can be incorporated in the
trajectory model. In other words, the vertical variability
of the horizontal wind is suppresssed in the trajectory model-
ing approach. As we show later, the effect of suppressing
vertical variations of horizontal wind on the predicted con-
centrations can be quite substantial.
In the preceding discussion, we have identified the sources of errors
associated with the trajectory modeling approach; the remaining task is to
establish the magnitudes of errors so that the range of validity of the
trajectory model can be determined. In principle, this task can be accom-
plished by evaluating the magnitudes of the respective error terms under
commonly occurring circumstances. In practice, the multiplicity of possible
conditions or combinations of conditions that can take place in an urban
atmsophere render this approach impracticable. Furthermore, only qualita-
tive, or order-of-magnitude, estimates can be obtained. For these reasons,
we propose an alternative approach in Section C: assessment of the tra-
jectory model through numerical experiments.
-------
77
2. The Grid Model
Although the representations of the various terms in the atmospheric
diffusion equation can all be accommodated in a grid model, the model can
be considered as the discrete analog of Eq. (III-l). In the process of dis-
cretization,* inaccuracies are unfortunately introduced. These inaccuracies
are usually discussed in terms of the order of truncation terms. As an
illustration of this type of analysis, consider the following simple form
of Eq. (III-l) containing only the time-dependent and x-direction advective
terms:
with a constant reference velocity UQ. Suppose we choose the following
difference equation, from a first-order scheme, to approximate Eq. (III-8):
Then we can carry out a Taylor series expansion about the time-space point
* In the present investigation, we focused our attention on only the finite
difference method, in which the coordinates are discretized. In the
particle-in-cell technique, the pollutants masses are discretized, and
different types of difficulties are introduced. Nevertheless, this method
is a viable procedure that can also be classified as a grid modeling
approach.
-------
(n,j) to obtain
78
.
3t 0 9X
UQAX
1 "
x
+ Higher Order Terms.
(111-10)
Comparing Eq. (III-8) with Eq. (111-10), we find that all the terms on the
right-hand side are introduced through discretization. The lowest order
term of these, having the form of a second-order spatial derivative, can be
mathematically characterized as a diffusion process. This term is therefore
often called "numerical," "pseudo," or "artificial" diffusion.,
For the finite difference scheme to be stable, the diffusion coefficient
must be positive. This restriction leads to the famous Courant condition for
stability:
AX
(III-ll)
Thus, it is apparent from this simple analysis that the primary source of
errors arising from adoption of the grid model is associated with the
introduction of an undesirable diffusion term, which is always positive.
The presence of this additional diffusion term masks the true diffusion
and, of course, introduces inaccuracies.
Although the type of analysis present above is useful in revealing some
insights into the critical problems involved in adopting finite difference
approximations, its use in practical problems is, nevertheless, very limited.
In the first place, extension of such an analysis to the full three-dimensional,
nonlinear problem would probably be too complicated to lead to useful conclusions
-------
79
For example, even in the simple case we discussed above, a similar expansion
for a variable velocity would generate terms proportional to 3lL/3x in the
set of artificial diffusion terms. Whether the additional terms would tend
to alleviate the artificial diffusion problem, however, would depend on the
sign and magnitude of the acceleration term. Furthermore, the result of this
type of analysis can be stated, at best, in terms of order-of-ma,gnitude ex-
pressions. Although the analysis shows that the higher order truncation terms
always vanish when higher order finite difference schemes are used, this
phenomenon does not necessarily imply that these schemes are more suitable.
A notorious example can be found in airshed modeling: Near localized sources,
higher order schemes predict unreasonable negative concentrations, whereas
simple first-order schemes do not. In view of these deficiencies, we concluded
that the inaccuracies in the grid model could be more profitably assessed through
the numerical experiments dicussed in the next section.
C. ASSESSING THE VALIDITY OF AIRSHED MODELS THROUGH NUMERICAL EXPERIMENTS
We have explored the validity of both the trajectory model and the grid
model by examining the formulae from which the two classes of models are de-
rived. As shown in Figure III-l, the atmospheric diffusion equation was the
common basis for comparison. By recognizing mathematical terms that have been
incorrectly (though, in some cases, unavoidably) introduced or neglected in each
of these two modeling approaches, we can identify the sources of errors. We
list these sources below:
> Trajectory model sources
- Neglect of horizontal mixing across the boundaries of the parcel.
- Neglect of the vertical component of the wind velocity (the move-
ment of the parcel is two-dimensional).
- Assumption that the entire parcel moves with a wind velocity that
is invariant with height.
> Grid model source--"Numerical" diffusion introduced by finite differencing
-------
80
Although errors of each type can be estimated by evaluating the magnitudes
of the relevant terms under commonly occurring conditions, quantitative esti-
mates are, nevertheless, difficult to obtain. The remaining portion of this
section is, therefore, devoted to a description of a numerical experiment
that provides a means for assessing the absolute errors.
The major components of the numerical experiment consist of the follow-
ing three steps:
(1) Find the exact solution of the atmospheric diffusion equation
for some well-defined hypothetical cases. These cases should
be carefully chosen so that they are as general and as realistic
as possible. In addition, they must include, at a minimum, one
or more of the key ingredients noted earlier. However, these cases
should be sufficiently simple that analytic solutions to the atmos-
pheric diffusion equation can be obtained.
(2) Exercise the trajectory or the grid model for these hypothetical
cases, and compare the differences between the analytic solutions
and the predictions of each model.
(3) Compare these results for variations in each parameter over a range
of values that may occur in a real atmosphere, so that the range of
validity of each of these two modeling approaches can be ascertained.
To examine the importance of the various effects that we mentioned earlier,
we need to include, where possible, in the atmospheric diffusion equation the
following terms, taken one or more at a time for evaluative purposes:
Term
Horizontal diffusion
Vertical convection
Vertical variations of the
horizontal wind speed
Time-dependent and
advection terms
Effect to be Evaluated
Neglect of horizontal diffusion
Convergent or divergent flow
Wind shear
Numerical errors
-------
81
It should be evident from the preceding discussion that the hypothetical
cases can be of no greater complexity than a two-dimensional, time-dependent
formulation if analytical solutions are to be derived. Unlike the modeling
of turbulent flow, where fundamental differences can exist between two- and
three-dimensional turbulence, this assumption of two-dimensionality does
not unduly affect the conclusions of a validity study. Furthermore, since
meteorological parameters are of primary importance in the validity study of
different modeling approaches, we also assumed that the chemical-reaction
and volume-source terms are absent in Eq. (III-l). Thus, we considered the
following equation in the present study:
3C 3C _ 8 /,, 9C \ , 3 / 3C
Although Eq. (111-12) is a considerably simplified form of the atmos-
pheric diffusion equation, general solutions still cannot be found for
arbitrarily specified wind speeds, diffusivities, and boundary conditions.
We carried out a limited effort to examine existing analytic solutions
(usually special cases) that were relevant to the present study. Table
III-l summarizes the results. As Table III-l shows, none of the cases
for which analytical solutions had been obtained contains all the ingredients
that are necessary to assess both the trajectory and the grid models. Com-
promises must thus be made, such as considering only special cases that
isolate certain effects that are neglected in either the trajectory model or
the grid model. Table III-2 summarizes the cases considered in this study.
D. THE VALIDITY OF THE TRAJECTORY MODEL
This section examines, to the extent possible, the individual errors
committed through the neglect of horizontal diffusion, vertical wind, and
wind shear in the trajectory model. As a basis of evalution, we compare
the analytic solution for each of the first six cases listed in Table III-2
with the corresponding prediction of the trajectory model.
-------
82
TABLE III-l
EXACT SOLUTIONS TO THE DIFFUSION EQUATION
Investigator
Roberts (1923)
'
Calder (1949)
Smith (1957)
Smith (1957)
Smith (1957)
Monin (1959)
Yordanov (1965)
Yordanov (1968)
Halters (1969)
Dllley and
Yen (1971)
Calder (1971)
3C 3C 9C f 3 !„ 3C\ ^ 3 /j, 3C\
Tf 3X 32 3X V H3x./ 32 \ V32/
Type of Source 3/at u w *H KV
Line and point Yes 0 -0 K KV
source '
Line source No U 0 K Ky
Line source No U2m 0 0 Ky?"
,-. Elevated point Ho U(2 + h)1/2 0 K(z + h)1/2 Kjz t h)1/2
source "
Elevated line Ho U(z + h)a 0 o K (2 + h)1""
source , "
Case 1
Ground-level Ho U Wo KjH 2)2, 0 < 2 < H
line source " ~
|| = 0, 2 H
Case 2
Kyz, 0 £ z 5 1/2H
KV(H - z), 1/2H <_ 2 <^ H
|j = 0. 2 H
Ground-level No U WO Kv2~a, 0 < 2 < H
line source " ~ ""
|f 0. 2 H
Elevated line Yes 0 00 Kvtz, z < ||_|
source
KvtL, 2 < |L|
Elevated point No U,2m, z v |L| 0 0 K,z, z < |L|
source
L. C.
Case 1
Elevated point Yes 0 00 K^z", z < a |L|
source
Case 2
K/. z < a |L|f
K^aL)". z > a M1"
Ground-level No U 0 K-z KVZ
line source
Ground-level No (U-ax)(-\ -^frf1 ) ° Kv(~)
line source ' I/ m \ 1' * V
Line and point No Uzm 0 K-z" KyzB
source
Stable atmosphere.
-------
TABLE II1-2
SUMMARY OF THE CASES CONSIDERED IN THE VALIDITY STUDY
83
Type of Assessment
Trajectory model
The effect of neglecting
horizontal diffusion
Type of Sources
Instantaneous, ground-level
line source
at
Hodel Equation
,,
^u 9 T-T
3x
- constant
« constant
Comments
See Section C-l-a
Continuous, ground-level
line source
ac .
"
Ic
3X2
3 L ac\
3Z I V azJ
> '
See Section C-l-b
and Walters (1969)
Time- and space-varying
ground-level area source
3t
3X
U constant
K,, constant
See Section C-l-c
The effect of neglecting
vertical wind
Continuous, ground-level
crosswind line source
The effect of neglecting Continuous, ground-level
wind shear crosswind line source
Time- and space-varying
ground-level area source
Grid model
The effect of numerical
errors
Time- and space-varying
ground-level area source
3Z 3Z
u - (U1 ax)
az z
V l z
It - IF
S If)
7
u(z) u
3c _ 3
3x az
(«,£)
l£
at
l£ K '
ax " ^ ~.
V a constant
K,, constant
K.r e constant
i- /K ^
az ^v azj
See Section C-2 and
Dilley and Yen (1971)
See Section C-3-a
See Section C-3-b
See Section D
-------
1. The Effect of Horizontal Diffusion
We first consider two physical situations that provide, in effect, upper
and lower bounds on the errors induced in the trajectory model. The first
is that of an instantaneous line source at ground level in an atmosphere with
a uniform wind blowing in the x-direction. The second is that of a continu-
ous line source at ground level in a similar atmosphere. In the dispersion
of a puff from an instantaneous release, horizontal dispersion, in the absence
of wind shear, can be expected to play a key role in spreading out the cloud.
For a continuously emitting source, concentration gradients in the direction of
the mean wind are substantially smaller than those for an instantaneous release.
Thus, a comparison of the concentrations predicted by the trajectory model
with the actual concentrations provides upper and lower bounds on the errors
committed by not including horizontal diffusion. To provide a more realistic
assessment, we consider next the impact on the induced errors of an urban-type
source distribution—a time- and space-varying area source.
a. Instantaneous Line Sources
We can derive the mean concentration resulting from an instantaneous line
source under the conditions of a constant crosswind, U, and a constant hori-
zontal diffusivity, K,,, from a simplified form of Eq. (111-12):
2 / \
ac . I, 3c _ K 9_^ + JL(K ic.) (111
at + U T " KH 2 + sz KV 3z ' U11
By invoking the coordinate transformations,
£ = x - Ut
P = z ,
T = t
-------
85
we obtain for Eq. (111-13)
_
2 9p \V 8p
d c, \
(III 14)
We can write the appropriate initial and boundary conditions as
c(?,p,0) = Q£6(5)«5(p) , (111-15)
C(?,P,T) = 0 , C -> ±M , (111-16)
-Kv -^ = 0 , p = 0 , (111-17)
C(?,P,T) = 0 , p -> - , (111-18)
_3
where c is the mean concentration, in g-m , and Q is the mass of
-1
pollutant emitted per unit width in the y-direction, in g-m . The
problem defined by Eqs. (111-14) through (III 18) describes the two-
dimensional dispersion of a puff of inert contaminant relative to its
horizontal center of mass in a horizontally and vertically homogeneous
atmosphere.
We can express the solution of Eq. (111-14) in the form
f r2!
cU.p.O = X(P,T) exp -?r-f-y • (111-19)
I ' \ P 9 L / I
We define the zeroth and second moments, respectively, of C(£,P,T) as
follows:
oo
CQ(P.T) = / c(?,p5T) d£ , (111-20)
-------
86
(111-21)
_o
where cn and c? have units of g-m and g, respectively. We can express
f(p,i) and X(P,T) in terms of CQ and c^,
2c,
f =
(111-22)
The zeroth moment, CQ (P,T), satisfies
3c
0 _
< !!°
V 3p
(111-23)
(111-24)
C0(p,0)
-K,
V 3p
CQ(P,T)
Q£6(p)
- 0 , p = 0
= 0 , p
(111-25)
(111-26)
(111-27)
The solution of Eq. (111-24), subject to Eqs.(111-25) through (111-27) and
, is
CQ(P,T) =
n 1 c~p
/0 \2-n /I \ /IT \2-n
2-n
p
(2
-n)2^
T_
(111-28)
,2 - n
-------
The second moment, c2, satisfies
c2(P,o)
9 I v
7— KI
ac
+ 21(H C0
= 0
-3c,
-K.
•V 3p
C2(p,r)
In addition to K
V
= 0 , p = -0
= 0 , p
, we set KM = constant. The solution of
Eq. (111-29) subject to Eqs. (111-30) through (111-32) is
1-n
,(P,T) =
2KA
n
(2 - n)2"n
2-n
r
-i cXp
1 \ ^ 2-n
2-n/ Kl
2-n
P
(2 - n)2 KlT
87
(111-29)
(111-30)
(111-31)
(111-32)
(111-33)
Using Eqs.(111-22), (111-23), (111-28), and (111-33), we can obtain
the solution of Eq. (111-14):
C(?,P,T) =
I
1
/ \ / V
9 n / 1 \/ \ 9 n
fr> ~\
-------
The ground-level concentration at the centroid of the puff is
c(0,0,T) =
2-n
We can consider Eqs. (111-34) and (111-35) as the "exact" expressions for the mean
concentration in the puff, and we can compare them with corresponding expres-
sions derived from the trajectory model.
We now proceed to develop the form of the trajectory model applicable
to the description of the dispersion of an instantaneous release. We have
denoted the actual mean concentration of a pollutant from such a release by
_2
C(?,P,T) in the case of a line source; c is expressed in units of g-m , and
the instantaneous source, in units of g-m' . The concentration C(P,T), de-
_3
rived using the trajectory model, is also in the customary units of g-m
Thus, for inert contaminants and no elevated sources, the governing equation
and associated initial and boundary conditions become, for an instantaneous
release,
0 O v 0 3 T /
o X
c(p,0)
-Kvff
C(P,T)
- 9 k 8c
3p \^V 9p/
= QA6(p)
= 0 , p
= 0 , p
'
'
= 0
->- CX)
(111-37)
(111-38)
(111-39)
-------
89
where the proper source strength, Q „, in Eq. (Ill 37) is expressed in units
-2
:A
of g-m , i.e., an instantaneous area source. The key problem, then, in
formulating the trajectory model for an instantaneous source is to relate the
true source strength Lj(g-m in the case of a line source) to the source
_o
strength Q* (g-m ) in Eq. (111-37). For an instantaneous line source, the
source strength (L in Eq. (111-37) is related to the actual strength Q^ by
Q,
(111-40)
where SL is a length in the x-direction over which the actual source is averaged,
o
Because of the necessity of using g-m as the concentration unit in both des-
criptions, the true instantaneous strengths must be spatially averaged in the
trajectory model. We show subsequently that this averaging is unnecessary for
continuous sources.
We can obtain the solution of Eqs. (111-36) through (111-39), with
KV - "KlPn, from Eq. (111-28), with Q£ from Eq. (111-40):
C(P,T) =
exp
2-n
(2 - n
(111-41)
We can express the measure of the deviation of c(0,T) from c(0,0,T) in
Eqs. (111-35), (111-40), and (III-41) by their ratio:*
*A similar analysis of an instantaneous, ground-level point source reveals
that the ratio of the ground-level concentration predicted by the trajectory
model to the actual concentration is
Y =
A
-------
90
(111-42)
o
Figure III-2 presents a plot of this ratio for three values of 4TTKu/£
n
commonly encountered in an urban scale problem. As Figure III-2 shows, for
small T, y < U the trajectory model underpredicts the ground concentrations,
whereas for large T, y > 1, the trajectory model overpredicts the ground con-
centrations. The explanation for this result is as follows. The ratio y, as
given by Eq. (111-42),, can be viewed as the ratio of two length scales: that
associated with horizontal turbulent diffusion and that characteristic of the
spatial averaging of the emissions. Initially, the spreading of the pollutant
cloud varies approximately as the square root of the time. Therefore, at this
stage, the spread is not sufficient to compensate for the influence of the
artificial spatial averaging of emissions; consequently, near the source,
the trajectory model tends to predict concentrations that are too low. After
a substantial amount of time has elapsed, the effect of horizontal diffusion
overtakes the effect of the spatial averaging of the emissions, and the
trajectory model begins to overpredict. In summary, for an instantaneous
release, the trajectory model is most accurate when the two characteristic
lengths are comparable.
In most applications of the trajectory model, continuous (rather than
instantaneous) sources are considered. Thus, the above analysis represents
an unnecessarily severe test of the validity of the trajectory model insofar
as the effect of horizontal diffusion is concerned.
b. Continuous Line Sources
We now consider the case of a continuous ground-level crosswind line
source with a constant mean wind speed that is independent of height. Since
-------
10
FIGURE III-2. THE EFFECT OF NEGLECTING HORIZONTAL DIFFUSION ON THE TRAJECTORY
MODEL PREDICTIONS (FOR INSTANTANEOUS LINE SOURCES)
-------
our main purpose in considering this case is to assess the effect of horizontal
diffusion, the basic equation governing the concentration distribution, in
(x,z) coordinates, is
3c(x,z) _ ., 3C J)_
~ K
3x
3X
3C
az
c(x,z) - 0
X -> ±00
(111-43)
(111-44)
c(x,z) = 0 ,
(111-45)
c ->
, x,z -* 0
(111-46)
If
f°V K ^
J I v 3Z-
-oo»' \
dx - q.
z = 0 , x f 0
(111-47)
(111-48)
_0
where c(x,z) is expressed in g-nf and q .--epresents the pollutant flux for
-1 -1
a line source (in g-m sec ).
Walters (1969) solved the case in which KH = KQz and Ky
c(x,z) = -
K
exp
-A tan (y f-,
\ A
1
(111-49)
2 2
J zS
1/2
where A = U/(KnK,) , and y = (Kn/K.
U 1 U
neglected in Eq. (111-43), the result is
1/2
. When horizontal diffusion is
c(x,z) = -— exp I-
(111-50)
-------
Equations (111-49) and (111-50) provide a comparison of the effect of
neglecting horizontal diffusion with the effect of including it (see Walters,
1969). When y(z/x) « 1, i.e., when y(z/x) is sufficiently close to the ground,
the functional dependence of c on x and z is the same in the two cases. How-
ever, the ratio of the predicted magnitudes of the concentration varies from
unity (when KQ = 0) to 2 (when KQ ->- <*>). Walters also determined the
conditions under which horizontal diffusion cannot be neglected when predic-
ting the mean concentration from a continuous line source.
To employ the trajectory model for a continous source, we must convert
the downwind distance x into travel time T. In this case, x = UT, since
the velocity is uniform. The trajectory model is defined by Eqs. (111-36)
through (111-39). In relating the trajectory model to the continuous source
problem, we note that the proper source strength Qfl in Eq. (111-37) is in
-2 -1 -1
units of g-m , whereas the actual source strength q0 is in units of g-m sec
J6
Thus, it is necessary to convert the steady-state diffusion problem into an
unsteady-state problem to employ the trajectory model. If we let
QA = , (111-51)
then QA has the appropriate units of an instantaneous source. In effect, we
need not define an area associated with the column, since Q. represents the
mass of material emitted over the time it takes the wind to travel 1 meter.
Using Eq. (111-51) as the emission strength in Eq. (111-37), we obtain
the following solution of Eqs. (111-36) through (111-39) for the simple case
of n = 1:
C(P,T) = —L. exp f- --£_ _ (111-52)
KjT
Equation (111-52) is merely Eq. (111-50) with x replaced by UT and z
replaced by p. The ratio of the mean ground-level concentration predicted
by the trajectory model to the "exact" value can be obtained by replacing
-------
94
x and z in Eq. (111-49) by UT and p, respectively, and by dividing the
resulting expression into Eq. (111-52) after setting p = 0 in each ex-
pression. Thus,
Y = 1 + e"X7T . (111-53)
Since typical values of X in the atmosphere range from 0.75 to 500, a
reasonable upper bound on the magnitude of the error introduced into
trajectory model predictions when horizontal diffusion is not included
is
Y - 1 < 10%
It is therefore clear that, for the case of linearly varying diffusivities,
we can neglect horizontal diffusion with little error.
c. Time- and Space-Varying Line Sources
The previous two sections highlight certain omissions in the trajectory
model. However, these examples are somewhat idealized when compared with
situations in which one might actually use a trajectory model. Conditions
that one might commonly encounter include:
> Distributed sources that emit continuously and vary with time.
> Diffusivity-height relationships that are not linear.
To assess the performance of the trajectory model in situations other
than those explored thus far, we consider here a continuous, ground-level
area source under conditions of constant wind speed and vertical turbulent
diffusivity varying as a power law function of altitude.
Equation (111-34) gives the mean concentration from an instantaneous
ground-level line source with a uniform mean wind. In that equation, Q
-1
is in units of g-m . We now wish to consider the case in which the ground-
-------
level emissions are distributed over the strip 0 - x - L. We examine first
an instantaneous area source. If we assume that the original source strength
is distributed over the strip 0 f x - L, then the instantaneous area source
_2
strength QA, in g-m , is
dx
(111-54)
The concentration resulting from an infinitesimal line source of strength
Qfl da located at x = a is, from Eq. (111-34),
QAda
dc(s,z,t) =
--
2-n
exp -
(x
- Ut -
4KHt
«)2
(2 -
z2-n 1
2 -
(111-55)
Thus, the concentration resulting from the instantaneous area source of width
L and unit length is •,
c(x,z,t) =
(
n
to ^2-n r
^w2
/ 1 ^
]_ <-AF
1/77 ,\2-n
z2-n
L (2 -
n)2 KltJ
exp
0
(x - Ut - a)'
4KHt
(111-56)
where we have allowed QA to vary with location. In addition, for a continuous
ground-level area source, where qA = AQA/At and varies in time (g-m sec ),
-------
the concentration is
96
c(x,z,t) =
•* |4TKH(t - 3)]
(2 - n)2- r(^
2-n
exp
.2-n
(2 - n)
2
- 3)
exp
J [X -(Ut - 3)- a]'
4KH(t - 3)
qA(a,3) da d|3
(111-57)
Note we can derive this equation from Eq. (111-56) by applying the principle
of superposition.
We can obtain the corresponding solution for the trajectory model from
Eq. (111-41), which gives the mean concentration from an instantaneous source
-2 -1
of strength QA- To consider sources of strength qA(0 (in g-m sec ), we
invoke the principle of superposition. Integrating over time the concentration
resulting from an instantaneous source qA dg released at time T = 3 gives
C(P,T) =
(2 - n)2-n r/
qA(3)
exp
2-n
(2 - n
,2 ir
- 3)
d3
(111-58)
We obtained the spatial and temporal distributions of the emissions used
for the present study from the emission pattern of carbon monoxide for a hori-
zontal strip of Los Angeles (see Figures III-3 and III-4). Mathematically
expressed, this type of pattern is the summation of a series of rectangular
step functions:
-------
Q-
Q.
I
CO
c
o
•r-
CO
LoJ
20
15
10
MALIBU
f
DOWNTOWN LOS ANGELES
POMONA
I
0
10
20
30
40
50
Miles
FIGURE III-3. SPATIAL DISTRIBUTION OF CARBON MONOXIDE EMISSIONS (10:00 A.M. PST)
-------
1.5 -
tn
n
-M
cr
S-
o
nj
U_
CO
O
CO
CO
1.0
0.5
8
Time—hours
10
12
14
oc
FIGURE III-4. TEMPORAL DISTRIBUTION OF CARBON MONOXIDE EMISSIONS
-------
yy
where
(111-59)
1=1
•I
E.j(t), a constant, for I'AX > x > (ial)Ax,
0 elsewhere,
and where
N = the total number of grid points (25),
Ax = the grid spacing,
E-(t) = the magnitude of emission flux strength from
grid point i at time t.
Under the assumption of a step function for the spatial distribution of
emissions, as described above, we can reduce the double integral on the
right-hand side of Eq. (111-57) to a single integral:
c(x,z,t) =
4- exp
/" ?-n
/ o n \ C- \\ /•
•>TI M /
z2-n
o
(2 - n) K,(t - g)
r 1 l
2 - n
'v^
2J Ei(6)
i = l
erf
x - (i - I)AX - U(t - p)
K(t - 3)
- erf
- JAX - u(t - e)
(t - 3)
(111-60)
-------
100
The integrals in Eq. (111-58) and (111-60) unfortunately exhibit
singular behavior near the upper limit of the integration. Thus, the achieve-
ment of results that are acceptably accurate would require extremely fine
meshes. For example, using numerical techniques to evaluate either integral
[Eq. (111-58) or Eq. (111-60)] with a relative error less than e, we find that
the total number of mesh points required is on the order of
M
2-n
1-n
For n = 0 and e = 0.1 percent, the required number is a staggering 10 .
However, we can remove this difficulty by elongating the time axis in accor-
dance with the following coordinate transformations:
A = (T - B)
1-n
2-n
for Eq. (111-58)
or
x = (t -
1-n
v2-n
for Eq. (111-60).
Equations (111-58) and (111-60) subsequently become
(2 - n) 2-"
77 2-n
1-n
2-n
qAU)
exp
2-n
2-n
dA
(2 - n)
2
(111-61)
-------
101
and
c(x,z,t) =
(2 - n)
2-n
1-n
L2-n
exp
2-n
(2 - n)
2-n
- erf
erf
2-n
x -• (i - I)AX - Ux
~
1-n
x - TAX - UX
<1 2-n
2-n
1"11
dX
(111-62)
We evaluated these integrals using Simpson's rule.
We performed calculations for several different sets of conditions
representing typical or extreme conditions observed in urban atmospheres.
We computed the ratios y of the ground concentrations predicted by the "exact1
solution to the ground concentrations predicted by the trajectory model.
Figure III-5 shows these ratios as a function of release time of the air
parcels in the trajectory model. Although Figure III-5 shows that the effect
of neglecting horizontal diffusion increases with increasing horizontal
diffusivity and vertical wind shear and with decreasing wind speeds, the
absolute magnitudes of the errors are rather small--apparently less than
10 percent.
-------
KH
U "
Symbol n (mph) (m /sec)
O 0 4 50
© 0 4 500
*~ D 0 2 50
A \ 4 50
A
O 8 ® A Time After
0 ?1 12 |3 14 | 5 96 $7 jj ^9 Release-hours
1.1
Q 1-1 £JgJ A
° A D
o &
n
A
0.9
FIGURE III-5. THE EFFECT OF NEGLECTING HORIZONTAL DIFFUSION ON THE
TRAJECTORY MODEL PREDICTIONS (FOR URBAN-TYPE SOURCES)
o
1X3
-------
103
Thus, we concluded that, for all practical purposes, the neglect of horizontal
diffusion in the trajectory model is unimportant when compared with other
uncertainties in airshed modeling.
2. The Effect of Vertical Hinds
Convergent and divergent flows are not uncommon in many urban areas.
Channeled by local topography, two air flows having opposite direction can
clash to produce a strong convergent flow, such as that characterizing the
famous San Fernando convergent zone in Los Angeles. Hot or cool spots in an
urban area can also create local convergent or divergent flows. As we indi-
cated earlier, existing trajectory models invoke the assumption that the
vertical component of the wind field can be neglected. (We note that although
this assumption is commonly made, it is not necessary.) This section assesses
the errors committed as a result of neglecting the vertical wind in the tra-
jectory model.
Dilley and Yen (1971) studied a continuous ground-level crosswind
line source emitting into an atmosphere in which the wind consists of a
local convergent flow (with both horizontal and vertical components) super-
imposed on a horizontal wind, which also varies with height. We chose this
case, illustrated in the following sketches, as a basis for studying the effect
of neglecting the vertical wind on the predictions of a trajectory model.
ll!£ SWRCt OF
POILUTAMS
n'tSOSCALIWIND
ISLAND
-------
104
The equation governing the pollutant concentration can be written as
!£. + az /_z_Y" 3c_
3x m + 1 lz, I 3z
3Z
3Z
(111-63)
subject to the usual condition for a continuous ground-level line source at
x = -x x > 0. As shown by Dilly and.Yen (1971), the solution of Eq. (111-63)
^ o
is
c(x,z) =
(m - n + 2)
a
(m + l)(m - n + 2) KI
1
(u: - axs)S - (i
I.
^ - ax)s .
exp
l_
f xS m-n+2
a(u, - ax) z
, , - w , Ox m-n .,
(m + l)(m - n + 2)z1 K-^
I 1
(u, - ax )s - (u, - ax)
where m - n + 1 > 0, u, - ax > 0, and
(111-64)
v* —
m + n
m-n + 2
s =
m + 1
m-n + 2
-------
105
Equation (111-41) gives the trajectory model applicable to this situation.
The basic difficulty, however, in comparing Eqs. (111-41) and (111-64) is to
relate the parcel travel time, T, to the downwind distance, x. Let us assume
that the parcel velocity, U, for the trajectory model is u1 - ax, i.e., the
parcel velocity at the reference altitude z^. It follows that the location
of an air parcel that was at s at T = 0 is
A
(111-65)
Thus, the travel time, T, is related to the downwind distance, x, as follows:
ul - axs
ul - ax
(111-66)
As in the previous line source examples, the appropriate source strength;
QA , for use in the trajectory model is related to the actual continuous
emission rate, q., as follows:
The solution is thus given by Eq. (111-41) with QA replaced by q£/U.
Therefore, the mean concentration predicted by the trajectory model is
c(x,z) =
u, - ax
n / v 1
(2 - n)2-" r|
1 I \Y 2-n
^ " 7 :
•/ \ l
I /ul " aM 2-n
a I u, - ax j
exp
az
2-n
(2 - n)
2
ul - axs
u, - ax -J
(111-67)
-------
106
Given Eqs. (111-64) and (111-67), we can now complete the ratio of the
predicted ground-level concentrations in which we are interested. The ratio
is
Y =
c(x.O)
c(x,0)
r(s)(m
1)S(m- n + 2)2"1 -2p 1
rf
2-n
(111-68)
where
P =
m(l - n)
(m - n + 2)(2 - n)"
and
u, - x
" x
-------
107
This equation shows that the deviation -of the trajectory model predictions
from the exact solutions is a function of two dimensionless distances. The
first, T, is a nondimensional form of the reference height, the elevation at
which the wind is used to compute the trajectory, the second, $, is a nondimen-
sional horizontal distance. The values of $ vary continuously from 1 to 0
as the air parcel travels from the source point x - x to a location where
a reverse flow begins to appear (x = Uj/a) and beyond which the solutions
developed above no longer apply. We computed the ratio. Y for a family of
the parameters m and n and for three values of <;. Figure III-6 presents the
results.
From a study of Figures III-6(a) through III-6(h), many interesting
observations emerge concerning the effect of neglecting the vertical wind
component in a trajectory model. As shown in Figures 1 1 1-6 (a) and 1 1 1-6 (b),
the ratio y is independent of t, if either m = 0 or n = 1, i.e., the choice
of a reference height becomes immaterial if the atmosphere possesses either
a constant diffusivity profile or a wind field that varies linearly with
height. For either case, it is clear that the trajectory model always over-
predicts the ground-level concentrations, and the deviations increase as the
trajectories move downwind of the source. Furthermore, the deviations increase
with a decreasing exponent in a power law diffusivity profile or an increasing
exponent in the power law wind profile. For a general combination of m and n,
Figures III-6(c) through III-6(h) show that the ratio y depends on the choice
of the reference height. The trajectory model predictions increase with de-
creasing reference height. The ratio y, however, always increases as the
distance from the source point increases, indicating an accumulation of pollutants
due to the lack of vertical transport by the vertical component of the wind
in the trajectory model .
The most significant conclusion. that .can be drawn from the above
analysis is that, for meteorological conditions typical of those observed in
urban environments, the value of y can vary greatly (i.e., by an order of
magnitude), particularly at distances far from the point of release. This
implies that, with the exception of the special case of a vanishing vertical
-------
10
2.5
1.0
0.5
0.25
0.1
0.1
n = 0
m = 0
I I I i I I I I I I I I
0.25
0.5
.0
(a)
FIGURE III-6. THE EFFECT OF VERTICAL WIND ON
THE TRAJECTORY MODEL PREDICTIONS
-------
y
10
n = 1
= 1
2.5
1.0
0.5
0.25
0.1
0.1
0.25
0.5
1.0
(b)
FIGURE III-6. THE EFFECT OF VERTICAL WIND ON
THE TRAJECTORY MODEL PREDICTIONS (Continued)
-------
10
2.5
1.0
0.5
0.25
0.1
n = 0
m = TT
Code
A
B
C
10
10
-2
0.1
0.25
0.5
1.0
(c)
FIGURE III-6. THE EFFECT OF VERTICAL WIND ON
THE TRAJECTORY MODEL PREDICTIONS (Continued)
-------
10
2.5
1.0 -
0.5 ~
0.25
0.1
0.1
0.25
0.5
1.0
(d)
FIGURE 111-6. THE EFFECT OF VERTICAL WIND ON
THE TRAJECTORY MODEL PREDICTIONS (Continued)
-------
10
2.5 h
1.0 h
0.5 h
0.25 h
0.1
0.1
0.25
0.5
(e)
1.0
FIGURE III-6. THE EFFECT OF VERTICAL WIND ON
THE TRAJECTORY MODEL PREDICTIONS (Continued)
-------
0.25 -
0.1
(f)
FIGURE III-6. THE EFFECT OF VERTICAL WIND ON
THE TRAJECTORY MODEL PREDICTIONS (Continued)
-------
10
2.5
1.0
0.5
0.25
0.1
n - TT
m = 7T
Code
A
B
C
10
1
10
-2
I I I I
0.1
0.25
0.5
1.0
(g)
FIGURE III-6. THE EFFECT OF VERTICAL WIND ON
THE TRAJECTORY MODEL PREDICTIONS (Continued)
-------
0.25 -
0.1
0.25
0.5
1.0
(h)
FIGURE III-6. THE EFFECT OF VERTICAL WIND ON
THE TRAJECTORY MODEL PREDICTIONS (Concluded)
-------
116
wind* (i.e., a ->- 0 and $ ->- 1), the neglect of the vertical wind can
cause gross errors in the predictions of pollutant concentrations.
3. The Effect of Wind Shear
As we mentioned earlier, the case of a horizontal wind that varies
with height cannot be properly handled by a trajectory model because, in-
trinsically, only one horizontal wind at any location can be used to com-
pute the movement of the air column. We assess here the errors incurred
as a result of this assumption. We consider two different cases: (1) a
simplejcrosswind continuous line source and (2) a more realistic urban-
type (distributed) source. As shown in Figure III-7, we allowed the wind
speed and vertical turbulent diffusivity in both cases to vary with altitude
according to the following power laws:
(111-69)
(111-70)
Convergence in an urban area, as we emphasized earlier, is by no means
small. For example, a change of more than 1 mph over a 1 mile distance
is quite common. If we use the two-dimensional continuity equation to
estimate the convergence, we obtain
a = AW = _ AU . 1 mph = 1 h -1 „ 3 x 1Q-4 se(fl
AZ AX 1 mile
which agrees with measurements made by Ackerman (1974).
-------
Height
SOURCE
K = K,
Diffusivity
Wind Speed
J
POLLUTANT
CLOUD
TYPICAL
CROSS SECTION
Downwind Distance
FIGURE III-7. ASSESSING THE EFFECT OF WIND SHEAR
-------
118
a. A Continuous Line Source
The well-known Roberts solution (Monin and Yaglom, 1971) gives the
mean concentration downwind of a continuous ground-level line source in
an atmosphere with a power law wind speed and vertical diffusivity profile:
c(x,z) =
(m - n + 2)z^Q£
u1r(s)
ul
(m - n
+ 2)2 K x
' (— J i\-i A
exp
_ s
u,z
m-n+2
(m - n + 2)
(111-71)
where
m - n + 1 > 0*
and
r =
m + 1
m-n + 2
s =
m + 1
m-n + 2
*Monin and Yaglom (1971) appear to have incorrectly quoted m-n+2 > 0.
A more stringent condition, m - n + 1 >_ 0, is needed to satisfy the
condition that the flux is zero along the x-axis (except at the origin).
-------
119
For the trajectory model, we assume that the air parcel moves with
the wind speed at the reference height,u, as we have defined before. Then,
from Eq. (111-41), the solution is
C(P,T) =
(2 -
exp
2-n
(2 - n)
2
(111-72)
To compare the two solutions, we let x = UT and z = p in
Eq. (111-71) and set p = 0 in both equations. We then obtain the ratio
of the trajectory model solution to the exact solution:
= r(s)(m - n + 2)
25'1
(111-73)
where
P =
m(l - n)
(m - n + 2)(2 - n)
We evaluated the ratio y over a wide range of values of m and n as a
function of the dimensionless time n (see Figure III-8). The results show
that the predictions of a trajectory model that neglects the variation of
-------
120
10.0
5.0
2.0
1.0
n = 0
m = 0
0.5
0.2
0.1
10
-2
10
-1
10
FIGURE III-8.
(a)
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR LINE SOURCES)
-------
121
10.0
5.0
2.0
1.0
0.5
0.2 _
0.1
10
-2
10
-1
10
(b)
m = 0
FIGURE III-8.
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR LINE SOURCES) (Continued)
-------
10.0
5.0
2.0
1.0
0.5
0.2
0.1
10
-2
10'
-1
10
(c)
m - 0
FIGURE III-8.
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR LINE SOURCES) (Continued)
-------
10.0
5.0
n = 1
2.0
1.0
m = 1.0
m = 0.8
m = 0.6
m = 0.4
m = 0.2
m = 0
0.5
0.2
0.1
10"2 10"1
10
FIGURE III-8.
(d)
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR LINE SOURCES) (Continued)
-------
124
10.0
5.0
2.0
1.0
m = 0
0.5
0.2
0.1
-•I
10
,-2
10
-1
10
10V
FIGURE III-8.
(e)
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
• (FOR LINE SOURCES) (Concluded)
-------
125
horizontal wind with height can be affected not only by the shape of the wind
profile (as expressed by the exponent m), but also by the shape of the diffusivity
profile (as expressed by the exponent n). With the exception of a linear dif-
fusivity profile, Figure III-8(d), the performance of the trajectory model
deteriorates with increasing time. The effect of a nonuniform wind profile
on the trajectory model predictions is such that the atmosphere acts as if
an imaginary emission source were introduced when its diffusivity variation
is less than linear and as if an imaginary sink were introduced when its
diffusivity variation is greater than linear.
To obtain a quantitative assessment of the trajectory model in real
situations, we conducted a literature survey to estimate the possible values
of m and n in an urban atmosphere. As discussed in Appendix B, we found that
the ranges of values for m and n likely to occur in an urban atmosphere are
0.2 ~ 0,;'4 for m and ~ 1 for n.
2 -1
If we let K, = 1 m sec and z, = 10 m, a real time ranging from
1-1/2 minutes to 3 hours corresponds to a change of 1 to 100 in r\. Figure
III-8 indicates that over the ranges of m and n specified above, the trajectory
model can be in error by more than 50 percent for the time span indicated as
a result of the neglect of shear effects alone.
b. A Continuous Areal Source
The second case we consider here is an emission source that varies
with location—a more realistic situation in an urban area.
Using assumptions identical to those made in the first case, we can obtain
the solutions for the exact model and the trajectory model by applying the
principle of superposition to Eqs. (111-71) and (111-72):
-------
\Zb
c(x,z) =
(m
- n +
2)
r
zl
lyts)
ul
(m.-
n +
2
)
2
/
"'"'
exp
V
m-n+2
(m - n + 2)'
and
m-n,
- a)
da ,
C(P,T) =
2"n
2 - n
f qA(x)
1
-2-n '„ <-*)2""
exp
2-n
(2 - n
- A)
(111-74)
Again, singular behavior exists near the upper limit of the integration, and
coordinate transformations similar to the ones we discussed earlier must be
invoked. We evaluated the resulting integrals using Simpson's rule. We used
the same step-wise emissions pattern described in Section D-2. Figure III-9
shows the results for a wide range of values of m and n.
The significance of the ordinate y is the same as that defined earlier;
it is the ratio of the prediction of the trajectory model to that of the exact
model. Figures III-9(a) through III-9(e) clearly demonstrate that, under con-
ditions that are likely to occur in an urban atmosphere, the errors incurred
as a result of the neglect of variations in horizontal wind with height can
be quite substantial. For instance, using the set of values
-------
127
10.0
5.0
2.0
1.0
0.5
0.2
0.1
m. = 0.2
n = 0
Ooo°o0000
°AAAAAAAA
1=1—n D—9-
1J I 1 t^t s—"^
n n n
_a
D u
10
20
A A A A A A
n D a D a D
Symbol
O
A
D
U121
0.01
0.1
1
30
40
50
(a)
FIGURE III-9.
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR AREAL SOURCES)
-------
10.0
5.0
2.0
1.0
0.5
0.2
0.1
m = 0.4
n = 0
o o
o o o
o o o o o °
n
A & A A
D D n D
A A
,-,
n a
ooooooo00
AAAAAAAAA
annQ°aaDD[P
a u a a
Symbol
O
A
a
Vi
K.,Ax
0.01
0.1
1
0
FIGURE III-9.
10
20
30
40
(b)
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR AREAL SOURCES) (Continued)
-------
10.0
5.0
2.0
1.0
• m = 0.2
n = 0.25
0OOOOo999A AAAA^^^^"
_£_!** ^nSoaooaaooaaa
OOoooOo° °O(
a a a o a
a a n D D
0.5
Symbol
O
A
D
Vl
0.01
0.1
1
0.2
0.1
10
20
30
'40
50
FIGURE III-9.
(c)
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR AREAL SOURCES) (Continued)
-------
IdU
10.0
5.0
2.0
1.0
0.5
0.2
0.1
m = 0.4
n = 0.25
0 o o o o o
A A A A A A
n D D ° D D
o o o o o °
A A A A A A
a D
n D a
o
D a
o o o o
o o o
A A A
A A
n n a a a n a
A A A A
C D
O O
A A
[
Symbol
"
O
A
a
uizi
0.01
0.1
1
10
20
30
40
50
(d)
FIGURE III-9.
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR AREAL SOURCES) (Continued)
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131
10.0
5.0
2.0
1.0
0.5
0.2
0.1
m = 0.2
n = 0.5
D D D D °
D o
D D D
£66666666666A
Symbol
O
A
D
Vl
0.01
0..1
1
10
20
30
40
(e)
FIGURE III-9.
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR AREAL SOURCES) (Continued)
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132
10.0
5.0
2.0
1.0
D
D
u
D a
D a
A A A A
m = 0.4
n = 0.5
a
D
D
0 a a D a
D D
°°00oOOOoOA
o
ff.5
0.2
0.1
Symbol
O
A
D
Vl
0.01
0.1
1
10
.20
30
40
50
FIGURE III-9.
(f)
THE EFFECT OF WIND SHEAR ON TRAJECTORY MODEL PREDICTIONS
(FOR AREAL SOURCES) (Concluded)
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133
= 1.5 m sec"
= 10 m
2 -1
K, = 1 m sec ,
Ax = 3 x 103 m
p
we obtain a dimensionless number of 0.1 for u,zf/(l/2)K,Ax; therefore, the
set of points represented by triangles in Figures III-9(a) through III-9(e)
is applicable. The corresponding values of y then show that the trajectory
model can overpredict the ground-level concentrations by more than 50 percent.
E. THE VALIDITY OF THE GRID MODEL—THE EFFECT OF NUMERICAL ERRORS
As we have pointed out in Section B, numerical errors generated using
a grid model arise primarily in the process of discretizing the atmospheric
diffusion equation, which is generally in a differential form. Two major
aspects of a grid model are crucial in the determination of the magnitudes
of numerical errors. The first is the choice of the cell size and the
corresponding time interval. Although a decrease in the cell size and time
interval ideally results in a decrease in the magnitudes of numerical errors
generated using a grid approach, a compromise must be made between the cell
size and time interval on one hand and the computing time and availability of
data on the other (Seinfeld, 1970). Unfortunately, this results in a set of
cell sizes producing numerical errors that are by no means insignificant. The
second aspect of a grid model that affects the numerical errors is the type
of numerical scheme that is used to represent the governing equation. There
are probably as many numerical schemes as there are numerical models, and
their performances vary greatly as the conditions of their applications change.
However, they have one thing in common: None of them are perfect representations
under all conditions. The objective here is not to evaluate the relative
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134
merit of each of these numerical schemes. Rather, it is to estimate quantitatively
the magnitudes of numerical errors generated using a grid model under realistic
conditions. To avoid compounding the issue, we did not use fancy or sophisticated
numerical schemes. Instead, we used a second-order difference scheme developed
by Price et al. (1966) and a simple first-order difference scheme. Thus, the
results derived in this section can be viewed as the upper bounds on numerical
errors committed as a result of discretization.
The methodology we describe here to assess the numerical errors generated
by a grid model is the same as the one we used to assess the inaccuracies of
trajectory models. First, we obtained analytical solutions for certain specific
but realistic cases. Then, we exercised a grid model to provide the correspond-
ing predicted concentrations under similar conditions. Finally, we compared
the two sets of numbers. All of the cases considered in this study, as pre-
scribed by Eq. (111-12), were two-dimensional (crosswind) and time-dependent.
We assumed that the wind speed and diffusivities (both horizontal and vertical
components) were constant. Furthermore, we adopted step-wise emission patterns,
as described in Eq. (111-59), under these conditions. The exact solution is
c(x,z,t) -
- erf
exp
4K1(t - B)
erf
x - (i - I)AX - U(t - B)
4KH(t - 3)
X - JAX - U(t - g)
4KH(t - B)
(111-75)
Again, the removal of the singularities at the upper limit of the integral
required coordinate transformations.
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135
The grid model that was the subject of the present investigation uses two
differencing schemes to represent Eq. (111-12}: a first-order differencing
scheme,
rn+l _ rn UAt (r r \ H
Ci - Ci - "AT (Ci " Ci-l} + ~T
and the second-order differencing scheme developed by Price (1966),
rn+l _ rn UAt /,rn 9rn rn \ . KHAt /rn 9rn . Pn \ ,TTT 77x
Ci - Ci - 2A7l3Ci -2Ci-r Ci-2j + —r(Ci-l - 2Ci + Ci+lJ • (IH-77)
The first-order scheme is the simplest and the most primitive of all finite
difference schemes. Thus, the results probably represent the worst case insofar
as the generation of errors as a result of discretization is concerned. Since
Price's scheme is a higher order method, it is presumably more accurate. In
addition, it has the desirable property of suppressing the prediction of negative
concentrations of reactive pollutants in regions where sharp concentration gradi-
ents exist. We computed the ratios of the ground concentrations predicted using
the grid model, i.e., Eq. (111-76) or Eq. (111-77), to those predicted using
the exact solution, i.e., Eq. (111-75).
Using a realistic spatial and temporal emission pattern as shown in
Figures III-3 and III-4, along with the second-order finite difference scheme
(Price et al., 1966) in the grid model, we plotted the ratio y in Figure III-
10 for the following case:
u = 4 mph
Ku = 50 m2 sec'1
n
Several interesting observations concerning the accuracy of the grid
model emerge from a close scrutiny of Figure 111-10. First, the numerical
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I3b
10.0
5.0
2.0
1.0
Symbol Hour
O 3
A
D
° ° ° n n
D D
a ^ o « g A A
A
D
0.5
0.2
0.1
O
O
O
O
o o
O
A D
O 00 A O ° D
O
o
2
O A
A
D
10
20
30
40
50
FIGURE 111-10. THE EFFECT OF NUMERICAL ERRORS ON GRID MODEL PREDICTIONS:
RESULTS USING THE FIRST-ORDER FINITE DIFFERENCE SCHEME (WIND SPEED = 4 MPH)
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137
errors associated with the grid model apparently exhibit a wave-like
behavior. The wave trains emanating from the upwind edge of the modeling
region (which experiences a step jump in emissions) appear to be amplified
both in amplitudes and in phase angles as the simulation time increases.
After a nine-hour simulation, numerical errors can be an unbearable ±50 per-
cent for many downwind locations. However, the difference scheme selected
strongly influences the numerical errors generated using a grid model. To
demonstrate this aspect, we tested the simpler—and thus more inaccurate--
first-order difference scheme. As shown in Figure III-ll, the resultant
wave-like error propagation has amplitudes significantly higher than those
for the second-order case.
As we discussed earlier, numerical errors also.depend upon a complex
matrix of physical parameters in the simulation. In the present study, we
explored some of the more important ones. Since the numerical errors generated
by the grid model originate primarily from inhomogeneities in the concentration
distributions, spatial variations of the emissions undoubtedly have a strong
effect on the performance of the model. Instead of using the realistic spatial
emission pattern shown in Figure III-3, we used the smooth pattern shown in
Figure 111-12. With all other conditions identical to those in the numerical
studies discussed in the preceding paragraph, we tested the second-order
finite difference scheme. The results, plotted in Figure 111-13, show that
the errors are bounded by ±20 percent, a value more tolerable than the ±50 per-
cent variation mentioned above.
We then explored the effect of the physical horizontal diffusion on
the accuracy of the grid model. In this study, we increased the value of
2 1
the horizontal diffusivity, KH, by a factor of 10 (to 500 m sec ) over that
used in previous cases. This value probably represents an upper bound for
KU for urban-scale airshed models. The effect of this change, as established
by comparing the results shown in Figure 111-10 and 111-14, was minimal. Fig-
ure 111-15 shows the effect of varying the wind speed, under the same conditions
as those used for Figure 111-10, except that the wind speed was decreased
to 2 mph. This change resulted in considerable improvement (the error bounds
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138
10.0
5.0
2.0
1.0
Symbol Hour
O 3
A 6
D 9
008
a
D D
D
A A
O
O
D
A D
A
D
O
o o
A
O
0
D
O
A
O
0.5
D A O
A E3 A O A
a D
a
0.2
0.1
10
20
30
40
50
FIGURE III-ll. THE EFFECT OF NUMERICAL ERRORS ON GRID MODEL PREDICTIONS
RESULTS USING THE SECOND-ORDER FINITE DIFFERENCE SCHEME
AND REALISTIC SPATIAL AND TEMPORAL EMISSION PATTERNS
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139
10
I
E
ex
D-
i
i
t/>
c
o
•r—
t/1
10
10
20
30
40
50
Miles
FIGURE 111-12. A SMOOTH PATTERN OF POLLUTANT EMISSIONS
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140
10.0
5.0
2.0
1.0
Symbol Hour
O 3
A 6
D 9
0
D D
n
U
D
A
o a p
-Q-
a
A U
n O O
D
ooooooooooo oo°oo o°ooc
0.5
0.2
0.1
10
20
30
40
50
FIGURE 111-13. THE EFFECT OF NUMERICAL ERRORS ON GRID MODEL PREDICTIONS
RESULTS USING THE SECOND ORDER FINITE DIFFERENCE SCHEME
AND SMOOTH SPATIAL AND TEMPORAL EMISSION PATTERNS
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141
10.0
5.0
2.0
1.0
Symbol Hour
O 3
A
D
n D a D
o D D
A"a@nAAA
" A A
a
0.5
0.2
0.1
O ^ A
O D
O
O n O
O o O ° °
O Q
o
D
O 8 O O C
D A A
10
20
30
40
50
FIGURE 111-14 THE EFFECT OF NUMERICAL ERRORS ON GRID MODEL PREDICTIONS:
RESULTS UNDER THE SAME CONDITIONS AS THOSE OF FIGURE 111-10,
EXCEPT FOR AN INCREASE IN HORIZONTAL DIFFUSION
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142
10.0
5.0
2.0
1.0
0.5
0.2
0.1
Symbol Hour
O 3
A 6
D 9
y A
o o
o o
^ A
O O
8 o
A D g
O D n
A
O
D
A D
9 o
H
O
-10
20
30
40
50
FIGURE 111-15. THE EFFECT OF NUMERICAL ERRORS ON GRID MODEL PREDICTIONS
RESULTS UNDER THE SAME CONDITIONS AS THOSE OF FIGURE 111-10,
EXCEPT FOR A REDUCTION IN WIND SPEED
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143
are ±25 percent, a reduction of one-half from the 4 mph case), in agreement
with the qualitative conclusions drawn from the linear analysis we described
earlier.
F. CONCLUSIONS AND RECOMMENDATIONS
We investigated the validity of urban airshed models that use either a
trajectory approach or a grid approach through comparisons of exact solutions
of the atmospheric diffusion equation, for simple but realistic cases, with
the corresponding predictions of the trajectory or grid model. Despite our
lengthy exposition in the previous sections, the question of the validity of
urban airshed models is by no means completely resolved. For example, we
only superficially treated the effect of numerical methods used in solving the
grid model equations. Because of a lack of time, we did not explore the use of
sophisticated, higher order finite difference techniques and the variation
of time and spatial step sizes. For example, we did not include the particle-
in-cell method, a viable numerical scheme that can be classified as a grid
model (Sklarew, 1971), in this study at all. With regard to the trajectory
model, we did not address such questions as the uncertainties in obtaining
a Lagrangian wind velocity in a turbulent atmosphere (see Appendix A) or the
inaccuracies arising from the variation of the horizontal shape of the air
*
column (conventional trajectory modelers assume the shape is invariable).
Nevertheless, many useful quantitative estimates concerning tha range of
applicability of these two classes of models under various conditions of
atmospheric stability, wind shear, and source configuration emerged from this
study.
In our assessment of the trajectory model, we found (as expected) that
the exchange of material at the boundary of the air parcel as a result of
horizontal diffusion is not important. However, we showed that the effect
* In the present study, we did not consider this aspect because, rigorously
speaking, we assumed that the trajectory model has a zero base area.
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144
of neglecting the vertical wind component is quite significant, depending on
(among other things) the strength of the convergence. This limitation does
not severely restrict the utility of the trajectory model because, conceptually,
we can easily include the vertical transport term in a model that also includes
the vertical diffusion term (e.g., see Eschenroeder, 1972). The most detri-
mental conclusion drawn regarding the trajectory model relates to the neglect
of wind shear: For wind profiles typically observed in an urban area, the
trajectory model predictions can be in error by an order of magnitude when
shear effects are not taken into account.
We also assessed the inaccuracies in the grid model due to finite differ-
encing under realistic conditions. Many interesting results emerged, including
c) 11 '^ *"J
the wave-like propagation of numerical errors. The shape of the error wave,
which grows with simulation time, depends on such parameters as the spatial
variability of the concentration field, the wind speed, the spatial (or temporal)
step sizes, and the differencing scheme used in the grid model. Our study has
further shown that, using an uncentered second-order difference scheme, the
resu-Tts of a nine-hour simulation produced by a grid model are probably accep-
table if the spatial variations in emissions are relatively modest, the wind
relatively low (~2 mph), or both. For more demanding situations, in terms of
the conditions that apply and the length of time being simulated, a search for
a more suitable finite difference scheme is warranted [e.g., the material-
conserving computation procedure developed by Egan and Mahoney (1972) or the
spectral method that has been actively developed over the past few years by
Orszag (1970, 1971) and Orszag and Israeli (1974)].
We note, however, that the judgment we have made above regarding the
effect of numerical errors on long-period grid model predictions may be too
severe for inert species, such as CO. In urban air pollution, concentrations
of most of the primary air pollutants characteristically drop to insignif-
icantly low levels during the early afternoon (1 p.m. to 2 p.m.) as a result
of extensive ground heating by sunlight and the ensuing inversion breakup.
If the model simulation were to start at sunrise (about 5 a.m. or 6 a.m.),
it would reach the nine-hour mark around 2 p.m. Although the effect of
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145
numerical errors would be intolerable by then, the low levels of the primary
air pollutants, fortunately, would make the consequence more bearable. For
instance, the CO levels in Los Angeles in the afternoon are typically 3 to
5 ppm; a 50 percent error in the concentration levels amounts to merely
2 ppm, an acceptable error. This may explain why error generation and prop-
agation did not strongly affect the multiple-day runs recently performed
by Systems Applications.
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146
IV SENSITIVITY STUDY OF THE SAI URBAN AIRSHED MODEL
A. INTRODUCTION
The formation of air pollution in an urban airshed is the result of a
complex chain of events. First, for all air pollutants, the levels of
contaminant concentrations depend on the emissions sources, which, in an
urban area, consist of a matrix of ground-based area! sources and volumetric
sources aloft, with widely varying effluent compositions, emissions rates,
and other emissions characteristics. Second, after the pollutants are
released into the air, they are influenced by the turbulent motion of the
atmosphere: They are carried downstream by a mean transport wind, and they
are diffused in all directions by turbulent eddies. Concentration levels
at any receptor point depend not only on the synoptic-scale and mesoscale
meteorology, but also on the modifications of air motion caused by the
local topography in the vicinity of the receptor point. Thus, it is diffi-
cult to enumerate all of the parameters that may affect the eventual distri-
bution of air pollutants. Finally, compounding the difficulties, secondary
pollutants are produced through chemical reactions of primary (or emitted)
pollutants in an urban atmosphere. The rate of transformation depends on
the intensity of solar radiation and on local concentration levels and,
consequently, on all of the parameters delineated above. It is therefore
clear that the primary task of modeling photochemical air pollution consists
of sifting through the myriad of physical parameters and selecting only
several of the more important ones to include in the model.
The process of developing an airshed model thus inevitably involves
a trial-and-error approach. First, a primitive model is developed. Its
predictions are then evaluated, either by comparing them with observations,
if adequate data are available, or by theoretically assessing their validity,
The objective of these evaluation processes is to identify the sources of
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147
errors or inadequacies in the primitive model. Hopefully, based on
these findings, improvements to the model can be made.
In this process of model development, the study of the sensitivity
of the model plays a vital role. Ideally, the sensitivity study should
be preceded by a validity study to affirm that all parameters necessary
to simulate urban air pollution have been included in the model. The
sensitivity study should be followed by an analysis of the model predictions
to establish the cause-effect relationship between the input conditions and
the model predictions. Through its variations of input parameters within
the range of physical reality, the sensitivity study serves as a vehicle
for examining the responses of the model. The goal of such an analysis is
to assess the influence of each parameter on the prediction of air quality.
More specifically, such a study seeks to achieve the following objectives:
> To assess the importance of a given input parameter so that
decisions can then be made as to whether this parameter should
be retained in the model. In the event that this parameter is
to be neglected, the analysis can provide an error bound for ne-
glecting it.
> To determine the necessity of including the temporal or spatial
variation of a physical parameter once its importance has been
established.
> To estimate the required accuracy of a given parameter so that
appropriate arrangements can be made to meet these requirements,
or, correspondingly, to assess the effect of a parameter with a
given level of uncertainty.
> To enhance existing knowledge of the role played by each parameter
so that explanations can be offered in those cases in which model
predictions differ from observational data.
> To aid in the construction of a repro-model by choosing a proper
combination of input parameters.
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148
In view of these diversified objectives and the complexities of the
urban airshed model itself, the successful execution of a sensitivity study
is not a simple or straightforward undertaking. A well-planned program
is needed to extract the maximum possible information at a given level of
effort. This chapter describes our sensitivity study of the SAI urban air-
shed model.
Section B presents the design of the sensitivity study. Section C
summarizes our effort to extract useful information from the unavoidably
massive collection of computer printout comprising the output of the study.
Section D discusses the overall sensitivity of the SAI urban airshed model
and presents our conclusions and recommendations.
B. DESIGN OF THE SENSITIVITY STUDY
A sensitivity study can be defined as a numerical experiment to assess
the effect of varying one or more input parameters in a model under con-
trolled but realistic conditions. Thus, in essence, the execution of a
sensitivity study may be no more than a series of modeling exercises with
different sets of input data. The complexities of SAI's urban airshed model,
however, make this job rather tedious, if not difficult. Careful plans
were therefore necessary to ensure the achievement of the intended goals
summarized above. In this section, we discuss two key elements in carrying
out our sensitivity analysis: the detailed planning of the cases we con-
sidered and the selection of the criteria we used in the analysis of the
sensitivity of the SAI model.
1. Plans for Carrying Out the Sensitivity Study
As the subject for analysis, we chose the SAI urban airshed (grid) model
(Reynolds et al., 1973; Roth et al., 1974; Reynolds et al., 1974), primarily
because the limited resources available for this project prohibited explora-
tion of the sensitivities of both a grid model and a trajectory model. Of
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149
these two approaches, the grid model offers the advantage of conveniently
providing basin-wide coverage and, consequently, a richer information base.
Furthermore, resource limitations also prevented us from studying more than
one base case.
We selected a late summer day in Los Angeles, September 29, 1969, as
the base case for the study. This day was not only a typical smoggy day,
but also one that has been extensively measured and studied. For a compre-
hensive description of the aerometric data collected on this day and of
the corresponding predictions obtained using the SAI urban airshed model,
we refer the reader to Reynolds et al. (1974).
Having chosen an appropriate model and a base case, we then faced the
problem of selecting the test cases. In the following subsections, we briefly
describe the input parameters chosen, the reasons for their selection, and the
ways in which they were varied. Table III-l presents a summary of the cases
explored in this study.
a. Surface Wind
Our interest in varying the surface wind field was twofold: to examine
both the accuracy and the sensitivity of the model. First, we wanted to
assess the effect of uncertainties in the wind speed and wind direction
measurements on the airshed model predictions. Second, we wished to
determine the response of the airshed model predictions to systematic changes
in wind speed. For the first task, we randomly varied both the wind speed
and the wind direction by an amount characteristic of the uncertainty in
measuring or reporting this quantity: 1 mph for the wind speed and 1 point
(= 22.5°) for the wind direction. For the second task, we systematically
increased or decreased the measured wind speed by a fixed percentage. We
explored four cases: -50 percent, -25 percent, +25 percent, and +50 percent.
b. Diffusivity
We considered both the horizontal and the vertical diffusivities. For
2 -1
the horizontal diffusivity, we used two extreme values: 0 and 500 m sec .
-------
Table IV-1
SUMMARY OF THE CASES INVESTIGATED IN THE SENSITIVITY STUDY
Input Parameter
Wind direction
Wind speed
Variations
Comment
Horizontal diffusivity
Vertical diffusivity
Mixing depth
Radiation intensity
Emission rate
Station measurements* randomly perturbed by 0 or +22.5°
Values at each grjd point"*" randomly perturbed by 0 or +_22.5°
Station measurements* randomly perturbed by 0 or +1 mph
Values at each grid point"1" randomly perturbed by 0 or +1 mph
Station measurements* decreased by 50%
Station measurements* decreased by 25%
Station measurements* increased by 25%
Station measurements* increased by 50%
Decreased5 to 0
iy -I
Increased5 to 500 m sec"
2 -1
Decreased** to 0.5 m sec
2 -1
Increased** to 50 m sec
Decreased by 25%
Increased by 25%
Decreased by 30%
Increased by 30%
Decreased by 15%
Increased by 15%
See Section C-l-a
See Section C-l-a
See Section C-l-b
See Section C-l-b
See Section C-2
See Section C-2
See Section C-2
See Section C-2
See Section C-3-a
See Section C-3-a
See Section C-3-b
See Section C-3-b
See Section C-4
See Section C-4
See Section C-5
See Section C-5
See Section C-6
See Section C-6
* The station measurements were subsequently interpolated, using techniques described in Liu et al. (1973).
f These values were obtained from manually prepared wind data; see Reynolds et al. (1974).
? 1
§ A value of 50 m sec~ was used in the base case.
2 -1
** A value of 5 m sec was used in the base case.
en
o
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'51
Our objective was to determine whether the horizontal diffusion term should
be retained in the model and whether numerical diffusion is an important
source of error. We decreased and increased the value of the vertical
diffusivity from the one used in the base case by an order of magnitude to
examine the effect on the predicted air quality.
c. Mixing Depths
Our objective in this case was to assess the effect of varying the
mixing depth. Toward this end, we uniformly decreased or .increased the
values used in the base case, which change with time and location, by 25
percent, the amount that may represent the error bounds in the determination
of the mixing depth.
d. Radiation Intensity
For photochemical air pollutants, the light intensity, or the clos.?ly
related photolysis rate constant, is the most important parameter in delineat-
ing the chemical evolution of these species (Hecht et al., 1973). We de-
creased and increased this rate constant by 30 percent.
e. Emissions Rate
Contaminant concentrations in an airshed undoubtedly depend directly
on the rates of emissions from pollutant sources. We uniformly varied the
emissions rates used in the base case by +15 percent and -15 percent,
values chosen because they may be characteristic of uncertainties in the
emissions rate derivation.
2. Criteria for Assessing the Sensitivity of the SAI Model :
The second problem we faced in designing the sensitivity study
was the selection of appropriate criteria for assessing the impact on the
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152
predicted air quality of variations in parameter magnitudes for each of
the cases delineated above. In view of the large amount of output from the
SAI urban airshed model, the choice was neither unique nor straightforward.
For example, an 11-hour simulation of the full SAI model furnished nearly
30,000 data points for the ground-level concentrations alone. Consequently,
even if we had been concerned only with the ground-level concentrations, we
would have had to consider 30,000 pairs of data for each of the cases dis-
cussed earlier. It would have been not only impossible, but also overly
specific to analyze them individually. Thus, we had to find ways to sort
out this large collection of data so that proper cause-effect relationships
among the variables could be established. Many conceivable schemes for
aggregating predictions exist. In the following subsections, we describe
those we considered to be the most suitable.
a. Basin-Wide Averages
The first type of criterion we considered was basin-wide averages. We
designated the ground-level concentrations from the base-case and test-case
predictions (both were one-hour averages) by c^'n. and c.'., respectively,
i ,j i >j
where m denotes the pollutant species; i and j are the horizontal location
(N is the total number of locations); and n represents the hour. Then
we formulated the following criteria to assess the difference between the
two sets of data.
> Average deviation:
D'"'1
W ?cm'"
(iv-i)
Average absolute deviation:
lDim'n =
(IV-2)
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153
> Average relative deviation:*
ITIm'n - 1
ldl = N"
(IV-3)
> Standard deviation:
m,n
(IV-4)
As is clear from these definitions, the criteria given above provide
an average measure of the effect to be assessed. Thus, they are most likely
to be successful in detecting systematic trends between the two sets of data.
For instance, they can be used if the effect of increasing the wind speed
by some amount (say, 50 percent) is to be assessed. However, these criteria
are obviously inadequate for assessments of local changes. For example,
using these averages, it would be difficult to detect the large local changes
that are observed when the wind direction is randomly disturbed. In this
case, the local maxima might have been shifted markedly, but the overall
average effect would be very small.
* Although we label this the "average relative deviation," it should more
properly be called the "average relative absolute deviation." We made
no attempt to calculate an actual average relative deviation,
Tm»n = I
a N
II
i j
~m,n cm>n
i , j i ,j
because we felt that it would not add significantly to the set of
criteria listed.
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154
b. Local Maximum Deviations
The second type of criterion that we considered was derived by identi-
fying the largest deviation at any of the N locations in the grid. Mathe-
matically, this can be expressed as
Dm,n = ~m,n _ m,n
max ca,(3 Ca,3
where (a,3) is located by finding max ( |cm'n - cm'"|),
i,j \ 1>J 1>J /
or as
~m,n _ cm,n
dm,n y,v " y,v
max m,n
where (y,v) is located by finding max |^'" - c™*"] } /c1?'1?
It is apparent that these criteria are considerably more sensitive than
the basin-wide averages. For example, the following trivial inequalities
can be written:
ldm'nl ^ ldlm'n
1 max' ' '
-------
155
c. Deviations at the Local Maxima
A third type of criterion was obtained by first locating the maxima
in the base case and then calculating the deviation at that location.
Formally, such criteria are defined by
rm,n _ ~m,n m,n
~ CI,J " CI,J '
~m,n _ cm,n
gm'n = -LJ _ Li ' (IV-8)
where
~ »
CI,J
,m,n _ m,y /Jn>n\
-T I - max ic. • ] ,
*• iV i •; \ ' sJ /
and
CZ
One would expect that the third type of criterion would be more sensitive
than the first, and certainly less sensitive than the second.
Of these three types of criteria, we used one or more as indices to
measure the effect in each of the individual cases we delineated earlier.
We based our choice upon the appropriateness and significance of a particular
criterion for the specific case of interest. The following section discusses
the nature of these considerations.
-------
15L
C. ANALYSIS OF THE SENSITIVITY OF THE SAI MODEL
In this section, we report the results of our analysis of the sensitivity
of the SAI urban airshed model. We explored each of the cases listed in
Table IV-1 using the criteria discussed in Section B-2 as the basis for evalu-
ation and comparison. We describe below the effects on the model predictions
of varying the input parameters.
1. The Effect of Random Perturbations in the Wind Field
We investigated three types of variations in the wind field: random
changes in the wind direction, random changes in the wind speed, and systematic
changes in the wind speed. We used the first two types to assess the effect
of errors in measuring wind direction and wind speed and the third, to evalu-
ate the response of the model to variations in wind speed (i.e., assessment
of sensitivity).
a. Wind Direction
In the past, we have used two different methods to prepare the wind field,
one of the primary input parameters in the SAI urban airshed model. Although
both of them use measurements from a network of ground-level monitoring stations,
the first method involves manually smoothing, interpolating, and processing
measurement data, whereas the second one provides for automatic performance of
these tasks (Liu et al., 1973). In the present study, we varied both manually
and automatically prepared wind fields to test the effect of random changes
in wind direction. For the manually prepared wind field, we randomly varied
wind direction in each cell of the grid by -1, 0, or +1 point.* For the
automated wind field, we randomly varied the measured wind direction at each of
the monitoring stations by -1, 0, or +1 point; we then automatically interpolated
the data to provide the wind direction for each cell.
*0ne point is equivalent to 22.5°, which is typically the unit used in reporting
wind direction.
-------
Using the new wind field, we carried out simulations for each of
these two variable cases and compared the results with the base case.
We considered only carbon monoxide, an inert species, because the accuracy
of meteorological parameters was of primary concern in this particular
investigation. Figures IV-1 and IV-2 show the results of the calculated
average deviations, D, and the standard deviations, a. The deviations
tend to be negative, as is quite evident in the case where we varied the
manually prepared wind field, because randomly disturbing the wind direc-
tion is tantamount to incorporating an artificially created horizontal
diffusion (see Appendix C). Thus, one would expect the effect to be
much greater than it is for the automated wind field, since the manually
prepared wind field is characterized by greater randomness. This expecta-
tion is confirmed for the latter case by the larger magnitudes of both
the average and standard deviations (see Figures IV-1 and IV-2).
The magnitudes of the average and standard deviations also indicate
that, on a basin-wide average, the effect is minimal. For example, the
maximum values are tabulated below:
Id" a
1 max max
Type of Wind Field (percent) (ppm)
Manually prepared 6.9% 0.62
Automated 4.9 0.33
However, the deviations in individual cells may not be small. For
instance, the largest deviations in the grid in terms of concentration
units and percentage are shown in Table IV-2. As this table shows, the
local maxima can be quite significant. Thus, although the net effect of
randomly varying the wind direction does not greatly influence the basin-
wide average concentrations, large local deviations can arise because the
random changes in wind direction result in the shuffling of the peak con-
centrations within the basin. The magnitudes of these local deviations
-------
0.30
0.30
0.20
0.10
c
o
d)
o
O)
to
S-
0)
-0.1
-0.2
-0.3
O VARIATION OF THE STATION VALUES
D VARIATION OF THE GRID VALUES
I I I L
5 6 7 8 9 10 11 12 13 14 15 16
Time—hour
FIGURE IV-1. THE EFFECT--EXPRESSED AS AVERAGE DEVIATIONS—OF
RANDOM PERTURBATIONS IN WIND DIRECTION
-------
O VARIATION OF THE STATION VALUES
D VARIATION OF THE GRID VALUES
5 678 9 10 11 12 13 14 15 16
Time—hour
FIGURE IV-2. THE EFFECT—EXPRESSED AS STANDARD DEVIATIONS--OF
RANDOM PERTURBATIONS IN WIND DIRECTION
-------
Table IV-2
THE LARGEST DEVIATIONS IN THE GRID GENERATED BY RANDOMLY VARYING THE WIND DIRECTION
(a) |Dmax| in ppm
Hour
Type of Wind Field 6 7 8 9 K) 11 12 _ ]3_ _J4 _15_
Manually prepared
Automated
(b) Ml in Percentages
Hour
Type of Nind Field 6 7 8 9 10 11 12 13 14 15 16
Manually prepared
Automated
0.52
0.93
1.11
1.55
1.09
1.26
1.88
1.31
1.92
0.92
2.07
0.86
2.02
1.75
2.75
1.15
2.81
0.77
2.31
0.87
1.44
0.88
5.9%
8.7
10.1%
15.1
8.6%
10.5
13.
14.
8%
8
17.2%
13.6
20.6%
20.7
22.4%
29.4
28.9%
20.8
35.3%
34.3
34.0%
40.6
49.3%
32.4
CTi
o
-------
161
(which generally increase with time) can reach 40 to 50 percent after 11
hours of simulation; these values appear to match the magnitude of numerical
errors typically found in grid airshed models (Liu and Seinfeld, 1974).
b. Wind Speed
To test the effect of uncertainties in the wind speed measurements on
the model predictions, we carried out an analysis similar to that discussed
above. However, this time we randomly varied the wind speed (in each cell
in the case of the manually prepared wind field and at each monitoring site
in the case of the automated wind field) by -1, 0, or +1 mph.
As shown in Figures IV-3 and IV-4, although the magnitudes are generally
lower compared with the corresponding cases of variations in wind direction,
the following trends are still noticeable:
> The average deviations are always negative, manifesting a
smoothing process. In other words, a random variation in
wind speed is equivalent to an artificially created horizontal
diffusion.
> The changes due to varying the manually prepared wind field
are more pronounced than those caused by varying the automated
wind field. Again, the reason for this effect is that the former
case has a higher degree of randomness.
> The effects of changes in wind speeds on the basin-wide average,
as shown in Figures IV-3 and IV-4 are very small. The maximum
values are as follows:
Id] a
1 'max max
Type of Wind Field (percent) (ppm)
Manually prepared 4.9% 0.45
Automated 2.6 0.27
-------
0.2 -
O VARIATION OF THE STATION VALUES
n VARIATION OF THE GRID VALUES
5 67 8 9 10 11 12 13 14 15 16
Time—hour
FIGURE IV-3. THE EFFECT—EXPRESSED AS AVERAGE DEVIATIONS-
OF RANDOM PERTURBATIONS IN WIND SPEED
-------
163
c
o
ro
• i —
-------
164
> Because of the dislocation of the peak concentrations, the
local maximum deviations, as tabulated in Table IV-3, remain
large; the magnitudes are considerably smaller than those
that resulted when the wind direction was randomly varied.
To demonstrate the effect of random variations in both wind direction
and wind speed, we plotted Figure IV-5, which shows the relative changes
at the locations of maxima for the base case. Although there appears to
be no discernible trend in the signs (either positive or negative) of the
deviations, changes due to random perturbations in wind direction (indicated
by circles) are strongly related to changes due to perturbations in wind
speed (indicated by triangles). This is indeed surprising, since the magni-
tudes of the random perturbations (one point for wind direction and one
mile per hour for wind speed) were somewhat arbitrarily chosen.
2. The Effect of Variations in Hind Speed
To assess the response of the SAI airshed model predictions to systematic
changes in wind speed, we conducted four simulations, uniformly varying the
wind speed used in the base case by +50, +25, -25, and -50 percent. We then
compared the predicted ground-level concentrations with those of the base case.
These comparisons provided a rich information base that sheds light on certain
characteristics of the SAI airshed model predictions.
Figures IV-6 through IV-21 present the average deviations and the local
maximum deviations (both in absolute units and relative percentages) for the
following species: CO, NO, 03, and N02> As these figures show, the changes
increase with time until they reach their peaks, which generally occur around
early afternoon. This may be attributable to the large residue of pollutants
present in the airshed at the beginning of the simulation and the consequent
delay in the response of the model to changes in wind speed. Furthermore,
since time is required for the reactions of substances to proceed, the rates
of increase with time can be different for different species. This difference
is evident in Figures IV-14 through IV-21.
-------
Table IV-3
THE LARGEST DEVIATIONS IN THE GRID GENERATED BY RANDOMLY VARYING THE WIND SPEED
(a)
Dmaxl 1n ppm
Type of Wind Field
Manually prepared
Automated
Type of Wind Field
Manually prepared
Automated
Hour
6789
0.24 0.89 1.73 1.63
0.40 0.98 1.05 0.82
^ dm
6789
10 11 12 13 14 15 16
1.13 1.45 1.46 1.61 1.62 1.22 0.68
1.31 1.24 1.02 0.80 0.72 0.70 0.68
in Percentages
ax
Hour
10 11 12 13 14 15 16
4.3% 7.8% 12.0% 12.9% 15.6% 18.9% 18.2% 17.7% 20.2% 17.6% 20.7%
3.9 10.2 14.3 9.7
18.0 20.7 19.3 14.2 18.7 10.1 15.1
CTl
en
-------
166
O WIND SPEED
A WIND DIRECTION
n 12 is
Time—hours
FIGURE IV-5.
RELATIVE CHANGES IN WIND SPEED AND DIRECTION AT THE
LOCATIONS OF MAXIMA FOR THE BASE CASE
-------
c
o
Ol
0 •••
OJ
en
X
1 +50%
2 +252
3 -25%-
4 -502
1»«. no. Tee. »««. »eo. looe. noo. 1310. uc». uoo. noo. isoo.
Time--hour
FIGURE IV-6.-THE EFFECT—EXPRESSED AS AVERAGE DEVIATIONS-
OF VARIATIONS IN WIND SPEED FOR CO
c
o
o
u
en
ra
t-
1 +50%
Z +25%
3 -25%
4 -50%
T1me--hour
FIGURE IV-7.THE EFFECT—EXPRESSED AS AVERAGE DEVIATIONS-
OF VARIATIONS IN WIND SPEED FOR NO
-------
e
o
ro
•r—
o
O)
en
fO
CJ
1 +50%
2 +25%
3 -25%
4 -50%
«». »00. . TOO. «»«. »00. 1000. lino. 1JOO. 1.1««. l»«0. 1500. 1*01.
Time—hour
FIGURE IV-9.THE RFFFCT—EXPRESSED AS AVERAGE DEVIATIDNS-
OF VARIATIONS IN WIND SPEED FOR N02
-------
169
1 +50%
2 +25%
3 -25%
4 -50%
Time—hour
FIGURE IV-10. THE EFFECT—EXPRESSED AS
PERCENTAGE DEVIATIONS—OF VARIATIONS IN
WIND SPEED FOR CO
o
o>
o
CD
C1
(O
4->
0)
01
Q.
Time--hour
FIGURE IV-11. THE EFFECT-EXPRESSED AS
PERCENTAGE DEVIATIONS—OF VARIATIONS IN
WIND SPEED FOR NO •
-------
170
c
o
. tee.
Time—hour
FIGURE IV-12. THE EFFECT — EXPRESSED AS
PERCENTAGE DEVIATIONS— OF VARIATIONS IN
WIND SPEED FOR 0
c
o
0}
01
o
10
O)
D-
1 +502
2 +25%
3 -25%
4 .-50%
Time—hour
FIGURE IV-13. THE EFFECT—EXPRESSED AS
PERCENTAGE DEVIATIONS--OF VARIATIONS IN
WIND SPEED FOR NO.,
-------
171
c
o
•f—
CD
a
/ V
\
1 +50%
2 +25%
3 -25%
4 -50%
»•«. »g«. 7«o. «no. »no, 1000. iioo. uoo. lloo. l*oc. isoo. if.no.
Time—hour
FIGURE IV-14,. THE EFFECT--EXPRESSED AS MAXIMUM
DEVIATIONS-OF VARIATIONS
IN WIND SPEED FOR CO
o
+->
as
OJ
a
1 +50%
2 +25%
3 -25%
4 -50%
Time—hour
FIGURE IV-15. THE EFFECT—EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS
IN WIND SPEED FOR NO
-------
172
c
o
01
o
E
3
X
(O
1 +50%
2 +25%
3 -25%
4 -502"
«g«. tit. 791. «»0-. lot. 1000. 1110. 1300. 1300. 1*00. 1400. 1600,
Time—hour
FIGURE IV-16. THE EFFECT--EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS
IN WIND SPEED FOR 00
c
o
01
o
E
E
•^~
X
CO
1 +50%
2 +25%
3 -25%
4 -50%
MA* 601. TQ«. . •»•. VOO. 1000. HOB. 1200. 1.100. 1400. 1500. 1600.
Time—hour
FIGURE IV-17. THE EFFECT—EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS
IN WIND SPEED FOR NO,
-------
173
O) -
o
10
(J
O)
D.
X
(O
1 t$Q%
2 +25%
3 -25%
* -50%
'_!L *—
««0. HI. Too. 801. • «
-------
174
1 +50%
2 +25%
3 -25%
4 -50%
«.»•——I—-I 1 1 1 1 1 1 1 1 1
Time—hour
FIGURE IV-20. THE EFFECT-EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS—OF VARIATIONS IN
WIND SPEED FOR 0
01
a
a)
o>
ta
+j
c
(U
u
^ »§.»
1 +502
2 +25%
3 -25%
4 -50%
«. too. TIO. «»». «"». 1D». HOD. 1201. 1100. 1*00. 1500. ItOt.
Time—hour
FIGURE IV-21. THE EFFECT--EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS—OF VARIATIONS IN
WIND SPEED FOR N00
-------
175
The most interesting conclusion that can be drawn from Figures IV-6
through IV-21, however, is based on the fact that the changes do not follow
the simple one-over-wind-speed law as predicted by a box model (Hanna, 1972).
To illustrate this point, we consider a simple box model based on
c = k £ , (IV-9)
where
c = concentration,
Q = emissions strength,
u = wind speed,
k = proportional constant.
If the wind speed is changed by, say, x percent, then according to Eq. (IV-1),
the box model would consequently predict
c - k « • • (IV-10)
Thus, the relative change in percentage obtained using the box model would
be
d = c - c
UV-11)
100
For the four values of x used in our simulations, d has the following values
(in percentages):
+50%
+25
-25
-50
33.3%
20.0
33.3
100.0
-------
176
A comparison of the above tabulation with Figures IV-10 through IV-17
results in the observations listed below.
> The response of the SAI airshed model is time-dependent, whereas
that of a simple box model is time-invariant. This difference is
apparently due to the following deficiencies in the simple box model:
- The invocation of the steady-state assumption.
- The improper treatment of the initial conditions.
> The response of the SAI airshed model varies with each chemical
species, whereas that of a simple box model does not. However,
Hanna (1973) has extended the simple box model to chemically re-
active substances. His results show that the concentrations of
reactive species depend on the wind speed in a way that is more
complicated than the simple one-over-wind-speed law.
> An anomaly was observed between 800 and 900 PDT in that an increase
in the wind speed tended to induce an increase in the maximum
deviation in CO concentrations. This effect may be attributable to
the shift from a land breeze to a sea breeze regime during this period
> Measured in terms of the basin-wide average (see Figures IV-10
through IV-13), the responses of the SAI airshed model, even at
the peaks, are considerably smaller in magnitude than those pre-
dicted using the box model. For example, the maximum responses
for carbon monoxide, and inert species, are (in percentages):
x d
+50% 19.6%
+25 11.8
-25 20.2
-50 51.7
These values are much smaller than the corresponding entries in
the previous tabulation. However, Figure IV-10 shows that the
+50 percent curve is very close to the -25 percent curve, as pre-
dicted by the simple box model.
> Measured in terms of the local maximum (see Figures IV-14 through
IV-21), the response of the SAI airshed model can be significantly
higher than that of the box model. Therefore, depending on the
spatial average, the more elaborate SAI model provides a spectrum
of responses, whereas the simple box model provides only one.
-------
177
In view of these observations, we can conclude that, although a simple
box model may retain some of the most important features of sophisticated
airshed models, it also lacks many other ingredients that are more complex
but are indispensable to a successful airshed model. The arguments for
returning to the use of the simple box model have been under attack from a
different direction by Hameed (1974). However, we believe that the evidence
presented here, which is in line with the arguments expressed by Lamb and
Seinfeld (1974), is more direct and fundamental.
As a final note, we would like to point out the potential application
of the results of our sensitivity study to the development of repro-models
(Horowitz et al.,1973). In particular, the cause-effect relationship between
wind speeds and air quality that has been established in the sensitivity analy-
sis could be very useful in generating the approximate functions needed in tne
repro-models.
3. The Effect of Variations in Turbulent Diffusivity
Eddy diffusion due to turbulent motions of the atmosphere is the
principal mechanism for the dispersion of air pollutants. In the SAI
airshed model, the treatment of turbulent diffusion is imbedded in the
so-called K-theory, which involves the use of horizontal and verti.cal
diffusion coefficients. In this section, we investigate the effect of
varying these coefficients. Since we anticipated that interactions be-
tween diffusion and chemical reactions would not predominate under these
conditions, we considered only the simplified case of an inert species.
a. H ori z o n t a 1 Pi f f u s i o n
A simple order-of-magnitude analysis of the diffusion equation
would show that horizontal diffusion, under typical conditions, is dwarfed
by horizontal advection. However, to determine the importance of this term
quantitatively, we chose carbon monoxide as the base case, using a constant
p 1
(physical) horizontal diffusion coefficient of 50 m sec" , which is a typical
value for this situation. Next, we examined the effect of varying this
-------
178
value by setting the coefficient first at zero and then at 500 m2 sec \
These numbers undoubtedly represent the extreme values of the physical
horizontal diffusivity.
Table IV-4 presents the results of these two sensitivity runs, which show
that the effect of changes in (physical) horizontal diffusivity from 50 m2 sec"1
to zero is minimal (less than 0.4 percent for basin-wide averages and less
than 3 percent for maxima). Because of this result and because the horizontal
diffusivity in the model is the sum of two components,
(KH}P
where
(KH)M = horizontal diffusivity in the model,
(KH)N = horizontal diffusivity due to numerical diffusion,
(Kn)p = horizontal diffusivity due to physical diffusion.,
either or both of the following two conditions must prevail:
> The advection term is much greater than the (numerical and physical)
diffusion term.
2 -1
> The magnitude of the numerical diffusion is much greater than 50 m sec
In contrast, Table IV-4 shows that noticeable effects begin to emerge as the
2 -1
(physical) diffusivity is changed from 50 to 500 m sec (about 2 percent for
basin-wide averages and 13 percent for maxima*). Logically, this implies that
both of the following two considerations must apply:
* The effect on the average concentration of varying the magnitude of the
diffusion coefficient is highly disproportionate. For instance, we showed
in the next section that an order-of-magnitude change in the vertical dif-
fusivity results in a change of only about 10 percent in average concentration.
-------
Table IV-4
THE LARGEST DEVIATIONS IN THE GRID GENERATED BY RANDOMLY VARYING THE HORIZONTAL DIFFUSION
(a) |d| in Percentages
Hour
KH
KH
Case
^0
9
•> 500 m /sec
6
0.07%
0.6
7
0.16%
1.3
8
0.23%
1.8
9
0.27%
2.1
10
0.30%
2.4
11
0.34%
2.6
12
0 . 34%
2.7
13
0.29%
2.3
14
0.22%
1.7
15
0.20%
1.6
16
0.17%
1.4
(b) |dmaxl in Percentages
Hour
Case 6 7 8 9 10 11 12 13 14 15 16
Ku ->- 0 0.52% 0.88% 0.91% 1.39% 1.16% 1.38% 1.93% 2.02% 1.71% 1.48% 1.21%
n
Ku -> 500 m2/sec 4.4 5.9 7.6 11.4 9.2 10.4 12.6 12.9 11.3 9.6 8.2
H
-------
180
> The advection term is not much greater than the diffusion term
2 -1
when the latter is 500 m sec .
> The magnitude of the numerical diffusion is on the order of
2 -1
500 m sec or less.
Therefore, according to linear analysis, the estimated magnitude of numerical
diffusion in the SAI airshed model is
1 UAX - TT • 4 m sec"1 • 3000 m - 6 • 103 m2 sec"1
The results of our analysis demonstrate that this estimate is at least an
order of magnitude too high.
In summary, the sensitivity study reveals that the horizontal diffusion
term becomes competitive with the advection term only when the magnitude of
2 -1
the former exceeds 500 m sec . Furthermore, this study implies that the
2 -1
magnitude of numerical diffusion is on the order of 500 m sec or less.
Thus, these findings constitute evidence that numerical errors in the grid
model are not as severe as one may think. Liu and Seinfeld (1974) have reached
the same conclusion via a different approach.
b. Vertical Diffusion
Vertical diffusion is an important process in determining the distribution
of air pollutants in the atmosphere. The magnitudes of vertical diffusivities
depend strongly on atmospheric stability and on mixing depth and mildly depend
on wind speed and other parameters. Because of the wide range of values for
vertical diffusivity (typically from 10"1 to 102 m2 sec'1) and the lack of reli-
able means for measuring it directly, vertical diffusivity is the most difficult
parameter to determine or estimate. Since the present state of the art allows
only an order-of-magnitude determination, we varied vertical diffusivity by
2 1
either increasing or decreasing the reference values in the base case (5 m sec~ )
by a factor of 10.
-------
181
The results of the calculated average deviations and maximum deviations,
presented in Figures IV-22 through IV-33, show the following:
> As is true in all cases considered in the sensitivity study, the
residue air pollutants are responsible for the gradual buildup of
the effect of changing the vertical diffusivity.
> Roughly speaking, the effect on the ground-level concentrations of
varying the wind speed by 25 to 50 percent is about the same as that
of varying the vertical diffusivity by an order of magnitude.
> The effect of decreasing the vertical diffusivity is not linearly
proportional to that of increasing the vertical diffusivity; the
former is significantly larger. Furthermore, this discrepancy is
more pronounced for secondary pollutants, such as ozone.
> The effect of varying the vertical diffusivity is not the same for
local maxima and basin-wide averages; the effect for the latter is
considerably smaller.
In addition to the usefulness of these observations in determining the
response of the SAI urban airshed model to changes in atmospheric stability,
these results also present additional evidence that a simple box model will
not suffice. Succeeding sections present still further evidence in support
of this conclusion.
4. The Effect of Variations in Mixing Depth
The height of the mixed layer is also an important meteorological
parameter, since variations in it affect pollutant concentration levels.
The mixing depth is usually determined from vertical temperature soundings.
Since ±25 percent can be taken as an approximate measure of the uncertainty
in determining the mixing depth, we varied the values used in the base case
by these amounts. Figures IV-34 through IV-49 present the results of the
sensitivity calculations, which show the following:
-------
182
c
o
OJ
o
CJ
C'l
CU
-------
183
c
o
ctf
-------
12 -
10 11 12 13
Time—hour
14 15 16
FIGURE IV-24. THE EFFECT--EXPRESSED AS AVERAGE DEVIATIONS-
OF VARIATIONS IN VERTICAL DIFFUSIVITY FOR 0.
-------
185
10
OJ
QJ
CD
-2
-4
-6
-8 _
-10
O 0.1 * K
10.0 * K.
56789
11 12 13 14 15 16
Time—hour
FIGURE IV-25. THE EFFECT—EXPRESSED AS AVERAGE DEVIATIONS-
OF VARIATIONS IN VERTICAL DIFFUSIVITY FOR N00
-------
186
50
40
tr
o
-------
187
50
40
30
c
o
r«
-------
188
100
90
80
70
c 60
o
o>
Q
CD
rtf
Q>
U
at
a,
50
40
30
20
10
O 0.1 * K
I I I I
8 9 10 11 12 13 14 15
Time—hour
16
FIGURE IV-28. THE EFFECT-EXPRESSED AS PERCENTAGE DEVIATIONS
OF VARIATIONS IN VERTICAL DIFFUSIVITY FOR 00
-------
189
10 1 12 13 14 15 16
FIGURE IV-29. THE EFFECT--EXPRESSED AS PERCENTAGE DEVIATIONS-
OF VARIATIONS IN VERTICAL DIFFUSIVITY FOR N00
-------
190
o
• r—
fO
•i—
O)
O .
E:: -
rs 3
E
•i—
X
O
0.1
D 10.0 * K
J \ I L
_L
_L
10 11 12 13
Ttme—hour
14 15 16
FIGURE IV-30. THE EFFECT--EXPRESSED AS MAXIMUM DEVIATIONS-
OF VARIATIONS IN VERTICAL DIFFUSIVITY FOR CO
-------
191
30
25
cr
o
O)
E
'r-
X
s:
20
15
10
O
0.1 * K
D 10.0 * K
J L
J I I L
10 11 12 13
Ttme—hour
14 15 16
FIGURE IV-31. THE EFFECT—EXPRESSED AS MAXIMUM DEVIATIONS-
OF VARIATIONS IN VERTICAL DIFFUSIVITY FOR NO
-------
192
50
40
o
•r—
03
*r—
>
X
(vJ
30
20
10
O 0.1 * K
I
I
I
10 11 12 13
Ttme—hour
14 15 16
FIGURE IV-32. THE EFFECT--EXPRESSED AS MAXIMUM DEVIATIONS-
OF VARIATIONS IN VERTICAL DIFFUSIVITY FOR 00
-------
193
10 11 12 13 14 15
Time—hour
FIGURE IV-33. THE EFFECT--EXPRESSED AS MAXIMUM DEVIATIONS-
OF VARIATIONS IN VERTICAL DIFFUSIVITY FOR N02
-------
O
en
ra
QJ
-------
195
c
o
O)
01
«
S-
0)
V+252
2 -25J
•II. »OI. TOO. 100. «00. 1000. 1)00. UOO. 1300. 1400. ISOO. 1400.
Time—hour
FIGURE IV-36. THE EFFECT-EXPRESSED AS AVERAGE
DEVIATIONS—OF VARIATIONS IN
MIXING DEPTH FOR 0,
c
o
-------
196
Time--hour
FIGURE IV-38. THE EFFECT-EXPRESSED AS
PERCENTAGE DEVIATIONS—OF VARIATIONS
IN MIXING DEPTH FOR CO
c
o
OJ
o
<0
01
CL
1 +25X
Z -25i
"«tf. »00. 700. *<>9. «00. 1090. 1100. 1200. 1100. I'OO. 1^00. 1600.
Time—hour
FIGURE IV-39. THE EFFECT—EXPRESSED AS
PERCENTAGE DEVIATIONS—OF VARIATIONS
IN MIXING DEPTH FOR NO
-------
c
o
IO
f
>
o .
u
Time—hour
FIGURE IV-40. THE EFFECT—EXPRESSED AS
PERCENTAGE DEVIATIONS—OF VARIATIONS
IN MIXING DEPTH FOR 00
c
o
QJ
a
cr
o
u
01
a.
"vOO. *00« T<0. . AOO. 9*0. 1000. 1100. 1200. 1300. 14*0. 1500. IfrOO.
Time—hour
FIGURE IV-41. THE EFFECT—EXPRESSED AS
PERCENTAGE DEVIATIONS—OF VARIATIONS
IN MIXING DEPTH FOR N00
-------
c
o
E
i
x
ro
1 +25Z
2 -25%
Time—hour
FIGURE IV-42. THE EFFECT—EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS IN
MIXING DEPTH FOR CO
o
•r-
••->
ro
-------
X
-------
200
01
o
01
Oi
10
4->
C
O>
o
j_
01
O-
e
E
1 +251
2 -25X
90«. 400. TOO. *ftO. 900. 1000. 1101. ]200. 1300. 14«0. 1SOO. 160<
Time--hour
FIGURE IV-46. THE EFFECT. EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS—OF VARIATIONS IN
MIXING DEPTH FOR CO
d)
Cn
ia
01
o
i.
QJ
O.
X
re
I +25X
2 -251
»««. to>. rot. ooo
Time--hour
FIGURE IV-47. THE EFFECT—EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS—OF VARIATIONS IN
MIXING DEPTH FOR NO
-------
201
o
£ U.J
CT>
to
s_
01
Q-
e
I ».
X
IO
1 +25X
2 -25t
>-=;
«»». t09. 780. BOO. "00. 1000
Time—hour
FIGURE IV-48. THE EFFECT-EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS—OF VARIATIONS IN
MIXING DEPTH FOR 0^
c
0
O)
o
d)
en
OJ
o
s_
-------
202
> The buildup of the mixing depth variation effect is time-dependent.
> Decreasing the mixing depth has a greater effect on the ground con-
centration than increasing it. This result is more pronounced for
reactive pollutants.
> The effect of changing the mixing depth is not uniform over the
modeling region; it varies from place to place.
> The effect on ground-level concentrations of changing the mixing
depth is roughly the same as that of changing the wind speed, as
one would expect from a dimensional analysis.
5. The Effect of Variations in Radiation Intensity
Since the objective of airshed modeling is to simulate photochemical
air pollution, concentration levels of both primary and secondary air pol-
lutants that participate in the photochemical reactions are of particular
interest. Hecht, Roth, and Seinfeld (1973) have analyzed the sensitivity
of the kinetic mechanism used in the SAI airshed model. In particular,
they estimated the sensitivities of predicted concentration histories in
a smog chamber to variations in the magnitudes of the primitive and inter-
mediate parameters in the kinetic model, such as the reaction rate constants
and- the stoichiometric coefficients. Hecht et al. concluded that the rate
constant for the photolysis of N02, a function of UV light intensity, is
one of the most sensitive parameters. Thus, in the present investigation,
we varied the photolysis rate constant (or, equivalently, the radiation in-
tensity) to determine the effect of photochemistry on the ground concentrations
predicted by the SAI urban airshed model. In the base case, the radiation
intensity varied with the hours of the day; for our two sensitivity runs, we
increased and decreased the base-case values by 30 percent.
The results of these calculations, presented in Figures IV-50 through
IV-65, show that, as expected, changes in the light intensity do not affect
carbon monoxide. The following comments can be made about the three photo-
chemically reactive species, NO, Og, and N0,>:
-------
203
o
•I—
•!->
IO
a>
01
10
t-
ai
«=:
I +30*
2 -30%
»»«. «••. 799. ««0. Dim. 1000. »00. 1290. 1100. 1*00. 1^00. 1600
Time—hour
FIGURE IV-50. THE EFFECT—EXPRESSED AS AVERAGE
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR CO
c
o
at
o «.«
(D
a>
ro
S-
O)
1 +30X
2 -301
*M«. *•«. TOt. «00. 100. 1000. 1100. 1ZOO. 1301. MOO. 1^00. 1600
Time—hour
FIGURE IV-51. THE EFFECT-EXPRESSED AS AVERAGE
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR NO
-------
204
o
QJ
O
CD
a
1 +302
' 2 -30X
M«. tit. lit. lot. 100. 1000. lilt. 1201. 1.100, 1»0». 1500.
Time—hour
FIGURE IV-52. THE EFFECT-EXPRESSED AS AVERAGE
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR 00
Time--hour
FIGURE IV-53. THE EFFECT-EXPRESSED AS AVERAGE
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR N00
-------
205
c
o
I ••
O)
O1
fO
+J
C
01
(J
ftOd. TOO. ftOO. 900. 1000. 1100. 1200. 1300. MOO. 1500. 1600
Time—hour
FIGURE IV-54. THE EFFECT EXPRESSED AS PERCENTAGE
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR CO
CO
•r-
>
G>
O
rd
C)
o
a! «*•»-
1 +30Z
2 -30S
'. 4|(. 401* TOO. BOO. tno. 1000. 1100. 1ZOO. 1300. 14CO. 1900. 149*
Time—hour
FIGURE IV-55. THE EFFECT—EXPRESSED AS PERCENTAGE
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR NO
-------
206
o
•r—
+J
(O
QJ
01
O)
U
L.
QJ
Ou
1 +30X
2 -30%
M»« AOO. TOfl. BflO. «(10. 1000. 1100. 1?00. 1300. 1*90. 1500. 1A09
Time—hour
FIGURE IV-56. THE EFFECT—EXPRESSED AS PERCENTAGE
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR 0.
o
O)
a
01
en
ia
4->
ai
o
S-
ai
o.
1 +30*
2 -30Z
'«•• »»l. 701. KOI. •««. 1000. 1109. 1200. 1300. 1«00. 1^00. 1400.
Time—hour
FIGURE IV-57. THE EFFECT-EXPRESSED AS PERCENTAGE
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR N00
-------
c
o
03
*r—
>
g ••'
E
E
•r-
X
ID
1 +302;
2 -30%
Time—hour
FIGURE IV-58. THE EFFECT-EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR CO
207
O
*^-
•M
-------
208
01 «.
o
E
E
•r—
X
jo
\
1 +302
/
«so». 600. TOO. HOO. -*no. 1000. 1100. i?oo. noo. i4oo. 1400. i«oo
Time--hour
FIGURE IV-60. THE EFFECT-EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS Ifl RADIATION
INTENSITY FOR .0,
o
1 1 1 1 1 1 1 1 —I — -1
Time—hour
FIGURE IV-61. THE EFFECT—EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS IN RADIATION
INTENSITY FOR JJO,
-------
209
c
o •.«
c
dl
o
s-
0)
O-
I
X
CO
1 +30Z
2 -302
••••——i 1-
««». «00. TOO
-•I 1 1 , , ,.
Time—hour
FIGURE IV-62. THE EFFECT—EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS—OF VARIATIONS IN
RADIATION INTENSITY FOR CO
101. too. 700. no. t»a. !°°o. llo«. laoo. 1100. !»««. 1500. 1*00
Time—hour
FIGURE IV-63. THE EFFECT—EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS—OF VARIATIONS IN
RADIATION INTENSITY FOR NO
-------
210
c
° 1*1.J-
-------
211
> As in all of the other sensitivity experiments, the effects of
varying the radiation intensity are time-dependent, reflecting
the fact that the atmosphere plays the role of a reservoir.
> The effect of changing the light intensity is as significant as
that of changing the wind speed. Both of these parameters are
highly influential in determining ground-level concentrations.
> The effect of increasing the light intensity is to increase the
N02 concentration levels in the morning and to decrease them in
the afternoon. The reverse is true when the light intensity is
decreased. These trends, clearly shown in Figures IV-53 and IV-61,
can be explained simply as follows. According to the results of
smog chamber experiments, the net effect of increasing the light
intensity is to accelerate the conversion of NO to NCL, thereby
shifting the peak N02 concentration as illustrated in Figure IV-66(a).
Consequently, the computed absolute difference shows a crossover at
a certain point in time [Figure IV-66(b)j; this result appears both
in the average deviation (Figure IV-53) and in the maximum deviation
(Figure IV-61). Furthermore, as shown in Figure IV-66(c), a double
peak variation occurs if the relative difference (in terms of per-
centage) is computed. Again, this effect can be observed in Figures
IV-57 and IV-65.
6. The Effect of Variations in Emissions Rate
The last parameter we explored in the sensitivity study was the rate of
emissions from the various pollutant sources. (Incidentally, this is also
the only nonmeteorological parameter we investigated.) The emissions rate in
the base case varies both temporally and spatially. Although many interesting
sensitivity runs could have been made, we examine'd only the simplest possible
case: We uniformly increased the emissions rate by 15 percent in one run and
decreased it by 15 percent in another.*
* Although reactive hydrocarbons are not considered here, we also varied their
emissions rate by the same percentages.
-------
c
o
c
O)
o
c
o
o
CM
o
BASE CASE
INCREASING
RADIATION INTENSITY
Time
(a) Conversion of NO to N02
o
4J
-------
213
Figures IV-67 through IV-82 show "the following interesting phenomena:
The effects of increasing and decreasing the emissions rate are almost
identical, particularly in the relative maximum deviations (Figures IV-
79 through IV-82), which exhibit nearly perfect matches. This phenome-
non is, of course, related to the fact that, as a first approximation,
the concentration level is proportional to the emissions rate [see Eg.
(IV-9)]. Therefore, among all of the parameters we considered, the
emissions rate is probably the only one that may be amenable to a simple
linear approach.
> The effect of changing the emissions rate varies not only with time,
but also with chemical species. Even at their peaks, the basin-wide
average percentage changes in the ground-level concentrations of CO
and N02 (approximately 6 to 8 percent) are about the same. However,
the corresponding maximum changes, with the exception of that for CO
(approximately 10 percent), are all greater than the percentage change
in the emissions rate (see Figures IV-79 through IV-82).
D. DISCUSSION AND CONCLUSIONS
In the work discussed in this chapter, we studied the sensitivity of the
SAI urban airshed model to changes in the input parameters. The variables we
explored were random perturbations in wind speed and wind direction, systematic
variations in wind speed, horizontal and vertical diffusivities, mixing depth,
radiation intensity, and emissions rate. Despite the arbitrary selection of
the base case, a close scrutiny of the results of this study shows that they
conform qualitatively to what one would expect physically. Thus, we believe
that the conclusions drawn from these results have rather general validity.
As we attempted to demonstrate in Section C, many interesting findings
or observations can be unearthed from the voluminous data set generated during
this study. Two items, briefly discussed below, appear to be the most significant
conclusions.
-------
O
•r-
•M
ia
OJ
Q .«.•
O)
en
IO
s-
01
1 +151
2 -15X
Mi, 6fl»» TOO, 490. 40fl* 1000. HBO. 1ZOO. 1300. t«00. 1*00. l«00.
Time--hour
FIGURE IV-67. THE EFFECT-EXPRESSED AS AVERAGE DEVIATIONS-
OF VARIATIONS IN EMISSIONS RATE FOR CO
Time—hour
FIGURE IV-68. THE EFFECT—EXPRESSED AS AVERAGE DEVIATIONS-
OF VARIATIONS IN EMISSIONS RATE FOR NO
-------
I.I
o
»r—
4->
-------
216
c
o
+
re
-------
217
I
OJ
C)
KS
J->
01
o
-------
218
I ••"
E
3
X
ro
1 +15%
2 »15Z
Time--hour
FIGURE IV-75. THE EFFECT-EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS IN
EMISSIONS RATE FOR CO
a
o
OJ
o «.«
X
-------
219
c
o
X
CU
1 -USX
2 -15%
»•«. tO>. TOO. «»0. 490. 1091. 1110. 1200. 1.100. 1*00. 1500. 1690.
Time—hour
FIGURE IV-77. THE EFFECT—EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS IN
EMISSIONS RATE FOR CL
c
o
I ...
X
It)
1 +15J
2 -15X
tee. »»«. TOO. «»o. 100. 1000. 1100. uoo. 1300. itoo. 1^00. i»oo.
Time—hour
FIGURE IV-78. THE EFFECT-EXPRESSED AS MAXIMUM
DEVIATIONS—OF VARIATIONS IN
EMISSIONS RATE FOR N00
-------
220
cj
i.
0)
D_
X
ra
1 +15*
Z -15%
100. «00. 70
Time—hour
FIGURE IV-79. THE EFFECT—EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS—OF VARIATIONS
IN EMISSIONS RATE FOR CO
c
o
•"-> ft.J-
01
-------
221
e:
OJ
s.
01
o.
c *H.A
3
2 -15%
Time—hour
FIGURE IV-81. THE EFFECT—EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS--OF VARIATIONS
IN EMISSIONS RATE FOR 0,
C
O J»-3
O
O
OJ
o
IO
+J
c
OJ
o
E
E
•r-
X
ta
1 +152
2 -15X
*••• 40«. TOO. 600. «*0. 1020. 1100. 1?00. 1300. 1*00. 1500. 1600.
Time—hour
FIGURE IV-82. THE EFFECT—EXPRESSED AS MAXIMUM
PERCENTAGE DEVIATIONS—OF VARIATIONS
IN EMISSIONS RATE FOR N00
-------
222
1• Justification for a Complex Model
Our study has demonstrated that the sensitivity of the SAI urban airshed
model, a complex grid model, to variations in meteorological or emissions par-
ameters is generally characterized by the following features:
> The effects are time-dependent.
> The effects vary spatially.
> The effects differ for different chemical species.
> The effects do not follow a simple rule, such as the inverse
proportionality for wind speed.
With the possible exception of the effect due to variations in chemical
species [as we stated earlier, Hanna (1973) has developed a box model to
account for reactive species], none of the simple box models appear to be able
to reproduce these essential features. The reasons for this failure are rather
obvious. For example, inadequate or improper handling of the initial concen-
trations in >;he box model is responsible for its failure to produce the time-
dependent behavior of the effects. Oversimplification in the simulation of
advective and diffusive processes apparently makes the simple box model incapable
of recreating the detailed spatial distribution of the pollutant concentrations
or the correct dependence on the various meteorological and emissions input
parameters. Thus, our sensitivity study provides direct evidence that a simple
box model is not sufficient to simulate urban air pollution.
2. The Sensitivity of the SAI Model
The results of our assessment of the relative sensitivities of the SAI
airshed model to various meteorological and emissions parameters can be used
for several purposes:
-------
223
> To give a rule-of-thumb estimate of the response of the model
predictions to changes in the input parameters.
> To provide insights into the expected behavior of the SAI model
in particular and into urban airshed models in general.
> To facilitate the development of a repro-model.
This section discusses the overall sensitivities of the SAI urban airshed
model. As shown in Figures IV-83 through IV-86, we plotted the response (the
peak values of the basin-wide averages) of the model predictions against changes
in the input parameters for the four species considered in this study. The
following observations emerge from an examination of these curves. First, with
the exception of ozone, the slopes of the responses are less steep than those
that are inversely proportional to the corresponding changes. Second, the
responses of CO and NO^ tend to vary linearly in the log-log plot, whereas
those of NO and 03 tend to be nonlinear.
Using the slopes of the reponses as indices, we ranked the various input
parameters according to their importance in affecting the model predictions.
Table IV-5 presents this ranking.
In the cases of radiation intensity and emissions rate, the slopes are
directly proportional.
-------
224
10.0
5.0
2.5
0.5
0.25
o WIND SPEED
A VERTICAL DIFFUSIVITY
D MIXING DEPTH
A EMISSIONS RATE
I I I I I I I
0.1
0.25
0.5 1.0
Relative Change
2.5
5.0
10.0
FIGURE IV-83. THE AVERAGE EFFECT OF CHANGES IN INPUT
PARAMETERS ON CO CONCENTRATION
-------
225
10.0
5.0
9 K
L. • -J
10
0)
to
§ 1.0
Q.
C/l
(U
0.5
0.25
I i
0.1
|| I I 111!
o WIND SPEED
A VERTICAL DIFFUSIVITY
a MIXING DEPTH
» RADIATION INTENSITY
A EMISSIONS RATE
J I
J I
I I I I
0.25
0.5 1.0 2.5
Relative Change
5.0
10.0
FIGURE IV-84. THE AVERAGE EFFECT OF CHANGES IN INPUT
PARAMETERS ON NO CONCENTRATION
-------
226
10.0
5.0
o 'WIND SPEED
A VERTICAL DIFFUSIVITY
D MIXING DEPTH
RADIATION INTENSITY
.EMISSIONS RATE
0.5 1.0
Relative Change
5.0
10.0
FIGURE IV-85. THE AVERAGE EFFECT OF CHANGES IN INPUT
PARAMETERS ON 03 CONCENTRATION
-------
10.0
5.0
2.5
V}
01
to
§ 1.0
Q.
(/)
0)
0.5
227
0.25
o WIND SPEED
A VERTICAL DIFFUSIVITY
a MIXING DEPTH
® RADIATION INTENSITY
A EMISSIONS RATE
I 1 1 J I l I I
0.1
0.25 0.5 1.0
Relative Change
2.5
5.0
10.0
FIGURE IV-86. THE AVERAGE EFFECT OF CHANGES IN INPUT
PARAMETERS ON N02 CONCENTRATION
-------
Table IV-5
RANKING OF THE RELATIVE IMPORTANCE
OF THE INPUT PARAMETERS
Parameter or Variable
Wind speed
Horizontal diffusivity
Vertical diffusivity
Mixing depth
Radiation intensity
Emissions rate
CO
A
D
C
B
D
B
N0_
A
D
C
B
A
A
°3
A
D
C
B
A
B
N02
A
D
C
B
B
B
A = most important.
D = least important.
-------
229
APPENDIX A
THE NONUNIQUENESS OF LAGRANGIAN VELOCITIES
-------
APPENDIX A
THE NONUNIQUENESS OF LAGRANGIAN VELOCITIES
This appendix demonstrates that Lagrangian velocities, as conven-
tionally defined, may not be unique in an atmospheric turbulent flow.
We adapted the derivation we present here from a paper by Dyer (1973).
Consider an air column that is moving through a two-dimensional
turbulent atmosphere, as illustrated below:
Time Elapsed = T
We can obtain the actual velocity of the air column, say, from
tracer data, as a function of time (t) and location (x,y); we denote
such velocities by vAC(x,y;t).
We can then obtain two types of Lagrangian averages:
a space average,
=!/
-------
and a time average,
1 r
= Jj
VAC = v(x'y;t) dt
0
Since urtr = dx/dt, it follows that
T
=x/ VvACdt
0
T
= x UACVAC
= JL
UAC
In a turbulent atmosphere, if we assume that
UAC UAC + UAC
VAC = VAC + VAC
then
=:f4
UAC
or
UACVAC
-------
232
Therefore, depending on the turbulence statistics, the two types of
average Lagrangian velocities defined earlier can be different.*
*Phillips (1966) and Longuet-Higgins (1969) have pointed out the
nonuniqueness of the Lagrangian and Eulerian velocities in studies
of ocean currents.
-------
233
APPENDIX B
WIND AND DIFFUSIVITY PROFILES IN THE LOWER ATMOSPHERE
-------
234
APPENDIX B
WIND AND DIFFUSIVITY PROFILES IN THE LOWER ATMOSPHERE
In this appendix, we present a review of the literature dealing with
the study of wind profiles and vertical diffusivity profiles in the lowest
layer of the atmosphere. The purpose of this review was to estimate the
magnitudes and exponents of the wind or vertical diffusivity profiles that
commonly occur in an urban atmosphere.
1. Hind Profile
Micrometeorologists have intensively studied the variations of
horizontal wind with height. In the 1940s, for example, Deacon (1949) in
England and Laikhtman (1944) in the U.S.S.R. carried out comprehensive
investigations of vertical profiles of mean wind velocity. They verified
that in the surface layer (approximately a few meters above the ground),
th-e logarithmic law for the mean velocity, under the condition that the
atmosphere is neutrally stratified, is
where u* is the friction velocity, ZQ is the roughness parameter, and
k is the yon Karman constant. Departures from adiabatic conditions tend
to increase the rate of change of wind speed with height under unstable
conditions and to decrease the rate under stable conditions. Therefore,
for these adiabatic cases, Laikhtman and Deacon proposed the following
formula:
J IL-] ' . i . (B-2)
kit- g) UQ
-------
235
They found that the coefficient, a function of atmospheric stability,
ranges as follows:
8 > 1 for unstable conditions,
8 = 1 for neutral conditions,
8 < 1 for stable conditions.
Note that in the limiting case of 8 -> 1, the above formula reduces to
the logarithmic law.
At greater heights within the planetary boundary layer (less than
a few hundred meters above the ground), the vertical wind profiles are
generally characterized by a power law. For example, the following
general relationship has often been used.
UR . , . - (B-3)
where UR is the wind speed at a reference height ZR. The exponent m
depends on the stability of the atmosphere and the roughness of the
ground surface. In a neutrally stable atmosphere in open country, the
value of m is normally about 1/7, and it increases as the stability
increases. DeMarrais (1959) obtained the following values for m:
Stability
Class m
1 0.1
2 0.15
3 0.20
4 0.25
5 0.25
6 0.30
-------
236
In a recent study, Jones et al. (1971) also confirmed the dependence
of m on the temperature lapse rate at urban sites. They found the
following quantitative relationships between m and A6, the potential
temperature difference in °C between 530 and 30 feet:
m a 0.2 , A6 <:0
m - 0.21 , A6 = 0
m = 0.33 A0 + 0.21 , 0 < A6 < 0.75
The surface roughness depends on the type of terrain. Typical values
of m for different types of terrain, estimated from experimental data
collected under fair weather conditions, are as follows:
Davenport (1965) Shellard (1965)
Type of Terrain Estimate Estimate
Open country 0.16 0.16
Suburbs 0.28
Metropolitan 0.40
In summary, in an urban environment, the wind speed increases with
height according to the power law. The exponent in the power law,
being a function of atmospheric stability, is likely to be in the range
0.4 > m > 0.2.
2. Vertical Diffusivity Profile
According to similarity analysis, the vertical diffusivity profile
in the surface layer is intimately related to the wind profile we discussed
above. In this layer, the inertia force is usually small compared with
the viscous force. And, if we further assume that the horizontal pressure
gradient is also negligible, the momentum equation finally reduces to
IzM)"0 • (B-4)
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It then follows that, if the horizontal wind velocity is prescribed
by a power law, such as that shown in Eq. (B-3), the vertical diffusion
coefficient for momentum, KM, can be described by a conjugate power law
of the following form (with the subscript M suppressed):
K=K 't
where n = 1 - m. There is considerable evidence that the Reynolds
analogy holds in the surface layer; i.e., the vertical diffusion
coefficient, KC, is proportional to the coefficient of vertical
momentum transfer, K^. Thus, the vertical diffusion coefficient for
species can be described by the same expression, i.e., Eq. (B-5).
Many field experiments carried out recently (Ikebe and Shimo,
1972; Cohen et al., 1972) have verified the power law profile and
have shown that vertical diffusivity varies with atmospheric stability
in a manner similar to that of the horizontal wind, except that the
effect of stability on vertical diffusivity appears to be stronger.
This may result in the observed increases in mass diffusivity at sunrise
and decreases at sunset that are much more pronounced than the observed
diurnal variations in the momentum diffusivity (Israel and Herbert,
1970). Theoretically, the exponent in the power law, n, would be
expected to be less than, equal to, or larger than one, depending on
whether the atmosphere is stable, neutral, or unstable (Deacon, 1949).
However, using field measurements of the vertical thoron profiles in the
lowest meter of the atmosphere, Ikebe and Shimo (1972) estimated that
n = 1.2 for neutral conditions, and n = 1.3 to 1.5 for unstable conditions,
The above results agree with values derived from hourly average radon-gas
concentration measurements made at greater heights (up to 271 meters)
by Cohen et al. (1972). They found that n = 1.4 for stable conditions,
n = 1.2 for neutral conditions, and n = 1.5 for unstable conditions.
In summary, the range of variations in n, according to available
experimental results, appears to fall within 1.5 > n > 1.2.
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238
APPENDIX C
A THEORETICAL ANALYSIS OF THE EFFECT OF RANDOM
PERTURBATIONS OF THE MEASURED WIND
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239
APPENDIX C
A THEORETICAL ANALYSIS OF THE EFFECT OF RANDOM
PERTURBATIONS OF THE MEASURED WIND
In this appendix, we briefly examine the effect of random perturbations
of the wind speed and wind direction measurements on the concentration
field from a theoretical point of view. For the sake of simplicity, we
use the following two-dimensional diffusion equation to illustrate this
effect:
at ax vu"'
where the last term represents the vertical diffusion.
Conceptually, we can see that the measured wind, u, used in
Eq. (C-l) consists of a small and random part superimposed on the
true wind. The random part can arise either as a result of instru-
mentation errors or data reduction procedures. Let
UT = the true wind,
U = the random error in the wind speed, wind direction,
or both,
= the resultant concentrations if the true wind is used,
c = the deviation from the true concentration due to random
error in wind speed, wind direction, or both.
Then,
u = UT + u , (C-2)
c = + c . (C-3)
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240
Substituting Eqs. (C-2) and (C-3) info Eq. (C-l) and taking the
ensemble average, we obtain
(C-4)
where we have assumed that
= ~KH 15T ' (C-5)
as we did in the treatment of eddy diffusion. Thus, the net effect of the
random perturbations of the wind is to introduce a diffusion-like term in
the atmospheric diffusion equation. This finding is, of course, not
surprising because diffusion processes are characterized by their ability
to promote randomness, as clearly demonstrated in our numerical experi-
ments (Section C) by negative average deviations.
To estimate the magnitudes of the artificial diffusion coefficients
created by the random perturbations, we interpret Prandtl's mixing length
theory here as
KH ^ (Ax) ' (C-6)
where L is the separation between the two stations where the winds have
been randomly perturbed For the case in which wind speed is randomly
varied at every grid point,
u = 1 mph ,
L = AX = 2 miles
Thus, we obtain
MJC 10 V sec"1
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241
For random perturbations of the measured wind speed at every monitoring
station, we have
Ax = 2 miles
L ~ 10 miles
Thus,
KH ~ 0.28 x 103 m2 sec"1
The effect of random perturbations of the measured wind angles can
be similarly estimated if we assume an average wind speed. As shown
in the following sketch,
uaverage
the equivalent random wind speed is approximately
D~uaverage:$1n Ae -
Since A9 = 22.5° in our numerical experiment, using an average wind
speed of 5 mph yields
u ~ 2 mph
Consequently, the artificial diffusivity should be twice as high as what
we estimated before. This apparently explains the reason why, in our
numerical experiments, the effect of random perturbations in wind direction
is more pronounced than that of random perturbations in wind speed.
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242
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246
TECHNICAL REPORT DATA
(Please read IniLrnctions on the reverse before completing)
. REPORT NO.
EPA-600/4-76-016 a
3. RECIPIENT'S ACCESS! Of* NO.
.TITLE AND SUBTITLE CONTINUED RESEARCH IN MESOSCALE AIR
'OLLUTION SIMUALTION MODELING. VOLUME I. Assessment
of Prior Model Evaluation Studies and Analysis of Model
Validity and Sensitivity ______
5. REPORT DATE
May 1976
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
M.K. LIU, D.C. WHITNEY, J.H. SEINFELD, AND P.M. ROTH
8. PERFORMING ORGANIZATION REPORT NO.
EF75-23
9. PERFORMING ORG '\NIZATION NAME AND ADDRESS
SYSTEMS APPLICATIONS, INC.
950 NORTHGATE DRIVE
SAN RAFAEL, CALIFORNIA 94903
10. PROGRAM ELEMENT NO.
1AA009
11. CONTRACT/GRANT NO.
68-02-1237
12. SPONSORING AGENCY NAME AND ADDRESS
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, N.C. 27711
13. TYPE OF RE PORT AND PERIOD COVERED
FINAL REPORT 6/74-6/75
14. SPONSORING AGENCY CODE
EPA-ORD
15. SUPPLEMENTARY NOTES
16. ABSTRACT
This report summarizes three independent studies: an analysis of prior
evaluative studies of three mesoscale air pollution prediction models (two trajectory
models and one grid model), an examination of the extent of validity of each type of
model, and an analysis of the sensitivity of grid model predictions to changes in the
magnitudes of key input variables. The analysis of prior studies showed that the three
models evaluated generally reproduced measured ground-level pollutant concentrations
with less than acceptable accuracy. This outcome is the result partly of problems of
inadequacies in the models themselves and partly of the nonrepresentativeness of the
measurement data. In the validity study, the results indicate that numerical diffusion
can introduce significant error in the grid model, whereas neglect of wind shear and
vertical transport are most detrimental in the trajectory approach. The sensitivity
analysis assessed the change in magnitude of predicted atmospheric pollutant concen-
trations due to variations in wind speed, diffusivity, mixing depth, radiation inten-
sity, and emissions rate. The results of the sensitivity analysis showed that varia-
tions in these key input variables influence predictions according to the following
order of decreasing influence: wind speed, emissions rate, radiation intensity, mix-
ing depth, vertical diffusivity, and horizontal diffusivity. Moreover, the responses
of CO and N0? tend to vary linearly with the meteorological and emissions parameters,
whereas those of NO and 0 tend to1be nonlinear.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
*Air Pollution
^Photochemical Reactions
*Reaction Kinetics
*Numerical Analysis
^Mathematical Models
*Atmospheric Models
*Sensitivity
^Verifying
13B
07E
07D
12A
14B
14G
'13. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
19. SECURITY CLASS (This Report)
UNCLASSIFIED
246
20. SECURITY CLASS (This page}
UNCLASSIFIED
22. PRICE
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