The Monte Carlo method -s one way of estimating the magn tuci of model uncertainties due to input data errors. The method involves running the model multiple times, with the input parameters slightly perturbed each time (see Volume I, Section 1x1. It is necessary to implement the Monte Carlo
sensitivity analyses on a platform where the user can easily run the model repeatedly, efficiently extract the information of interest, and not be overwhelmed by the amount of the output generated. The MDA (Modeler Data Archive) software package previously described in this volume serves as an ideal choice for this platform in that the execution of most of the dispersion models has been automated, and in that the extraction of useful information from the outputs can be achieved by the post-processors that have already been developed.
implements the Monte Carlo method MDAMC.
In the following, we shall call the software package that
B. CHOICE OF MODELS AND INPUT PARAMEEM
There are some important criteria that should be heeded in choosing specific dispersion models for application of the Monte Carlo sensitivity analyses. First, it is desirable that the input, the execution and the
post-processing of the model be fully automated. Second, it is desirable that the model can execute reasonably fast (say, less than 10 seconds for each run), since it is necessary to run the model hundreds to thousands of times. Last, as a somewhat less stringent requirement, the model should have a simple I/O structure, such as a small number of compact input and output files are involved. Based on these criteria, the SLAB model was chosen for
testing of MDAMC. The AFTOX, DEGADIS, GASTAR and GPM models also satisfy these criteria, but were not used in the sensitivity study reported in this section.
The input parameters accepted by the models can be classified as primary and secondary. Secondary input parameters are derived from the primary input parameters. Wind and temperature measurements, and surface roughness are the examples of primary input parameters. Monin-Obukhov length and stability class are the examples of secondary input parameters. In the Monte Carlo study, only
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variations in the primary input parameters are considerec. The following seven primary input parameters are pertiirbed at ezch Monte Czfla sisulation in Qur example:
domain averaged wind speed (u),
difference in wind speed between the domain-average and a tower (du), difference in temperature between two levels on a tower (dT1,
relative humidity (RH), surface roughness (zol,
* source emission rate (QI, and source diameter (DI.
The first four parameters are related to the meteorology, the fifth parameter is related to the site condition, and the last two parameters are related to the source condition.
is no correlation among the primary input parameters.
variables such as Monin-ûbukhov length, friction velocity, and stability parameter are calculated from the above seven primary parameters.
In this application, it is assumed that there Other secondary
Currently, the MDAMC package uses concentrations and cloud widths at
certain downwind distances as indicators of model uncertainty due to input data errors.
Perhaps the most difficult problem encountered in Monte-Carlo sensitivity analyses is the specification of the distributions of the primary input
parameters. The Gaussian distribution ( f o r example, Reference 71) and the log-normal distribution (for example, Reference 721 are common choices for many ambient measurements.
distributions for some parameters.
roughness and the source emission rate, the need for a detailed description of their distributions becomes less clear. O'Neill et al. (Reference 731 found out the results of a Monte Carlo analysis of their stream ecosystem model were not sensitive to the choice of parameter distributions. Therefore, it was decided that a simple uniform distribution would be used for all parameters in this example. For a uniform distribution the probability of occurrence of the parameter is the same at a l l points within an upper and lower bound.
of these bounds, the probability of occurrence is zero.
However, there is a lack of knowledge about the Moreover, in the case of the surface
Outside
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The range of a parameter is the only information needed to fully define a uniform distribution. The ranges of uncertainties associated with
meteorological observations depend on the kind of the instrument used, the averaging time, the orientation with the wind direction, and the atmospheric stability (see Volume III of this report). For simplicity, the MDAMC package assumes the following default values for the ranges of uncertainties for the input parameters; however, the user always has the option of specifying his own ranges.
wind speed (u and du):
temperature difference (dT):
relative humidity (RH): the mean f 10 percent
surf ace roughness ( z o 1 : the mean f 1/2 order of magnitude source emission rate (QI: the mean f 1/2 order of magnitude source diameter (DI: the mean k 1/2 order of magnitude
the mean f larger of O. 5 m/c and cU the mean k 0 . 2 O C
For example, if the observed domain-averaged wind speed, u, is 5.6 m í s and the standard deviation, t~ is 0.9 d s , is the wind speed for each Monte Carlo simulation will be drawn randomly from the range between 4.7 and 6.5 d s . the reported surface roughness is 0.0316 m, the surface roughness for each Monte Carlo simulation will be drawn randomly from the range between 0.01 and O. 1 m.
U’
If
C. IMPLEMENTATION
During the execution of MùAMC, the user has to specify: 1 ) a dispersion model whose uncertainty due to data input errors is to be investigated, 2 ) a trial from which perturbations on the primary input parameters will be created, 3) the number of Monte Carlo simulations to be made, and 4) the ranges for the primary input parameters, if the default values provided by the program were not des i red.
The output file created by MDAMC echoes most of the user inputs just described previously.
output parameters for each Monte Carlo simulation. Finally, the file includes the minimums, maximums, means and standard deviations for all the parameters
As an option, the file lists the values of the input and
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based on all the simulations, so that the user can analyze the relationship of input data errors to model uncertainty. An example of this output file is shown in Table 17.
D. RECULTS
In the following, the Desert Tortoise 3 experiment and the SLAB model are chosen to demonstrate the use of the MDAMC package.
to complete 500 simulations using the SLAB model on a PC with 80386 CPU and 80387 math co-processor, both running at 2SMHz.
the primary input parameters that were previously described were used.
Desert Tortoise 3 experiment the following observed values were listed in the
MDA: u = 7.4 m/s, du = 0.2 mis, dT = -0. OSOC, RH = 14.8 percent, zo = 0.003 rn, Q = 130.7 kgis, D=0.0945 m, and u= 1.0 m/s. For a uniform distribution with
the default ranges of uncertainties, the ratios of standard deviation to mean f o r u, du, dT, RH, zo, Q, and D are 0.078, 2.89, 5 . 7 7 , 0.39, 0.47, 0.47, and 0.47, respectively.
input parameters perturbed simultaneously. In order to isolate the influence of each parameter, MDAMC was run seven more times, each time varying only one
of the primary input parameters. Table 18 summarizes the results when all seven parameters were perturbed, and the corresponding probability density functions (pdf) of the concentrations and widths are shown in Figure 20.
19 through 25 summarize the results when only one of the parameters was
perturbed. Model results using the original input data without any
perturbation were also included in the tables and referred to as the "reference value. "
It takes roughly two hours The default uncertainties for
For the
The MDAMC package was first run with all seven primary
Tables
Table 26 summarizes the ratio of the relative model uncertainties,
and "IJ/", to the relative input data uncertainties, ri/T (C = concentration,
w = width, i = input parameter) for this particular example of Monte Carlo sensitivity analysis. Note that the relative sensitivities are less than unity f o r all variables and that the predictions are the most sensitive to variations in wind speed and source strength.
From Figure 20 it is clear that even though all the primary input parameters were given a uniform distribution, the distribution of the
subsequent model results is far from being uniform. It is evident from Table
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TABLE 17. AN EXAMPLE OF THE OUTPUT FILE GENERATE3 BY THE MDMC PACKAGE, WHE3E 20 MONTE CARLO SIMULATIONS OF THE SLAB MODEL FOR THE DESERT
TORTOISE 3 EXPERIMENT ỹEủE PERFORMED.
Trial name: Et3
Ro. o f simulation: 20
3:iq. value, 1.5.. u . b . , man, sigma, and siqxa/nean €or each variable:
Note that the ceans ana aianas here are based on the THE0fLErIC;IL WIFORH cis:ribu=ion
.I 7.40 6 . 4 0 8.40 7 - 4 0 0.577 0.7808-21
EU 0 . 2 0 0 - 0 . 8 0 0 1.20 3.200 0.577 2.99
dT -0.200E-01 -0.220 0.180 -0.ZOOE-01 0.115 -5.77
XH 14.8 4.80 24.8 14.8 5.77 0.200
2 0 0.300E-02 0.9488-03 0.948E-02 0.521E-02 0.246E-02 0.472
Q 131. 41.3 413. 227. 107. O. 472
0.164 0.776E-01 0.472 Rdiarn 0.945E-01 0.2991-01 0.299
AFTOX - n
DEGAOIS - n
GASTAR - n a n - n
s m - y NDIST - 2
And the downwind d i s t a n c e s iml are:
100. 800.
U du di! RH 20 Q Rdiam 5 PG concippml ... s i q y ( m ) . . .
Following are the values of the parameters for each simulation:
7.180E+00-4.577E-01 3.693E-02 2-218E+OI 2.403343 2.990E+02 9-45I.E-O2 4.433E+02 4 3.050E+05 Z.OLOE+Ol 1.389E+01 1.067E+02 6.864E+00 1.131E+00 4.853E-02 1.107E+01 7.1123-03 4.0923+02 1.123E-01 7.997E+02 4 3.4391+05 2.1333+04 1.485E+Ol 1.165E+02 7.804E+00 3.245E-01 1.086E-01 Z-O81E+O1 6.7108-03 3-527E+02 2-61OE-01 6.27SE+Ot 4 3.233E+05 1.7738+04 2.917E+01 9.7638+01 7.811E+00-3.991E-01 1.369E-01 1-971E+01 4-210E-03 2-209E+02 2-54I.E-O1 4.139E+Ot 4 2.35lE+05 1.2588+04 2.892E+01 8.658E+01 7.385E+00-4.086E-01 3.8861-02 7-263E+00 3-628E-03 2-4723+02 2-154E-O1 5.2733+02 4 2.631Et05 1.484E+O4 2.990E+OL 9.3661+01 7.260E+00-5.513E-02 3.835E-O2 2.471E+01 9-208E-03 2.439EcOZ 2-61lE-01 7.467E+02 4 2.427E+05 1.22JE+O4 3.068E+01 9.046Ec01 6.756E+00-6.007E-01-1.978E~Ol 1-236Ec01 8-63OE-O3 2.2148+02 1.248E-01-2.045E+03 4 2.7033+05 1.156E+04 2.007E+Ol 9.6433+01 7.371E+00 5.474E-01 1.230E-01 1.302EL01 1.997E-03 1-7358+02 1-133E-01 4.028E102 4 2.554E+05 1.246E+04 1.743E+01 8.82OE+O1 7.080E+00 6.179E-01-3.019E-02 2.278E+01 1.084E-03 9.177E+01 1.131E-01 7-224E+02 4 1.72ZE+05 7.799E+03 1.9423+01 7.7708+01 8.364Et00-6.5131-01 5.882E-02 8.0733+00 1.572E-03 1.757E+02 2.180E-01 4-656Z+02 4 2.1053+05 1.152E+04 2.507E+Ol 7.962E+01 6.662E+00 1.603E-01 8-6171-02 1.420E+01 7.3773-03 2.309EI02 5-549E'-O2 5.028E+02 4 1.8143+05 1.219f+04 1.363E+01 1.041E+02 7.1681+00 4.973E-01-3.6368-02 1.331E+01 4-051E-03 2.577E+02 2.089E-O1 1.069E+03 4 2.910¿+05 :.557Z+04 2.7038+01 9.725E+01 7.159E+00-3.6573-01 1.005E-01 1-489E+01 6-732Z-O3 3.555E+02 2-741E-01 4.581E+02 4 3.266ở+05 1.886E+O4 3.2548+01 1.0528+02 7.559E+00 1.063E+00-1.039E-01 6-34@E+00 7-3733-03 1.567E+02 2.055E-01 3.937E+03 4 1.8743+05 7.885B+03 2.534E+Ol 7.902E+01 5.433E+O0-5.475E-01 1.349E-01 2-175E+01 4.992503 3.64SE+02 1-11ZE-01 2.826E+02 4 3.4161+05 2.292E+04 1.553E+Ol 1.204E+02 7.844E+00 6.039E-01-2.1723-01 2-071E+01 4.2368-03 4.359E+01 7.307E-02-2.181Z+03 4 8.527E+04 2.506E+03 1.482E+Ol 6.115E+01 8.110Et00-2.902E-01-4.297E~O2 l.OZlE+Ol 9.127B-03 3-012E+Ot 2-712E-01 1.52CZ+03 4 2.693E+05 1.323E+04 2.882E+01 8.913E+O1 8.133EtOO 1.044E+00 3.487E-02 1.403E+01 4.713s-O3 ?.018Et02 4.743E-02 9.934Et02 4 1.321ở+05 5.430Z+03 1.346E+O1 7.261E+01 7.879E+00 5.178E-01-4.2318-03 2.049E+01 8.5421-03 2.229B+O2 2.872E-01 1.2633+03 4 2.1249+05 ?.OSCE+04 2.930E+01 8.3433+01 8.045E+00 7.9095-02 1.600E-01 l.lOBE+Ol 4.2853-03 2.4558+02 3-045E-02 4.606Z+02 4 9.948E+04 1.06Z+04 9.998E+OO 9.029E+01 Following are the min., m a . , means and standard deviations of the parameters for all s i m l a c i o n s :
6.433E+00-6.513E-01-2.172E~Ol 6-3403+00 1-084E-03 4.3591+01 3-0453-02-2.181E+03 8.527E+04 2.50=+03 9.998E+00 6.115E+01 8.364Ec00 1.131E+00 1.600E-01 2.471E+01 9-2093-03 4.092E+02 Z.87tE-01 3.9371+03 3.439%+05 2.2928+04 3.254E+01 1.204E+02 7.443E+00 1.405E-01 2.3691-02 1-545E*01 5.402E-03 2.358E+02 1-681E-01 5.70EE+OZ 2.3745+05 1.309E+04 2.3998+01 I).lBOE+Ol 5.291E-01 5.764E-01 1.OZJ.E-O1 5-561E+OO 2.513E-O3 9-3191+01 8.577E-O2 1.1803+03 7.503E+04 5.115Ee03 7.150E+00 1.416E+01
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TABLE is. MODEL UNC3TAINTIES FOR THE SLAB MODEL UHEìì ALI SLyE?4 ?RIHARY :!!?Si' PARAMETERS [ S E TEST) UERE PRWRBED SIMULTANECUSLY IN SOO MONTS CARLO SIP!üLATIONS FCR THE: DESERT TORTOISE3 EXPBIMENT. (S. D. :