ML20212P743

Uncertainty & Sensitivity Analysis of Environ Transport Models

ML20212P743
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Issue date:	03/12/1987
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ML20212P709	List: IA-86-849, Paper Entitled, Effectiveness of Early Evacuation of Small Areas,Shelter & Relocation in Reducing Severe Accident Consequences, Presented at Fifth Intl Meeting on Thermal Nuclear Reactor Safety on 840909-13.Related Info Encl ML20212P706 ML20212P714 ML20212P743 ML20212P770 ML20212P823
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UNCERTAINTY AND SENSITIVITY ANALYSIS OF ENVIRONMENTAL TRANSPORT MODELS Timothy S. Margulies and Leslie E. Lancaster U.S. Nuclear Regulatory Comission, Washington, D.C. 20555 l

ABSTRACT An uncertainty and sensitivity analysis has been made of the CRAC-2 (Cal-culations of Reactor Accident Consequences) atmospheric transport and deposi-tion models. Robustness and uncertainty aspects of air and ground deposited material and the relative contribution of input and model parameters were sys-tematica1 F studied. The underlying data structures were investigated using a multiway layout of factors over specified ranges generated via a Latin hyper- l cube sampling scheme. The variables selected in our analysis include: weather bin, dry deposition velocity, rain washout coefficient / rain intensity, duration of release, heat content, sigma-z (vertical) plume dispersion parameter,  !

sigma-y (crosswind) plume dispersion parameter, and mixing height. To deter-mine the contributors to the output variability (versus distance from the site) step-wise regression analyses were performed on transformations of the spatial concentration patterns simulated. I I

INTRODUCTION l' An important part of performing a radiological risk assessment for an

, electric generating facility is quantifying the uncertainties - those associ-

! ated with the probabilities of accident scenarios and those associated with the (off-site).consequenceestimates. This paper describes the process of perfoming an integrated uncertainty and sensitivity analysis of environmental transport and deposition models in the atmosphere. The study objectives are (1) to measure the overall uncertainty of the output variables of interest (2) to rank and quantify the inputs (or model parameters) according to their contributions to output uncertainty and (3) to compare several uncertainty approaches and ways of displaying results (e.g., stochastic and ignorance uncertainties). The analysis will also indicate modeling and research needs.

8703160234 870312 h,hRAN 849 PDR 46 A

d STATISTICAL APPROACH A variety of techniques are available to quantify ithe uncertainty t that in c plex models for assessing radiological impact upon These man include - the

)

and the en mayincludenonlinearitiesandtime-varying [ phenomena [1].

Monte Carlo [2), fractional factorial design 3), Latin hypercube s response surface [7-8) and differential sensitivity analysis (e.g adjoint [9.103) methodologies.

ble, economical to use, easy to implement, provide a capability to output distribution function and rank input variables by differen implemented onThean environmental advantages transport and properties of the Latin sequence code (called CRAC-2).

hypercube sampling techniques are:

l o

The full range of each input variable is sampled and correlatio l cients between all pair-wise input variables can be specified.

o It provides unbiased estimates of cumulative distribution fu means for model output under moderate assumptions.

ld The LHS method is a member of the class of sampling technique Monte Carlo and stratified random sampling. Furthemore, LHS has clear waste repositories [113 have applied LHS techniques. [12)andrepre-i f the recently been applied to a multicomponent ident calcu- aeroso new integrated risk code (called MELCOR) for severe reactor acc lations (in-plant andthat It is noted off-site) being developed the weather time seriesby thesampling data Nucleart Re sch Comission.

CRAC-2 is essentially a random sampling scheme without However, many other repla dl of weather bins (i.e., a Latin hypercube sampling thod in fixed at a single, best estimate value. atmospheric dispersio d sensitivity CRAC-2, will be presented in a following section. vide a approach.

We remark that one may wish to distinguish sbetween in differ uncertainty associated with modelling of physico-chemical processe ,

particular:

1.

The statistical uncertainty due to inherent random nature of the processes, and j

2.

The state (perhaps " lack-of") knowledge uncertainty.

47 i

This latter state of knowledge uncertainty may be further subdivided into model and parameter uncertainty. The parameter uncertainty is due to insufficient knowledge about what the input to the encoded model should be. This study documented herein deals with parameter uncertainty. The modeling uncertainty is due to simplifying assumptions and the fact that the models used may not accurately model the true physical process.

The process of conducting an uncertainty and sensitivity analysis starts with an identification of a set of key parameters in the model understudy. For each chosen variable, a set of quantitative information is developed regarding the range of variation, probability distribution, as well as, correlations among the variables. In our study infomation was primarily based upon expert opinion. Secondly, this infomation is used as input to the Latin hypercube samplingcode[13,14]. LHS is used to generate what is called a design matrix.

Specifically, if N computer runs are to be made of the computer code to be analyzed (e.g., the CRAC-2 dispersion model) with k parameters under study, the design matrix has dimensions N x k. Each row of this matrix contains the input valuations of the each of the chosen k parameters.-

The next step in the process involves perfoming a sensitivity analysis on the calculated results of ground (or air) concentration versus distance from the power plant, for example. geaimistodetermineandquantifythe l relative contributions of the k variable toward the output variability versus distance. This may be achieved by perfoming step-wise regression analyses on the concentration patterns simulated. Alternatively, one may perfom a regression on the ranks of the data, replacing the " raw" data values by their ranks. This may be preferred when highly nonlinear relationships are present between the model outputs and inputs. Both graphical analyses and statistical distribution fitting procedures may also be extremely useful in identifying patterns in the data.

WEATHER DATA SAMPLING METHOD The transport and dispersion of material associated with a postulated accidental release of radioactivity at a nuclear power plant depends upon the weather conditions at the start of the release from containment through a period of tens of hundreds of hours. The CRAC (Calculations of Reactor Accident Consequences) model which was developed during 1975 Reactor Safety Study (WASH-1400)[15] and subsequently modified (and called CRAC-2)[16] to estimate the offsite health and economic portion of reactor risk to society has several weather data sampling capabilities. The consequence analysis is typically given in tems of a CC0F (cumulative complementary distribution I

function) representing the range of weather scenaries. The weather data is usually taken from annual records of onsite meteorological tower measurements or nearby National Weather Stations. The data required for CRAC and CRAC-2 consists of hourly averages of windspeed, atmospheric stability class and precipitation. The recomended "importance" sampling method in CRAC-2 was i

l l

48

j i

Tat designed to reduce the variability due to either 1) random 2) stratified random samplingorstratifiedsamplingtechniques[17]. For example, risk curves o generated with CRAC used a stratified sagling technique that selects data every 4 days + 13 hours1.50463e-4 days <br />0.00361 hours <br />2.149471e-5 weeks <br />4.9465e-6 months <br /> in an attempt to account for diurnal, seasonal and 4-day weather cycles. That is, CRAC for WASH-1400 selected 91 weather n sequences to represent the 8760 hours0.101 days <br />2.433 hours <br />0.0145 weeks <br />0.00333 months <br /> of annual data. Such a sampling scheme is sensitive to the chosen start time and can miss infrequent meteorological conditions (including rain and windspeed slowdowns) with possible corresponding high dose-rate consequences. .:

In CRAC-2, the annual weather data is sorted into " weather bins" before ] -

C1.

performing any consequence calculations. The weather bins are defined in Table q a

1. For example, the first bins refer to whether it begins to rain at a certain distance from the reactor or that the windspeed drops. If neither of these conditions occur the weather sequence is categorized by the stability and J windspeed at the start of the accident. Also, a probability is assigned to each bin according to its frequency of occurrence (directional information is "

available for the nonrain cases only). Four random samples with equally spaced ~

l

, probability are typically chosen from each weather bin to perform the atmo-spheric transport and deposition calculations in CRAC-2. ,

ATMOSPHERIC DISPERSION MODEL J 9

Both CRAC and CRAC-2 use similar atmospheric transport and dispersion [r models - a standard Gaussian-plume fomulation[18-20]. Currently, the Gaussian plume atmospheric dispersion model is employed in most consequence-modeling (2) '

codes for the following reasons: (1) economy in tems of computer time and o -

general lack of availability of the meterological parameters necessary for 4 input to more complicated models. It is recognized that the simple Gaussian model is not appropriate for complex terrain (e.g., mountain-valley) or land (sea) breeze flows. Assuming the material is reflected at the ground, the -

ground-level, time-integrated concentration for a source of strength Q is given  ;

by

((x,y,0)= 0 exp [ - y2 + -h 2 g(1)

T T (m) y 0"a ( *) W 2 C"[ f *) 2C['(4 l~ where distrib5(x) tions, and g;ip(x),

re ectively, arethe standard functions of thedeviations downwind distance, of the crosswind

x. u is and ve ,

l l themeanwindspeed,andhjsthesourcereleaseheight. If Q is in curies, I the units of M are Ci-sec/m . In CRAC 2 and CRAC, equation (1) is simplified l

by replacing the Gaussian crosswind profile ( y direction) with a rectangular (or " top hat") function of width 3 gj; i.e., in equation (1) the term 1

49 J

--c-,-

, - , , , - - - , s _y, , _ _ _ - . , , - - - _ . - _ , _ _ , , , -,,-

,___w _--_, n,-

Table 1: Weather Importance Bins Applied to One Ye'ar 'of New York City Meteorlogical Data Weather Bin Definitions R - Rain starting within indicated interval (miles).

5 - Slowdown occurring within indicated interval (miles).

A-C D E F - Stability categories.

1(0-1), 2(1-2), 2(2-3), 4(3-5), 5(GT 5) - Wind Speed intervals (m/s).

Number of Classification (C) Weather Bin Sequences Percent C1 1R 0) 697 7.96 2R 0-5) 12 .14 3R 5-10) 62 .71 4R 10-15 102 1.16 5R 15-20 75 .86 6R 20-25 67 .76 7R 25-30 61 .70 C2 8S 0-10) 24 .27 95 10-15) 16 .18

.21 10 S 15-20) 18 11 S 20-25 14 .16 12 S 25-30 18 .21 C3 13 C 3 168 1.92 14 C 4 892 10.18 C4 15 D 1 0 0.00 16 D 2 61 .70 l) 17 0 3 226 2.58 18 D 4 948 10.82 19 D 5 3325 37.96 C5 20 E 1 0 0.00 21 E 2 27 .31 22 E 3 167 1.91 23 E 4 682 7.79 24 E 5 270 3.08 C6 25 F 1 0 0.00 26 F 2 116 1.32 l 27 F 3 310 3.54 28 F 4 402 4.59 29 F 5 0 0.00 W 1UO.EI 50 l

l.

i 4

1 exp I -y 2

) (2) role (fi g(x) y 2 k(x) ) )

1

- in C is replaced by 1/(3 ry(x)). With this substitution, equation (1) becomes

'X(x.o) = 0 exp[ -h 2 (3) g,,,

k com -

3/2 N gy(x) gg(x)u 2 6"2 (x) )

J The amplitude of the top hat is 0.836 of the Gaussian peak: however, the area  ;

under the top hat curve is identical to the area under the Gaussian crosswind a trit profile. cho!

be !

For each start-hour selected by the meteorological sampling technique, the CRAC 2 dispersion model uses the subsequent meteorological conditions to pre-dict the dispersion and transport of the released cloud of radtoactive equ' material. The sequence of hourly recordings is used to account for changing '

aa weather conditions; 1.e., wind speed, atmospheric stability, and precipitation 4 san may change during plume passage. The wind direction, however, is assumed to be

invariant. T in Based on the windspeed in each hour, the stability, windspeed, and accu. cap

,' . mulated precipitation are assigned to all spatial intervals which the plume passes during the hour. If the windspeed for an hour is not sufficient for the i plume to fully traverse an interval, the windspeed, stability and accumulated '

of

! precipitation are averaged for all hours the plume is within that interval (the the.

j average of A and C stability is B, of A and B is 8). Values of r (x) and g,(x) ob!

i are calculated for each spatial interval using Pasquill-Gifford c rves as bir The empirical, best-fit functions for the curves given set :

5 Successive growth rates of r (x) providedinTurner[21]22]areused.foreachspatialinterva by Martin and Tikvart[

1 nut ,

us spati Ci'

i interval by calculating the virtual-source distance at the current stability i Si class necessary to give that value, and extrapolating growth to the end of the current spatial interval.

Equation (3) is modified for radioactive decay of each isotope during downwind transport, including build-up of any daughter products, e.g.,

ra -

Q(x) = Qexp(-Xx/u) rt CC where his the radioactive decay constant. Further modifications of ecuation (3) are incorporated in CRAC 2 to account for the effects of (1) duration of i 4 51 I

i.

I 1

- - - - - > - . ...,,,n,.-- n_ . , , . - - - - - - . . - . ,.,---,,,-,,---,-__,---.n . . _ - , . , _ , _ . , _ . , , - - - ,-,----n,-------,- ,.n .-,,r

1 l

release, (2) surface roughness, (3) mixing layer depth, (4) dry and wet removal l processes, (5) building wake, and (6) plume rise caused by sensible heat buoyancy. Refer to reference [16] for a sumary of these effects and submodels l

in CRAC 2.

! Computer Packages In order to perform the analysis six computer packages were either used, developed or modified. By using an input and output file for each of these computer packages, they were run independently. A brief description of these l

computer packages will now be given.

Latin Hypercube Sampling (LHS) l l The LHS computer packages [13,14] were used to generate probability dis-tributions for each of the eight selected independent variables. For each chosen parameter, probability density functions and any rank correlations must l be specified so that the design matrix can be generated.

1 For a N x k design matrix, each of the k parameters is divided into N equi-probable intervals. Then a random sample is chosen from each interval in a manner that preserves the individual probability density function. In this sanner, k N-tuples (an N x 1 vector) of input settings are detemined.

, The parameter LHS input matrix for our CRAC 2 dispersion study is given in Table 2. The quantitative infomation was mainly used for demonstration of capability purposes and based primarily upon expert judgement.

CRAC 2 has the weather data sampling option of looking at a discrete set

, of weather bins as previously discussed. We studied the stabilization within these bins; that is, we performed two calculational cases: the typical four observations per bin and twelve observations per bin. Besides the weather l 3 bins, eight other independent variables were selected to perfom the CRAC 2 l ,

sensitivity and uncertainty analysis. The sample size was detemined by the number of observations per bin over the nonzero (e.g., 25 for the New York City data set used) weather bins. Therefore, for the two cases, the sample sizes were 100 and 300, respectively.

CRAC 2 The CRAC 2 computer package [16] was modified to handle the eight indepen-dent variables as random variables, with values given by the output of LHS, rather than as deterministic parameters. An output file was created to collect 2400(7200 for case two) records of the nine independent variables repeated for each distance, the ground and air concentrations, and the 24 corresponding distances chosen.

52

- _ _ . _ uan_ _ _

_ . _ _ w_. z. .

. l Table 2: CRAC 2 Variable Ranges and Distributions (Used As Input to LHS code 1 for an SST 1 Release Scenario)

C

, a Range Distribution Units o Variable 7 t 104 - 3.6 x 10 Loguniform Cal / see w Sensible Heat ,

0 Uniform p = .5 hour 1 Duration of Release .25 - 1.0 C 1.0 - 3.0 Uniform p = .5 ,

Loguniform m/sec Dry Deposition Velocity 10~3 - larl

-1 10 10-3 Loguniform sec Rain Coefficient (mmbr~I) S a

8 Rain Intensity Exponent 0.75 - 1.0 Uniform 1 l

Uniform m Mixing Height 250. - 3000.

Loguniform 1 Sigma-Y Multiplier .333 - 3.0

.333 - 3.0 Loguniform 1

{

Sigma-1 Multiplier t Weather Bins (1.-2.2.-3..... Uniform 1 28.-29.) I 9

1 4

i

,i i

t 53 l

1

Data Base Program (DBPR) 3 The DBPR computer package was written to prearrange the data file output created by CRAC 2. The file was partitioned on six weather classifications and, within each of these classifications (Table 1) and, the records were rank ordered on distance. For each classification and distance (144 sets in total),

the minimum, mean, maximum, and standard deviation were computed and tabled as well as the overall minimum, mean, maximum, and standard deviations. This ,

output file wLs created to serve as input to the BMDP step-wise-regression computer package and the Weibull computer program.

BIOMEDICAL PROGRAMS (BMDP) - PROGRAM 2R The UCLA biomedical computer package [233 for performing a step-wise re-gression was used to get an importance ranking of the eight independent vari-ables. For these runs the dependent variable was the logarithm of ground exposure risk.

Weibull Distribution (WHYD)

To check within and between variatiors of weighted ground concentrations over the six classifications over the 24 distances,144 Weibull distributions [24,25] were fitted to corresponding sets of ground concentration values. WHYD was developed and programed within NRC by using Menon's method of moments to estimate the two-parameter Weibull distribution. Using an iterative scheme Pittman's estimate was used to estimate the location parameter.

Tell-A-Graph The fitted Weibull distributions and data were plotted using the Inte-gratedSoftwareSystemCorporationgraphicspackage[26]inordertodoavisual analysis of the within and between variations of ground exposure risk over the six classifications over the chosen distances.

CALCULATED RESULTS For illustrative purposes, calculations have been perfomed using CRAC 2 and the LHS design matrix input (based on Table 3) and assuming a severe l reactor accident scenario [27] called siting source tem one (SSTI). This corresponds to the largest releases postulated in the Reactor Safety Study [15]

where essentially all installed safety features are assumed to fail. The uncertainty occurrence of inthatthe ground weather concentratiog) class weighted (Ci/m as described by thedistributions by Weibull annual frequency of at 1.25 miles is displayed in Figure 1. The Weibull distributions are 54

+~ -

f _,

.o.---

4 i

l l

l I --

! UnceduWy Analysis of CRAC2 t

% g,N i o.s - .N,\

! w f

N h

o.s-u t.egend a ct.p.121 g x c2.o= tas

,,_ g%

] K j o cle .t.22 j s ca.o=12s i

N c5.p=_t.n_,

e.2-M cs,p= t.s-z i ...,...,h"1(Th ' ~ ~#l'"i(f w #* , W C

(c

.4

, M . . . , . .W. , 3W* . . ., . . . _omund ce~ .:..::

!)

i-I F) of Ground Cumulative Complementary Distributionf Each FunctionWeather (CCD

[f..

..+ Figurel: Concentration Values (Weighted by Annual Frequency o SSTI Release

")

~

J Class))At (C1/m .

D = 1.25 miles from , --.. -

the .Reactor c...... c. -.. ca. give an

,e

• -- -.- e-.e.....-.

j -

'~'4- . . . . . , , , _ , , ,

k @

k' b L h shown for each weather class designated C1 through C6 (corresponding to rain,The windspeed slowdowns, and stability classes C, C, E ar:d F, respectively).

graphed median values indicate, for example, that the D and F weather classes would result in a higher probability of exceeding a given level of ground deposited radioactive exposure risk than the rain class. For the rain and F -

stability weather classes the weighted ground concentratior.s range over several orders of magnitude less than (between CC0F values of 0.1 and 0.9, say) the ranges shown for the other. weather classes. Several step-wise regression analysis results are sumarized in Table 3. These were perfomed on 0

.c logarithmically transfomed ground concentration values by weather class.

E Regression analyses were also perfonned on transfomed ground concentration 2 values at each selected distance as wel1.

5 The calculated R values provide a measure of importance of each variable to the uncertainty and may be used to rank the variables. It is also seen in s Table 3, for example, that approximately half of the variab{lity in weather

" class F has been accounted for (based on interpreting the R value). The The relative 2 contributions of each of the variables studied are shown.

" rankings of the important variables depends upon the weather class'under h consideration. Further analysis could consider including interaction tems or i other independent variables. It is remarked that there are several ways to R display uncertainty - of various types. It may be useful to display uncertainty due to the stochastic weather data as well as uncertaintySeedue to i

[o both meterological data variations and parameter lack-of-knowledge.

e Figure 2 which shows the mean and peak values of these two types of uncertainty The E in the weighted ground concentration versus distance fror release point.

3 mean values of the distributions differ by approximately 3 to 5 while the

" worst-case" values are (not surprisingly) about an order of magnitude apart.

[c CONCLUSIONS E

The process of conducting an integrated uncertainty and sensitivity anal-ysis was described and applied to the CRAC 2 atmospheric transport and deposi-tion model using a suite of mainly existing codes. These include codes to 3 implement Latin hypercube sampling of prescribed distributions and corre-3 r lations for the variables understudy, to fit statistical distributions andAll

- graphically display the infomation and to perfom regression analyses. The 3 in all, the experience was worthwhile and results infomative and useful.

g , present uncertainty and sensitivity study may be built upon to encompass the y y complete radiological consequence assessment, while still focusing on the dis-3 p play of different uncertainties. Also, other models that specifically address complex terrain and land-sea breeze flows should be examined along with avail-able experimenta1' data.

l l

i 56

I -. . . . . . . . . . . .

ff i

l Table 3: Sensitivity Analysis of CRAC 2 Ground Exposure Results Vic sus We l

' results; numbers in parentheses indicate parameter ranking)

Rain Duration Weather Dry Rain Sigma-r of Sensible Mixing l CRAC 2 Intensity Sigma-Y Height Frequency Deposition Intensity Release Heat l

Weather Velocity Exponent Coefficient Multiplier Multiplier Bins (Percent) 0.42(3) l 0.44(8) 0.44(7) 0.44(6) 0.44(5) 12.29 0.44 (4) 0.38 (1) 0.40(2) l Rain l (Bins 1-7) 0.28 (1) 0.43(7) 0.43(5) 0.43(4) 0.43(6) 0.43(8) 0.39(3)

Windspeed 1.03 0.35(2) l i Slowdowns

! (8-12) 0.86(7) 0.45(1) 0.83(4) 0.80(3) 0.75 (2) 0.86(5)

C Stability 12.10 0.86 (6) -

l (13.14) 0.66(5) 0.69(8) 0.68(7) 0.58 (3) 0.68(6) 0.64(4)

U D Stability 52.06 0.33(1) 0.50(2)

(15-19) 0.68(7) 0.64(4) 0.39(1) 0.59(2) 0.67(6)

E Stability 13.09 0.66(5) 0.68(8) 0.61(3) i (20-24) 0.70(4) 0.67(2) l 0.71(5) 0.72(6) 0.74(8) 1 F Stability 9.45 0.74 (7) 0.7 (3) 0.58 (1) ll (25-29) l 4

m m A _ _

4

i l

CRAC2 Ground Exposure Risk n rm.. A uncerkinty Ardi c a Given SST1 Rabw==

t -:

,j d

1 to l

fa e

y e w

%.g

! U 0.1 g* -

0.01  %.%, x rout * ,

1 u O MEAN," ,,

es PEAK **

) ,

l 0.001 , , , . . .

o 5 to 15 20 25 30 35 Distance (miles)

Figure J2: Mean and Peak Values for Stochastic Variation (due to Meteorology) and Full Latin Hypercube (Stochastic and Parameter Lack-of-Knowledge)

Uncertainty Analysis of Ground Exposure Risk Versus Distance From Reactor Given an SST1 Release.

• Stochastic V' ariation

• • Stochastic and Lack-of-Knowledge Variation

l i

' . . i

l. 1 REFERENCES '

o 1.

D. C. Cox and Paul Baybutt, Risk Analysis, Vol. 1, No. 4, p. 251, 198 .

2. Halton, J. H. SIAM Review, 12,,1(1961).

l 3.

Cochran, William G., Sampling Techniques, John Wiley and Sons, 19 4.

Cranwell, R. M. and Helton, J.131-144 C., " Uncertainty Analysis for Geologi Disposal of Radioactive Waste," pp.

on Uncertainties Associated with the Regulation of the Geologi of High-Level Radioactive Waste, Gatlingburg, Tennessee, March ,

1981, CONF-810372, ed. 0.C. Kocher, 5.

Iman, R. L. and W. J. Conover "The Use of the Rank Transfonn in R gression," Technometrics 21, p. 499-509, 1979.

l

! 6. McKay, M. D., Conover, W. J. and Beckman, R i

from a Computer Code," Technometrics 21, p. 239-245,1979.

7.

R. H. Myers, Response Surface Methodology, Allen and Bacon, In (1971).

8.

P. Baybutt, D. C. Cox, and R. E. Kurth, " Methodology for Uncerta Analysis of Light Water Reactor Meltdown Accident Conseq Report from Battelle Columbus Laboratories to U.S. Nuclear R Connission (May 1981).

4 Oblow, E. W.,

Nuclear Science and Engineering, Vol. 6_5,, p.187,1978.

9.

10.

D. G. Cacuci, C. F. Weber, E. M. Oblow, and J. H. Marable, " S i Theory for General System of Nonlinear Equations," Nuclear Engineering 75,88(1980).

11.

J. E. Campbell, R. L. Iman andModel Transport M. Reeves, " Risk Methodol Sensitivity Analysis,"

3 Disposal of Radioactive Waste:Sandia National Laboratories. Albuquerque, g NUREG/CR-1377 SAND 80-0644, New Mexico, l J. C. Helton, R. L. Iman, J. D. Johnson, C. D. Leigh, " Uncert r

I

12. Sensitivity Sandia Analysis of a Model for Multicomponent Aerosol D National Laboratories Albuquerque, New Mexico.

SAND 84-1807, i

.I Iman, R. I. and M. J. Shortencarier, "A FORTRAN 77 Progra a

e

13. Guide for the Generation of Latin Hypercube and Random S with Computer Models," U. S. Nuclear Regulatory Commission, HUREG/CR-3624, March 1984.

59 c _. -

x , - . .. _ _

n...=

e i

e

. i i

14. Iman, R. L. and Davenport, J. M., "An Iterative Algorithm to Produce a 1 Positive Definite Correlation Matrix from an Approximate Correlation 8101376, Ma- i.

trix (With a Program User's Guide)." Technical Report SAND ;7 Sandia National Laboratories Albuquerque, New Mexico, 87185.

• 15.

U.S. Nuclear Regulatory Commission, " Reactor Safety Study, Appendix VI: 1 Calculation of Reactor Accident Consequences NUREG-75/014,1975. l'

?

Ritchie, L. T., J. D. Johnson, R. M. Blond, " Calculations of Reactor

16. U.S. Nuclear Accident Consequences Version 2, CRAC 2: Computer Code g Regulatory Comission, NUREG/CR-2326 February 1983.

Ritchie, L. T., Aldrich, D. C. and Blond, R. M., " Weather Sequence 38, 1981. ,

l 17.

Sampling for Risk Calculations," Trans. Am. Nuc Soc., Vol.

18.

U.S. Nuclear. Regulatory Commission, " Environmental Transport and Conse-  ;

quence Analysis," Chapter 9 in "PRA Procedures Guide." Vol. 2, ,

NUREG/CR-2300, 1982. '

19. Gifford, F., " Atmospheric Dispersion Models for Environmental Pollution ,f Applications," in Lectures on Air Pollution and Envircrvnental Impact Analysis, D. A. Hanghen, ed. AMS, Boston,1975.

20. Slade, D. H. (Ed.) Meterology and Atomic Energy, U.S. Atomic Energy .

Comission TID-24190,1968. -

Turner, D. B., " Workbooks of Atmospheric Dispersion Estimates," U.S. EPA '

21. '

Office of Air Programs, Publication No. 999-AP-26, Research Triangle Park, NC, 1970. l 22.

Martin, D. O. and J. A. Tikvart, "A General Atmospheric Diffusion Model for Estimating the Effects of Air Quality of one or More Sources,"

presented at the 61st Annual Meeting of the Air Pollution Control Asso- .

ciation.

23. Dixon, W. J. (Chief Editor), BMDP Statistical Software, Berkley, >

University of California Press, 1981.

Sarah, A. E. and B. G. Greenberg, Contributions to Order Statistics, John 24 Wiley and Sons, Inc.,1982.

25. Cohen, A. C. , Jr. , Technometrics , p. 579, 1965.

26.

Tell-A-Graf User's Manual, Version 4.0, Integrated Sof tware System Corpo-ration (ISSCO), San Diego, California, 1981.

27.

Blond, R. M., M. Taylor, T. Margulies, M. Cunningham, P. Baranowsky, R.

Denning and P. Cybulskis, "The Development of Severe Reactor Source Tems: 1957-81,"

November 1982.

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