Uncertainty and Sensitivity of Accident Consequence Assesments on by dfhdhdhdhjr

VIEWS: 5 PAGES: 33

									        An Application to Nuclear Safety
- UA/SA Using An Accident Consequence Assessment Code -




                       T. Homma
          Japan Atomic Energy Research Institute


           SAMO2004 Venice, Sept 12 - 17, 2004
                                                        Safety Goal for Nuclear Installations
                                                    Level 3 PSA for a reference plant
                                                    due to internal accidents                   The NSC of Japan issues the
                                                                                                interim report on safety goal
                                            10 -5                                               (2004)
Average individual risk (reactor year -1)




                                            10 -6                                                  Individual early fatality risk: the
                                            10 -7               Safety Goal (draft): -6
                                                                                    10              expected (average) value for
                                                                                                    average individual early fatality
                                            10 -8
                                                                                                    risk near the site boundary due
                                            10 -9
                                                                                                    to nuclear accidents will be less
                                            10 -10              Cancer Fatality                     than about 1×10-6 year-1
                                            10 -11              Early Fatality
                                                                                                   Individual latent cancer fatality
                                            10 -12                                                  risk: the expected (average)
                                            10 -13                                                  value for average individual
                                            10 -14                                                  latent cancer fatality risk in the
                                            10 -15                                                  some region from site boundary
                                                 0.1             1               10       100
                                                                                                    due to nuclear accidents will be
                                                       Distance from the release point (km)         less than about 1×10-6 year-1


                                                                                                                                      2
                                                      Two Types of Uncertainty
Stochastic (aleatory) Uncertainty
                                  known as randomness or variability of the system under study
                                  variability in environmental conditions (e.g. weather condition)
                                  physical variability will not decrease
Subjective (epistemic) Uncertainty
                                  results from the existing state of knowledge
                                  modeling uncertainty and input parameter value uncertainty
                                  as we gain more knowledge, uncertainty will decrease

                                      Stochastic Uncertainty                                                                     Subjective uncertainty
                                 1.0E+00                                                                               1.0E+00




                                                                                        Conditional Probability >= X
  Conditional Probability >= X




                                 1.0E-01                                                                               1.0E-01




                                 1.0E-02                                                                               1.0E-02



                                                                                                                                       mean
                                 1.0E-03                                                                               1.0E-03         95%
                                                                                                                                       5%


                                 1.0E-04                                                                               1.0E-04
                                      1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05                                       1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

                                                 Number of Consequences, X                                                             Number of Consequences, X
                                                                                                                                                                              3
                The Problem Settings
 How do we deal with the stochastic uncertainty
  (weather conditions) in accident consequence
  assessments and how much is the statistical
  variability?

 How much of the overall uncertainty about individual
  risk is attributable to stochastic uncertainty and how
  much to parameter uncertainty?

 What are the main contributors to uncertainty in
  individual risk of early and latent cancer fatality?

                                                           4
              OSCAAR Code System
 Off-Site Consequence Analysis of Atmospheric
   Releases of radionuclides
                                     CURRENT
           Meteoro-
            logical                                 HEINPUT
             data
                                     Population
         Meteorological   DOSDAC     Agricultural
                                        data          Health
           sampling
                                                      effect
              MS
                                                        HE
                            Early     Protective
Source    Atmospheric     exposure    measure
 term     dispersion
                           EARLY         PM
          Deposition                                 Economic
                                                       loss
             ADD
                           Chronic                    ECONO
                          exposure     HINAN

                          CHRONIC


                                                                5
       Atmospheric Dispersion and Deposition
Multi-puff Trajectory Model
                                  N              N                                    ri2              ( zi  z 0 ) 2 
                                    (r, z, t )  (2 )
                                                            Qi (t )
              ( x, y, z, t )                                                   exp            exp                
                                                                   r2,i z ,i          2 r2,i              2 z ,i 
                                         i                  2/3                                                  2
                                  i 1           i 1                                
                                                                                                      
                                                                                                                        
     Dry and wet deposition                                                                          Z
                                                        d ( x, y )  vd  X ( x, y )     X ( x, y, z )dz
                                                                                                      0




                                                                                                                             6
                      Dose Calculation Models
   Total dose for a specific organ from different exposure pathways
                                RF j : Reduction factors (shielding and filtering factors)
             
  Dtot ,i  RFj  DCi , j   j DCi, j : Dose coefficients
           j
                                 j : Time-integrated concentration, contamination,
    i: organ                                  intake
    j: pathway                                                            Cloud




Atmospheri       Atmospheric
                                                                         Inhalation    Dose to man
 c release        dispersion




                                                                       Contamination




                                 Deposition                               Ground




                                                                        Resuspension




                                                         Foodstuff
                                                       contamination     Ingestion

                                                                                                 7
         Health (deterministic) Effects Model
Early and Continuing effects(Early mortality and
  morbidity)
    Hazard function (two-parameter Weibull function) approach
    Early fatal effects comprise haematopoietic, pulmonary, and
     gastrointestinal syndrome. Those depend on the level of medical
     treatment received
    Effectiveness of a specified dose for induction of early effects
     depends on dose rates.


 r  f m  r m  f s  r s  fi  r i                                     1


                                           
                                                Risk of Early Fatality
                                                                         0.8
 r  1  exp  ( H  H  H )
  m
                         1
                          m      m
                                 2
                                        m
                                        3
                                                                         0.6
                                 
                  
                   d                                                                             minimal
 H  ln 2           ) dt 
                                                                         0.4                      supportive
             D50 (d                                                    0.2
                                                                                                  intensive

                      
 D50 (d )     1 / d                                                  0
                                                                               0   2   4      6     8          10
                                                                                       Dose(Gy)

                                                                                                                8
         Health (stochastic) Effects Model
Late Somatic Effects (Cancer mortality and morbidity)
    Linear or linear-quadratic dose-response model and DDREF
                              1        
                         R        bD cD
                            DDREF      
    For estimating the life-time risk in the population, the absolute or
     relative risk projection models are available
    Data of Hiroshima and Nagasaki
       – Reassessment of the radiation dosimetry
       – Life span study on atomic bomb survivors
                                                          Life time risk(/104 person-Gy)
           Effect       Projection Latency    Plateau    OSCAAR NUREG             ICRP60
       Leukemia         Absolute   2         39          59         49         50
       Bone cancer      Absolute   2         25          1.1        4.5        5
       Breast cancer    Relative   10        Life-time   31         54         20
       Lung cancer      Relative   10        Life-time   110        78         85
       G.I. cancer      Relative   10        Life-time   230        168        240
       Thyroid cancer   Absolute   5         Life-time   4.6        7.2        8
       Skin cancer      Absolute   10        Life-time   0.4        -          2
       Other cancers    Relative   10        Life-time   56         138        90
       Total                                                 490        499       500      9
                Meteorological Sampling
Aims of Meteorological Sampling
   Strong dependence of the magnitude of the consequences on the
    weather after an accident
   Huge computer resources using a full year of hourly data
   Select a representative sample of weather sequences which
    adequately produce the range of consequences
Sampling Techniques
   Random sampling of the specified number of sequences
   Cyclic sampling (sequences are selected with a set time interval
    between them)
     – but, these tend to sample the commonly occurring groups
        frequently, while overlooking more unusual sequences
   Stratified or bin sampling (sequences are grouped into a number of
    categories, which give rise to the similar consequences)


                                                                         10
    General Consideration for Met. Sampling
Completeness
     The consequences calculated would reflect the full spectrum of the
      consequences related to the postulated accident under investigation.
Consistency
     The parameters selected for classification of weather sequences and the
      sampling scheme itself should be seamlessly associated with the models,
      parameters and methods used in the code system.
Stratification
     The sampling scheme could divide the entire set of meteorological
      sequences in such a way that the members in each single stratum or
      group would be very similar.
Practicability
     A practicable number of samples should be predetermined according the
      models used in the consequence assessment code.
Optical Allocation
     A fixed number of samples need to be optically allocated among the
      groups in order to “maximize” the precision of consequence assessment.

                                                                                11
 Sensitivities of Early Fatality
to Meteorological Parameters




SPD0 : initial wind speed                STABi:mean stability to i km
STPi : travel time to i km               DURi : period of rain to i km
I.SPDi : Inverse of wind speed to i km   RAINi : total rainfall to i km

                                                                          12
Classification of New Sampling Scheme

            Wind direction                           9 groups
   Wet             Rain < 5mm                          G1
(to 10km)          Rain ≧ 5mm                          G2
                     Travel time
                      (to 20km)        Unstable      Neutral    Stable
    Dry
(to 10km)             < 2.5 hr           G3            G6        G9
                      2.5 - 5 hr         G4            G7       G10
                       5 hr ≦            G5            G8        G11


             11 Groups x 9 (wind directions) = 99 Groups


                             144 Weather sequences


                                                                         13
                                        Performance of New Sampling Scheme
                                        New stratified sampling scheme                                                      Cyclic sampling scheme
                              100                                                                                 100
Conditional Probability, ≧C




                                                                                    Conditional Probability, ≧C
                              10-1                                                                                10-1

                                         1000 sets of 144 sequences
                                   -2
                              10                                                                                  10-2


                              10-3                                                                                10-3
                                                        8760 sequences

                              10-4                                                                                10-4
                                 10-3       10-2     10-1     100     101     102                                    10-3    10-2    10-1     100     101       102
                                           Early Fatalities (normalized), C                                                  Early Fatalities (normalized), C

                               The statistical variability of the probability distribution of the early health effect is not
                                large and the performance of this scheme is better than other conventional schemes.
                               The advantage of the stratified sampling scheme is to give the rare cases of
                                catastrophic health effects when we use the same number of sequences.
                                                                                                                                                                 14
          Steps in UA/SA on Input Parameters
    Identify uncertain model parameters
    Assign upper and lower bounds,
     distribution, and correlation
1. PREP
    Perform parameter value sampling
                                               パラメータ X
                                              ParameterX1 1       パラメータ X
                                                                 ParameterX2 2               パラメータ X
                                                                                            ParameterXKk
       – Simple random sampling
       – Latin hypercube sampling
       – Sobo'l quasi-random sampling
2. Run OSCAAR with the Sampled Input                      Y  f ( X 1 , X 2 ,  , X K )
   Values
3. SPOP
    Estimate output distribution functions
     (UA)
    Examine relationships between input
     and output variables (SA)                                  Prediction Y
                                                                   予測値Y




                                                                                                     15
             Expert Judgement Elicitation
Joint EC/USNRC project 「Uncertainty Analysis of Accident
   Consequence Models for Nuclear Power Plants 」(1993-1996).

    Objectives : to develop credible and traceable uncertainty
      distributions for the respective ACA code input parameters.

    Two important principles for the application of formal expert
      judgement elicitations:
        – The elicitation questions would be based on the existing
          models used in their codes such as COSYMA and MACCS.
          (A library of information can be of use to other models and
          codes.)
        – The experts would only be asked to assess physical quantities
          which could be hypothetically measured in experiments.


                                                                          16
       Expert Judgement Elicitation (Cont.)
   Uncertainty distributions for physically observable quantities
    were provided by experts at each expert panel formed for the
    following areas of codes: atmospheric dispersion, deposition,
    external doses, internal dosimetry, food-chains, early health
    effects and late health effects.
   Combine these uncertainty distributions into a single joint
    distribution and translate distributions over physically observable
    quantities into distributions on code input parameters.

    Information about 5%、50% and                                                 Uncertainty distributions of
    95% quantiles on the uncertainty                                             the code input parameter values
    distribution from expert judgement
                                                                                                   Parameter A
 Expert A                                  Single joint distribution
                                                                            Obtain
                            Combine
                                                                       distributions on
Expert B                 the uncertainty                                                             Parameter B
                                                                          code input
                          distributions
                                                                          parameters

 Expert C                                                                                           Parameter C


                                                                                                                   17
   Target Variables and Elicitation Variables
 Case 1: code input parameters correspond to measurable quantities
  (e.g. deposition velocity)
 Case 2: some analytical functional dependence (e.g. dispersion
                 x yP  )x ( y 
  parameter             )          yQ



 Case 3: some numerical relationship (e.g. retention of material is
  modelled using a set of first-order differential equations with code
  input parameters)

          A                             kAB 、kAC :transfer coefficient(target variable)
   kAB            kAC                   Yi、Zi :retention of material in compartments, B and C
                                        (elicitation variable)
    B           C

                 Case 2 and 3 need probabilistic inversion


                                                                                                18
                Example for Dose Coefficient
Metabolic model of Caesium

      ST                                           Quantile information from experts

                            Blood                    Retention of Cs-137 in Body tissue from a
      SI                                                            unit intake
                 0.1    TBlood      0.9                            5%          50%         95%
                                                     1 day    8.70E-01    9.62E-01     9.92E-01
     ULI           Body               Body
                                                    1 week    7.45E-01    8.59E-01     9.43E-01
                 tissue A           tissue B
                      TBodyA              TBodyB    1 month   5.45E-01    7.24E-01     8.93E-01
                                                     1 year   2.38E-03    6.48E-02     2.64E-01
     LLI
                                                    5 years   1.21E-10    1.08E-05     6.30E-03
                          Bladder

 In internal dosimetry panel, 8 experts were asked about the retention of
 materials in the human body.
 Estimate the distributions of the biological half life TBlood,TBodyA and TBodyB
 from the distributions of the retention of Cs-137 in Body tissue from a unit
 intake by using probabilistic inversion technique.
                                                                                             19
                        Result of Probabilistic Inversion
Distributions of the target variables obtained from probabilistic inversion
                                                           5%
                                                          5%値         50% 95%値
                                                                      50%値 95%
                                              1.0




                                       累積確率
                                              0.8                          ●            ICRP値:0.25
                                                                                        ICRP




                                        CDF
                                              0.6                    394
                                              0.4
                                              0.2
                                              0.0
                                                1.0E-05         1.0E-02           1.0E+01            1.0E+04
                                                           Biological half life TBlood (d)
                                                      5% 50% 95%値
                                                   5%値 50%値 95%                                          5% 50% 95%値
                                                                                                       5%値 50%値 95%
       1.0                                                      Bloodの生物的半減期(d)
                                                                          1.0
       0.8                                                                        0.8
累積確率




                                                                           累積確率
       0.6                                                                        0.6         ICRP値:110
 CDF




                                                                                                               ●




                                                                            CDF
                                                                                               ICRP
       0.4                                                  1 8 .5                0.4
       0.2                     ICRP値:2.0
                              ICRP                                                0.2                               6 .3
       0.0                                           ●                            0.0
         1.0E-04                      1.0E-01                   1.0E+02             1.0E-01          1.0E+01        1.0E+03         1.0E+05
                    Body tissue Aの生物的半減期(d)
                   Biological half life TBodyA (d)                                                Biological half life TBodyB (d)
                                                                                                 Body tissue Bの生物的半減期(d)

              Comparison of distributions of elicitation variables
                                              Fraction of amount reaching blood retained in whole body
                   1 day                       1 week                1 month                    1 year                      5 years
               DM           Pred.          DM         Pred.       DM         Pred.         DM          Pred.             DM        Pred.
    5%       8.70E-01      2.86E-01      7.45E-01 7.45E-01 5.45E-01 5.45E-01 2.38E-03 2.38E-03                         1.21E-10 1.20E-10
   50%       9.62E-01      9.62E-01      8.59E-01 9.26E-01 7.24E-01 7.24E-01 6.48E-02 2.56E-02                         1.08E-05 3.18E-07
   95%       9.92E-01      9.92E-01      9.43E-01 9.69E-01 8.93E-01 8.93E-01 2.64E-01 4.71E-01                         6.30E-03 3.33E-02
                                                                                                                                              20
       Uncertainty Distribution of Dose Coefficients
                          Uncertainty distributions of the biological half lives
  1                           1                                        1


                                                                                                                   Rank correlation coefficients
 0.5                        0.5                                       0.5
                                                                                                               +   extracted from the distribution
  0                           0                                        0
                                                                                                                   among target variables
                                                                      1.0E-01          1.0E+02       1.0E+05
 1.0E-05   1.0E-01   1.0E+03 1.0E-04         1.0E-01        1.0E+02




Calculate inhalation and ingestion                                                                         ICRP metabolic models
dose coefficients.                                                          DSYS                                    +
                                                                                                              Dosimetry data


Uncertainty on effective dose coefficient for Cs-137 from ingestion
                                       1.0
                                       0.8
                                                                                                        ICRP :1.3E-8
                              累積確率




                                       0.6                                              ●
                               CDF




                                       0.4                                      7.3

                                       0.2
                                       0.0
                                                             4 .0 E- 0 9          1 .2 E- 0 8    2 .9 E- 0 8
                                         1.0E-09
                                         1.0E-09                                                                   1.0E-07
                                                                                                                   1.0E-07
                                                                5%値                50%値           95%値
                                                              5%          50%
                                                       Cs-137の経口摂取による一般公衆成人の          95%
                                                                実効線量係数(Sv/Bq)
                                                          Effective dose coefficient (Sv/Bq)                                                    21
                                  Input Parameters
  Variable                         Meaning                            5%        95%       95%/5%
Atmospheric dispersion and deposition: 19 parameters
VG           Deposition velocity for particulates (m/s)             2.2×10-5   1.3×10-2     570
RA           Washout coefficients (hr/mm/s)                         5.1×10-3     4.8        941
PY_D         Horizontal dispersion coefficient Py for stability D     0.17       0.36       2.2
QY_D         Horizontal dispersion coefficient Qy for stability D     0.77       1.03       1.3
PZ_D         Vertical dispersion coefficient Pz for stability D       0.23       3.06       13
QZ_D         Vertical dispersion coefficient Qz for stability D       0.31       0.87       2.8
Dose model: 33 parameters
BRATES       Breathing rate (m3/s)                                  1.5×10-3   3.2×10-3     2.3
FFI1         Filtering factor for wood building (-)                  0.037       0.96       26
FFI2         Filtering factor for concrete building (-)              0.015       0.39       26
INH_CS       Inhalation effective dose coefficient (Sv/Bq)          4.0×10-9   2.7×10-8     6.8
Health effects model: 13 parameters
LD50_PULM LD50 for pulmonary syndrome (Gy)                            7.68       156        20
BETA_PULM Shape factor for pulmonary syndrome (-)                     5.44       10.1       1.9
L_LUNG       Life-time risk for lung cancer (104 person-Gy)         0.00020      453      2.3×106
L_OTHERS Life-time risk for other cancer (104 person-Gy)             0.0011      947      8.6×105

                                                                                                    22
                   OSCAAR Calculations
Site Data
    A model plant is assumed to be located at a coastal site facing the
     Pacific Ocean.
    Population and agricultural production data from the 1990 census
Source Term
                      Item                              Value
    Time before release                                  3.0 h
    Duration of release                                  1.0 h
    Warning time                                         2.0 h
    Release height                                       40 m
    Energy content of release                           0 MW
    Reference inventory                              1100 MW(e)
    Chemical Xe-Kr Organic-I         I     Cs-Rb   Te-Sb   Ba-Sr, Ru   La
    Group
    Release       0.56      0.004   0.07   0.01    0.03      0.01      0.01
    Fraction


                                                                              23
                  OSCAAR Calculations (cont.)

   Countermeasures Strategy                            Countermeasures Timing


                                          Accident               Release start
Sheltering zone (>10 mSv/w)                                             Duration
                                                Time before release
                                                        3h                of release



                                                         Warning time
          30 km                                            2h


                                                                      Sheltering
                      10 km                          Time for     Duration
                                                     direction
                                                                              24 h
                                                        1h


                  Evacuation zone (>50 mSv/w)           Sheltering in concrete building                Evacuation
                                                  Time for    Time for       Duration     Time for           Duration
                                                    direction completion                  completion
                                                        1h        1h            2h               2h        168 h = 7 d



   Relocation zone (>140 mSv/y)




                                                                                                                  24
             Uncertainty Analysis Procedure
Subjective Uncertainty                    Stochastic Uncertainty
                                           M weather sequences
                     K parameters
                                              p1   p2  pM 
                                                                        pi  Y1i 
                 X 11   X 12    X 1K    Y11 Y12     Y1M          i          
                                                                  
                 X 21   X 22    X 2K    Y21 Y22     Y2 M           pi  Y2i 
                                                                       i          
  N runs     X                Y                            
                                                                              
                                                              
                X               X NK   Y            YNM                     
                 N1     X N2             N 1 YN 2                   pi  YNi 
                                                                       i          

Average Individual Risk
    Individual risk as a function of distance
   ri , j ( x ) : risk at x km, j th sector                                   P ( x)  r ( x)
                                                                                       j       i, j

                                                             R( x)   p  (
                                                                                j
                                                                                               )
                                                                                P ( x)
                                                                          i
   Pj (x )   : population at x km, j th sector                        i                    j
    pi       : probability of i th weather sequence                                    j




                                                                                                      25
                                         Example of CCDFs for Individual Risk

                              100                                                                                1

                                                                                                                0.9
                                                                                                                             M EA N
                                                                                                                0.8
Conditionalof exceeding >=X




                                                                                                                             99%
                              10-1
Probability probability, X




                                                                                      Cumulative distribution
                                                                                                                0.7

                                                                                                                0.6

                              10-2                                                                              0.5
                                         99th percentile                                                        0.4

                                                                                                                0.3
                                   -3
                              10
                                                                                                                0.2

                                                                                                                0.1

                                                                                                                 0
                              10-4
                                  10-5     10-4     10-3     10-2      10-1     100                                   10-4       10-3        10-2        10-1        100
                                        Individual risk of early fatality, X
                               Average individual risk of early fatality at 1 km, X                                   Average individual risk of early fatality at 1 km




                                                                                                                                                                   26
             Uncertainty of Average Individual Risk
                                      (Expected Values due to weather variability)
        Conditional probability of early fatality                           Conditional probability of cancer fatality
1E 0
                                                                 1E 0
                                          95%
                                          5%
1E-1
                                          75%
                                          25%
1E-2                                      Central Es tim ates    1E-1


1E-3

                                                                 1E-2
1E-4


1E-5
                                                                                                                   95%
                                                                 1E-3                                              5%
1E-6                                                                                                               75%
                                                                                                                   25%
                                                                                                                   Central Estimates
1E-7
    0                         5                             10   1E-4
                                                                        0       5          10          15     20           25          30

                  Distance from the site (km)                                         Distance from the site (km)

                                         Ratio of 95% to mean value
                  x (km)      0.5 1.5 2.5 3.5 4.5 5.5  7                             9          12.5   17.5   22.5
               Early Fatality 2.3 3.1 4.0 5.5 8.6 5.7 12.2                          10.5         -      -      -
              Cancer Fatality 4.0 3.2 3.3 3.2 2.9 2.6 3.0                           3.3         3.1    3.6    3.1
                                                                                                                                  27
     Contribution of Stochastic Uncertainty
         (weather scenario variance)
Overall variance
                                                      M                    M
     V ( y )  VS [ E ( y | S )]  ES [V ( y | S )]   pi ( i   )   pi i2
      ˆ             ˆ                   ˆ                    ˆ ˆ     2
                                                                             ˆ
                                                      i 1                 i 1

                                                                                          M
            beween-scenario
               variance
                                    within-scenario
                                       variance
                                                             E ( y )  ES [ E ( y | S )]   pi i  
                                                             ˆ              ˆ                   ˆ    ˆ
                                                                                          i 1
Early fatality
                                      0.5km     1.5km        2.5km       3.5km    4.5km          5.5km
     overall mean individual risk   1.02E-02 5.90E-03 1.68E-03 6.09E-04 3.12E-04 2.77E-04
        overall variance V(y)       1.06E-03 7.16E-04 1.91E-04 6.52E-05 4.01E-05 2.90E-05
              Vs[E(y|S)]            2.56E-04 1.46E-04 2.26E-05 5.18E-06 2.26E-06 2.02E-06
              Es[V(y|S)]          8.09E-04 5.70E-04 1.68E-04 6.00E-05 3.79E-05 2.70E-05
   % of variance between scenarios 24.0      20.4     11.8      8.0      5.6      6.9

Latent cancer fatality
                                      0.5km     1.5km        2.5km       3.5km    4.5km          5.5km
     overall mean individual risk    7.92E-02 1.12E-01 8.99E-02 6.84E-02 5.05E-02 4.99E-02
        overall variance V(y)        2.65E-01 3.98E-01 5.83E-01 4.00E-01 1.12E-01 1.06E-01
              Vs[E(y|S)]             1.30E-02 2.31E-02 2.50E-02 1.47E-02 6.25E-03 5.12E-03
              Es[V(y|S)]           2.52E-01 3.75E-01 5.58E-01 3.86E-01 1.06E-01 1.01E-01
   % of variance between scenarios   4.9      5.8      4.3      3.7      5.6      4.8                    28
                        Sensitivity of Early Fatality
         Number of early fatality           Average individual risk of early fatality

                                            1
         1              R2=0.81

                                            5
                                           0.
                                                                                        R -square
        5
       0.
                                                                                        VG
                                    PRCC                                                Q Y_D
SRRC




                                            0                                           B R A TES
         0
                                                                                        FFI 2
                                                                                        LD 50_P U L
                                                                                        B ETA _P U L
                                        5
                                      -0.
         5
       -0.



                                           -1
        -1
                                                0                5                10
                                                    Distance from the site (km)

                                                                                               29
                  Sensitivity of Latent Cancer Fatality
            Number of cancer fatality           Average individual risk of cancer fatality

                                                 1
        1
                        R2=0.73

                                               0.
                                                5
                                                                                             R -square
        5
       0.                                                                                    VG
                                                                                             Q Z_D
                                                                                             FFI 1
SRRC




                                        PRCC
                                                 0                                           FFI 2
        0
                                                                                             L_LU N G
                                                                                               N
                                                                                              I H _C S

                                                 5
                                               -0.
     5
   -0.




                                                -1
       -1
                                                     0     5      10      15      20   25
                                                         Distance from the site (km)

                                                                                                   30
                     Sobol’ Sensitivity Indices
– A model output f ( x)  f ( x1 ,, xn ) can be decomposed into
  summands of different dimensions:
                             n
  f ( x1 ,  , xn )  f 0   f i ( xi )                   f       ij   ( xi , x j )    f12n ( x1 ,  , xi , x j ,  , xn )
                            i 1                        1i  j  n


– the variance D of f (x) can be decomposed as:
             n
      D   Di              D          ij      D12n s
            i 1           1 i  j  n



– Sensitivity measures                          S i1 ,,is     can be introduced:
                                   Di1 ,,is
                   S i1 ,,is 
                                      D

                                   Dj                                                        Di  Di ,ci       D  Dci      D
   First-order : S j                                                 Total : ST                                      1  ci
                                   D                                                   i
                                                                                                 D               D           D


                                                                                                                                     31
                                            Sobol’ Sensitivity Indices
                                       for a Specific Weather Sequence
                                Dry weather sequence                                                 Wet weather sequence

                       S1            FSIGY         FFI2              BR                         S1           FSIGY       WCA         SFG2
                       ST            FSIGY         FFI2              BR                         ST           FSIGY       WCA         SFG2
                       1                                                                        1

                      0.8                                                                       8
                                                                                               0.




                                                                              Sensi vi i ces
Sensitivity indices




                                                                                  ti ty ndi
                      0.6                                                                       6
                                                                                               0.

                      0.4                                                                       4
                                                                                               0.

                      0.2                                                                       2
                                                                                               0.

                       0                                                                        0
                            0    1         2      3      4       5        6                          0   1       2       3     4     5        6
                                     Distance from the site (km)                                              stance f
                                                                                                             Di                te km
                                                                                                                      rom the si ( )

                                                                                                                                         32
                              Summary
 The uncertainty factors (a ratio of 95% to mean )for the expected values
  is less than about four for both average individual risks of early and latent
  cancer fatality near the site boundary.
 The contribution of stochastic uncertainty to the overall uncertainty for
  average individual risk of fatality is only dominant close to the site
  boundary at about 20%, and that for average individual risk of cancer
  fatality is quite stable about less than 6% at all distances.
 When considering the computational costs, the correlation/regression
  measures are useful for understanding the sensitivity of the expectation
  value and some percentile of the CCDFs to the input parameters.
 For specific weather conditions, the Sobol’ method with total effect
  indices is effective in identifying the important input parameters.




                                                                                  33

								
To top