Unit 3: Identification of Hazardous Events

Transcription

1 Unit 3: Identification of Hazardous Events H.P. Nachtnebel Institut für Wasserwirtschaft, Hydrologie und konstruktiver Wasserbau

2 Objectives Risk assessment is based on the estimation of probabilities of hazardous events estimation of a loss (damage) function for env. risk or dose-response function for health risk his modules analysis probabilities of hazardous events ime Series Analysis page 2

3 Risk Definition A hazardous event A probability distribution function (pdf) he consequences (damages, victims,..) R( X*) X* f ( X) D( X) dx f (Q) Potential Damages D (Q) X* old Q Q Environmental Risk Analysis and Management H.P. Nachtnebel

4 Definition and application he main task is to analyse data sets (in time) Definition time series emporal sequence of measurements with some statistical properties ype of the measurements Continuous X(t) X i ( t i ) Discrete ime Series Analysis page 4

5 ime Series of Runoff and Precipitation continuous discrete ime Series Analysis page 5

6 An Example A critical load has to be analysed What is the probability that this level is exceeded? hreshold Q* ime Series Analysis page 6

7 An Example: temporal variability of water availability Demand Run length of decrease ime Series Analysis page 7

8 An Example: surplus and deficit Surplus S i Deficits D i ime Series Analysis page 8

9 Extremes Annual series: each largest value in a year ime Series Analysis page 9

10 Partial Duration Series All independent values above the threshold level e.g check the time distance and the minimum in between hreshold Q* ime Series Analysis page 10

11 A comparison of an annual with a partial series Extremwertstatistik Seite 11

12 Distribution of selected flood peaks Probability of occurrence (%) Q* Peak discharge Q ime Series Analysis page 12

13 Example of 2 distributions Normal distribution 2-parameters Symetric Unbounded on both sides Jahresniederschlag Jahrestemperatur Extremwertstatistik Seite 13

14 Example of 2 distributions Normal distribution 2-parameters Symetric Unbounded on both sides Gumbel-distribution 2-parametric Double exponentiel F( x) Asymetric with fixed skewness c s 1,1396 a Location parameter, Modalwert a x 0,5772* c c scale parameter 1,28255 c Unbounded at the right side s x e e ax c Jahresniederschlag Jahrestemperatur Extreme values: floods low flows heavy rainfall Extremwertstatistik Seite 14

15 Useful Distributions for Extremes Log-Normaldistribution Gumbeldistribution Log-Gumbeldistribution Pearson III-distribution Log-Pearson III distribution Weibull distribution Wakeby distribution Gamma distribution. ime Series Analysis page 15

16 Quantils and distribution Relation between Q and P(Q>Q ) F(Q) is the distribution function f(q) is the density function F(Q) 1 f(q) 0 Q Area right from Q = 1/ Q Q ime Series Analysis page 16

17 Gumbel distribution Gumbel distribution 2-parametric a, c and double exponential Left sided bounded, right sided unbounded ax e c 1 F( x ) e 1 e ax c ln 1 1 a x 1 ln ln 1 c log arithmieren log arithmieren *( 1) Equation of a straight line ime Series Analysis page 17

18 Example: Estimation of a rare event Jahr Q max Rank Weib ,7 1, ,5 1 2,8 1,1 Plotting Positions ,3 5 1,4 1,4 5,5 1,6 often Weibull plotting is used , ,1 1, ,3 3,7 ime Series Analysis page 18

19 Graphical Representation Wahrscheinlichkeitspapier für Gumbel-Verteilung Unterschreitungswahrscheinlichkeit [%] Wiederkehrintervall Q(m 3 /s) 1100?? X Modus Mittel reduzierte Variable y ime Series Analysis page 19

20 Computational Approach Gumbel distribution F(x): a nd c are related to s x and x ax e c 1 F( x ) e 1 c a x 0,5772* c 6 s x Estimation of a rare event with an average return period once in years x x u( ) * s x x x K( ) * s x Normal distribution Gumbel distribution ime Series Analysis page 20

21 Example: Flood Protection Jahr Q max Rank / 1,7 2,5 1 1,3 5 1, ,1 Weib. 1,8 2,8 1,1 1,4 5,5 1,6 2,2 11 1,2 y y K( )* y s y ln ln 1 x x K( ) * 100 sx 373 4,323*128,3 928 m ³ / s ,3 3,7 ime Series Analysis page 21

22 Sampling uncertainty Assumptions: Perfect observations Perfect model 1000 years observed ake sub-samples each of 25 years Extreme value analysis for each sub sample ime Series Analysis Using all data Seite 22

23 Estimation uncertainty he shorter the observation length n the larger is uncertainty he larger the variance s x the larger is uncertainty he longer the estimation period the larger is uncertainty he larger the confidence the larger the uncertainty x u( )* s x u( ) * 2 11,14* K 1,1* K s x n 11,14* 4,3231,1* 4, , ,960*208, m ³ / s 409 m³ / s ime Series Analysis page 23

24 Comparison of different pds (models) fitted to the same data set HQ 250 HQ Statistik Ill - Vandans Jährliche Reihe [m³/s] Q sortiert nach WEIBULL 450 P III 95% Konfidenzintervall P III 400 Gumbel 95% Konfidenzintervall Gumbel 350 LP III 95% Konfidenzintervall LP III 300 AEV 95% Konfidenzintervall AEV Pu (((Nachtnebel und Stanzel, 2007) Extremwertstatistik Seite 24

25 Summary and conclusions A time series has been observed A critical threshold is being defined annual or partial series is obtained A model is chosen and fitted to the extremes Extrapolation and estimation of rare events (X 100, X 500, ) Assessing the uncertainty ime Series Analysis page 25