Hydrological extremes. Hydrology Flood Estimation Methods Autumn Semester

Transcription

1 Hydrological extremes droughts floods 1

2 Impacts of floods Affected people Deaths Events [Doocy et al., PLoS, 2013] Recent events in CH and Europe Sardinia, Italy, Nov Central Europe, 2013 Genoa and Liguria, Italy, 2011, 2014 Central Switzerland, 2007, 2005 Tessin, 2014! 2

3 Design values for flood protection/mitigation Federal Office for Water and Geology: Flood Control at Rivers and Streams. Guidelines of the FOWG, 2001 (72p.) 3

4 Flood estimation methods (Flood Frequency Analysis) Lecture content Skript: Ch. VII.12 à 16.2, VII.17 problem formulation hydrological models specific to flood frequency analysis deterministic methods probabilistic methods direct at site statistical regionalisation indirect methods practice in Switzerland Objective: to learn how to estimate a design flood, Q p (R) 4

5 Flood estimation methods problem formulation Coping with floods require to investigate four main questions how frequently a flood of a given magnitude (i.e. a flood characterised by a return period R) occurs how high is the water level in the river for a flood of given magnitude does the river overflow the levees and what is the extent of the inundated areas what are the best prevention and mitigation measures hydrology, flood estimation methods (flood frequency analysis) hydraulics, 1D / 2D unsteady flow propagation models hydraulics, 2d unsteady flow propagation models structural and non structural measures, civil protection plans 5

6 Flood estimation methods problem formulation (example) the bridge needs to be designed to withstand a water level corresponding to a 100 year return period flood ê by means of Flood Frequency Analysis (FFA) we can estimate Q 100 to feed a 1D steady flow hydraulic model estimating the discharge for a given return period is one of the most frequent design problems in hydrology 6

7 Flood hazard probability - definitions Q p = peak discharge Hazard = probability of occurrence, within a specific period of time in a given area, of a potentially damaging flow à 1-F Qp, where F Qp is the non-exceedence probability Vulnerability = degree of damage in probability terms inflicted on a structure by a natural phenomenon of a given magnitude. Risk = Hazard Vulnerability Potential Loss à it includes explicit assessment of the system, i.e.river, protection system, goods and infrastructures at risk of damage L = time horizon for hazard/risk assessment HAZARD ABSOLUTE PROBABILITY ê Pr[Q p >q(r)] ßà Q p (R) absolute risk : 1 1 R = = Pr Q p > q 1 F Qp HAZARD RELATIVE PROBABILITY ê Pr[Q p >q(r) at least once in L years] HYDROLOGIC RISK relative risk r = L R = 1 F Qp L HYDRAULIC RISK ê Pr[system failure in L years] 7

8 Relative risk absolute risk : relative risk F Qp R L r F Qp R L r % % 10 65% 10 18% 20 88% 20 33% 50 99% 50 64% % % % % 10 40% 10 10% 20 64% 20 18% 50 92% 50 39% % % % 10 5% 20 10% 50 22% % 8

9 Flood estimation methods overview (1) Deterministic methods based on local analysis of empirical historical evidence envelope curves empirical equations Q p = c Ab concept of probable maximum precipitation (PMP) à probable maximum flood (PMF) WMO publ on PMP 9

10 Flood estimation methods overview (2) Q p Probabilistic methods direct à statistical analysis at site indirect derived distribution techniques statistical regionalisation Q p Q p site 1 R Q p site 3 site 2 R f p,t f Q p p t R rainfall-runoff simulations i(t) Q(t) t t R Q i Q p /Q i A regional curve R index flood Q p 10

11 Flood estimation methods reference space-time scales Flood estimation methods are suitable for ranges of space and time scales depending on their characteristics direct at-site statistical methods à length of data record statistical regionalisation à broad range of scales indirect R-R methods à from small to very large scales depending on model complexity and rainfall input! à SEE SLIDES SPECIFIC TO EACH METHOD 11

12 Deterministic methods 12

13 Deterministic methods (1/3) PROBABLE MAXIMUM FLOOD reference concept for other methods (e.g. envelop curve) greatest flood to be expected assuming concurrent occurrence of those factors that lead to maximum rainfall (probable maximum precipitation, PMP) and to favourable conditions to generation of maximum flood discharge à simultaneous max rainfall and max soil moisture PMP = greatest depth of precipitation for a given duration meteorologically possible for a design watershed or a given storm area at a particular location at a particular time of year based on atmospheric physics theoretical considerations for the maximisation of local storm rainfall à see WMO, Manual on Estimation of Probable Maximum Precipitation (PMP), publ PROS CONS established procedure (in the past) widely used for design of safety organs of large dams ê controversial concept contradictory concept: probable ßà maximum high uncertainty associated with methods to estimate PMP ê NO FREQUENCY CHARACTERISATION (i.e. no R associated with the estimate) 13

14 Deterministic methods (2/3) ENVELOPE CURVE concept similar to that of PMF, but based on observed floods in a region greatest observed floods in a region are normalized with respect to the basin area à specific discharge à q = Q p / A and plotted against the basin area a curve is drawn, which envelopes all the observed specific discharges the typical equation for envelope curves is q = ˆq + b A ν where ˆq, b, and ν are empirical parameters q = Q p / A [m 3 /s km 2 ] REMARKS lower limit of validity envelope curve the curve does not provide a frequency characterisation for the predicted q observed values A [km 2 ] the observed values used to build the curve (may) have different return periods àestimates obtained from the envelope curve are not characterised byhomogeneous return periods 14

15 Deterministic methods (3/3) EMPIRICAL EQUATIONS concept similar to that of the envelope curve, based on a broad sample of large observed floods in a region they typically provide the value of the largest flood without a frequency characterisation Q p is expressed typically as a function of baisn area à typical forms of the equations are Q p = c A + b d or Q p = a A b where A is the basin area and a, b, c and d are empirical parameters examples of emp. eq. used in CH Kürsteiner (1917) à Q max = c A 2 3 c à where A is the basin area, c a coefficient specific for the basin and Q max in [m 3 /s] and A in [km 2 ] [Spreafico et al., B.amt f. Wasser und Geologie, rep. 4, 2003] Müller-Zeller (1943) à Q max = α ψ A 2 3 ψ à where A is the basin area, α is a zonal coefficient, ψ is the runoff coefficient, Q max in [m 3 /s], and A in [km 2 ] à α =

16 Probabilistic methods a) direct at site 16

17 Probabilistic methods at site analysis (1/6) two type of analysis based on ANNUAL FLOOD SERIES (AFS), i.e. based on the time series of the maximum values of the peak flows observed in each year PEAK OVER THRESHOLD SERIES (POT), i.e. based on the time series of the values of the peak flows observed in each year, which exceed a pre-defined threshold, Q* Q(t) Q * peaks over a threshold Q 1 Q 2 Q 3 Q 4 annual peak flow t 1 t T = 1 year 2 t 3 t 4 t 17

18 Probabilistic methods at site analysis (2/6) Q(t) peaks%over%a%threshold% ANNUAL FLOOD SERIES based estimation e.g. the observed annual maxima are treated as discrete random variable, Q i, extracted from a continuous process Q i = max Q( t), ( i 1)T t T { } the AFS series is used to fit a theoretical (or empirical) probability distribution EV1 or Gumbel distr. EV2 or Frechet distr. à à GEV (General Extreme Value)à F Q F Q F Q q u e ( q) = e ( ) α ( q) = e ( q 0 q)β ( q) = e ( ) α 1 k q u 1 k peak flow [m 3 /s] Q * Q 1 Q 2 Q 3 Q 4 annual%peak% flow% t 1 t T = 1 year 2 t 3 t 4 t for a selected return period R the peak flow can be estimated by substituting F Q = R 1 R and then computing Q by solving the CDF equation for Q e.g. EV1 à Q R return period [years] ( ) ( ) = u α ln ln( R R 1) 18

19 Probabilistic methods at site analysis (3/6) PEAK OVER THRESHOLD based estimation Q(t) Q * peaks%over%a%threshold% Q 1 Q 2 Q 3 Q 4 annual%peak% flow% Hypotheses t 1 t T = 1 year 2 t 3 t 4 t the observed peak o.t. flows are assumed to be characterised by the same probability distribution à pdf of the exceedances the number of exceedances N(0, t) follows a Poisson distribution à Pr N [ 0,t) = n = λt n e λt n! for t = T = 1 year à Λ = λt = average number of exceedances per year the annual maximum is the CDF of the annual maxima is: where F Z ( ) f Z ( Z t1 ) = f Z Z t2 F Qp is the CDF of the exceedances ( ) =... = f Z ( Z tn ) { } Q p = max Z ti, 1 i N [ 0,t) ( q) = Pr Q p q = e Λ 1 F Z q ( ) characterising F Z ( ) and λ specifies entirely the CDF of the annual maxima 19

20 Probabilistic methods at site analysis (4/6) AFS ESTIMATION SUITABLE DISTRIBUTIONS Distribution Domain Mean μ x Variance σ x 2 Skewness γ Lognormal 2 par. z = lnx f ( x) = Exponential Gumbel 1 xσ z 2π e f ( x) = 1 e ( x x 0 ) β F X x b e ( x) = e ( ) a 1 2 ln x µ z σ z General Extreme Value F X ( x) = e ( ) α 1 k x u 1 k 2 (0, ) (x 0, ) (-, ) (-, ) e µ z +σ 2 z 2 µ z e σ 2 z 1 ( ) e σ 2 z ( 1) 1 2 e σ 2 z ( + 2) x 0 + β β 2 2 b a π 2 6a f ( k) f ( k) f ( k) Pearson Type III Log Pearson Type III 20

21 Probabilistic methods at site analysis (5/6) POT ESTIMATION SUITABLE DISTRIBUTIONS CDF of the exceedances à CDF of annual maxima Exponential F Z = 1 e x q 0 ϑ x q 0 Gumbel (EV1) F X = e Λe x q0 ϑ Pareto F Z = 1 q 0 x 1 k Generalized Pareto F Z = 1 1 k β x c ( ) 1 k x q 0 Frechet (EV2) F X = e Λ q 0 x 1 k General Extreme Value F X = e 1 k x ε α 0 0 ( ) 1 k 21

22 Probabilistic methods at site analysis (6/6) REMARKS the parameters of the distributions are estimated by menas of the usual methods (method of moments, least squares, maximum likelihood, ) the extrapolation of the estimates for high return periods should be consistent with the numerosity of the data record (e.g. max R 2N, with N = record length the POT method is best suitable for shorter observational records (to extend the number of observations) for highly variable basin flood response (to avoid missing secondary peak flows that are higher than some of the annual maxima) flood frequency distributions should always be plot together with their confidence interval (see statistics books) Q p R 22

23 Probabilistic methods b) statistical regionalisation 23

24 Statistical regionalisation replaces at-site direct analysis for ungauged sites and/or for sites with short observational records can be used to extend at site analysis to higher return periods it is based on the concept that basins with similar characteristics have similar flood response à HOMOGENEOUS REGION Q p 1 A 1 A2 A 3 A 4 N = N 1 F Q p Q p 2 F Q p Q p 3 N = N 2 N = N 3 F Q p Q p x,a x Q p 4 Homogeneity quantified with respect to hydrologic or statistical similarities, or both N = N 4 F Q p Q p i, A i, respectively AFS and drained areas of the i-th hydrometric stations in the homogeneous region Q p x, A x, peak flow and drained areas of the station for which Q p (R) is to be computed N i, record length at each station i REQUIRED DATA AFS from flow gauging stations of a homogeneous region hydrologic, thematic or statistical information to quantify the homogenity of a set of basins 24

25 Statistical regionalisation Index Flood method (1/3) Procedure to compute the peak flow of an ungauged basin of area A for a return period R A) Homogenous region 1 identify a homogenous region for which AFS data are available Q p 1 A 1 A2 A 3 A 4 N = N 1 F Q p Q p 2 Q p 3 N = N 3 N = N 2 F Q F Q p p Q p x, A x Q p 4 N = N 4 F Q p 2 estimate for each AFS its index flood, Q i e.g. mean of the population à mean of the AFS à Q i = Q pi Q i 3 find a suitable relationship between the estimated index floods and some basin characteristics typically à Q i = f ( A i ) Area A 1 A 2 A 3 A 4 25

26 Statistical regionalisation Index Flood method (2/3) B) AFS analysis and computation of Q p x(r*) 1 transform each AFS to a dimensionless series by dividing each AFS value by the index flood Q p i/q i 2 pool the dimensionless AFS to obtain a dimensionless data set of size N = N 1 + N N m where N i is the numerosity of each of the m AFS available in the homogeneous region Q* indexed values 3 4 fit a suitable probability distribution to the dimensionless AFS à dimensionless flood growth curve for a given return period R*, compute from the dimensionless growth curve the value Q*=Q p i/q i } Q i R* F Q p, R 5 6 from the relationship Q i = f(a i ) compute for A x the index flood for the ungauged site X, Q i x multiply the dimensionless value Q* by Q i of the ungauged site to obtain Q p x(r*) } Q i x A 1 A 2 A 3 A 4 A x Area 26

27 Statistical regionalisation Index Flood method (3/3) REMARKS Estimation of the index flood the most frequent assumption is that the index flood corresponds to the mean of the population of the observed AFS à a reasonable estimator is the mean of the observed AFS data the relationships to estimate the index flood from watershed characteristics can use any of the climatological and morphological watershed features (e.g. area, slope, average permeability, max daily precipitation, etc.) à m the scaling of the index flood with the watershed area is most frequently used à Q i A i Q i can be alternatively expressed as function of the POT series underlying the dimensionless growth curve Q i = ε + α k m QPOT ( 1 Λk ) + αλk 1+ k where the POT series is Generalized Pareto Distributed, the resulting AFS distribution is GEV, m QPOT is the mean of the POT series, α, k and ε are the GEV parameters and Λ is the mean number of exceedences per year Evaluation of the homogeinity of the region often based on quantitative analysis of basin characteristics or AFS statistics, without rigorous testing tests exist, which provide a quantitative measure of the homogeneity generally based on some statistics (e.g. CV, statistical moments, ) strong dependence on the numerosity of the sample 27

28 Probabilistic methods c) indirect R-R methods 28

29 Flood estimation methods based on R-R modelling R-R models can be used to estimate flood peaks for a given return period R of Q p is the same of the rainfall input if the R-R model is linear Flood estimation based on R-R models is denoted as DESIGN FLOOD APPROACH 1 2 H i R ê DDF T Design hyetograph t ì f 3 h 4 ê Infiltration model IUH t t ì Q 5 Q p (R) Flood hydrograph REMARKS FEM based on R-R methods provides also flood volume and duration useful for design of protection systems 29 t

30 Simplified R-R model for flood estimation Rational Method (1/3) Concept Peak flow for a given return period, R, is linearly proportional to a coefficient, c, which accounts for spatial distribution of rainfall infiltration properties across the basin the network response the watershed area, A the (constant) rainfall intensity, i, computed from the IDF for a return period R and a duration t c, which is specific to the watershed RATIONAL METHOD Q R = c i A Use small basins ( km 2 ), highly homogeneous low return periods, R 50 years 30

31 Simplified R-R model for flood estimation Rational Method (2/3) General parameter dependence on process and basin characteristics i = i(r, t c ) and i = H(R, t c )/t c c = k φ ψ, with k, parameter accounting for the spatial distribution of rainfall φ parameter accounting for infiltration across the basin ψ accounting for the network response and, in turn k = k(ω t, t c ) with Ω t, parameters characterising the space-time rainfall patterns t c, watershed characteristic time φ = φ(θ t, t c, i) with Θ t, parameters characterising the watershed infiltration capacity ψ = ψ (Ξ t, t c, i) with Ξ t, parameters characterising the network structure and response à linear model à ψ = ψ (Ξ t, t c ) 31

32 Simplified R-R model for flood estimation Rational Method (3/3) Simplified parameterisation Q R = k(ω t, t c ) φ(θ t, t c, i) ψ (Ξ t, t c ) A H / t c à Q R = φ ψ (Ξ t, t c ) A H / t c neglected, k=1 lumped value, independent from t c and i dependent on the underlying linear model with φ estimated from tables, literature, joint records of event rainfall and runoff t c estimated as the critical time leading to the max Q for the return period R kinematic transfer (linear channel) à ψ (Ξ t, t c ) = A(t)/A with A(t) area drained up to time t and A= watershed area à max Q R is reached for t c =time of concentration, when with A(t)/A=1 max Q R = φ 1 A H / t c linear reservoir à ψ (Ξ t, t c ) = 1-exp(-t c /K) with K = linear reservoir storage constant à max Q R is reached for max Q R = max {φ A (H / t c ) [1-exp(-t c /K)]} à dq R / dt à t c, which maximizes Q R 32

33 Practice in Switzerland 33

34 Practice of flood estimation in Switzerland No prescribed method at the federal and cantonal level Some Cantons issued guidelines The Confederation issued in 2003 a comprehensive guideline Consolidated habits generally dictate preference for a specific method among statistical methods (direct, regionalisation) lumped R-R models empirical formulas depending on design target publikation/00565/index.html?lang=de 34

35 BAFU guideline Flow chart to decide among statistical methods lumped R-R models regional methods Decision tree based on flow and rainfall data availability 35