Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems

Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood

Control Systems

B. Klein, M. Pahlow, Y. Hundecha, C. Gattke and A. Schumann

Institute of Hydrology, Water Resources Management and Environmental Engineering

Ruhr-University Bochum, Germany

ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems

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Outline

• Introduction

• Theory of Copulas

• Bivariate Frequency Analysis

• Research Area

• Application

• Conclusions

Outline – Introduction – Theory of Copulae – Bivariate Frequency AnalysisResearch Area – Application -Conclusions


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Introduction

• To analyze flood control systems via risk analysis a lot of different hydrological scenarios have to be considered

• Probabilities have to be assigned to these events

• Univariate probability analysis in terms of flood peaks can lead to an over- or underestimation of the risk associated with a given flood.

Multivariate analysis of flood properties such as flood peak, volume, shape and duration

• Considerably more data is required for the multivariate case

In practice the application is mainly reduced to the bivariate case.

• Traditional bivariate probability distributions have a large drawback: Marginal distributions have to be from the same family

Analysis via copulas

Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions


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Theory of Copulas

Outline –Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions

X,Y X YF x, y C F x ,F y C u, v

Copulas enable us to express the joint distribution of random variables in terms of their marginal distribution using the theorem of Sklar (1959):

where: FX,Y(x,y) is the joint cdf of the random variables

Fx(x), Fy(y) are the marginal cdf‘s of the random variables

C is a copula function such that: C: [0,1]² [0,1]

C(u,v) = 0 if at least one of the arguments is 0

C(u,1)=u and C(1,v)=v


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Archimedian Copulas

1C u, v u , v

1C u, v exp ln u ln v


A large variety of Copulas are available to model the dependence structure of the random variables (Nelson, 2006; Joe, 1997), such as Archimedian copulas:

where: is the generator of the copula

One-parameter Archimedian copula Gumbel-Hougaard Family:

where: Parameter 1,


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2-Parameter Copulas

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C u, v 1 u 1 v 1

0,


2-Parameter copula BB1 (Joe, 1997):

2-Parameter copulas might be used to capture more than one type of dependence, one parameter models the upper tail dependence and the other the lower tail dependence.

where: Parameter

Parameter models the upper tail dependence 1,


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Parameter Estimation & Evaluation

n

i i,...

i 1

R Sl ,... log c , max

n 1 n 1


Other estimation methods: Spearman‘s Rho, Kendalls Tau, IFM- (Inference from margins) method

Rank-based Maximum Pseudolikelihood:

Evaluation of the appropriate family of copulas, comparison of parametric and nonparametric estimate of: (Genest and Rivest, 1993)

2CK (t) (u, v) [0,1] : C(u, v) t

Archimedian copulas: C

tK t t

t


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Bivariate Frequency Analysis

X,Y X YP X x,Y y F x, y C F x ,F y


X Y X,Y

X Y X Y

P X x Y y 1 F x F y F x, y

1 F x F y C F x ,F y

X,Y X YP X x Y y 1 F x, y 1 C F x ,F y

Non-exceedance probability:

Exceedance probability exceeding x and y :

X,Y X Y

X Y X Y

1 1T Max T ,T

P(X x Y y) 1 F x F y C F x ,F y

Return period:

X,Y X Y

X Y

1 1T Min T ,T

P(X x Y y) 1 C F x ,F y

Return period:

Exceedance probability exceeding x or y :


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Research Area


Flood Retention System:Volume: ~ 100 Mio. m3

Watershed of the river Unstrut:

2 Reservoirs

Polder system

Highly vulnerable to floods

RIMAX joint research project:“Flood control management for the river Unstrut” Analysis, optimization and extension of the flood control system through an integrated flood risk assessment instrument

Catchment area: 6343 km²


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Stochastic Rainfall Generator Spatially Distributed Rainfall Data

Hydrological Model

Reservoir ModelHydrological Load Scenarios Copula Analysis

Technical/Operational Solutions

Hydrodynamic Model Socio-Economic Analyses

Damage FunctionsInundation Areas

Decision Support System

Hydrological Loads

with probabilistic assessments

Socio- economic consequences

Methodology


RIMAX joint research project: “Flood control management for the river Unstrut”


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Generation of Flood Events for Risk Analysis


Stochastic generation of 10x1000 years daily precipitation

Daily water balance simulation with a semi-distributed model (following the HBV concept)

Selection of representative events with return periods between 25 to 1000 years

Disaggregation of the daily precipitation to hourly values for the selected events

Simulation of hourly flood hydrographs via an event-based rainfall-runoff model


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Bivariate Analysis Flood Peak-Volume


Bivariate analysis of flood peak and volume

Univariate probability analysis in terms of flood peaks can lead to an over- or underestimation of the risk associated with a given flood:

Peak Return Period T = 100 a


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Marginal distributions of the flood peaks:

Generalized Extreme Value (GEV) distribution

Parameter estimation method:Reservoir Straußfurt: L-Moments Reservoir Kelbra: Product moments



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Marginal distributions of the flood volumes:

Generalized Extreme Value (GEV) distribution using the method of product moments as parameter estimation method


Reservoir Straußfurt Reservoir Kelbra


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Parametric and nonparametric estimates of


Archimedian copulas: 2-Parameter copula BB1:

2CK (t) (u, v) [0,1] : C(u, v) t


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1000000 simulated random pairs (X,Y) from the copulas


Only the Gumbel-Hougaard copula and the BB1 copula can model the dependence structure of the data

BB1 copula provides a better fit to the data


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Joint return periods:


A large variety of different hydrological scenarios is considered in design

E.g. return period of flood peak of about 100 years at reservoir Straußfurt, the corresponding return periods of the flood volumes ranges between 25 and 2000 years


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Critical Events at the reservoir Straußfurt


Waterlevel > 150.3 m a.s.l. Outflow > 200 m3s-1 Severe damages downstream

TvX,Y>40 years: all selected events are critical events Hydrol. risk is very high

25<TvX,Y<40 years: 3 of 5 selected events are critical events

TvX,Y<25 years: 2 of 12 selected events are critical events Hydrol. risk is low


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Spatial Variability


Catchment area with two main tributaries:

What overall probability should be assigned to events for risk analysis?

Two reservoirs are situated within the two main tributaries Reservoir operation alters extreme value statistics downstream

Gages downstream can’t be used for categorization of the events

Bivariate Analysis of the corresponding inflow peaks to the two reservoirs to consider the spatial variability of the events


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Bivariate Analysis of corresponding Flood Peaks

1000000 simulated random samples from the copulas


Parametric and nonparametric estimates of KC(t)

Gumbel-Hougaard copula is used for further analysis


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Bivariate Analysis of corresponding Flood Peaks

Joint return periods:


A large variety of different hydrological scenarios is considered in design

E.g. Return period of about 100 years at reservoir Straußfurt, the return periods of the corresponding flood peaks at the reservoir Kelbra ranges between 10 and 500 years


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Conclusions

• A methodology to categorize hydrological events based on copulas is presented

• The joint probability of corresponding flood peak and volume is analyzed to consider flood properties in risk analysis

• Critical events for flood protection structures such as reservoirs can be identified via copulas

• The spatial variability of the events is described via the joint probability of the corresponding peaks at the two reservoirs



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BMBF (Federal Ministry of Education and Research) / RIMAX

Unstrut-Project: TMLNU, MLU LSA, DWD

Acknowledgments

Thank you very muchfor your attention!

[email protected]/hydrology

Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems

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