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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Archimedian Copulas
1C u, v u , v
1C u, v exp ln u ln v
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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,
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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2-Parameter Copulas
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C u, v 1 u 1 v 1
0,
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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,
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Parameter Estimation & Evaluation
n
i i,...
i 1
R Sl ,... log c , max
n 1 n 1
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Frequency Analysis
X,Y X YP X x,Y y F x, y C F x ,F y
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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 :
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Research Area
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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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
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
RIMAX joint research project: “Flood control management for the river Unstrut”
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Generation of Flood Events for Risk Analysis
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Analysis Flood Peak-Volume
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Analysis Flood Peak-Volume
Marginal distributions of the flood peaks:
Generalized Extreme Value (GEV) distribution
Parameter estimation method:Reservoir Straußfurt: L-Moments Reservoir Kelbra: Product moments
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Analysis Flood Peak-Volume
Marginal distributions of the flood volumes:
Generalized Extreme Value (GEV) distribution using the method of product moments as parameter estimation method
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
Reservoir Straußfurt Reservoir Kelbra
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Analysis Flood Peak-Volume
Parametric and nonparametric estimates of
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
Archimedian copulas: 2-Parameter copula BB1:
2CK (t) (u, v) [0,1] : C(u, v) t
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Analysis Flood Peak-Volume
1000000 simulated random pairs (X,Y) from the copulas
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Analysis Flood Peak-Volume
Joint return periods:
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Analysis Flood Peak-Volume
Critical Events at the reservoir Straußfurt
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Spatial Variability
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Analysis of corresponding Flood Peaks
1000000 simulated random samples from the copulas
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
Parametric and nonparametric estimates of KC(t)
Gumbel-Hougaard copula is used for further analysis
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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Bivariate Analysis of corresponding Flood Peaks
Joint return periods:
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
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
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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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
Outline – Introduction – Theory of Copulas – Bivariate Frequency AnalysisResearch Area – Application -Conclusions
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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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