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

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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-Univ
ersity Bochum, Germany
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Outline

• Introduction
• Theory of Copulas
• Bivariate Frequency Analysis
• Research Area
• Application
• Conclusions

Outline Introduction Theory of Copulae
Bivariate Frequency Analysis Research Area
Application -Conclusions
3
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 Analysis Research Area
Application -Conclusions
4
Theory of Copulas
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 cdfs of
the random variables C is a copula function
such that C 0,1² ? 0,1 C(u,v) 0 if at
least one of the arguments is 0 C(u,1)u and
C(1,v)v
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
5
Archimedian Copulas
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 j is the generator of the
copula One-parameter Archimedian copula
Gumbel-Hougaard Family
where Parameter
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
6
2-Parameter Copulas
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.
2-Parameter copula BB1 (Joe, 1997)
where Parameter Parameter models the
upper tail dependence
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
7
Parameter Estimation Evaluation
Rank-based Maximum Pseudolikelihood
Other estimation methods Spearmans Rho,
Kendalls Tau, IFM- (Inference from margins) method
Evaluation of the appropriate family of copulas,
comparison of parametric and nonparametric
estimate of (Genest and Rivest, 1993)
Archimedian copulas
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
8
Bivariate Frequency Analysis
Non-exceedance probability
Exceedance probability exceeding x and y
Return period
Exceedance probability exceeding x or y
Return period
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
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Research Area
Watershed of the river Unstrut
Catchment area 6343 km²
Highly vulnerable to floods
Flood Retention System Volume 100 Mio. m3
2 Reservoirs
Polder system
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
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
10
Methodology
RIMAX joint research project Flood control
management for the river Unstrut
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
11
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
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
12
Bivariate Analysis Flood Peak-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
? Bivariate analysis of flood peak and volume
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
13
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 Analysis Research Area
Application -Conclusions
14
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
Reservoir Straußfurt
Reservoir Kelbra
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
15
Bivariate Analysis Flood Peak-Volume
Parametric and nonparametric estimates of
Archimedian copulas
2-Parameter copula BB1
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
16
Bivariate Analysis Flood Peak-Volume
1000000 simulated random pairs (X,Y) from the
copulas
? BB1 copula provides a better fit to the data

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

Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
17
Bivariate Analysis Flood Peak-Volume
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
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
18
Bivariate Analysis Flood Peak-Volume
Critical Events at the reservoir Straußfurt
Waterlevel gt 150.3 m a.s.l. ? Outflow gt 200 m3s-1
? Severe damages downstream
TvX,Ygt40 years all selected events are critical
events ? Hydrol. risk is very high
25ltTvX,Ylt40 years 3 of 5 selected events are
critical events
TvX,Ylt25 years 2 of 12 selected events are
critical events ? Hydrol. risk is low
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
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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 cant 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

Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
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Bivariate Analysis of corresponding Flood Peaks
Parametric and nonparametric estimates of KC(t)
1000000 simulated random samples from the copulas
? Gumbel-Hougaard copula is used for further
analysis
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
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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
Outline Introduction Theory of Copulas
Bivariate Frequency Analysis Research Area
Application -Conclusions
22
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 Analysis Research Area
Application -Conclusions
23
Acknowledgments

• BMBF (Federal Ministry of Education and Research)
  / RIMAX
• Unstrut-Project TMLNU, MLU LSA, DWD

Thank you very much for your attention!
bastian.klein_at_rub.de www.ruhr-uni-bochum.de/hydrol
ogy