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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-University Bochum, Germany
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
Outline
• Introduction
• Theory of Copulas
• Bivariate Frequency Analysis
• Research Area
• Application
• Conclusions
Outline – Introduction – Theory of Copulae – Bivariate Frequency Analysis
Research Area – Application -Conclusions
2
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
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ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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):
FX,Y  x, y   C  FX  x  , FY  y    C  u, v 
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
Outline –Introduction – Theory of Copulas – Bivariate Frequency Analysis
Research Area – Application -Conclusions
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ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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:
C  u, v    1   u  ,   v  
where:  is the generator of the copula
One-parameter Archimedian copula Gumbel-Hougaard Family:



C  u, v   exp     ln u     ln v 

where: Parameter

1



  1,  
Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis
Research Area – Application -Conclusions
5
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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):


 1

C  u, v   1   u   1   v   1





1 
where: Parameter    0,  
Parameter   1,  
models the upper tail dependence
Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis
Research Area – Application -Conclusions
6
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
Parameter Estimation & Evaluation
Rank-based Maximum Pseudolikelihood:

S 
 R
l  ,...   log  c ,...  i , i    max
 n 1 n 1  
i 1

n
Other estimation methods: Spearman‘s 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)
K C (t)  (u, v)  [0,1]2 : C(u, v)  t
Archimedian copulas: K C  t   t 
 t
 t 
Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis
Research Area – Application -Conclusions
7
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
Bivariate Frequency Analysis
Non-exceedance probability:
P  X  x, Y  y   FX,Y  x, y   C  FX  x  , FY  y  
Exceedance probability exceeding x and y :
P  X  x  Y  y   1  FX  x   FY  y   FX,Y  x, y 
 1  FX  x   FY  y   C  FX  x  , FY  y  
Return period:

TX,Y 
1
1

 Max TX ,TY 
P(X  x  Y  y) 1  FX  x   FY  y   C FX  x  , FY  y 
Exceedance probability exceeding x or y :
P  X  x  Y  y   1  FX,Y  x, y   1  C  FX  x  , FY  y  
Return period:

TX,Y 
1
1

 Min TX ,TY 
P(X  x  Y  y) 1  C FX  x  , FY  y 
Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis
Research Area – Application -Conclusions
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ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
9
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
Methodology
RIMAX joint research project: “Flood control management for the river Unstrut”
Stochastic Rainfall Generator
Hydrological Model
Reservoir Model
Spatially Distributed Rainfall Data
Copula Analysis
Hydrological Load Scenarios
Hydrological Loads
Technical/Operational Solutions
with probabilistic assessments
Hydrodynamic Model
Socio-Economic Analyses
Inundation Areas
Damage Functions
Decision Support System
Socio- economic consequences
Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis
Research Area – Application -Conclusions
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ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
11
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
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ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
13
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
14
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
Bivariate Analysis Flood Peak-Volume
2
Parametric and nonparametric estimates of K C (t)  (u, v)  [0,1] : C(u, v)  t
Archimedian copulas:
2-Parameter copula BB1:
Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis
Research Area – Application -Conclusions
15
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
16
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
17
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
Bivariate Analysis Flood Peak-Volume
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
Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis
Research Area – Application -Conclusions
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ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
Outline – Introduction – Theory of Copulas – Bivariate Frequency Analysis
Research Area – Application -Conclusions
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ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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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ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
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ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
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
22
ISFD4: Probabilistic Analysis of Hydrological Loads to Optimize the Design of Flood Control Systems
Acknowledgments


BMBF (Federal Ministry of Education
and Research) / RIMAX
Unstrut-Project: TMLNU, MLU LSA,
DWD
Thank you very much
for your attention!
[email protected]
www.ruhr-uni-bochum.de/hydrology
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