bivariate flood frequency analysis using copula with parametric and nonparametric marginals...
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Bivariate Flood Frequency Analysis using Copula with
Parametric and Nonparametric Marginals
Subhankar KarmakarSlobodan P. Simonovic
The University of Western OntarioThe Institute for Catastrophic Loss
Reduction
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Presentation outline
Introduction Objectives of the study Study area
Data and flood characteristics Marginal distributions of flood
characteristics Parametric and nonparametric estimation
Joint and conditional distributions using copula
Conclusions
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Introduction
Flood management (design, planning, operations) requires knowledge of flood event characteristics
Flood characteristics
Peak flow DurationVolume
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Introduction
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Introduction
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Study objectives To determine appropriate marginal distributions
for peak flow, volume and duration using parametric and nonparametric approaches
To define marginal distribution using orthonormal series method
To apply the concept of copulas by selecting marginals from different families of probability density functions
To establish joint and conditional distributions of different combinations of flood characteristics and corresponding return periods
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Study area Red River Basin 116,500 km2 (89% in
USA 11% in CDN) Flooding in the basin
is natural phenomena
Historical floods: 1826; 1950; 1997
Size of the basin and flow direction
No single solution to the flood mitigation challenge
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Study area - data
Daily streamflow data for 70 years (1936-2005)
Gauging station (05082500) - Grand Forks, North Dakota, US Location - latitude 47°55'37"N and longitude
97°01'44"W Drainage area - 30,100 square miles Contributing area - 26,300 square miles
http://waterdata.usgs.gov
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Study area – flood characteristics
Dependence between P, V and D
P and V highly correlated All the correlations positive
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Marginals - parametric
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Marginals - nonparametric
Nonparametric kernel estimation of flood frequency
Orthonormal series method
n
1ll
1 }h/)xx{(K)nh()x(f̂
js0dx)x()x( js j1dx)}x({ 2j
1)x(0 )jxcos(2)x(j
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Results
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Results
P follows gamma distribution (parametric) and V and D follow distribution function obtained
from orthonormal series method (nonparametric)
Mixed marginals
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Results
Second test
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Copula
An alternative way of modeling the correlation structure between random variables.
They dissociate the correlation structure from the marginal distributions of the individual variables.
n - dimensional distribution function can be written: ))x(F),......,x(F(C)x,......,x(F nn11n1
1,.....,0)),(),......,((),......,( 11
11
11 nnnn uuuFuFFuuC
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Copula
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Results – joint distributions
Peak flow - Volume
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Results – joint distributions
Volume - Duration
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Results – joint distributions
Peak flow - Duration
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Results – conditional distributions
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Results – conditional distributions
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Results – conditional distributions
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Results – return period
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Conclusions Concept of copula is used for evaluating joint
distribution function with mixed marginal distributions - eliminates the restriction of selecting marginals for flood variables from the same family of probability density functions.
Nonparametric methods (kernel density estimation and orthonormal series) are used to determine the distribution functions for peak flow, volume and duration.
Nonparametric method based on orthonormal series is more appropriate than kernel estimation.