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ICWRER 2013 | Flood Frequency Analysis at River Confluences …
Flood Frequency Analysis at River Confluences – Univariate vs. Multivariate Extreme Value Statistics
Jens Bender1,2 · Thomas Wahl2,3 · Christoph Mudersbach1 · Jürgen Jensen1,2
1 University of Siegen, Research Centre for Water and Environment · 2 University of Siegen, Institute of Advanced Studies · 3 University of South Florida, College of Marine Science
AbstractIn this study we analyze the combined flood probability at a confluence using different
statistic procedures. The study is exemplarily carried out at the confluence of the rivers
Ilz and Wolfsteiner Ohe in Germany, where long time series of the hourly discharge
are available at both rivers upstream of the confluence as well as downstream. On
the one hand we perform a univariate statistical flood frequency analysis upstream
and downstream of the confluence as it is applied commonly at major German rivers.
The aim is to determine the statistical relevant inflow of the tributary for several given
design discharges at the main stream. On the other hand we perform a bivariate
statistical analysis using Archimedean copula functions at both streams upstream of
the confluence. Comparing the results highlights the limited capability of the univariate
approach to determine the statistical relevant inflows from the tributary. In particular
for higher return period discharges at the main stream, the resulting inflows from the
tributary differ from the results of the bivariate statistical analysis.
1. Introduction
Design discharges at rivers are required for many engineering purposes, e.g. numerical
inundation modeling, design of flood protection structures, etc. As an alternative to
rainfall-runoff modeling, a broad variety of univariate statistical methods are available to
determine the design discharges (e.g. HQ100) along rivers based on recorded discharge
data (see e.g. Rao and Hamed, 2000). However, gauge data upstream of a confluence does
not provide information on the statistical relevant inflow from the tributary. Especially
for flow modeling along a river the inflow boundary conditions of tributaries are of great
interest. Not at least because inflows affect the water levels downstream of the confluence
and can cause hydraulic effects, like turbulences and tailbacks. In Germany only few
approaches exist to estimate the statistical relevant discharge boundary conditions at
a tributary for a given design flood at the main stream and vice versa. One of those
approaches is the so called "Confluence Formula" (German: Mündungsformel). Today, the
Confluence Formula is a widely used approach to estimate the discharge conditions at
river confluences in Germany. The formula was developed by the Federal Department
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of Environment of the state Baden-Wuerttemberg on basis of gauge records at a single
confluence of two rivers. Due to the lack of available methods, however, the Confluence
Formula is often applied to other ungauged rivers neglecting different topographical and
hydrological boundary conditions of the catchments.
Several studies have been carried out considering the bivariate nature of this concern.
Morris and Calise (1987) estimated the joint flood probability at a tributary, which is
influenced by the water level in the main stream. They used bivariate density functions to
describe the dependence between the water level in the main stream and the tributary.
Raynal and Salas (1987) were the first who used the bivariate General Extreme Value
(bGEV) distribution in the context of joint flood risk at the confluence of the rivers Bear
and Dry Creek in California, USA. They compared four approaches to estimate the total
discharge downstream of a confluence: (i) with the sum of the recorded discharges, (ii)
under the assumption of a perfect linear dependence (i.e. correlation of one) between
the main stream and tributary discharges, (iii) under assumption of independence
(correlation of zero) between the discharges, and (iv) with the use of the bGEV considering
a correlation of r = 0.86. They suggested that the correlation coefficient was a suitable
measure to describe the dependence and they proposed the application of the bivariate
Normal distribution function for estimating the joint probability of floods at confluences.
Over the last years copula functions have been used for several multivariate hydrological
analyses. They were applied for rainfall frequency analysis (e.g. De Michele and Salvadori,
2003, Grimaldi and Serinaldi, 2006, Zhang and Singh, 2007), flood frequency analysis
considering peak flow and flood volume (e.g. Favre et al., 2004, Zhang and Singh, 2006,
Karmakar and Simonovic, 2009), drought frequency analysis (e.g. Shiau, 2006, Kao and
Govindaraju, 2010, Song and Singh, 2010a, b), storm surge modelling (e.g. Wahl et al.,
2012), and for several other multivariate problems.
For flood risk analyses at river confluences copulas were already used by Wang et al.
(2009). They presented a copula-based algorithm to determine the joint probability at a
river confluence using copulas of the Archimedean family. Using Monte-Carlo simulations
they determined the joint probability of a concurrent occurrence of flood events in both
streams.
NCHRP (2010) first presented a general approach to estimate the joint flood risk at
ungauged rivers for the design of road drainage structures. The study was based on 83
homogeneous distributed gauge pairs throughout the USA. Basically three multivariate
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methods were applied: bivariate distribution functions, copula functions and the total
probability method. For the generalization of the results from individual gauge pairs they
used the ratio of the confluent catchments.
Chen et al (2012) used 4-dimensional Copula functions to model the coincident risk of
concurrent occurring flood magnitudes and dates at confluences. Case studies were the
upper Yangtze River in China and the Colorado River in the United States.
At German major rivers, where a dense network of discharge gauges exists, this bivariate
issue is however still reduced to a univariate problem. This allows the application of
common extreme value models to the data sets from all gauges along the stream. The
difference between two design discharges derived by two neighbored gauges is then
considered as the statistical relevant discharge of all tributaries between the gauges.
If there is more than one tributary between two gauges the inflow of the individual
tributaries is weighted according to the catchment sizes (BfG, 2009).
In the present study we conduct two analyses of the joint flood occurrence at a gauged
river confluence in Germany. First, the simplified univariate approach is applied as it is
done at the German major rivers. In a second step we use Archimedean copula functions
to determine the joint flood risk. The main intention of this study is to compare the
results of both approaches to determine their capability of assessing the joint flood risk
at the confluence.
2. Data
2.1. General This study is exemplarily carried out at the confluence of the river Ilz and its tributary
Wolfsteiner Ohe in Bavaria, Germany. Hourly discharge data are available at river Ilz
some 2.2 km upstream (gauge Schrottenbaummühle) as well as 3.6 km downstream
(gauge Kalteneck) of the confluence. The river Wolfsteiner Ohe has a gauge (Fürsteneck)
approx. 2.0 km upstream the confluence with the river Ilz (see Figure 1). All time series
were provided by the Federal Environment Authority of Bavaria and cover the period
from 1972 to 2011, i.e. 40-yr long time series without gaps or discontinuities are available
for the analyses.
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Figure 1 Schematic illustration of gauge locations at the confluence of the rivers Ilz and Wolfsteiner Ohe.
River Ilz drains at gauge Schrottenbaummühle a total area of 364 km² upstream of the
confluence, whereas river Wolfsteiner Ohe contributes at gauge Fürsteneck an almost
equal drainage area of 370 km². The mean discharge at gauge Schrottenbaummühle
amounts to MQs = 7.6 m³/s with a standard deviation of ss = 7.7 m³/s. At gauge Fürsteneck and
gauge Kalteneck the mean discharges and the standard deviations amount to MQF = 8.5 m³/s, sF = 8.0 m³/s, and MQK = 16.6 m³/s, sK = 16.3 m³/s, respectively. The highest discharges
at all three gauges were observed on 21st December 1993 with HHQS = 208.1 m³/s at
gauge Schrottenbaummühle and HHQF = 192.4 m³/s at gauge Fürsteneck which in turn
resulted in a maximum discharge at gauge Kalteneck of HHQK = 416.9 m³/s. Although the
highest measured flood peaks occurred at the same day, thorough investigations show
that 13 of the total 40 annual maximum flood peaks at rivers Ilz and Wolfsteiner Ohe did
not occur concurrently within a time frame of ±7 days.
2.2. Measurement Inaccuracies
A general problem in flood frequency analysis based on discharge data sets is related to
inaccuracies of the measurements. In particular during extreme events the extrapolated
water level-discharge function (discharge curve) often leads to large errors since discharge
curves are calibrated on measurements during low or medium runoffs (Maidment, 1992).
In the case at hand considerable discrepancies between the sum of the gauge data
upstream and the gauge data downstream of the confluence can be found, especially
during flood events. Although no information is available on incorrect operation of any
of the gauges, these discrepancies would affect the results of the statistical analysis. For
that reason and by having the exemplary nature of this study in mind, the discharge
data at gauge Kalteneck is replaced by a synthetic time series derived by the sum of the
discharges observed at the two gauges located upstream:
Q t Q t Q tKalteneck syn Schrottenbaummühle Fürsteneck( ) ( ) ( )= +
,��� (1)
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3. Methods
3.1. Simplified univariate ApproachThe univariate approach only uses the discharge time series at the main stream, up- and
downstream of the confluence, which are in this case the gauges Schrottenbaummühle
and Kalteneck (synthetic). A common univariate flood frequency analysis is carried out
for both time series using extreme value distribution functions. The annual maximum
discharges (AMAX) of the hydrological years (in Germany from 1 November until 30
October) is considered as the relevant flood indicator. For catchments smaller than
50,000 km², as it is the case here, Svensson et al. (2005) suggested considering a minimum
time interval of five days as independence criterion in order to separate consecutive
flood events.
In a next step, univariate distribution functions are fitted to the AMAX series of the gauges
at the main river upstream and downstream of the confluence. Although the General
Extreme Value distribution (GEV) is often being treated as one of the main distribution
functions for extreme value analyses (e.g. Coles, 2001), other distribution functions are
additionally fitted to the data set. The most appropriate distribution function is identified
by the minimum root mean square error (RMSE) of the empirical distribution and the
parametric distribution function. The distribution parameters are estimated with the
maximum likelihood approach (e.g. Rao and Hamed, 2000) and the plotting positions are
derived by following the approach proposed by Gringorten (1963). Further Information
about flood frequency analyses can be found e.g. in Rao and Hamed (2000).
Next, all relevant quantiles of the parametric distribution functions are determined. In
this study we focus on the common quantiles in flood frequency analyses for the non-
exeedance probabilities of P = 0.5, 0.8, 0.9, 0.95, 0.98 and 0.99 with the corresponding
annual return periods of T = 2, 5, 10, 20, 50 and 100 years. Since higher return periods
then these are usually of minor interest for design purposes, they are not considered in
this study. The difference of the downstream quantiles and the upstream quantiles are
treated as the statistical relevant inflow of the tributary for a given flood event at the
main stream with an annual return period of T.
3.2. Bivariate Approach using Copula Functions
3.2.1. Theoretical Background
Copulas are flexible joint distributions for modeling the dependence structure of two or
even more random variables. First mentioned by Sklar (1959), the joint behavior of two (or
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more) random variables X and Y with continuous marginal distributions u = FX(x) = P(X ≤ x) and v = FY(y) = P(Y ≤ y) can be described uniquely by an associated dependence function
or copula-function C. In the bivariate case, the relationship between all (u,v) Є [0,1]² can
be written as
F x y C F x F y C u vX Y X Y, , , ,( ) ( ) ( )[ ] ( )= = (2)
where FX,Y (x,y) is the joint cumulative distribution function (cdf) of the random variables
X and Y.
A copula function with a strictly monotonically decreasing generator function
φ: [0,1] → [0,∞] with φ(1) = 0 belongs to the Archimedean Copula family. The general
form of one-parametric Archimedean copulas is
C u v u vθ ϕ ϕ ϕ, -( ) ( ) ( )[ ]= +1 (3)
where θ denotes the copula parameter. In this study three Archimedean copulas, namely
the Clayton, Frank, and Gumbel copulas are considered. They are relatively easy to
construct, flexible and capable to cover the full range of tail dependencies. The Clayton
copula has lower tail dependence, while the Frank copula has no tail dependence and
the Gumbel copula has strong upper tail dependence (Schölzl and Friedrichs, 2008). The
copula parameters are estimated based on the inversion of Kendall’s τ. This is possible as
there exists an expression for τ as a function of θ for Archimedean copulas (see Table 1).
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Table 1: Archimedean copula functions considered for the present study and their generator functions, ranges for the copula parameters θ and functional relationship to Kendall’s τ.
Copula function Cθ Generator φ(t)**
Range of θ Functional relationship of θ to τ
Clayton or Cook-Johnson
u v− −−
+ − θ θ θ1
1
1θt − − 0,∞[ ) 2θ
θ +
Frank
( ) ( )1 11ln 1
1
θu θv
θ
e e
θ e
− −
−
− −− +
−
1ln1
θt
θee
−
−
−− −
( ) { }0, \−∞ ∞ ( )[ ]1
41 1 D θ
θ− − *
Gumbel or Gumbel-Hougaard
( ) ( )1
exp ln lnθ θ θu v− − + −
( )ln θt− [ )1,∞ 11 θ −−
* 1. Debye Function ( )1
0
1
1
θ
t
tD θ dt
θ e=
−∫** t = u or t = v
Further important features of copulas and information about the theoretical background
can be found e.g. in Nelsen (1999), who provided a detailed introduction to the subject.
4. Results
4.1. The univariate approachIn the univariate case we fitted distribution functions to the AMAX series of gauges
Schrottenbaummühle and the synthetic gauge Kalteneck. In both cases the GEV
distribution provides the best fit. Figure 2 shows the result from fitting the GEV including
the upper and lower 95% confidence levels.
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Figure 2 Fitted GEV distributions to AMAX series of gauge Schrottenbaummühle upstream of the confluence (A) and the synthetic gauge Kalteneck downstream of the confluence (B) including the upper and lower 95% confidence levels.
Table 2 shows the numeric quantiles of both fitted distributions. It can be seen that the
analyzed quantiles of the downstream located synthetic gauge Kalteneck varies between
151.1 m³/s and 390.7 m³/s for P = 0.5 (T = 2 a) and 0.99 (T = 100 a), respectively. At gauge
Schrottenbaummühle the values vary between 80.6 m³/s and 214.7 m³/s for the same
quantiles.
Table 2: Quantiles of the fitted distributions at gauge Schrottenbaummühle and the synthetic gauge Kalteneck and their differences
P 0.5 0.8 0.9 0.95 0.98 0.99 [ - ]
Return period 2 5 10 20 50 100 [a]
HQT - Kalteneck (syn.) 151.1 207.8 248.2 289.1 345.6 390.7 [m³/s]
HQT – Schrottenbaumm. 80.6 111.1 133.3 156.2 188.3 214.5 [m³/s]
Difference 70.5 96.7 114.9 132.9 157.3 176.2 [m³/s]
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The difference of the same quantiles is than treated as the statistical relevant inflow of
river Wolfsteiner Ohe for the corresponding flood events at river Ilz. These values range
between Q = 70.5 m³/s (P = 0.5, T = 2 a) and Q = 176.2 m³/s (P = 0.99, T = 100 a).
4.2. The bivariate approach using Copulas
In contrast to the univariate approach, the time series of gauge Kalteneck does not
play a role in the bivariate analysis. In this case, only the discharge series of both
gauges upstream of the confluence are considered (gauges Schrottenbaummühle and
Fürsteneck). Before choosing a suitable copula type, the marginal distributions need
to be fitted. Here, in turn, the question arises which data should be modeled. In many
bivariate statistical investigations mostly the AMAX series of both variables are modeled,
e.g. the annual maximum peak flow series and the annual maximum flood volume series,
despite considering possible different background of their genesis and physical relations.
Here, we model the AMAX series of the main stream at gauge Schrottenbaummühle
together with the concurrent flows of the tributary at gauge Fürsteneck. The fact that
13 of the total 41 annual maxima values did not occur within a time frame of ±7 days
confirms that this procedure is reasonable. For the sake of minimizing the randomness
of the concurrent discharges, we consider the maximum discharge at gauge Fürsteneck
within a time frame of ±7 days with reference to the occurrence time of the AMAX
values at gauge Schrottenbaummühle. We already showed that the GEV fits the AMAX
series at gauge Schrottenbaummühle best (see Figure 2). For the gauge Fürsteneck, the
2-parametric Weibull distribution provides the best fit (see Figure 3).
Figure 3 Fitted 2-parametric Weibull distribution to concurrent discharge values at gauge Fürsteneck with reference to the occurrence of the AMAX values at gauge Schrottenbaummühle
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In order to choose the most appropriate Archimedean copula function, we applied the
Bayesian method for copula selection as suggested by Huard et al. (2005). The Gumbel
copula appears to the best model to describe the dependence structure between the
AMAX values at gauge Schrottenbaummühle and the concurrent discharges at gauge
Fürsteneck. With a given value of Kendall’s τ = 0.567 the copula parameter θ can be
calculated using the functional relationship of τ and the Gumbel copula parameter as
outlined in Table 1 to θ = 2.3077.The results from fitting the Gumbel copula (with the
above mentioned marginal distributions) to the data set are illustrated in Figure 4.
Figure 4 Isolines of equal return periods derived by the bivariate analysis compared with the results of the univariate approach.
The black crosses in Figure 4 show the observed AMAX values at gauge Schrotten-
baummühle and the concurrent discharges at gauge Fürsteneck. The grey dots are 10,000
synthetic values derived from the fitted copula and the marginal distribution functions
giving an optical impression of the goodness of the fit. The black lines illustrate the
bivariate isolines of equal quantiles or return periods, respectively. Here, the bivariate
return period of a certain quantile is defined as the inverse value of the non-exeedance
probability. Other approaches defining the return period in multivariate cases can be
found in Gräler et al. (2013).
For comparison purposes the blue crosses illustrate the results of the univariate method
as described in section 5.1. It can be seen, that for small return periods (up to T = 20 a),
the probabilities of the concurrent flows (on the ordinate) of both approaches agree
well. However, the values for higher return periods, i.e. T = 50 and 100 a, deviate by
up to 20 m³/s (12.5 %). This might be caused by the use of the Weibull distribution in
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the bivariate case and the GEV in the univariate case; both distributions have different
curvatures in the range of higher return periods (see Figures 2A and 3).
Although the statistically relevant inflows of the tributary derived with the univariate
approach generally correspond with the results using copulas, the return periods of the
univariate discharge combinations are higher as compared to the bivariate approach.
This is highlighted in Figure 4 by the fact that the results from the univariate analysis
lie generally in the upper right of the isolines from the bivariate analysis. The discharge
with a return period of T = 10 a in the univariate case would be classified, despite almost
equal marginal values, as an event with a return period of T " 15 a in the bivariate case
(considering the above mentioned definition of bivariate return period). Using the
univariate approach assumes independent variables which is not valid in this case where
the modeled variables have a rank correlation of τ = 0.567.
5. Conclusion
The main objective of this study was to compare the results from univariate and bivariate
statistical extreme value analysis at a river confluence. The application of both methods
to a case study in Germany with long discharge time series at both confluent rivers shows
that the simplified univariate approach is capable to determine the statistical relevant
discharges of the tributary given design discharges for the return periods of T = 2, 5, 10
and 20 years at the main stream (i.e. the results of the univariate and bivariate approach
are similar). However, significant differences are found for higher return periods, i.e.
T = 50 and 100 a. Whether this result is transferable to other confluences, in particular
to major rivers, will be further investigated. Moreover, classifying the results of the
univariate approach according to bivariate probabilities shows that the return periods
are generally overestimated. This is due to the fact that using univariate approaches
does not allow for modeling the dependence structure of the two variables.
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DOI: 10.5675/ICWRER_2013
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DOI: 10.5675/ICWRER_2013
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