8. Forum DKKV/CEDIM: Disaster Reduction in Climate Change 15./16.10.2007, Karlsruhe University
Estimation of upper bounds using envelope curves
Björn Guse1,2, Annegret Thieken2, Bruno Merz2
1
Center for DisasterManagement and Risk Reduction Technology (CEDIM),
Section Engineering Hydrology, GeoForschungsZentrum Potsdam, Telegrafenberg, Potsdam, Germany,
E-Mail:[email protected], phone: +49-0331-288-1896
2
1 Motivation
In the last years flood events have caused high damages in Germany. The most destructive event was
the Elbe flood in 2002. The estimated costs amounted to 11.6 billion € for Germany alone.
The return period of flood events can be estimated by flood frequency analysis. Vice versa, a flood
frequency analysis allows the calculation of flood discharges at distinct return periods. Due to short
time series and rare extreme events the results of a flood frequency analysis are uncertain, especially
for return periods of more than
100 years (e.g. Institute of
Hydrology, 1999). Fig. 1
illustrates that the differences
between
six
traditional
distribution functions are rising
with increasing return period.
Commonly used extreme value
distributions are unbounded and
yield
unreasonably
high
discharge value for low
probabilities.
To improve the flood frequency
analysis for high return periods,
additional
information
is
necessary
(Merz,
2007).
Therefore this work is aimed at
deriving upper bounds and to
include them in the distribution
function.
Fig.1: Comparison of different distribution functions at the gauge Bad Dueben/ Vereinigte Mulde
2 Envelope curves
Empirical envelope curves are a traditional method to appraise the upper bound of flood events (e.g.
Castellarin et al., 2005; Mimikou, 1984). An envelope curve shows the relationship between the flood
of record of a gauge site and its catchment area in a log-log-diagram.
Since the idea of envelope curves emerged (Jarvis, 1925), this method has been applied worldwide at
different scales. Envelope curves were derived for Europe and the World (Herschy, 2002) or at lower
scale, for example, for part of Greece (Mimikou, 1984).
8. Forum DKKV/CEDIM: Disaster Reduction in Climate Change 15./16.10.2007, Karlsruhe University
Using the principle “trading space for time” the floods of record at more than hundred gauges in
Saxony were collected (green dots in Fig. 4). This data set was complemented by data from
comparable regions. In the World Catalogue of Maximum Observed Flood (Herschy, 2003) Saxony is
classified as a mountainous region with humid continental climate and a short summer. Assuming that
all gauge sites which also belong to this climate region have similar flood behaviour, gauge sites from
northern Bavaria, the Czech Republic and the southwest of Poland were added to the data set (red dots
in Fig. 2).
On the basis of these data an envelope curve was estimated with an empirical approach: First, the
flood of record of a specific interval of catchment area (for example between 100 and 200 km²) was
selected and a line between two adjacent floods of record was drawn. In the next step, the mean slope
of all these lines was calculated. Finally, the envelope curve was moved upwards until it is located
above all data.
Fig. 2 shows an envelope curve based on floods of record from Saxony and the same climate region.
Especially for a catchment size between 1000 and 50.000 km² the additional data supports the
envelope curve.
100000
Unit discharge [L/(s*km²)]
10000
1000
100
10
1
1
10
100
1000
10000
100000
1000000
Area [km²]
Flood of records in identical climate region
Flood of records in Saxony
Envelope Curve for Saxony and identical climate region
Fig. 2: Empirical envelope curve for Saxony including floods of record from the same climate region
3 Distribution function with upper bounds
To improve flood frequency analysis at high return periods, the upper bound was derived from Fig. 2
for each gauge and was integrated as supplementary information in a distribution function (red line in
Fig. 3). For this purpose, the extreme value distribution with four parameters (EV4) was chosen. The
parameters were calculated by the Maximum-Likelihood-method with the AFINS-Tool (Botero and
Francés, 2006).
8. Forum DKKV/CEDIM: Disaster Reduction in Climate Change 15./16.10.2007, Karlsruhe University
Fig. 3 shows a comparison between
the EV4 and the unbounded
generalized extreme value distribution
(GEV) at the gauge of Bad Dueben/
Vereinigte Mulde. The comparison
shows that the discharge is similar for
short and middle return periods. For
return periods of more than 100 years
the EV4 estimates lower flood
discharge because of the asymptotic
approach to the upper bound.
Fig.3: Comparison of flood quantiles between the generalized extreme value distribution (GEV) and the extreme
value distribution with four parameters (EV4) at the gauge Bad Dueben/ Vereinigte Mulde
4 Summary and conclusion
In this article the idea to improve the extreme value statistic for high return periods using a distribution
function with upper bound is illustrated for floods in Saxony. The upper bound of flood discharges
was determined on the basis of observed floods of record in Saxony. To fill gaps in the data collective
additional data from the same climate regions were used. The Saxonian envelope curve provides an
upper bound for each gauging station, which can be integrated into a flood frequency analysis with
EV4. By this approach, the estimation of discharge for high return periods seems to provide more
realistic discharge estimation for high return periods
Acknowledgement
We thank the State Agency of Environment and Geology of the Free State of Saxony for the
permission to use the discharge data at the gauge sites.
Literature
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