Simulation methods for design flood analyses require estimates of extreme precipitation for simulating maximum discharges. This article evaluates the multi-exponential weather pattern (MEWP) model, a compound model based on weather pattern classification, seasonal splitting and exponential distributions, for its suitability for use in Norway. The MEWP model is the probabilistic rainfall model used in the SCHADEX method for extreme flood estimation. Regional scores of evaluation are used in a split sample framework to compare the MEWP distribution with more general heavy-tailed distributions, in this case the Multi Generalized Pareto Weather Pattern (MGPWP) distribution. The analysis shows the clear benefit obtained from seasonal and weather pattern-based subsampling for extreme value estimation. The MEWP distribution is found to have an overall better performance as compared with the MGPWP, which tends to overfit the data and lacks robustness. Finally, we take advantage of the split sample framework to present evidence for an increase in extreme rainfall in the southwestern part of Norway during the period 1979–2009, relative to 1948–1978.
Flood estimation is important for design and safety assessments,
flood risk management and spatial planning. It aims to assess the
probability of occurrence of large events, e.g., discharges with return periods
of 100 to 10 000 years. Estimation of events with such low probability is particularly arduous.
It can only be based on a few data points representing the most extreme events in a time series of a
limited length. Thus extrapolation to long return periods is usually needed. In dam safety analyses,
for example, return period estimations of
In Norway, a simple event-based rainfall–runoff model, PQRUT, has been used
since the 1980s as a simulation method for dam safety analyses for which the
magnitude of low frequency events (e.g., 500-, 1000-year peak inflow) and the
probable maximum flood are required. Recently, a semi-continuous model,
SCHADEX
Daily data from 368 precipitation stations in Norway were extracted from
the European Climate Assessment and Dataset (ECA&D), a database of daily
meteorological stations across Europe. From these 368 stations, 192 stations
with at least 50 years of record with less than 10 % missing data per
year over the period 1948–2009 were selected for further analyses. Years
with more than 10 % missing data are entirely replaced by 'NA',
representing missing values. Figure
Left: location and altitude (m a.s.l.) of the stations. Right: histogram of altitude (m a.s.l.).
As already stated in Sect.
Let
In the previous section, we implicitly assumed that central rainfall,
Use of the EXP, GPD, MEWP and MGPWP models
requires the choice of high enough thresholds such that EVT can be applied.
Selection of an adequate threshold gives rise to a bias-variance tradeoff:
the higher the threshold, the better the approximation
of the tail of
Given
Parameters
The
The goal of this evaluation is to assess which model performs better at the
regional scale, i.e., for a set of
The SPAN criterion evaluates the stability of the return level estimation,
when using data for each of the two subsamples. More precisely, for a given
return period
Graphical tools for model evaluation based on
The
The
We wish to evaluate and compare the performance of EXP, GPD, MEWP and MGPWP
for estimating central rainfall values across Norway. To apply the split
sample procedure described in Sect.
As is always the case for extreme value analysis, threshold choice is
uncertain. We therefore considered a large set of thresholds with
The estimation scheme can be summarized as follows. For each of the
considered EXP MEWP MEWP MEWP
and the six corresponding models with the GPD distribution. This gives in
total
For the cases involving the use of WP, we employ the weather-type (WT)
classification described in
Weather pattern classification with four classes (denoted WT1 to WT4 above) and eight
classes (WP1 to WP8 above). This is Fig. 5 of
In cases where subsampling is also undertaken by season, we impose a
restriction of Step 1: compute the 12 mean monthly maxima of
central rainfall. Step 2: set Step 3: compute the mean of these values over moving windows of size Step 4: select the Step 5: fit the considered model (e.g., MEWP Step 6: compare the monthly fits to the monthly empirical distributions. This comparison
is made with the KGE score (Kling–Gupta efficiency, where Step 7: compute a global KGE score as a weighted mean of these 12 KGE scores, with
weights proportional to the mean monthly maxima, in order to force the model to have the best
fits for the months with the highest risk. Step 8: set Step 9: set Step 10: compare the three global KGE scores obtained respectively for
This procedure is applied for each station and each model separately. This
implies that, for a given station, the choice of season may vary among
models. However, it was found that changes in the definition of the
season-at-risk for a given station are very minimal (i.e., a few percent
difference, and always pertain to the intermediate months that could well be
classified into either of the two periods). We believe that these
differences have very little influence on the evaluation of the model fits.
For illustration, Fig.
Length of the season-at-risk (shapes) and first month of the
season (color code in the inset) for each station, with model MEWP
The SPAN,
Scores are reported in Figs.
Scores of evaluation for MEWP models, for
Same as Fig.
Comparison of the 100-year and 1000-year return levels (in mm)
estimated on C
Left: estimated
Figure
Figure
Figure
Regarding the choice of threshold, MEWP distributions give relatively stable
scores for
It is interesting at this point to compare large return levels obtained with
the selected MEWP
Box plot of the difference (in mm) between the 100-year return
levels of MEWP
Scores of evaluation for the local and regional definition
of the seasons. Better scores have values closer to
Divergence in density between
We have already mentioned in Sect.
The split sample procedure can be used to give insight about potential change
in extreme rainfall in Norway over the period represented by the rainfall
time series. For this we split the observed years of each station into two
subsamples: C
Box plot of the difference in 100-year return level estimated for C
Left: map of 100-year return level estimated on C
We see that
This article evaluates a compound model based on weather pattern classification, seasonal splitting and exponential distributions, the so-called MEWP model, for its suitability for use in Norway. The MEWP model is the rainfall probabilistic model used within the SCHADEX method, which is currently being tested in Norway as an alternative simulation method for flood estimation. We show in particular the benefit gained by subsampling the heavy rainfall data according to season and weather pattern. Our results also indicate that models based on the exponential distribution perform better than those based on the more flexible generalized Pareto distribution, which tends to overfit the data and lacks robustness. We have also demonstrated that a regional definition of seasons in MEWP is possible. Finally, we give evidence for an increase in extreme rainfall intensities in Norway in recent years, particularly in the southwestern region.
Our analysis has also shown that the GPD distribution better models the bulk of the distribution of extremes, but fails to robustly estimate the tail, and therefore fails in extrapolation to large return levels. The reason for this failure is twofold: firstly, the lack of data for estimating such a flexible distribution when using a local approach; secondly, the inherent nature of the GPD, which is a heavy-tailed distribution when the shape parameter is positive, and can therefore tend to give unrealistic return levels for very long return periods. To address this issue, a regional approach allowing the use of neighboring stations to infer MEWP distributions at local sites is of interest. Finally, there are also other, more flexible, distributions which may be more robust than the GPD distribution and could be used within the MEWP approach. This also represents an important topic for future work.
This study is the first extensive evaluation of MEWP in Norway. It has also
been applied successfully in France
Deborah Lawrence acknowledges support from the project “FLOMQ – Robust framework for estimating extreme floods in Norway” supported by the ENERGIX program of the Norwegian Research Council, as well as internal research funds from NVE which have made the collaboration with LTHE possible. The authors acknowledge support from the COST Action ES0901 FloodFreq which has made the collaboration between NVE and EDF possible, including a Short Term Scientific Mission for Anne Fleig to EDF. Edited by: P. Tarolli Reviewed by: two anonymous referees