INTERNATIONAL JOURNAL OF CLIMATOLOGY
Int. J. Climatol. 31: 1457–1472 (2011)
Published online 13 July 2010 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/joc.2182
Comparison of regional and at-site approaches to modelling
probabilities of heavy precipitation
Jan Kyselý,a * Ladislav Gaála,b and Jan Picekc
a
Institute of Atmospheric Physics AS CR, Prague, Czech Republic
b Slovak University of Technology, Bratislava, Slovakia
c Technical University, Liberec, Czech Republic
ABSTRACT: Several approaches to estimating distributions of precipitation extremes are compared by means of simulation
experiments, and their applications into observed data in the Czech Republic are evaluated. Regional frequency models,
which take into account data in fixed or flexible ‘regions’ when fitting a distribution at any site, lead to estimates with much
smaller errors compared to a single-site analysis, and efficiently reduce random and climatologically irrelevant variations
in the estimates of the model parameters and high quantiles. The region-of-influence (ROI) methodology with a built-in
regional homogeneity test is recognized as a useful approach, with the model based on proximity of sites outperforming
the Hosking–Wallis regional frequency analysis. Comparison of estimates of the return period of a heavy precipitation
event on 24 June 2009, which triggered a disastrous local flash flood, illustrates that the at-site analysis leads to unrealistic
and extremely uncertain estimates that strongly depend on whether or not a single outlying observation is involved in the
sample, while all regional methods yield return periods in the same order of several hundreds of years, notwithstanding
whether the 2009 data are included in the sample. The ROI method may be found useful for modelling probabilities of
other meteorological variables, extremes of which are strongly influenced by sampling variability, and may also represent
an efficient tool for ‘smoothing’ random variations in the estimates of model parameters and high quantiles of precipitation
in high-resolution regional climate model simulations. Copyright  2010 Royal Meteorological Society
KEY WORDS
extreme value analysis; regional frequency analysis; precipitation extremes; region-of-influence method; central
Europe
Received 8 January 2010; Revised 1 May 2010; Accepted 3 May 2010
1.
Introduction
Extreme precipitation events are associated with large
negative consequences for human society, mainly as they
may trigger floods and landslides. The recent series of
flash floods in central Europe between 24 June and 4
July 2009, the worst one over several decades in the
Czech Republic as to the number of persons killed and
the extent of damage to buildings and infrastructure, may
serve an example. The series started with a local flash
flood in the Odra River basin of the Nový Jičı́n district
in the late evening of 24 June 2009, leaving ten people
dead and causing extensive damage. The precipitation
amount recorded at lowland site Bělotı́n, 124 mm in 24 h,
far exceeded the previous record-breaking value in this
location.
Estimates of growth curves and design values of
precipitation amounts (corresponding to 50- and 100-year
return periods) and their uncertainties are important in
hydrological modelling, design of hydraulic structures,
urban and landscape planning, etc. The interest in high
quantiles of precipitation distributions is also related
* Correspondence to: Jan Kyselý, Institute of Atmospheric Physics AS
CR, Bočnı́ II 1401, 14131 Prague 4, Czech Republic.
E-mail: [email protected]
Copyright  2010 Royal Meteorological Society
to possible climate change effects, as climate model
simulations – although not capable of reproducing some
important features of observed patterns, which reduces
their credibility – tend to project increased severity of
precipitation extremes over central Europe in a warmer
climate (Christensen and Christensen, 2004; Beniston
et al., 2007; Kyselý and Beranová, 2009).
The present study compares several up-to-date methods
of modelling probabilities of precipitation extremes that
make use of regional approaches (Burn, 1990; Hosking
and Wallis, 1997). Unlike a single-site analysis (termed
‘at-site’ throughout the paper), which estimates probability distributions from observations at individual locations
separately, a regional method makes inference by taking into account data in a ‘region’ (pooling group), in
which one may assume that the distributions of extremes
are identical apart from a site-specific scaling factor. The
condition is referred to as regional homogeneity and its
validity is evaluated by statistical tests (Hosking and Wallis, 1997; Viglione et al., 2007). In other words, all data in
a region – often weighted in some way – are taken into
account when estimating the distribution of extremes at
a given site. The advantage is that sampling variations
in the estimates of model parameters and high quantiles
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J. KYSELÝ et al.
may be substantially reduced compared to the single-site
analysis, and the inference becomes more robust.
The region-of-influence (ROI ) method, originating in
hydrology (Burn, 1990; Castellarin et al., 2001), is one of
the variants of the regional methods: it identifies unique
(flexible) pooling groups for each site under study, which
form the database for the estimation. The similarity of
sites is evaluated in terms of a set of site attributes that
are supposed to be linked to the probability distributions
of extremes, and the assumption of regional homogeneity
of data may be verified by a built-in regional homogeneity
test (Zrinji and Burn, 1994). Weights are assigned to
data from individual sites in a pooling group according
to a dissimilarity measure, and all (weighted) data are
employed in the estimation of model parameters and
high quantiles at a given site. Most studies that made
use of the ROI method (or, in a broader sense, pooling
methodologies) dealt with hydrological data (high flows)
while applications into precipitation data have been rare
(Schaefer, 1990; Di Baldassare et al., 2006; Gaál et al.,
2008).
Another concept of the regional frequency analysis,
the Hosking–Wallis (HW ) regional approach (Hosking
and Wallis, 1993, 1997), is based on delineating fixed
regions (instead of flexible pooling groups) and assigning
the same weights to all sites in a region (provided that
the record lengths are also identical; for details refer to
Section 3.3). The methodology has been widely applied
into modelling probabilities of heavy precipitation in
many European regions, including the United Kingdom
(Fowler and Kilsby, 2003), Belgium (Gellens, 2002),
Slovakia (Gaál, 2006), Italy (Norbiato et al., 2007) and
the Czech Republic (Kyselý and Picek, 2007). In the latter
study, the area of the Czech Republic was divided into
four homogeneous regions; this regionalization is revised
in the present paper (Section 3.2), after data from a dense
network of rain-gauges became available, since some of
the original regions were found heterogeneous with more
data involved in the tests.
The particular goals of the present study are twofold.
In the first part (Section 4), we compare performance
of individual regional frequency models (several variants of the ROI method that differ in site attributes
used to measure the similarity of sites, and the HW
analysis) with the at-site estimation in terms of Monte
Carlo simulation experiments, and attempt to identify
the method of the regional frequency analysis that leads
to superior results [measured by the root-mean-square
error (RMSE) of the estimates]. The comparison of the
performance of the individual regional models makes
use of data on annual maxima of 1-day (1D) and
5-day (5D) precipitation amounts at 209 stations covering the Czech Republic over 1961–2007 (Section 2).
This part of the study resumes results of an extensive
simulation study which evaluated a larger set of the ROI
models (Gaál and Kyselý, 2009), and enhances them
by comparing the ROI method against the HW analysis. The dataset has also been updated with respect to
Gaál and Kyselý (2009) by the inclusion of years 2006
and 2007. In the second part (Section 5), the regional
methods are applied into estimating high quantiles of
1D and 5D precipitation amounts at individual sites and
return periods associated with the recent heavy precipitation event on 24 June 2009. The estimates and their
uncertainty are compared with those based on a singlesite analysis, and conclusions concerning advantages and
drawbacks of the individual methods are drawn. The two
parts are referred to as ‘simulation’ and ‘application’
hereafter.
2.
Data
Daily precipitation totals measured at 209 stations mostly
operated by the Czech Hydrometeorological Institute
(CHMI) were used as the input dataset (Figure 1). The
Figure 1. Locations of stations employed in the study, and the nine homogeneous regions of the HW regional analysis.
Copyright  2010 Royal Meteorological Society
Int. J. Climatol. 31: 1457–1472 (2011)
REGIONAL AND AT-SITE MODELS FOR HEAVY PRECIPITATION
altitudes of the stations range from 150 to 1490 m
a.s.l., and the observations at most sites span the period
1961–2007. The dataset is based on that described in
detail in the work by Kyselý (2009) except for that
additional shorter records have been included, and the
series have been extended to the more recent past (2007).
The stations approximately evenly cover the area of the
Czech Republic.
A large majority of the station records (167) comprise
the whole period of 1961–2007; 42 of the 209 stations
have daily data over shorter sub-periods of at least
33 consecutive years (mostly as the stations started
to operate after 1961 or closed before 2007). If a
significant relocation of a station occurred, exceeding
50 m in altitude, either the longer subperiod (of at least
33 consecutive years) before or after the relocation has
been preserved in the dataset, or all data have been
omitted. The overall average record length is 45.8 years.
Occasional minor gaps in the daily records occurred at
45 stations, not exceeding 3 months in total at any of the
sites; we decided to preserve these stations in the analysis
because of their locations in areas insufficiently covered
by rain gauges with complete records. The missing data
were supplemented by interpolating measurements at two
to five nearest locations available in the climatological
database of the CHMI using the methodology described
by Kyselý (2009). The percentage of the missing daily
records in the entire dataset was only 0.05%. Compared
to the regional frequency analysis by Kyselý and Picek
(2007), the current dataset involves a much larger number
of sites (209 against 78), more evenly covers the area
of the Czech Republic, and extends to the very recent
past.
Samples of annual maxima of 1D and 5D precipitation
amounts at each station are further examined. The percentage of stations with a trend significant at the 0.05
level is low and close to the nominal value for both
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characteristics, so the data do not violate the assumption
of stationarity.
Basic features of the precipitation regime of the Czech
Republic, with a focus on extremes, are described in
Kyselý and Picek (2007) and Kyselý (2009).
3.
Methodology
3.1. ROI method
The ROI method consists in identifying unique pooling
groups of stations, which form the database for the
estimation, for each site under study. The method is
presented in detail by Burn (1990) and Gaál and Kyselý
(2009); here the description is confined to key ideas and
settings of the methodology.
The similarity of sites is evaluated in terms of a set
of site attributes; in the present study, the attributes
are (1) geographical characteristics (latitude, longitude,
altitude) and (2) climatological characteristics describing
long-term precipitation regime (mean annual precipitation, mean ratio of precipitation totals in warm and cold
seasons, and mean annual number of wet days; cf Gaál
and Kyselý, 2009). Note that the set of climatological
characteristics makes the latter variant of the ROI method
consistent with the HW regional approach (Section 3.2),
in which the delineation of fixed regions makes use of
cluster analysis based on the same climatological characteristics (Kyselý and Picek, 2007). The site attributes
are used to calculate the Euclidean distance between each
pair of sites in the attribute space. Nevertheless, before
determining the distance metric, the site attributes are
standardized, i.e. divided by their sample standard deviation, in order to account for different units/magnitudes;
for details refer to Gaál and Kyselý (2009). Individual
variants of the ROI models are summarized in Table I; the
hybrid model combines geographical and climatological
characteristics, weights of which were determined in an
Table I. Summary of the ROI models involved in the simulation study.
Model
abbreviation
Similarity of sites measured in terms of
ROIgeo2
ROIgeo3
Geographical distance
Distance in the attribute space of geographical
characteristics (longitude, latitude, altitude)
Distance in the attribute space of
climatological characteristics (mean annual
precipitation, ratio of warm/cold season
precipitation, number of wet days)
Distance in the attribute space of weighted
geographical and climatological characteristics
(see Gaál and Kyselý, 2009 for details)
Distance in the attribute space of site statistics
derived from samples of extremes (the
coefficient of variation, Pearson’s second
skewness coefficient, the 10-year growth factor
estimated using the GEV distribution; Burn,
1990; Gaál and Kyselý, 2009)
ROIcli3
ROIhyb
ROIsta
Copyright  2010 Royal Meteorological Society
Note
Used only for estimating the reference model
Int. J. Climatol. 31: 1457–1472 (2011)
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J. KYSELÝ et al.
extensive simulation study (Gaál and Kyselý, 2009). We
avoid using site statistics derived from the examined
samples of extremes when forming the regions; this is
in accord with the idea that testing regional homogeneity – which makes use of site statistics – should be independent on the process of forming the regions (cf Hosking and Wallis, 1997; Castellarin et al., 2001; Smithers
and Schulze, 2001).
The key issue of the ROI method is to decide on the
number of similar sites that form the pooling group of
any given site, i.e. how to draw the borderline between
sites belonging and not belonging to a pooling group
employed for the estimation. We found useful to link this
issue with a built-in test on regional homogeneity of data,
which evaluates whether the distributions of extremes
may be considered identical after scaling by the at-site
mean. Analogously to Castellarin et al. (2001), Cunderlik
and Burn (2005) and Gaál and Kyselý (2009), the target
(optimum) size of a region is determined according to
the 5T rule (Jakob et al., 1999), which suggests using
five times T station-years of data for the estimation of
a quantile corresponding to return period T . The longest
return period T , estimation of which may be useful, is set
to 200 years (which exceeds return levels usually referred
to in practical applications). This leads to the target size
of the pooling group Nt equal to 22 stations at most sites
(Nt may slightly vary depending on the actual length of
samples involved in a pooling group: at 39 sites, the target
size is 23). Note that for a fair comparison of the ROI
method with the HW analysis (Section 3.2), the target
size is fixed and does not change with T .
We implement the regional homogeneity test of Lu and
Stedinger (1992) when forming the pooling groups in
order to test their regional homogeneity (Appendix). The
procedure of forming the pooling groups is as follows:
1. The initial pooling group with target size Nt corresponding to the 5T rule for T = 200 years is identified
for a given site and distance metric in a given variant of the ROI method (ROIgeo2, ROIgeo3, ROIcli3,
ROIhyb; see Table I for the explanation of acronyms).
2. Homogeneity of the pooling group is tested. If homogeneous, it is the final pooling group and the procedure
terminates. If heterogeneous, the procedure continues
with adding the next closest site to the pooling group
according to a given distance metric, and step (2) is
repeated until homogeneity is reached or all sites (209)
are included.
3. In case that no homogeneous stage is reached by
successively adding sites to the initial pooling group,
the program code returns to the initial stage with
Nt sites and starts searching for a homogeneous
composition by removing the least similar sites from
the proposed pooling group. The first homogeneous
stage then defines the final composition of the pooling
group for the given site.
4. In the worst case when neither building-up nor removing sites leads to a homogeneous pooling group, the
ROI consists of nothing but the target site.
In practice, point (4) is reached extremely rarely
(Table II); for the ROIgeo2 method, which is found superior in the simulation study (Section 4), there is no occurrence of a single-site pooling group at all. Table II also
shows that for majority of sites (66% when averaged over
all variants of the ROI method and the two durations of
precipitation), the initial pooling group is homogeneous;
for another 20% of stations, homogeneity is reached in
the building-up procedure (by adding the next closest
sites, step 2), and for the remaining 14% of stations,
homogeneity is reached by removing sites from the initial
pooling group (step 3).
Once a pooling group is delineated, weights based on
a dissimilarity measure are assigned to individual sites
involved in the pooling group (Section 3.3), and all
(weighted) pooled data are employed in the estimation
of the model parameters and high quantiles at a given
site.
3.2.
HW regional analysis
The regional analysis of Hosking and Wallis (1993, 1997)
consists in delineating fixed homogeneous regions in
Table II. Statistics of the size of the pooling groups formed by the ROI method for 1D and 5D precipitation amounts.
Duration
1D
5D
Pooling scheme
ROIcli3
ROIgeo3
ROIgeo2
ROIhyb
ROIcli3
ROIgeo3
ROIgeo2
ROIhyb
Homogeneity
Hom
N/A
207
209
209
207
209
208
209
209
2
0
0
2
0
1
0
0
Number of sites with
Nt
22/23
22/23
22/23
22/23
22/23
22/23
22/23
22/23
N
N = Nt
N > Nt
N < Nt
104
132
130
143
155
142
144
156
38
55
52
49
26
41
39
33
67
22
27
17
28
26
26
20
19.6
22.4
24.5
23.8
22.8
21.2
22.6
22.7
‘Hom’ denotes the number of homogeneous pooling groups for the 209 stations in the analysis; ‘N/A’ labels the number of cases when regional
heterogeneity/homogeneity cannot be defined, i.e. the single-site pooling groups. N (Nt ) stands for the actual (target) size of pooling groups, N
is the average size of the pooling groups at the 209 stations.
Copyright  2010 Royal Meteorological Society
Int. J. Climatol. 31: 1457–1472 (2011)
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REGIONAL AND AT-SITE MODELS FOR HEAVY PRECIPITATION
which the at-site distributions of precipitation extremes
are identical except for a site-specific scaling factor.
The delineation of homogeneous regions for 1D and
multi-day precipitation extremes in the Czech Republic
has been updated and modified with respect to the original
one presented in Kyselý and Picek (2007). The reason for
revisiting the original results of the homogeneity tests
was the extension of the dataset of daily precipitation
amounts, which now consists of 209 stations (compared
to 78 in Kyselý and Picek, 2007), includes more recent
past (up to 2007 compared to 2000), and altogether covers
9573 station-years (compared to the original 3120). The
two largest regions of the original regionalization became
heterogeneous when the new data were included, and
have been split into three and two smaller areas. The
present study recognizes nine regions (Figure 1) that are
considered homogeneous according to the test of Lu
and Stedinger (1992) and the H1 test of Hosking and
Wallis (1993) with respect to the statistical distributions
of annual maxima of 1- to 7-day precipitation amounts.
It should be noted that the average size of the fixed
regions in the HW analysis (23 stations) almost exactly
matches the target size of pooling groups in the ROI
method based on the 5T rule (usually 22 stations). The
average actual size of pooling groups (between 20 and
24 sites in all variants of the ROI method and for both
durations; Table II) differs only little from the target size.
Therefore, the comparison of the ROI and HW methods
is straightforward as there is little difference between the
two approaches with respect to the amount of regional
data typically involved in the estimation.
3.3. Estimation of model parameters and quantiles
For estimating pooled cumulative distribution functions
and high quantiles of 1D and 5D precipitation amounts,
the generalized extreme value (GEV) distribution is
applied (Coles, 2001). The distribution parameters are
estimated using the L-moment-based index storm procedure (Hosking and Wallis, 1997) in both the ROI and
HW models. Dimensionless data xj k are calculated by
rescaling the original data Xj k with the sample mean µj
(index storm):
xj k =
Xj k
,
µj
j = 1, . . . , N,
k = 1, . . . , nj
(1)
where N denotes the number of sites, and nj is the sample
size at the j th site. The dimensionless data xj k at site j are
(j ) (j )
then used to compute the sample L-moments l1 , l2 , . . .
and L-moment ratios (Hosking, 1990):
(j )
t (j ) =
and
l2
(j )
(2)
l1
(j )
(j )
t3 =
l3
(j )
l2
Copyright  2010 Royal Meteorological Society
(3)
where t (j ) is the sample L-coefficient of variation (L-CV)
(j )
and t3 is the sample L-skewness at site j . The pooled
L-moment ratios t (i)R and t3(i)R for the target site i are
derived from the at-site sample L-moment ratios as their
weighted averages:
N
t (i)R =
Wij t (j )
j =1
N
(4)
Wij
j =1
where Wij are the weights associated with the j th site
in the analysis. An analogous relationship holds true also
for t3(i)R .
In the ‘traditional’ HW regional analysis, weighting
coefficients Wij are proportional only to the record length
of the sites in a given region (nj ). An additional factor,
the reciprocal value of the distance metric that measures
the dissimilarity between sites, Dij , is introduced in the
ROI method (Castellarin et al., 2001; Gaál and Kyselý,
2009):
nj
∀j ∈ ROIi
∗
Wij = Dij
(5)
0
∀j ∈ ROIi
where ROIi stands for the ROI of site i, and
Dij
if i = j
∗
Dij =
Dij,min if i = j
(6)
where Dij,min is the lowest non-zero value of the distance
metric between the target site i and all other sites
j (Castellarin et al., 2001). (Note that Dii = 0, which
would lead to Wii = ∞ if Dij was applied instead of
Dij∗ .) Using the reciprocal value of the distance metric
element Dij as the pooled weighting factor is equivalent
to assigning higher weights to sites that lie in the
proximity of the target site in a given attribute space.
The (weighted) pooled L-moment ratios t (i)R and t3(i)R
are then used to estimate the parameters of the GEV
distribution and the pooled growth curve. A quantile
corresponding to return period T is calculated as a
product of the dimensionless T -year growth factor xiT
and the index storm µi .
In the application part of the study, 90% confidence
intervals (CIs) of high quantiles and return periods are
derived using the parametric bootstrap. The procedure is
as follows (cf Hosking and Wallis 1997, p. 94–95):
1. For all stations, random numbers from interval [0,1]
are generated; the size of the random sample at a given
station equals the corresponding record length.
2. Taking into account intersite correlations for the
observed data in the region of interest, the random
numbers are transformed to have multivariate normal
distribution with the given covariance matrix.
3. An inverse transformation is applied in which the
fitted pooled distribution function (GEV) is considered
the parent distribution at a given station.
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J. KYSELÝ et al.
4. Having simulated a data sample that resembles the
real world in terms of the number of sites in a region,
length of the observations, and spatial correlations
between the sites, the regional models are applied
to estimate the T -year return values at each site,
calculated as the product of the pooled (dimensionless)
T -year growth factors and the at-site index value (both
obtained from the actual loop of simulation).
5. The above-mentioned loops of the bootstrap resampling are repeated 5000 times.
6. The bounds of the 90% CI are given by the 5 and 95%
quantiles of the empirical distribution of the simulated
T -year return values.
4.
Simulation study
4.1. Setting
In this section, individual models are compared by means
of Monte Carlo simulation experiments. The models
include several variants of the ROI method that differ
in the site attributes used to measure the similarity of
sites – ROIgeo2, ROIgeo3, ROIcli3 and ROIhyb (for the
explanation of acronyms see Table I), the HW regional
analysis, and the at-site analysis.
The way the unknown parent (true) at-site distributions
of extremes are estimated is adopted from Castellarin
et al. (2001) and Gaál et al. (2008). We use a reference ROI approach in which the similarity of sites is
determined according to statistical properties of the atsite samples of annual maxima of 1D/5D precipitation
amounts (ROIsta, Table I). The selected statistics characterize location, scale and shape of the empirical distribution of extremes. The reference ROI pooling group for
estimating the ‘true’ growth curve is constructed in the
same way as the examined ROI pooling schemes (except
for different site attributes; Section 3.1), i.e. the target
size is the same and the regional homogeneity is tested
by the Lu-Stedinger test.
In each loop of the Monte Carlo simulation procedure,
samples of annual maxima of 1D and 5D precipitation
amounts that resemble the real world in terms of the
number of sites in a region, length of the observations, and spatial correlations between the sites are
drawn for each site from the parent GEV distribution. In most features, the Monte Carlo simulation procedure resembles the list of steps of the parametric
bootstrap described in detail in Section 3.3; nevertheless, there are two important differences. The first one
consists in the inverse transformation of the multivariate normal data using the GEV distribution, parameters
of which are given by the pooled L-moments according to the ROIsta pooling scheme. The second difference is that having applied the regional models on
the simulated at-site samples, the dimensionless growth
factors (not the quantiles themselves) corresponding to
T -year return levels for T = 10, 20, 50, 100 and 200
are estimated and compared with the ‘true’ growth factors obtained from the ROIsta pooling scheme. The
loops of the Monte Carlo simulations are repeated 5000
times.
The performance of the individual models is compared in terms of the RMSE of the estimated dimensionless quantiles corresponding to the 10- to 200year return values for both 1D and 5D precipitation
amounts.
4.2.
Results
Average values of RMSE (Table III) as well as the boxplots (Figure 2) reveal that any of the regional methods
clearly outperforms the at-site model even for lower
quantiles (10-year return values in Table III; 20-year
return values in Figure 2), and the differences increase
towards the upper tail of the distributions for both 1D and
5D precipitation amounts. Among the regional models,
three variants of the ROI models (except for ROIcli3
which is based purely on climatological site attributes)
have lower RMSE than the HW analysis, which is again
true for all quantiles and both 1D and 5D amounts.
Although differences among the regional models are not
large, the single model with the superior performance
Table III. Average RMSE (in %) of dimensionless quantiles corresponding to the T -year return level (T = 10, 20, 50, 100 and
200) of 1D and 5D precipitation amounts.
T (years)
ROIcli3
1D precipitation amounts
10
2.688
20
4.886
50
8.155
100
10.817
200
13.611
5D precipitation amounts
10
2.359
20
4.370
50
7.513
100
10.109
200
12.848
ROIgeo3
ROIgeo2
ROIhyb
HW
At-site
2.444
4.466
7.502
9.979
12.580
2.342
4.322
7.288
9.708
12.252
2.408
4.432
7.478
9.970
12.592
2.618
4.755
7.932
10.514
13.222
3.535
6.494
11.514
16.002
21.134
2.301
4.076
6.898
9.257
11.764
2.236
3.966
6.692
8.958
11.361
2.260
4.012
6.763
9.055
11.494
2.447
4.360
7.348
9.818
12.424
3.471
6.428
11.487
16.031
21.245
The lowest value of RMSE in each row is marked in boldface.
Copyright  2010 Royal Meteorological Society
Int. J. Climatol. 31: 1457–1472 (2011)
REGIONAL AND AT-SITE MODELS FOR HEAVY PRECIPITATION
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benefit of taking regional data into account when estimating distributions of extremes is larger for multiday amounts. Note that the average size of the regions
formed in the ROI method is similar for 1D and 5D
amounts (slightly larger for 1D amounts, except for
ROIcli3 ; Table II) but the distributions of extremes tend
to be more homogeneous for 5D amounts, which is
reflected in a larger percentage of pooling groups that
are homogeneous in the initial stage of forming, i.e. for
N = Nt .
5. Application study
5.1. Comparison of high quantiles estimated using
the regional models and the at-site analysis
Two variants of the ROI method (ROIgeo2 and ROIhyb),
the HW analysis and the at-site analysis are employed
for estimating high quantiles of 1D and 5D precipitation amounts in this section. Since the number of sites
is large, we focus on two areas that differ in orography and synoptic-climatological influences on heavy precipitation: the northeast region and the central lowland
region.
Figure 2. Box-plots of the RMSE of dimensionless quantiles corresponding to the 20-, 50- and 100-year return levels, estimated by
individual regional models and the at-site analysis (simulation study).
Top: 1D precipitation amounts, bottom: 5D precipitation amounts. This
figure is available in colour online at wileyonlinelibrary.com/journal/joc
is the ROIgeo2 pooling scheme that makes use of the
actual proximity of sites; for 5D amounts, ROIgeo2 and
ROIhyb are comparable but the former still leads to
slightly better results. The evaluation of the models in
terms of bias yields a similar pattern, with ROIgeo2
superior at most return levels, but the differences between
the models are smaller and less consistent than for
RMSE.
It is also worth noting that the regional models are
somewhat more beneficial in the frequency analysis of
5D than 1D amounts; for all return levels, the average/median RMSE of any given regional model is lower
for 5D than 1D amounts, while the errors are virtually independent of the duration of precipitation events
for the at-site analysis (Table III, Figure 2). This is
in accordance with a climatological expectation, since
1D annual maxima are mostly associated with convective phenomena that may affect relatively small areas
without a relationship to regional patterns, while for
5D extremes the links to regional patterns (e.g. those
that describe spatially varied Atlantic and Mediterranean
influences, the latter triggering long-lasting heavy precipitation; Štekl et al., 2001) are stronger. Hence the
Copyright  2010 Royal Meteorological Society
5.1.1. Northeast region
The northeast region (Figure 3) is an area with complex orography and enhanced role of cyclones of the
Mediterranean origin in producing heavy precipitation
events. (It also covers the area most affected by the
2009 flash floods.) Figure 4 shows the 50-year return
levels of 1D precipitation amounts, together with their
uncertainty (90% CIs), estimated by means of the atsite analysis, the HW analysis, and two variants of the
ROI method. Differences between the at-site analysis
and any regional model are very pronounced; the at-site
analysis leads to estimates that are strongly influenced
by random sampling variability (large site-to-site fluctuations, which reflect particular sample properties and
not climatological patterns) and suffer from substantial
uncertainty (measured by the width of the 90% CIs).
In any regional method, the width of the 90% CIs is
reduced to around 40% of the at-site values on average, and the random site-to-site fluctuations in the estimates (including fluctuations in the width of the CIs) are
suppressed. It is worth noting that differences between
the HW analysis and the two ROI models are minor
at most sites (compared to the at-site analysis), in spite
of different concepts of pooling regional information. A
marked difference between the two ROI models, especially in the width of the 90% CIs, appears at one
site only (the highest elevated station in the analysis,
Praděd, 1490 m a.s.l.), and it is related to the fact that
the ROIhyb method leads to an extremely small homogeneous region (with two sites only), which contrasts with
typical numbers of sites in pooling groups (24.5/22.3
on average for the 33 sites in the northeast region in
ROIgeo2 /ROIhyb).
Figure 5 compares the best regional method ROIgeo2
(as recognized by means of the simulation study,
Int. J. Climatol. 31: 1457–1472 (2011)
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J. KYSELÝ et al.
Figure 3. Locations of 33 rain-gauge stations in the northeast region. The Bělotı́n station is marked in red.
Figure 4. Fifty-year return levels of 1D precipitation amounts and their 90% CIs, estimated by means of the at-site analysis, the HW analysis,
and two variants of the ROI method, at stations in the northeast region. The stations are ranked in ascending order with respect to the mean
annual maximum (index storm). This figure is available in colour online at wileyonlinelibrary.com/journal/joc
Copyright  2010 Royal Meteorological Society
Int. J. Climatol. 31: 1457–1472 (2011)
Figure 5. Twenty-, 50- and 100-year return levels of 1D precipitation amounts and their 90% CIs, estimated by means of the at-site analysis (narrow black bars) and the ROI method based on actual proximity
of sites (ROIgeo2 ; wide grey bars), at stations in the northeast region.
REGIONAL AND AT-SITE MODELS FOR HEAVY PRECIPITATION
Copyright  2010 Royal Meteorological Society
1465
Section 4) with the at-site analysis for three return levels, the 20-, 50- and 100-year values. It is obvious that
the at-site estimation becomes more and more unreliable
when approaching the extreme upper tail of the distribution, which is reflected again in both random site-to-site
fluctuations and huge uncertainty of the estimates; the
differences between ROIgeo2 and the at-site analysis are
much larger for the 100-year return level than for the 20year return level. This demonstrates that the ‘added value’
of the regional models becomes particularly important for
very high quantiles. Unreliability of the at-site estimation
of high quantiles is illustrated by the fact that at 52% of
the stations in the northeast region, the 100-year return
values estimated from the single-station samples lie outside the 90% CIs estimated by means of the ROIgeo2
method.
Very similar results as to the comparison of the
individual models are obtained for 5D precipitation
amounts (Figures 6 and 7). Both random site-to-site
fluctuations and uncertainty of the estimates are reduced
when a regional model is employed; the width of the 90%
CIs of the 50-year return values is again reduced in the
regional models to around 40% of the at-site estimated
uncertainty on average. However, unrealistically narrow
at-site CIs may become wider in a regional model; this
is the case of station Praděd, for which the 90% CIs
of high quantiles are extremely narrow in the at-site
analysis, markedly underestimating the true uncertainty.
The underestimation of the width of the CIs reflects
specific sample characteristics (as discussed below), and
is quite obvious in the upper left panel of Figure 6 in
comparison to the other sites. The estimated 90% CIs are
wider in the HW and ROIhyb models.
The suspicious behaviour of the width of the CIs for 5D
precipitation amounts at the Praděd station in the at-site
analysis, and particularly the unrealistically low upper
bound of the CIs of high quantiles (50- and 100-year
return levels; Figure 7), is related to the estimated value
of the shape parameter k of the GEV distribution, which
is positive at the Praděd station (k = 0.15; corresponds
to light bounded upper tail) while negative at all other
stations in the area (heavy unbounded upper tails). (The
parameterization of the GEV distribution is the same as
in Hosking and Wallis (1997).) Histograms in Figure 8
reveal that such random fluctuations in the estimates of
the shape parameter, which governs the tail behaviour of
the distribution of extremes, are efficiently reduced in the
regional methods, and the estimates at individual sites
tend to cluster around values typical for the examined
variable. The histogram is not shown for the HW analysis
which takes all data in the (fixed) region together with
the same weights in the estimation at any location, so
the estimated shape parameter is identical at all sites
(k = −0.27).
5.1.2. Central lowland region
While the northeast region examined in the previous
section is an area with complex topography (Figure 3),
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J. KYSELÝ et al.
Figure 6. Same as in Figure 4 except for 5D precipitation amounts. This figure is available in colour online at wileyonlinelibrary.com/journal/joc
the central lowland region in the Elbe River basin is a flat
area without distinct orographic and/or other features that
may significantly influence characteristics of precipitation
extremes. One may therefore assume that differences
in high quantiles of precipitation amounts are minor
between individual locations, particularly those that lie
close to the central parts of the area (11 stations in
Figure 9).
Figure 10 illustrates that the at-site analysis may lead
to very uncertain and climatologically irrelevant estimates
of high quantiles. Site-to-site fluctuations of the estimates
of the 50-year return level of 1D precipitation as well as
the width of the 90% CIs are extremely large; in one case,
the upper bound of the 90% CI is smaller than the lower
bound of the 90% CI at another (nearby) site, which
suggests that the estimated 50-year return levels significantly differ at about p = 0.01 (cf Kharin and Zwiers,
2005). However, there is no climatological justification
for such conclusion, and the ‘significant difference’ is
purely a statistical feature reflecting properties of particular – relatively small – samples.
The large fluctuations in the estimates of high quantiles and their uncertainty in the at-site analysis again
Copyright  2010 Royal Meteorological Society
stem from random variations in the estimates of the
model parameters, particularly the shape parameter k;
the at-site analysis leads to values of k at the 11 sites
between −0.35 and +0.13, covering a wide range of
tail behaviours, while all regional methods agree on
estimates of k between −0.19 and −0.10 at all stations. This results in much more levelled-off estimates
of high quantiles and their uncertainty (Figures 10 and
11), in accordance with a climatological expectation.
One may again observe that at one site (Dubá-Panská
Ves), the width of the 90% CI of the 50-year return
value becomes larger when estimated with any regional
method than the at-site analysis. This further illustrates
that the at-site analysis may result in severely biased
estimates not only of the high quantiles themselves but
also of their uncertainty. The differences between the
at-site and regional analysis become increasingly important towards the tail of the distribution of extremes
(Figure 11). We also note that the similar characteristics of extremes in this area are supported by close
values of the index storm (mean annual maximum of
1D precipitation) at the sites, which range from 33 to
38 mm.
Int. J. Climatol. 31: 1457–1472 (2011)
Figure 7. Same as in Figure 5 except for 5D precipitation amounts.
REGIONAL AND AT-SITE MODELS FOR HEAVY PRECIPITATION
Copyright  2010 Royal Meteorological Society
1467
5.2. Estimates of the return period associated
with heavy precipitation event on 24 June 2009
The flash flood in the late evening of 24 June 2009
in the Odra River basin (the Nový Jičı́n district) in
the northeast region was one of the most disastrous
flash floods in the Czech Republic over several decades.
It caused massive damage to human settlements and
infrastructure, and left ten people dead. Several other
flash floods followed in central Europe during the period
lasting to July 4, characterized by a persistent inflow of
warm and extremely moist air from southeast, at the north
side of a cyclone residing over the Balkan Peninsula.
The inflow of the moist air was associated with recurring
convective rainfall and thunderstorms in the afternoon
and evening hours.
The precipitation amount recorded at lowland site
Bělotı́n (Figure 3), 124 mm in 24 h (114 mm during 3 h,
19–22 LT in the evening of June 24), is quite unusual at
a site located below 300 m a.s.l., and far exceeded the
previous record-breaking value in this location. This is
demonstrated in Figure 12 which plots annual maxima
of 1D precipitation amounts over 1961–2007 together
with the value observed on 24 June 2009. In 2 years
only did the maximum 1D amount at Bělotı́n exceed
60 mm, and the previous maximum was 76.2 mm (1967).
A specific feature of the flash flood on 24 June 2009
was its very local nature; some stations located less
than 50 km away recorded precipitation amounts between
0 and 1 mm during the event, and at no other site
in the database available for the present study did the
precipitation amount on that day exceed 50.2 mm.
Estimates of the return period associated with this
event based on different methods are compared in
Table IV. The at-site estimation completely fails; the
return period estimated from the historical (1961–2007)
data is ∼45 000 years, and the lower and upper bounds
of the 90% CI differ by seven orders of magnitude. All
regional models agree on return periods in the order of
several hundreds of years; although there are no means
of validating the estimates, the fact that the differences
between the individual regional methods are relatively
minor imparts some credibility to them. Another indication that the regional estimates are climatologically much
more relevant stems from the finding that they change
only moderately when the data are supplemented with
the single extreme observation on 24 June 2009 (at the
Bělotı́n station, for which the estimates are calculated,
while all other data samples remain unchanged; the bottom row of Table IV). In the at-site analysis, on the other
hand, the estimated return period declines by two orders
of magnitude if the single new observation is included,
but the associated uncertainty is still extremely large (the
upper bound of the 90% CI is ∼70 000 years).
This example highlights that the at-site estimation
may be extremely affected by the inclusion of a single
(outlying) observation, which makes it highly unreliable.
The regional methods, on the other hand, provide tools
for reducing the uncertainty and obtaining more robust
estimates. In this case they strongly suggest that the return
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J. KYSELÝ et al.
Figure 8. Histograms of estimates of shape parameter k of the GEV distribution from the at-site analysis and two variants of the ROI method
at 33 stations in the northeast region.
Figure 9. Locations of 11 rain-gauge stations in the central lowland region.
period of such event is in the order of several hundreds of
years at a given location, provided that the precipitation
extremes are stationary.
6.
Discussion and conclusions
The present study shows that the regional methods
for modelling probabilities of precipitation extremes are
clearly superior to the at-site analysis. They efficiently
‘trade space for time’ and reduce random sampling variability in the estimates of the model parameters and high
quantiles. Sampling variations in the estimates based on
Copyright  2010 Royal Meteorological Society
the at-site analysis may also affect the width of the CIs,
which are much wider in the at-site than regional analysis
at most sites, but occasionally may become too narrow (if
a light-tailed bounded distribution is estimated for a given
sample). The regional methods reduce climatologically
irrelevant variations in the estimates and under typical
conditions also their uncertainty.
The ROI methodology with a built-in regional homogeneity test is recognized as a useful approach by means
of the Monte Carlo simulations, and the ROI model based
on proximity of sites outperforms the ‘conventional’
HW regional analysis for both 1D and 5D precipitation
amounts. This may partly be related to the fact that fixed
Int. J. Climatol. 31: 1457–1472 (2011)
REGIONAL AND AT-SITE MODELS FOR HEAVY PRECIPITATION
1469
Figure 10. Fifty-year return levels of 1D precipitation amounts and their 90% CIs, estimated by means of the at-site analysis, the HW analysis,
and two variants of the ROI method, at 11 stations in the central lowland region. The stations are ranked from west to east. This figure is
available in colour online at wileyonlinelibrary.com/journal/joc
regions are difficult to delineate in a stable and robust
manner in an area with complex orography such as the
Czech Republic. The lack of robustness of the traditional
regionalization was manifested in the need for revisiting
the original regionalization by Kyselý and Picek (2007),
because some of the former regions were found heterogeneous when more data were involved in the tests. This
points to another major advantage of the ROI approach:
subjective decisions – unavoidable when fixed regions in
the HW analysis are formed – may efficiently be suppressed, and most settings of the ROI method may be
justified by results of the simulation experiments. Similar
Copyright  2010 Royal Meteorological Society
comparative studies are needed in other European regions
in order to arrive at a more complete picture.
The regional methods were applied to estimate the
recurrence probability associated with the unusual heavy
precipitation event that triggered the flash flood on
24 June 2009, and the estimates and their uncertainty
were compared with those from a single-site analysis.
This example illustrates that the at-site estimation leads
to unrealistic and extremely uncertain estimates that
strongly depend on whether or not a single outlying
observation is involved in the sample. The regional
methods, on the other hand, tend to agree on return
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J. KYSELÝ et al.
Figure 11. Twenty-, 50- and 100-year return levels of 1D precipitation amounts and their 90% CIs, estimated by means of the at-site analysis
(narrow black bars) and the ROI method based on actual proximity of sites (ROIgeo2 ; wide grey bars), at 11 stations in the central lowland
region.
Figure 12. Annual maxima of 1D precipitation amounts at the Bělotı́n station over 1961–2007, together with the value recorded on 24
June 2009. The horizontal line shows the mean annual maximum over 1961–2007 (index storm). This figure is available in colour online
at wileyonlinelibrary.com/journal/joc
periods in the order of several hundreds of years,
notwithstanding whether the 2009 data are included in
the sample. It should be emphasized that the return period
refers to a given single location for which the estimation
is made (the Bělotı́n station), and such event may be
expected more frequently in a wider area. Although the
regional models are clearly beneficial, it is also obvious
that uncertainty associated with recurrence probability of
such an extreme observation remains large due to an
unavoidable lack of data (short records), and increases
if possible climatological uncertainties (like the issue of
long-term stationarity) are considered.
The ROI methodology may be extended in a number of
ways in future studies on probabilities of extreme events.
It may become a useful tool for modelling probabilities
of other meteorological variables, extremes of which are
strongly influenced by random sampling variability (e.g.
wind gusts); it may be extended towards estimation at
ungauged locations, provided that the site characteristics
(which are simply geographical coordinates in ROIgeo2 )
Copyright  2010 Royal Meteorological Society
and the index storm values are obtained using mapping
techniques (Brath et al., 2003; Caporali et al., 2008);
and it may be applied into the frequency analysis of
short-term precipitation amounts (hourly data), provided
that such high-quality and long-term data are available
(which is not the case for the area under study at
the moment). The methodology may also benefit from
the ‘peaks-over-threshold’ analysis (Ding et al., 2008),
provided that the issue of regional homogeneity is bridged
(the regional homogeneity tests have been designed for
the ‘block maxima’ approach). Last but not the least,
non-stationarity may easily be incorporated in the ROI
method of the frequency analysis using time index as
a covariate (Coles, 2001), in order to address possible
effects of climate change on probabilities of precipitation
extremes (which are, however, relatively uncertain in
central Europe according to recent regional climate model
simulations; Kyselý and Beranová, 2009).
We also note that the ROI methodology may be transferred to the analysis of precipitation extremes in climate
Int. J. Climatol. 31: 1457–1472 (2011)
1471
REGIONAL AND AT-SITE MODELS FOR HEAVY PRECIPITATION
Table IV. Return periods and their 90% CIs associated with the 1D precipitation amount observed at the Bělotı́n station on 24
June 2009 (123.8 mm), estimated by different methods.
Return period (90% CI)
estimated from the 1961–2007
data (years)
Return period (90% CI)
estimated from the data
supplemented with the single
observation on 24 June 2009
(years)
At-site
ROIgeo2
ROIhyb
HW
45110 (605–2.123 × 109 )
657 (458–1340)
695 (392–1568)
483 (359–920)
419 (333–864)
415 (279–944)
353 (275–641)
283 (85–69 730)
model outputs. As it efficiently reduces (random) variations in the estimates of parameters of the extreme value
distributions at individual locations (or gridboxes) that
result from large spatial variability of heavy precipitation, it represents a straightforward tool for ‘weighting’
data from neighbouring gridboxes within the estimation procedure. The HW regional frequency analysis has
already been incorporated in the evaluation of precipitation extremes in climate models over the UK (Fowler
et al., 2005, 2007) as well as for the construction of
their future scenarios (Ekström et al., 2005; Fowler and
Ekström, 2009). A similar regional approach to modelling precipitation extremes in climate model simulations, which incorporates also non-stationarity of the
model parameters, was developed by Hanel et al. (2009)
and applied in the Rhine River basin. The ROI methodology may represent a useful alternative or even a step
forward, particularly in areas in which the delineation of
fixed regions is less clear. Another possible drawback
of using fixed regions for the estimation of extremes
in future scenarios is the fact that boundaries between
regions may change, and regions that were drawn according to present climatological conditions may become
heterogeneous and inadequate for the estimation of precipitation extremes in a changing climate. Difficulties like
that are eliminated in the ROI approach since pooling
groups are constructed in a flexible way, according to
similarity of sites measured in terms of the attributes
of at-site (or gridbox) data, and their homogeneity is
warranted by the built-in test. The need for ‘weighting’
data and reducing random sampling variability becomes
more important with increasing spatial resolution of
the regional climate model simulations, like those carried out within the framework of the EU-FP6 project
ENSEMBLES.
two anonymous reviewers helped improve the original
manuscript in several points. The study was supported
by the Grant Agency of AS CR (young scientists’ project
B300420801) and the Czech Science Foundation (project
P209/10/2265). J. Picek was supported by the Czech Science Foundation under project P209/10/2045.
Appendix
The Lu-Stedinger test of regional homogeneity
Suppose that a region has N sites, with site i having
record length ni and sample L-moment ratios t (i) (LCV) and t 3 (i) (L-skewness) of a given variable (maximum
annual 1D/5D precipitation amounts).
The test statistic is (Lu and Stedinger, 1992; Fill and
Stedinger, 1995)
χR2 =
N
(xiT − xRT )2
Var xiT
i=1
where
N
xRT =
ni xiT
i=1
N
,
ni
i=1
xiT
t (i)
=1+
1 − 2−k
[− ln(1 − 1/T )]k
1−
(1 + k)
,
and k = 7.8590C + 2.9554C 2 ,
2
ln 2
ln 3
Acknowledgements
C=
Thanks are due to J. Hošek, Institute of Atmospheric
Physics, Prague, and P. Skalák, CHMI, Prague, for their
assistance in drawing maps and updating precipitation
data. Special thanks are due to P. Štěpánek, CHMI,
Brno, for preparing the basic dataset of daily precipitation, performing quality checks and supplementing
missing daily observations. Comments of M. Hanel and
where stands for the gamma function and k denotes
the shape parameter of the GEV distribution (in paramaterisation according to Hosking and Wallis, 1997).
Var xiT , i = 1, . . . , N was determined from tables
in Lu and Stedinger (1992) as the asymptotic variance of the growth factors xiT of a given variable
Copyright  2010 Royal Meteorological Society
t3(i)
+3
−
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J. KYSELÝ et al.
at site i corresponding to the return period of T =
10 years.
2
2
(where χ0.95,N−1
is the 95% quantile
If χR2 < χ0.95,N−1
of χ 2 distribution with N − 1 degrees of freedom) we do
not reject the null hypothesis (the region is homogeneous)
R
the null hypothesis is
at p = 0.05; if χR2 ≥ χ0.95,N−1
rejected and the region is heterogeneous.
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