This paper is structured as follows: Section 2 describes the theoretical framework; Section 3 gives some remarks about the interpretation of the theory; Section 4 describes the extreme value analysis of the applications mentioned and Section 5 , the discussion and conclusions.

As an elaboration of Van den Brink and Können ( 2008 ), we here apply the same method to five quantities related to severe weather events. First, to extreme ERA40 wind speeds over the Northern Hemisphere. Second, to extreme significant ERA40 wave heights over the Northern part of the Northern Hemisphere (NH). Third, to extreme daily precipitation in Europe, using the ECA‐D dataset (Klein Tank et al. 2002 ). Fourth, to extreme daily precipitation in the Netherlands. Fifth, to extreme surges in the Southern North Sea, by merging observations with data generated by the WAQUA surge model (de Vries 2000 ) driven by the ECHAM5‐MPI climate model (Jungclaus et al. 2006 ).

In an earlier article (Van den Brink and Können 2008 ) we developed an alternate approach to the problem. It is based on the empirical distribution of the highest value in an observational series (the ‘outlier’). The method provides a diagnostic whether extrapolation to high return values is justified, and is able to yield an estimate of that return value with a higher precision than the traditional methods. By applying the method on the ERA40 wind data, we illustrated (Van den Brink and Können 2008 ) the potentials of that approach showing that the transformed ERA40 extreme wind speeds over the North Atlantic area are well described by a Gumbel distribution up to return periods of 10 4 years.

Much scientific research has been done on finding the best estimates of weather (related) extremes, like precipitation, wind speed, sea surges (e.g. Cook 1982 ; Koutsoyiannis 2004a ; Van den Brink et al. 2004b ), as well as other geophysical hazards, like earthquakes or tsunamis. An important issue in any of these studies is the choice for the distribution to be made to fit the observed extremes. Mostly, the generalized pareto distribution (GPD) is applied to the (independent) exceedances over a high threshold, or the generalized extreme value (GEV) distribution to annual maxima. However, estimates of high return values depend strongly on the value of the shape parameter of these distributions, which is hard to estimate with sufficient accuracy from the observational records. A method to bypass the problem is to try to lengthen the observational series by means of combining. This approach is applied by Van den Brink et al. ( 2005 ) by treating the sequence of operational seasonal ECMWF (European Centre for Medium‐Range Weather Forecasts, Reading UK) forecasts as observational series of the current climate. It also forms the basis of the regional frequency analysis, where spatially homogeneous records are combined to reduce the statistical uncertainty (Buishand 1991 ; Hosking and Wallis 1997 ).

The probability of an extreme event can be expressed in its return periodwiththe cumulative distribution function of the annual maxima of the variable. The probability that a‐year eventhappens to occur in a certain year is given by:The value ofis called the return level or the return value. The probability that a‐year event happens to occur at least once in a‐year period is (assuming independence):in whichis the highest value (the outlier) in a‐year period. It then follows:If we define:then it follows:in whichis the standardized Gumbel distribution:For all practical situations, Equation ( 5 ) can be approximated by:Van den Brink and Können () gives a graphic visualization of Equation ( 6 ), showing how the horizontal ‘distance’ on a Gumbel plot between the plotting position of the highest event in a record and its theoretical distribution is Gumbel distributed.

In this section we present a short outline of the theory. For a more general formalism, reference is made to Van den Brink and Können ( 2008 ), who gives a slightly different derivation. The Appendix proves the equivalence of the two derivations.

The method indicates whether the fitted distributions lead to systematic biases in the extrapolation or not, but do not provide information about the statistical uncertainty in the estimate of extreme return values.

While a Gumbel plot of Δ X̂ n gives information about its statistical properties, a spatial representation of the values of Δ X̂ n informs about the areas where the assumptions about F may fail. If low (or high) values of Δ X̂ n are clustered in certain areas, this indicates that the assumptions about F fail for its extremes. (e.g. over sea, over mountains or latitude‐bounded)

If the m values are Gumbel distributed according to Equation ( 6 b), it can be concluded that the assumption about the type of F̃ is confirmed, and that the extrapolated fit of is unbiased for all m records, up to return periods of years, i.e. the total number of years in the m records.

The analysis of the distribution of Δ X̂ n can be performed by plotting the ordered values of Δ X̂ n , which is expressed in Gumbel‐variate units, for m independent records on a Gumbel plot. In such a representation, Δ X̂ n should be at the ordinate and the Gumbel‐variate‐transformed m at the abscissa. Note that m in this case represents the number of records, and not the number of years in a record. The distribution of Δ X̂ n can thus easily be compared with the theoretical distribution (Equation ( 6 b)), which is represented by the diagonal on such a Gumbel‐Gumbel plot.

The largest value of Δ X̂ n in a set of records does not necessarily imply that it also corresponds to the highest event in that set but that, in the perspective of the location‐specific climatologies, the event is most exceptional to occur in the given n ‐year period.

Special attention has to be paid to obtain independent values of Δ X̂ n , as required by Equation ( 3 ). The method has no restrictions due to statistical (in)dependence of the elements in the underlying. records, as it only requires that the events that determine the values of Δ X̂ n , i.e. the most exceptional events in every record are independent. In case of meteorological‐related events this is guaranteed when the most exceptional events originate from distinct meteorological systems. A simple criterion is to require a minimum temporal interval between the dates that these events occurred. If an extreme event determines Δ X̂ n for multiple records, then the event should be considered only at its most exceptional moment, i.e. only that record where the event has the highest (estimated) return period. In the case of spatially distributed time series (like gridded data) where the area of interest is very large, one can require that either the time interval exceeds a certain threshold, or the spatial distance is large enough. These thresholds depend on the variable of interest: extreme hourly precipitation will have a much smaller spatial and temporal correlation than daily temperature extremes. In the first situation, a time interval of 1 day or a spatial distance of 500 km will satisfy, whereas for temperatures, a time interval of 14 days or a distance of 2000 km will be more appropriate. Too large values for these thresholds only marginally influence the empirical distribution of Δ X̂ n , as the probability that multiple independent record extremes happen to occur within the time interval is very small.

The method can thus be considered as a pooling technique on the frequency of the variable (which is more general than pooling on its amplitude).

Equation ( 6 ) is valid for every F ( y ) and n , as it analyses the probability of occurrence of the observed highest event (its ‘exceptionality’) and not its value. This implies that the records on which the method is applied, neither need to run simultaneously in time (like in gridded data) nor need to be of equal length. Also, records of different distributions (e.g. for land and sea or tropics and extratropics) could be combined.

In case of applying the method to time series of a given element observed on different locations, the first step is usually in the time domain and the second step in the spatial domain. In case of application to one single time series, the second step can be achieved by splitting the record into multiple (independent) shorter records and then by bringing the set of Δ X̂ n of the sub‐series to the standardized Gumbel distribution.

For convenience, we assumed in Equation ( 1 ) F to be the distribution of the annual maxima. However, the derivation by Van den Brink and Können ( 2008 ) shows that Equation ( 6 ) holds for any arbitrary parent distribution.

While the standard goodness‐of‐fit tests are focused on testing how well the fit behaves in interpolation, this method focuses on extrapolation, which is much more appropriate in extreme value analysis.

Equation ( 6 ) focuses on the highest event in a record, and is thus especially of interest for the analysis of extremes, and for extrapolation purposes. Although not necessary, it is often convenient to use one of the extreme value distributions for F . In that case, Equation ( 6 ) represents the application of extreme value theory twice: first by bringing F to one of the extreme value distributions, and then by bringing Δ X̂ n to the standardized Gumbel distribution (Equation ( 6 b)).

Equation ( 6 ) becomes of practical relevance in the case that multiple records are available, as every record ends up with a single value of Δ X̂ n . Combination of all values of Δ X̂ n enables to test whether they are Gumbel distributed, and thus whether the fit to the data is appropriate for extrapolation. See Van den Brink and Können ( 2008 ) for a summary of the interpretation of the statistical distribution of Δ X̂ n .

Equation ( 6 ) is exact if and only if F ( y ) is known. In practical situations, F has to be empirically determined. Then, F is assumed to be of a certain distribution‐type F̃ , and its parameters are estimated from the data. Given that situation, Equation ( 6 ) can serve as a goodness‐of‐fit measure for F , as a wrong choice for results in a wrong estimate of Δ X n (denoted as Δ X̂ n ) and hence in Δ X̂ n being not Gumbel distributed. Note that no free parameters are left in Equation ( 6 ). We refer to Van den Brink and Können ( 2008 ) for the influence of sampling effects on the distribution of Δ X̂ n .

4. Five Applications

4.1. Extreme NH wind speeds In our work Van den Brink and Können (2008), we showed that the ERA40 annual extreme wind speed u over the North Atlantic area is Gumbel distributed if uk is fitted instead of u, with k the locally determined Weibull shape parameter—a hypothesis originally put forward by Cook (1982). Here we extend this analysis to the entire NH (latitude > 10°N). We use the 44 annual maxima for every grid point of the 10 m wind speed from the ERA40‐dataset (Uppala et al. 2005) for the period 1958‐2001. The T159 resolution is interpolated to a spatial resolution of 1° for the whole NH. Note that inhomogeneities in the wind may exist (especially on the Southern Hemisphere, Wang et al. 2006). We calculated the Weibull shape parameter k from the upper 36% of all 6‐hourly wind speeds for every grid point on the NH, fitted a Gumbel distribution (using maximum likelihood estimation, MLE) to the annual maxima of uk, and calculated ΔX̂ n according to Equation (5). Then the NH was subdivided into 24 boxes, each of size 20° in latitude and 60° in longitude (Figure 1). Figure 1 Open in figure viewer PowerPoint Extreme wind speed according to the 1958‐2001 ERA40 data. The NH is divided into 24 boxes. The blue circles in them give the grid points where the biggest outliers in wind speed took place. The Gumbel plot in each box shows the distribution of the outliers ΔX̂ n in that box We required a minimum interval of 3 days between the 1200 outliers in each box in order to ensure mutual independence. This yielded 11 up to 256 independent values of ΔX̂ n , depending on the box. The Gumbel plots of the distribution of ΔX̂ n for every box are included in Figure 1, together with the positions of the largest outliers. The size of the circles corresponds to the value of ΔX̂ n . Figure 1 shows that outside the tropics (latitude > 30°N), the assumption that the extremes of uk can be described by a Gumbel distribution is confirmed, but that in the tropics the assumption fails. The kinks in the Gumbel plots of ΔX̂ n in that region can be attributed to the occurrence of tropical cyclones, which generate a second population in the distribution of extreme winds (Van den Brink et al. 2004a).

4.2. Extreme waves Caires and Sterl (2005) present global estimates of 100‐year return values of significant wave heights, based on the ERA‐40 reanalysis data. Their calculation of return values is based on the peaks‐over‐threshold (POT) method, with a threshold on the 93% level of all 6‐hourly data. They conclude that ‘the large amount of data used in this study provides evidence that the distribution of significant wave height belongs to the domain of attraction of the exponential’. Here, we test this conclusion by calculating the distribution of ΔX̂ n for the waves in the 1958‐2000 period between 30 and 70°N. To ensure independence in the set of calculated values of ΔX̂ n we required each pair of outliers with a mutual distance less than 7500 km to be separated by more than 4 days (>96 h). This selection procedure results in 192 independent values for ΔX̂ n , representative for a total of 192 × 43 = 8256 years. Figure 2 shows the Gumbel plot of the 192 independent values of ΔX̂ n , for three choices of the distribution F̃: an exponential distribution (using L‐moments, like in Caires and Sterl 2005), a GEV and a Gumbel distribution (using MLE). It shows that the exponential fit clearly underestimates the distribution of ΔX̂ n , and thus overestimates the extremes. According to Equation (7), the average ‘distance’ ΔX̂ n between the highest event in a 43‐year record and the 100‐year return value should be ln(100)− ln(43) = 0.84. Instead, the estimated value ΔX̂ n = 0.84 corresponds in Figure 2 to a theoretical value ΔX n = 2.04, i.e. a return period of 330 years (Equation (7)). Fitting a GEV distribution leads to underestimation of high values of ΔX n , a general feature of the GEV distribution, see Van den Brink and Können (2008). However, fitting a Gumbel distribution to the annual maxima gives good results for the distribution of ΔX̂ n . This result is in accordance with Bouws et al. (1998) who state that “The Fisher‐Tippett Type I (i.e. Gumbel) distribution often seems to give a good fit to 3‐hourly data from the North Atlantic and North Sea” (p. 106). Figure 2 Open in figure viewer PowerPoint Gumbel plot of the statistical distribution of ΔX̂ n for the significant wave heights in the 1958‐2000 period between 30 and 70°N for three assumed extreme value distributions of the wave heights (Gumbel, GEV and exponential). Assuming them Gumbel distributed performs best. The exponential to POT is the distribution applied by Caires and Sterl (2005). POT stands for peak‐over‐threshold We conclude that the proposal of Caires and Sterl (2005) to use an exponential distribution for the distribution of extreme wave heights must be rejected. The consequence of using a Gumbel distribution instead is that the 100‐year return values become on average 15% lower than the values of Caires and Sterl (2005).

4.3. Extreme precipitation rates in Europe 2005 R is Weibull distributed: (8) k equal to 2/3. Analogously to the argument that justifies the transformation of the wind speed in Section R2/3 is exponentially distributed, with a fast convergence of its normalized maxima to the Gumbel distribution. Wilson and Toumi () argue on theoretical grounds that the extreme precipitationis Weibull distributed:with the shape parameterequal to 2/3. Analogously to the argument that justifies the transformation of the wind speed in Section 4.1 , this implies thatis exponentially distributed, with a fast convergence of its normalized maxima to the Gumbel distribution. We extracted daily precipitation of 2482 European stations from the ECA‐D dataset Klein Tank et al. 2002 of different lengths running in the period 1951‐2008. We required that the records are more than 20 years in length. This leads to 2147 records with a total of 88 470 annual maxima. We fitted a Gumbel distribution to the annual maxima of R2/3 for the 2147 records (using MLE), and calculated ΔX̂ n according to Equation (5). To ensure independence in the set of calculated values of ΔX̂ n we required each pair of outliers with a mutual distance less than 1000 km to be separated by 2 or more days. This leads to 1379 independent values for ΔX̂ n , representing a total of 56 261 years. The Gumbel plot of the 1379 independent values of ΔX̂ n is shown in Figure 3, together with the locations of the stations used. Only the 1379 mutually independent stations are shown. The colours correspond to the values of ΔX̂ n . Figure 3 Open in figure viewer PowerPoint Gumbel plot of the statistical distribution of ΔX̂ n and the spatial distribution of ΔX̂ n for the annual maximum 1‐day precipitation sums R, obtained by fitting a Gumbel distribution to R2/3 to the 2147 records of the ECA‐D dataset (1951‐2008). The colour coding in plot and map indicate the magnitude of ΔX̂ n and relate the points on the graph with their positions on the map. Only the 1379 independent values of ΔX̂ n (out of 2147) are shown Figure 3 shows that the distribution of ΔX̂ n corresponds well with theory, which confirms the assumption of Wilson and Toumi (2005) that extreme precipitation is Weibull distributed with shape parameter k = 2/3 up to return periods of about 50 000 years. The highest ΔX̂ n value of 10.02 is found in the 1974‐2005 record of Meknes, Morocco (5.53°W, 33.88°N) on 24 October 1977, with 249.9 mm. Its estimated return period of 584 500 years has a probability of almost 10% to occur within 56 261 years. Note that the precipitation amount of this event is only the 25th highest in the whole record, but nevertheless the most exceptional event given its climatology (i.e. its distribution ) and its length n of 26 years. The even higher value for ΔX̂ n that is present in the set (10.89 for the 204 mm event in Lien i Selbu, Norway on 17 March 2003, with an estimated return period of 2.8 × 106 years) showed after detailed inspection to be erroneous. 2004b (9) 2008 (10) k the Weibull shape parameter, and i the number of independent events in a year or season. Our conclusion, above, that the (transformed) precipitation is Gumbel distributed, seems to be contradictory to many papers that state that precipitation is heavy tailed (e.g. Koutsoyiannis, and references therein), i.e. is distributed according the GEV distribution with its shape parameter θ> 0:The answer can be found in the work of Furrer and Katz (), who give an expression for the GEV shape parameter if the GEV distribution is used as a pen‐ultimate distribution for normalized maxima from the Weibull distribution:withthe Weibull shape parameter, andthe number of independent events in a year or season. To compare both approaches, we also fitted a GEV distribution to R itself, assuming the GEV shape parameter to be constant over Europe, and estimated its value by iteratively fitting a GEV distribution to all 88 470 annual maxima (normalized by the local estimates of the location and scale GEV parameters, see Buishand 1991, Appendix A for details). This yields a value for θ of 0.1008, in good agreement with literature (e.g. Gellens 2002) and consistent with the value according to Equation (10) for k = 2/3 and i = 150. Figure 4 shows that the distribution of ΔX̂ n obtained by fitting a GEV distribution to R with a constant shape parameter of θ = 0.1008 to the 2147 records, is as good as in the case that a Gumbel distribution is fitted to R2/3 (Figure 3). This empirically proves that the GEV distribution with Equation (10) can indeed (Furrer and Katz 2008) serve as a very good pen‐ultimate distribution for the Gumbel distribution. Figure 4 Open in figure viewer PowerPoint Comparison of two Gumbel plots of the statistical distribution of the 1379 independent values of ΔX̂ n over Europe for the annual maximum 1‐day precipitation sums R. Red: as obtained when a Gumbel distribution is fitted to R2/3 (identical to the Gumbel plot in Figure 3). Blue: as obtained by fitting a GEV distribution with constant shape parameter θ = 0.1008 to R Figure 5 shows the Gumbel plot for the 1951‐1998 record of Mont‐Aigoua, France (44.1°N, 3.583°E), which is the record with the highest absolute precipitation amount in the ECA‐D dataset, namely 520 mm on 24 February 1964. Figure 5 Open in figure viewer PowerPoint Gumbel plot for the annual maximum 1‐day precipitation sums R in Mont‐Aigoua, France (1951‐1998). The fits are the Gumbel distribution to R2/3 (red), the GEV distribution with θ = 0.1008 to R (blue), and the GEV distribution with free shape parameter (green). The vertical axis is linear in R2/3. The largest observed 1‐day amount (520 mm) is indicated by a dashed horizontal line. The horizontal distances of the maximum observed value to the three fits (the magnitude of ΔX̂ n ) of 1.05, 1.94 and 2.15 indicate return periods of this event of 137, 335 and 413 years, respectively. The vertical bars indicate the 95% confidence intervals. The green bar runs till R = 2361 mm/day The fits are the Gumbel distribution to R2/3, the GEV distribution to R with the shape parameter fixed to θ = 0.1008 and the GEV distribution to R. The values for ΔX̂ n of 1.05, 1.94 and 2.15, respectively, indicate a return period of the largest event of 137, 335 and 413 years, depending on which of the three respective fits is adopted. The vertical axis is linear in R2/3, which transforms the GEV fit with shape parameter 0.1008 almost into a straight line. The GEV fit with a fixed shape parameter and the Gumbel fit are almost similar, both in the estimates and in the uncertainty ranges. They outperform the GEV fit with a free shape parameter, which is too sensitive to the outlier. In most practical cases fitting a Gumbel distribution to R2/3 will thus be preferred over fitting a GEV with fixed shape parameter to R, as the more symmetrical confidence intervals of the former leads to better estimations of the upper confidence interval of extrapolated return values.

4.4. Extreme precipitation in the Netherlands We applied the GEV distribution with fixed shape parameter to 1‐day precipitation maxima to the four seasons separately, using 294 station records in the Netherlands with a length of at least 20 years. The total sets contains 16 524 years and covers the period 1906‐2007. The records can be downloaded from http://www.knmi.nl/klimatologie/monv/reeksen/. The GEV shape parameter is fixed and is like before empirically determined. To ensure independence in the set of calculated values of ΔX̂ n , we required each pair of outliers to be separated by 2 or more days. The JJA 1‐day precipitation maxima are well described by a GEV distribution with a fixed shape parameter of 0.109. This value is almost the same as the value derived from the ECA‐D dataset and is in accordance with the k = 2/3 Weibull shape parameter predicted by Wilson and Toumi (2005). The distribution of the 119 independent values of Δ X̂ n representing a total of 6672 JJA maxima, closely coincides with the diagonal of the Gumbel‐Gumbel plot (Equation (6b)).

The fixed GEV shape parameter for DJF is almost zero. This deviates considerably from the value of 0.1 expected for a Weibull shape parameter k = 2/3. This suggest that the conditions behind the Wilson and Toumi (2005) formula are violated. We speculate that this failure is related to the absence of deep convection in winter, combined with the lack of orographic forcing in this flat country.

The DJF maxima are much more spatially correlated than the JJA maxima, with only 30 independent values of Δ X̂ n , representing 1562 years. This reduction can be attributed to the fact the DJF maxima are due to large‐scale precipitation, with a larger spatial correlation than the convective precipitation that causes the JJA maxima. The precipitation on 4 December 1960 caused the record extreme for 178 stations (indicated by the filled black circles in Figure 6), with a maximum amount of 83.9 mm in Joure (5.82°E, 52.98°N). This event also caused the highest Δ X̂ n value of 8.95 in Den Helder (4.75°E, 52.97°N) with 83.3 mm. Its estimated return period of 5 × 10 5 years is very unlikely to happen in a 1562‐year period, and hints on underestimation of the DJF extremes by a GEV distribution with fixed shape parameter and hence on rejection of the Wilson and Toumi (2005) hypothesis. Evaluation of the DJF 1‐day precipitation extremes in the ECA‐D dataset (1379 independent values, figure not shown) leads to the same conclusion; due to the larger dataset the deviation of the most extreme outliers from the diagonal in the Gumbel‐Gumbel plot is more pronounced. The outlier in the plot in Figure 6 suggests that our method is able to detect this bias in a dataset consisting of as few as 30 independent values.

The independent values of Δ X̂ n in DJF are clustered in the coastal zone. The coastal clustering can be attributed to the relatively warm sea and the cold land, resulting in nocturnal convection over sea which occasionally affects the coastal zone.

The MAM and SON seasons show a mix of summer‐ and winter features, which is apparent in values of the GEV shape parameter that are in between the summer and winter values. It is also be read from the Gumbel plots of ΔX̂ n as the dots in them less perfectly follow the diagonal, particularly in SON. Figure 6 shows the following features: Figure 6 Open in figure viewer PowerPoint Gumbel plot of the statistical distribution of ΔX̂ n and the spatial distribution of ΔX̂ n for the 1‐day precipitation sums for the Netherlands in MAM, JJA, SON and DJF (1906‐2007). The colour coding in plot and map indicate the magnitude of ΔX̂ n and relate the points on the graph with their positions on the map. The independent values of ΔX̂ n are shown in colour; the dependent values are indicated by the open circles. The number of independent values m is indicated for each season. The total number of stations is 294; the closed circles in the DJF graph are the 178 stations where the event on 4 December 1960 caused the outlier