1. Introduction
Extreme precipitation events have a large impact on society through damage caused by floods, dike breaches, landslides, or sewer system overload. The total cost of natural disasters in the European Environment Agency (EEA) member countries during 1980–2017 is estimated at EUR 557 billion (EEA 2019). Around 63% of all the economic losses were the result of storms and rainfall-induced floods.
Subdaily extreme rainfall, caused by convective events, is known to impact many sectors such as agriculture, forestry, tourism, health, and water management. As an example, subdaily extreme rainfall can induce flash floods that may develop in time scales shorter than an hour in an urban area. In recent years, there has been a growing body of evidence that short-duration rainfall intensity has increased at global scale (Westra et al. 2014), and that future increases are to be expected (Ban et al. 2015; Prein et al. 2017; Martel et al. 2020).
Lately, the effect of climate change on short-duration extreme precipitation has been under strong investigation through the use of climate modeling. In fact, current global climate models (GCMs) operate on a spatial resolution in the range of 50–100 km [e.g., phase 6 of the Coupled Model Intercomparison Project (CMIP6)], or even on a finer scale of 25 km (Vannière et al. 2019), and are therefore unable to explicitly resolve small-scale physical processes such as convection. Regional climate models (RCMs) allow to downscale the GCM results by increasing the spatial resolution in a small, limited area of interest. As a result, significant international efforts, such as the Coordinated Regional Downscaling Experiment (CORDEX), have been made to downscale GCMs using RCMs over various regions in the world (Giorgi et al. 2009). For the European domain, the simulations are conducted at resolutions of 0.44° ≈ 50 km (EUR-44) and 0.11° ≈ 12.5 km (EUR-11) (Jacob et al. 2014, 2020).
Several studies have proved that a clear added value of regional downscaling can be found for extreme precipitation [Prein et al. (2015) and references therein]. For the European domain of CORDEX (EURO-CORDEX), for instance, recent works found that the fine-gridded (12-km resolution) RCMs add value to the coarser-gridded (50 km) RCMs in terms of both mean and extreme precipitation for almost all seasons and all regions throughout Europe, especially due to improved orographic representation but in part also due to a better representation of convective processes (Torma et al. 2015; Prein et al. 2015, 2016).
While coarser-resolution RCMs (grid spacing above 5 km) are found reliable for projecting seasonal changes and frequency in precipitation, simulations with a horizontal grid spacing below 5 km are generally considered to resolve (at least partly) convective phenomena, which is essential to capture rainfall extremes. Kendon et al. (2017) found that the explicit representation of the convective storms is necessary for capturing changes in the intensity and duration of summertime rain on daily and shorter time scales. Climate change projections at these convection-permitting scales exist for different limited regions worldwide (e.g., Pan et al. 2011; De Troch et al. 2014; Argüeso et al. 2014; Kendon et al. 2014; Ban et al. 2015; Brisson et al. 2016b; Fosser et al. 2017; Reszler et al. 2018) but are mostly single-model efforts due to the huge computational costs. Coordinated ensemble simulations at convection-permitting scale have been very rare (Helsen et al. 2020; Met Office 2021) but have recently been set up over different areas in Europe with promising first results (Coppola et al. 2020). Large climate simulations at the 1-km scale are currently being considered but face data-handling issues (Schär et al. 2020).
Prior to climate change assessments, it is important to compare the runs with observations to find potential added value of high-resolution simulations and to gain confidence in climate projections. The evaluation of precipitation extremes in RCMs is traditionally based on a comparison of the extreme value statistics of observations and simulations. In a novel approach, Westra et al. (2014) and Cortés-Hernández et al. (2016) formulated a range of statistical metrics to evaluate the capability of RCMs in capturing the dominant spatiotemporal characteristics of short-duration rainfall extremes. This includes the assessment of a correct simulation of the seasonal cycle, the diurnal cycle, the relationship between extreme intensity and temperature/humidity [e.g., Clausius–Clapeyron (CC) scaling], temporal scaling [intensity–duration–frequency (IDF) information], and the spatial organization of rainfall extremes. A positive evaluation of these metrics should increase confidence in RCM simulations and could give more insights in its limitations with potential directions for model improvements.
In addition to model evaluation, the metrics can also support the relationship between climate science and impact studies, the latter of which is highly relevant to climate services and decision-making. For instance, a correct simulation of the diurnal cycle is required for quantifying outdoor thermal comfort conditions because the timing of a heavy rainfall event is important. Similarly, a correct simulation of the seasonal cycle concerns many sectors such as agriculture, forestry, tourism, and health because it is important to determine the correct season in which the most extreme rainfall events occur. However, different sectors are interested in different rainfall accumulation periods. For instance, while urban water resources engineers are mainly interested in information on subdaily rainfall extremes to protect various engineering systems (e.g., sewer systems, urban drainage systems) against floods, farmers or tour operators are rather interested in monthly or seasonal rainfall extremes. One of the most commonly used tools in hydrologic engineering are IDF or depth–duration–frequency (DDF) curves and give an overall and consistent picture of rainfall extremes across different accumulation periods. Note that future projections of DDF curves are recently investigated in a EURO-CORDEX ensemble by Berg et al. (2019).
Our main aim in this paper is to provide a diagnostic framework for RCMs that is partly based on the metrics of Westra et al. (2014) and Cortés-Hernández et al. (2016), but it goes beyond in terms of different aspects of extreme value analysis. More specifically, our work provides inference for the statistical performance metrics, including uncertainty estimation and significance testing, and is implemented in freely available R codes. The new evaluation tools are tested on a small ensemble at EUR-11 and convection-permitting resolutions [high resolution (H-Res)] of climate simulations over Belgium, produced within the context of the CORDEX.be project (Termonia et al. 2018b). Observed precipitation extremes over time scales of 1–24 h were derived by aggregating 10-min pluviograph data. We evaluate the added value of H-Res with respect to EUR-11 simulations.
The paper is organized as follows. Section 2 contains the definition of extremes often used in climate studies. This includes annual maxima, peaks over threshold, and high quantiles. In section 3, the extremes definitions are integrated in the evaluation metrics, and statistical models for these metrics are proposed. In section 4, we describe the data (RCMs and observations) used in this study. The results of the application of the evaluation metrics are discussed in section 5. In section 6, some conclusions are drawn.
2. Extremes definition
a. Extremes in hourly precipitation series
- Annual maximum (AM):
- Peaks over threshold (POT):
Note that POT values regularly appear in clusters, and the series is then declustered by selecting the highest value in each cluster. Willems (2000) recommended using an interevent time equal to the rainfall duration d, with a minimum value of 12 h. Practice shows that the optimal threshold is such that we have an average of 3–5 exceedances per year (Coles 2001).
- High τ quantiles: if nonzero d-hourly precipitation is characterized by its distribution function F, the τ quantile is
Of particular importance is the conditional τ quantile, which is conditioned on a certain predictor variable Y: Qτ(y) = inf[x:F(x) > τ|Y = y]. An example is the modeling of the influence of the temperature/humidity on extreme hourly rainfall; see section 3a. Note that the present quantile-based definition of extremes, relative to wet periods, might be sensitive to changes in the fraction of wet hours (Ban et al. 2015). This does not, however, pose a problem for the current study, which is an evaluation of RCM performance and not a study on climate change.
There exists a powerful mathematical theory of AM and POT extremes, which provides extreme value models and inferential techniques. The statistical modeling of extreme values may be considered as a well-established area of investigation (Leadbetter et al. 1983; Coles 2001; Beirlant et al. 2004). An essential difference with unconditional quantiles is that these extreme value models are able to estimate the probability of events that are more extreme than previously observed events.
b. Clock-hourly aggregated maxima versus subhourly aggregated maxima
3. Methods
We selected a range of metrics that were proposed in the review of Westra et al. (2014) and Cortés-Hernández et al. (2016) to examine the ability of RCMs to reproduce the most important features of extreme precipitation, and we tabulated them in Table 1 (first column). The seasonal cycle is a particularly useful evaluation tool because convectively driven rainfall extremes occur in Belgium mainly in the warm season, and are typically of short duration (e.g., hourly). On the other hand, longer-duration extreme precipitation events (associated with stratiform weather types) are more likely to occur in autumn and winter. The diurnal cycle is commonly used to prove the triggering physical mechanisms of convective events. In this section, we propose an overall framework for the statistical modeling of various aspects based on extreme value analysis, as tabulated in Table 1, for example, extreme rainfall IDF relationships (Van de Vyver 2015a, 2018), spatial extremes (Cooley et al. 2012; Davison et al. 2012), or extremes of dependent time series (Leadbetter et al. 1983; Smith and Weissman 1994).
Summary of the evaluation metrics AM is annual maxima, POT is peaks over threshold, and CQ is conditional quantiles.
We compare RCM output at the nearest grid points of observation stations (see section 4 for data description). Evaluation of models against station observations often raise issues of spatial-scale mismatch since the spatially averaged values of the corresponding gridcell values underestimate the point-scale precipitation. The coarser the resolution is, the larger is the underestimation. However, the daily summer extremes of ALARO at 10- and 4-km resolution are shown to be in good agreement with the observations (De Troch et al. 2013). In the context of subdaily summer extremes, Olsson et al. (2015) demonstrated that RCMs at 6 km are nearly unbiased, whereas a negative bias still exists at 12 km. In any case, the decrease in magnitude is unlikely to affect the evaluation metrics (Cortés-Hernández et al. 2016).
a. Temperature and humidity dependency
Extreme precipitation intensity is known to exponentially increase with daily mean (dewpoint) temperature at a rate of roughly 7% °C−1, the so-called CC rate (Trenberth et al. 2003; Westra et al. 2014). Recent studies using hourly precipitation observations from various locations in western Europe, showed that for temperatures above ~10°–12°C, 1-h precipitation extremes increase approximately 2 times as fast as the CC rate, a phenomenon that is usually called super-CC scaling (Lenderink and van Meijgaard 2008, 2010; Westra et al. 2014).
The aim here is to test to what extent the RCMs reproduce the super-CC scaling features. A statistical method was recently proposed to study the scaling of subdaily rainfall extremes against the (dewpoint) temperature (Van de Vyver et al. 2019) and is briefly outlined here. We denote by P(d) the nonzero d-hourly precipitation, and, unless otherwise stated, T is the daily mean dewpoint temperature. For brevity, we write P instead of P(d).
Note that the scaling rates and the changepoint are simultaneously estimated. The inference of the regression models is further explained in appendix A. In addition, we used a goodness-of-fit (GoF) criterion to examine the influence of the predictor (temperature or dewpoint temperature) to extreme precipitation.
b. Temporal scaling
A fundamental property of precipitation extremes is temporal scaling invariance, which implies that the statistical properties of extremes across different rainfall durations can be related by a scaling factor. The intention is to investigate to what extent the RCMs are capable of reproducing the correct scaling relationships.
Let It(d) = Pt(d)/d (mm h−1) be the average intensity. For a 1-yr time window [0, D], the maximum of the continuous process It(d) is MD(d) = max[It(d)|0 ≤ t ≤ D], which we briefly denote by M(d).
Bayesian inference and information criterion-based model selection [i.e., the Akaike information criterion (AIC) and Bayesian information criterion (BIC)] for temporal scaling GEV models was developed in Van de Vyver (2015a, 2018), and we refer to these works for the technical details. In particular, it was shown in Van de Vyver (2018) that there is a very strong evidence that the extreme intensities exhibit the multiscaling property, at least for the Belgian pluviograph dataset (see section 4b).
Since classical extreme value theory is based on the assumption that the series under study is stationary (Leadbetter et al. 1983), it is important to examine a possible nonstationary behavior of the rainfall data. When pronounced nonstationarity in the data is present, a nonstationary GEV model can be considered by introducing time covariates in the parameters (Coles 2001). For example, the constant μ can be replaced by a time-dependent function μ(t) = μ0 + μ1 Y(t), with, for example, Y(t) being the global mean temperature or a climate oscillation index at year t. To our knowledge, Ouarda et al. (2018) presented the only nonstationary extension to date of the temporal scaling GEV model, Eq. (5). Although, their IDF models will provide a better fit for nonstationary data, they are of limited use for the current validation framework because the temporal scaling parameter η is the crucial metric here, and they considered it as constant since η does not show any trend in their data. To the best of our knowledge, nonstationary IDF relationships with a time-varying scaling parameter do not exist so far. Other nonstationary IDF curves are also developed in Cheng and AghaKouchak (2014) and in Agilan and Umamahesh (2016), but they fitted a nonstationary GEV distribution to the individual durations without using a scaling relationship.
c. Spatial structure
The aim is to examine the spatial structure by a dependence measure for spatial extremes, the madogram (Cooley et al. 2006; Vannitsem and Naveau 2007; Naveau et al. 2009), which is the first moment version of the well-known variogram for general nonextreme events. The statistics of spatial extremes naturally extend the classical univariate extreme value models, and also include basic concepts of Gaussian geostatistics. A popular approach for modeling the maxima observed at different sites, is based on fitting max-stable processes (Davison and Gholamrezaee 2011; Davison et al. 2012; Cooley et al. 2012). Specifically, a max-stable process Z(x) at location x has the GEV distribution (F) throughout the spatial domain, includes flexible correlation functions, and can therefore be thought of as the “extreme value analog” of Gaussian random fields.
The use of spatial statistical measures for the validation of RCMs is very limited. To our knowledge, the only applications can be found in (Cortés-Hernández et al. 2016; Hobæk Haff et al. 2015; Rasmussen et al. 2012), but there are some important differences with the present method. First, Rasmussen et al. (2012) and Hobæk Haff et al. (2015) modeled spatial dependence over the full range of the precipitation distribution, without explicitly accounting for extremes. By contrast, the madogram focuses solely on the extremal spatial dependence, without regard to the main body of the precipitation distribution. Second, the binned madogram estimates provide a local average that gives a clear view on the extremal dependence as a function of the distance, which is not the case with the pairwise extremal coefficient approach in Cortés-Hernández et al. (2016).
d. Temporal clustering
Since extreme precipitation events have the tendency to occur in temporal clusters, we examine how well RCMs reproduce the period of clustering. We introduce a measure of short-term temporal dependence for extremes in a series.
4. Model and observation data
a. Regional climate models
One of the goals of the CORDEX.be project was to contribute to EURO-CORDEX with three RCMs (Termonia et al. 2018b). Four Belgian climate modeling groups provided climate simulations for the EUR-11 domain corresponding to a 12.5-km horizontal resolution, in line with the EURO-CORDEX prescriptions. Additional to the 540 simulation years for the EUR-11 domain, 780 simulation years at convection-permitting resolution (H-Res) were produced, where the horizontal resolution ranges from 2.8 to 5 km [see Table 4 in Termonia et al. (2018b)]. The EUR-11 and H-Res runs are performed on a limited geographical domain using a one-way nesting approach, that is, by imposing meteorological conditions at the boundaries from model simulations at lower resolution. To validate the EUR-11 simulations, we consider the evaluation runs that use ERA-Interim as lateral boundary conditions over the EURO-CORDEX region. In turn, these simulations are used as boundary conditions for a second nesting over Belgium at convection-permitting scale. Precipitation is typically stored with an hourly frequency for the EUR-11 simulations, whereas the H-Res simulations are archived at subhourly frequency. There are four models used: the combined ALADIN and AROME model (ALARO-0), the Consortium for Small-Scale Modeling Community Land Model (COSMO-CLM) (two versions), and the Modèle Atmosphérique Régional (MAR). The MAR simulations are only run at H-Res resolution. The evaluation runs cover the period starting from 1979 up to 2009, 2010, 2014, or 2017 (according to the model). In Table 2 we provide their main features.
The RCMs contributed to the CORDEX.be project (Termonia et al. 2018b) and their characteristics.
Prein et al. (2015) define a “convection-permitting model” as featuring a horizontal resolution of less than 4 km, and without a deep-convection parameterization scheme. Therefore, in strict terms, only the H-Res simulations of COSMO-CLM qualify.
1) ALARO-0
The ALARO model (version ALARO-0) is a hydrostatic RCM that is based on ALADIN, a numerical weather prediction system developed by the international ALADIN consortium for operational weather forecasting and research purposes (Bubnová et al. 1995; Termonia et al. 2018a). The ALADIN model is the limited area model (LAM) version of the Action de Recherche Petite Echelle Grande Echelle Integrated Forecast System (ARPEGE-IFS). The ALARO model includes the precipitation and cloud scheme Modular Multiscale Microphysics and Transport (3MT), which is a parameterization of deep convection optimized for resolutions in the so-called gray zone (4–10 km). The main strength is scale awareness, that is, the parameterization itself works out which processes are unresolved at the resolution used (Gerard et al. 2009). This opposes more conventional parameterization practices that either enable or disable parameterization schemes. This allows 3MT to generate consistent results across spatial scales, as shown by De Troch et al. (2013).
2) CCLM
The COSMO-CLM (CCLM) model is based on the COSMO model, which is a nonhydrostatic limited area model of the Deutsche Wetterdienst (DWD) for operational purposes. The limited-area modeling (CLM) community implemented the COSMO model for climate simulations (Rockel et al. 2008). The EUR-11 simulation does not explicitly resolve convection and uses the Tiedtke convection scheme, while the H-RES simulation dynamically resolves deep convection. More model settings and technical recommendations for simulations at convection-permitting scale are given in Brisson et al. (2016a), and the added value was confirmed in Brisson et al. (2016b).
In this work we include two versions of CCLM: CCLM-UCL, which is a two-moment scheme with hail parameterization implemented by Van Weverberg et al. (2014), and CCLM-KUL, which is the computationally efficient urban land surface scheme TERRA_URB implemented by Wouters et al. (2016).
3) MAR
The MAR model is a hydrostatic primitive equation model. The atmospheric part of MAR is fully explained in Gallée and Schayes (1994), and the surface vegetation–atmosphere interface [Soil Ice Snow Vegetation Atmosphere Transfer (SISVAT)] is described in De Ridder and Gallée (1998). Although being originally developed for the polar regions, MAR was recently adapted and applied to the temperate climate of Belgium (Wyard et al. 2017). In this paper, version 3.9 of MAR is used, which explicitly resolves a large part of precipitation (98%) at 5-km resolution, without a convective scheme. In future developments of the MAR model, more account must be taken of convective precipitation.
b. Observations
The observational dataset comprises 18 Belgian pluviograph series with 10-min precipitation and cover the period of 1967–2004. These data were obtained by Hellmann–Fuess pluviographs, that were part of the hydrometeorological network of the Royal Meteorological Institute (RMI) of Belgium (Fig. 1, along with Table S1 in the online supplemental material). The observation period does not entirely overlap with that of the RCM evaluation runs for different reasons. First, there is no clear trend in the extremes (see section 4c). Second, we are not looking at the one-to-one time correspondence between model and observations. Last but not least, it is unlikely that the evaluation metrics, related with the underlying physics of convective events, differ for the nonoverlapping periods. As extremes are, by definition, rare, as many data as possible are used for a reliable inference.
Elevation map (m) of Belgium, together with the location of the 10-min pluviograph stations. Geospatial information of the locations (latitude, longitude, and elevation) is listed in Table S1 in the online supplemental material.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
For each station, we extracted the d-hourly precipitation annual maxima, for rainfall durations d = 1, 2, 6, 12, 24, 48, and 72 h. For each duration, we consider an annual maximum value as “missing” if (i) it is below the 40th percentile of the annual maximum series and (ii) the number of missing values of that year is larger than one-third. We consider a particular year as missing if at least three of the seven durations are missing. The same data have been included in previous studies concerning spatial extremal dependence (Vannitsem and Naveau 2007), trends in historical extremes (Ntegeka and Willems 2008), spatial extreme value models (Van de Vyver 2012), sliding 24-h maxima (Van de Vyver 2015b), IDF relationships (Willems 2000; Van de Vyver 2015a, 2018), and climate model validation (Tabari et al. 2016).
Satellite-based precipitation products, such as MSWEP (Beck et al. 2019) and GPM (Hou et al. 2014), provide complete spatial coverage and can be potentially interesting for the analysis of spatial extreme rainfall. However, none of these products have a sufficiently fine temporal resolution and/or a sufficiently long common evaluation period.
c. Trend testing
Most statistical methods assume that the series under investigation are stationary, but in reality, this is usually not the case. To investigate a possible nonstationarity in the annual maximum series, we performed the Mann–Kendall monotonic trend test (Chandler and Scott 2011), at the 5% significance level (Table S2 in the online supplemental material). The d-hourly maximum time series are found to be sufficiently stationary. Indeed, among all RCMs and observations, at most 20% of the station or grid locations are found to feature (significant) increasing trends.
5. Results and discussion
We first illustrate that systematic and significant differences exist between observed and RCM-simulated rainfall extremes. We compute the bias as the difference between the modeled and observed mean annual maxima at each location (18 in total). Figure 2 shows a boxplot of the bias, for different durations and spatial resolutions. None of the RCMs reproduced the observed extremes at all durations. In particular, the 1- and 2-h extremes are poorly represented by the EUR-11 simulations and are underestimated at almost every station. This is partly due to the scale mismatch between EUR-11 resolution and the point observations. The H-Res simulations show better performance for these short-duration extremes. On average, ALARO also performs better for longer-duration extremes, but the spatial variability in the estimations is higher. On the other hand, CCLM tends to overestimate the longer-duration extremes. The Kolmogorov–Smirnov test (Fig. 3) confirms that the difference between the modeled and observed annual maximum distribution is statistically significant in many cases.
Boxplot of the bias in the mean annual maxima for different durations and model resolutions. Every box includes the 18 pluviograph stations/grid points. Positive or negative bias indicate overestimation or underestimation, respectively.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
Number of locations with p value > 0.05, for the Kolmogorov–Smirnov test statistic, under the hypothesis that the observed and modeled annual maxima follow the same distribution. Horizontal dashed line: total number of locations (i.e., 18).
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
Closely related to this preliminary result, Fosser et al. (2015) and Kendon et al. (2017) reported that convection-permitting models did not improve the simulations of daily mean precipitation in comparison with large-scale climate models, although they provided more accurate simulations of heavy hourly precipitation. In the context of extremes, this was also observed in Tabari et al. (2016).
A brief overview of the overall conclusions for the different evaluation metrics results is given in Table 3. The results are discussed in more detail in the rest of this section.
General overview of the RCM evaluation results for EUR-11 vs H-Res simulations.
a. Seasonality
We considered the AM and POT series and we recorded the seasons when the extreme events occurred, see Fig. 4 (EUR-11) and Fig. 5 (H-Res) for the AM series. Similarly, Figs. S1 and S2 (found in the online supplemental material) show the results for the POT series, which consist of cluster maxima of excesses above the 0.99 quantile (the quantile computation is considered for nonzero rainfall, viz., ≥0.1 mm). We observe that the seasonal cycle was very well reproduced by all models and all seasons. Notable exceptions can be found for MAR due to strongly underestimated rainfall extremes in summer and large overestimations in winter. The results of the hourly summer POT extremes of the CCLM models are a bit improved when using a higher spatial resolution (H-Res).
Seasonal-cycle analysis with the average seasonal occurrence of observed annual maximum d-hourly rainfall of the 18 pluviograph stations and the 18 grid points of the EUR-11 simulations.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
As in Fig. 4, but for the H-Res simulations.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
b. Diurnal cycle
Figure 6, along with Fig. S3 in the online supplemental material, shows the discrete probability distributions of the time of occurrence (UTC) of the AM and POT hourly rainfall, respectively, and are further referred to as the diurnal cycle. In addition to these graphs, we have calculated the Kullback–Leibler divergence (see appendix C) to measure how close the observed and simulated diurnal cycles are to each other. The results are shown in Table 4. The diurnal cycle of the EUR-11 simulations is poorly represented by all models, except ALARO. Using a higher spatial resolution strongly improved the results for the CCLM models. The conclusion is about the same for AM and POT extremes, except that the peaks in the observed daily cycle are less pronounced for POT than for AM.
Diurnal-cycle analysis with the average occurrence at each time of day (UTC) of observed annual maximum hourly rainfall of the 18 pluviograph stations and the 18 grid points of the RCM simulations (EUR-11 and H-Res).
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
c. Temperature and humidity dependency
To reduce the uncertainty in the estimation, the inference of the extreme precipitation/dewpoint–temperature relationship is based on a joint evaluation of all the locations in the study region (15 in total). Figure 7 shows the binned high quantiles of hourly precipitation against daily mean dewpoint temperature, for observations and RCMs at both spatial resolutions. The application of the quantile regression models is demonstrated in Tables S3 and S4 (found in the online supplemental material) for the following predictors: air temperature and dewpoint temperature, respectively. The estimations, together with the 95% confidence interval, are plotted in Figs. S4 and S5 (found in the online supplemental material) for the 0.95 and the 0.99 quantiles (predictor: dewpoint temperature). All RCMs except MAR are able to reproduce the super-CC scaling behavior. The 0.95 quantiles of the EUR-11 simulations (Fig. 7) have a CC scaling rate β1 that is significantly smaller than that of the observations (around 4% against the observed 6%–7%). The H-Res simulations have disagreeing values for β1 equal to around 4%, 6%, and 7.5%. For ALARO, again the difference in β1 between both spatial resolutions is less pronounced. For the 0.99 quantile (see Fig. S5), the β1 values for observations and RCMs agree well generally, and the other parameters (β2 and Tc) are satisfactorily reproduced by the high-resolution models. Note that Figs. S4 and S5 show estimates for which there are relatively large differences between EUR-11 and H-Res, but that the confidence intervals are so large that it cannot be concluded that these differences are statistically significant.
High τ quantiles of hourly precipitation as a function of the daily mean dewpoint temperature. Solid lines are estimated with binning; dotted lines are estimated with piecewise linear quantile regression [Eq. (2)]. Light-gray solid lines are piecewise regression lines of the observed quantiles, which serve as a reference (cf. top left).
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
Next, the predictive skill of the (dewpoint) temperature was compared by means of the GoF criterion; see Fig. 8. Most important, the influence of the predictors is of the same order of magnitude as in the observations for all the models, except for MAR. Increasing the model resolution leads to increased predictive skill for CCLM but slightly decreased skill for ALARO.
The strength of the relationship between the predictor (air temperature or dewpoint temperature) and 0.99 quantiles of hourly precipitation. Bars: the goodness-of-fit criterion GoF [Eq. (13)]. The vertical lines indicate the 95% confidence intervals of GoF.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
d. Temporal scaling
1) Scaling invariance of simulated extremes
First, we checked whether the temporal scaling GEV models, Eq. (5)–(6), are suitable for the simulated extreme intensities, where we relied on the log–log linearity of the GEV parameters (location and scale) with duration d, and in which the slopes correspond to the temporal scaling exponents (i.e., η or η1,2). As explained in section 2b, we considered the clock-hourly aggregated maxima of the EUR-11 simulations, whereas for the observations and the H-Res simulations, the subhourly aggregated maxima are used. In Fig. 9, we have plotted the maximum likelihood estimators
GEV parameters (location/scale) plotted against rainfall duration d for station Uccle: (top) location and (bottom) scale. Points: maximum likelihood estimators
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
Having verified the log–log linearity of the GEV parameters, we investigated further which scaling model is the most likely for extreme intensities. Two model selection procedures were used: (i) based on AIC and (ii) based on Bayesian hypothesis testing. The latter involves the zero hypothesis,
Number of stations (of 18) for which the multiscaling GEV model is better than the simple scaling GEV model. Model selection is based on AIC and evaluation of the posterior odd R = p(
Demonstrating the statistical significance of the multiscaling property is complicated here for several reasons: (i) the subjective choice of the number of series that should pass a particular test, (ii) different tests may give different results, and (iii) different time series lengths. However, instead of this determination, the aim here is to validate the models and therefore we limit ourselves to comparing the percentages of the RCMs with those of the observations. In general, we can observe that the RCM-simulated extremes (except MAR) broadly share the multiscaling property of the observed extremes.
2) Scaling model parameters for observed/simulated extremes: A comparison
We investigate to what extent the temporal scaling GEV parameters of the simulations correspond to the observed ones. Recall that the model consists of two types of parameters: (i) the GEV parameters of the 1-h extremes, (μ, σ, ξ), where μ and σ describe the main body of the GEV distribution, and ξ describes the tail of the distribution; (ii) the temporal scaling parameters, η or η1,2.
For each location, we plot the posterior mean and the 95% credible intervals of the shape parameter ξ and the temporal scaling parameters η1,2 in Fig. S6 (EUR-11) and Fig. S7 (H-Res) of the online supplemental material, and Fig. S8 shows the corresponding IDF curves. We have chosen to display ξ because this parameter is critical for the estimation of high return levels. It can be already seen that for ALARO, the η1,2 values are systematically underestimated, a feature that is also retrieved below.
The significance of the parameter preservation by the RCMs was again tested with AIC and Bayesian hypothesis testing (Table 6). For example, for the shape parameter ξ, the hypotheses are formulated as
Number of stations (of 18) for which the parameters of the multiscaling GEV model of the observed and the RCM-simulated extremes are found to agree. Testing for equality of model parameters ψ = (μ, σ, ξ, η1, η2) is based on AIC and evaluation of the posterior odd R = p(
From Table 6, it can be seen that the shape parameter ξ is generally one of the best reproduced parameters. Although Figs. S6 and S7 in the online supplemental material seem to reveal differences in the ξ values between models and with observations, it was found here that the differences are largely not statistically significant, highlighting the importance of a statistical framework for model validation. The scale parameter σ, on the other hand, was reproduced the least well, by all RCMs and for each spatial scale. Since σ describes the size of the deviation around the location parameter μ, we may conclude that the variability of the 1-h extremes is, therefore, not very well captured by the RCMs. The EUR-11 simulations represented the temporal scaling parameters η1,2 better than the GEV parameters concerning the moderate 1-h extremes, that is, (μ, σ). For the H-Res simulations, the same is true for CCLM, while this is not immediately clear to ALARO.
The effect of moving to a finer spatial scale is remarkably different between ALARO and CCLM. The benefit for ALARO is mainly the relatively good presentation of the μ parameter, which means that the 1-h extremes are well simulated to some extent. This agrees with the preliminary analysis of the bias in the annual maxima (Fig. 2). The temporal scaling parameters η1,2 are not as well represented as CCLM (for both spatial scales). As mentioned earlier, the η1,2 values are indeed slightly underestimated by ALARO, and consequently the resulting IDF curves are generally less steep. Opposed to ALARO, CCLM does not simulate the 1-h extremes very well, but it has the advantage that the temporal scaling parameters η1,2 are better reproduced.
e. Spatial structure
Figure 10 shows the spatial madogram, Eq. (7), for 1-, 6-, 12-, and 24-h extremes of observations and ALARO at both spatial resolutions (EUR-11 and H-Res). The madogram estimations of the observed 1-h extremes agree well with a similar study on the same dataset (Vannitsem and Naveau 2007). We first note that the empirical estimates may be higher than the theoretical upper bound, that is,
Spatial extremal dependence: the madogram υF(h) [Eq. (7)] as a function of distance h for the observed annual maxima at the 18 pluviograph stations and for the 18 grid points from ALARO. Dots: binned empirical madogram [Eq. (8)]. Dashed lines: 95% confidence bounds of the empirical madogram. Red line: independent case, υF = 1/6.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
The decorrelation length has to increase for longer rainfall durations, because the short-duration extremes are more likely to be caused by small-scale convective events, while longer-duration events are associated with large-scale precipitation patterns (Westra et al. 2014). For distances larger than 50 km, it can be seen in Fig. 10 that the red horizontal line (i.e., the independent case, υF = 1/6) falls in the 95% confidence interval of the observed madogram for d = 1 h and d = 6 h. Similarly, the decorrelation lengths for d = 12 h and d = 24 h are higher in comparison with d = 1 h.
It is clear that strong spatial correlations in the climate model simulated extremes are present within distances smaller than 50 km, and that the confidence intervals of the observed and simulated madograms do not overlap that much within this distance. This is consistent with the known overestimation of the spatial extent of rainfall events by RCMs (Maraun et al. 2010; Hobæk Haff et al. 2015). Using a finer spatial resolution only slightly reduced the correlations, but the overlap of the confidence intervals for the different resolutions is fairly small. Similar results were obtained for the other RCMs (not shown).
f. Temporal clustering
In Fig. 11, we plot the mean cluster size θ−1, estimated with the runs estimator Eq. (9), as a function of a high threshold. Series of d-hourly precipitation were considered, for d = 1, 3, 6, 12 h, and clusters of extremes were identified. The main conclusion is that hourly EUR-11 extremes gather in clusters of larger temporal extent than the observed extremes. The bias in the cluster size tends to be smaller for higher precipitation durations d, except for ALARO at d > 6 h. There is little improvement for hourly H-Res extremes, but the benefit for ALARO is threshold dependent. For d > 6 h, ALARO did not produce improved cluster sizes when using a finer spatial resolution. The temporal clustering in MAR is too high and features biases up to 20%, in particular for short durations, but the difference with the observations decreases for increasing d values.
Mean cluster size (h) against the threshold u, expressed in probability level, for (top) EUR-11 and (bottom) H-Res simulations. The runs estimator [Eq. (9)] was used.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0004.1
6. Conclusions
In this paper, we applied statistically based metrics to evaluate whether RCMs (nonconvection and convection permitting) capture the spatiotemporal characteristics of heavy subdaily rainfall events. The metrics provide a general picture of the RCM’s performance, in contrast to the current evaluation strategy, which usually only compares observed and modeled high quantiles of the rainfall distribution. The metrics can also be useful in the context of climate change impact modeling to determine whether extreme precipitation simulations can be used directly as input for a specific impact model.
Since these metrics are usually calculated empirically from the data, they may suffer from large uncertainties. We attempted to overcome this problem by assuming various statistical extreme value models for a range of aspects of the performance metrics. Our strategy includes a better inference of the metrics such as uncertainty quantification and model selection, which could indicate whether RCMs differ significantly.
Our main conclusions are as follows:
The EUR-11 simulations poorly reproduced the hourly extremes, whereas the longer-duration extremes (d ≥ 6 h) were much better presented. Conversely, the hourly extremes were well simulated at finer spatial resolution, H-Res, but only one model succeeded to reproduce satisfactorily the longer-duration extremes. Consequently, the H-Res simulations are particularly useful for impact studies related with short-duration extreme rainfall events, such as local urban flooding risk assessment. On the other hand, there is no spatial resolution that consistently improved the simulation of extreme rainfall events across all the durations, so that it is not clear which spatial resolution is the most advantageous for an IDF analysis.
The mean cluster length of extreme values in the hourly EUR-11 simulations is overestimated. Using a finer spatial resolution improves this estimate significantly.
The spatial clustering is overestimated by every RCM, at all durations. Using a finer resolution only slightly improved the spatial dependence estimation. We conclude that the RCMs may be not suited for flood risk estimation over a catchment area, which often requires information of the joint probability between extreme rainfall at multiple sites (Westra et al. 2014).
Except for the spatial extremes aspects, there is an acceptable performance of ALARO and the CCLM models in terms of the physically meaningful metrics. In contrast, MAR is not able to correctly represent extreme rainfall in subdaily resolution because it lacks a parameterized convective scheme.
This study can be extended in various ways. First, the application of extreme value theory could be further explored for metrics such as the seasonal/diurnal cycle. An example of statistical modeling of the annual cycle of extreme 1-day precipitation events was proposed in Maraun et al. (2009) and Schindler et al. (2012), where GEV distribution parameters of monthly maxima were modeled by oscillating functions with a 1-yr period. Future research should indicate whether the method can be extended to subdaily precipitation extremes. Second, evaluation metrics could be involved to examine the similarity of the synoptic situation in observed and simulated extreme events (Westra et al. 2014). An example of a GEV model to investigate the influence of the synoptic-scale atmospheric circulation on extreme precipitation was given in Maraun et al. (2012). Future research should aim to search for the most appropriate correction of future IDF relationships.
Acknowledgments
This work is supported by URCLIM and has received funding from EU’s H2020 Research and Innovation Program under Grant Agreement 690462. The CORDEX.be project was financially supported by the Belgian Science Policy (BELSPO) under Contract BR/143/A2. CICADA is the valorization project of CORDEX.be and has been supported by BELSPO. The computational resources and services for the ALARO-0 regional climate simulations were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation–Flanders (FWO) and the Flemish government department EWI.
Data availability statement
The hourly and subhourly RCM simulations used in this study are openly available from the World Data Center for Climate (WDCC):
ALARO-0 (https://doi.org/10.26050/WDCC/CORDEX.be_RMIB-UGent_ALARO-0),
CCLM-UCL (https://doi.org/10.26050/WDCC/CORDEX.be_UCLouvain_CCLM6-0-6),
CCLM-KUL (https://doi.org/10.26050/WDCC/CORDEX.be_KULeuven_CCLM5-0-6), and
The observed subdaily precipitation extremes are openly available from Zenodo (http://doi.org/10.5281/zenodo.4741178).
The code availability is listed in Table 1. The “temperature and humidity dependency” metric code, associated with the quantile regression models in Van de Vyver et al. (2019), is written in R and is available via Zenodo (http://doi.org/10.5281/zenodo.4644567). The “temporal scaling” metric code, associated to the temporal scaling GEV models and IDF statistics in Van de Vyver (2015a, 2018), is written in R and is available via Zenodo (http://doi.org/10.5281/zenodo.4644184). The code for the other metrics, “spatial structure” and “temporal clustering,” is not written by the authors and is available in existing CRAN packages.
APPENDIX A
Quantile Regression
APPENDIX B
Simple Scaling Versus Multiscaling
APPENDIX C
Kullback–Leibler Divergence
APPENDIX D
Bayesian Hypothesis Testing
Two applications are given below.
a. Choosing between temporal scaling GEV models
0: η1 = η2 (simple scaling), with associated model parameters . 1: η1 ≠ η2 (multiscaling), with associated model parameters .
b. Testing for equality
We perform hypothesis testing to investigate the plausibility that a certain statistical parameter is different for observations and RCMs. We denote by ψ(obs) = [μ(obs), σ(obs), ξ(obs), η(obs)] and ψ(mod) = [μ(mod), σ(mod), ξ(mod), η(mod)] the set of parameters of the simple scaling GEV model for the observed and RCM-simulated extremes, respectively. The data used are made of the observed and simulated extremes,
0: , with associated model parameters . 1: , with associated model parameters , where is the parameter vector ψ(.) with the element ψj removed.
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