The REGNIE gridded daily precipitation dataset , developed by the German weather service specifically for hydrological applications and with a grid spacing of roughly 1 km, is used to compute separate time series of observed daily precipitation area-averaged over the Wupper catchment (Fig. 2) for each full winter and summer in the period 1979–2015. From these time series the 99th precipitation percentiles of all days are computed separately for each season, and all days above their seasonal 99th percentile are defined as “extreme”. The areal extent of the Wupper catchment contains 753 REGNIE grid cells; precipitation-recording stations of the German weather service are marked in Fig. 2. An advantage of the REGNIE dataset is that measured totals are conserved, so that observed events (dry or wet) can be found preserved in the gridded field, which is in contrast to other methods on coarser grids, which use smoothing . Despite this, the usual warnings about using gridded observations to study heavy precipitation events must be recalled. In the absence of a sufficiently dense rain-gauge network in and around the catchment, the spatial variability and local intensity maxima of heavy precipitation events will not be captured in the gridded product, leading to precipitation extremes that are both underestimated and too spatially homogeneous, in particular in areas of complex topography and for convective events e.g.. The rain-gauge network underlying the gridded dataset must thus be sufficiently dense so that catchment-relevant extremes are acceptably captured. Alternatively, individual station(s) known to be broadly representative could be used for small- to medium-sized catchments.

To identify the large-scale circulation patterns associated with the heavy rainfall days, the corresponding 500 hPa geopotential height (Z500) anomalies are extracted from the ERA-Interim reanalysis . REGNIE precipitation has a measurement period of 07:30–07:30 LT (local time), equating to 05:30–05:30 UTC in summer and 06:30–06:30 UTC in winter. Z500 anomalies are thus averaged over the timesteps 12:00, 18:00 and 00:00 UTC, i.e. the middle of the accumulation period, and are relative to their 1979–2015 seasonal means.

The extracted Z500 anomaly patterns next undergo a cluster analysis via the simulated annealing and diversified randomization (SANDRA) method . SANDRA has been shown to overcome many of the limitations of standard k-means clustering algorithms, greatly reducing the role of stochastic effects in the final cluster partitions and thus providing clusters much closer to the “global optimum” . It is also less numerically costly than model-based clustering algorithms such as Gaussian mixture models e.g.. Relevant software for meteorological applications has been developed in the EU COST Action 733 , and we use this software in our study. Geopotential height is a standard variable for cluster analyses of atmospheric circulation patterns e.g.. Following , the spatial extent of the clustering domain is subjectively chosen such that the typical synoptic patterns associated with extreme precipitation in the Wupper catchment can be captured within the domain when present (Figs. 3 and 4), which is easily identifiable from historical extremes. Prior to the cluster analyses, outliers that would have little chance of being assigned to an appropriate cluster are removed from the datasets. Outliers are identified by computing, for each day, the Pearson pattern correlation of each Z500 anomaly pattern with that on all other extreme days; any day whose maximum pattern correlation (i.e. across all days) is more than 2 standard deviations below the sample mean of the same is excluded from the cluster analysis. In our case, this results in just 1 day being removed from each of the winter and summer input data, leaving 31 and 33 days respectively. As a stability criterion, the number of clusters K is increased until the minimum intra-cluster pattern correlation – that is, the Z500-anomaly pattern correlation between each cluster member and its own cluster mean – is not less than 0.5. This way all days are assigned to a cluster with which they have genuine similarities, rather than simply the error-minimized “least bad” cluster, as is typically the case in clustering large datasets of meteorological variables.

The resulting Z500 anomaly clusters and any outliers are considered as “reference” extremal circulation patterns against which candidate days from a given dataset can be classified as PEDs, based on their similarity to these references. To this end, the area-weighted Pearson pattern correlation ρi,j (uncentred) between the Z500 anomaly fields of the candidate day i and the cluster centroid j is used; for our clustering domain (Figs. 3 and 4) this encompasses 1935 data points (i.e. grid cells). A perfect ρi,j would have a value of 1. With the guiding aim of correctly classifying as many extreme days (i.e. P≥P99D) and rejecting as many non-extreme days as possible, a ρ threshold (ρjt) is chosen for each cluster centroid j and days with a ρi,j below this threshold are rejected. ρjt for each cluster is simply the minimum intra-cluster pattern correlation, reduced by 10 % so that days with a ρ comparable to the lowest intra-cluster ρ are not rejected. To account for clusters with a particularly high ρjt due to few members, ρjt is capped at two-thirds.

All remaining days not rejected based on their ρi,j are next assessed in terms of relevant meteorological predictors at the local-scale, i.e. in the vicinity of the catchment. The choice of meteorological predictor and the area around the catchment in which it is assessed are flexible. In general, they may depend on the catchment, season and variables available from the coarser parent model. advise choosing predictors that are easy to diagnose from coarse-resolution models and consistent with meteorological knowledge of precipitation extremes, e.g. circulation and stability metrics. Guidance may also be sought from statistical downscaling techniques that have been successfully applied in the region.

For the Wupper catchment in summer (JJA) and winter (DJF) we select daily maxima (06:00–05:59 UTC) of relative humidity (700 hPa JJA, 300 hPa DJF) as an indicator of (near-)saturated air masses in the troposphere, 500 hPa horizontal divergence (JJA, DJF) as an indicator of tropospheric vertical ascent (of a frontal or convective nature), convective available potential energy (CAPE; JJA) as an indicator of atmospheric instability, and daily accumulated coarse-model precipitation (JJA, DJF). As with the Z500 data, variables are extracted from ERA-Interim on a Gaussian N128 grid (∼0.7∘). To account for the transient nature of many extreme weather systems and the often low temporal resolution of reanalysis or model data (e.g. 6-hourly in the case of ERA-Interim), it is not only the nearest ERA-Interim grid cell to the catchment centre that is considered, but an entire area of 7×7 grid cells around it (3×3 in the case of coarse-model precipitation). With the guiding aim of “catching” the highest number of observed precipitation extremes (i.e. P≥P99D) while excluding as many other days as possible, exceedance thresholds for each variable are empirically chosen, either as exceedances of a given percentile (divergence, CAPE, coarse-model precipitation) or as absolute values (relative humidity). The thresholds used for the Wupper catchment are summarized in Table 1. To account for different model climatologies and thus facilitate transferability to other models, the (absolute) relative humidity threshold (RHthresh) determined from the training data can be redefined as a function of the model's climatological mean (RH‾), i.e. RHthresh=A⋅RH‾, with A a constant; this function can be applied to another model's RH‾ to get RHthresh for that model.

In order to be classified as a PED, each threshold must be exceeded at any one of the grid cells (not necessarily the same cell) around the catchment. A schematic summarizes the full two-step selection algorithm (Algorithm 1). Extremal patterns identified for the Wupper catchment are presented in Sect. 3.1.

The combination of variables, thresholds and clusters for detecting observed precipitation extremes and excluding non-extreme days is, as discussed above, empirically determined on the basis of the ERA-Interim and REGNIE datasets. Once this has been achieved, the method is applied identically to 0.11∘ (∼12.2 km) evaluation simulations over the pan-European EURO-CORDEX domain , roughly 25–72∘ N, 20∘ W–50∘ E, covering the period 1979–2015. Simulations were performed with the regional climate model COSMO-CLM CCLM; version 4.8, with ERA-Interim reanalysis as lateral boundary forcing. CCLM is the community model of the German regional climate research community jointly further developed by the CLM-Community. The years 1989–2008 were simulated by the CLM-Community as part of the EURO-CORDEX experiment . The years 1979–1988 and 2009–2015 (up to 31 July 2015) were simulated by the present authors using identical model version and settings.

Z500 CCLM data are interpolated to the clustering domain and the selected meteorological variables are conservatively regridded to a grid of similar spatial resolution to that used in the training stage, i.e. 0.7∘, and centred on the Wupper catchment. All winter and summer days are then either classified as PEDs for further dynamical downscaling with CCLM to a convection-permitting resolution of 0.02∘ (∼2.2 km) or rejected; the nesting ratio of 5.5:1 is in line with that recommended in the literature . The enhanced performance of CCLM at convection-permitting resolution (relative to coarser resolutions) in reproducing precipitation statistics, particularly extreme statistics, over central Europe has been extensively documented .

The additional downscaling step is performed using the same version of CCLM with a 221×221 grid cell domain centred on the Wupper catchment (Figs. 3 and 4), giving sufficient spatial spin-up upstream. A total of 161 of the CCLM grid cells fit inside the catchment. The simulations are carried out in “weather forecast mode”, i.e. initialized with interpolated values from the parent model. The multi-year simulations of the parent model ensure that soil moisture and temperature are spun-up at the 12 km scale, though not necessarily at the scale of the CPM. The soil moisture climatology tends to be drier in CPMs due to the sparser nature of their precipitation events . While studies suggest that the transient boundary conditions are of first-order importance for the occurrence of precipitation e.g., precipitation extremes highly sensitive to localized soil-moisture anomalies may be inadequately represented under such a procedure.

Lateral boundary conditions are updated every 3 h and 50 unevenly spaced terrain-following vertical levels are used. For each identified PED, the 0.02∘ simulation is initialized at 12:00 UTC the preceding day to allow abundant precipitation spin-up time; as little as 3–6 h are typically sufficient in CPMs though . PEDs on consecutive days are downscaled continuously to save resources. For validation, the precipitation statistics of the dynamically downscaled PEDs from the CCLM evaluation runs are compared with those of the observed PEDs identified from ERA-Interim. Area averages of daily precipitation over the Wupper catchment are considered, using REGNIE and 0.02∘ model output. The REGNIE and CCLM grids are of similar spatial resolution (1 and 2.2 km, respectively). Users should nonetheless be cognizant that datasets of different resolution may exhibit differing statistical characteristics simply because of their different resolutions, e.g. for the area mean. The evaluation and validation of the identified PEDs is presented in Sect. 3.2.

To provide a sterner test of the method, we additionally perform two sets of 30-season convection-permitting time-slice simulations over the Wupper catchment so that the method can also be assessed in reverse – of the actually simulated 0.02∘ extreme days (P≥P99D), how many would have been identified as PEDs from the 0.11∘ coarse model?

A different GCM – the Max Planck Institute's Earth System Model (MPI-ESM-LR) – at the start of the modelling chain provides a new challenge for the method from the previous ERA-Interim-driven simulations. The MPI-ESM-LR runs are continuous transient simulations performed as part of the CMIP5 project , using observed greenhouse gas concentrations from 1949–2005 (historical) and Representative Concentration Pathway 8.5 RCP8.5; from 2006 to 2100. One MPI-ESM-LR member (1949–2100) has been continuously downscaled with CCLM over the EURO-CORDEX domain to 0.11∘ resolution by the CLM-Community ; model settings are as in the previously discussed ERA-Interim-driven evaluation runs, greenhouse gas concentrations excepted.

For the present study, we have dynamically downscaled the aforementioned 0.11∘ CCLM transient simulations to 0.02∘ over 30 summers from the historical and RCP8.5 periods, 1970–1999 and 2070–2099 respectively. The 0.02∘ model domain and set-up are the same as in Sect. 2.3 (greenhouse gas concentrations aside); simulations are initialized in April and run continuously until the end of August each year, with analysis restricted to JJA. Summertime extreme precipitation in the Wupper basin tends to be of a convective and more catchment-scale nature than its wintertime counterpart, with small-scale variability and chaotic processes playing an enhanced role in event intensity. In addition to this, potential differences in large-scale circulation found in a future climate pose an additional challenge for the classification algorithm. The choice of 30 summers, historical and future, is thus intended to ensure a robust testing of our method. The performance testing via seasonal time-slice simulations is presented in Sect. 3.3.

The greater diversity of synoptic patterns that can lead to extreme precipitation in the Wupper catchment in summer, compared to winter, can be seen in the number of clusters K necessary before our stability criterion (see Sect. 2.1) is reached (Figs. 3 and 4). The higher K also hints at the generally more challenging nature of forecasting summertime intense precipitation, when synoptic forcing tends to be weaker and small-scale chaotic processes play an increased role. In winter (Fig. 3), precipitation extremes in the Wupper catchment are most often associated with a dipole-like structure characteristic of a strong positive phase of the North Atlantic Oscillation , with various shifts of the dipole centres (clusters 1–3). Such a synoptic pattern gives a strong zonal forcing, sweeping deep low-pressure systems and associated frontal precipitation across the catchment; similar clusters have previously been identified for north-eastern Switzerland . For the remaining cluster (cluster 4) and the outlier, shallower depressions become embedded in a relatively weak zonal flow, depositing their albeit less intense precipitation over a more prolonged period; these patterns account for less than one-sixth of all extreme days (P≥P99D) though. In summer, a dipole-like pattern can also be seen on some extreme days (cluster 1), though accounting for just over one-seventh of all extremes; such events in summer can also be expected to include enhanced frontal convection. The remainder of the summertime extremes are associated with a weaker zonal forcing. High pressure over eastern Europe often advects warm, moist air from the Mediterranean into central Europe (clusters 2 and 4), enhancing instability and increasing the chance of deep convection; also identified such a pattern as bringing intense precipitation to south-west Germany during summer. Another common pattern is that of a front, often with a small embedded low, extending across the catchment (clusters 3 and 8) in the wake of an eastward moving ridge and triggering frontal lifting as it passes. Quasi-stationary mid-tropospheric cut-off lows (clusters 5–7) are the most common cause of summertime extremes in our catchment, allowing slow-moving surface lows to advect a persistent moisture stream, within which intense convective cells can develop, across the catchment. A not dissimilar pattern was also identified by in their study of extreme precipitation in Austria.

While still capturing more or less all observed extreme days, the constraints derived from ERA-Interim variables enable the classification algorithm to reduce the number of PEDs to well below 10 % of all days (Table 2). In the process, the number of “redundant days” (i.e. P<P90D) falls from about 3000 to 48 in winter and 126 in summer. The “redundant days” thus occupy a much smaller fraction in the PEDs than in the set of all days. Such a good performance in the training dataset is, however, no surprise.

Applying the same methodology to the 0.11∘ CCLM evaluation runs (ERA-Interim driven) over the same period, a similar number of PEDs are identified for dynamical downscaling to 0.02∘ (Table 2). The PEDs again represent well below 10 % of all days, slashing the computational expense against a continuous simulation of the whole period by an order of magnitude. Of note is that although the 0.11∘ CCLM simulations are forced at the lateral boundaries by ERA-Interim, only 123 of the 320 dates identified as PEDs in CCLM in summer are also found amongst the ERA-Interim PEDs. This is attributable to the fact that RCMs without interior constraints (i.e. some form of internal nudging) cannot synchronously reproduce the local-scale day-to-day variability of their parent model . RCMs of sufficiently large domain size thus often generate large internal variability e.g., often comparable to that found in GCMs , which can cause the local RCM solution to significantly deviate from that of its parent model. In the presence of a strong zonal throughflow, e.g. in winter, the growth of differing internal solutions is limited due to small-scale perturbations being more rapidly swept out of the domain . This explains the higher fraction of common days that we find in winter (150 / 220).

Comparing the empirical cumulative distribution functions (ECDFs) for catchment-averaged precipitation (observed) of all days and PEDs from the training dataset (ERA-Interim), the greatly increased probability of heavy precipitation on a randomly selected PED becomes apparent (Fig. 5, blue curve): in the set of PEDs, the probability of exceeding the climatological winter (a) 99th (90th) percentile is about 20 % (80 %), whereas in the set of all days it is only 1 % (10 %). For summer (b), the situation is less pronounced but the climatological (JJA) 99th (90th) percentile is exceeded on about 15 % (60 %) of the days in the PED set. Taking all days, the ECDF can be well described by a typical gamma distribution; the gamma distribution is known to represent the bulk of the daily precipitation distribution well, though it performs less well for the tails . The form of the ECDF of the observed PEDs (blue curve), however, is far removed from that of the set of all days (red curve), as the probability is shifted towards more intense precipitation. The change in form of the ECDF – from one dominated by dry to light-rain days, to one dominated by heavy- to extreme-rain days – results from the classification algorithm's removal of days with a low potential for intense precipitation.

Dynamically downscaling all CCLM 0.11∘ PEDs to 0.02∘ produces ECDFs of daily precipitation that closely resemble those of the observed PEDs, for both seasons (Fig. 5, green curve); both ECDFs are again dominated by heavy to extreme precipitation events, with dry days (PD<0.1 mm) almost completely eliminated. Indeed, many of the seemingly dry to light-rain days counted over the Wupper catchment in the convection-permitting simulations do still feature heavy precipitation, though displaced to neighbouring regions of the 0.02∘ simulation domain (Fig. 6); this occurs most often in summer, owing to the more small-scale and chaotic nature of convective precipitation. The good match between the ECDFs of observed and downscaled PEDs shows that, with skilful classification of the PEDs, selective downscaling can be relied on to realistically reproduce the same range of precipitation events over the catchment as expected from the training dataset and observations, allowing of course for known model biases e.g.. In the process, computational expense is reduced by over 90 % (Table 2) compared to the computational efforts that would be required for a continuous simulation over the same period at such high spatial resolution. While the spatial resolutions of REGNIE and CCLM are similar (1 and 2.2 km, respectively), users should beware that area means in datasets with considerably different grid resolutions may differ simply because of the different sample sizes, i.e. the number of grid cells contained within the averaging area, in particular for small catchments and large differences in the grid-box area.

The dynamical-downscaling of two sets of 30-summer time slices – historical (1970–1999) and RCP8.5 (2070–2099) – from 0.11 to 0.02∘ provides an additional set of tests for the classification algorithm, namely: what fraction of the actually simulated extreme days in the 0.02∘ model would the method have identified as PEDs from the 0.11∘ model? In addition, is classification performance degraded in a future climate? The summer season is chosen to ask these questions due to the greater challenges in predicting summertime intense precipitation (see Sects. 2.4 and 3.1).

Applying the classification algorithm, identically to in Sect. 3.2, to the 0.11∘ historical and RCP8.5 simulations again yields selections of PEDs representing less than 10 % of the total days (Table 3). Amongst these PEDs, at least 75 % of 0.02∘-simulation extreme days are captured in both time slices. In the case of the historical simulations, the fraction of redundant days (i.e. P<P90D) climbs by almost six percentage points relative to the training dataset; for the RCP8.5 simulations, the fraction falls marginally. The former increase may simply be an artefact of the fewer summers (30 vs. 37) present in this testing dataset. The similarity of performance between the historical and future simulations is noteworthy, particularly in light of RCP8.5 2070–2099 representing the end of the most extreme RCP scenario. Projected changes in large-scale extratropical circulation can be considerable e.g. and are known to exert strong control on regional precipitation climatologies . In the case of the MPI-ESM-LR model used in this study, for example, a northward shift of the annual-mean jet in the Atlantic sector and reduction in the frequency of both North Atlantic and Eurasian summertime anticyclonic blocking are projected under the RCP8.5 scenario. Despite this, the classification algorithm performs more or less the same in historical and future climates. While the classification algorithm unsurprisingly fails to capture all extreme days in either the historical or RCP8.5 simulations, the fact that the performance is the same across both climates indicates the added value of employing our downscaling methodology, allowing more robust conclusions to be drawn from the output. Of the extreme days that are not captured, 6 out of 7 (historical) and 4 out of 5 (RCP8.5) are lost due to their circulation patterns not matching any of the pre-defined extremal clusters well. This could indicate that the training period for identifying the extremal patterns is too short to encompass sufficient diversity or, more likely, that the GCM in question (MPI-ESM-LR) does not adequately represent the frequency and/or persistence of the large-scale circulation patterns that lead to observed extremes in our catchment. For example, CMIP5 GCMs are known to underestimate the frequency of Eurasian blocking and GCMs in general often underestimate the persistence of blocking systems e.g.; the poleward flank of such blocking anticyclones often transports warm moist air into central Europe, enabling intense convective precipitation (see Sect. 3.1). In the case of MPI-ESM-LR during summer, a southward bias in the storm-track axis and over-active North Atlantic blocking are also evident in the CMIP5 historical simulations .

The similar performance of the classification algorithm across climates, as well as the evaluation period, is confirmed by looking at the historical and RCP8.5 ECDFs (Fig. 7). As in the training dataset, the ECDFs of the PEDs are shifted towards more intense precipitation compared to the ECDFs for the sets of all days. For the PEDs, the probability of exceeding the respective climatological (JJA) 99th (90th) percentile in the historical and RCP8.5 simulations is similar to that found in the training dataset and the dynamically downscaled PEDs of the evaluation period, roughly 10 % (55 %) (compared to 1 % (10 %) for all days), and the ECDFs are dominated by heavy to extreme events with dry days almost absent. To quantify differences in the distributions of precipitation events amongst all days and the PEDs for discrete intensity ranges, we compute the relative likelihoods (R) of finding a precipitation event within a given intensity range for the historical and RCP8.5 simulations (Fig. 8); this is simply the ratio of the respective probabilities, e.g. P(E|PED):P(E), with the smaller of the two probabilities used as the denominator for plotting purposes.

The greatest difference between all days and the PEDs is found in the relative likelihoods of a randomly sampled day being dry, which is an order of magnitude lower in the PEDs. Indeed, considering the set of non-PEDs, the probability density function exhibits an even higher density of dry days than that found for all days (not shown). Focusing on just wet-day percentiles, a regime shift from a higher R for all days to a higher R for PEDs occurs above the median wet-day event. The higher R for the PEDs grows further as event intensity nears the most extreme precipitation events, consistent across historical and RCP8.5 experiments and approaching a factor of 10 in places (Fig. 8). For the most extreme events (PD≥PW99.9), more variability between historical and RCP8.5 R values emerges as the number of days involved limits towards zero. Future changes in the fraction of wet days, and the sensitivity of wet-day percentiles to such changes , likely contribute to some of the small differences in relative likelihood between the historical and RCP8.5 experiments.

The preconditioning of PEDs on known extremal circulation patterns does not just reduce the total number of days to dynamically downscale. Importantly, it also allows conclusions to be drawn about changes in catchment-relevant precipitation between two periods, e.g. present and future climates, for these circulation patterns. A classification method that does not guarantee the capture of all extreme days clearly cannot be used to draw overall conclusions about precipitation changes in a given catchment. Preconditioning on circulation types does, however, allow conclusions to be drawn about changes in specific classes of precipitation extreme (Fig. 9), e.g. as identified via the clustering methodology outlined in Sect. 2.1. For example, for a known extremal circulation pattern, will the likelihood that the accompanying precipitation exceeds some catchment-relevant threshold be higher or lower in the future? This approach is in a way analogous to the framework used in conditional event attribution e.g., where the question is posed: for some observed circulation pattern, how is the event's intensity affected by known thermodynamic changes in the Earth's climate system? A major advantage of such an approach is the relative robustness of projected thermodynamic changes in the climate system compared to projected dynamical changes . From a catchment-hydrology perspective, one could imagine this being particularly useful for catchments vulnerable to specific compound extremes, for example intense precipitation in an estuarine catchment compounded by a shoreward moving low-pressure system with strong onshore winds e.g.. Beyond the extremal patterns identified from the training period, however, there remains the possibility that a future climate may also contain new extremal circulation patterns that were previously either not associated with extreme precipitation or simply not present at all. Such systematic effects could only be explored with continuous dynamical downscaling of the different climates.