The ensemble contains regional climate model simulations of the ReKliEs-De and EURO-CORDEX projects (Table ). Besides state-of-the-art dynamical regional downscaling models (RCMs) also empirical-statistical downscaling models (ESDs) were used. The total ensemble consists of 6 different RCMs (WRF, CCLM, HIRHAM5, RACMO22E, REMO and RCA4) and 2 ESDs (WETTREG2013 and STARS3) driven by 7 different Global Climate Models (EC-EARTH, CNRM-CM5, CanESM2, HadGEM2-ES, MPI-ESM-LR, IPSL-CM5A-MR, MIROC5). Table  displays all combinations of RCMs and GCMs that were analyzed and allocates them to their respective project. Within ReKliEs-De, GCMs were selected with the aim to cover the spread of anticipated near-to-midterm (until 2100) temperature and precipitation changes in the area Germany drawn from all available CMIP5 models.

Following the CORDEX-EUR11 protocol all RCM simulations cover the European continent on a 0.11∘ (approx. 12 km) grid. The ESD simulations use the same grid, but covering just the Central European part of the domain, due to their inherent methodological restrictions. Our analysis encompasses the historical and RCP8.5 model runs for a total period of 1971–2100.

In some cases, specific GCM and RCM versions are used for different combinations. See for a detailed model matrix, which specifies the model names and versions.

We focus our analysis on a Central European domain according to the ReKliEs-De project definitions. This domain is defined by all grid boxes over land areas that belong to river catchments discharging into German territory. The eight main river catchments are Danube, Elbe, Ems, Main, Mosel, Neckar, Rhine and Weser. The resulting mask covers mainly Germany and parts of the Czech Republic as well as parts of the alpine region. Figure  shows a map of all grid boxes considered in our analysis.

Within the ReKliEs-De project 24 standard climate indicators were calculated, including those which are used in this study:

tasmax [∘C]: daily maximum near surface (2m) air temperature;

The quality of the bias adjustment depends on the quality of the reference data set. The reference dataset used in this study is based on a combination of two data sources interpolated onto the same grid as the model data (the CORDEX-EUR11 grid): (1) the climate station network provided by the German Weather Service (DWD) and (2) the European gridded dataset EOBS-0.22deg-rot-v15.0 . The interpolation is based on which utilizes a distance and directional weight. Hence two stations lying in the same direction of the grid point (e.g. both north of the grid point) will have a lower weight than two stations lying in opposite directions (e.g. north and south of the grid point). As for the reference orography, we selected an orography based on SRTMv3 which was bilinearly interpolated onto the 0.11∘ CORDEX-EUR11 grid. For the interpolation of the temperature fields a constant lapse rate of 0.65K/100m was applied. We used no height adjustment for the precipitation fields.

The method was designed to adjust the bias for climate extremes, e.g. the number of hot days (su30) and the number of very wet days (r20mm) in regional climate model simulations. It is important to note that it had been a priority not to alter the simulation data, themselves. A basic assumption is that climate indicators using fixed thresholds must be applicable for the entire area of interest, encompassing mountainous as well as lowland regions. The underlying idea is to identify thresholds in the simulations of the reference period (1971–2000) which are specific to the individual GCM-RCM combinations and compare them to the defined fixed thresholds in the observed climate for the same period.

An overview of the work flow for the temperature indicator su30 is given in Fig. . The algorithm is as follows: We start by calculating the percentile values Psu30 and Pr20mm in the gridded daily observation data (1971–2000). Subsequently, the historical simulations for the period 1971–2000 by the GCM-RCM combinations are used to determine the values related to the percentiles calculated in the observation data. This is performed for every GCM-RCM combination. In most cases the resulting thresholds are exhibiting a bias, i.e., the thresholds are not matching those from the observation data, with model-specific deviations towards higher or lower thresholds. Therefore the thresholds need to be adjusted. The indicators for the period 1971–2000 are calculated a second time, using the bias-adjusted thresholds instead of the fixed thresholds (30 ∘C for su30 or 20 mm for r20mm, respectively). Their values are then very close to the observations, meeting the aim of the bias adjustment. For intercomparison purposes, threshold matrices are given in Table  for the model ensemble with su30 in the upper and r20mm in the lower row.

To further illustrate the steps of the method, details from an analysis using a simulation of the global model MPI-ESM r1i1p1 (MP1) forcing the regional model CCLM (CLM) are presented here for su30. Within the reference period (1971–2000) and for the ReKliEs-De domain, the simulated long-term annual mean of tasmax is 1.8K lower than the observed annual mean of the maximum temperature, cf. the boxes in the upper left part of Fig. c and a. As a consequence of the simulated lower mean of the maximum temperature, the number of hot days (su30), averaged over the whole domain is much lower in the climate simulation by MP1–CLM (0.7 days, Fig. d) than in the observed climate (4.5 days, Fig. b). According to our method, we adjust the threshold of 30 ∘C, so that the count of hot days approaches the observed 4.5 days. This is achieved by using all grid points of the ReKliEs-De domain from the years 1971–2000 of the gridded observation data to calculate the percentile which belongs to the fixed threshold of 30 ∘C. For this paricular threshold, a percentile of 98.80 is determined. In the next step the tasmax value belonging to the percentile of 98.80 is identified in the MP1–CLM simulation for the period 1971–2000 and all grid points of the ReKliEs-De domain. It is found to be 26.6 ∘C. This constitutes the new bias adjusted threshold for the recalculation of su30 for MP1–CLM which turns from a su30 to a su26.6. As Fig. e shows, the resulting area average for the indicator su30 simulated by the bias-adjusted MP1–CLM (4.4 days) is very close to that from observed data in Fig. b. Moreover, the spatial patterns of modelled and observed data have a close resemblance, too (cf. Fig. b and e).

In the frame of the ReKliEs-De project, further climate indicators based on fixed thresholds (e.g., id, su, r10mm, gsl) are subjected to the same bias adjustment process. With the bias adjustment for those indicators established, the model specific-thresholds are used to determine the indicators in model projections for the entire 21st century.

The method, described in Sect.  has been applied to the entire ReKliEs-De ensemble in order to assess climate extremes and their climate sensitivity. In the following paragraphs we discuss the individual patterns (maps) for su30 and r20mm with and without threshold adjustment. We also discuss the underlying threshold matrix and present the results of future projections.

The left-hand tables in Table  show the simulated su30 and r20mm values averaged over the RekliEs-De domain for the historical period (1971–2000) without threshold adjustment. The colors indicate the direction of the model bias, negative (light grey) and positive (light yellow) compared to the climate indicators derived from observations. Similar values are colored in light green. Since most of the RCMs have a cold bias , the su30 numbers are frequently underestimated, e.g., for EC-EARTH forcing REMO (ECE–REM), 1.4 occurrences of su30 are computed for the ReKliEs-De area in the simulation of the period 1971–2000, whereas the measurements yield a count of 4.5 days. There is a reversed situation for r20mm. Here, the RCMs frequently overestimate the mean precipitation patterns for the ReKliEs-De domain and also the r20mm indicator.

The right-hand tables of Table  depict the adjusted thresholds for su30 and r20mm, respectively. They range from 25.6 ∘C (ECE–HIR) to 31.4 ∘C (CA2–W13 and MP1–W13) for su30 and from 25.1 mm (ECE–HIR) to 17.9 mm (ECE–W13 and MI5–W13) for r20mm. Applying these threshold adjustments, the values of su30 and r20mm amount to nearly 4.5 and 5.1 days, respectively, for all ensemble members.

The resulting patterns for the 1971–2000 period with and without adjusted thresholds are shown in Fig.  (su30) and Fig.  (r20mm) for three GCMs (MP1, ECE and HG2). After adjustment, all patterns show similar regional characteristics with just minor differences. Without threshold adjustments most of the model members underestimate the averaged number of su30 by almost 4 days (e.g. MPI-CLM, MPI-RCA, ECE-CLM, ECE-RCA). Simulations with HG2-REM and HG2-WRF, on the other hand, exhibit a rather close match to the climate conditions. The comparison of the r20mm patterns with and without threshold adjustments reveals positive bias up to about 3 days (e.g. MPI–CLM and MPI–WRF).

In order to assess the future development of su30 and r20mm for the ReKliEs-De domain the climate indicators were calculated for every model member in an RCP8.5 simulation by using the adjusted threshold, respectively. Figure  depicts the changes for each model combination. It shows the climate signals between the periods 2041–2070 and 1971–2000. The RCMs are represented in Fig.  by red lines and the ESDs by magenta lines. The observed state for the baseline period is indicated by thick blue lines. The conditions of the period 1971–2000 for all ensemble members after bias adjustment are gathered at the 4.5;5.1 days point, i.e., they closely approximate the observations. Since the signals are determined from projections using an RCP8.5 scenario it can be inferred that without climate protection the number of hot days (su30) will change until 2041–2070 between 8.8 and 21.5 days – the range is determined using the 10th and the 90th percentile of the ensemble. This behaviour is corroborated by all models. For r20mm two modelling families can be distinguished: RCMs and ESDs. They exhibit different trend directions with a decrease in r20mm for the ESDs and a clear increase for the RCMs. Model combinations which simulate a strong increase of su30 show a smaller increase of r20mm, and vice versa. All RCMs in the ReKliEs-De ensemble simulate an increase in the number of extreme weather days (su30 and r20mm).

Simulations of regional climate models suffer from model bias and users should be aware of this fact. The simulated climate average for the historical period may be 1–2K below the observed average whereas, e.g., the number of hot days (su30) is clearly lower. This also strongly varies depending on the reference dataset used.

By the approach described in this paper we adjusted the point of view for each climate indicator in order to arrive at a similar mean level. This enhances the comparability of regional features and climate sensitivity considerations.

This approach does not aim at replacing established bias adjustments as an important intermediate stage for regional impact assessments. Yet, it improves the qualitative evaluation of features in regional climate ensembles without injecting too much complexity.

Such an application has its spatial limitations. The ReKliEs-De area is rather large and stretches the concept of obtaining feasible area statistics for climate indicators. However, the method could also be applied to single grid boxes or sub-domains. In that case the area must be of sufficient size and appropriate location, so that there are enough events defined by the threshold in the model simulations and the observations. In addition, border effects might occur, once the area in which the amount of threshold correction is determined differs from that in which the threshold correction is applied. It should be added that the quality of the results highly depends on the quality of the observation data. A final remark: An investigation to what extent the derived adjusted threshold matrix can be transferred to other variables would have been beyond the scope of this study.