Extreme precipitation statistics are often described as a function of the timescale involved as intensity–duration–frequency or depth–duration–frequency curves (e.g. Overeem et al., 2008). We consider two timescales or durations. One is a duration of 1 h, which is simply the time series of hourly precipitation sums available in each RCM grid point. The other is a duration of 24 h, where a 24 h sum is calculated in a sliding window with a 1 h time step. We will refer to these as hourly and daily duration, respectively. Our daily duration corresponds to the traditional climatological practice of reporting daily sums but allows heavy precipitation events to occur over 2 consecutive days. We also emphasise that the duration, as defined here, is not the actual length of the precipitation events in the model data but merely a concept for defining timescales.

Extreme value analysis (EVA) provides methodologies to estimate high quantiles of a statistical distribution from observations. The theory relies on the fundamental convergence properties of the time series of extreme events; for details, we refer to Coles (2001).

There are two main methodologies in EVA for obtaining estimates of the high percentiles and the corresponding return levels. In the classical or block maxima method, a generalised extreme value distribution is fitted to the series of maxima over a time block, which is usually 1 year. Alternatively, in the peak-over-threshold (POT) or partial-duration series method, which is used here, all peaks with maximum above a (high) threshold, x0, are considered. The peaks are assumed to occur independently at an average rate per year of λ0. To ensure independence between peaks, a minimum time separation between peaks is specified. Theory tells us that, when the threshold goes to infinity, the distribution of the exceedances above the threshold, x-x0, converges to a generalised Pareto distribution for which the cumulative distribution function is as follows: Gx-x0=1-1+ξx-x0σ-1ξ,x>x0. The parameter σ is the scale and a measure of the width of the distribution. The parameter ξ is the shape and describes the character of the upper tail of the generalised Pareto distribution (GPD); ξ>0 implies a heavy tail, which usually is the case for extreme precipitation events, while ξ<0 implies a thin tail. Note that, quite confusingly, an alternative sign convention of ξ occurs in the literature (e.g. Hosking and Wallis, 1987).

If we now consider an arbitrary level x with x>x0, the average number of exceedances per year of x will be the following: 1λx=λ01-G(x-x0). The T year return level, xT, is defined as the precipitation intensity, which is exceeded on average once every T years, as follows: λxTT=1, and by combining with Eq. (1) we obtain an expression for the return level, xT, as follows: λ01-G(xT-x0)T=1, from which we calculate the following: 2xT=G-11-1λ0T+x0. Data points to be included in the POT analysis can be selected in two different ways. Either the threshold x0 is specified and λ0 is then a parameter to be determined or, alternatively, λ0 is specified and x0 is determined as a parameter. We choose the latter approach, since it is most convenient when working with data from many different model simulations.

Choosing λ0 is a point to consider. Too high a value would include too few data points in the estimation, and too low a value implies the risk that the exceedances xT-x0 cannot be considered as being distributed according to GPD. We choose λ0=3 in accordance with Berg et al. (2019), which gives 75 data points for an estimation of the 25-year-long time slices. Hosking and Wallis (1987) investigated the estimation of parameters of the GPD and, based on this, warn against using the often-applied maximum likelihood estimation for a sample size below 500. Instead, they recommend probability-weighted moments, and we have followed this advice here.

We required a minimum of a 3 and 24 h separation between peaks for a 1 and 24 h duration, respectively. This is in accordance with Berg et al. (2019), and furthermore, synoptic experience tells us that this will ensure that neighbouring peaks are from independent weather systems. We found only a weak influence of these choices on the results of our analysis.

In practical applications of EVA, the parameters are estimated with large uncertainties due to the limited length of the time series. The threshold has the smallest relative uncertainty, the scale has a larger relative uncertainty and the shape has the largest relative uncertainty. Therefore, the relative uncertainty of the return levels also increase with increasing T, as can be seen from Eq. (2).

The delta change and bias correction approaches were introduced in general terms in Sect. 1. Now we will formulate EVA-based analytical quantile mapping based versions of the two approaches. In what follows, OT is the T year return level estimated from present-day pseudo-observations, while CT (control) and ST (scenario) denote the corresponding return levels, estimated from present-day and end-21st-century model data, respectively. Finally, PT (projection) denotes the end-21st-century return level after a bias adjustment has been applied.

The inter-model cross-validation goes in detail as follows. Each of the N models are successively regarded as being pseudo-observations. The individual adjustment methods are calibrated on the present-day parts of the pseudo-observations and model return levels (present-day and end-21st-century) as appropriate, depending on whether it is a bias correction or delta change method. The calibration is done as described above. The adjustment methods are then applied to present-day observations and model data, again as appropriate, to obtain the adjusted return levels for end-21st-century. These are then validated against the return levels for end-21st-century derived from pseudo-observations.

The basic validation metric will be the relative error of the return levels for end-21st-century for a given duration and return period T, as follows: RE=PT-VT/VT. Thus, this determines the absolute difference between the projected return level PT obtained from using an adjustment and the validation return level VT estimated from pseudo-observations for end-21st-century divided by the validation return level. This metric is calculated for every grid point and for every combination of model or pseudo-observations. Since we have N=19 model simulations in the ensemble, we have N×N-1=342 different combinations for validating each adjustment method and can make statistics of the relative error. This quantifies the average performance of the different methods.

End-user scenarios are often constructed as the median or mean from ensembles. We also tested this in the inter-model cross-validation set-up. The calibration is performed, as before, on each of the remaining models and adjusted return levels for end-21st-century are calculated. But then the median of these adjusted future return levels is calculated, and this is validated against the future pseudo-observations. Note that this gives only N=19 different combinations and, therefore, less robust statistics compared to the above.

Figure 2 displays the geographical distribution of the 10-year return level for precipitation intensity of a 1 h duration, calculated as the median return level over all 19 model simulations. The smallest return levels are mainly found in the arid North African region and, to some extent, in the Norwegian Sea, while the largest return levels are found in southern Europe and in the Atlantic northwest of the Iberian Peninsula. Mountainous regions, such as the Alps and western Norway, stand out as having higher return levels than their surroundings. This supports the idea that the models are not totally unrealistic in modelling extreme precipitation.

There is a general increase in the range of 20 %–40 % in climatic conditions from the present-day to end-21st-century. The relative changes are geographically quite uniform across the area. For instance, no evident difference between the land and sea appears. Moreover, the mountainous regions do not stand out from their surroundings.

We also show, in Fig. 3, the median 10-year return level for a 24 h duration. Again, the largest return levels are found in southern Europe and northwest of the Iberian Peninsula. Also, the mountainous regions stand out with higher return levels that are even more pronounced than for a 1 h duration. The return levels generally increase conditions from the present-day to end-21st-century by around the same percentage as for a 1 h duration, and they are also geographically homogeneous.

To obtain a more detailed impression of the data, Fig. 4 shows the return levels and their changes from the present-day to end-21st-century for a grid point in northern Germany for all 19 model simulations. For a 1 h duration (Fig. 4a), return values increase from the present-day to end-21st-century in all cases. For a 24 h duration (Fig. 4b), the return levels typically increase from the present-day to end-21st-century but with some exceptions. This behaviour is common to all regions. For both durations, we also note the large spread in return levels within the ensemble. The spread is much higher than the change between the present and future for most models; in other words, there is a poor signal-to-noise ratio. This is probably a combined effect of different climate signals in different models and natural variability (Aalbers et al., 2018).

In the following, we will present results using two different types of displays. First, we will use spatial maps of the median relative error, calculated from all combinations of model/pseudo-observations. Second, we will, for each adjustment method and for each combination of model/pseudo-observations, calculate the median relative error over each of the eight PRUDENCE subregions defined in Christensen and Christensen (2007) and shown in Fig. 1. For each region, we will illustrate the distribution of the relative error across all combinations of model/pseudo-observations by showing the median and the 5th and 95th percentiles of this distribution.

To obtain further insight into the difference in performance between hourly and daily precipitation, we consider, for a given return period, the relationship between the bias factor for present-day BP,T=CTOT and BF,T=STVT end-21st-century for all model/pseudo-observation combinations (see Fig. 10).

In this figure, the relationship between bias factors in the present-day and end-21st-century appears more pronounced for 1 h duration than for 24 h duration. That said, it must be borne in mind that if the point x,y is in the plot, then so is the point 1/y,1/x, and this implies an inherent tendency towards a fan-like spread of points from 0,0, as seen on both plots.

To quantify the strength of the above relationship, we define an index as follows: R=BF-BPBF+BP/2, where 〈⋅〉 means averaging over combinations of model/pseudo-observations. This index is an extension of the index introduced by Maurer et al. (2013). It is the ensemble average of the relative absolute difference between the present-day and future bias. A value of R=0 means that these biases are equal, i.e. perfect stationarity, and the smaller the value of R, the closer to stationarity (in an ensemble sense).

Values of R are given in the upper left corner of each panel of Fig. 10, and they also support the partial relationships described above and a stronger one for hourly duration. These relations are important since they could explain the generally good performance of the FAC method seen in the previous section. Suppose that BP,T=BF,T, then, in the following: PT=STCTOT=STOTCT=STBP=STBF=STVTST=VT, and the FAC method will therefore adjust perfectly.

We also note that daily data, due to the summation, would have less erratic behaviour than hourly data, and therefore, we would expect any relationship to be less masked by noise for daily data than for hourly data from purely statistical grounds. Therefore, any explanation as to why it is opposite should probably be found in physics or the details of modelling. We will discuss this further in Sect. 5.3.

The study by Räty et al. (2014) touches upon issues related to ours. However, our study includes smaller temporal scales (hourly and daily) and higher return periods (up to 100 years vs. the 99.9th percentile of daily precipitation corresponding to a return period of around 3 years). Nevertheless, the two studies agree in their main conclusion, namely that applying a bias adjustment seems to offer an additional level of realism to the processed data series, including in the climate projections, as compared to using unadjusted model results. The two studies both support the somewhat surprising conclusion that using present-day (pseudo-) observations as the scenario gives a skill comparable to that of the bias adjustment methods.

Kallache et al. (2011) proposed a correction method for extremes, i.e. cumulative distribution function transfer (CDF-t), and obtained good validation result with the calibration/validation split of historical data from southern France. The CDF-t method was applied by Laflamme et al. (2016) on daily New England data, who concluded that “downscaled results are highly dependent on RCM and GCM model choice”.

The grid spacing of present state-of-the-art RCMs available in large ensembles, such as CORDEX, is around 10 km, and at this resolution, it is necessary to describe convection through parameterisations. This is obviously an important deficit for our purpose, since this could represent a systematic bias in all our simulations and, therefore, violate our underlying assumptions that the individual model simulations and the real-world observations behave similarly in a physical sense. Thus, we do not promote naively applying the presented adjustment methods to hourly data from these models. Instead, the present work should be seen as a statistical exercise, and the methods can, in the future, be applied to convection-permitting model simulations that better represent the convective process. The results from the present work would apply equally to that case.

With the advent of convective-permitting models, a more realistic modelling of convective precipitation events is within reach, and a change in the characteristics of such events is seen (Kendon et al., 2017; Lenderink et al., 2019; Prein et al., 2015). This next generation of convection-permitting RCMs with a grid spacing of a few kilometres allows a much better representation of the diurnal cycle and convective systems as a whole (Prein et al., 2015). With that in mind, we foresee redoing the analysis when a suitable ensemble of convective-permitting RCM simulations becomes available.

The success of applying bias adjustment to climate model simulations is linked to the biases being stationary, i.e. present and future biases being more or less identical. In Sect. 4.3, we showed (in Fig. 10) that this was the case for a 1 h duration and less so for a 24 h duration in our pseudo-reality setting. Such a relationship is an example of an emergent constraint (Collins et al., 2012). This is a model-based concept, originally introduced to explain that models which have too warm (cold) a present-day climate tend to have a relatively warmer (colder) future climate. The reason for this is that it is the same underlying physics which generates the present-day and future temperatures (Christensen and Boberg, 2012).

We suggest that our observed emergent constraints could be explained in a similar manner, namely as a result of the Clausius–Clapeyron relation linking atmospheric temperature changes to changes in its humidity content and, thereby, precipitation changes. The change prescribed by the Clausius–Clapeyron equation is usually termed the thermodynamic contribution. In addition to this, there is a dynamic contribution, and this may explain the differences between the hourly and daily relation seen in Fig. 10. The rationale is that hourly extremes are entirely due to convective precipitation events with almost no dynamic contribution (Lenderink et al., 2019), while daily extremes are a mixture of convective events and large-scale strong precipitation, of which the latter has a more significant dynamic contribution (Pfahl et al., 2017), causing the less marked emergent constraint for the daily timescale. This interpretation is also supported in Fig. 4, in which daily precipitation sees some crossovers (future return level smaller than the present), whereas hourly precipitation does not have any crossovers.

In the definition of model bias, it is tacitly assumed that the observational data set has the same spatial resolution as the model data. In practice, however, it is rarely possible to separate the bias from a spatial-scale mismatch. For instance, if we compare modelled precipitation, which represents averages over a grid box, with rain gauge data, which represent a point, there can be a quite substantial mismatch for extreme events (Eggert et al., 2015; Haylock et al., 2008). Therefore, if the bias is adjusted towards such point values, it may lead to further complications (Maraun, 2013).

Sometimes, though, it is desirable to include the scale mismatch in the bias adjustment. Many impact models, e.g. hydrological models, are tuned to perform well with local observational data as input. This presents an additional challenge if this impact model is to be driven by climate model data for climate change studies, since the climate model will have biases in its climate characteristics (mean, variability, etc.) compared to those of the observed data. Applying the adjustment step, the hydrological model can rely on its calibration to observed conditions (Haerter et al., 2015; Refsgaard et al., 2014).

Only the basic adjustment methods have been included in our study. The simple climate factor approach has been applied in numerous hydrological applications (DeGaetano and Castellano, 2017; Sunyer et al., 2015 and other sources). We also wanted to test quantile-mapping approaches, which in extreme value theory takes the form of a parametric transfer function. This we have applied in two flavours in the spirit of Räty et al. (2014). Finally, we wanted to benchmark against the canonical benchmark methods, namely observations and raw model output.

There is a myriad of more specialised methods which are each tailored to account for a particular deficit of the simpler methods. First, there is the issue of whether it is, for precipitation, more reasonable to map relative quantile changes rather than absolute ones (Cannon et al., 2015). It has also been argued that a bias correction method should preserve long-term trends, i.e. the climate signal, and only adjust the shorter timescales, as extensively discussed in Cannon et al. (2015). Then multivariate methods have been argued for and applied in order to preserve relationships between variables (Cannon, 2018) and nested methods to account for different biases for different timescales (Mehrotra et al., 2018). Also, methods to correct for systematic displacement of variable features in complex terrain have been suggested and applied (Maraun and Widmann, 2015). Finally, Li et al. (2018) adjust stratiform and convective precipitation separately instead of adjusting the total precipitation. In this way, any future change in the ratio between the two types of precipitation is accounted for.

It could be interesting to examine the above methods in future studies, though we acknowledge it would be a quite extensive work. We can, at present, only guess at the outcome of such work, but the more refined methods may not perform too well in the inter-model cross-validation setting. The reason for this suspicion is that these methods, while being more elaborate, in most cases also have more parameters to be estimated, implying a higher risk of overfitting. An argument in favour of this is that the present study shows that the more elaborate quantile-mapping methods of DC or BC do not outperform the simpler FAC method.