The first statistical downscaling method applies a bias correction to the statistics that describe the precipitation time series. Consequently, this method does not return a precipitation time series, unlike the three other downscaling methods. This method can be regarded as a BC method applied directly to statistics (indicators) instead of to a daily precipitation time series. The BC factor is calculated as the ratio of the observed indicator to the model indicator of historical simulations, and then applied on the model indicator of scenario simulations to derive projected indicator. The indicators used in this study, to which the BC is applied, are discussed later on in Sect. 2.4.

An important assumption of all BC methods is that the climate model precipitation bias is time invariant, which might not be the case (Leander and Buishand, 2007). Furthermore, BC methods assume that the temporal structure of wet and dry days of the scenario-projected precipitation by the climate model is accurate. Successive days are also assumed to be independent.

The change factor of mean method or delta change method is frequently applied in the literature. The same simple rationale of the BC method can be applied by using a change factor approach instead. Here, no correction is applied to GCM precipitation projections. Instead, the relative change between the historical and scenario simulations of the GCM is used to calculate a change factor that can then be applied to the observed time series (Sunyer et al., 2012, 2015). The method applied to the precipitation P of day t in month m can be summarised by Eq. (1). 1aPm,tProj=amPm,tObs, in which the following applies: 1bam=Pm,…GCMScenPm,…GCMHis.

In this notation, the precipitation is given for month m and time step t in the observations (Obs), and GCMScen and GCMHis refer to the scenario and historical simulations of GCMs, respectively. For this implementation, the change factor is calculated per month.

CFM does not change the number of dry days directly. However, since the change factor is applied to all precipitation in a given month, days in the Uccle time series with precipitation values close to the wet day threshold (P<1.0 mm – dry day; P≥1.0 mm – wet day) can change the state, depending on the change factor am. The Uccle precipitation time series has a resolution of 0.1 mm. The wet days nearest to the threshold have a value of 1.0 mm, while the closest dry days have a value of 0.9 mm. Consequently, wet days are changed into dry days for am<1.0, while a transformation of dry days into wet days requires am>1.00.9=1.11. In conclusion, CFM is expected to show slight changes in terms of dry days, with a bias towards rising the number of dry days and, thus, the dry spells they compose. The mean monthly total precipitation changes projected in this method can be used as a reference for the other methods.

An important assumption made in all CF methods is that the changes at the local (weather station) level are the same as the changes described at the spatial, grid-averaged scale of climate models. Different from the BC methods, the CF methods assume the temporal structure of the observed time series is preserved. Furthermore, it is assumed in the CFM method that all precipitation in a given period (i.e. month or season) is changed by the same factor, regardless of the time step considered or the precipitation intensity observed. In addition, the method assumes consecutive days are independent.

QP methods form a more advanced approach to the application of change factors. The core principle of the methods is that the change factors are calculated and allocated based on the exceedance probability of the precipitation intensities. More precisely, the observed daily precipitation with exceedance probability p is modified by a change factor obtained by comparing the scenario and historical simulations of climate models for the same exceedance probability p. This is opposed to the idea of applying the same change factor to observed precipitation amounts ranging from zero to the most extreme values, as is done in CFM.

The QP version applied by Ntegeka et al. (2014) is used here in which the empirical exceedance probabilities pk are estimated by making use of the formula (kn+1) for Weibull plotting positions, where k is the quantile rank (1 for the highest), and n is the number of wet days. This approach can change the exceedance probabilities strongly in comparison to the linear interpolation of the cumulative density function represented by (kn), especially for extreme ranks. This approach was shown to be best suited for estimating return periods of extreme events (Makkonen, 2006).

In QP, the dry day frequency is perturbed by making use of a two-step perturbation process. In a first step, change factors are calculated to determine the relative change in dry day frequency between the scenario and historical simulations of climate models. These determine whether dry days in a given month should be converted to wet days or the other way around. This is done randomly using a stochastic approach. However, the following assumption concerning the clustering of dry days is made: only wet days preceded or followed by a dry day are eligible for the conversion, or only dry days both preceded and followed by a wet day can be converted. After the wet/dry day perturbation step, the precipitation intensity of remaining wet days is perturbed by change factors derived from comparing the scenario and historical simulations of climate models. Due to the randomness introduced by the dry day perturbation step, multiple time series are generated. A sensitivity analysis is executed by varying the number of simulations (see Sect. S2 and Figs. S1–S3). The selection of the best simulation is based on the following four indicators that can be derived from a precipitation time series: the mean (M), coefficient of variability (CV), skewness (S) and average monthly autocorrelation coefficient for a lag of 1 d (ρ1). Using these four indicators, the distance D between the climate change signals of the generated series and the GCM time series, for a given month m, is calculated as follows: 2Dm=∑i=14Ii,mgIi,mObs-Ii,mGCMScenIi,mGCMHis2, where g denotes the generated series for the indicator I, and Obs, GCMScen and GCMHis have the same meaning as in Eq. (1). The simulation corresponding to the smallest distance is selected as the best one.

The CF assumptions remain in place for the QP method and the assumption regarding consecutive days as independent. Unlike the CF method, it is now assumed that extreme and non-extreme precipitation amounts can change with different factors. The temporal structure of the observed time series is not explicitly changed. Furthermore, it is assumed that the highest relative changes are applied to the days with the highest daily precipitation. The method allows for an explicit perturbation of the temporal structure of the observed time series.

The fourth selected statistical downscaling method in this study is the stochastic and event-based approach developed by Thorndahl et al. (2017), which is not directly based on change factors but generates stochastic time series instead. Consequently, it belongs to the category of the weather generators. The method constructs a stochastic time series by alternating wet and dry events. Wet events are sampled from an observed point precipitation time series. Observed dry event durations are fitted to a two-component mixed exponential distribution (three parameters, i.e. λa, λb and pa, in Eq. 3) from which dry events durations (also called inter-event durations; tie) are sampled. Both sampling operations are performed for each season separately. 3ftie=paλa,ieexp⁡-λa,ietie+(1-pa)λa,ieexp⁡-λb,ietie, where λa,ie and λb,ie are the rate parameters for two populations, a and b, with different exponential distributions, and pa is the weight of population a. More information about the two-component mixed exponential distribution can be found in the Supplement (Sect. S1). A popular way to fit this type of distribution to data points is by applying iterative expectation–maximisation algorithms (Yilmaz et al., 2015). An implementation of this algorithm for fitting mixed exponential distributions is included in the R package of Renext (Deville and IRSN, 2016).

Figure 1 shows the two-component mixed exponential density functions that are fitted to the empirical probabilities of the observed (Uccle) dry event lengths. The fitted distributions underestimate the proportion of inter-events with a duration of 1 d. This underestimation is countered when sampling since the complete range [0, 1.5] of sampled durations is rounded to 1 d. Figure 2 shows the two-component mixed exponential cumulative density functions that are fitted to the seasonal empirical cumulative density functions of observed (Uccle) daily precipitation intensities. The fitted distributions are very close to simple exponential distributions since pa≈pb≈0.5(pb=[1-pa]) and λa≈λb.

When the sampling processes are performed, the three parameters (λa, λb and pa) of the two-component mixed exponential distribution are converted into stochastic variables (sampled from a uniform distribution) in order to accommodate for climate change. A similar approach is used for extreme precipitation, requiring the sampling of two parameters. In total, five parameters are sampled from uniform distributions for each season.

The stochastic nature of this method requires a large number of simulations. These are evaluated using several target variables and the corresponding change factors, which are calculated using the GCM ensemble. For each climate change scenario, one simulation is picked from the accepted simulations as the best simulation based on the performance it shows for different target variables. This method requires making an arbitrary choice on several parameters, i.e. the boundaries of the uniform sampling intervals, the number of simulations, target variables and their weights. The sampling boundaries for the dry spell parameters and the number of simulations are the subject of a sensitivity analysis (see Sect. S2 and Fig. S4). The other parameters are further discussed in detail hereafter.

The two parameters (slope α and intercept β) of a linear change factor function (Eq. 4), used to alter event precipitation amounts in function of its exceedance probability, are sampled from uniform distributions. 4ci=αFi+β, in which the following applies: 5Fi=pa1-exp⁡λai+1-pa[1-exp⁡λbi], where ci is the change factor as a function of intensity i, and Fi is the probability of a given rainfall intensity i being less than or equal to i using the same two-component mixed exponential distribution used for fitting the inter-event durations (Eq. 3). λa and λb are the rate parameters for populations a and b with different exponential distributions, and pa is the weight of population a.

Thorndahl et al. (2017) specify that the sampling boundaries are empirically selected by executing the method for very broad sampling ranges and iteratively narrowing them down based on the simulations that are accepted. When applying this strategy, a test run comprising 50 000 simulations did, however, not show clear boundaries for these parameters. Instead, sampling ranges are chosen at 0.000–0.050 and 0.80–1.20 for α and β, respectively, for all seasons. These values correspond well to the parameter ranges found by Thorndahl et al. (2017) for the accepted runs in their study.

The performance of a simulation is evaluated based on a set of target variables. The target values for these variables are determined by application of change factors to the corresponding variables of the observed time series H. The value for the target value i for the simulation j is denoted as Mi,j, and the climate change factor for target value i as cfi. The performance P is then calculated using Eq. (6a). Assuming a Gaussian distribution of the target variables, the acceptance criterion Pcrit for each target variable is taken as its 95 % confidence interval (Eq. 6b). A simulated time series j is accepted when Pi,j>Pcrit,i for all target variables i. 6aPi,j=1-cfi⋅Hi-Mi,jcfi⋅Hi6bPcrit,i=1-2⋅σcf,icfi. For all accepted simulations, the overall performance is calculated as a weighed sum of all individual target variable performances. For n target variables and weights wi, this becomes Pj=∑i=1nwiPi,j.

The set of target variables in the original implementation is altered in order to fit the specific needs of this study better. In total, two target variables relating to precipitation with T=2 years and T=5 years are removed. Instead, five new target variables are added, assuring the annual and seasonal number of dry days is adequately reproduced in the accepted simulations (Table 2). The weights, attributed to each target variable for calculation of the overall performance, are attributed in favour of the dry days target variables in order to reflect their importance for this study. The largest weights are assigned to the target variables that are expected to undergo the largest changes, which are expected to be the hardest to simulate.

Like the other statistical downscaling methods, some assumptions are made in the WG method. It makes assumptions similar to change factor methods due to the selection procedure. The changes found for climate model grid-averaged spatial scales are treated as targets for the stochastic simulations. Furthermore, this weather generator assumes wet event durations will not change, while dry event durations will. In addition, it is assumed that observed time steps with larger precipitation amounts will have a relatively larger increase in precipitation in comparison to time steps with lower precipitation amounts.

The first method, BC, applies a bias correction directly to the research indicators. This means no underlying time series is created. A first consequence is that not all projections are necessarily compatible with each other if the indicators are interdependent. This is the case for a number of dry spells since there are only a limited number of dry days to be distributed over the different classes of dry spells.

Second, the number of extreme events, such as long and very long dry spells, is limited. In the 30-year period of observations in Uccle, only 20 and 11 long and very long dry spells occurred, respectively, while the number of these events varies substantially among CMIP6 projections (15–100 and 11–88 under SSP5–8.5). This leads to very large bias-correction factors, which in turn lead to (over)spectacular results after downscaling (see Fig. 6). The same problem holds true for the dry spell length indicator. An absolute bias correction approach instead of a relative one might be more appropriate. In the same spirit, Raymond et al. (2019) discuss changes in extreme dry spell lengths in absolute terms (days) rather than percentages.

Note that these concerns do not take away from this method's ability to qualitatively downscale indicators such as number of dry days or total precipitation. These indicators are often projected by making use of relative change factors, as is also the case for the other statistical downscaling methods.

CFM does not account directly for changes in the number of dry days. This method applies a change factor to the observed time series in order to match the changes in total precipitation. For this specific research indicator, the result should consequently be no different than the one obtained using BC. The slight differences between these methods in Fig. 7 might be attributed to rounding differences.

The rationale behind the application of this method for assessing changes in drought finds its roots in the definition of the dry day threshold at 1 mm. As mentioned earlier, this is done to counter the so-called drizzle problem that GCMs are affected by, meaning that they overestimate the number of days with low numbers of precipitation. Consequently, days with precipitation amounts just below this threshold are classified as dry, while they might very well be lifted above this threshold in months where total precipitation is increased by the statistical downscaling method. Inversely, the wet days with precipitation just over the limit might convert to dry days in months with a decreasing total precipitation. Figure 7 shows this effect quite clearly for the summer months, where total precipitation is projected to decrease. The relative change in the number of dry days under the SSP5–8.5 scenario (+5.5 % in August) remains, however, rather small in comparison to BC (+17.5 %) or QP (+14.7 %), which both account for the number of dry days directly. The relative error of the drought-related indicators obtained here for CFM is far smaller than that reported for extreme precipitation in the mid-Europe region (Schmith et al., 2021).

The most interesting aspect in applying CFM, however, is the lack of vital assumptions as to how changing the number of dry days affect the dry spells. All required information is contained within the time series created by the GCM. In this light, the general trends for the number of dry days and dry spell length indicators as projected by QP are interesting to examine, while keeping in mind that the underlying changes in the number of dry days are considerably smaller than one would find through a direct change factor approach.

An important aspect for drought assessment in the QP method is in the form of the separate dry day perturbation step. Here, the time series is perturbed to match the projections of the number of dry days. Consequently, the QP method should be equal to BC in terms of the number of dry days projections. This is not exactly true, as shown in Fig. 5, but the differences are small enough to attribute them to rounding off the results differently. As dry days are the building blocks of dry spells, a solid downscaling approach towards the number of dry days is vital for downscaling the number of dry spells and dry spell length. Out of the four methods considered in this study, QP is the best-performing method in downscaling the number of dry days.

In several ways, WG seems to be the odd one out among the considered statistical downscaling methods. The original implementation of this method (Thorndahl et al., 2017) does, for instance, not downscale each member of the CMIP6 GCM ensemble separately as is the case for the other methods. Instead, WG aims to create one time series that corresponds well to the mean of the ensemble, at least in terms of the selected target variables. In theory, an implementation that downscales each member of the GCM ensemble separately is possible. Tests executed in this direction uncovered a practical problem related to the sampling boundaries for parameters governing the dry event duration distribution. As shown in the sensitivity analysis (see Sect. S2), WG struggles to deal with large changes in the number of dry days, e.g. under SSP5–8.5. While the changes in the sensitivity analysis are averaged out over the GCM ensemble, they are not when downscaling each ensemble member separately. The much larger changes that would have to be tackled by the WG would require much larger sampling boundaries. The largest change found in the GCM ensemble (one of the CanESM5 runs under SSP5–8.5) is a decrease of 40 % in the number of dry days. To accommodate this change, sampling boundaries upwards of 70 % are required in theory. It is expected that an even larger sampling range is needed, in combination with large numbers of simulations, to generate a comfortable number of accepted simulations. Testing at 40 % and 30 000 simulations showed that, for many members in the GCM ensemble, no accepted simulations could be generated. This is especially true for the SSP5–8.5 scenario.

For the monthly indicators, the number of dry days and total precipitation, BC, CFM and QP more or less match the temporal structure found in the Uccle observations. This is not, however, the case for WG. In total, two reasons can be identified for this. First, the method is implemented on a seasonal basis, following the original implementation (Thorndahl et al., 2017). Therefore, the method does not try to match changes in the number of dry days or total precipitation for every month but rather for the season as a whole. A comparison between a seasonal and a monthly implementation might be interesting to further investigate this method. A monthly implementation is expected to require larger numbers of simulations in order to achieve similar numbers of accepted simulations. This is due to the larger number of research indicators present (monthly instead of seasonal). Second, the downscaled time series do not necessarily match the mean of the GCM ensemble exactly for each research indicator. On the contrary, the method accepts all simulated time series that remain within the maximum deviation for each target variable (Table 7). These maximum deviations can be very large, e.g. ≃48 % for extreme precipitation and ≃15 % for total precipitation in summer (both under SSP5–8.5). Consequently, simulations that are far from the mean projections for some of the key research indicators (e.g. number of dry days) enter into the pool of accepted simulations and might be selected as the best simulation due to the high performance of the simulation for other target variables. This explains the difference of WG for the number of dry days (Fig. 5) and total precipitation (Fig. 7) in comparison to the downscaling methods that accurately downscale these indicators, even when grouping the results per season (DJF – December–February; MAM – March–May; JJA – June–August; SON – September–November).

The inaccurate simulation of the number of dry days affects the dry-spell-related indicators. It was concluded earlier that this is also the case for CFM. An additional concern for this downscaling method is that only one data point (best simulation) is available for comparison in Fig. 6, instead of the 28 data points (size of the ensemble) for the other downscaling methods. While this concern also holds true for the other indicators, it is mitigated by using these indicators (or similar) as target variables. In order to prevent the problems encountered with a relative bias correction applied directly to the dry spell indicators (see BC), this strategy cannot be followed for dry-spell-related indicators.