The proposed methodology relies on – or can be seen as an extension of – the Rank Resampling for Distributions and Dependences (R2D2) bias correction method . R2D2 consists of two steps: first, a univariate BC is performed to adjust the marginal distributions, and then the empirical copula function (i.e., the dependence structure between the variables of interest, rid of their marginal distribution) is adjusted. Thus, R2D2 belongs to the “marginal/dependence” family of multivariate bias corrections seefor a description of the other families: “successive conditional” and “all in one”.

For the first step, any 1d-BC method can be employed. In and in the following details of the present study, the cumulative distribution function – transform (CDF-t) approach, e.g., , is used to adjust the marginal properties for temperature. A specific version of CDF-t is used to correct precipitation data. This version relies on a stochastic singularity removal SSR; approach to manage dry time steps: first, zeros (from both references and simulations) are randomly transformed to positive but very small values (<10-6); then CDF-t is applied onto the whole set of data (i.e., transformed data and initially positive values altogether), and the correction results are thresholded such that values <10-6 are set to 0.

For the second step, R2D2 uses a “conditioning dimension” called “reference variable” in from the 1d-BC results. This univariate 1d-BC time series – and more precisely its ranks – serves as a conditioning to find, within the other 1d-BC variables, the values that have the same rank associations as those in the training reference dataset details and examples on R2D2 can be found in. Hence, this method relies on a univariate conditioning dimension to generate rank associations, in the same way as an analogue technique initially developed by relies on its predictors to generate values. By doing so, this MBC approach allows us to reproduce observed multivariate (spatial and multi-variable) dependence structures, while preserving some temporal properties of the initial simulations via the conditioning dimension.

However, if the temporal features of the conditioning dimension (i.e., one physical variable at one given location) is preserved by construction, this is not necessarily the case for the other variables (i.e., different physical variables and/or spatial locations) and even not the case at all for variables having a weak rank correlations with the conditioning dimension. Therefore, taking advantage of the analogue-based philosophy of R2D2, several extensions are here proposed to improve the temporal properties of the corrections brought by the initial R2D2.

The main idea of the proposed extensions consists of seeing the R2D2 approach as an analogue-based method. Indeed, in Sect. , it is clear that the resampling of the multivariate ranks is conditional to a single rank value of the conditioning dimension. In analogue techniques used in the climate literature e.g.,among others, the conditioning (i.e., predictor) variable can be multivariate. In our case, since the purpose of R2D2 is to correct the dependence structure, we want the notion of analogue situations to only account for the dependence structure and not for the marginal distribution. Hence, the distance between two situations is not computed based on the raw values of the conditioning dimensions but based on their ranks. The best analogue is thus defined as the situation (e.g., day) having the association of ranks the closest to that of the conditioning dimension in terms of Euclidean distance. Here, an extension of R2D2 is proposed and allows different configurations, all relying on R2D2 applied conditionally on a multidimensional conditioning dimension:

R2D2 conditional on a multivariate information at a given time t. The conditioning dimensions in R2D2 can be chosen freely. They can belong to the set of variables to be corrected, provided as exogenous variables or be a combination of both. There is no restriction either on the spatial scales of the conditioning dimensions. For instance, as a bivariate conditioning dimension, one could combine a daily North Atlantic Oscillation (NAO) index, to provide large-scale information, with the temperature at one grid point as a source of small-scale information. Other choices could be the temperature at two given locations, the temperature and the precipitation at one location, etc.

Whatever the configuration, the choice of the conditioning dimension is however not trivial, as it conditions the temporal properties of the model that will be conserved after correction. In the case of a configuration accounting for the rank sequence, the length of the sequence to search the analogues has to be chosen. This length will be referred to as “Block-A” (for “block analogue”) hereafter. Moreover, in order to avoid discontinuities in the reconstructed final sequence of ranks, the whole sequence of the best analogue is not fully kept but only a subsequence corresponding to a given number of elements at the end of the complete sequence. This kept subsequence is referred to as “Block-K” (for “block kept”) hereafter, and its length also has to be chosen as shorter than or equal to Block-A. Searching for the best analogue with a length Block-A and then keeping only a length Block-K – shorter than Block-A – allows us to not only avoid discontinuities in the (rank and correction) time series but also give flexibility to the proposed BC method to adapt to the temporal dynamics of the climate model to correct. Preliminary tests (not shown) indicate that Block-A = 9 and Block-K = 7 are reasonable choices and that the results are only weakly influenced by a slight change of those values.

In the case of ties for the choice of best analogues, the proposed R2D2 algorithm selects the first time step having the minimal Euclidean distance. Hence, the order in which the dataset is searched to find the analogue ranks will influence the selected time step and may thus potentially also influence the results. For example, ties might occur when including daily precipitation as a conditioning variable, particularly when precipitation at a single location is the only conditioning dimension. Indeed, there are potentially many time steps where it did not rain at the conditioning location. All of these time steps would all have the same rank (in this case, 1). This means that some rank combinations that are also compatible with the rank of the conditioning dimension (in this example, no rain at the conditioning location) will not be present in the corrected dataset. Hence, some rank associations can be either under-represented or over-represented in the corrected dataset compared to the reference dataset because of ties in the conditioning dimensions. However, when using multiple conditioning dimensions (i.e., multiple conditioning locations and/or variables), the number of candidates for the best analogue with the same exact minimal Euclidean distance decreases, hence reducing the effect of the ties in the time series of the conditioning dimensions. Other choices could be made to deal with tied ranks: last or median time step (instead of first) could be selected or even a randomly drawn time step (among those with a common minimal distance to the condition information). Nevertheless, this is not investigated in the present paper and left for future work.

In the following, 20 different R2D2 configurations are applied and compared to the reference WFDEI dataset, the plain simulations and the univariate BC results obtained from CDF-t. Those 20 configurations and their notations are given in Table . For the versions including five grid points in the conditioning, the locations are chosen to characterize five cities – Paris, Madrid, Stockholm, Rome and Warsaw – spread out over the region. In the same manner – but more automatically – the N grid points (N= 100 or 400) in the other versions are chosen to uniformly cover the region of interest.

Note that the configuration using a conditioning with only one physical variable at a single location without accounting for lags (i.e., R.1.1.0) exactly corresponds to the initial R2D2 method.

Moreover, in practice, the R2D2 configurations with 400 grid points or with 4167 (i.e., all) grid points for the conditioning dimension provided results equivalent to those from the same configurations but with only 100 grid points (not shown). This emphasizes a preliminary result: taking a large number of spatial information is not necessarily needed once a sufficient information is provided. Hence, in the following, experiments R.400.1.0, R.400.2.0, R.400.1.1 and R.400.2.1 will not be presented, nor will experiments R.4167.1.0, R.4167.2.0, R.4167.1.1 and R.4167.2.1, as they provide results similar to R.100.1.0, R.100.2.0, R.100.1.1 and R.100.2.1, respectively.

The different configurations of the R2D2 extensions, as well as the CDF-t univariate BC (referred to as BC1D in the following), are applied and evaluated according to the following 2-fold cross-validation approach. First, the methods are calibrated over the 1979–1997 period and applied to correct the 1998–2016 climate projections for evaluation. Then, they are also applied the other way around, i.e., calibrated on 1998–2016 and applied for evaluation on 1979–1997. Finally, the two 19-year evaluation periods are gathered to dispose of the whole 38-year time period for evaluation.

Every method is applied on daily values but on a monthly basis, i.e., for each month separately that are joined afterwards. However, evaluations are performed on a seasonal basis – i.e., for each season (DJF, MAM, JJA, SON) separately – to reduce the number of figures and to group similar behaviors.

Here, we first look at the ability of R2D2 to reproduce the short-term temporal dependences of the conditioning dimensions, through the first-order autocorrelation ρ, corresponding to the coefficient of a first-order auto-regressive model (AR1).

When the conditioning dimension is univariate and continuous, with unique ranks (i.e., no repetitions of values) and belongs to the variables to be corrected, it is the only variable (from the BC1D dataset) that the R2D2 resampling scheme does not modify. Therefore, in this case, if the number of days is the same in the reference and model dataset, each time step is sampled exactly once and they are uniformly selected. Hence, in this specific case, R2D2 reproduces exactly the inter-site and inter-variable empirical copula of the reference but not the temporal dependences of the data.

However, when the conditioning has a dimension equal to or greater than 2, there is no guarantee that the exact same rank associations exist in the reference dataset. Indeed, the higher the dimension of the conditioning, the less probable it is to find the exact rank association in the reference and in the BC1D dataset. This can come from either (i) a sampling issue: the higher the dimension, the more points are needed to uniformly sample the space or (ii) from biases in the dependence structure (biases in the rank associations) of the conditioning dimension in the dataset to be corrected. In this case, R2D2 uses the rank associations of the reference dataset that is the closest in terms of Euclidean distance. Hence, the rank association of the conditioning dimension in the resulting R2D2 dataset can be different from that of the BC1D dataset. One consequence is that some time steps (i.e., days in our case) can be resampled several times, while others might not be sampled at all. This can obviously have consequences on the properties (marginal, inter-site, spatial, temporal) of the multivariate corrections.

Therefore, we now analyze the distributions of the time steps that have been selected, since it is an indicator of potential biases introduced by the analogue-resampling scheme in R2D2.

To reproduce exactly the empirical copula of the reference dataset, each time has to be selected only once. The more uneven the distribution of selected time steps, the more likely it is that the correction has modified the frequency of some situations with respect to the reference dataset. However, there is not a direct relationship between the unevenness of the distributions and the biases introduced in the correction. For instance, if some rank associations do not appear in the correction, they could have been substituted by a very similar association. In this case, the bias introduced would be very small.

The distributions of time steps selected in the reference dataset in January by the different configurations of R2D2 are shown in Fig.  (the distributions for April, July and October are provided in Figs. S7–S9). As expected, R.1.1.0 presents a uniform histogram, since it uses a univariate conditioning that permits the sampling of the whole reference time steps. However, this is not the case for the other configurations of R2D2, which all have dimensions of the conditioning equal to or greater than 2 (see Table ). For the configurations of R2D2 with only temperature as the conditioning dimension and without time lags (R.5.1.0, R.100.1.0), the sampling is quite uniform. This suggests that the spatial properties of the temperature are quite similar between the reference dataset and the BC1D dataset. When adding the precipitation as the conditioning dimension without time lags (R.2.1.0, R.2.5.0, R.2.100.0), the histograms are slightly less uniform. This indicates that there can be discrepancies between the references and the BC1D dataset for the spatial dependence of precipitation or the dependence between temperature and precipitation. Finally, when adding time lags in the conditioning dimensions, both for temperature and precipitation (R.1.1.1, R.1.2.1, R.1.100.1, R.5.1.1, R.5.2.1, R.5.100.1), the distributions of selected times also appear to be less uniform. This is especially true for R.5.2.1, where we observe a trough in the distribution between days 600 and 700. It indicates that the modeled rank sequences in this period rarely appear in the BC1D dataset.

Those elements can help us to interpret the performance of the different configurations of R2D2 with respect to the reconstruction of temporal, spatial and marginal properties of the temperature and precipitation fields.

Moreover, in order to see how much the different R2D2 configurations change the temporal structures of the original raw simulations, for all sites and climate variables, we have computed the correlation between the ranks of the initial raw simulations and the ranks of the multivariate corrected time series. The closer the correlation is to 1, the less R2D2 has modified the temporal structures of the raw simulations. The correlation coefficients for the different sites in winter are shown in Figs.  for temperature and for precipitation. The other seasons are shown in Figs. S10–S15 in the Supplement. By construction, the CDF-t BC1D mostly conserves the ranks of the raw simulations.

For temperature (Fig. ), we see that the time series of ranks have been modified substantially by R.1.1.0 (Fig. b). When the number of geographical sites increases (R.5.1.0 and R.100.1.0; Fig. c and d), we observe the transitive effect and the rank time series are more correlated to those from raw simulations. It is made possible because the variations of temperature are spatially smooth and because the references and BC1D data seem to have similar temperature temporal properties.

The transitivity effect is also seen when precipitation is added as a conditioning dimension (R.1.2.0, R.5.2.0, R.100.2.0; Fig. e–g) or when time lags are added (R.1.1.1, R.5.1.1, R.100.1.1, R.1.2.1, R.5.2.1, R.100.2.1; Fig. h–m). However, fewer changes are made in the rank time series when the number of conditioning sites increases. However, for those versions of R2D2 with a high number of conditioning sites, the resulting rank time series are slightly more modified (i.e., rank correlation further away from 1) than with just the temperature as the conditioning dimension. It may come from the fact that the higher the dimension of the conditioning, the more likely the rank sequence of the conditioning dimension has to be modified.

For precipitation (Fig. ), similar observations can be made. However, the changes in the rank time series are larger than for temperature. It can be partially explained by the fact that the transitivity effect is weaker for precipitation. Indeed, precipitation events occur at local scale and with a spatial correlation radius smaller than for temperature.

Globally, due to the transitivity effect, sites strongly correlated with the conditioning dimension in the reference dataset have their rank sequences mostly conserved after the correction if the conditioning dimension has similar temporal properties in the reference and the model. As a consequence, adding more sites in the conditioning dimension generally leads to more regions that mostly preserve the rank sequences of the model. However, to some extent, this effect can be counteracted by the fact that, as the dimension of the conditioning grows (e.g., adding rank lags in the conditioning), it becomes harder to find the exact rank associations in the reference data. It leads to alterations in the rank sequences for the conditioning dimension and for the sites that are correlated with it, and finally to a potential decrease of the rank correlation between the raw simulations and their corrections.

As seen previously, some of the proposed R2D2 extensions allow us to adjust temporal dependence structure. However, as the initial R2D2 method was designed to bias correct multi-site and inter-variable dependences in addition to marginal distribution, one can wonder how the temporal structure improvements – as well as the time-sampling features – made by the R2D2 extensions impact the corrections performed on the other dependences, i.e., whether or not they degrade the marginal, spatial and inter-variable properties. This is the purpose of Sect.  (marginal), (inter-variable) and (spatial).

To fill some needs of the climate change impact community, an MBC method has been proposed in this study. In addition to marginal properties, this MBC is designed to adjust both the inter-site and inter-variable dependence structures of climate simulations and at the same time to improve the temporal properties of the corrections. Our approach is based on the previously existing R2D2 method that relied on a univariate “conditioning dimension” to sample ranks from a reference dataset and therefore reconstruct the copula-based spatial and inter-variable dependences. The suggested R2D2 extensions allow resampling ranks given a multivariate conditioning dimension, which could be ranks of multiple physical variables at a time step t, ranks from a single physical variable but over a sequence of N time steps (t-(N-1),…,t) or ranks of multiple physical variables over a sequence of N time steps.

Several configurations (i.e., different conditioning dimensions including different sites and climate variables, with or without lagged information) have been applied to correct daily precipitation and temperature simulations over Europe from a single climate model run, the IPSL-CM5 Earth system model , with respect to the WFDEI data as references. As the initial R2D2 approach by was able to properly adjust spatial and inter-variable structure but not the temporal properties of the simulations, the underlying question of the present study was to understand (i) if the proposed multidimensional conditioning in R2D2 improves the temporal aspects of the corrections and (ii) the impact of this conditional resampling on the adjustment quality of the other (i.e., marginal, spatial, inter-variable) properties. Hence, the various R2D2 configurations have been evaluated and compared to the raw simulations as well as to corrections from the univariate BC method CDF-t e.g.,, first in terms of autocorrelation to characterize the main temporal aspects and then in terms of marginal properties, spatial dependences and temperature vs. precipitation correlations.

For temporal properties, although the R2D2 variants strongly improve the initial R2D2 approach, they do not reach the rather good level obtained for temperature with the tested 1d-BC (due to the correct auto-correlations from the raw temperature simulations), while the results are more debatable for precipitation. In general, the main conclusions were that including more information (sites and/or lagged ranks) in the conditioning dimension generally improves the reconstruction of the autocorrelation coefficients, both for temperature and precipitation. However, when the dimension of the conditioning (i.e., the number of variables, sites and lags to condition the resampling) increases, the distribution of the sampled time steps can be quite different from the uniform one. This has then consequences mostly on the marginals (i.e., univariate properties), where the mean and standard deviation can have stronger biases for non-uniform sampling. For the other evaluations (spatial and inter-variable properties), although variations in the results are visible depending on the conditioning dimension used, the main conclusion is that the proposed R2D2 configurations are relatively stable. Thus, in general, the introduction of additional conditioning information into R2D2 allows improving temporal properties with a good preservation of the capability of the initial R2D2 to adjust both the spatial and inter-variable dependences of the raw simulations. Hence, in practice, we recommend a “compromise setting” in the choice of the conditioning dimension, with not too few and not too many sites. Such a choice would prevent both from missing transitivity effects and from having an uneven time step distribution.

The method suggested in this study is of course upgradable along different axes.

First, as our goal was not to test the various R2D2 configurations on several climate models but rather to establish a proof of concept of the R2D2 extension on an illustrative simulation run, only one climate model has been used for application and evaluation in the present study. Although we hypothesize that the main general findings obtained on this single model application will still be valid for other models and simulations, this will need to be confirmed to generalize and refine our results to more model simulations.

Moreover, the fundamental assumption of R2D2 is that the spatial and inter-variable copulas (i.e., rank association) is stationary in time, even for future climate projections. This assumption – considering that rank associations act as proxies of physics and that physics does not change in time – is nevertheless debatable since it needs to be verified in further works. However, it highlights the fact that the conditioning dimension has to be carefully chosen to be relevant, both to drive (i.e., condition) the correction of the properties of interest and to translate the potential changes that may happen in future climate and that would impact the corrections. Hence, the potential non-stationarity of the dependences between climate variables (i.e., rank association) may be worth exploring further. This could be done via several questions; e.g., how stable is the rank association with time? How long does the considered period have to be in order to faithfully estimate the association? How can climate change alter it? Those questions, however, are not specific to our suggested method. They concern many methods and studies relying on dependences of climate variables in climate change contexts. Such fundamental questions deserve to be investigated on their own and are thus left for future works.

More generally, the choice of the conditioning dimension is a key element of the R2D2 method. Indeed, as seen in this study, what is corrected or not by R2D2 is partially driven by the chosen conditioning information. Thus, testing alternative conditioning dimensions could also be of interest for future work, to bring additional physical/geographical information, valuable to generate proper multivariate corrections. This alternative conditioning – e.g., including NAO or other indices, characterization of the circulation or other covariates – nevertheless has to be determined according to the specific region of interest, the climate variables to be corrected, etc. This adaptation of the conditioning to the application is a requirement to inject the relevant and suited physical information into R2D2.

Of course, if a “good” conditioning must optimize the R2D2 correction of some statistical properties, it mainly has to optimize the properties that are the most useful for the users of the corrections. In other words, the choice of the R2D2 configuration has to be tailored for the end users of the simulations. It is thus very important for the end users to know which properties are essential to be corrected in order to design the R2D2 configuration that is most appropriate for their specific application. Indeed, if many statistical features of the simulations are to be corrected, it is not clear that a single configuration will be able to correct all properties. For some regions and sets of climate variables, this can happen, but in other cases it might be needed to prioritize the most essential ones and then choose the associated R2D2 configuration.

Finally, trying to correct multiple statistical properties at the same time remains a difficult challenge, as adjusting one often modifies another one. Additionally, one can wonder what is kept from the raw climate simulations if a correction is performed to adjust many statistical aspects. Hence, when applying a multivariate bias correction method with a configuration allowing us to modify (explicitly or implicitly) several properties, a compromise has to always be searched in order to balance, on the one hand, the level of correction needed to make the simulations useful for the application of interest and, on the other hand, the climate model signal preserved by the applied correction method. This is the only way to make the (M)BC useful in practice and physically reliable.