Over the last few years, multivariate bias correction methods have been developed to adjust spatial and/or inter-variable dependence properties of climate simulations. Most of them do not correct – and sometimes even degrade – the associated temporal features. Here, we propose a multivariate method to adjust the spatial and/or inter-variable properties while also accounting for the temporal dependence, such as autocorrelations. Our method consists of an extension of a previously developed approach that relies on an analogue-based method applied to the ranks of the time series to be corrected rather than to their “raw” values. Several configurations are tested and compared on daily temperature and precipitation simulations over Europe from one Earth system model. Those differ by the conditioning information used to compute the analogues and can include multiple variables at each given time, a univariate variable lagged over several time steps or both – multiple variables lagged over time steps. Compared to the initial approach, results of the multivariate corrections show that, while the spatial and inter-variable correlations are still satisfactorily corrected even when increasing the dimension of the conditioning, the temporal autocorrelations are improved with some of the tested configurations of this extension. A major result is also that the choice of the information to condition the analogues is key since it partially drives the capability of the proposed method to reconstruct proper multivariate dependences.
Climate model simulations are and will remain the main source of numerical projections to understand and anticipate climate change consequences.
Those projections are performed under various greenhouse gas emission scenarios, prescribed for instance within the fifth international Coupled Model Intercomparison Project
To get robust impact estimations, the climate projections have thus to be as precise and informative as possible. However, even simulations of the current climate often present statistical biases: their mean, variance or more generally their distributions can more or less largely differ from observational reference datasets
However, if many statistical aspects can be adjusted with such methods, all are only univariate, i.e., related to only one physical variable at a single location. If multiple variables and/or at multiple locations have to be corrected, the independent applications of several one-dimensional bias correction (1d-BC) methods will not modify the intrinsic dependence structure of the simulations to be corrected
Consequently, some multivariate bias correction (MBC) methods have been recently designed to tackle the issues of the biases in multivariate dependences.
The goal is basically the same as for univariate corrections: find a transformation that makes climate model simulations have the same targeted statistical features as a reference in the calibration period. In this case, the target statistical features include not only univariate features but also multivariate statistical features such as correlations or the empirical copula.
The various MBCs developed so far can be categorized into three main families the “marginal/dependence” approaches, correcting separately univariate distributions and dependences before joining them to provide multivariate corrections the “conditional successive” methods, adjusting one variable at a time but conditionally on the previously corrected variables to ensure proper multidimensional relationships the “all-in-one” models, which do not separate the multivariate distribution, neither in marginal/dependence, nor in conditional distributions, but directly transform one multidimensional distribution into another multidimensional distribution
A first intercomparison and critical review of MBC methods has been carried out by
In the present study, we rely on a recently developed MBC method named R
The rest of this article is organized as follows:
Sect.
To perform tests and analyses of the proposed correction method, we will rely on daily temperature at 2 m (T2) and precipitation (PR) from one run of a global climate model to be corrected on one hand and from an observation-based reference dataset on the other hand.
The latter corresponds to WFDEI data, which are the WATCH Forcing Data
The climate model data to be corrected are extracted – for the same region – from simulations performed by the IPSL-CM5A-MR Earth system model
Note that only one climate model is used for application and evaluation purposes in the present study. Of course, other models will have other biases that must be corrected differently. However, our goal is not to test the proposed approach on many climate models but rather to establish a proof of concept of the R
The proposed methodology relies on – or can be seen as an extension of – the Rank Resampling for Distributions and Dependences (R
For the first step, any 1d-BC method can be employed. In
For the second step, R
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 R
The main idea of the proposed extensions consists of seeing the R R R R
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
In the case of ties for the choice of best analogues, the proposed R
In the following, 20 different R
Summary of the 20 R
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 R
Moreover, in practice, the R
The different configurations of the R
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.
In this section, we examine the effects of R
Here, we first look at the ability of R
For winter temperature (Fig.
Maps of temperature autocorrelations of the order of 1 d for winter over the 1979–2016 period, for
For R.1.1.0 (conditioning dimension is temperature in Paris; Fig.
However, when increasing the number of sites (R.5.1.0 and R.100.1.0; Fig.
Adding precipitation in the conditioning dimension (R.1.2.0, R.5.2.0 and R.100.2.0, respectively, Fig.
When using lags in the conditioning dimensions, all configurations with lags give similar results in terms of RMSE computed on the AR1 coefficient (RMSE
Moreover, the configurations using more sites (R.100.1.1 and R.100.2.1) give slightly better results. The spatial variations of the AR1 coefficients are qualitatively better respected, with lower values of autocorrelation in Spain, the UK and Libya compared to the rest of the map. Quantitatively, however, there is a negative bias of about
In the end, as the initial temperature simulations have AR1 coefficients similar to those from the references, the IPSL and BC1D simulations show the best temporal properties (best R
For winter precipitation (Fig.
Same as Fig.
When applying R.1.1.0 – the configuration of R
Adding precipitation in the conditioning dimensions helps to improve the precipitation AR1 coefficient since it is likely that the correlation between precipitation in two close sites is stronger than the correlation between temperature in one site and precipitation in the other site. With 100 conditioning sites, geographical features present in the reference dataset start to be visible, for instance, higher AR1 coefficients on the coasts of northern Africa and on the northern coasts of Scandinavia. Nevertheless, the first-order autocorrelations are still biased negatively with respect to the reference dataset.
In terms of RMSE, R.1.100.0 performs slightly better than the BC1D dataset (RMSE(BC1D)
As for the temperature, the configurations of R
Hence, depending on the choice of the conditioning dimensions, R
Generally, as seen in this section, although the proposed extensions clearly improve the initial R
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 R
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, R
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 R
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 R
Distributions of time steps selected in the reference dataset in January by the different R
Those elements can help us to interpret the performance of the different configurations of R
Moreover, in order to see how much the different R
For temperature (Fig.
Maps of Spearman (rank) correlations calculated for each grid point in winter over 1979–2016 between the initial climate model temperature simulations and their corrections by
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.
For precipitation (Fig.
Same as Fig.
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 R
We first check whether the R
For each season and each grid point, biases in mean temperatures have been computed and are shown in Fig.
Boxplots of differences in mean temperature per grid point with respect to WFDEI, i.e., mean(model or BC) minus mean(WFDEI):
R.1.1.0 provides similar performance. Since the conditioning dimension is univariate, R
On average, going from one conditioning site to five, with R.5.1.0, increases the biases in mean (0.13
Similar observations can be made when looking at R
Somehow similar patterns of biases also occur when looking at the standard deviation of the temperatures (Fig. S20).
For precipitation, univariate biases are investigated in separating occurrences of rainfall and conditional intensities given rainfall occurrences. Hence, Fig.
Biases in standard deviations for conditional precipitation values are also given for information (Fig. S29) and coincide with results for means.
Generally, for both temperature and precipitation marginal properties, the biases tend to be stronger for R
We now evaluate the capability of the different R
Boxplots of differences in temperature vs. precipitation Pearson correlations between WFDEI and the different datasets (IPSL, 1d-BC IPSL and the R
In the IPSL model and in the BC1D correction, the correlation between temperature and precipitation is weaker than in the reference dataset.
We expect R.1.1.0 to have the best performance with regards to inter-variable rank correlation. Indeed, it has a univariate conditioning dimension, implying that the empirical copula between temperature and precipitation of the reference data observed during the calibration periods is reproduced almost exactly.
In practice, in Fig.
Finally, we evaluate the spatial correlation by computing the loading values of the first empirical orthogonal function (EOF) obtained from a principal component analysis (PCA) applied on temperature and precipitation separately.
For each dataset, we compare the associated loading values with those obtained for the references. The results for winter and summer are summarized in Fig.
Boxplots of differences in loading values for the first EOF (EOF1) between model or corrected data and WFDEI (i.e., EOF1(model or BC) minus EOF1(WFDEI)). Panels
For both temperature and precipitation, and for all seasons, the raw IPSL simulations have loading values well centered around those of WFDEI since the median of the differences is close to 0.
Simply by correcting the marginal distribution, BC1D improves the agreement with the reference dataset. Indeed, EOFs are computed from the variance–covariance matrix, which is sensitive to the change in the marginal distributions.
In the R
Spatial correlograms are not shown but clearly indicate similar results.
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 R
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
For temporal properties, although the R
The method suggested in this study is of course upgradable along different axes.
First, as our goal was not to test the various R
Moreover, the fundamental assumption of R
More generally, the choice of the conditioning dimension is a key element of the R
Of course, if a “good” conditioning must optimize the R
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.
The R
The supplement related to this article is available online at:
MV had the initial idea for the method, developed the associated code and made the figures. ST wrapped the R package. MV and ST performed the analyses and wrote the article.
The authors declare that they have no conflict of interest.
We acknowledge the World Climate Research Programme's Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the IPSL climate modeling groups for producing and making available their model output. For CMIP, the US Department of Energy's Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. Finally, Mathieu Vrac would like to thank Pascal Yiou (LSCE) for fruitful discussions about analogues.
This work has been supported by the EUPHEME and CoCliServ projects, both part of ERA4CS, an ERA-NET initiated by JPI Climate and co-funded by the European Union (grant no. 690462).
This paper was edited by Simone Marras and reviewed by Sylvie Parey and Verena Bessenbacher.