**4. Discussion and Conclusions**

This paper presents the results from a study in which two joint bias corrections methods applied in the bi-variate mode are compared against quantile mapping applied both traditionally and in the delta change mode using five EURO-CORDEX GCM-RCM simulations as a proxy for the future climate. The evaluation is two-fold: first, cross-validation is performed to obtain quantitative estimates for the relative method performance in the early and late 21st century conditions, when applied to construct future projections of the joint distribution of daily mean temperature and precipitation; second, these projections were fed to a hydrological model to assess, whether or not the bi-variate adjustments improve future hydrological simulations in comparison to univariate quantile mapping.

The main results of these exercises are summarized as follows:


• For the hydrological variables, the bi-variate approaches offered no substantial advantage over the univariate methods with M4 often having similar performance to M2. Only marginal improvements in comparison to methods M1 and M2 are seen in the cross-validation statistics for high flows and for the monthly mean and annual maximum snow water equivalent in Tornio and Sava. Although quantile mapping applied as a delta change method (M1) has slightly poorer performance in projecting marginal distributions of temperature and precipitation than quantile mapping-based bias correction (M2) and its bi-variate version (M4), the cross-validation statistics indicate that it has a relatively good ability to capture the future hydrological conditions. Nevertheless, for the hydrological variables studied (apart from snow), there were only small differences in cross-validation statistics between the tested methods, indicating that care should be taken when selecting MOS methods for particular purposes and (ideally) several methods should be used in parallel. Overall, the results highlight the difficulty to illustrate the added value of more complex methods, when applying them in producing projections for daily mean temperature and precipitation.

The main shortcoming of this study is the limited number of GCM-RCM simulations available for cross-validation tests, and optimally a larger set of model simulations should be used. Furthermore, the response to different bias correction and delta change algorithms is likely dependent on the hydrological model and the used parameterisations. For example, earlier studies have shown e.g., [44] that projections for evapotranspiration based on parameterising potential evapotranspiration using only temperature are not suitable for all climatic conditions. Furthermore, snow processes were parameterized in the HYPE simulations using a simple degree-day algorithm, which does not take solar radiation and other meteorological factors into account. More complex parameterizations, which require multiple variables as input, should be evaluated in further studies. If bias correction of a higher dimensional joint distribution were required, more sophisticated bias correction methods could, at least in principle, provide larger improvements in comparison to univariate methods, depending on the available data for robust calibration. This has been demonstrated by Cannon [17], who showed that M4 performs very well when adjusting a higher dimensional distribution for Canadian Forest Fire Weather Index calculations in the present-day climate. Moreover, different implementations of M3 could also be studied in future research. Most importantly, Gaussian copula is unlikely to be the optimal choice for describing the temperature and precipitation dependence structure in some cases and the use of other copulas should be further explored.

The presented framework allows to make some inferences about the ability of bias correction and delta change methods in constructing projections for future climate and their applicability from hydrological modeling perspective, an information of major interest to the impact modeling community. However, these tests are not sufficient alone to determine whether a particular method is suitable for climate change assessments. Additional tests such as those implemented in the VALUE framework [24] should be conducted to obtain a complete picture of benefits and limitations of bias correcting inter-variable correlations. As discussed above, the used approach does not easily allow to evaluate the potential benefits/adverse effects of the modification of temporal sequencing to the hydrological model results, which would require temporally synchronized model simulations and reference data. Furthermore, the effect of errors in the spatial representation of climate model output caused (e.g.,) by differing topography should be studied comprehensively to see how sensitive future hydrological simulations are to the correct representation of spatial fields.

**Supplementary Materials:** Figures S1–S7 are available online at http://www.mdpi.com/2225-1154/6/2/33/s1.

**Author Contributions:** C.D. selected and provided the hydrological sub-models and O.R. ran the hydrological simulations. O.R. and J.R. implemented the bias adjustment methods. All authors contributed to the planning of the study, the analysis of the results and writing the manuscript.

**Acknowledgments:** The study is supported by the Academy funded Center of Excellence (project No. 307331). Olle Räty is funded by the Vilho, Lauri and Yrjö Väisälä Foundation and by NordForsk through project number 74456 "Statistical Analysis of Climate Projections" (eSACP). Further funding was received from the projects HazardSupport and AQUACLEW. HazardSupport is financed by the Swedish Civil Contingencies Agency, MSB (grant No. 2015-3631). AQUACLEW is part of ERA4CS, an ERA-NET initiated by JPI Climate and funded by FORMAS (SE), DLR (DE), BMWFW (AT), IFD (DK), MINECO (ES) and ANR (FR) with co-funding by the European Union (Grant 690462). We would also like to thank the climate modeling groups participating the European branch of Coordinated Regional Climate Downscaling Experiment (EURO-CORDEX) for producing and making their model output publicly available.

**Conflicts of Interest:** The authors declare no conflict of interest.
