**1. Introduction**

In recent years, bias adjustment has become the de facto standard for preprocessing global (GCM) and regional (RCM) climate model simulations for climate change impact studies, hydrological modeling being no exception. The use is driven by practical needs. Due to systematic errors in climate model simulations with respect to the observed climate, GCM and RCM output usually cannot be directly used in impact modeling, as impact models require unbiased, high-resolution information as their input. This is because of non-linear and threshold processes within impact models. For example, a cold bias in forcing data to a hydrological model could lead to an impact result indicating no change in snow depths if the cold bias kept temperatures below 0 degrees.

Numerous methods belonging to so-called model output statistics (MOS) have been developed to adjust biases in temperature and precipitation data from climate models. These range from simple scaling of time-mean climate to more sophisticated methods addressing biases in the daily variability. This group also covers the widely used quantile mapping techniques of which there are a number of different variations [1–8]. Studies have illustrated that bias correction methods are able to reduce biases in climate model output [6] and also to provide noticeable improvements to hydrological simulations in the present-day climate [9,10]. However, most of these methods are restricted to the independent adjustment of biases in the marginal aspects of GCM-RCM simulations and do not take biases in inter-variable correlation structures into account. For example, studies have indicated that highest precipitation intensities co-occur with high surface temperatures in winter, as indicated by Clausius–Clapeyron relation, while mostly negative relationships between temperature and precipitation have been observed in summer in Europe [11,12]. In case a GCM-RCM has difficulties to reasonably capture such relationships, a bias correction method that does not explicitly take inter-variable correlations into account might not be sufficient for certain applications such as hydrological modeling.

To address this issue, different types of bi- and multivariate bias correction algorithms have been recently proposed [13–17]. These studies have given evidence that jointly bias correcting multiple variables improves the multivariate aspects of bias corrected model simulations when compared against the observed climate, and might outperform their univariate counterparts in further applications, such as in the calculation of Canadian Forest Fire Weather Index [17]. However, most of the intercomparison studies have concentrated on evaluating the relative performance of bias correction methods in the present-day climate, which does not inform on their ability to predict climate variables in changing climatic conditions. In other words, it is not known how well the adjustment of inter-variable correlations, which is inherently constrained by biases in the present-day, is able to capture the inter-variable correlations in the future climate. This information is crucial for reliably assessing potential climate change impacts, particularly as concerns have been expressed on the shortcomings and potentially unjustified use of bias correction in non-stationary conditions [18,19].

Due to the lack of an observational basis, surrogate data emulating the future observations has been proposed to be used as proxy data (hereafter referred to as pseudo-realities) to assess the ability of bias adjustment to improve projections in a changing climate. The pseudo-reality approach has been used in recent studies [20–23] and was also considered in the European Concerted Research Action ES1102 VALUE (Validating and Integrating Downscaling Methods for Climate Change Research) framework as an important, although not sufficient step, when evaluating bias adjustment method performance [24].

Most of the pseudo-reality studies have concentrated on the analysis of the application of bias adjustment directly to climate model output, which does not give direct information on their usability to construct future projections for the purposes of hydrological climate change impact studies. One of the first attempts to extend the pseudo-reality approach to hydrological simulations was made by Velázquez et al. [25], whose study evaluated implications of non-stationarity to bias correction for future conditions and how they affect the estimation of future changes in river discharges. Their results showed that although monthly mean river discharges were improved in some cases after bias correcting the hydrological model input, biases still remained in the results. In their study the pseudo-reality approach was applied without taking pseudo-reality biases in the present-day climate into account. If a hydrological model is sensitive to absolute biases in climate model outputs, the hydrological model behavior and its response to the projected changes might be unrealistic, which would hamper the evaluation of bias adjustment methods in the pseudo-reality framework. Furthermore, the study used only two GCM-RCM combinations and one bias correction method that did not take inter-variable correlations directly into account in the bias correction step. From the hydrological modeling perspective, a physically plausible description of co-variations of temperature and precipitation might be important to reasonably describe the surface fluxes such as evapotranspiration and processes affecting water stored in soil and snow pack, which together regulate the river discharge generation. It is also important to assess the performance of the hydrological modeling of low frequency impacts

(e.g., high and low flows) as these impacts are often of interest to users, but could be subject to different biases than mean flows.

Here, we extend the study of Velázquez et al. [25] to assess the relative performance of four bias adjustment methods from the hydrological impact modeling perspective. More specifically, the aim is to address the following questions:


We use five GCM-RCMs produced in the European branch of the Coordinated Regional Climate Downscaling Experiment (EURO-CORDEX) initiative [26]. In addition to the separate adjustment of temperature and precipitation distributions, two methods [14,17] which take biases in their inter-variable relationships into account are compared against univariate quantile mapping to assess the extent to which hydrological simulations benefit from the additional correction of temperature and precipitation correlations in the changing climate. We perform cross-validation tests in a pseudo-reality framework broadly similar to the one used in Velázquez et al. [25] and extend the analysis by taking into account GCM-RCM biases in the calibration data in order to bring the hydrological simulations closer to the observations (i.e., by bias adjusting the pseudo-reality data).

The paper is structured as follows. In the next section we introduce the GCM-RCM simulations used in this study together with the hydrological model used to conduct the hydrological simulations. In addition, the bias adjustment methods, the pseudo-reality framework and the cross-validation statistics used to assess the relative method performance are also discussed in Section 2. The results are shown in Section 3 and the discussion together with conclusions are presented in Section 4.
