## 1. Introduction

[2] Bayesian model averaging (BMA) has been introduced by *Raftery et al*. [2005] for postprocessing of meteorological forecasts. It generates a single forecast PDF for a quantity like temperature or rainfall by combining several forecasts from different models. The goal of statistical postprocessing is to obtain calibrated and sharp predictive continuous PDF's of the quantity to be forecast given uncalibrated raw predictive ensembles [*Raftery et al*., 2005; *Gneiting et al*., 2007].

[3] In hydrology probabilistic runoff forecasts are typically generated by running a hydrologic model several times using atmospheric variables from different meteorological models as forcing. In recent years BMA has been used increasingly for the combination of multiple rainfall-runoff models [*Ajami et al*., 2007; *Duan et al*., 2007; *Vrugt and Robinson*, 2007; *Diks and Vrugt*, 2010; *Parrish et al*., 2012]. Besides the work of *Bogner et al*. [2013] there has so far been no similar BMA study accounting for the correlation structure between different lead times in the field of hydrologic forecasting. Note that *Yuan and Wood* [2012] successfully applied a different Bayesian postprocessing approach to probabilistic hydrologic forecasts. On the basis of *Berrocal et al*. [2007], combining BMA with the geostatistical output perturbation (GOP) method [*Gel et al*., 2004] in order to achieve spatially correlated BMA forecasts, we will use this method to account for the correlations among lead times. That is, space is replaced by lead times. For simplicity this BMA method is called multivariate BMA from now on.

[4] While multivariate BMA is supposed to improve probabilistic forecasts in terms of multivariate verification measures like the energy score *Gneiting et al*. [2008], it seems also to slightly improve the marginal predictive distributions at single lead times. *Pinson et al*. [2008] and *Pinson and Girard* [2012] introduced a similar method used for postprocessing of probabilistic forecasts of wind power. Like multivariate BMA, it resorts to the multivariate Gaussian distribution and estimates covariance matrices from the series of prediction errors.

[5] For this study, the meteorological forcing consists of a mixture of different models. On the one hand, both global and regional models are used. On the other hand, the models can be divided into deterministic forecasts and ensemble prediction systems (EPS). Each member of an EPS stands for a model run with different initial states, boundary conditions or model formulations [*Palmer*, 2000]. Ideally, these model runs are exchangeable, i.e., they are supposed to be statistically identical. One of the main goals of this study is to combine forecasts stemming from such a bunch of different models into a sound probabilistic density forecast.

[6] In the following, we apply the methods introduced above to river discharge forecasts for Thur River in Eastern Switzerland. Hourly runoff forecasts forced by a mixture of deterministic models and EPS's that cover different ranges of lead times are combined to one single multivariate PDF using multivariate BMA. Thus, multivariate BMA features modifications that allow on the one hand to account for temporally correlated prediction errors and on the other hand to produce time consistent forecasts though the different forcings cover unequal ranges of lead times. Note that any analysis has been performed using the statistical software package R [*R Development Core Team*, 2011].

[7] Section 2 presents the data set used in this study along with an example of a particular forecast. BMA, and especially multivariate BMA, are explained in detail in section 3. The forecasts obtained by univariate and multivariate BMA over a period of two and a half years are verified in section 4. Lastly, we discuss the results shortly in section 5 and make some concluding remarks in section 6.