## 1 Introduction

[2] The work described herein is motivated by the needs of the short-term optimization methodology for hydropower production. Stochastic optimization, i.e., optimizing when accounting for uncertainties in available resources (water) and utility (prices) in the future, is about to be introduced. This uncertainty must be quantified, and in this paper, we focus on water resources availability expressed as inflow forecasts. In the following, we use the more general term flow since the methodology can be applied to any flow forecast. When optimizing a water system, all catchments and several lead times must be considered simultaneously. Depending on the hydropower system in question, we may be dealing with a set of headwater catchments, or a system of upstream-downstream reservoirs in which water from one catchment arrives perhaps days later in a lower catchment, or a combination of both. Thus, there is a need to construct a multivariate predictive distribution for the forecasts which accounts for between-catchment and between-lead time dependencies. Furthermore, short-term stochastic optimization methods require ensembles, i.e., samples of flow forecasts, as input. Based on the ensembles, the optimization methods build scenario trees for future flows. An overview of scenario tree algorithms is given in *Dupacova et al*. [2000] and a recent development is described in *Rasol et al*. [2013]. It is therefore desirable to have probabilistic forecasts that can be sampled from.

[3] Traditionally numerical physically based hydrological models are used to provide deterministic forecasts of flow, and we refer to them as hydrological forecasts. The driving forces of these dynamic models are precipitation (water in) and temperature (evaporation, snow accumulation, and snow melt are temperature-driven processes). The hydrological forecast is found by running the numerical hydrological model using observed temperature and precipitation values as input to obtain initial states, whereas precipitation and temperature forecasts are used as inputs during the forecasting period. This gives a deterministic flow forecast which contains errors for several reasons [*Refsgaard and Storm*, 1996]. These include errors in input data, especially in precipitation and temperature forecasts (e.g., catch deficit and interpolation of precipitation volumes); errors in internal states when the forecast starts (e.g., too much snow); errors in model parameters (e.g., the conductivity parameter is overestimated); errors in model structure (e.g., the absence of important processes); and errors in data used for calibration (e.g., errors in the rating curve).

[4] For almost three decades, the estimation of uncertainties in hydrological modeling in general has been a major field of interest [e.g., *Kuczera*, 1983; *Beven and Binley*, 1992; *Yapo et al*., 1998; *Thiemann et al*., 2001; *Kavetski et al*., 2006]. However, only a few of these studies have focused in particular on uncertainties linked to forecasts [e.g., *Krzysztofowicz*, 2002, 1999; *Todini*, 2008; *Montanari and Grossi*, 2008; *Reggiani and Weerts*, 2008]. For recent publications, see *Cloke et al*. [2013] and references therein. Probabilistic forecasts may be either purely statistical or based on a hydrological forecast. An example of a pure statistical model is the climatology or persistent forecast. In order to benefit from our understanding of hydrological processes as well as our observations, it is appealing to base probabilistic forecasts on both hydrological and statistical forecasts.

[5] Conceptually, there are two ways of using a hydrological model to make probabilistic forecasts [e.g., *Renard et al*., 2010]; endogenous and exogenous (pre/postprocessing) methods. Whereas endogenous methods are based on a physical model and aim to make parts of this model stochastic, exogenous methods involve the construction of an uncertainty model for the deterministic forecast based on the joint distribution of forecast and observations. Endogenous methods can be used to make the internal states of the hydrological model stochastic, e.g., using an Ensemble Kalman Filter [e.g., *Moradkhani et al*., 2005], or make the model parameters stochastic [e.g., *Reichert and Mieleitner*, 2009], or replace the hydrological model with a probability density function describing precipitation-runoff processes [e.g., *Bulygina and Gupta*, 2009], or to produce seasonal flow forecasts [e.g., *Wang et al*., 2009]. Exogenous methods keep the numerical hydrological model deterministic. Pure postprocessing methods build a stochastic model for flow forecasts from either one [e.g., *Krzysztofowicz*, 1999; *Montanari and Grossi*, 2008; *Weerts et al*., 2011; *Brown and Seo*, 2013], or several hydrological models using either Bayesian Model Averaging [e.g., *Vrugt et al*., 2007] or the Model Conditional Processor [*Todini*, 2008; *Coccia and Todini*, 2011]. Most postprocessing methods aim to build a model which provides a predictive distribution of the flow forecast. The major challenges associated with this approach are linked to accounting for temporal dependencies, specification of the distribution, and accounting for heteroscedasticity of forecast errors. An attractive solution to the last two issues is to apply quantile regression [*Koenker*, 2005] to directly estimate the flow forecast quantiles [*Weerts et al*., 2011]. However, this method requires many parameters, since a unique regression equation has to be established for each quantile. Preprocessing might be included in a form of a stochastic model for weather forecasts [e.g., *Krzysztofowicz*, 2002; *Todini*, 2008], or probabilistic calibrated ensemble weather forecasts may be used [e.g., *Cloke and Pappenberger*, 2009; *Marty et al*., 2013]. Most of the postprocessing studies listed above focus on forecasting lead times of up to a few days for one site. Nevertheless, probabilistic methods for the seasonal forecasting of streamflow [e.g., *Wang et al*., 2009, 2011] and multisite forecasting [e.g., *Reggiani and Weerts*, 2008; *Wang et al*., 2009] are also developed.

[6] The aim of this work was to construct reliable and sharp joint predictive distributions for flow forecasts for several lead times in a system of catchments. The models introduced here can be seen as extensions of the single-catchment one lead time model introduced by *Engeland et al*. [2010]. These extensions are based on two working hypotheses:

[7] 1. The sliding window climatology forecast, the persistent forecast, and the hydrological forecast all contain predictive information about the future flow, and the relative importance of these forecasts varies with lead time.

[8] 2. The between-catchment and between-lead time residual dependencies vary with lead time.

[9] The probabilistic forecast models proposed here can utilize information from the three operationally available forecasts (the hydrological, the sliding window climatology, and the persistent). Furthermore, our models account for the dependencies in forecast errors both between catchments and between lead times. This information is important for how joint management of several hydropower reservoirs may be optimized for durations longer than 1 day. The working hypotheses and the predictive performance of the proposed models were tested for five catchments and ten lead times on a headwater system form part of the of the Ulla-Førre hydropower complex in southern Norway. The predictive performance of the models was evaluated according to the reliability (correctness of the distribution), sharpness (width of the forecast intervals), and efficiency (how well the median values fitted with observed flows). The unique contribution of this paper is to provide an extension of the statistical model of *Engeland et al*. [2010] to incorporate several lead times and catchments, and to use energy score (ES) [*Gneiting et al*., 2008] as a measure of the quality of a multivariate probabilistic flow forecast.