## 1. Introduction

[2] Hydrological models have become essential tools for flood hazard mitigation and water resources management. Increasingly, there is a demand for probabilistic predictions to reflect the fact that model predictions are subject to errors and that prediction uncertainty needs to be taken into account in decision making.

[3] Various methods have been developed over recent decades to quantify hydrological prediction uncertainty. The methods range from lumping all errors into only prediction errors [e.g., *Sorooshian and Dracup*, 1980; *Kuczera*, 1983; *Vrugt et al*., 2005], to implicitly specifying model input, output, parameter and structural errors through the likelihood function on the total error [e.g., *Beven and Binley*, 1992; *Freer et al*., 1996], through to explicitly characterizing each source of errors [e.g., *Moradkhani et al*., 2005; *Kuczera* *et* *al*., 2006; *Huard* *and* *Mailhot*, 2008; *Reichert* *and* *Mieleitner*, 2009; *Renard* *et* *al*., 2010; *Salamon* *and* *Feyen*, 2010; *Renard* *et* *al*., 2011]. In nearly all cases, statistical models are used to represent the structure of the prediction errors. Some statistical models assume homoscedastic error distributions [e.g., *Diskin and Simon*, 1977], while others assume heteroscedastic error distributions either explicitly [e.g., *Sorooshian and Dracup*, 1980; *Schoups and Vrugt*, 2010] or through data transformation [e.g., *Thiemann et al*., 2001; *Thyer et al*., 2002; *Wang et al*., 2012a]. They also differ in the way they represent the temporal dependence of the prediction errors. The most commonly used are independent error models [e.g., *Diskin and Simon*, 1977] and autoregressive error models [e.g., *Kuczera*, 1983; *Bates and Campbell*, 2001; *Engeland and Gottschalk*, 2002].

[4] We attempt to devise error models that can be applied to real-time hydrological forecasts. To make these models easy to apply, we have sought to make them as simple as possible and to minimize computation. To achieve this, we have chosen to focus only on errors in the response variable—i.e., the streamflow prediction.

[5] The performance of hydrological models varies with flow magnitudes and soil moisture [*Freer et al*., 2003; *Choi and Beven*, 2007], and because flow magnitudes and soil moisture often vary with season it is useful to consider errors as being dependent on season. It is possible to attempt to reduce seasonally dependent prediction errors by calibrating hydrological models differently for different seasons (or months), and then apply a generic error model. Alternatively, or in addition, the error model can be varied seasonally. For example, *Yang et al*. [2007] considered a continuous-time autoregressive error model and used different asymptotic standard deviations and characteristic correlations for dry and wet seasons. Their case study of the Chaohe Basin in northern China showed that using seasonally dependent parameters leads to more accurate probabilistic streamflow prediction than using constant parameters throughout the year. *Engeland et al*. [2010] created 15 seasonally dependent weather classes for a catchment in northern Norway and evaluated hydrological prediction errors with an autoregressive model using weather class specific parameters. They demonstrated the usefulness of seasonally dependent parameters for accounting for high uncertainties linked to snow cover formation and snowmelt processes.

[6] In this study, we consider a hydrological model that is calibrated to all available data (i.e., not calibrated conditionally for each season or month). We then rely on the error model to cope with seasonal dependence in the prediction errors. Our approach to seasonal error modeling is similar to postprocessing: we treat the hydrological model parameters (once calibrated) as fixed, and then devise seasonally dependent error models without revising the hydrological model parameters.

[7] Varying error models by month or by season introduces a large number of additional parameters. Any model that has a large number of parameters may be prone to overfitting. One approach to guarding against overfitting is to apply constraints on parameters. In one of our error models (the seasonally variant model), we allow the parameters to be specified for each month without constraints. To guard against overfitting, we devise a hierarchical error model that connects the parameters of different months through a Bayesian prior (we refer to this as the *hyper-distribution*). The extent of parameter variation with month is then inferred from data.

[8] Bayesian hierarchical modeling has been applied to hydrological prediction errors previously, notably in the Bayesian Total Error Analysis methodology (BATEA) introduced by *Kavetski et al*. [2002]. BATEA is based on a Bayesian hierarchical model but is different from this study in several respects: First, BATEA uses a Bayesian hierarchical model to introduce latent variables describing uncertainties in observations and model structure. The hierarchical error model in this study attempts to connect model parameters from different months and avoid possible overparameterization. Second, BATEA applies a Markov-Chain Monte Carlo (MCMC) method to evaluate the posterior distribution of model parameters. In this study, we estimate only the single best value of model parameters, and parameter uncertainty is not explicitly considered. Third, BATEA explicitly treats different sources of error, while the hierarchical error model in this study aggregates all sources of error into the prediction residual.

[9] In section 2, we describe the hydrological model used in this study and present methods relating to the error models, their estimation and evaluation. A case study of five catchments in Australia to demonstrate model calibration and verification is given in section 3. We discuss and summarize our findings in section 4.