## 1. Introduction

[2] Assessment of parameter and predictive uncertainty of hydrologic models is an essential part of any hydrologic study. Uncertainty analysis forms the basis for model comparison and selection [*Schoups et al.*, 2008], allows identification of robust water management strategies that take account of prediction uncertainties [*Ajami et al.*, 2008], and provides an impetus for targeted data collection aimed at improving hydrologic predictions and water management [*Feyen and Gorelick*, 2004]. Furthermore, accurate parameter uncertainty estimation is often required for regionalization and extrapolation of hydrologic parameters to ungauged basins [*Vrugt et al.*, 2002; *Zhang et al.*, 2008].

[3] Uncertainty analysis is commonly based on a regression model, whereby observations are represented by the sum of a deterministic component, i.e., the hydrologic model, and a random component describing remaining errors or residuals. These residual errors typically consist of a combination of input, model structural, output, and parameter errors. Model parameter inferences are then based on a likelihood function quantifying the probability that the observed data were generated by a particular parameter set [*Box and Tiao*, 1992]. The mapping from parameter space to likelihood space results in the identification of a range of plausible parameter sets given the data and allows estimation of parameter and predictive uncertainty.

[4] In recent years, much debate has focused on the use of either a formal or informal approach for specifying the likelihood function [*Mantovan and Todini*, 2006; *Beven et al.*, 2008; *Vrugt et al.*, 2008b; *Stedinger et al.*, 2008; *McMillan and Clark*, 2009]. In the formal approach, one starts from an assumed statistical model for the residual errors, i.e., the functional form of the joint probability density function (pdf) of the residual errors is specified a priori. This statistical model is then used to derive the appropriate form for the likelihood function [*Box and Tiao*, 1992]. For example, assuming that the errors are independent and identically distributed according to a normal distribution with zero mean and a constant variance *σ*^{2}, results in the standard least squares (SLS) approach for parameter estimation. An advantage of the formal approach is that error model assumptions are stated explicitly, and their validity can be verified a posteriori [e.g., *Stedinger et al.*, 2008].

[5] The formal approach has been criticized for relying too strongly on residual error assumptions that do not hold in many applications [*Beven et al.*, 2008]. In many cases, residuals errors are correlated, nonstationary, and non-Gaussian [*Kuczera*, 1983]. A common form of nonstationarity is heteroscedasticity, which in many studies is observed as an increase in error variance with streamflow discharge [*Sorooshian and Dracup*, 1980]. Violation of SLS assumptions may introduce bias in estimated parameter values and affect parameter and predictive uncertainty [*Thyer et al.*, 2009]. Alternatively, informal likelihood functions have been proposed as a pragmatic approach to uncertainty estimation in the presence of complex residual error structures. A well-known example is the generalized likelihood uncertainty estimation methodology of *Beven and Freer* [2001]. Here the likelihood function is specified a priori without explicitly linking it to an underlying error model. The modeler has flexibility in specifying the form of the likelihood function, which makes the informal approach attractive in situations where traditional error assumptions are violated. For example, *Beven et al.* [2008] have advocated the use of a flat likelihood function to avoid overconditioning of the statistical error model on a single calibration data set. However, since the informal approach makes no explicit reference to the underlying error model, its assumptions are implicit and cannot be checked a posteriori. Further discussion and comparison of formal and informal approaches are given by *Mantovan and Todini* [2006], *Beven et al.* [2008], *Vrugt et al.* [2008b], *Stedinger et al.* [2008], and *McMillan and Clark* [2009].

[6] The main goal of this paper is to extend the applicability of the formal approach by deriving and applying a formal likelihood function based on a general error model that allows for model bias and for correlation, nonstationarity, and nonnormality of model residuals. As such, we preserve advantages of the formal approach (theoretical basis and possibility of diagnostic checking of error model assumptions), while gaining flexibility and reducing the need for unrealistic assumptions about the residual errors.

[7] We build on previous formal approaches that have been used to relax some of the SLS error assumptions [*Sorooshian and Dracup*, 1980; *Kuczera*, 1983; *Thiemann et al.*, 2001; *Bates and Campbell*, 2001]. In particular, we follow *Bates and Campbell* [2001] and account for serial dependence of residual errors using a general autoregressive (AR) time series model. The main contribution of our work lies in the treatment of heteroscedasticity and nonnormality, whereas previous approaches have used data and model response transformations, e.g., Box-Cox transformations [*Box and Tiao*, 1992], to induce homoscedasticity (constant variance) and remove skewness, we instead rely on an explicit statistical model to account for heteroscedasticity and nonnormality. Error standard deviation is modeled as a linear function of simulated streamflow, and nonnormality is accounted for with a parametric error distribution that allows for separate control of kurtosis and skewness in the model residuals. As discussed below, our approach is both more flexible and more intuitive compared to the transformation method. Focus is on correct simulation and representation of total residual errors, i.e., measurement, model input, and model structural errors are treated in a lumped manner, as opposed to recent attempts at separating the various error sources in hydrologic modeling [*Kavetski et al.*, 2006a; *Kuczera et al.*, 2006; *Gotzinger and Bardossy*, 2008; *Vrugt et al.*, 2008b; *Reichert and Mieleitner*, 2009; *Thyer et al.*, 2009; *Renard et al.*, 2010]. The lumped approach provides less insight into the sources of error but yields practical estimates of parameter and total prediction uncertainty.

[8] The next section presents the statistical modeling approach, derives a new likelihood function for parameter inference, and outlines a method for predictive simulation. The methodology is applied in section 3 to estimate parameter and predictive uncertainty of a spatially lumped rainfall-runoff model, using synthetic and real data from a humid and a semiarid basin. Following *Thyer et al.* [2009], we assess effects of assumptions in the error model on parameter and prediction uncertainty. Section 4 discusses and summarizes our findings.