## 1. Introduction

[2] The use of conceptual hydrologic models necessitates some type of calibration procedure to determine optimum values for the models' parameters such that the predicted outcome most closely matches the observed. Typically this will be done either manually or automatically, with automatic methods becoming the primary method of use recently [*Madsen et al.*, 2002]. Automatic calibration can be performed by a variety of methods such as global optimization algorithms [e.g., *Duan et al.*, 1992], Monte Carlo methods [e.g., *Beven and Binley*, 1992], and Bayesian inferential approaches [e.g., *Kuczera and Parent*, 1998], and many techniques have the ability to incorporate multiple objective measures into their calibration logic [e.g., *Vrugt et al.*, 2003].

[3] As interest in quantifying the uncertainty in model predictions has grown, the use of Monte Carlo (such as the generalized likelihood uncertainty estimation (GLUE) methodology) and Bayesian inferential approaches have gained interest, as well. Much debate has centered on the differences between these methods [see *Beven et al.*, 2007; *Mantovan and Todini*, 2006; *Mantovan et al.*, 2007]; in actuality these methods can be looked at as special cases of one another (i.e., GLUE is a less statistically strict form of the formal Bayesian approach).

[4] Bayesian inference is based on the application of Bayes' theorem, which states that

where *Q* is the data, θ is the parameter set, *P*(θ∣*Q*) is the posterior distribution, *P*(θ) is the prior distribution, and *P*(*Q*∣θ) is the likelihood function summarizing the model (for the input data) given the parameters. Even a cursory examination of equation (1) reveals that the choice of the likelihood function will play an important role. Because of its importance to the resultant posterior distribution, the likelihood function must make explicit assumptions about the form of the model residuals [*Stedinger et al.*, 2008].

[5] It is in the likelihood function where the GLUE method largely deviates from the formal Bayesian approach. The GLUE methodology typically opts for informal likelihood (objective) functions rather than risking violating the strong assumptions that arise from the use of formal likelihood functions, with critics of the Bayesian approach holding the overriding view that the appropriateness of such assumptions is inherently difficult to satisfy in hydrologic problems, and thus the use of Bayesian methods is problematic [*Beven*, 2006a].

[6] Despite this criticism, the use of Bayesian methods has become increasingly common [e.g., *Kuczera and Parent*, 1998; *Marshall et al.*, 2004; *Samanta et al.*, 2007; *Smith and Marshall*, 2008, 2010; *Vrugt et al.*, 2009]. However, the predominant usage of Bayesian methods in hydrology has been under the assumption of uncorrelated, Gaussian errors, as evidenced by a plethora of studies [e.g., *Ajami et al.*, 2007; *Campbell et al.*, 1999; *Duan et al.*, 2007; *Hsu et al.*, 2009; *Marshall et al.*, 2004; *Samanta et al.*, 2008; *Smith and Marshall*, 2010; *Vrugt et al.*, 2006]. Although there are undoubtedly situations (perhaps catchments with a very simple hydrograph) in which the errors are adequately represented by such a likelihood function, these scenarios are more likely to be the exception than the rule in hydrology.

[7] Because the posterior distribution is proportional to the prior distribution multiplied by the likelihood function (equation (1)), it is important to consider the form of the errors (assumed by the likelihood) when implementing the Bayesian method. Given the tendency for errors arising from hydrologic models to be (at least) heteroscedastic [e.g., *Kuczera*, 1983; *Sorooshian and Dracup*, 1980], the application of likelihood functions should not be chosen on the basis of computational simplicity. The ultimate effectiveness of Bayesian inference relies on properly characterizing the form of the errors via the formal likelihood function and holds untapped potential for improvement in uncertainty estimation.

[8] *Xu* [2001, p. 77] points out that “in the field of hydrological modeling, few writers examine and describe any properties of residuals given by their models when fitted to the data”. This provides a potential limiting constraint on more appropriate and widespread use of Bayesian inference. The need to address the lack of normality in hydrologic modeling errors is not new, but there is little guidance on exactly how to go about selecting an appropriate likelihood for a particular data-model combination.

[9] Although the assumption of normality is common to hydrological modeling studies, several alternatives aimed at addressing typical characteristics of rainfall-runoff modeling errors have been made. For example, *Bates and Campbell* [2001] considered multiple likelihood functions that aimed to address the issues of nonconstant variance (via a Box-Cox transformation) and correlation (via an autoregressive model) in the modeled errors. *Marshall et al.* [2006] considered a Student's *t* distribution with four degrees of freedom due to its higher peaks and heavier tails. More recently, *Schaefli et al.* [2007] attempted to address the nonconstant variance problem through the use of a mixture likelihood corresponding to high and low flow states. In arid regions, there is further difficulty in appropriately capturing the form of the errors caused by extended periods of no-flow conditions. This can lead to errors that are severely zero inflated (i.e., a significant proportion of the residuals are zero) due to the ability of the model to correctly predict a zero flow at a zero flow observation.

[10] In spite of the variations in the likelihood function outlined above, the issue of residuals originating from distinct flow states, as is the case with ephemeral catchments worldwide, has not been addressed at length in the literature. In this paper, we focus on the development and application of a likelihood function that specifically focuses on addressing the potential problems uniquely caused by zero-inflated errors under a Bayesian inferential approach. This paper is divided into the following sections: section 2 introduces a formal likelihood function that addresses the zero-inflation problem common in arid catchments, section 3 introduces two test applications for the formal likelihood function to be analyzed, and section 4 provides a discussion of the results and the important conclusions of the study.