## 1. Introduction

[2] With a growing desire to better quantify watershed processes and responses, many modeling studies have been undertaken ranging from attempts at developing models based completely on physical process understanding to simple black box methods [e.g., *Atkinson et al.*, 2003; *Beven*, 1989]. Recently, there has been a movement toward a combination approach, including as much of the known physics of the system as possible, while maintaining model structure parsimony [*Littlewood et al.*, 2003]. A driving force in this has been the advent of more sophisticated algorithms capable of automatic parameter estimation and uncertainty quantification, given such model parameters are “effective,” nonmeasureable values.

[3] Model calibration techniques have changed with the availability of ever-faster computing systems, from simple trial-and-error methods to fully computerized algorithms designed to completely investigate the parameter space [*Vrugt et al.*, 2003]. Automatic calibration techniques are varied in how they attempt to implement objective mathematical procedures and search the parameter space to optimize the model simulation. Commonly used calibration methods range from stochastic global optimization techniques [e.g., *Duan et al.*, 1992; *Sorooshian and Dracup*, 1980; *Thyer et al.*, 1999] to Monte Carlo methods [e.g., *Beven and Binley*, 1992; *Freer et al.*, 1996; *Uhlenbrook et al.*, 1999] to Markov chain Monte Carlo routines [e.g., *Bates and Campbell*, 2001; *Campbell et al.*, 1999; *Kuczera and Parent*, 1998]. In a push to characterize the predictive uncertainty associated with estimated parameter values, Monte Carlo–based approaches have moved to the forefront of automatic calibration routines [*Feyen et al.*, 2007]. The most frequently implemented variants of Monte Carlo methods include uniform random sampling (URS) (often implemented in the popular generalized likelihood uncertainty estimation (GLUE) approach) and Markov chain Monte Carlo (MCMC) schemes [*Bates and Campbell*, 2001; *Marshall et al.*, 2004].

[4] While these methods have been successfully implemented in hydrologic studies, in many cases they also suffer from a variety of problems. In general, all Monte Carlo–based techniques suffer from inefficiency in the exploration of the parameter space [*Bates and Campbell*, 2001]. This is especially true for highly parameterized models, where parameter interactions can be very involved and not adequately explored by the algorithm without an extremely large number of samples [*Kuczera and Parent*, 1998]. Markov chain Monte Carlo–based approaches are more adept at exploring the parameter space in an “intelligent” manner. However, such approaches often suffer greatly from initialization problems associated with the variance of the proposal being either too large or too small, preventing the algorithm from efficiently reaching convergence [*Haario et al.*, 2006]. The additional problem of convergence to the posterior distribution is of significant concern for hydrologic models, as their nonlinear nature often leads to a complex parameter response surface with many local optima [*Duan et al.*, 1992].

[5] The study presented here provides a comparison of three recently developed Markov chain Monte Carlo algorithms intended to overcome some of the inefficiencies common to other well established MCMC methods. This paper is divided into the following sections: section 2 presents a brief description of Bayesian methods in hydrology and introduces the MCMC algorithms featured in this research; section 3 presents two case studies with their results; and section 4 offers relevant conclusions gleaned from this research and applicable to a variety of modeling problems.