## 1. Introduction

### 1.1. Bayesian Inference

[2] Estimation of hydrologic quantities with computer simulation models has greatly advanced in recent years. Through the realization that uncertainty is persistent in all layers of hydrologic prediction, the problem of streamflow forecasting has been reevaluated by much of the scientific community, leading hydrologists to generate hydrologic forecasts within a probabilistic framework [*Najafi et al.*, 2012; *Madadgar et al.*, 2012]. Most often, this is performed through Bayesian inference. Bayesian methods are attractive in hydrology because they have been proven effective, not only in statistical research but also in applications to hydrologic modeling [*Kuczera and Parent*, 1998; *Marshall et al.*, 2004; *Moradkhani et al.*, 2005a, 2005b; *Kavetski et al.*, 2006; *Bulygina and Gupta*, 2010; *Renard et al.*, 2011; *DeChant and Moradkhani*, 2012]. Although it is often common to base probabilistic estimation in hydrology from the Bayesian perspective, specifics about varying implementations differ greatly. These differences stem from the sources of uncertainty accounted for in the analysis, assumptions about the form of the errors, and whether the estimation is performed within a batch or sequential framework. In the current study, the focus is on combining the strengths of sequential and batch Bayesian methods for improved state-parameter estimation.

[3] Sequential Bayesian estimation, often referred to as data assimilation, is a class of methods that seek to estimate the uncertainty associated with the input-state-output relationships of a given model, at every model evaluation in which an observation of the system, state or output, is available. Of these techniques, currently the ensemble Kalman filter (EnKF) is the most commonly used technique in the hydrologic community [*Moradkhani et al.*, 2005a; *Zhou et al.*, 2006; *Hendricks Franssen and Kinzelbach*, 2008; *DeChant and Moradkhani*, 2011a; *Leisenring and Moradkhani*, 2011; *Montzka et al.*, 2011; *Nie et al.*, 2011; *Li et al.*, 2012; *Liu et al.*, 2012]. The EnKF and its several variants have been widely used throughout the hydrologic literature; however, several studies have highlighted problems owing to the limiting assumptions within this technique [e.g., *Moradkhani et al.*, 2005b; *Weerts and El Serafy*, 2006; *Moradkhani et al.*, 2006; *Salamon and Feyen*, 2009; *Matgen et al.*, 2010; *Montzka et al.*, 2011; *Plaza et al.*, 2012; *DeChant and Moradkhani*, 2012]. Recent research has suggested that the particle filter (PF) is a viable alternative to the EnKF in cases where the underlying assumptions are violated [*Moradkhani et al.*, 2005b; *Moradkhani and Sorooshian*, 2008; *Leisenring and Moradkhani*, 2011; *DeChant and Moradkhani*, 2012; *Rings et al.*, 2010; *Plaza et al.*, 2012]; however, the viability of using the PF in certain applications has been questioned throughout the broader data assimilation literature. These concerns are highlighted in the following sections, and the ways to move forward in hydrologic data assimilation are proposed.

### 1.2. Bayesian Filtering Effectiveness and Efficiency

[4] Although the PF technique has been shown to be effective in many hydrologic modeling applications, this method has received criticism because of its large computational demand in comparison with EnKF-based approaches [*Zhou et al.*, 2006; *van Leeuwen*, 2009; *Snyder et al.*, 2008]. Often described as “the curse of dimensionality,” high-dimensional filtering requires a large ensemble size to avoid collapse of the filter, a problem that the PF is more susceptible to than the EnKF. Although the EnKF is better suited to avoid ensemble collapse at lower ensemble sizes than the PF, when the Gaussian error assumption of the EnKF is violated, the performance is suboptimal at all ensemble sizes [*DeChant and Moradkhani*, 2012]. As the assumption of Gaussian error structure will be violated in nearly all hydrologic applications, the PF can be an attractive alternative.

[5] All PFs are based on the Sequential Importance Sampling (SIS) algorithm [*Liu et al.*, 2001]. Although SIS alone can be an effective PF, it is highly subject to collapse, with only a few of the samples having significant weight. This is referred to as weight degeneration. To avoid this problem, resampling methods have been suggested in the statistical literature. Resampling is the process of replicating ensemble members with significant weight, while discarding samples with insignificant weight, to maintain an effective sample that represents the system probability distribution. These techniques include residual resampling [*Liu and Chen*, 1998; *Douc et al.*, 2005; *Weerts and El Serafy*, 2006], multinomial resampling [*Douc et al.*, 2005], weighted random resampling [*Leisenring and Moradkhani*, 2011], stratified resampling [*Hol et al.*, 2006], and systematic resampling [*Moradkhani et al.*, 2005b]. All of these methods have been proven to be effective for building a posterior density but have small differences in their implementation.

[6] Another potential strategy to improving posterior estimation through the PF is with multimodel analysis, through a combination of PF and Bayesian model averaging (BMA) [*Parrish et al.*, 2012]. This method is particularly suited to manage errors resulting from model structural imperfections. Unlike model averaging studies, the current study focuses on posterior estimation within a single-model structure; however, advancements made in this study are compatible with PF and BMA combinations. To improve single-model analysis within filtering, it is necessary to create the most representative posterior distribution possible. This study focuses on enhancing sampling of the posterior with Markov chain Monte Carlo (MCMC) moves.

[7] MCMC refers to several techniques that estimate a posterior density through simulation. Unlike the PF, which is based on the law of large numbers, MCMC is based on ergodic theory and estimates the posterior with a single or multiple chains, which explore to the posterior distribution [*Kuzcera and Parent*, 1998; *Marshall et al.*, 2004; *Kavetski et al.*, 2006; *Smith and Marshall*, 2008; *Vrugt et al.*, 2009; *Jeremiah et al.*, 2011]. This methodology has complementary benefits to PF techniques and may be used to more efficiently sample from the posterior. Several studies in the statistical literature have suggested using MCMC techniques for rejuvenating particles at each observation time step to improve the diversity of each sample, leading to a more complete characterization of the posterior distribution [*Andrieu et al.*, 2010; *Doucet and Johanson*, 2009; *Kantas et al.*, 2009]. This study will expand on these ideas suitable for application to hydrologic models. Recently, we noticed a parallel study [*Vrugt et al.*, 2012] applying similar methods within the context of hydrologic modeling. This study was accepted for publication while the current study was under review. To avoid confusion with that study, we note that the current study proposes a new adaptation of MCMC to the PF for improving joint state-parameter estimation in an entirely sequential framework in the case of stationary parameters. The idea and preliminary results of the current work were presented by*Moradkhani et al.* [2010].