Smoothing Problems in a Bayesian Framework and Their Linear Gaussian Solutions

Abstract

Smoothers are increasingly used in geophysics. Several linear Gaussian algorithms exist, and the general picture may appear somewhat confusing. This paper attempts to stand back a little, in order to clarify this picture by providing a concise overview of what the different smoothers really solve, and how. The authors begin addressing this issue from a Bayesian viewpoint. The filtering problem consists in finding the probability of a system state at a given time, conditioned to some past and present observations (if the present observations are not included, it is a forecast problem). This formulation is unique: any different formulation is a smoothing problem. The two main formulations of smoothing are tackled here: the joint estimation problem (fixed lag or fixed interval), where the probability of a series of system states conditioned to observations is to be found, and the marginal estimation problem, which deals with the probability of only one system state, conditioned to past, present, and future observations. The various strategies to solve these problems in the Bayesian framework are introduced, along with their deriving linear Gaussian, Kalman filter-based algorithms. Their ensemble formulations are also presented. This results in a classification and a possible comparison of the most common smoothers used in geophysics. It should provide a good basis to help the reader find the most appropriate algorithm for his/her own smoothing problem.

A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Applications

“…In contrast, filtering is a process of estimating the current state of a dynamical system using only the past measurements. The theoretical basis of smoothing has been well established for some time (e.g., see Gelb 1974;Jazwinski 2007;Särkkä 2013) and has been used in multivariate geophysical applications (e.g., see Cohn et al 1994;Bennett 1992;Wunsch 1996;Lermusiaux et al 2002;Cosme et al 2012;Lolla 2016). For linear systems with Gaussian measurement noise, the Kalman smoother is optimal in a Bayesian sense (Gelb 1974).…”

“…However, ocean and atmospheric dynamics are highly nonlinear and can develop far-from-Gaussian distributions (Miller et al 1999;Lermusiaux 1999a,b). Practical smoothing schemes for such systems are commonly limited to linearized and/or low-dimensional models (e.g., see Särkkä 2013;Lolla 2016), or to schemes that respect the nonlinearities in the dynamics, but are limited to Gaussian smoother updates, such as ensemble smoothers (Lermusiaux and Robinson 1999;Evensen and Van Leeuwen 2000;Bocquet 2005;Cosme et al 2012). Hence, the utilization of efficient and accurate smoothing in high-dimensional nonlinear models, using fully Bayesian updates, is the challenge addressed here.…”

Obstacles and benefits of the implementation of a reduced-rank smoother with a high resolution model of the tropical Atlantic Ocean

“…For the smoothers, see also Cosme et al (2011). Here we only provide an overview of the sequential algorithms, close to the EnKS, and an intuitive interpretation of the equations.…”

Potential of an ensemble Kalman smoother for stratospheric chemical-dynamical data assimilation

“…Using posterior observations thus has the potential to impose an additional constraint on the model state by performing a retrospective analysis. This procedure is often referred to as the 'smoothing problem' (see Cosme et al, 2012, for a complete overview of the different algorithms). The smoothing problem may offer an additional potential to improve the stratospheric analysis, for two main reasons.…”