Smoothing problems in a Bayesian framework and their linear Gaussian solutions

Description

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 di erent smoothers really solve, and how. We start addressing this issue from a Bayesian viewpoint. The ltering problem consists in nding 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 other formulation is a smoothing problem. The two main formulations of smoothing are tackled here: the joint estimation problem ( xed-lag or xed-interval), where the probability of a series of system states conditioned to observations is to be found, and the marginal estimation problem, that 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 lter-based algorithms. Their ensemble formulations are also presented. This results in a classi cation and a possible comparison of the most common smoothers used in geophysics. It should be a good basis to help the reader nd the most appropriate algorithm for his/her own smoothing problem.

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