Abstract. This paper deals with a new data assimilation algorithm, called Back and Forth Nudging. The standard nudging technique consists in adding to the equations of the model a relaxation term that is supposed to force the observations to the model. The BFN algorithm consists in repeatedly performing forward and backward integrations of the model with relaxation (or nudging) terms, using opposite signs in the direct and inverse integrations, so as to make the backward evolution numerically stable. This algorithm has first been tested on the standard Lorenz model with discrete observations (perfect or noisy) and compared with the variational assimilation method. The same type of study has then been performed on the viscous Burgers equation, comparing again with the variational method and focusing on the time evolution of the reconstruction error, i.e. the difference between the reference trajectory and the identified one over a time period composed of an assimilation period followed by a prediction period. The possible use of the BFN algorithm as an initialization for the variational method has also been investigated. Finally the algorithm has been tested on a layered quasi-geostrophic model with sea-surface height observations. The behaviours of the two algorithms have been compared in the presence of perfect or noisy observations, and also for imperfect models. This has allowed us to reach a conclusion concerning the relative performances of the two algorithms.

International audienceIn this Note, we introduce a new algorithm for data assimilation problems, called the back and forth nudging (BFN) algorithm. The standard forward nudging algorithm is first studied for a linear ODE model. The backward nudging algorithm is then introduced in order to reconstruct the initial state of the system. These two algorithms are combined in the new BFN algorithm. The mathematical proof of its convergence is given for a linear ODE system

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.

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