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In data assimilation for geophysical problems, the increasing amount of satellite data to analyze makes it more and more challenging to guarantee near real time forecasting. Thus, low time and memory consuming data assimilation methods become very attractive. The back-and-forth nudging (BFN) method is a non-classical data assimilation method that can be seen as a deterministic and smoothing version of the Kalman filter. From a practical point of view, the BFN method is very valuable for its simplicity of implementation (no optimization, no differentiation, ...) and its rapidity of convergence. Under observability conditions, we prove the mathematical convergence of BFN at deep layers for a multi-layer quasi-geostrophic (MQG) ocean circulation model using an infinite dimensional variant of LaSalle's invariance principle. We also extend the BFN to the problem of joint state-parameter identification. The numerical experiments, performed on 120km large swath sea surface height (SSH) simulated data of the Surface Water Ocean Topography (SWOT) satellite, show the high robustness of the algorithm to uncertainties and the few iterations needed to reach convergence, whereas some problems remain due to non-reversibility properties in time. We also give a strategy to improve geophysical model accuracy, considering the large number of uncertain parameters inherent to models and their impacts on state estimation performance. We propose here a joint state-parameter estimation, tested on the baroclinic wavenumber as an unobserved parameter.

discrete and continuous dynamical systems. series s

Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations