On adaptive filtering for high dimensional systems under parameter uncertainty and its application to satellite data assimilation in oceanography

In this paper, the adaptive filtering theory, recently proposed and developed the authors of present work [1-9] for stochastic, encountered in the field of data as simulation in meteorology and oceanography, is reviewed. Several important questions on numerical estimation og the gain matrix, model reduction, structural choices for the gain, filter stability… are discussed. We show the connections of present approach with a standard Kalman filtering. Adaptive filter is implemented along with a Kalman filtering. Adaptive filter is implemented along with a Kalman filter and standard Newton relation method on the four-layer adiabatic Miami Isopycnical Co-ordinate Ocean Model (MICOM) to produce the estimate for the deep oceanic circulation using assimilate synthetic observations of surface height. Numerical results justify high efficiency of the adaptive filter whose performance is slightly better than that of a Kalman filter due to impossibility to correctly specify the error statistics in a Kalman filter