PHYSICALLY-BASED APPROACH TO COVARIANCE MODELING

The specification of covariance models often uses the implicit assumption that the error variance and error correlation can be specified independently. Yet, in most Kalman filter experiments of geophysical fluid flows, the spatial variation of the error variance and the error correlation appears to be related throughout their evolution in time. In an attempt to explain these relationships, and to provide a basis to develop more general covariance models for data assimilation, covariance model formulations for non-homogeneous error statistic have been reexamined. Starting from first principles of probability theory and considering non-homogeneous non-isotropic error statistics, a number of fundamental relationships between the statistics of a (scalar) field and of the statistics of its derivative have been found. These relationships have been applied to a number of phenomenon of geophysical interest such as the linear balance, balance about a non-trivial state, and tropical waves, and have revealed several connections (some intriguing) between the error correlation and spatial variation of the error variance. Somewhat more general covariance models for data assimilation have thus been developed and some statistical validation will be presented.