On the approximation of the inverse error covariances of high-resolution satellite altimetry data

High-resolution (swath) altimeter missions scheduled to monitor the ocean surface in the near future have observation-error covariances (OECs) with slowly decaying off-diagonal elements. This property presents a challenge for the majority of the data assimilation algorithms which were designed under the assumption of the diagonal OECs being easily inverted. In this note, we present a method of approximating the inverse of a dense OEC by a sparse matrix represented by the polynomial of spatially inhomogeneous differential operators, whose coefficients are optimized to fit the target OEC by minimizing a quadratic cost function. Explicit expressions for the cost function gradient and the Hessian are derived. The method is tested with an OEC model generated by the SWOT simulator.