Facilitating Inversion of the Error Covariance Models for the Wide-Swath Altimeters

Abstract

Wide-swath satellite altimeter observations are contaminated by errors caused by the uncertainties in the geometry and orientation of the on-board interferometer. These errors are strongly correlated across the track, while also having similar error structures in the along-track direction. We describe a method for modifying the geometric component of the error covariance matrix which facilitates accuracy in the removal of the respective error modes from the signal and improves computational efficiency of the data assimilation schemes involving wide-swath altimeter observations. The method has been tested using the Surface Water and Ocean Topography simulator. We show substantial computer cost savings in the pseudo-inversion of the respective error covariance matrix. This efficiency improvement comes with a few per cent error in the approximation of the original covariance model simulating uncertainties in the geometry and orientation of the on-board interferometer.

Block-Circulant Approximation of the Precision Matrix for Assimilating SWOT Altimetry Data

“…However, with the recent advent of wide-swath interferometry such as SWOT, the validity of this assumption could be questioned due to the substantial contribution of errors correlated with the SSH field observed at the centimeter level of accuracy. Therefore, in recent years, considerable efforts have been made to find numerically efficient algorithms for approximating the action of R −1 and/or G on the innovation vector (e.g., [24][25][26][27]).…”

“…The respective SSH observation grid points will be enumerated by indices 1 ≤ i ≤ n x and 1 ≤ j ≤ n y . In contrast to our previous studies [25,28] focused on the approximation of the SWOT precision matrix at submesoscale wavelengths below 100 km, here we consider scale-independent approximations of R −1 and G based on their block-circulant representation. In addition, since the residual atmospheric error R a is several times smaller than R s in magnitude [29,30], in this study, we neglect its contribution to R.…”

“…In [25,28], it has been shown that R −1 and its symmetric square root R −1/2 could be well-approximated by sparse block-diagonal matrices if the target SSH features are at the submesoscale and are characterized by flat spectra at spatial frequencies above 0.01 km −1 . This approximation is suitable for use in multi-scale DA methods (e.g., [31,32]) at the second stage, after mesoscale features have already been assimilated.…”

The Effect of Spatially Correlated Errors on Sea Surface Height Retrieval from SWOT Altimetry

“…In recent years, various approaches to denoising the SWOT signal from spatially correlated errors were considered (e.g., [11][12][13][14][15][16][17]). In particular, Ref.…”

“…In a different approach, Ruggiero et al [14] and Yaremchuk et al [15] developed heuristic sparse approximations of the inverse error covariance matrix. More recently, separable [16] approximation to the SWOT error covariance matrix and block-diagonal (hereinafter BD) [17] approximation to its inverse square root have been developed.…”