Accounting for observation errors in image data assimilation

This paper deals with the assimilation of image-type data. Such kinds of data, such as satellite images, have good properties (dense coverage in space and time), but also one crucial problem for data assimilation: they are affected by spatially correlated errors. Classical approaches in data assimilation assume uncorrelated noise, because the proper description and numerical manipulation of non-diagonal error covariance matrices is complex. This paper proposes a simple way to provide observation error covariance matrices adapted to spatially correlated errors. This is done using various image transformations: multiscale (Wavelets, Fourier, Curvelets), gradients, and gradient orientations. These transformations are described and compared to classical approaches, such as pixel-to-pixel comparison and observation thinning. We provide simple yet effective covariance matrices for each of these transformations, which take into account the observation error correlations and improve the results. The effectiveness of the proposed approach is demonstrated on twin experiments performed on a 2-D shallow-water model.