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On the mapping of multivariate geophysical fields: Error and variability subspace estimates

A basis is outlined for the first‐guess spatial mapping of three‐dimensional multivariate and multiscale geophysical fields and their dominant errors. The a priori error statistics are characterized by covariance matrices and the mapping obtained by solving a minimum‐error‐variance estimation problem. The size of the problem is reduced efficiently by focusing on the error subspace, here the dominant eigendecomposition of the a priori error covariance. The first estimate of this a priori error subspace is constructed in two parts. For the ‘observed’ portions of the subspace, the covariance of the a priori missing variability is directly specified and eigendecomposed. For the ‘non‐observed’ portions, an ensemble of adjustment dynamical integrations is utilized, building the non‐observed covariances in statistical accord with the observed ones. This error subspace construction is exemplified and studied in a Middle Atlantic Bight simulation and in the eastern Mediterranean. Its use allows an accurate, global, multiscale and multivariate, three‐dimensional analysis of primitive‐equation fields and their errors, in real time. The a posteriori error covariance is computed and indicates complex data‐variability influences. The error and variability subspaces obtained can also confirm or reveal the features of dominant variability, such as the Ierapetra Eddy in the Levantine basin.