An Optimal Linear Transformation for Data Assimilation

Abstract Linear transformations are widely used in data assimilation for covariance modeling, reducing dimensionality (such as averaging dense observations to form “superobs”), and managing sampling error ensemble assimilation. Here we describe a linear transformation that is optimal the sense that, transformed space, state variables have uncorrelated errors, diagonal gain matrix update step. We conjecture, provide numerical evidence, best possible precede localization an Kalman filter. A central feature of this step scalars, which term canonical observation operators (COOs), relate pairs rank‐order those by their influence update. show idealized problem sample‐based estimates COOs, conjunction with sample covariance, can approximate well true values, but practical implementation high‐dimensional applications remains subject future research. The COOs also completely important properties step, such observation‐state mutual information, signal‐to‐noise degrees freedom signal, so give new insights, including relations among reduced‐rank approximations variational schemes, particle‐filter weight degeneracy, local transform