On Cost Functions in the Hybrid Variational–Ensemble Method

Abstract

Abstract In the hybrid variational–ensemble data assimilation schemes preconditioned on the square root of background covariance , is a linear map from the model space to a higher-dimensional space. Because of the use of the nonsquare matrix , the transformed cost function still contains the inverse of . To avoid this inversion, all studies have used the diagonal quadratic form of the background term in practice without any justification. This study has shown that this practical cost function belongs to a class of cost functions that come into play whenever the minimization problem is transformed from the model space to a higher-dimension space. Each such cost function is associated with a vector in the kernel of (Ker), leading to an infinite number of these cost functions in which the practical cost function corresponds to the zero vector. These cost functions are shown to be the natural extension of the transformed one from the orthogonal complement of Ker to the full control space. In practice, these cost functions are reduced to a practical form where calculation does not require a predefined vector in Ker, and are as valid as the transformed one in the control space. That means the minimization process is not needed to be restricted to any subspace, which is contrary to the previous studies. This was demonstrated using a real observation data assimilation system. The theory justifies the use of the practical cost function and its variant in the hybrid variational–ensemble data assimilation method.