[p2] Mathematical aspect
[p228] 
Cost functions in the hybrid variationalensemble method

L. Duc (JAMSTEC), and K. Saito (Meteorological Research Institute) 
Abstract 
In the hybrid variationalensemble data assimilation schemes preconditioned on the square root U of the background covariance B, U is a linear map from the model space to a higher dimensional space. Due to the use of the nonsquare matrix U, the transformed cost function still contains the inverse of B. 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 which 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 U (KerU), 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 KerU to the full control space. That means they are as valid as the transformed one in the control space. The theory justifies the use of the practical cost function and its variant in the hybrid variationalensemble data assimilation method. 