Multiscale Algorithm for Atmospheric Data AssimilationPart I

A multiscale algorithm for the problem of optimal statistical interpolation of the observed data has been developed. This problem includes the calculation of the vector of the \analyzed" (best estimated) atmosphere ow eld w a by the formula w a = w f + P f H T y; where the quantity y is deened by the equation using the given model forecast rst guess w f and the vector of observations w o. H is an interpolation operator from the regular grid to the observation network, P f is the forecast error covariance matrix, and R is the observation error covariance matrix. At this initial stage the case of univariate analysis of single level radiosonde height data is considered. The matrix R is assumed to be diagonal, and the matrix P f to be given by the formula P f ij = f i ij f j , where ij is a smooth decreasing function of the distance between the ith and the jth points. Two diierent multiscale constructions can be used for eecient solving the problem of optimal statistical interpolation: a technique for fast evaluation of the discrete integral transform P i P f ij v j , and a fast iterative process which eeectively works with a sequence of spatial scales. In this paper we describe a multiscale iterative process based on a multiresolution simultaneous displacement technique and a localized variational calculation of iteration parameters.