Multiscale Algorithm for Atmospheric Data Assimilation Part I. Multiscale Iterative Process

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 flow field wa by the formula wa = wf+PfHTy, whre the quantity y is defined by the equation (HPfHT+R)y = wo-Hwf, using the given model forecast first guess wf and the vector of observations wo. H is an interpolation operator from the regular grid to the observation network, Pf 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 Pf to be given by the formula Pfij = sfimijsfj, where mij is a smooth decreasing function of the distance between the ith and the jth points. Two different multiscale constructions can be used for efficient solving the probelm of optimal statistical interpolation: a technique for fast evaluation of the discrete integral transform ?i Pfijvj, and a fast iterative process which effectively 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.