Variational Analysis Using Spatial Filters

In this paper the standard variational analysis scheme is modified, through a simple transform, to avoid the inversion of the background error covariance matrix. A close inspection of the modified scheme reveals that it is possible to use a filter to replace the multiplication of the covariance matrix. A variational analysis scheme using a filter is then formulated, which does not explicitly involve the covariance matrix. The modified scheme and the filter scheme have the advantage of avoiding the inversion or any usage of the large matrix for analyses using gridpoint representation. To illustrate the use of these schemes, a small-sized and a more realistic analysis problem is considered using real temperature observations. It is found that both the modified scheme and the filter scheme work well. Compared to the standard and modified schemes the storage and computational requirements of the filter scheme can be reduced by several orders of magnitude for realistic atmospheric applications.