Numerical Variational Methods for Data Filtering and Interpolation

Abstract

An objective representation of an observed meteorological field is obtained by minimizing a quadratic functional that measures both the smoothness (and regularity) of the objective field and the closeness to the observed data. This is a particular form of the general model for numerical variational analysis suggested by Wahba and Wendelberger to generalize the idea introduced by Sasaki. The solution of this minimization problem can be obtained by using homogeneous splines and cross validation or by using the finite element method to determine an approximate solution. Finite elements can provide data compression for large N. Testud and Chong proposed such a method, based on bilinear finite elements for 3-dimensional wind field analysis. We go further with the same principle and study a more regular approximation, which can be analytically differentiated, obtained by bicubic splines. The numerical simulations proposed by Chong and Testud are used to compare the capability of all these methods (splines, bilinear elements, and bicubic splines) to restitute the wind field and its divergence. The finite elements give quite accurate results, especially the bicubic splines, and are very easy to compute.