A CONVOLUTION ALGORITHM WITH APPLICATION TO DATA ASSIMILATION

A computationally efficient algorithm to approximate a convolution w f is derived when the real-valued weighting functions w are dimension ally separable, i.e., w(x(1),x(2)) = w(1)(x(1))w(2)(x(2)) and w(x(k)) = w(-x(k)), and the function f(i, j) is defined on a discrete integer lattice in R2, The algorithm consists of a product of operator polynom ials P-k(D-k), k = 1, 2, which are composed of simple averaging operat ors D-k operating on f. It generalizes the smoothing algorithm of Good rich, Passi, and Limber [Pror. 24th Symposium on the Interface: Comput ing Science and Statistics, 1992] for data constrained to a uniformly spaced, large-dimensioned orthogonal grid lying in a two-dimensional s pace, using a Gaussian weighting function w(r) = exp(-r(2)/b(2)). The new algorithm has a direct application to data assimilation problems i n meteorology and oceanography. Data assimilation optimally combines o utput of a numerical model with observational data to derive more accu rate initial conditions needed for numerical integration of the model equations. Computer simulations with the algorithm were found to be ab out ten times faster than a comparable algorithm of Derber and Rosati [J. Phys. Ocean, (1989), pp.1333-1347].