  Computational kernel of the three-dimensional variational data assimilation
(3D-Var) problem is a linear system, generally solved by means of an iterative
method. The most costly part of each iterative step is a matrix-vector product
with a very large covariance matrix having Gaussian correlation structure. This
operation may be interpreted as a Gaussian convolution, that is a very
expensive numerical kernel. Recursive Filters (RFs) are a well known way to
approximate the Gaussian convolution and are intensively applied in the
meteorology, in the oceanography and in forecast models. In this paper, we deal
with an oceanographic 3D-Var data assimilation scheme, named OceanVar, where
the linear system is solved by using the Conjugate Gradient (GC) method by
replacing, at each step, the Gaussian convolution with RFs. Here we give
theoretical issues on the discrete convolution approximation with a first order
(1st-RF) and a third order (3rd-RF) recursive filters. Numerical experiments
confirm given error bounds and show the benefits, in terms of accuracy and
performance, of the 3-rd RF.
