Handling boundaries with the one‐dimensional first‐order recursive filter

Abstract

In variational data assimilation, a crucial task is to determine the background‐error covariance matrix B. The effect of B on a field is often modelled through a series of operators among which is a correlation operator. The recursive filter, when properly normalized to ensure the maximum of the solution is 1, is a convenient correlation operator and is widely used as such. Often, multi‐dimensional operators are constructed from the product of one‐dimensional operators. When their coefficients are calculated appropriately, the normalized one‐dimensional first‐order recursive filter applied N times models an autoregressive function of order N.In ocean data assimilation, an extra difficulty is to handle, within the correlation operator, the boundaries formed by the different coastlines or the bathymetry. Handling the east/west wrapping or polar folds in global configurations are also part of the issue. The purpose of this article is to set up a mathematical framework for using the recursive filter in ocean data assimilation and properly accounting for boundaries. Generally, these problems are dealt with by calculating proper normalization factors using costly methods, or extending the grid near boundaries. It is shown that the normalization factors can be calculated inexpensively through an analytical formula with a corrective term to handle properly the boundary when Neumann or Dirichlet boundary conditions are used.In its classical formulation however, the recursive filter uses Robin (third type) boundary conditions. This formulation can be slightly modified in order to account for Neumann, Dirichlet or periodic boundary conditions. To do so, extra coefficients are calculated through a simple recursive formula. In a global 1/4° analysis framework, this new formulation is shown to be less expensive and more accurate than the classical formulation associated with an appropriate grid extension.