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

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.