Stochastic Integration for the Heterogeneous Correlation Modeling Using a Diffusion Equation

Abstract

In this note, a stochastic integration scheme is proposed as an alternative to a deterministic integration scheme, usually employed for the diffusion operator in data assimilation. The stochastic integration scheme is no more than a simple interpolation of the initial condition in lieu of the deterministic integration. Furthermore, this also presents a potential in high performance computing. For the classic preconditioned minimizing problem, the stochastic integration is employed to implement the square root of the background error covariance matrix, while its adjoint is obtained from the adjoint code of the square root code. In a first part, the stochastic integration method and its weak convergence are detailed. Then the practical use of this approach in data assimilation is described. It is illustrated in a 1D test bed, where it is shown to run smoothly for background error covariance modeling, with nearest-neighbor interpolations, and O(100) particles.