Fast Anisotropic Gaussian Convolution in 3D-Var Data Assimilation

This paper presents an efficient method for modeling horizontal covariances in three-dimensional variational data assimilation (3D-VAR) using a fast anisotropic Gaussian convolution. Unlike conventional isotropic Gaussian convolution, which assumes uniform spatial correlation scales, the proposed approach introduces an adaptive anisotropic diffusion tensor that accounts for spatial heterogeneities. The method is implemented through a non-orthogonal coordinate transformation, enabling an efficient recursive filtering approximation of anisotropic Gaussian convolution. By incorporating spatially varying anisotropic correlation structures, this approach improves background error covariance representation and enhances information propagation in data assimilation. Numerical experiments show that the method effectively captures directional dependencies while maintaining computational efficiency comparable to isotropic convolution. This advancement aligns 3D-VAR with four-dimensional variational assimilation (4D-VAR) by introducing flow-dependent anisotropic corrections within a computationally feasible framework.