Mapping and pseudoinverse algorithms for ocean data assimilation

Among existing ocean data assimilation methodologies, reduced-state Kalman filters are a widely studied compromise between resolution, optimality, error specification, and computational feasibility. In such reduced-state filters, the measurement update takes place on a coarser grid than that of the general circulation model (GCM); therefore, these filters require mapping operators from the GCM grid to the reduced state and vice versa. The general requirements are that the state-reduction and interpolation operators be pseudoinverses of each other, that the coarse state define a closed dynamical system, that the mapping operations be insensitive to noise, and that they be appropriate for regions with irregular coastlines and bathymetry. In this paper, we describe three efficient algorithms for computing the pseudoinverse: a fast Fourier transform algorithm that serves for illustration purposes, an exact implicit method that is recommended for most applications, and an efficient iterative algorithm that can be used for the largest problems. The mapping performance of 11 interpolation kernels is evaluated. Surprisingly, common kernels such as bilinear, exponential, Gaussian, and sinc perform only moderately well. We recommend instead three kernels, smooth, thin-plate, and optimal interpolation, which have superior properties. This study removes the computational bottleneck of mapping and pseudoinverse algorithms and makes possible the application of reduced-state filters to global problems at state-of-the-art resolutions.