Mapping and pseudo-inverse algorithms for 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 pseudo-inverses of each other, that the coarse state defines 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 investigate a variety of approaches, including computing the pseudoinverse by brute force, using the FFT, subsampling methods, implicit methods, and finally develop a novel iterative approach. We also evaluate the mapping performance of eleven interpolation kernels; surprisingly, common kernels such as bilinear, exponential, Gaussian, and sinc, performed only moderately well. This comprehensive study greatly reduces the computational bottleneck and guesswork of pseudo-inverse algorithms, making possible the application of reduced-state filters to global problems at state-of-the-art resolution.