Krylov Subspace Algorithms for Space-time Oceanography Data Assimilation

A challenging problem in many remote sensing applications is the assimilation of large quatities of sparse and irregularly sampled space-time data into a dynamic model. Kalman filtering, while providing a conceptual solution to such problems, cannot typically be applied exactly due to the extreme dimensionality of such assimilation problems. Particularly challenging is one of the most critical aspects of the filter, namely, computation of the error covariances. These are of interest both in their own right and on the computation of the gain matrix for assimilating new data. In other work, we have developed a method for static estimation problems that allows efficient computation of both estimates and error statistics. This method is based on the conjugate gradient algorithm, which generates a sequence of estimates on so-called Krylov subspaces of increasing dimension. The principal novelty of our approach is that we have been able to use the conjugate search directions generated by this algorithm to produce a sequence of increasingly accurate approximations to the error covariances. In this paper, we extend this machinery to space-time problems by developing methods for propagating estimates and error statistics through both temporal prediction and measurement assimilation. We demonstrate the power of this algorithm in the context of assimilating TOPEX/POSEIDON altimetry into a linearized Rossby wave dynamic model.