A multiresolution optimal interpolation scheme is described and used to map the sea level anomaly of the Mediterranean Sea based on TOPEX/Poseidon and ERS-1 data. The principal advantages of the multiresolution scheme are its high computational efficiency, the requirement for explicit statistical models for the oceanographic signal and the measurement errors, and the production of error variances for all estimates at multiple scales. A set of MATLAB-callable routines that implement the multiresolution scheme have been made available via anonymous FTP. The oceanographic signal is here modeled as a stationary 1/k process, where k is the horizontal wavenumber. Measurement noise is modeled as the sum of two separate random processes: a Gaussian white noise process and a correlated process of a low wavenumber representing the uncertainties in the orbital position of the satellite and in the atmospheric load corrections. The efficiency of the multiresolution scheme allowed the testing of more than 16 000 sets of hypothesized statistical prior model parameters to determine the most likely parameters. Mapping results with and without low wavenumber error corrections are presented and compared.

The successful application of a recently developed treebased multiscale estimation framework [l] to large-scale static estimation problems has proven the statistical flexibility and the computational efficiency of the framework. However, the estimation of processes evolving in time remains a considerable challenge. In this paper, we address this challenge by investigating multiscale models for the steady-state estimation of dynamic systems. In particular, we build such models for 1-D and 2-D heat diffusion processes, making use of a canonical correlations realization technique.

Conventional optimal estimation algorithms for distributed parameter systems have been limited due to their computational complexity. In this paper, we consider an alternative modeling framework recently developed for large-scale static estimation problems and extend this methodology to dynamic estimation. Rather than propagate estimation error statistics in conventional recursive estimation algorithms, we propagate a more compact multiscale model for the errors. In the context of 1-D di!usion which we use to illustrate the development of our algorithm, for a discrete-space process of N points the resulting multiscale estimator achieves O(N log N) computational complexity (per time step) with near-optimal performance as compared to the O(N) complexity of the standard Kalman "lter.

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