Multiresolution optimal interpolation and statistical analysis of TOPEX/POSEIDON satellite altimetry

Abstract

Abstruct-A recently developed multiresolution estimation framework offers the possibility of highly efficient statistical analysis, interpolation, and smoothing of extremely large data sets in a multiscale fashion. This framework enjoys a number of advantages not shared by other statistically-based methods. In particular, the algorithms resulting from this framework have complexity that scales only linearly with problem size, yielding constant complexity load per grid point independent of problem size. Furthermore these algorithms directly provide interpolated estimates at multiple resolutions, accompanying error variance statistics of use in assessing resolutionlaccuracy tradeoffs and in detecting statistically significant anomalies, and maximum likelihood estimates of parameters such as spectral power law coefficients. Moreover, the efficiency of these algorithms is completely insensitive to irregularities in the sampling or spatial distribution of measurements and to heterogeneities in measurement errors or model parameters. For these reasons this approach has the potential of being an effective tool in a variety of remote sensing problems. In this paper, we demonstrate a realization of this potential by applying the multiresolution framework to a problem of considerable current interest-the interpolation and statistical analysis of ocean surface data from the TOPEXPOSEIDON altimeter.

Fractal Estimation using Models on Multiscale Trees

“…An example of such an application to the estimation of non-isotropic fractal parameters for a 2-D random field based on irregular, nonstationary data is given in [2].…”

A multiscale approach for estimating solute travel time distributions

“…This requirement re¯ects our desire to use a commonly accepted model of log conductivity variability, rather than any fundamental property of heterogeneous porous media. The performance advantage o ered by the multiscale approach would have been more dramatic if we had modeled the log conductivity ®eld as a 1af process, as was done with sea surface elevation data in [6], precipitation data in [16], and soil moisture data in [10]. In this case d 4 at each node, and the storage requirement and number of¯oating point operations are both only O(N f ).…”

“…This is the case when the tree describes a one-dimensional Gauss± Markov process, as in the example discussed above. It is also the case for certain two-dimensional random ®eld models, such as the 1af fractal model used in [6] to process satellite altimetry data. When the process is twodimensional Gauss±Markov, the state vector needs to include enough ®nest scale values to partition the nodes on the tree according to the multiscale Markov criterion cited above [4].…”

“…The second pass moves upward from the leaf nodes, merging estimates from the children of each node and performing a second update at each measurement node. The complete estimation algorithm is described in [3,6]. It is a generalization of the RauchTung-Striebel smoother used to estimate the states of time series models.…”

Realization of Multi-Scale Stochastic Process on Multiple Tree

“…They can represent a fractional Brownian mo tion (fBm) (3), 1-D wide-sense reciprocal process and 2-D Markov Random field (MRF) (6). Other appli cations include an efficient image processing algorithm (7), smoothing problem on remote sensing (8), and es timation of random data on geophysics (9) (10). The readers are suggested to follow the references therein for discussions on the merit and demerit of multi-scale system.…”