Mathematical Frontier Research as Conceived in Space-Geodesy

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The major new results of this research project can be briefly summarized as follows : (i) We systematically discuss the estimation and hypothesis testing problems in a mixed integer linear model, which directly motivated by space geodesy and is the basis for precise GPS positioning. We have first mathematically solved the problem of constructing Voronoi cells for an arbitrary positive definite matrix associated with mixed integer linear models by using interval mathematics to eliminate an infinite number of redundant linear constraints. We have also provided spherical and ellipsoidal enclosures to bound the Voronoi cell. By re-formulating the problem of bounding the Voronoi cell as a mathematical programming model, we have successfully found the tightest possible bounds of the Voronoi cell from inside and outside, respectively. As a result, we are able to find the tightest possible probabilistic bounds. The error bound has been shown to perform much better than the one proposed by Shan … More non (1959). In fact, the hypothesis testing problem is, for the first time, addressed in this research project and published in IEEE Transactions on Information Theory. By combining GPS with acoustic technology, we have first shown that ocean-bottom crustal deformation could be measured at the accuracy of sub-centimeters in all the three components. This is also the first theoretical demonstration that ocean bottom crustal deformation could be measured as easily and routinely as on land ; (ii) We have proposed a new robust estimation method, which is named as sign-constrained robust least squares. The method has been shown to be able to tolerate up to 50 percent of outliers, and even much more than 50 percent when used iteratively. As a by-product, we have proved that neither the plus sign nor the square is responsible for the non-robustness of least squares, as otherwise has been claimed in the literature on robust estimation. We have further proved that both Huber's M-estimation and weighted L1 method are not robust if measurements are of different accuracy ; (iii) We have first proposed the bias-corrected estimation technique to estimate variance components in linear inverse ill-posed models and applied it to satellite gravity fields, together with our multiple parameter regularization method. We have also proved that not all the variance covariance components are estimable. Less