Geodesy and Mathematics: Interactions, Acquisitions, and Open Problems
The paper highlights arguments that, coming from Mathematics, have fostered the advancement of Geodesy, as well as those that, generated by geodetic problems, have contributed to the enhancement of different branches in Mathematics. Furthermore, not only
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Abstract
The paper highlights arguments that, coming from Mathematics, have fostered the advancement of Geodesy, as well as those that, generated by geodetic problems, have contributed to the enhancement of different branches in Mathematics. Furthermore, not only examples of success are examined, but also open questions that can constitute stimulating challenges for geodesists and mathematicians. Keywords
Collocation Generalized random fields Geodetic boundary Geodetic inverse Ill-posed problems Integer estimation Multiscale mollifier Multiscale tikhonov and truncated singular value frequency regularization Space regularization Value problems
There is no branch of Mathematics, however abstract, which may not some day be applied to phenomena of the real world. Nikolai Ivanovich Lobachevsky
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Introduction
The paper highlights arguments that, coming from Mathematics, have fostered the advancement of Geodesy, as well as those that, generated by geodetic problems, have contributed to the enhancement of different branches in Mathematics. Furthermore, not only examples of success are examined, but also open questions that can constitute stimulating challenges for geodesists and mathematicians. We perform a general overview, without any pretence of completeness, of areas like geometry of the gravity field (GF), boundary value problems (BVP) for the Laplace operator, Runge approximation (RA), probability theory (in particular, Generalized Random Fields), and statistics (in partic-
W. Freeden University of Kaiserslautern, Kaiserslautern, Germany e-mail: [email protected] F. Sansò () Politecnico di Milano, DICA, Milan, Italy e-mail: [email protected]
ular integer parameters estimation and rank deficient problems). In Sects. 6–8 we turn the attention to novel applications to Geodesy in the context of multiscale approximation (MA). In fact, multiscale reconstruction and decorrelation methods are a research field originated in geophysics for, e.g., earthquake modeling some decades ago, in which today’s Geodesy and Mathematics show mutual influences, especially on the subject of spectral and space data sampling. Then we focus the attention on the inverse problems of Geodesy and their regularization strategies. Two examples are studied in more detail: Downward continuation of gravitational information to the Earth’s surface via satellite gravitational gradiometry (SGG) is seen to be adequately realized in the tensorial frequency framework of non-bandlimited Tykhonov and bandlimited truncated singular value regularization. The inverse gravimetry (IG) problem is shown to be appropriately regularized by use of space multiscale mollifiers, to detect fine particulars of geological relevance. Finally, the authors want to stress again that the choice of arguments by no means can cover the whole area; it rather reflects the background of the authors and has to be taken as illustrative of a general process of interaction between sciences. In addition, neither extreme depth to explain all facets of the ge
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