Applicazioni di metodi computazionali in geodesia

Computing the gravitational effects due to given mass distributions is a fundamental problem in the applications of geophysics and geodesy. Potential Theory can only be partially used in geophysics: in gravity, magnetism, methods of calculating electromagnetic, electrical heat flux, and computing the flux of fluids. Therefore, Potential Theory, with the use of complex mathematical tools, constitutes the basis for the solution of various geophysical problems, managing in fact to be almost indispensable to better understand some geophysical data. Potential theory addresses the mathematics of equilibrium and, in particular, the study of harmonic functions, given their fundamental role in equilibrium problems in a homogeneous medium. The advent of Inverse Theory has revolutionized the whole procedure for interpreting geophysical data. This was made possible thanks to the very rapid development of computer science, technology that has improved the computing software and the numerous methods used in mathematical modelling. In the thesis work I will present new expressions for the gravitational potential that involve alternative and less expensive computational capabilities than those reported in the specialized literature. I will show that the singularities that can influence the computation of the effects of gravity (potential, gravity and tensor gradient fields) can be systematically addressed by invoking the theory of distribution with suitable differential calculus formulas. The general approach will be led with reference to the case of models of polyhedral bodies, regular or not, having either a constant or a depth–relative mass density. The validated analytical formulas have been fully confirmed by applications with Matlab® programs, coded and carefully tested by calculating the effects of gravity induced by attractive bodies positioned in arbitrary observation points. The formulas illustrated in the thesis have been numerically checked with the alternative ones derived on the basis of different approaches, already established in scientific literature, intensively and repeatedly testing the effects of gravity induced by real attractive bodies with arbitrarily assigned density variations. The efficiency of the proposed formulas lies in the ability to correctly evaluate the singularities that arise in cases where the attracted points occupy different positions of the attractive mass. The research activity was further implemented with the introduction of analytical formulations based on the approximation of the ground masses with shapes of vertical prisms considered as prisms with polynomial density. The gravitational anomaly associated with a polyhedral body of arbitrary geometric shape and with different values of polynomial density in both horizontal and vertical directions is analytically evaluated. The gravity anomaly is evaluated at an arbitrary point that does not necessarily coincide with the origin of the reference system in which the assigned density function is located. It has been established that the density contrast is comparable to exponential polynomial functions of higher order than the third. By invoking the recent results of Potential Theory, the solutions have proved to be devoid of singularities and are expressed as the sums of algebraic quantities that depend only on the vertices of the polyhedron and on the density function of the polynomial. The accuracy, robustness and effectiveness of the approach can be demonstrated with numerical calculations on the basis of examples derived from the existing literature.