Potential fields data modeling: new frontiers in forward and inverse problems

Since the '50, potential fields data modeling has played an important role in analyzing the density and magnetization distribution in Earth's subsurface for a wide variety of applications. Examples are the characterization of ore deposits and the assessment of geothermal and petroleum potential, which turned out to be key contributors for the economic and industrial development after World War II. Current modeling methods mainly rely on two popular parameterization approaches, either involving a discretization of target geological bodies by means of 2D to 2.75D horizontal prisms with polygonal vertical cross-section (polygon-based approach) or prismatic cells (prism-based approach). Despite the great endeavour made by scientists in recent decades, inversion methods based on these parameterization approaches still suffers from a limited ability to (i) realistically characterize the variability of density and magnetization expected in a study area and (ii) take into account the strong non-uniqueness affecting potential fields theory. The prism-based approach is used in linear deterministic inverse methods, which provide just one single solution, preventing uncertainty estimation and statistical analysis on the parameters we would like to characterize (i.e, density or magnetization). On the contrary, the polygon-based approach is almost exclusively exploited in a trial-and-error modeling strategy, leaving the potential to develop innovative inverse methods untapped. The reason is two-fold, namely (i) its strongly non-linear forward problem requires an efficient probabilistic inverse modeling methodology to solve the related inverse problem, and (ii) unpredictable cross-intersections between polygonal bodies during inversion represent a challenging task to be tackled in order to achieve geologically plausible model solutions. The goal of this thesis is then to contribute to solving the critical issues outlined above, developing probabilistic inversion methodologies based on the polygon- and prism-based parameterization approaches aiming to help improving our capability to unravel the structure of the subsurface. Regarding the polygon-based parameterization strategy, at first a deep review of its mathematical framework has been performed, allowing us (i) to restore the validity of a recently criticized mathematical formulation for the 2D magnetic case, and (ii) to find an error sign in the derivation for the 2.75D magnetic case causing potentially wrong numerical results. Such preliminary phase allowed us to develop a methodology to independently or jointly invert gravity and magnetic data exploiting the Hamilton Monte Carlo approach, thanks to which collection of models allow researchers to appraise different geological scenarios and fully characterize uncertainties on the model parameters. Geological plausibility of results is ensured by automatic checks on the geometries of modelled bodies, which avoid unrealistic cross-intersections among them. Regarding the prism-based parameterization approach, the linear inversion method based on the probabilistic approach considers a discretization of target geological scenarios by prismatic bodies, arranged horizontally to cover it and finitely extended in the vertical direction, particularly suitable to model density and magnetization variability inside strata. Its strengths have been proven, for the magnetic case, in the characterization of the magnetization variability expected for the shallower volcanic unit of the Mt. Melbourne Volcanic Field (Northern Victoria Land, Antarctica), helping significantly us to unravel its poorly known inner geophysical architecture.