This thesis has developed two new methodologies for processing and interpreting potential field data. The first one is the Polynomial Equivalent Layer, which is a new cost-effective method for processing large potential-field data sets via the equivalent-layer technique. In this approach, the equivalent layer is divided into a regular grid of equivalent-source windows. Inside each window, the physical-property distribution is described by a bivariate polynomial. While the classical approach directly estimates the physical-property distribution within the equivalent layer, the method Polynomial Equivalent Layer deals with a linear system of equations with dimension based on the total number of polynomial coefficients within all equivalent-source windows. This new method drastically reduces the number of parameters to be estimated in the inverse problem if compared with the classical approach. By comparing the total number of floating-point operations required to estimate an equivalent layer via our method with the classical approach, both formulated with Cholesky's decomposition, we verify that the computation time required for building the linear system and for solving the linear inverse problem can be reduced by as many as three and four orders of magnitude, respectively. Applications to both synthetic and real data show that our method performs the standard linear transformations of potential-field data accurately. The second new methodology developed in this thesis is a non-linear method for inverting gravity-gradient data to estimate the shape of an isolated 3-D geological body located in subsurface. This method assumes the knowledge about the depth to the top and density contrast of the source. The geologic body is approximated by an ensemble of vertically juxtaposed 3-D right prisms, each one with known thickness and density contrast. All prisms have a polygonal horizontal cross-section whose vertices are equally spaced from 0o to 360o and have their horizontal locations described in polar coordinates referred to an origin inside the polygon. The method recovers the geometry of the geological body by estimating the radii of all vertices and the horizontal Cartesian coordinates of all origins defining the horizontal cross-sections of all prisms. This problem is formulated as a constrained non-linear optimization and we also used a preconditioning strategy in order to improve the convergence. Although the proposed inverse method can obtain a stable estimate that fits the observed data, different estimates with different maximum depths can produce equally acceptable data fits. To deal with this ambiguity, we produced a set of estimates with different maximum depths and used a criterion based on the L1 norm of the residuals and the estimated volume for choosing the estimate with optimum maximum depth. This criterion allows the analysis of the in-depth resolution of the observed gravity-gradient data. We confirmed the ability of our method to recover the source geometry entirely if the data have sufficient in-depth resolution. If not, our method is able to recover only the upper part of the source. By inverting the data from a survey over the Vinton salt dome (Louisiana, USA) with a density contrast of 0.55 g/cm3, we estimated a massive cap rock whose maximum depth attains 460 ± 10 m and its shallowest portion is elongated in the northeast-southwest direction, according to the direction of the main geological fault in the study area.
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