Representações harmônicas esférica e elipsoidal e correção bouguer para capa elipsoidal aplicada à análise de dados gravimétricos satelitais do estado da Bahia

Abstract

Harmonic modeling is used for gravimetric field representation of massive bodies, such as the Earth. The solution of Laplace’s equation based on spherical coordinates, the spherical harmonic series, is widely used in bibliographies on physical geodesy. Taking into account the Earth’s ellipsoidal shape, it would be more adequate to use the solution of the Laplace equation via ellipsoidal coordinates, reaching the ellipsoidal harmonic series. In the present study, the harmonic series - spherical and ellipsoidal - are applied to gravimetric satellite data over the state of Bahia. For the calculation of harmonic coefficients of spherical and ellipsoidal harmonic modeling, the ordinary and weighted least squares methods are used. For the weighting matrices of the weighted least square method, diagonal matrices with principal diagonal values are used based on the functions: related to the inverse of the residue minus the variance and the second related to the inverse of the geometric means of the residue minus the variance. Bar charts provide evaluative metrics for comparing the different models. Theoretical gravity at the level of the Earth’s reference ellipsoid is used for gravimetric corrections and its formula can be conceived through the derivative of the gravitational potential added to the centrifugal potential of rotation of the Earth. The present work presents two theoretical gravity maps calculated through the spherical and ellipsoidal harmonics. After harmonic analysis, it will present the gravimetric corrections, mainly the Bouguer correction. Bouguer’s simple correction, commonly used, is based on approximating the subsurface to an infinite horizontal plate of thickness equal to the elevation of the observation point, but this conception lacks realism to the terrestrial format. Based on the Bullard B correction for a spherical cap, the thesis proposes a new Bouguer correction for an ellipsoidal cap establishing an appropriate approximation for the terrestrial shape. In this way, the Bouguer corrections for the infinite plate, spherical cap and ellipsoidal cap are computed and arranged in maps for the state of Bahia. To support the geological interpretation, a gravimetric inversion was performed to estimate the depth of the Moho through the data processed with the ellipsoidal harmonics. The obtained results allowed to sketch three east-west geological profiles for the state of Bahia with interpretations of the geological blocks together with a model of the crust and the mantle.