Abstract
During the last 20 years, geophysicists have developed great interest in using gravity gradient tensor signals to study bodies of anomalous density in the Earth. Deriving exact solutions of the gravity gradient tensor signals has become a dominating task in exploration geophysics or geodetic fields. In this study, we developed a compact and simple framework to derive exact solutions of gravity gradient tensor measurements for polyhedral bodies, in which the density contrast is represented by a general polynomial function. The polynomial mass contrast can continuously vary in both horizontal and vertical directions. In our framework, the original three-dimensional volume integral of gravity gradient tensor signals is transformed into a set of one-dimensional line integrals along edges of the polyhedral body by sequentially invoking the volume and surface gradient (divergence) theorems. In terms of an orthogonal local coordinate system defined on these edges, exact solutions are derived for these line integrals. We successfully derived a set of unified exact solutions of gravity gradient tensors for constant, linear, quadratic and cubic polynomial orders. The exact solutions for constant and linear cases cover all previously published vertex-type exact solutions of the gravity gradient tensor for a polygonal body, though the associated algorithms may differ in numerical stability. In addition, to our best knowledge, it is the first time that exact solutions of gravity gradient tensor signals are derived for a polyhedral body with a polynomial mass contrast of order higher than one (that is quadratic and cubic orders). Three synthetic models (a prismatic body with depth-dependent density contrasts, an irregular polyhedron with linear density contrast and a tetrahedral body with horizontally and vertically varying density contrasts) are used to verify the correctness and the efficiency of our newly developed closed-form solutions. Excellent agreements are obtained between our solutions and other published exact solutions. In addition, stability tests are performed to demonstrate that our exact solutions can safely be used to detect shallow subsurface targets.
Highlights
Gravity exploration methods try to identify the anomalous mass bodies in the Earth (Blakely 1996)
When the observation site is located inside the mass body, a local spherical coordinate system can be introduced at the observation site, so that a factor of distance squared ( R2 ) would be introduced, which can further weaken the singularity appearing in the integrands of the gravity signals (Jin 2002)
From the mathematical point of view, the gravity field can be evaluated without mathematical singularities, as O(R−2 × R2) = O(1), but the mathematical singularities always remain in the gravity gradient tensor formulation, when observation sites approach the causative body, that is O(R−3 × R2) = O(R−1)
Summary
Gravity exploration methods try to identify the anomalous mass bodies in the Earth (Blakely 1996). Pohánka (1998), Holstein (2003), Hansen (1999), D’Urso (2014b) and Ren et al (2017a) have successfully derived closed-form solutions for the gravity field of a polyhedral body, in which the mass contrast linearly varies in both horizontal and vertical directions. Only Holstein (2003) and D’Urso (2014b) have successfully derived closed-form solutions for gravity gradient tensor of a polyhedral body, in which the mass contrast varies linearly in both horizontal and vertical directions. To the best of our knowledge, closed-form solutions for gravity gradient tensor signals of a polygonal body with high-order polynomial mass contrasts (such as quadratic and cubic orders) varying in both horizontal and vertical directions have not been previously reported. Results from high-order Gaussian quadrature are used as reference solutions for the tetrahedron model
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