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 * Jingtian Tang

We have developed a new analytical expression for the magnetic-gradient tensor for polyhedrons with homogeneous magnetization vectors. Instead of performing the direct derivative on the closed-form solutions of the magnetic field, it is obtained by first transforming the volume integrals of the magnetic-field tensor into surface integrals over polyhedral facets, in terms of the gradient theorem. Second, the surface divergence theorem transforms the surface integrals over polyhedral facets into edge integrals and structure-simplified surface integrals. Third, we develop analytical expressions for these edge integrals and simplified surface integrals. We use a synthetic prismatic target to verify the accuracies of the new analytical expression. Excellent agreements are obtained between our results and those calculated by other published formulas. The new analytical expression of the magnetic-gradient tensor can play a fundamental role in advancing magnetic mineral explorations, environmental surveys, unexploded ordnance and submarine detection, aeromagnetic and marine magnetic surveys because more and more magnetic tensor data have been collected by magnetic-tensor gradiometry instruments.

A new singularity-free analytical formula has been developed for the gravity field of arbitrary 3D polyhedral mass bodies with horizontally and vertically varying density contrast using third-order polynomial functions. First, the observation sites are moved to the origin of the coordinate system. Then, the volume and surface integral theorems are invoked successively to transform the volume integrals into surface integrals over polygonal faces and into line integrals over the edges of the polyhedral mass bodies. Furthermore, singularity-free closed-form solutions are derived for these line integrals over the edges. Thus, the observation sites can be located inside, on, or outside the 3D distributions. A synthetic prismatic mass body is adopted to verify the accuracy and singularity-free property of our newly developed analytical expressions. Excellent agreements are obtained between our solutions and other published closed-form solutions with relative errors in the order of [Formula: see text] to [Formula: see text]. In addition, an octahedral model and a near-Earth asteroid model are used to verify the accuracy of the presented method for complicated target structures by comparing the results with those from a high-order Gaussian quadrature approach.

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Magnetization is a natural property of magnetic materials which is widely used to analyze paramagnetic data, locate unexploded ordnance, explore mineral deposits and gas resources, and image anomalous magnetic structures in the Earth's crust. An essential and challenging task is to compute accurate magnetic anomalies caused by realistic magnetic 3D structures. In this study, for the first time, we derive the analytical expressions of magnetic potential, magnetic field and magnetic gradient tensor for magnetic materials with 3D polyhedral shapes and magnetization vectors that may vary in the space following polynomial trends. The order of polynomials can vary from one to high positive integers in both horizontal and vertical directions. Synthetic and realistic deposit models are used to validate our analytical solutions. We release the open‐source code in C++.