Surveys in Geophysics

, Volume 38, Issue 2, pp 479–502 | Cite as

Gravity Anomalies of Arbitrary 3D Polyhedral Bodies with Horizontal and Vertical Mass Contrasts

  • Zhengyong Ren
  • Chaojian Chen
  • Kejia Pan
  • Thomas Kalscheuer
  • Hansruedi Maurer
  • Jingtian Tang


During the last 15 years, more attention has been paid to derive analytic formulae for the gravitational potential and field of polyhedral mass bodies with complicated polynomial density contrasts, because such formulae can be more suitable to approximate the true mass density variations of the earth (e.g., sedimentary basins and bedrock topography) than methods that use finer volume discretization and constant density contrasts. In this study, we derive analytic formulae for gravity anomalies of arbitrary polyhedral bodies with complicated polynomial density contrasts in 3D space. The anomalous mass density is allowed to vary in both horizontal and vertical directions in a polynomial form of \(\lambda =ax^m+by^n+cz^t\), where mnt are nonnegative integers and abc are coefficients of mass density. First, the singular volume integrals of the gravity anomalies are transformed to regular or weakly singular surface integrals over each polygon of the polyhedral body. Then, in terms of the derived singularity-free analytic formulae of these surface integrals, singularity-free analytic formulae for gravity anomalies of arbitrary polyhedral bodies with horizontal and vertical polynomial density contrasts are obtained. For an arbitrary polyhedron, we successfully derived analytic formulae of the gravity potential and the gravity field in the case of \(m\le 1\), \(n\le 1\), \(t\le 1\), and an analytic formula of the gravity potential in the case of \(m=n=t=2\). For a rectangular prism, we derive an analytic formula of the gravity potential for \(m\le 3\), \(n\le 3\) and \(t\le 3\) and closed forms of the gravity field are presented for \(m\le 1\), \(n\le 1\) and \(t\le 4\). Besides generalizing previously published closed-form solutions for cases of constant and linear mass density contrasts to higher polynomial order, to our best knowledge, this is the first time that closed-form solutions are presented for the gravitational potential of a general polyhedral body with quadratic density contrast in all spatial directions and for the vertical gravitational field of a prismatic body with quartic density contrast along the vertical direction. To verify our new analytic formulae, a prismatic model with depth-dependent polynomial density contrast and a polyhedral body in the form of a triangular prism with constant contrast are tested. Excellent agreements between results of published analytic formulae and our results are achieved. Our new analytic formulae are useful tools to compute gravity anomalies of complicated mass density contrasts in the earth, when the observation sites are close to the surface or within mass bodies.


Gravity Singularity-free Polyhedral body Prism Horizontal and vertical mass contrasts 



This study was supported by Grants from the National Basic Research Program of China (973-2015CB060200), the Project of Innovation-driven Plan in Central South University (2016CX005), the National Science Fundation of China (41574120, 41474103, 41204082), the State High-Tech Development Plan of China (2014AA06A602), and an award for outstanding young scientists by Central South University (Lieying program 2013).


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment MonitoringMinistry of Education (Central South University)ChangshaChina
  2. 2.School of Geosciences and Info-PhysicsCentral South UniversityChangshaChina
  3. 3.School of Mathematics and StatisticsCentral South UniversityChangshaChina
  4. 4.Department of Earth SciencesUppsala UniversityUppsalaSweden
  5. 5.Department of Earth Sciences, Institute of GeophysicsETH ZurichZurichSwitzerland

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