<p>Understanding the behavior of the first and second order derivatives of the gravitational potential is crucial for gravity related applications, especially when individual element contributions are considered. Kernel functions, entering the different gravity functional expressions, determine their magnitude and sign. They establish a link between gravity signal and modelled shape geometry, by quantifying the relative position between computation point and source mass. To test the spatial distributions of sign alterations in the computed signal, the analytical method of a right rectangular prism is implemented as an independent solution. Thus, absolute values of the potential derivatives are calculated using prismatic modelling and then the appropriate gravitational functional sign is applied based on the prismatic and point mass representation using kernel functions for six case scenarios. Specifically, a simple rectangular prism and five irregularly shaped distributions are considered, namely a melted ice layer of a glacier in the Austrian Alps and four simulated subsidence surfaces. The gravitational signal of the last irregular cases is computed by dividing the entire volume into smaller prismatic elements. The only observed differences refer to components <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$V_{xx} , V_{yy} ,V_{zz}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>V</mi> <mrow> <mi mathvariant="italic">yy</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>V</mi> <mrow> <mi mathvariant="italic">zz</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> for computation points located close to the sources, due to their inhomogeneous spatial distributions. The numerical investigations identify specific 3D geometric patterns for the first and second order derivatives of the potential, that are defined rigorously for the examined test cases.</p>

Kernel Function Analysis of Gravitational Potential Derivatives up to Second Order

期刊论文
发表日期：

2026年2月3日

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Dimitrios Tsoulis,
Georgia Gavriilidou

摘要

Understanding the behavior of the first and second order derivatives of the gravitational potential is crucial for gravity related applications, especially when individual element contributions are considered. Kernel functions, entering the different gravity functional expressions, determine their magnitude and sign. They establish a link between gravity signal and modelled shape geometry, by quantifying the relative position between computation point and source mass. To test the spatial distributions of sign alterations in the computed signal, the analytical method of a right rectangular prism is implemented as an independent solution. Thus, absolute values of the potential derivatives are calculated using prismatic modelling and then the appropriate gravitational functional sign is applied based on the prismatic and point mass representation using kernel functions for six case scenarios. Specifically, a simple rectangular prism and five irregularly shaped distributions are considered, namely a melted ice layer of a glacier in the Austrian Alps and four simulated subsidence surfaces. The gravitational signal of the last irregular cases is computed by dividing the entire volume into smaller prismatic elements. The only observed differences refer to components $V_{xx} , V_{yy} ,V_{zz}$ $V_{xx}, V_{yy}, V_{zz}$ for computation points located close to the sources, due to their inhomogeneous spatial distributions. The numerical investigations identify specific 3D geometric patterns for the first and second order derivatives of the potential, that are defined rigorously for the examined test cases.