Analytical Expressions of Gravitational field and its Vertical Gradient generated by 2-D and 3-D frustum with a Quadratic Density distribution

Summary

Advances in high-precision gravimetric instrumentation have significantly improved the quality of gravity data and increased the need for refined theoretical modeling of local gravitational fields. Although analytical expressions for vertical gravitation (VG) of variable density two-dimensional (2-D) frustums has been studied earlier, these formulas are relatively complex, and much less attention has been paid to three-dimensional (3-D) frustums. This study derives analytical solutions for VG and vertical gradient of the vertical component of gravitation (VGG) of 2-D and 3-D frustums with a quadratic density function in the z-direction. Validation through comparison with numerical methods and previous computational results confirms the correctness and effectiveness of the proposed algorithm. Analysis of the derived formulas enables the quantitative determination of the relationship between the number of VGG extreme points (possibly three or five), the frustum size, and the observation points location. Based on the multibeam data and gravity model, the RMS misfits are 4.1, 3.5, and 3.1 mGal for the three Necker Ridge profiles, and approximately 10.5 Eötvös for the Horizon Tablemount VGG profile. Additionally, we apply this formula to calculate VGG of Horizon Tablemount, and successfully verify the existence of five extreme points. This characteristic demonstrates that the occurrence of significantly smaller local VGG is an inherent property of the VGG itself, independent of terrain undulation and specific density distribution patterns. This study provides effective computational tools and a theoretical framework for high-precision gravitation modeling.