Exact solutions of the vertical gravitational anomaly for a polyhedral prism with vertical polynomial density contrast of arbitrary orders. Issue 3 (21st June 2018) - Chen, Chaojian; Ren, Zhengyong; Pan, Kejia; Tang, Jingtian; Kalscheuer, Thomas; Maurer, Hansruedi; Sun, Ya; Li, Yang
SUMMARY: We present general closed-form solutions for the vertical gravitational anomaly caused by a polyhedral prism with mass density contrast varying with depth. Our equations are the first ones to implement a polynomial vertical mass density contrast of arbitrary order. Singularities in the gravity field which arise when the observation site is close to or in the anomalous polyhedral prism are removed in our analytic expressions. Therefore, the observation site can be located outside, on the faces of or inside the anomalous mass bodies. A simple prismatic body of anomalous density is adopted to test the accuracy of our newly developed closed-form solution. Cases of constant, linear, quadratic, cubic and quartic polynomial orders of mass density contrast are tested. For cases of constant, linear, quadratic and cubic polynomial orders, the relative errors between our results and other published exact solutions are less than 10 −11 %. For the case of quartic polynomial order, relative errors less than 10 −10 % are obtained between our solutions and those computed by a high-order Gaussian quadrature rule (512 × 512 × 512=134 217 728 quadrature points), where our new analytic solution needs significantly less computational time (0.0009 versus 31.106 s). These numerical experiments not only verified the accuracy of our new formula but also demonstrated their potential in computing exact gravity anomalies for complicated mass density distributions in the Earth.