Gravitational attraction of a vertical pyramid model of flat top-and-bottom with depth-wise parabolic density variation

Introduction

Computation of theoretical gravity response of an arbitrarily-shaped 3D target body has been addressed by many authors. Different types of gravity models, like right rectangular prism, polygonal lamina and polyhedral models with constant and different depth-wise density functions have been used by several authors (Talwani and Ewing 1960; Nagy 1966; Banerjee and Das Gupta 1977; Chai and Hinze 1988; García-Abdeslem 1992, 2005; Rao et al.1993, 1995; Pohanka 1988, 1998; Hansen 1999; Tsoulis 2000; Chakravarthi et al.2002; Hamayun et al.2009; Martins et al.2010; Holstein et al.2013; Oliveira and Barbosa 2013; D’Urso 2014a, 2014b, 2015). Different types of density function, like exponential (Chai and Hinze 1988), parabolic (Rao et al. 1993) and hyperbolic density functions (Rao et al. 1995) have been used in sedimentary basin modelling. Chakravarthi et al. (2002) have proposed an analytical solution of right rectangular prism model with a depth-wise parabolic density contrast variation. Hansen (1999), Holstein (2003) and Hamayun et al. (2009) have used polyhedral model in gravity forward modelling with a depth-wise linear density variation.

A detailed study of singularities encountered in gravity forward modelling of different models have been addressed by Okabe (1979), Pohanka (1988), Kwok (1991), Petrović (1996), Tsoulis (2000), Tsoulis and Petrović (2001), Holstein (2002) and D’Urso (2013).

Starostenko (1978) has proposed an inhomogeneous vertical pyramid model with flat top-and-bottom and sloping sides with a depth-wise linear density variation. However, he was unable to derive a complete analytical expression for its gravity effect.

Here, we derive the complete gravity expression for the same pyramid model with a depth-wise parabolic density contrast function variation and illustrate its effectiveness through two synthetic examples after customary validation check of our forward problem solution. We also demonstrate the usefulness and effectiveness of our model in a case study (Chai and Hinze 1988; Chakravarthi et al. 2002).

Theory

Consider an isolated regular pyramid model with depth-wise parabolic density contrast, ABCDEFGH with flat top, ABCD and bottom surface, EFGH (figure 1a). The gravity effect (figure 1b) of such a model at any arbitrary point (x, y, z) in free-space is given by,

1

$$\begin{array}{lcr}{g}_{\text{pyramid}}\left(x,y,z\right)& =& \gamma \underset{\zeta =h_1}{\overset{h_2}{\int }}\underset{}{\overset{}{\int }}\end{array}$$