Elliptical harmonic method for gravity forward modelling of 2D bodies

The elliptical harmonic method for gravity forward computation of two-dimensional (2D) bodies with constant, polynomial and exponential density distributions is presented. This paper gives the elliptical harmonic expansions for the gravity field of 2D bodies mainly including the gravitational attractions and gradients, and the conversions from the surface integral expressions of the elliptical harmonic coefficients and the surface integral of the density function into line integrals using Gauss divergence theorem for evaluation of the gravity field. Due to the requirements of the Gauss divergence and Stokes theorems that the boundary surface or curve of the body is piecewise smooth and the vector field as well as its first-order partial derivatives are continuous, the effects of the discontinuities of the elliptical coordinates on the integral conversion using Gauss divergence and Stokes theorems are considered. The integrands of the converted line integrals are the analytical expressions with respect to the coordinates of the internal point of the 2D body for the constant and polynomial density models, but are not elementary functions for the exponential density model that the integrands can be solved by approximating the density function with polynomial function. The numerical experiments with two tested bodies including a rectangular cylinder body with quadratic density contrast varying with depth and a 26-sided polygon body with quadratic density contrast varying in both horizontal and vertical directions show the convergence, accuracy and stability of the elliptical harmonic algorithm. Compared with the circular harmonic approach, the elliptical harmonic approach has smaller non-convergence region between the boundary of the 2D body and the reference ellipse and is convergent faster for most of the external observation points.