A spectral-domain approach for gravity forward modelling of 2D bodies

We provide the spectral-domain solutions for the gravity forward modelling of a 2D body with constant, polynomial and exponential density distributions, including the gravitational attraction and its arbitrary-order derivatives. The gravity effects may be directly obtained from the derivatives of a constructed scalar quantity and the surface integral of the density function over the 2D body. Then, the surface integrals of the spectral expansion coefficients of the scalar and the density function are converted to the line integrals along the boundary of the mass body using the Gauss divergence theorem and thus can be evaluated by numerical integration. The surface integrals for the exponential density model are expressed as the infinite expansions of line integrals and converge fast. Approximating the mass body to a 2D polygon, the surface integrals can be evaluated by simple analytic formulas for the constant density model and by linear recursive relations for the polynomial density model. The numerical implementation shows that the spectral-domain algorithm of this paper can produce high accurate forward results, e.g. 13-16 digits achievable precision for the gravity vector. Although the spectral-domain method is only suitable for forward computation of the external gravity effects of the 2D body, it is numerically stable for arbitrary observation point outside the smallest enclosed circle. The closed-form solutions for the 2D body with constant or polynomial density distribution are high precision to evaluate the gravity anomaly at the point near or inside the body, but may be lower precision when the computation point moves away from the body. The spectral-domain algorithm can handle the polynomial and exponential density model in both horizontal and vertical directions.