Calculation of gravimetric density caused by three-dimensional structure

The difficulty with calculating the gravitational effect of three-dimensional structure is not in the calculation itself, but in the description of the structure such that it fills all space and does not contain voids. We have solved this problem by using contour maps of subsurface boundaries between media of different density. The contour lines are digitized, and the digitized data are used to construct a function that is then evaluated over a fine grid surrounding the borehole. The algorithm used to construct the approximating function is a modification of Hardy's multiquadric method for interpolating scattered data. The edges of the grid are smoothed, and then extended to distances great enough to be effectively infinite in the gravity calculation. The gravity calculation uses an algorithm by Banerjee and Gupta for the calculation of the gravitational effect of a right rectangular prism. For each rectangle of the grid, the depth of the corners is averaged and used for the depth of one end of the prism. At each grid rectangle there are two prisms: one with the top at the top surface and the bottom at the contact surface, the other with the top at the contact surface and the bottom at the bottom surface. Two contact surfaces (describing three layers) can be used, in which case a third prism, with its ends on the two contact surfaces, is calculated. Gravimetric density is then calculated from the vertical gradient of gravity.