We can easily convert the gravity problem with discrete objects of known position and unknown density into a more realistic gravity inversion problem by considering a cubic region of the Earth, with one face at the surface, divided into 8 smaller sub-cubes of equal dimension and unknown density (schematic above). Each sub-cube is 1 1 1 units in dimension with the lower left corner of the cube being at position (0, 0, 0). The gravitational anomaly (d) is measured at n locations on the surface of the Earth (where z = 2 in the coordinates). The gravitational anomaly at each location is simply the sum of the gravitational influences from the center of mass of each sub-cube, given by d(i) = Gp(j)/ r^2(ij) where di are the measured gravitational anomalies, p(j) are the density of each sub-cube, r(ij) is the distance from measurement location i to the location of the center of each sub-cube j, and G is a gravitational constant (assumed to be G = 1 for this problem)

Write out the matrix inverse problem mathematically that you would need to solve to determine the density of the Earth in each sub-cube, if you measured the gravitational acceleration at 8 locations at the surface of the Earth