On Discrete Schemes in Downward Continuation of Gravity

Two discretization schemes of the Poisson integral have been proposed to date. Although they are mathematically solvable, they produce different gravity on the geoid for the same input data. This discrepancy arises because of different discretization techniques of the Poisson kernel; still, this problem has not received adequate attention. The question is whether the system of the discretization is reasonable. Methods to discretize the Poisson integral are investigated in this study. For this purpose, a single mean scheme is presented to evaluate numerically the Poisson integral. A comparison between the point and mean schemes shows that, for a limit topographical grid size, the point discretization scheme results in a serious theoretical problem since it greatly underestimates gravity on the geoid, and even gives incorrect results for extreme cases. A careful construction of the coefficient matrix for the discrete system is much more important than using point gravity as input.