Stable computation of analytical expressions for the gravity and magnetic field response of distant polyhedral targets

Though implemented through different evaluation strategies, the analytical formulation of the polyhedral gravity and magnetic field response involves some decomposition of the primary volume integrals into a summation of integrals defined over individual polyhedral segments. The existing closed solutions for the latter lead to expressions that relate the computation point to each vertex defining the known polyhedral source. When the distance between the computation point and the polyhedral distribution increases significantly then the numerical evaluation of the analytical formulas fails to produce correct significant digits, due to destructive cancellation in summing that is large compared to the correct result. The present contribution proposes a concept as a solution to this well—documented numerical instability problem. Thereby we prove that the analytical cancellation of the dominant terms before the computation is crucial to decrease the order of complexity to an optimal numerical limit. Our investigations lead to an alternative vectorial method for the evaluation of the gravity anomaly of a polyhedral target. The method is compared to the existing analytical formulation of the vectorial method for different target sizes and distances of the computation point. The performed tests demonstrate the numerical stability of the new approach as well as its superiority in representing gravity anomalies at the \(10^{-11}\) ${10}^{-11}$ level for distances beyond 100 m.