Efficient Evaluation of Ellipsoidal Harmonics for Potential Modeling
This work provides a more efficient numerical method for computing ellipsoidal harmonics, which is relevant for potential modeling in geophysics and related fields.
The paper addresses the challenge of computing ellipsoidal normalization constants by applying tanh-sinh quadrature to decomposed singular integrals, proving optimality and demonstrating effectiveness compared to other quadrature methods.
Ellipsoidal harmonics are a useful generalization of spherical harmonics but present additional numerical challenges. One such challenge is in computing ellipsoidal normalization constants which require approximating a singular integral. In this paper, we present results for approximating normalization constants using a well-known decomposition and applying tanh-sinh quadrature to the resulting integrals. Tanh-sinh has been shown to be an effective quadrature scheme for a certain subset of singular integrands. To support our numerical results, we prove that the decomposed integrands lie in the space of functions where tanh-sinh is optimal and compare our results to a variety of similar change-of-variable quadratures.