Gravity field forward modelling using tesseroids accelerated by Taylor series expansion and symmetry relations

Abstract

SUMMARY In this study, we developed a new method that can significantly accelerate the forward modelling of gravity fields generated by large-scale tesseroids while keeping the computational accuracy as high as possible. The cost of the high efficiency is that the method only works under the assumptions that (1) all tesseroids in the same latitude band have the same horizontal dimension, (2) the computation points are located at the same surface level and aligned with the horizontal centres of tesseroids and (3) each tesseroid has a constant or linearly varying density. The new method first integrates the kernel function of the Newton’s volume integral analytically in the radial direction to eliminate its dependence on the vertical dimension of the tesseroid, and then expands the integrated kernel function into a Taylor series up to a certain order. Because the Taylor series expansion term of the integrated kernel function is an odd or even function of the difference between the longitudes of the tesseroid and computation point, there exist shifting or swapping symmetry relations among the gravity field of tesseroids. Consequently, the shifting or swapping symmetry is extended to the tesseroids with unequal vertical dimensions. Numerical experiments using the spherical shell model are conducted to verify the effectiveness of the new method. The results show that the computational speed of the new method is about 30 times faster than that of the traditional method, which employs the Gauss–Legendre quadrature rule and a 2-D adaptive subdivision approach, while keeping almost the same computational accuracy. When applying the new method to an ice shell with unequal thicknesses, the results reveal that the relative errors of calculating V, Vz and Vzz are smaller than 10−8, 10−6 and 10−4, respectively if the Taylor series expansion is truncated at order 4, while the computational time consumed by the new method is about 7 times less than that of the traditional method. Finally, the influence of the truncation order on the computational accuracy and the strategies for dividing the latitude band into several parts to further improve the accuracy are discussed.