Efficient 3D large-scale forward-modeling and inversion of gravitational fields in spherical coordinates with application to lunar mascons

Guangdong Zhao, Bo Chen, Leonardo Uieda, Jianxin Liu, Mikhail K. Kaban, Longwei Chen, Rongwen Guo



This new collaboration that came about in an unexpected way. I reviewed an initial version of this paper and it ended up being rejected by the journal. Originally, the paper focused solely on the forward modeling aspects and had no field data application (which was not possible without the inversion step). After the rejection, the authors reached out to me (I sign all of my reviews) and kindly asked if I wanted to help improve the paper further. We worked on this for the better part of a year, adding the inversion and lunar mascon application. I'm very happy the final result!

The paper introduces a method for accelerating 3D gravity inversion using tesseroids. It's fast due to the clever use of symmetry relations when computing the sensitivity/Jacobian matrix. This requires data laid out on a regular grid and a regular tesseroid mesh that aligns with the data grid. So there is a sacrifice of flexibility for performance. Nonetheless, the results are impressive.


An efficient forward modeling algorithm for calculation of gravitational fields in spherical coordinates is developed for 3D large‐scale gravity inversion problems. 3D Gauss‐Legendre quadrature (GLQ) is used to calculate the gravitational fields of mass distributions discretized into tesseroids. Equivalence relations in the kernel matrix of the forward‐modeling are exploited to decrease storage and computation time. The numerical tests demonstrate that the computation time of the proposed algorithm is reduced by approximately two orders of magnitude, and the memory requirement is reduced by N'λ times compared with the traditional GLQ method, where N'λ is the number of the model elements in the longitudinal direction. These significant improvements in computational efficiency and storage make it possible to calculate and store the dense Jacobian matrix in 3D large‐scale gravity inversions. The equivalence relations can be applied to the Taylor series method or combined with the adaptive discretization to ensure high accuracy. To further illustrate the capability of the algorithm, we present a regional synthetic example. The inverted results show density distributions consistent with the actual model. The computation took about 6.3 hours and 0.88 GB of memory compared with about a dozen days and 245.86 GB for the traditional 3D GLQ method. Finally, the proposed algorithm is applied to the gravity field derived from the latest lunar gravity model GL1500E. 3D density distributions of the Imbrium and Serenitatis basins are obtained, and high‐density bodies are found at the depths 10‐60 km, likely indicating a significant uplift of the high‐density mantle beneath the two mascon basins.


Zhao, G., Chen, B., Uieda, L., Liu, J., Kaban, M. K., Chen, L., & Guo, R. (2019). Efficient 3D large-scale forward-modeling and inversion of gravitational fields in spherical coordinates with application to lunar mascons. Journal of Geophysical Research: Solid Earth. doi:10.1029/2019JB017691.

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