# Fast inversion of gravity data using the symmetric successive over-relaxation (SSOR) preconditioned conjugate gradient algorithm

Zhaohai Meng^{1}

^{4}Fengting Li

^{2}Xuechun Xu

^{1}Danian Huang

^{3}Dailei Zhang

^{3}

^{1} College of Earth Sciences, Jilin University, Changchun, Jilin 130021, China.

^{2} College of Instrumentation and Electrical Engineering, Jilin University, Changchun, Jilin 130021, China.

^{3} College of GeoExploration Science and Technology, Jilin University, Changchun, Jilin 130021, China.

^{4} Corresponding author. Email: 526468457@qq.com

*Exploration Geophysics* 48(3) 294-304 https://doi.org/10.1071/EG15041

Submitted: 7 May 2015 Accepted: 13 January 2016 Published: 16 February 2016

## Abstract

The subsurface three-dimensional (3D) model of density distribution is obtained by solving an under-determined linear equation that is established by gravity data. Here, we describe a new fast gravity inversion method to recover a 3D density model from gravity data. The subsurface will be divided into a large number of rectangular blocks, each with an unknown constant density. The gravity inversion method introduces a stabiliser model norm with a depth weighting function to produce smooth models. The depth weighting function is combined with the model norm to counteract the skin effect of the gravity potential field. As the numbers of density model parameters is NZ (the number of layers in the vertical subsurface domain) times greater than the observed gravity data parameters, the inverse density parameter is larger than the observed gravity data parameters. Solving the full set of gravity inversion equations is very time-consuming, and applying a new algorithm to estimate gravity inversion can significantly reduce the number of iterations and the computational time. In this paper, a new symmetric successive over-relaxation (SSOR) iterative conjugate gradient (CG) method is shown to be an appropriate algorithm to solve this Tikhonov cost function (gravity inversion equation). The new, faster method is applied on Gaussian noise-contaminated synthetic data to demonstrate its suitability for 3D gravity inversion. To demonstrate the performance of the new algorithm on actual gravity data, we provide a case study that includes ground-based measurement of residual Bouguer gravity anomalies over the Humble salt dome near Houston, Gulf Coast Basin, off the shore of Louisiana. A 3D distribution of salt rock concentration is used to evaluate the inversion results recovered by the new SSOR iterative method. In the test model, the density values in the constructed model coincide with the known location and depth of the salt dome.

**Key words:** conjugate gradient, depth of the weighting function, gravity inversion, symmetric successive over-relaxation.

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