Three-dimensional inversion for sparse potential data using first-order system least squares with application to gravity anomalies in Western Queensland

Abstract

We present an inversion algorithm tailored for point gravity data. As the data is from multiple surveys, it is inconsistent with regards to spacing and accuracy. An algorithm design objective is the exact placement of gravity observations to ensure no interpolation of the data is needed prior to any inversion. This is accommodated by discretization using an unstructured tetrahedral finite element mesh for both gravity and density with mesh nodes located at all observation points and a first-order system least-squares (FOSLS) formulation for the gravity modelling equations. Regularization follows the Bayesian framework where we use a differential operator approximation of an exponential co–variance kernel, avoiding the usual requirement of inverting large dense co–variance matrices. Rather than using higher order basis functions with continuous derivatives across element faces, regularization is also implemented with a FOSLS formulation using vector-valued property function (density and its gradient). Minimization of the cost function, comprised of data misfit and regularization, is achieved via a Lagrange multiplier method with the minimum of the gravity FOSLS functional as a constraint. The Lagrange variations are combined into a single equation for the property function and solved using an integral form of the preconditioned conjugate gradient method (I–PCG). The diagonal entries of the regularization operator are used as the preconditioner to minimize computational costs and memory requirements. Discretization of the differential operators with the finite element method (FEM) results in matrix systems that are solved with smoothed aggregation algebraic multi–grid preconditioned conjugate gradient (AMG-PCG). After their initial setup, the AMG-PCG operators and coarse grid solvers are reused in each iteration step, further reducing computation time. The algorithm is tested on data from 23 surveys with a total of 6519 observation points in the Mt Isa–Cloncurry region in north–west Queensland, Australia. The mesh had about 2.5 million vertices and 16.5 million cells. A synthetic case was also tested using the same mesh and error measures for localized concentrations of high and low densities. The inversion results for different parameters are compared to each other as well as to lower order smoothing. Final inversion results are shown with and without depth weighting and compared to previous geological studies for the Mt Isa–Cloncurry region.