An Efficient Alternating Algorithm for the Lp-Norm Cross-Gradient Joint Inversion of Gravity and Magnetic Data Using the 2-D Fast Fourier Transform

An efficient algorithm for the $\mathrm {L}_{ \mathrm {p}}$ -norm joint inversion of gravity and magnetic data using the cross-gradient constraint is presented. The presented framework incorporates stabilizers that use $\mathrm {L}_{ \mathrm {p}}$ -norms ( $0\leq \mathrm {p} \leq 2$ ) of the model parameters, and/or the gradient of the model parameters. The formulation is developed from standard approaches for independent inversion of single data sets, and, thus, also facilitates the inclusion of necessary model and data weighting matrices, for example, depth weighting and hard constraint matrices. Using the block Toeplitz Toeplitz block structure of the underlying sensitivity matrices for gravity and magnetic models, when data are obtained on a uniform grid, the blocks for each layer of the depth are embedded in block circulant circulant block matrices. Then, all operations with these matrices are implemented efficiently using 2-D fast Fourier transforms, with a significant reduction in storage requirements. The nonlinear global objective function is minimized iteratively by imposing stationarity on the linear equation that results from applying linearization of the objective function about a starting model. To numerically solve the resulting linear system, at each iteration, the conjugate gradient algorithm is used. This is improved for large scale problems by the introduction of an algorithm in which updates for the magnetic and gravity parameter models are alternated at each iteration, further reducing total computational cost and storage requirements. Numerical results using a complicated 3-D synthetic model and real data sets obtained over the Galinge iron-ore deposit in the Qinghai province, north-west (NW) of China, demonstrate the efficiency of the presented algorithm.