An efficient algorithm for the Lp-norm joint inversion of gravity and magnetic data using the cross-gradient constraint is presented. The presented framework incorporates stabilizers that use Lp-norms (0≤ p≤ 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.

A fast algorithm for solving the under-determined 3-D linear gravity inverse problem based on the randomized singular value decomposition (RSVD) is developed. The algorithm combines an iteratively reweighted approach for L₁-norm regularization with the RSVD methodology in which the large scale linear system at each iteration is replaced with a much smaller linear system. Although the optimal choice for the low rank approximation of the system matrix with m rows is q = m, acceptable results are achievable with q ≪ m. In contrast to the use of the iterative LSQR algorithm for the solution of the linear systems at each iteration, the singular values generated using the RSVD yield a good approximation of the dominant singular values of the large scale system matrix. The regularization parameter found for the small system at each iteration is thus dependent on the dominant singular values of the large scale system matrix and appropriately regularizes the dominant singular space of the large scale problem. The results achieved are comparable with those obtained using the LSQR algorithm for solving each linear system, but are obtained at reduced computational cost. The method has been tested on synthetic models along with the real gravity data from the Morro do Engenho complex from central Brazil.