A local level-set method for 3D inversion of gravity-gradient data

We have developed a local level-set method for inverting 3D gravity-gradient data. To alleviate the inherent non-uniqueness of the inverse gradiometry problem, we assumed that a homogeneous density contrast distribution with the value of the density contrast specified a priori was supported on an unknown bounded domain D so that we may convert the original inverse problem into a domain inverse problem. Because the unknown domain D may take a variety of shapes, we parametrized the domain D by a level-set function implicitly so that the domain inverse problem was reduced to a nonlinear optimization problem for the level-set function. Because the convergence of the level-set algorithm relied heavily on initializing the level-set function to enclose the gravity center of a source body, we applied a weighted L-1-regularization method to locate such a gravity center so that the level-set function can be properly initialized. To rapidly compute the gradient of the nonlinear functional arising in the level-set formulation, we made use of the fact that the Laplacian kernel in the gravity force relation decayed rapidly off the diagonal so that matrix-vector multiplications for evaluating the gradient can be accelerated significantly. We conducted extensive numerical experiments to test the performance and effectiveness of the new method.