Evaluation of Solutions in a 2-D Gravity Inverse Problem

In this work we evaluate the solution quality of inverse problems in Exploration Geophysics when using singular value decomposition. The application is in the inversion of 2-D gravity data, in the linear case, with the incorporation of a priori information and also the incorporation of Gaussian noise to the gravity field data. In the simulations with synthetic data, both for the forward modeling and the inverse procedure, the area of interest was parameterized in uniform blocks, considering that the density is constant within each block. We performed the inversions with different numbers of singular values, where in each situation the number of singular values was associated with the condition number. Two approaches were employed to evaluate the quality of the estimated solution. In the first approach we studied the behavior of the resolution matrices: both data resolution matrix and model resolution matrix. We observed that when we used the optimal number of singular values the majority of the elements of the main diagonal where near one, for both matrices. When the values of the elements of main diagonal of the model resolution matrix are near one, the inversion is considered to be satisfactory. Also the main diagonal of the model resolution matrix can be plotted as a matrix or image in such a way that it is possible to evaluate not only the solution as a whole but also the quality of every element of the estimated model parameter vector. In the second approach we used the concepts of complementary vectors, both for data parameters and model parameters. The sum of the estimated model parameters and complementary estimated model parameters generates a third vector or image, which also allows evaluate the quality of the solution of the inverse problem, again as a whole and also the individual elements of the estimated model parameter vector.