Solving ill-posed magnetic inverse problem using a Parameterized Trust-Region Sub-problem

The aim of this paper is to find a plausible and stable solution for the inverse geophysical magnetic problem. Most of the inverse problems in geophysics are considered as ill-posed ones. This is not necessarily due to complex geological situations, but it may arise because of ill-conditioned kernel matrix. To deal with such ill-conditioned matrix, one may truncate the most ill part as in truncated singular value decomposition method (TSVD). In such a method, the question will be where to truncate? In this paper, for comparison, we first try the adaptive pruning algorithm for the discrete L-curve criterion to estimate the regularization parameter for TSVD method. Linear constraints have been added to the ill-conditioned matrix. The same problem is then solved using a global optimizing and regularizing technique based on Parameterized Trust Region Sub-problem (PTRS). The criteria of such technique are to choose a trusted region of the solutions and then to find the satisfying minimum to the objective function. The ambiguity is controlled mainly by proper choosing the trust region. To overcome the natural decay in kernel with depth, a specific depth weighting function is used. A Matlab-based inversion code is implemented and tested on two synthetic total magnetic fields contaminated with different levels of noise to simulate natural fields. The results of PTRS are compared with those of TSVD with adaptive pruning L-curve. Such a comparison proves the high stability of the PTRS method in dealing with potential field problems. The capability of such technique has been further tested by applying it to real data from Saudi Arabia and Italy.