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

Abstract

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.