||Due to the influence of noise, it is often difficult to obtain good depth resolution from magnetic survey data in three-dimensional geomagnetic studies. A perturbation analysis shows that noise leads to solutions with poor depth resolution, hence, the objective of this work is to use inverse regularization methods to optimize the depth resolution under the influence of noise. The Tikhonov, CGLS, and LSQR methods are employed. Our results show that, for large-scale problems it is an advantage to use the iterative methods, however for the numerical experiments in this thesis primarily Tikhonov are used.
In order to obtain depth resolution it is essential to work with multiple levels of data because this provides a large set of singular vectors for the reconstruction. These vectors hold information about the depth of the exact solution.
In the perturbation analysis we consider three sources of errors: White noise in the right-hand side, background signal in the solution, and model errors in the coefficient matrix. The analysis reveals that with realistic noise levels the reconstructed solutions display poor depth resolution. This is because only a fraction of the available SVD
components can be included in the regularized solutions, and these do not provide sufficient information to obtain depth resolution. Hence, a regularization approach where depth resolution can be obtained with few SVD components is needed. A way to achieve this is to apply the second derivative as opposed to the identity matrix in the regularization term.
Applying the second derivative improves depth resolution significantly under the influence of noise. Experiments with three different versions of the second derivative show that applying it to the x-, y-, and z-directions separately or solely to the z-direction yield solutions with good depth resolution. However, if the second derivative is applied
only to x and y, the obtained results resemble those found by use of the identity matrix.
The key to a successful reconstruction is to choose the optimal regularization parameter. Five parameter-choice methods are evaluated and both Akaike's information criterion and generalized cross validation give good estimates of the optimal regularization parameter.
Furthermore, a practical application of data measured at Mt. Vesuvio shows that the inversion approach treated in this work is applicable. The obtained results are in well agreement with surface measurements performed at Mt. Vesuvio.