Depth resolution in potential field inversion: theory and applications
Creators: name: Ialongo, Simone
Publisher: name: Università degli Studi di Napoli Federico II
Year: 2013

description: In this thesis we have implemented and studied on detail three different potential field inversion algorithms proposed by Li and Oldenburg  (2003), Portniaguine and Zhdanov (2002) and Pilkington (2009). We focused our attention on the dependency of the solution with respect to external constraints and particularly with respect to the depth weighting function. This function is necessary to counteract the natural decay of the data kernels with depth, so providing depth resolution to the inverse solution.
We derived invariance rules for either the minimum-length solution and for the regularized inversion with depth weighting and positivity constraints. For a given source class, the invariance rule assures that the same solution is obtained inverting the magnetic (or gravity) field or any of its kth order vertical derivatives. A further invariance rule regards the inversion of homogeneous fields: the homogeneity degree of the magnetization distribution obtained inverting any of the k-order vertical derivatives of the magnetic field is the same as that of the magnetic field, and does not depend on k. Similarly, the homogeneity degree of the density distribution obtained inverting any of the k-order vertical derivatives of the gravity field is the same as that of the 1st order vertical derivative of the gravity field, and does not depend on k. This last invariance rule allowed us using the exponent β of the depth weighting function corresponding to the structural index of the magnetic case, no matter the order of differentiation of the magnetic field. We also illustrated how the combined effect of regularization and depth weighting could influence the estimated source model depth, in the regularized inversion with depth weighting and positivity constraints. We found that too high regularization parameter will deepen the inverted source-density distribution, so that a lower value for the exponent of the depth weighting function should be used, with respect to the structural index N of the magnetic field (or of the 1st vertical derivative of the gravity field). In the attempt to keep the regularization parameter as low as possible, the GCV method yielded better results than the χ2 criterion. 
Furthermore we introduced a new approach to improve the resolution of the model, based on inversion of data with a differentiation order greater than that of the kernel. We analyzed also the case of a field generated by sources with different structural indices. This is a very important case, because it is the most common situation in real data. In this case, there isn’t a unique value for β allowing to obtain accurate estimations of depth to all the sources. Thus the depth weighting exponent β must be varied according to the structural index estimated for each source and according to the invariance rules. 
Furthermore we  studied the dependency of the model obtained by inversion on the depth weighting function when a priori information is included in the inversion. We presented a self-constrained inversion procedure based only on the constraints retrieved by previous potential field anomaly interpretation steps. We showed that adding, as inversion constraints, information retrieved by a previous analysis of the data has a great potential to lead to well-constrained solutions with respect to the source depth and to the horizontal variations of the source-density distribution. Our analysis on both synthetic and real data demonstrated that the more self-constraints are included in the inversion, the less important is the role of the tuning of the depth-weighting function through the actual value of the source structural index. 
Another type of a priori information regards the compactness of solution. This constraint can be imposed using the focusing inversion algorithm (Portniaguine and Zhadanov, 2002) or using sparseness constraints (Pilkington, 2009). In this case, imposing this type of constraint tends to decrease the importance of the depth weighting function.