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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.

A microgravity survey was completed to precisely locate and better characterize the near-surface geometry of a recent fault with small throw in a mountainous area in the Southern Apennines (Italy). The site is on a segment of the Irpinia fault, which is the source of the M6.9 1980 earthquake. This fault cuts a few meter of Mesozoic carbonate bedrock and its younger, mostly Holocene continental deposits cover. The amplitude of the complete Bouguer anomaly along two profiles across the fault is about 50 μGal. The data were analyzed and interpreted according to a self-constrained strategy, where some rapid estimation of source parameters was later used as constraint for the inversion. The fault has been clearly identified and localized in its horizontal position and depth. Interesting features in the overburden have been identified and their interpretation has allowed us to estimate the fault slip-rate, which is consistent with independent geological estimates.

Summary Gravity data inversion is a fundamental tool for geological exploration. A large amount of algorithms have been developed in the past, with different approaches. The choice of the inversion algorithm depends, mainly, on the geological contest, the kind of solution desired, its resolution at the end of the process, the availability of a priori information and on how they can be included in the inversion algorithm. A priori information has, often, a key-role in the inversion process. However, in case of reconstruction of salt and subsalt structures, seismic information may be poor and unfeasible. So we need to constrain the model with other kind of information. In this work, we present a part of recent Eni R&D activity focused on gravity data inversion. The shown results are mainly based on the Data Space Inversion algorithm (Pilkington 2009), originally presented for the magnetic case, and here extended to the gravity problem. We also putted a strong effort on the computational side of the problem, taking advantage from our experience in HPC (High Performance Computing), in order to speed up the inversion process and so enable its use at industrial level. In this paper we present and discuss some results regarding the application of the methodology to the SEAM (SEG Advance Modeling, 2007) demonstrating that the algorithm allows a consistent depth and density model.

In potential field inversion problems we usually solve underdetermined systems and this leads to a very shallow solution, typically known as minimum length solution. This may be avoided introducing a depth weighting function in the objective function (Li and Oldenburg, 1996). In this paper we derive invariance rules for either the minimum norm minimization and for the regularized inversion with depth weighting and positivity constraints. For a given source class, corresponding to a specific structural index N, the invariance rule assures that the same solution is obtained inverting the magnetic (or gravity) field or any of its nth order vertical derivatives. Although we demonstrate mathematically this invariance rule for the minimum norm minimization only, it is shown to occur also for the regularized inversion with depth weighting and positivity constraints. In this case, a source-class invariant form of depth weighting is derived, referring to that of the magnetic field, in the magnetic case, and to the 1st derivative of the gravity field, in the gravity case. We also illustrate how the combined effect of regularization parameter and depth weighting influences the estimated source model depth in the regularized inversion with depth weighting and positivity constraints.