3-D inversion of magnetic data using Boulanger and Chouteau algorithm: a case study on magnetic data of old Pompeii city

Inversion of magnetic data is the most important step in the interpretation of magnetic anomalies. Availability of 3-D inversion of magnetic data is required because earth material properties generally change in all three special dimensions. Magnetic data inversion has two main problems about non-uniqueness and instability of the solution which can be obviated by using constraints and a priori information. Non-uniqueness is the consequence of two ambiguities: I) following Gauss theorem, there are many equivalent sources that can produce the same known field at the surface (theoretical ambiguity), II) since the parameterization of the problem is such that there are more unknowns than observations, the system does not provide enough information in order to uniquely determine model parameters (algebraic ambiguity). Every measurement of data on the earth’s surface contains some noise which imposes large changes on the inverse solution, therefore the problem is also ill-posed. There are many constraints including compactness, minimization of inertia around an axis or a point, depth weighting and etc. Different combinations of these constraints in the objective function lead to different algorithms each of which are appropriate for some cases. In this paper, inversion algorithm proposed by Boulanger and Chouteau are utilized for the 3-D inversion of magnetic data. This technique was introduced for inversion of gravity data. Their algorithm takes the advantage of a model weighting matrix derived by multiplying compactness, hardness and depth weighting constraints. Furthermore, smoothness matrix is also inserted in the algorithm. Compactness constraint, introduced by Last and Kubic, try to minimize the volume of the anomalous body in 3-D. Hardness constraint, represented by P, is a diagonal matrix for which diagonal elements pii is fixed at 10-2 or 1 depending on whether the value of the ith initial susceptibility is fixed by geological information or not. Depth weighting function, introduced by Li and Oldenberg, is used to counteract the natural decay of the kernel, so all the cells have an equal probability during the inversion. The subsurface is discretized into a lot of cells for which the susceptibility of each cell is assumed to be constant. The model parameter, susceptibility contrast, is also limited to lower and upper bounds. This algorithm was programmed in MATLAB software, and its efficiency was investigated by applying it on synthetic and real data. The first synthetic model is a cube and inversion process was done for free-noisy and noisy data (5 % random noise) and in both cases recovered models were satisfactory. The second case is the model of vertical and dip dykes as a more complex synthetic example. Inverting free-noisy data leads to the exact recovering of true model. The reconstructed model obtained from noisy data actually represented an acceptable model. Therefore, results of synthetic cases were promising enough and convince us in order to apply the algorithm to real cases. Finally, the algorithm was applied two real profiles related to the archeological data sets of an area in old Pompeii city near Naples in Italy. Both profile lengths are 35.5 m with interval sampling of 10.4 cm. Inversion result of the data using this 3-D algorithm represents anomalies that are in a good agreement with subsurface anomaly positions.