Three-dimensional discrete magnetic inversion using graph theory to determine the framework of isolated sources

Applied geophysics is an important discipline for a prolific description of the subsurface. Commonly, the geophysical data is acquired, treated and inverted to produce interpreted geologic models. In this case, models can be composed by physical-property distributions and/or geometry of causative sources. In magnetic inverse problems, a challenging aspect is to reliably estimate both the geometry and magnetization of causative sources, once these parameters are estimated with tremendous amounts of uncertainties if no stabilization is considered. So, this study revisits graph-theory concepts applied to the equidistance function to stabilize the three-dimensional discrete magnetic inversion of the total-field anomaly using an ensemble of identical dipoles, which simultaneously estimates the magnetic framework and the magnetization direction of homogeneous sources. Our method consists of a Hierarchical Genetic Algorithm (HGA) to generate a population of candidate estimates. Each one comprises a set of identical magnetic dipoles, whose Cartesian coordinates and a single magnetization direction are estimated throughout the HGA generations. To produce unique and stable estimates, our HGA minimizes a single-objective function and a L-curve strategy guides for an ideal balance between data-misfit and equidistance functions. Thus, a compact ensemble of magnetic dipoles is obtained when the statistical variance of the distances of an minimum spanning tree is minimized jointly with the data-misfit function during the iterations of the HGA. Hence, the 3D compact distribution of dipoles can be associated with the skeleton of magnetic sources. An innovative searching for promising estimates is supplied by other geophysical methods applied to the magnetic data, such as Euler deconvolution and crosscorrelation. With these, the search spaces for each model parameter are defined in a semi-automatic basis, avoiding a strictly guess-and-check approach. The methodology is validated on two noise-corrupted synthetic data composed by vertical and dipping dike models. Results confirm the decent capability of the method in estimating a stable and compact ensemble of magnetic dipoles. Additionally, the magnetic properties are also recovered in a reliable way. A real test comprising the Morro do Forno magnetic anomaly show that the methodology works adequately in recovering the framework and orientation (i.e., Southwest–Northeast) of the main magnetic structures of the area. Nothing specific about the alkaline dykes of the region could be assertively mentioned, once a more refined study is required.