Imaging quadrupolar geophysical anomaly sources by 3D probability tomography. Application to near surface geoelectrical surveys

We present the theory of the source quadrupole occurrence probability tomography method within a general approach including the source pole and source dipole occurrence probability tomographies, developed and amply discussed and tested in previous papers. In this new formulation, poles and dipoles still have the original meaning to represent centres and boundaries, respectively, of elementary bodies with constant constitutive parameters, while quadrupoles are assumed to simulate sharp corner elements. The purpose of quadrupole analysis is to improve the resolution power of geophysical methods, using once more probability as a suitable paradigm allowing all possible equivalent solutions to be included into a unique 3D tomography image. The new approach is described by first assuming that any geophysical field dataset can be hypothesized to be caused by a discrete number of source poles, dipoles and quadrupoles. Then, the previously published theory of the source pole and dipole occurrence probability tomography is symbolically extended to include source quadrupoles. Finally, the new formalism is explicitly written for the geoelectrical method, for which a few simple synthetic examples are discussed in order to demonstrate how the combined pole, dipole and quadrupole tomographies can provide a complete resolution of the most probable buried sources of the anomalies detected within a datum domain. A field example related to a geoelectrical survey carried out in the archaeological park of Pompei is presented, aiming at detecting remains of ancient Roman urbanization.