Introduction to tensorial resistivity probability tomography
  The probability tomography approach developed for the scalar resistivity method is here extended to the 2D tensorial apparent resistivity acquisition mode. The rotational invariant derived from the trace of the apparent resistivity tensor is considered, since it gives on the datum plane anomalies confined above the buried objects. Firstly, a departure function is introduced as the difference between the tensorial invariant measured over the real structure and that computed for a reference uniform structure. Secondly, a resistivity anomaly occurrence probability (RAOP) function is defined as a normalised crosscorrelation involving the experimental departure function and a scanning function derived analytically using the Frechet derivative of the electric potential for the reference uniform structure. The RAOP function can be calculated in each cell of a 3D grid filling the investigated volume, and the resulting values can then be contoured in order to obtain the 3D tomographic image. Each non-vanishing value of the RAOP function is interpreted as the probability which a resistivity departure from the reference resistivity obtain in a cell as responsible of the observed tensorial apparent resistivity dataset on the datum plane. A synthetic case shows that the highest RAOP values correctly indicate the position of the buried objects and a very high spacial resolution can be obtained even for adjacent objects with opposite resistivity contrasts with respect to the resistivity of the hosting matrix. Finally, an experimental field case dedicated to an archaeological application of the resistivity tensor method is presented as a proof of the high resolution power of the probability tomography imaging, even when the data are collected in noisy open field conditions. 
Paolo Mauriello and Domenico Patella