Attività del Dipartimento

Extracting Unambiguous Information from Potential Field Anomalies

Maurizio Fedi

Inverse potential field problems are inherently difficult because of their ill-posedness and because they do not have a unique solution. This implies that the calculated solution may be extremely sensitive to errors and that, even if we had access to noise-free and continuous field data, we would face an ambiguity problem: by Green's third identity, any potential field in a sub-region can be reproduced by an infinite variety of surface distributions. In addition, any source distribution producing a null field, belonging to the so-called annihilator, cannot be determined from the data. In addition, other kinds of ambiguity must be considered: sampling ambiguity, algebraic ambiguity, and noise ambiguity. To face these ambiguities, some sort of incorporation of a priori knowledge is needed, to compute unique solutions, e.g. minimizing the density model weighted norm with respect to some reference model; constraining for upper and lower density bounds; searching for compact or smooth solutions; inserting an appropriate depth weighting function in the Tikhonov formulation. However, ambiguity may be faced in a completely different way, even when the data are incomplete, to determine which model parameters are common to all models fitting the data, or at least to wide classes of all the models. This leads to the unambiguous estimation of important source parameters, such as the total excess mass of the buried body, the source boundaries or lineaments, the scaling exponent of the depth weighting function, the maximum allowed depth for the sources, the fault dip. We will illustrate how these and other relevant source parameters may be unambiguously determined, often in a full automatic way, by the measured data. One important conclusion is that the use of homogenous spherical models for illustrating the ambiguity of potential fields, as commonly made in standard textbooks, is not appropriate: real-world anomalies refer to complex source distributions from which, fortunately, significant and unambiguous information may be extracted.