Solving the ambiguity in the potential field exploration of complex sources

It is theoretically demonstrated  that, even with perfectly complete and perfectly accurate data, there is a fundamental ambiguity in the analysis of potential field data. The ambiguity may be easily illustrated by computing some of the various kinds of structures that can give rise to the same anomaly field. To solve the ambiguity and yield reasonable geophysical models we must therefore supply a priori information. In gravimetry, the ambiguity comes from the fact that only the excess mass is uniquely determined by the anomaly, neither the density nor the source volume. However, not only the excess mass can be uniquely estimated. Examples are the center of a uniformly dense (or magnetized) sphere or the top of a deeply extended homogeneously-dense cylinder. A priori information may consist of direct information (e.g., depth, shape) and/or of assuming that the source distribution has some specified properties (e.g., compactness, positivity). If one tries to classify the physical source-distributions in terms of their complexity, we may however use two different scaling laws, based on homogeneity and self-similarity, which allow modeling of the Earth in its complex heterogeneity. While monofractals or homogeneous functions are scaling functions, that is they do not have a specific scale of interest, multi-fractal and multi-homogeneous models need to be described within a multiscale dataset. Thus, specific techniques are needed to manage the information contained on the whole multiscale dataset. In particular,  any potential field  generated by a complex source may be modeled as  a multi-homogeneous field, which typically present a fractional and spatially varying homogeneity degree. For a source of irregular shape, it may be convenient to invert not the  field but a related quantity, the scaling function, which is a multiscale function having the advantage of not involving the density among the unknown parameters. For density or magnetic susceptibility tomographies, the degree of spatially variable homogeneity can be incorporated in the model weighting function, which, in this way, does not require prior assumptions because it is entirely deductible from the data. We discover that difficult quantities, such as the bottom of the sources, or multiple source systems are reasonably well estimated by abandoning the analysis at a single scale and unraveling the scale-related complexity of geophysical signals. The inherent self-consistency of these new multiscale tools is a significant step forward, especially in the analysis of areas where there is scarce other information about the sources.