Scaling Laws in Geophysics: Application to Potential Fields of Methods Based on the Laws of Self-similarity and Homogeneity

We analyse two classes of methods widely diffused in the geophysical community, especially for studying potential fields and their related source distri- butions. The first is that of the homogeneous fractals random models and the second is that of the homogeneous source distributions called “one-point” distributions. As a matter of fact both are depending on scaling laws, which are used worldwide in many scientific and economic disciplines. However, we point out that their appli- cation to potential fields is limited by the simplicity itself of the inherent assumptions on such source distributions. Multifractals are the models, which have been used in a much more general way to account for complex random source distributions of density or susceptibility. As regards the other class, a similar generalization is proposed here, as a multi-homogeneous model, having a variable homogeneity degree versus the position. 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 are necessarily described within a multiscale dataset and specific techniques are needed to manage the information contained on the whole multiscale dataset.