DEXP imaging of potential fields with multi-homogeneity theory

One of the most interesting properties of the gravity fields generated by ideal sources (e.g., sphere, dyke, sill and contact)is that they are homogeneous functions of integer degree n, ranging from -2 to 1. It is to say that they satisfy the homogeneity equation in the harmonic region. However, when the source distribution is more complex than that of ideal sources, fields are not homogeneous. When analyzed at different distances, these fields will have different homogeneity degrees which can assume also a fractional and distance-dependent value. This results in the multi-homogeneity law which accounts for n varying at each observation site.

Accordingly, we may introduce the multi-homogeneity theory into the Depth From Extreme Points (DEXP) method. DEXP is an imaging method, which is based on field transformations, not involving any inverse matrix, so being faster and simpler to use. An important role in the scaling of the modelled field is played by the exponent N, the structural index. N is a parameter characterizing the type of source and is directly related to n as N=-n+q with q being the differentiation order of the Newtonian potential.

The proposed DEXP transformation for general sources is based on the multi-homogeneity theory so that the field is scaled by the inhomogeneous exponent N (x,y,z).

DEXP imaging of synthetic and real data demonstrated the ability to interpret complex bodies geometries which are brought by the DEXP method with the multi-homogeneity theory.