We present a method to determine the source properties of potential fields by exploring the field at multiple scales. At each scale the depth to the source and the structural index are determined by analyzing the scaling function t, a dimensionless quantity which defines the scaling properties of the field, and the differential scaling function ß. For homogeneous fields, the estimated results are scale invariant. The source properties are determined either by a geometrical method applied to the a single ridge or by solving for the zeros of ß.


Potential fields are related to often complex source distributions. One approach to interpret them is to assume simple source models which allow a simplified theory, since their fields are mathematically equivalent to those of sources defined by just one-point in the source region. This is true for continuous source distributions such as homogeneous spheres, contacts, cylinders and others. In the case of spheres, this result follows trivially from Gauss theorem, for the others from geometrical considerations taking also in the account the distance to sources. Euler Deconvolution is the classical method to handle with homogeneous fields (Reid et al., 1990).

Multiscale methods also have been proposed as valuable methods to interpret homogeneous potential fields. Among them we mention the Continuous Wavelet Transform (e.g., Sailhac and Gibert, 2003), the Multiridge analysis and reduced Euler Deconvolution (Florio and Fedi, 2006, Fedi et al., 2007). In this paper we refer to the methods based on the scaling function, such as SCALFUN (Fedi and Florio, 2006; Florio et al., 2007) and DEXP (Fedi 2007). A common characteristic of multiscale methods is the stability with respect to the noise and the possibility to study the field at different altitude ranges to try to separate effects of sources lying at different depths.

In this context, we develop here a new theory based on the vertical derivative of the scaling function, which we call the differential scaling function: it provides a local estimator, at each scale, of the source parameters of potential fields.

The scaling function

Source distributions corresponding to simple geological sources such as contacts, dykes, sills, spheres, cylinders, gained wide popularity since their fields obey to a scale invariance law, known as homogeneity relation (e.g., Thompson, 1982): The homogeneous fields generated by poles or dipoles, despite their simplicity, are important since the field of any complex source-density distribution is equivalent to that of a homogeneous density source of polar (or dipolar for magnetic fields) behavior when measured at some sufficiently high altitude. An example of homogeneous field (of degree n=-2) is the gravity field at any point P(x,y,z), due to a pole source at Q*(?*,?*,?*) :

? is the gravitational constant and m is the mass. For other sources, such as cylinders, dyke, contacts and others, the gravity field is still homogeneous, with integer values -2=ng=1 . As usual, geophysical fields are often referred not to their homogeneity degree n, but to the so-called structural index N0=-ng. The p-order vertical derivatives of the gravity field in equation (1) :

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