Scaling properties of potential fields due to scaling sources

The theoretical power spectrum of the 3‐dimensional potential field caused by an arbitrary 3‐dimensional source distribution is derived for gravity and magnetic data. A function with scale‐invariant features has a power spectrum, which is proportional to the frequency raised to minus the scaling exponent. For scaling source distributions, the power spectrum of the gravity and magnetic field is anisotropic and a specific scaling exponent exists only for lower‐dimensional cross sections of the field. We suggest an approach which allows, under certain conditions, to derive the power spectrum of a lower‐dimensional subset from the power spectrum of a 3‐dimensional function. For the special case where the 3‐dimensional function has an isotropic scaling exponent β3D, we confirm a known property, namely that a (3‐k)‐dimensional subset of the function has a scaling exponent of approximately k less than β3D. This property is not applicable to the anisotropic 3‐dimensional fields, but it can be applied to source distributions with isotropic scaling exponent. Summarizing our results, the scaling exponents of the density distribution and the gravity field are related by whereas the relationship between the scaling exponents of the susceptibility distribution and the magnetic field reduced to the pole can be stated as follows: