Potential-Field Forward Modeling and Inversion Using 3D Fast Fourier Transforms

Although 1D and 2D fast Fourier transforms (FFTs) have long been used for the filtering, interpretation, and modeling of potential-field data, 3D FFTs have not enjoyed similar popularity. This may change with the recent discovery (Caratori Tontini et al., in press, JGR) that simple 3D FFT filters can be used to transform distributions of density (or magnetization) within a box-shaped 3D volume into gravity (or magnetic) fields within the same volume. For example, the continuous 3D Fourier transform of the vertical gravity anomaly Δgz(x,y,z) in a volume is related to the 3D Fourier transform of the density ρ(x,y,z) in the volume by

ℑ[Δgz] = i4πG(kz/ | k | 2)ℑ[ρ]; | k | ≠0, (1)

where G is the gravitational constant, kx, ky, kz are wavenumbers, and | k | 2 = kx2+ ky2+ kz2. (2) Translating (1) into a digital FFT filtering operation requires careful consideration of the periodicity of the density distribution and the gravity field. Nevertheless, (1) provides a highly efficient way to calculate the vertical gravity anomaly of a 3D density distribution within a few minutes. The calculated gravity anomaly can be sampled at random points or on an arbitrary surface using tri-linear interpolation. An equivalent space- domain calculation of the gravity field on an arbitrary surface can take many hours. The inverse relation to (1) does not appear to offer a practical approach for calculating a reliable density distribution from observed gravity data. This is because the data would have to cover a substantial portion of the model volume, and because the transformation is undefined on the plane kz = 0, where both the denominator of the filter kernel and the Fourier transform of the gravity field are identically zero. Numerical experiments show that the density distribution resulting from the power on the plane kz = 0 represents a classic Parker annihilator. This annihilator can be calculated from the density distribution but not from the gravity field, as would be required for direct inversion. In forward modeling mode, equations such as (1) permit rapid testing of geological models against observed potential-field data. The calculation speed of the 3D FFT suggests that a practical iterative inversion algorithm could be developed from (1) using Markov perturbation of an initial density model. Such an algorithm would have immediate application to problems of hydrogeology, resource assessment, and tunnel detection.