Forward modelling of spectral depths using 3D Fourier convolution

3D Fourier convolution is a fast method for forward modelling irregular magnetic bodies. A kernel consisting of the field of a single dipole, convolved over a random distribution of dipoles, simulates the heterogeneity of bulk material. A demonstration studies the depth information collected by different survey parameters.

A procedure for creating depth transects across the Northern Territory (Australia) has been established using magnetic spectral depths. Although these transects showed depths to layers, they failed to show depths in the upper few hundred metres. Because the magnetic depths had been derived on the basis of heterogeneity of the bodies, a method of forward modelling based on heterogeneity is needed to explain this limitation and other issues. Spatial convolutions based on heterogeneity suffice for primitive models but are too slow for detailed work.

This paper demonstrates fast forward modelling using Fourier convolution, that is convolution of three-dimensional (3D) arrays via the frequency domain, to obtain total magnetic intensity grids and magnetic depth profiles for hypothetical structures. Randomly located dipoles are used to simulate the heterogeneity of the material of modelled bodies.

The loss of shallow depth signal in the magnetic transects is shown to arise mainly from the limitation of the line spacing of the underlying surveys. Depths of bodies at less than half the line spacing of the survey are not resolved at all and depths less than the line spacing itself appear deeper than the actual source depth.

Fourier convolution works equally well with non-layered, non-prismatic bodies. Modelling of an inclined, elliptical body is demonstrated by way of example. The associated depth profile shows a clear equivalent layer at a depth representative for such a body. The result allows interpretation of a characteristic pattern in magnetic depth transects as indicating the depth to a relatively compact non-layered body.

Fourier convolution showed a considerable speed advantage over spatial convolution at all array sizes used in the study. Convolutions of model arrays of 1000 × 1000 × 500 were calculated within a few minutes.