Edge detection of potential-field sources using scale-space monogenic signal; fundamental principles

We have developed a new phase-based filter to enhance the edges of geologic sources from potential-field data using the local phase in the Poisson scale-space monogenic signal. The Poisson scale-space representation of a potential-field data is equivalent to performing an upward continuation of the data. We created a band-pass filter by taking the differences between two Poisson scale-space representations of the data. The local phase was defined as the arctangent of the ratio of the magnitude of the x- and y-components of the first-order Riesz transform of the filtered data to these data. These components were computed in the wavenumber domain and then transformed back into the space domain by the inverse Fourier transform. In the wavenumber domain, we found that these components are the multiplication of the Fourier transform of the filtered data by a Fourier-domain kernel, which in turn is the multiplication of the first-order horizontal derivative filter by the first-order vertical integral filter. This operation is stable, making the local phase of the monogenic signal quite insensitive to noise. We proved that if the data were the vertical component f (sub z) of a conservative field F, the x- and y-components of the first-order Riesz transform of f (sub z) were the horizontal components f (sub x) and f (sub y) of F. Hence, the local amplitude of the monogenic signal of f (sub z) is the 3D analytic signal amplitude of the scalar potential of F and the local phase resembles the tilt angle (TILT). Tests on synthetic total-field anomalies and a real aeromagnetic anomaly over the Para-Maranhao Basin, Brazil, showed that the local phase in the scale-space monogenic signal had better performance than the TILT in delineating the geologic contacts that were not seen in the original data.