An alternate and effective approach to Hilbert transform in geophysical applications

The Hilbert transform defined via the Hartley transform in contrast with the well-known Fourier transform is mathematically illustrated with a couple of geophysical applications. Although, the 1-D Fourier and Hartley transforms are identical in amplitude with a phase difference of 45°, the Hilbert transform effectively differs when defined as a function of the Hartley transform in certain geophysical applications. It may be noted here that the Hilbert transform defined through Fourier and Hartley transforms while possessing the same magnitude differs in phase by 270°. It is derived and shown mathematically that the evaluation of depth of subsurface targets is directly equal to the abscissa of the point of intersection of the gravity (magnetic) field and the Hartley-Hilbert transform; however, it is not the case with the Fourier-Hilbert transform. The practical applications are illustrated with the interpretation of gravity anomaly due to an inclined sheet-like structure across the Mobrun ore body, Noranda, Quebec, Canada, and the vertical magnetic anomaly due to a cylindrical structure over a narrow band of quartzite magnetite, Karimnagar, India. The entire process can be automated.