Hilbert transform, Gabor deconvolution and phase corrections

Carlos A. Montaña, Gary F. Margrave

Constant Q theory is a simple and robust theoretical model for seismic waves attenuation which needs just two parameters to characterize anelastic attenuation in a medium: the quality factor Q and a reference frequency 0. for which the phase velocity is known. The convolutional model for a seismic trace can be extended to nonstationary attenuated traces using the constant-Q theory. Gabor deconvolution is a nonstationary extension of Wiener's deconvolution by factorizing the attenuated trace with the help of the Gabor transform. Both deconvolution methods are based on minimum phase assumptions and make use of the Hilbert transform for estimating the phase spectrum of the deconvolution operator. Gabor deconvolution compensates for the amplitude losses due to attenuation without necessity of any estimation of Q. However time shift and phase rotations between a recorded seismic trace and a synthetic trace generated from a well log remains after Gabor deconvolution is applied. One of the main causes of these phase differences is the fact that the digital Hilbert transform is an imperfect estimation of the analytical Hilbert transform. The phase shift can be removed partially either by adding a function, linear in time and quadratic in frequency, to the digital Hilbert transform estimation of the minimum phase function of the Gabor deconvolution operator or, by resampling the trace to a smaller sample rate.