Phase correction of Gabor deconvolution
Tianci Cui, Gary F. Margrave
Seismic data is always nonstationary due to ubiquitous anelastic attenuation modeled by the constant-Q theory. The stationary spiking deconvolution of stationary traces is extended to Gabor deconvolution of nonstationary traces, in which a seismic trace is decomposed into a time-frequency spectrum by the windowed Fourier transform and a nonstationary wavelet is estimated within each window. The amplitude spectrum of the nonstationary wavelet is accurately estimated by a smoothing process while its phase spectrum is calculated by the discrete Hilbert transform integrating within the seismic frequency band only. The Gabor deconvolved seismic trace ties the well reflectivity in amplitude and spectral content, but has phase being corrected respect to the seismic Nyquist frequency only. The phase error is the phase difference of the nonstationary wavelet with respect to the well logging frequency and the seismic Nyquist frequency. It can be calculated by knowing the Q values and the well logging frequency, to serve as a phase correction operator in the Gabor domain, which is equivalent to a time-variant residual drift time correction operator in the time domain. Without knowledge of Q or the well logging frequency, the residual drift time can be estimated by smooth dynamic time warping, which is more accurate than that estimated by time-variant crosscorrelation. The Gabor deconvolved nonstationary trace with phase or residual drift time correction ties the well reflectivity with little amplitude or phase errors.