Gabor deconvolution: Estimating reflectivity by nonstationary deconvolution of seismic data

Abstract

We have extended the method of stationary spiking deconvolution of seismic data to the context of nonstationary signals in which the nonstationarity is due to attenuation processes. As in the stationary case, we have assumed a statistically white reflectivity and a minimum-phase source and attenuation process. This extension is based on a nonstationary convolutional model, which we have developed and related to the stationary convolutional model. To facilitate our method, we have devised a simple numerical approach to calculate the discrete Gabor transform, or complex-valued time-frequency decomposition, of any signal. Although the Fourier transform renders stationary convolution into exact, multiplicative factors, the Gabor transform, or windowed Fourier transform, induces only an approximate factorization of the nonstationary convolutional model. This factorization serves as a guide to develop a smoothing process that, when applied to the Gabor transform of the nonstationary seismic trace, estimates the magnitude of the time-frequency attenuation function and the source wavelet. By assuming that both are minimum-phase processes, their phases can be determined. Gabor deconvolution is accomplished by spectral division in the time-frequency domain. The complex-valued Gabor transform of the seismic trace is divided by the complex-valued estimates of attenuation and source wavelet to estimate the Gabor transform of the reflectivity. An inverse Gabor transform recovers the time-domain reflectivity. The technique has applications to synthetic data and real data.

Gabor Deconvolution as Preliminary Method to Reduce Pitfall in Deeper Target Seismic Data

“…Dennis Gabor in 1946 brought up the idea of applying Fourier analysis in partition sense [2] . A series of non-stationary seismic trace is divided into smaller signal packet which was localized by a certain window.…”

Data Preconditioning With Gabor Nonstationary Deconvolution for Radar Imaging of Highly Dissipative and Dispersive Media

“…Therefore, a local analysis scheme is necessary to yield a time-frequency representation for the time-variant system. Among a variety of local analysis based algorithms for estimating the object reflectivity in the presence of nonstationary systems, Gabor nonstationary deconvolution has demonstrated success in attenuation compensation and phase correction for seismograms [34]. Through the partition of unity (POU) windowing scheme, the Gabor algorithm is able to minimize the energy loss during the forward and inverse Gabor transform and achieve an invertible dual-domain transformation.…”

“…There are various ways to estimate the attenuation function in the Gabor domain. In particular, Margrave et al [34,56] reported that the hyperbolic smoothing approach is robust and able to yield a consistent estimate of the Gabor magnitude spectrum of the propagating wavelet. Hyperbolic smoothing is essentially based on the theory that the amplitude spectrum of attenuation can be described by a constant Q model as…”

“…(14) isŜ(f, τ ). Therefore, the following assumptions are necessary in order to find the solution for R(f, τ ) [34,52]:…”

“…Various techniques have been developed in the field of Seismology to compensate for energy loss, to correct for wavelet distortion, and to produce an image with high resolution [33]. Among these, the Gabor deconvolution method [34] has been successfully tested with industrial seismic data and ground penetrating radar (GPR) data [35][36][37]. The results indicate that, compared to the industry standard approach, Gabor deconvolution provides improved amplitude and phase content of certain geological events.…”