Non-stationary deconvolution in presence of gaussian and spike-like noises using projected gabor deconvolution

Summary: Deconvolution is considered as a successful tool in seismic exploration for increasing the temporal resolution of the data. Gabor deconvolution is proposed to treat the non-stationarity issue of the problem by breaking it into several stationary sub-problems via a Gaussian window, solving them independently, and then, recombining/projecting the sub-solutions into an approximate solution to the original nonstationary problem. The projected Gabor deconvolution has recently been proposed by the second author as an improvement over Gabor deconvolution. In the projected Gabor deconvolution, the sub-problems are projected to a unified problem in time domain, and then, the resulting problem is solved. This modification brings useful advantages over the Gabor deconvolution including an improved convergence property, more efficiency for sparse deconvolution, more flexibility for incorporating prior information in the presence of noise, and more reflectivity structure via a least-squares method. In this paper, we propose a method for sparse and non-sparse deconvolution of non stationary seismic signals in the presence of Gaussian and spike-like random noises. Numerical tests using simulated and field data are presented to show high performance of the proposed method for generating accurate and stable reflectivity models from nonstationary seismograms.
According to the convolutional model of the Earth, a seismic signal can be modeled as convolution of the source generated wavelet with the Earth impulse response. The Earth impulse response contains the reflectivity information of the layer boundaries and the elasticity effects of the medium such as attenuation, absorption, etc. The aim of nonstationary seismic deconvolution is the recovery of the reflectivity series from such non-stationary signals. Gabor deconvolution, as an extension of stationary deconvolution, breaks the original problem into several stationary subproblems, then solves each sub-problem independently, and finally, recombines/projects the sub solutions into an approximate solution to the original non-stationary problem. A main property of the method is the treatment of the problem somehow in a blind fashion without requiring the attenuation model of the Earth as a priori. Projected Gabor deconvolution has been proposed as an alternative non-stationary deconvolution method. The projected Gabor deconvolution, compared to the Gabor deconvolution, shows an improved convergence property, and is more stable in the presence of noise, more efficient for non-linear optimizations like sparse recovery, and can better handle non- Gaussian noises in the data. The focus of this paper is on the latter property. Based on the projected Gabor deconvolution, we propose a method for non-stationary deconvolution of non-stationary seismic signals in the presence of spike-like noise.
Methodology and Approaches: Gabor deconvolution approximates a non stationary seismic deconvolution by several stationary sub-problems to be solved via stationary processes in the Gabor time-frequency domain. For random reflectivity sequences, it enables us to approximately determine the non-stationary operator via the Gabor time-frequency transform of the recorded trace. However, the inherent instability of the problem due to the approximations prevents the algorithm from converging to the solution. Furthermore, solving the Gabor deconvolution by nonlinear optimizations is rather time consuming. In the Gabor deconvolution, first an inverse operator is applied to each sub-problem, and then, the obtained sub-solutions are projected into the time domain to form a solution to the original problem. The projected Gabor deconvolution is similar to the Gabor deconvolution with the difference that the inverse operator is performed after the projection operator. There are many ways to estimate the Gabor and projected Gabor deconvolution operator. Among the many possibilities, we estimate the effect of attenuation with smoothing along hyperbolic trajectories in the time-frequency plane, and also, we estimate seismic wavelet with smoothing along the frequency axis. Thus, we can estimate the non-stationary deconvolution operator by these two procedures in an iterative manner. If we take inverse Fourier transform from columns of it and smooth it along main diagonal, we estimate the projected Gabor deconvolution operator. The projected Gabor deconvolution provides a solution to non-stationary deconvolution problem, which can be obtained by any regularization method that depends on the noise and reflectivity type.
Results and We have proposed a method, based on the projected Gabor deconvolution, for non stationary deconvolution of seismic signals in the presence of Gaussian and spike-like noises, and then, we have compared the results with those of the Gabor deconvolution method. Numerical examples from simulated and field data have confirmed that the projected Gabor deconvolution, compared to the Gabor deconvolution, has a better convergence property, generates more accurate results, and is more efficient to be solved via sparse optimizations.