Zero Memory Non‐linear Deconvolution*

Abstract

A type of iterative deconvolution that extracts the source waveform and reflectivity from a seismogram through the use of zero memory, non-linear estimators of reflection coefficient amplitnde is developed. Here, we present a theory for iterative deconvolution that is based upon the specification of a stochastic model describing reflectivity. The resulting parametric algorithm deconvolves the seismogram by forcing a filtered version of the seismogram to resemble an estimated reflection coefficient sequence. This latter time series is itself obtained from the filtered seismogram, and so a degree of iteration is required. Algorithms utilizing zero memory non-linearities (ZNLs) converge to a family of processes, which we call Bussgang, of which any colored Gaussian process and any independent process are members. The direction of convergence is controlled by the choice of ZNL used in the algorithm. Synthetic and real data show that, generally, five to ten iterations are required for acceptable deconvolutions. I N T R O D U C T I O NThe problem of removing the source waveform from a seismogram has traditionally been attempted using predictive deconvolution. The phase of the waveform, however, is indeterminable with this technique, and to assure a unique solution, minimum phase is usually assumed. But the concept that the earth's reflection coefficients are non-Gaussianly distributed allows both the color and phase of the waveform to be determined. Some recent non-Gaussian models have had significant success. Wiggins (1977) developed a deconvolution technique that minimized, heuristically, the entropy in a deconvolution. Since Gaussian models maximize the entropy, his use of a non-Gaussian model is evident. Gray (1979), following Wiggin's lead, proposed using a variable measure of entropy and developed algorithms to optimize these measures. Another approach, pioneered by Claerbout and Muir (1973) and subsequently refined by Taylor, Banks and McCoy (1979), uses the L1 versus L2 norm to minimize deconvolution residuals. In both papers the robustness property of the L1 *

Estimation of the Source Time Function Based on Blind Deconvolution with Gaussian Mixtures

“…The set of equation (6) can be solved using the iterative procedure (GODFREY and ROCCA, 1981). At each iteration, the energy of the estimated signal must be normalized to a fixed value and then the signal model is updated.…”

New Lagrange Multipliers for the Blind Adaptive Deconvolution Problem Applicable for the Noisy Case

“…As already was noted in [20,58], the described model for the convolutional noise p[n] is applicable during the latter stages of the process, where the process is close to optimality [38]. According to [38], in the early stages of the iterative deconvolution process, the ISI is typically large with the result that the data sequence and the convolutional noise are strongly correlated, and the convolutional noise sequence is more uniform than Gaussian [59]. However, satisfying equalization performance was obtained by [51] and others [20] in spite of the fact that the described model for the convolutional noise p[n] was used.…”

Deconvolution and Estimation of Transfer Function Phase and Coefficients for Nongaussian Linear Processes

“…It has been observed that in many geophysical contexts the observed data (the x t 's) are distinctly nonGaussian and a basic object is to deconvolve, estimating the a/s and v/s in the process. A discussion of related questions in the geophysical context can be found in Donoho (1981), Godfrey and Rocca (1981), and Wiggins (1978).…”