Non‐Gaussian reflectivity, entropy, and deconvolution

Abstract

Standard deconvolution techniques assume that the wavelet is minimum phase but generally make no assumptions about the amplitude distribution of the primary reflection coefficient sequence. For a white reflection sequence the assumption of a Gaussian distribution means that recovery of the true phase of the wavelet is impossible; however, a non‐Gaussian distribution in theory allows recovery of the phase. It is generally recognized that primary reflection coefficients typically have a non‐Gaussian amplitude distribution. Deconvolution techniques that assume whiteness but seek to exploit the non‐Gaussianity include Wiggins’ minimum entropy deconvolution (MED), Claerbout’s parsimonious deconvolution, and Gray’s variable norm deconvolution. These methods do not assume minimum phase. The deconvolution filter is defined by the maximization of a function called the objective. I examine these and other MED‐type deconvolution techniques. Maximizing the objective by setting derivatives to zero results in most cases in a deconvolution filter which is the solution of a highly nonlinear Toeplitz matrix equation. Wiggins’ original iterative approach to the solution is suitable for some methods, while for other methods straightforward iterative perturbation approaches may be used instead. The likely effects on noise of the nonlinearities involved are demonstrated as extremely varied. When the form of an objective remains constant with iteration, the most general description of the method is likelihood ratio maximization; when the form changes, a method seeks to maximize relative entropy at each iteration. I emphasize simple and useful link between three methods and the use of M-estimators in robust statistics. In attempting to assess the accuracy of the techniques, the choice between different families of distributions for modeling the distribution of reflection coefficients is important. The results provide important insights into methods of constructing and understanding the statistical implications and behavior of a chosen nonlinearity. A new objective is introduced to illustrate this, and a few particular preferences expressed. The methods are compared with the zero‐memory nonlinear deconvolution approach of Godfrey and Rocca (1981); for their approach, two distinctly different yet statistically comparable models for reflection coefficients are seen to give surprisingly similarly shaped nonlinearities. Finally, it is shown that each MED‐type method can be viewed as the minimization of a particular configurational entropy expression, where some suitable ratio plays the role of a probability.