Nonminimum-phase wavelet estimation using higher order statistics

Seismic data are often represented by the well known convolutional model: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ s(t)\ =\ w(t)\ {\otimes}\ r(t)\ {+}\ n(t),\] \end{document}(1) where w(t) denotes the seismic wavelet, r(t) the reflectivity series, and n(t) additive noise. The goal of seismic deconvolution is to design a filter f(t) capable of removing or compressing the wavelet. To understand the effect of removing the wavelet from the seismogram, convolve both sides of equation (1) with the filter \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ f(t)\ {\otimes}\ s(t)\ =\ f(t)\ {\otimes}\ w(t)\ {\otimes}\ r(t)\ {+}\ f(t)\ {\otimes}\ n(t). \] \end{document}(2) The last equation says that deconvolution can be successfully carried out if and only if two conditions are satisfied: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ f(t)\ {\otimes}\ w(t)\ {\approx}\ {\delta}(t)\] \end{document}(3) \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ f(t)\ {\otimes}\ n(t)\ =\ e(t)\ {\approx}\ 0. \] \end{document}(4) When these two conditions are simultaneously satisfied, one can write: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ f(t)\ {\otimes}\ s(t)\ {\approx}\ r(t). \] \end{document}(5) Equations (3) and (4) define a filter-design problem. However, in order to design f(t) , one must first know the wavelet, w(t) . Unfortunately, in most exploration scenarios, w(t) is not known and, therefore, needs to be estimated. This article focuses on estimating a wavelet with unknown amplitude and phase spectrum. Explicitly, the problem has one noisy observation (the seismogram) and two unknowns (the reflectivity and the seismic wavelet). We will examine how under certain conditions the stochastic nature of the reflectivity sequence can be exploited to estimate the wavelet. Stochastic wavelet estimation is often divided into two distinct problems: Problem 1 (estimation): Under proper assumptions, we can compute a functional of the wavelet directly from s(t) , the observed data. The functional must be capable of reducing the unknown random r(t) to a tractable quantity. It is also desirable to have a functional that annihilates the additive random noise n(t) . In mathematical terms, we seek an operator F such that \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[\ F[s(t)]\ =\ k_{r}\ {\bullet}\ G[w(t)]. \] \end{document}(6) where k r is a constant and G another functional to determine. Problem 2 (spectral factorization): This can be summarized as: Given an estimate of G[w(t)] , how do we estimate w(t) ? The latter leads to a new problem: …