A unified framework for the deconvolution of traces of nonwhite reflectivity

One of the fundamental assumptions of conventional deconvolution methods is that reflection coefficients follow the white-noise model. However, analysis of well logs in various regions of the world confirms that in the majority of cases, reflectivity tends to depart from the white-noise behavior. The assumption of white noise leads to a conventional deconvolution operator that can recover only the white component of reflectivity, thus yielding a distorted representation of the desired output. Various alternative processes have been suggested to model reflection coefficients. In this paper, we will examine some of these processes, apply them, contrast their stochastic properties, and critique their use for modeling reflectivity. These processes include autoregressive moving average (ARMA), scaling Gaussian noise, fractional Brownian motion, fractional Gaussian noise, and fractionally integrated noise. We then present a consistent framework to generalize the conventional deconvolution procedure to handle reflection coefficients that do not follow the white-noise model. This framework represents a unified approach to the problem of deconvolving signals of nonwhite reflectivity and describes how higher-order solutions to the deconvolution problem can be realized. We test generalized filters based on the various stochastic models and analyze their output. Because these models approximate the stochastic properties of reflection coefficients to a much better degree than white noise, they yield generalized deconvolution filters that deliver a significant improvement on the accuracy of seismic deconvolution over the conventional operator.