Lattice Filtering and Its Application to Seismic Data Deconvolution
Authors: Chen Weizhong , Dong Minyu 
Published: October 2008

Abstract
Lattice filtering, based on the Gram-Schmidt orthogonal decomposition theory, converts Least Square Filtering (LSF), a global optimization problem, into a series of local optimization problems. A striking feature of the lattice solution is that the filter coefficients are expressed as ratios of two summations. In this paper, we show that this feature leads to 1) a stable estimation of the lattice filter which is free of the data window boundary effects, and 2) a new formulation of an adaptive lattice filter for non-stationary signal processing. This adaptive lattice filtering is absolutely convergent and has a tracking speed that keeps pace with signal variations. It can also be implemented on array processors. A set of multi-pole windows are proposed for more efficient implementation of this adaptive lattice filtering. An application of 2) is the adaptive lattice deconvolution with a high tracking speed. To control the quality of the deconvolution, an effective method is developed to balance signal-to-noise ratio (SNR) against resolution with easy control in timespace domain. Field data processing indicates that this deconvolution is robust in revealing geological details because of its high tracking speed and good quality control. In computational efficiency, it is similar to the conventional least square deconvolution.

Introduction
Least Square Filtering (LSF) is a basic technique in seismic data processing, the mathematical model of which consists of choosing a filter (available in full paper), such that(available in full paper) where y(n) is the expected output, ;r(n) the input, and(available in full paper) the acutal output. In deterministic deconvolution, for example, the (available in full paper) is the inverse of a seismic wavelet, ;r(n) a seismic trace, y(n) a reflection coefficient sequence, and y(n) the result of the deconvolution. Equation (1) constitutes a global optimizat.ion problem.
In Equation (1), if the summation of n is done over (available in full paper), a Toeplitz matrix equation based on the autocorrelation of the input x(n) is obtained. This method is, thus, called the autocorrelation method. If the summation is over the interval on which the data of the expected output y(n) exists, a covariance matrix equation based on the autocovariance of x(n) is obtained. This method is generally called the covariance method.
For finite x(n) and y(n), the autocorrelation method assumes zeros for the data outside the data window. As a result, it suffers from the window boundary effects. A smooth window is usually used to reduce the boundary effects. The covariance method, however, does not make any assumption regarding the data outside the window. The optimization achieved is limited within the window over which the data of the expected output y(n) exist. In this case, the use of a smooth window does not make sense. Generally the covariance method is expected to yield a better result than that of the autocorrelation method. However, the covariance matrix equation is sometimes ill-conditioned, limiting its general use.