Sven Treitel

The Design of Multistage Separable Planar Filters

A two-dimensional, or planar, digital filter can be described in terms of its planar response function, which is in the form of a matrix of weighting coefficients, or filter array. In many instances the dimensions of these matrices are so large that their implementation as ordinary planar convolutional filters becomes computationally inefficient. It is possible to expand the given coefficient matrix into a finite and convergent sum of matrix-valued stages. Each stage can be separated with no error into the product of an m-length column vector multiplied into an n-length row vector, where m is the number of rows and n is the number of columns of the original filter array. Substantial savings in computer storage and speed result if the given filter array can be represented with a tolerably small error by the first few stages of the expansion. Since each constituent stage consists of two vector-valued factors, further computational economies accrue if the one-dimensional sequences described by these vectors are in turn approximated by one-dimensional recursive filters. Two geophysical examples have been selected to illustrate how the present design techniques may be reduced to practice.

The Design of High-Resolution Digital Filters

Seismic recordings made with standard filters often afford insufficient resolution for overlapping reflected events. In this treatment we apply least squares Wiener theory to the design of high-resolution digital time-domain filters. Under the assumption that estimates of the seismic pulse shape are available, we present techniques that allow one to calculate digital filters which transform this pulse into one which is sufficiently sharp so that it can be distinguished against a background of noise. The two design criteria governing filter performance are filter lag and filter memory function duration. The performance of a Wiener filter is numerically measurable by a quantity which we call the filter performance parameter P, where 0 ? P ? 1. The quality of the filter output improves as P approaches unity. We thus seek that combination of lag and memory function duration that maximizes P. This goal can be accomplished by the study of a two-dimensional display of P vs. lag and memory function duration. The proposed design techniques are illustrated by means of numerical examples.

Principles of Digital Multichannel Filtering

The transition from single‐channel to multichannel data processing systems requires substantial modifications of the simpler single‐channel model. While the response function of a single‐channel digital filter can be specified in terms of scalar‐valued weighting coefficients, the corresponding response function of a multichannel filter is more conveniently described by matrix‐valued weighting coefficients. Correlation coefficients, which are scalars in the single‐channel case, now become matrices. Multichannel sampled data are manipulated with greater ease by recourse to multichannel z‐transform theory. Exact inverse filters are calculable by a matrix inversion technique which is the counterpart to the computation of exact single‐channel inverse operators by polynomial division. The delay properties of the original filter govern the stability of its inverse. This inverse is expressible in the form of a two‐stage cascaded system, whose first stage is a single‐channel recursive filter. Optimum multichannel ...

The Stabilization of Two-Dimensional Recursive Filters via the Discrete Hilbert Transform

Two-dimensional recursive filters are useful only if stable, that is, if their outputs remain bounded for bounded inputs. The stability of a recursive filter depends on the phase spectrum of its denominator array. A two-dimensional generalization of the discrete Hilbert transform leads to a scheme producing stability with nominal distortions of the filters desired amplitude spectrum. The method is therefore an attractive alternate to a least-squares procedure recently described by Shanks et al.