The use of the conjugate‐gradient algorithm in the computation of predictive deconvolution operators

Abstract

A number of excellent papers have been published since the introduction of deconvolution by Robinson in the middle 1950s. The application of the Wiener‐Levinson algorithm makes deconvolution a practical and vital part of today’s digital seismic data processing. We review the original formulation of deconvolution, develop the solution from another perspective, and demonstrate a general and rigorous solution that could be implemented. By “general” we mean a deterministic time‐varying and multichannel operator design, and by “rigorous” we mean the straightforward least‐squares error solution without simplifying to a Toeplitz matrix. Also we show that the conjugate‐gradient algorithm used in conjunction with the least‐squares problem leads to a satisfactory simplification; that in the computation of the operators, the square matrix involved in the normal equations need not be computed. Furthermore, the product of this matrix with a column matrix can be obtained directly from the data as a result of two cascaded simple convolutions. The time‐varying deconvolution problem is shown to be equivalent to the multichannel deconvolution problem. Hence, with one simple formulation and associated programming, the procedure can be utilized for time‐constant single‐channel and multichannel deconvolution and time‐varying single‐channel and multichannel deconvolution.