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

Abstract

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.