An Efficient Algorithm for Impedance Reconstruction by the Modified Gelfand-Levitan Inverse Method
We present a fast approximate algorithm for reconstructing one-dimensional acoustical impedance profiles from reflection seismograms. Carroll and Santosa (1981) treated an equation different from the Schrodinger wave equation and derived the modified Gelfand-Levitan (MGL) integral equation. Their time domain interpretation (Carroll and Santosa, 1982) of the MGL equation was more natural than the usual Gelfand-Levitan equation in the context of exploration seismology. Santosa (1982) gave a numerical solution of the MGL based on Gaussian elimination, but that approach requires a large amount of computational time and memory.
We refine the MGL equation by a few changes of variables. The refined form of the equation yields a Toeplitz system which can be solved by Levinson recursion. The discrete version of the transformed impedance equation reconstructs the acoustical impedance profile of a Goupillaud type layered medium on an even grid. Our matrix solution is analogous to the Levinson recursion solution for the Wiener shaping filter. Since the systems developed in this paper can be solved more efficiently than general symmetric positive definite systems, our approach is a more efficient method for reconstructing impedance profiles.
KeywordsAcoustical Impedance Gaussian Elimination Toeplitz Matrice Inverse Filter Toeplitz System
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