Threshold algorithms for the prediction of reflection coefficients in a layered medium

Abstract

An algorithm is presented for the solution of the inverse problem of reflection seismology in the presence of noise. The algorithm is based on a new representation of reflection coefficients in terms of the recorded seismogram. This representation allows use of matrix perturbation methods for the analysis of error magnification in the recursive reconstruction of a stratified acoustic medium. Our analysis indicates that one of the main reasons for uncontrollable noise magnification is the assignment of significant values to very small reflection coefficients, most of which reflect only noise in the data rather than reflection information. Our analysis also allows one to decide when a small computed reflection coefficient should be set to zero. The strategy of setting small reflection coefficients to zero, which is called thresholding, has a stabilizing effect on inverse scattering algorithms. The threshold algorithm also permits adaptive change of noise barriers, which can be used for more detailed exposure of certain parts of a seismic section at the expense of its less important parts. These properties of the threshold algorithm are demonstrated on both synthetic examples and sets of seismic survey data. The general version of the threshold algorithms allows efficient implementation on modern computer architectures (such as parallel or pipelined computers). In particular, the algorithm can be implemented with linear complexity on parallel processors. Simplified versions of the general algorithm for special surface conditions are also presented.