Identification of seismic reflections using singular value decomposition

Abstract

Singular value decomposition (SVD) is applied to the identification of seismic reflections by using two different models: the impulse response lOdel, where a seismic trace is assumed to consist of a known signal pulse convolved with a reflection coefficient series plus noise and the delayed pulse model, where the seismic signal is assumed to consist of a small number of delayed pulses of known shape and with unknown amplitudes and arrival times. SVD clearly shows how least-squares estimation of the reflection coefficients may become unstable, since a division by the singular values is required. Two methods for stabilizing this procedure are investigated. The inverse of the singular values may be replaced by zeros when they are less than a given threshold. This is called the SVD cut-off method. Alternatively, we may use ridge regression which in filter design corresponds to assuming white noise. Statistical methods are used to compute an optimal SVD cut-off level and also to compute an optimal weighting parameter in ridge regression. Numerical studies indicate that the use of SVD cut-off or ridge regression stabilizes the least-squares procedure, but that the results are inferior to maximum-likelihood estimation where the noise is assumed to be filtered white noise.more » For the delayed pulse model, we use a linearization procedure to iteratively update the estimates of both the reflection amplitudes and the arrival times. In each step, the optimal SVD cut-off method is used. Confidence regions for the estimated reflection amplitudes and arrival times are also computed. Synthetic data examples demonstrate the effectiveness of this method. In a real data example, the maximum-likelihood method assuming an impulse response model is first used to obtain initial estimates of the number of reflections and their amplitudes and traveltimes. Then the iterative procedure is used to obtain improved estimates of the reflection amplitudes and traveltimes.« less