Maximum‐likelihood estimation of seismic impulse responses

Abstract

A seismic trace is assumed to consist of a known signal pulse convolved with a reflection coefficient series plus a moving average noise process (colored noise). Multiple reflections and reverberations are assumed to be removed from the trace by conventional means. The method of maximum likelihood (ML) is used to estimate the reflection coefficients and the unknown noise parameters. If the reflection coefficients are known from well logs, the seismic pulse and the noise parameters can be estimated. The maximum likelihood estimation problem is reduced to a nonlinear least-squares problem. When the further assumption is made that the noise is white, the method of maximum likelihood is equivalent to the method of least squares (LS). In that case the sampling rate should be chosen approximately equal to the Nyquist rate of the trace. Statistical and numerical properties of the ML- and the LS-estimates are discussed briefly. Synthetic data examples demonstrate that the ML-method gives better resolution and improved numerical stability compared to the LS-method. A real data example shows the ML- and LS-method applied to stacked seismic data. The results are compared with reflection coefficients obtained from well log data.