White-noise estimators for seismic data processing in oil exploration

Abstract
This paper is motivated by a problem from seismic data processing in oil exploration. We develop a Kalman filtering approach to obtaining optimal smoothed estimates of the so-called reflection coefficient sequence. This sequence contains important information about subsurface geometry. Our problem is shown to be equivalent to that of estimating white-plant noise for a linear dynamic system. By means of the equations which are derived herein, it is possible to compute fixed-interval, fixed-point, or fixed-lag optimal smoothed estimates of the reflection coefficient sequence, as well as respective error covariance-matrix information. Our optimal estimators are compared with an ad hoc "prediction error filter," (PEF) which has recently been reported on in the geophysics literature. We show that one of our estimators performs at least as well as, and in most cases, better than the prediction error filter. Simulation results are given which support and illustrate the theoretical developments.