An important first step in applying the Kalman filter to seismic data is the interpretation of the so called "state equations" on which the filter is based. We propose two new interpretations, both of which include the time-varying effects of attenuation and spherical divergence. We derive the first interpretation from a time-varying, auto-regressive, moving-average (ARMA) model.
The primary difference between the MA and the ARMA approaches is the way in which estimates of reflectivity are computed. With the MA model, fixed-lag smoothed, linear, minimum-error-variance estimates are obtained directly from the most basic Kalman filter algorithm. With the ARMA model, such estimates require costly extensions to the basic algorithm, implying that the ARMA approach may be preferred only when the MA model cannot parsimoniously represent a seismogram. We illustrate both the MA and ARMA interpretations by Kalman filtering synthetic seismograms.
STANFORD