Abstract

The normal incidence unit impulse reflection response of a perfectly stratified medium is expressible as an autoregressive-moving average (ARMA) model. In this representation, the autoregressive (AR) component describes the multiple patterns generated within the medium. The moving average (MA) component, on the other hand, bears a simple relation to the sequence of reflection coefficients (i.e., primaries only) of the layered structure.An alternate representation of the reflection response can be formulated in terms of a superposition of purely AR time-varying minimum-delay wavelets. Each successive addition of a deeper interface to the layered system gives rise to an AR wavelet whose leading term is equal to the magnitude of the primary reflection originating at this interface. We accordingly call these wavelets 'generalized primaries.' The AR component of every generalized primary contains only those multiple reflections that arise from the addition of its particular interface to the layered medium.Therefore, the impulsive reflection seismogram can be decomposed into a progressively delayed summation of as many generalized primaries as there are reflection coefficients, referred to here as a 'sum AR' representation. Because each generalized primary is a pure AR time-varying wavelet, it becomes meaningful to consider a short time gate of a seismogram to be approximately representable by an AR model. In turn, this means that maximum entropy spectral analysis (MESA) applied to a short time gate of a seismogram is justifiable on the basis of the one-dimensional (1-D) wave equation model.The conventional (ARMA) and the alternate (sum AR) representations of the impulsive reflection seismogram are entirely equivalent, yet they allow the study of this model from two different but complementary vantage points.

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