Hubraletal_Geophys

Hubraletal_Geophys - GEOPHYSICS VOL 45 NO 11(NOVEMBER 1980 P 1697-1705 7 FIGS A sum autoregressive formula for the reflection response Peter Hubral

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GEOPHYSICS, VOL. 45, NO. 11 (NOVEMBER 1980); P. 1697-1705, 7 FIGS. A sum autoregressive formula for the reflection response Peter Hubral*, Sven Treitel$, and Paul FL Gutowski$ 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 lay- ered structure. An alternate representation of the reflection response can be formulated in terms of a superposition of purely ARtime-varyingminimum~delay wavelets. Each~sll~veadditinn~nf_a~~int~~to thelayeredsystem 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 inter- face 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” repre- sentation. 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 justi- fiable 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 seis- mogram are entirely equivalent, yet they allow the study of this model from two different but complementary vantage points. INTRODUCTION The normal incidence reflection response of a hori- zontally stratified medium is by now a very familiar concept to the exploration seismologist. It describes the reflected spike sequence resulting from a unit spike plane wave normally incident upon a layered system. The theory is well established (Wuenschel, 1960; Goupillaud, 1961; Trorey, 1962; Kunetz, 1964; Sherwood and Trorey, 1965; Robinson, 1967). It has contributed much to a better understanding of the nature of the seismic reflection. For instance, the significance of short period peg-leg multiple reflec- tions (Anstey, 1960; O ’Doherty and Anstey, 1971; Schoenberger and Levin, 1974) was established on the basis of numerical simulations carried out with the normal incidence model. The discovery of the importance of peg-legs has, in fact, changed our concept of the classical primary reflection. Thus, it is useful to associate a given basic primary reflection with certain multiple reflections tending to reinforce the primary, thereby enhancing its detectability.
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This note was uploaded on 01/24/2011 for the course ERE 284 taught by Professor . during the Spring '10 term at Stanford.

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Hubraletal_Geophys - GEOPHYSICS VOL 45 NO 11(NOVEMBER 1980 P 1697-1705 7 FIGS A sum autoregressive formula for the reflection response Peter Hubral

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