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GEOPHYSICS,
VOL.
45, NO. 11 (NOVEMBER
1980); P. 16971705, 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
autoregressivemoving 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
ARtimevaryingminimum~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 timevarying 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 onedimensional (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 pegleg 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 peglegs 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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