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It is not very straightforward why oneway wavefield extrapolation
schemes would cause problems with modeling FEAVO, while they are fine
for the adjoint of modeling, migration. There are however differences.
Some operations can be irreversible (in the information
theory meaning of the word) even if they have an adjoint. For example,
summation of several values (``state 1'') into a single one (``state 2'')
has the spreading of the sum as an adjoint. But a large quantity of
information (all the frequencies except the zero one) was lost when
summing, and spreading cannot recover that. State 2 simply has less
information (and more entropy) than state 1, and anything we do to
state 2 cannot reverse that (i.e. spreading only recovers the zero
frequency). A related phenomenon happens during the imaging
condition. Shotprofile migration, in the example previously described, has information on the source wavefield and receiver wavefield before they are combined during the imaging condition. The new state (after the imaging condition) has less information than the old one, and when trying to go back, we cannot recover lost information without paying more in computational expenses. What errors were introduced by the loss?
Explodingreflector modeling with the oneway wave equation is a
popular way of generating seismic data. At each depth level, the
reflectivity values are spread to all offsets, added to the wavefield
being upward propagated from below, then the wavefield is marched
upwards to the next level. The fact that the wavefields travel
along only a single propagation leg
is accounted for by halving the velocity, effectively multiplying the
traveltimes by two. This produces correct traveltimes, correct geometry
of FEAVO patterns (Kjartansson V's), and the FEAVO in the resulting
data is eliminated by migration. The problem is that focusing, while
localized when compared to the size of the survey, is not a binary
condition. Figure
f5
Figure 5 Performing
a real experiment or twoway modeling, in which the energy travels
along leg (1) then along leg (2) produces a focused beam of width
w_{1}. This can be different from the beam of width w_{2}
produced by doubling traveltimes obtained by oneway modeling, even
if at a scale at which the width of the beams is negligible, the
travel paths are identical.

 
shows the details. If only pure
absorption is involved, it would not matter whether the heterogeneity
lied closer to the beginning than to the end of the travel path:
multiplication of amplitudes by an absorbtion factor is commutative.
But if velocity is at play, as it is often the case, then the
microscale of the effects (assumed to be divergent in the figure) will
look different if: (A) the wavefront goes along legs 1 and most of 2
and then encounters the velocity anomaly, as in the real experiment,
or (B) it travels only along leg 2 in the numerical experiment and the traveltimes are multiplied by two. The microcharacter of the FEAVO effects will look different than for real data. This is a secondorder effect only, but it is real. It can be ignored if the scope of the analysis is of a larger scale, but it can be important in particularly amplitudesensitive processes, such as WavefieldExtrapolation Migration Velocity Analysis, which inverts amplitude anomalies into velocity updates.
There are only two cases when the approximations of oneway modeling
are not a problem. The first is when the anomalies are purely due to
absorption. The second case is when migrating the modeled data with
the correct velocity and with an imaging algorithm close in accuracy
(adjoint if possible) to the one used in modeling. In other cases,
especially when studying the behavior of FEAVO itself, this effect
should be taken into account. Twopass one way or twoway wave
equation algorithms should be used for FEAVO modeling in such cases.
Next: Conclusions
Up: Vlad: Focusingeffect AVA
Previous: Migrating FEAVOaffected data
Stanford Exploration Project
5/3/2005