Modeling FEAVO-affected data

It is not very straightforward why one-way 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. Shot-profile 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?

Exploding-reflector modeling with the one-way 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 two-way modeling, in which the energy travels along leg (1) then along leg (2) produces a focused beam of width w1. This can be different from the beam of width w2 produced by doubling traveltimes obtained by one-way 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 second-order 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 amplitude-sensitive processes, such as Wavefield-Extrapolation Migration Velocity Analysis, which inverts amplitude anomalies into velocity updates. There are only two cases when the approximations of one-way 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. Two-pass one way or two-way wave equation algorithms should be used for FEAVO modeling in such cases.