Exploding-reflector modeling as a pseudo-unitary operator

In a constant velocity medium, the process of exploding-reflector modeling followed by exploding-reflector migration acts to preserve the amplitudes of flat-events. This is can be proved by considering the zero-offset case for the true-amplitude migration weights derived by Sava et al. (2001). Figure  confirms this by displaying the frequency-domain response of cascaded modeling and migration (${\bf A}_{\rm ER}'{\bf A}_{\rm ER}$). Panel (a) shows the input spectrum that has been band-passed and dip-limited to remove evanescent energy. Panel (b) shows the spectrum after exploding-reflector modeling and migration. Although high-spatial wavenumbers are attenuated slightly, the low-spatial wavenumbers associated with flat events remain essentially unchanged. Therefore, if a model consists of mostly flat-events, exploding-reflector modeling acts as a pseudo-unitary operator: ${\bf A}'_{\rm ER} {\bf A}_{\rm ER}\approx {\bf I}$.

Equation () decomposes exploding-reflector modeling by wavefield extrapolation into a spraying operator, ${\bf \Sigma}_{\omega}'$, followed by an extrapolation/recording operator, ${\bf B}$, so that

$$\begin{eqnarray} {\bf d} & = & {\bf A}_{\rm ER} \; {\bf m} \\ & = & {\bf B} \; {\bf \Sigma}_{\omega}' \; {\bf m}\end{eqnarray}$$ (80) (81)

where, in the terminology of Chapter , ${\bf B}={\bf Z}'_{N_\omega} \; ({\bf D}')^{-1}$. For a model consisting of mostly flat-events, this implies ${\bf B}' \, {\bf B} \approx {\bf I}$ subject to a multiplicative constant.

If the approximation that ${\bf B}' \, {\bf B} \approx {\bf I}$ is indeed valid, then amplitude problems on zero-offset migrations must be associated with the two-way wave propagation rather than the exploding-reflector migration. To test the validity of this conjecture, I can generate exploding-reflector and genuine zero-offset seismograms and compare the results of exploding-reflector migration on the two datasets.

Figure  compares exploding-reflector data with true zero-offset data. I generated the two datasets by running the adjoint of exploding-reflector and shot-profile downward continuation migration algorithms respectively. Kinematically they are similar; however, vertical amplitude streaking is much more apparent in the true zero-offset section. Field datasets, such as the stacked section shown in Figure , often contain vertical amplitude streaks. Such streaks often pose a dilemma for a processing geophysicist as to how they should be correctly treated.

 

marmERZOdata
Figure 4 Synthetic Marmousi datasets generated with one-way wave modeling: panel (a) shows the exploding-reflector dataset, and panel (b) shows the zero-offset dataset. The ellipses highlight the increased vertical streaking in the true zero-offset dataset.

 

illumexamples Figure 5 Stacked section from the Mobil AVO dataset Clapp (1999). Vertical streaking is visible throughout the section.

An even more interesting picture emerges when the two modeled datasets are migrated (Figure ) with an exploding-reflector migration algorithm. The exploding-reflector data migrates nearly perfectly. Whereas the high amplitude streaks in the true zero-offset dataset migrate into the commonly seen ``migration-smile'' artifacts.

 

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Figure 6 Marmousi reflectivity: panel (a) shows the initial model, panel (b) shows the reflectivity estimate by migrating the exploding-reflector data shown in Figure  (a), and panel (c) shows the reflectivity estimate obtained by migrating the zero-offset data from Figure  (b).