Migration

The experiment modeled following the procedure described above can be imaged by using conventional migration of aerial shot data. For example, the reverse time migration of the data can be expressed as:  

$$\widebar{I}_{i}\left(z_\xi,x_\xi,h_\xi\right)= \sum_{t} \wid... ...{P^g}_{_{\alpha}x_\xi^{i}}\left(t,z=z_\xi,x=x_\xi+h_\xi\right),$$ (9)

where the source and receiver data provide the surface boundary conditions to generate the source wavefield $\widebar{P^s}_{_{\alpha}x_\xi^{i}}$and the receiver wavefield $\widebar{P^g}_{_{\alpha}x_\xi^{i}}$.

To theoretically analyze the migration results we can express these wavefields as function of the data convolved, both in time and space, with the convolutional operator G₁, that represents the propagation from the surface into the subsurface. For the sake of simplicity, I limit this analysis to the flat-reflector case, where the wavefields at depth can be written as:

$$\begin{eqnarray} \widebar{P^s}_{x_\xi^{i}}\left(t,x,z\right) &=& G_1\left(x,z=... ... G_1\left(x,z=0;x,z\right) \ast D^g_{x_\xi^{i}}\left(t,x\right).\end{eqnarray}$$ (10) (11)

Similarly, the data themselves can be represented as the convolution of the image (exploding reflectors) with the operator G₀ representing the propagation from the subsurface up to the surface; that is, as:

$$\begin{eqnarray} D^s_{x_\xi^{i}}\left(t,x\right) &=& G_0\left(x=x_\xi^{i}-h_\x... ...i,z=z_\xi;x,z=0 \right) \ast I\left(z_\xi,x_\xi^{i},h_\xi\right).\end{eqnarray}$$ (12) (13)

Substituting equation 12 into equation 10, and equation 13 into equation 11, I obtain:

$$\begin{eqnarray} \widebar{P^s}_{x_\xi^{i}}\left(t,x,z\right) &=& G_1\left(x,z=... ...i,z=z_\xi;x,z=0 \right) \ast I\left(z_\xi,x_\xi^{i},h_\xi\right).\end{eqnarray}$$ (14) (15)

In general, the velocity models used for modeling and migration may be different, and thus the forward and backward propagators G₀ and G₁ are not the ``inverse'' of each other. Their convolution can be expressed as:  

$$\Delta G \left( x_0,z_0;x_1,z_1 \right) = G_0\left(x=x_0,z_0 ; x,z=0 \right) \ast G_1\left(x,z=0;x=x_1,z_1 \right).$$ (16)

When the modeling and migration velocity functions are the same, the convolutional operator $\Delta G$ is concentrated around $\left(x_0=x_1, z_0=z_1\right)$;otherwise, it shifts the wavefields proportionally to the differences between the velocities.

Substituting equation 14 and 15 into equation 9, I obtain the following expression for the image after migration

$$\begin{eqnarray} \lefteqn{\widebar{I}_{i}\left(z_\xi,x_\xi,h_\xi\right)} \nonumb... ...\right) \ast I\left(z_\xi',x_\xi^{i},h_\xi'\right), \nonumber \\ \end{eqnarray}$$

and then substituting equation 16 into equation 17 we get:

$$\begin{eqnarray} \lefteqn{\widebar{I}_{i}\left(z_\xi,x_\xi,h_\xi\right)} \nonumb... ...\xi'; \right) \ast I\left(z_\xi',x_\xi^{i},h_\xi'\right) \right].\end{eqnarray}$$ (17)

When the velocities are the same the migrated image $\widebar{I}_{i}$is an approximation of the square of the starting image I. When the migration velocity is different from the modeling velocity, the two images may substantially differ. Because we want to use migration results to estimate velocity, it is important to demonstrate that the velocity information contained in the prestack image obtained from the data modeled using the proposed procedure is consistent with the velocity information extracted from the prestack image obtained from migrating the whole data set.

Figure  shows the SODCIGs and the Angle Domain Common Image Gathers (ADCIGs) obtained by migrating the data sets shown in Figures  and , and compares these ADCIGs with the ADCIG (Figure e) computed from the SODCIG used as initial condition for modeling. The ADCIGs show the same kinematics (all three of them are flat), though the amplitudes in the ADCIGs obtained by migrating the data modeled by the proposed method is lower at wide aperture angles. This difference is likely to be related to the Jacobian of imaging, similarly to the phenomenon analyzed by Sava et al. (2004) for downward-continuation migration and modeling. It warrants further investigations.

Figure  shows the SODCIGs and the ADCIGs obtained by migrating the data sets shown in Figures  and  with a migration velocity too slow by 10%. The last panel (Figure e) shows the ADCIG computed from the prestack image obtained by source-receiver migration of the original 100-shots data set with a migration velocity too slow by 10%. The residual moveouts caused by the velocity error are the same for the ADCIG obtained by migrating the data modeled starting from an isolated SODCIG and by migrating the original data set.

 

Mig-Angs-INF-overn
Figure 6 SODCIGs and ADCIGs computed from the data shown in Figures  and : SODCIG from ''flat-reflector'' data (panel a), SODCIG from ''dipping-reflector'' data (panel b), ADCIG from ''flat-reflector'' data (panel c), ADCIG from ''dipping-reflector'' data (panel d), ADCIG computed from the SODCIG used as initial condition for modeling (panel e).

 

Mig-Angs-INF-SLOW-overn
Figure 7 SODCIGs and ADCIGs computed from the data shown in Figures  and  and migrated with a velocity too slow by 10%: SODCIG from ''flat-reflector'' data (panel a), SODCIG from ''dipping-reflector'' data (panel b), ADCIG from ''flat-reflector'' data (panel c), ADCIG from ''dipping-reflector'' data (panel d), ADCIG computed by a source-receiver migration of the original 100-shots data set with a velocity too slow by 10% (panel e).