Results

In order to test the anisotropic code based on equation (5), I use two different 2-D synthetic models. The first synthetic model is characterized by a vertical and lateral linear gradient of 0.5s^-1. Dipping and flat reflectors are embedded in the velocity field. The second model is the anisotropic Marmousi model generated by Alkhalifah (1997a).

The extended anisotropic split-step algorithm is based on a linear interpolation of the different downward continued wavefields with every reference velocity. Therefore, this algorithm by definition is able to handle linear lateral velocity gradient that characterized the synthetic seismic in Figure 1. In contrast, the Marmousi model has a complex velocity field that represents a challenge to the split-step anisotropic migration.

The seismic synthetic data for the first model was modeled using an analytic ray tracer for a factorized transversely isotropic medium, given by Alkhalifah (1995). The resulting zero-offset section modeled with this program is shown in Figure 1. The reflectors have a slope of $0^\circ$,$30^\circ$, $45^\circ$, $75^\circ$ and $90^\circ$, and the Thomsen's parameters are ${\delta=0}$ and ${\epsilon=0.2}$ and $\eta=\epsilon$(equation 6). Working with $\eta=\epsilon$ guarantees that the vertical velocity is equal to the migration velocity [equation (7)]

Figures 2 and 3 show the migrated zero-offset section with the extended anisotropic split-step depth migration for ${\eta=0}$ and ${\eta=0.2}$,respectively, and the correct migration velocity. This value of ${\eta}$correspond to a ratio between horizontal and vertical P-wave velocity of about $20\%$ Alkhalifah (1997a). As expected, an isotropic migration (${\eta=0}$) looks under-migrated and it needs higher migration velocities in order to correctly image the anisotropic seismic data (Figure 2). Figure 3 shows the resulting image using the correct migration velocity and ${\eta=0.2}$. In this case, I use five reference velocities to represent linear lateral change in velocity. It can be observed that the reflectors with dips $30^\circ$ and $45^\circ$ are well imaged. In contrast, reflectors with greater dips ($75^\circ$ and $90^\circ$) are not imaged because the number of reference velocities is insufficient to handle those dips.

Figure 4 shows the prestack migration image resulting from applying the extended anisotropic split-step migration for TI media. Like the zero-offset anisotropic migration (Figure 3), this prestack image was obtained by using 5 reference velocities. The first two reflectors are well imaged, although for deeper reflectors it is necessary to increase the number of reference velocities in the split-step migration.

Figure 5 shows the Marmousi velocity field used to model the anisotropic seismic data set Alkhalifah (1997a) . Overall, this finite difference modeling has the same survey geometry used by IFP. This includes the same source and receiver locations, an identical sampling interval and recording time, and the same minimum offset. The central frequency of this anisotropic data set is 30Hz and the maximum offset is 3575 m.

In the anisotropic Marmousi model, the parameter ${\eta}$ is a function of lateral coordinates and depth. The ${\eta}$ field was created following the original velocity field (Figure 5) and honoring a linear variation of the horizontal velocity in depth. Therefore, velocities greater than 2500m/s and lesser than the water velocity have associated a ${\eta=0}$.

Figure 6 shows the prestack anisotropic migration with 5 reference velocities and 5 reference ${\eta}$'s. The reference ${\eta}$'s are calculated in every depth step during the downward continuation. This prestack migrated section is a good image. It is important for imaging to take care of the vertical and lateral variation of ${\eta}$ is important because ${\eta}$ anomalies can cause small triplications in the wavefront Alkhalifah et al. (1997). In addition, if the anisotropic migration with constant ${\eta}$ is performed on this data set, dipping reflectors would be imaged with a smaller dip than in the original model