A transversely isotropic medium with a tilted symmetry axis normal to the reflector |

The angle decomposition discussed in the preceding section allows us to produce angle gathers after downward continuation in DTI media. Wavefield reconstruction for multi-offset migration based on the one-way wave-equation under the survey-sinking framework (Claerbout, 1985) is implemented by recursive phase-shift of prestack wavefields

followed by extraction of the image at time . Here, and represent the midpoint and half-offset coordinates, which are equivalent with the space and space-lag variables discussed earlier, but restricted to the horizontal plane. In equation 29, represents the acoustic wavefield for a given frequency at all midpoint positions and half-offsets at depth , and represents the same wavefield extrapolated to depth . The phase shift operation uses the depth wavenumber which is defined in 2D by the DSR equation 4 as follows:

where is equivalent to .

Figure 4 shows as a function of the midpoint wavenumber and the reflection angle for a DTI model characterized by (left). As expected, the range of angles reduces with increasing dip angle (or ). The phase shift per depth is maximum for horizontal reflector ( ) and zero offset (equivalent with ). The right plot in Figure 4 shows the difference between the for this DTI model and that for an isotropic model with velocity equal to km/s. As expected, for zero reflection angle, the DTI phase shift is given by the isotropic operator as we discussed earlier. For the non-zero-offset case, the difference increases with the reflection angle.

To use in this form we need to evaluate the reflection angle, , in the downward continuation process as the angle gather defines the phase angle needed for equation 30. Equation 28 provides a one-to-one relation between angle gathers and the offset wavenumber. However, to insure an explicit evaluation we formulate the problem as a mapping process to find the wavefield for a given offset wavenumber that corresponds to a particular reflection angle. As a result, we can devise an algorithm for downward continuation for a wavefield with sources and receivers at depth as follows:

- For a given reflection angle, use equation 28 to find the corresponding ( ).
- Using , map to (the angle decomposition).
- Apply the imaging condition by summing over frequencies to obtain imaged angle gathers.
- Apply phase shift to the wavefield to obtain by equation 29 using the depth wavenumber given by equation 30.
- Repeat the steps for depth .

kz
A plot of the vertical wavenumber,
, as a function of midpoint wavenumber and angle gather for a 2dip-constrained transversely isotropic
(DTI) DTI model with 2NMO velocity,
=2.0 km/s, 2titled direction velocity,
=1.8 km/s, and
(left)
and the difference in
between the DTI model and an isotropic
model with
=1.8 km/s (right).
The wave numbers are given in units of
.
Figure 4. |
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A transversely isotropic medium with a tilted symmetry axis normal to the reflector |

2013-04-02