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

Downward Continuation

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

$$W_{z+\Delta z}\left ({\bf m},{\bf h}\right)= e^{\textcolor{red}{- i k_z \Delta z}}W_z\left ({\bf m},{\bf h}\right)\;,$$ (29)

followed by extraction of the image at time $t=0$ . Here, ${\bf m}$ and ${\bf h}$ 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, $W_z\left ({\bf m},{\bf h}\right)$ represents the acoustic wavefield for a given frequency $\omega$ at all midpoint positions ${\bf m}$ and half-offsets ${\bf h}$ at depth $z$ , and $W_{z+\Delta z}\left ({\bf m},{\bf h}\right)$ represents the same wavefield extrapolated to depth $z+\Delta z$ . The phase shift operation uses the depth wavenumber $k_z$ which is defined in 2D by the DSR equation 4 as follows:

$$k_z = \sqrt{ {\omega^2 s^2(\theta) } - \left (k_m - k_h \right)^2} + \sqrt{ {\omega^2 s^2(\theta) } - \left (k_m + k_h \right)^2 } \;,$$ (30)

where $k_h$ is equivalent to $k_{\lambda_x}$ .

Figure 4 shows $k_z$ as a function of the midpoint wavenumber and the reflection angle for a DTI model characterized by $\eta =0.3$ (left). As expected, the range of angles reduces with increasing dip angle (or $k_m$ ). The phase shift per depth is maximum for horizontal reflector ($k_m=0$ ) and zero offset (equivalent with $\theta=0$ ). The right plot in Figure 4 shows the difference between the $k_z$ for this DTI model and that for an isotropic model with velocity equal to $v=1.8$  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 $k_z$ in this form we need to evaluate the reflection angle, $\theta$ , 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 $k_h$ 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 $z$ as follows:

• For a given reflection angle, use equation 28 to find the corresponding $k_h$ ( $=k_{\lambda_x}$ ).
• Using $k_h(\theta)$ , map $W(k_m,k_h,\omega ,z)$ to $W(k_m, \theta,\omega ,z)$ (the angle decomposition).
• Apply the imaging condition by summing over frequencies $\omega$ to obtain imaged angle gathers.
• Apply phase shift to the wavefield $W(k_m,k_h,\omega ,z)$ to obtain $W(k_m,k_h,\omega , z+\Delta z)$ by equation 29 using the depth wavenumber given by equation 30.
• Repeat the steps for depth $z=z+\Delta z$ .

The process provides imaged angle gathers in DTI media. This approach also allows us to better treat illumination as we downward continue while keeping the sampling in reflection angle uniform.

kz
Figure 4. A plot of the vertical wavenumber, $k_z$ , as a function of midpoint wavenumber and angle gather for a 2dip-constrained transversely isotropic (DTI) DTI model with 2NMO velocity, $v$ =2.0 km/s, 2titled direction velocity, $v_T$ =1.8 km/s, and $\eta =0.3$ (left) and the difference in $k_z$ between the DTI model and an isotropic model with $v$ =1.8 km/s (right). The wave numbers are given in units of $km^{-1}$ .

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