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

Dip-constrained TTI media

To appreciate the simplification attained from this constraint , we initially restrict our discussions to a homogeneous medium. In this case, the zero-offset isochron, which is representative of the equal traveltime surface, is spherical in shape, equivalent to the isotropic medium isochron, with a radius governed by the velocity in the tilt direction, $v_T$ , as follows:

$$r\left ({\bf x}\right)= v_T t\left ({\bf x}\right),$$ (1)

where $t$ is the time along the wavefront and ${\bf x}=\{x,y,z\}$ represents space coordinates. This convenient assertion is only true if we constrain the tilt axis to the direction normal to the reflector dip, and thus the group velocity equals the phase velocity equals the velocity along the tilt. Figure 1(a) shows a schematic plot of the zero-offset isochron with two representative examples of tilt direction that are constrained to be orthogonal to the isochron surface. 2Though such a medium do not physically exist, it is assumed here in the context of a process, and thus what matters is the local action of the isochron on the reflection, which is similar to the isotropic case.

For non zero-offset case, the traveltime isochron is constrained by the double-square-root (DSR) formula (Claerbout, 1995). Thus, the total traveltime, $t$ , is a combination of traveltimes from the source ${\bf s}$ located at ($s_x$ ,$s_y$ ), and the receiver ${\bf r}$ located at ($r_x$ ,$r_y$ ) to an image point in the subsurface at location ${\bf x}$ and is given by the expression

$$t = \sqrt{\frac{(s_x-x)^2+(s_y-y)^2+z^2}{v_g^2(\phi)} }$$

$$+ \sqrt{\frac{(r_x-x)^2+(r_y-y)^2+z^2}{v_g^2(\phi)} } \;,$$ (2)

where $v_g(\phi)$ is the group velocity as a function of group angle $\phi$ . From Figure 1(b), and considering, for simplicity, that the incident and reflected rays are confined to the vertical plane, $\phi$ can be evaluated geometrically as follows:

$$\phi = \frac{1}{2} \cos^{-1}{\frac{z}{\sqrt{(s_x-x)^2+(s_y-y)^2+z^2}}}$$

$$+ \frac{1}{2} \cos^{-1}{\frac{z}{\sqrt{(r_x-x)^2+(r_y-y)^2+z^2}}} \;,$$ (3)

otherwise we have to project the angles to the plane that constrains the incident and reflected rays. However, evaluating $v(\phi)$ 2$v_g(\phi)$ in complex media is complicated with no closed-form representation. An alternative is to rely on the phase angle by using plane waves and the Fourier decomposition.

If we reformulate the DSR equation in terms of changes in time, and thus, focus on the plane-wave relation we end up with the following DSR formula:

$$\frac{\partial t}{\partial z } = \sqrt{ {1 \over v^2(\theta) } - ... ... {1 \over v^2(\theta) } - \left (\frac{\partial t}{\partial s } \right)^2 } \;,$$ (4)

where now $v$ is the phase velocity and has a closed form representation in terms of the phase angle $\theta$ given by the acoustic approximation (Alkhalifah, 1998) as follows:

$$v^2(\theta) = \frac{1}{2} \left (v^2 (2 \eta +1) \sin^2 \theta + v_T^2 \cos^2 \theta \right)$$

$$+ \frac{1}{4} \sqrt{a \sin^4 \theta +b \sin^2 \left (2\theta\right) +c \cos^4 \theta} \;,$$ (5)

where $a=4 v^4 (2\eta +1)^2$ , $b=2 v^2 v_T^2 (1-2\eta )$ , $c=4 v_T^4$ , $v$ is the NMO velocity with respect to the tilted symmetry axis, and $\eta$ is the anisotropy parameter relating the NMO velocity to the velocity normal to the tilt. The angle $\theta$ in equation 4 is measured from the tilt direction and will also be given by the angle gather as part of the process of downward continuation as we will see later.

Thus, in the non-zero offset case the isochron depends on angle, but it is a single angle for both source and receiver rays and we do not have to worry about relating the two angles, as is the case in VTI and general TTI media. This provides us with analytical relations for plane waves at the reflection point. In this case, both the source and receiver waves have the same wave group velocity that differs along the non-zero offset isochron. In fact, for the zero-dip part of the isochron the reflection angle is at its maximum reducing to zero for a vertical reflector, as seen in Figure 1(b).

Next, we formulate the extended imaging condition, necessary for angle-gather development, for the DTI model. As shown in this section, angle gathers are also necessary for an explicit formulation of downward continuation in a DTI model.

Reflection
Figure 2. A schematic plot of the reflection geometry for a 2tilted transversely isotropic TTI medium with a tilt in the dip direction. The incident and reflection angles are the same given by the group angle $\phi$ . 2Here, ${\it s}$ and ${\it r}$ correspond, respectively, to the source and receiver locations, $d$ is the distance between the source and the reflector in the direction given by unit vector ${\bf n}$ normal to the reflector with direction described by unit vector ${\bf q}$ , and ${\bf n_s}$ and ${\bf n_r}$ are, respectively, the unit vector directions for each of the source and receiver rays with ray angle $\phi$ measured from the normal to the reflector.

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