Representing depth with VERTICAL TIME

In this section, I will summarize the derivation of the VTI eikonal equation for the (x-$\tau$)-domain, derived originally by Alkhalifah et al. (1997).

Obviously, the two-way vertical traveltime is related to depth,  

$$\tau(x,z) = \int_0^z \frac{2}{v_v(x,\zeta)} d\zeta,$$ (1)

where v_v is the vertical P-wave velocity, which can vary vertically as well as laterally. As a result, the stretch applied to the depth axis is laterally variant.

Alkhalifah (1997a) derived a simple form of the eikonal equation for VTI media, based on setting the shear wave velocity to zero. For 2-D media, it is

$$\begin{eqnarray} {v^2}\,\left( 1 + 2\,\eta \right) \,{\left(\frac{\partial t}{\p... ...\,\eta \,{\left(\frac{\partial t}{\partial x}\right)^2} \right)=1.\end{eqnarray}$$ (2)

This equation, based on the acoustic medium assumption in VTI media, though not physically possible, yields extremely accurate traveltime solutions that are close to what we find for typical elastic media.

Clearly, equation 2 includes first-order derivatives of traveltime with respect to position. In order to transform this eikonal equation from depth to time, we need to replace x with $\tilde{x}$, as well. Using the chain rule, $\frac{\partial t}{\partial x}$ in the eikonal equation 2 is given by  

$$\frac{\partial t}{\partial x} = \frac{\partial t}{\partial \tilde{x}} + \frac{\partial t}{\partial \tau} \sigma,$$ (3)

where $\sigma$, derived from equation (1),  

$$\sigma (x,z)= \frac{\partial \tau}{\partial x} = \int_0^z \... ...ial}{\partial x}\left(\frac{1}{v_v(x,\zeta)}\right)\,\, d\zeta.$$ (4)

Likewise, the partial derivative in z of the eikonal equation is  

$$\frac{\partial t}{\partial z} = \frac{2}{v_v} \frac{\partial t}{\partial \tau}.$$ (5)

Therefore, the transformation from (x, z) to ($\tilde{x}$, $\tau$) is governed by the following Jacobian matrix in 2-D media:

$$J = \left(\matrix{1& \sigma\cr 0& \frac{2}{v_v}\cr}\right)$$ (6)

Substituting equations (3) and (5) into the eikonal equation (2) yields an eikonal equation in the ($x-\tau$)-domain given by

$$\begin{eqnarray} {v^2}\,\left( 1 + 2\,\eta \right) \,{\left(\frac{\partial t}{\p... ...}+ \frac{\partial t}{\partial \tau} \sigma \right)^2} \right)=1,\end{eqnarray}$$ (7)

which is also indirectly independent on the vertical velocity. However, according to equation (4), $\sigma$ still depends on the vertical P-wave velocity. Alkhalifah et al. (1997) demonstrated that if the medium was factorized laterally ($v_v(x,z)=\alpha(z) v(x,z)$), then  

$$\sigma(\tilde{x},\tau)=\frac{-1}{v(\tilde{x},\tau)} \int_0^{... ...ilde{x},\tilde{\tau})}{ \partial \tilde{x}}\,\, d\tilde{\tau}.$$ (8)

which is independent of the vertical velocity. The departure of the medium from this special condition of laterally factorized media will cause errors in traveltime calculation; these errors, however, are generally small.

Using the method of characteristics, Alkhalifah et al. (1997) derived a system of ordinary differential equations that define the ray trajectories in the ($x-\tau$)-domain. Numerical solutions of the raytracing equations, as opposed to the eikonal equation, provide multi-arrival traveltimes and amplitudes, a feature that is key to properly image the Marmousi model Geoltrain and Brac (1993). Thus, in the Kirchhoff migration examples, I use ($x-\tau$)-domain traveltime maps extracted from ray tracing.