Representing depth with VERTICAL TIME

In this section, we derive the relation between the depth and vertical time axis for a general inhomogeneous medium. Using this relation, the VTI eikonal equation is represented in the new $(x-\tau)$-domain coordinate system. We have derived a similar relation in another paper Biondo et al. (1997); for isotropic media. Also Hatton et al. (1981) implemented a similar mapping to show the limitations of time migration.

Two-way vertical time is related to depth by the following relation,  

$$\tau(x,z) = \int_0^z \frac{2}{v_v(x,\zeta)} d\zeta,$$ (3)

where v_v is the vertical P-wave velocity, which can vary vertically as well as laterally. As follows from equation 3, the stretch applied to the depth axis is laterally variant.

Alkhalifah (1997b) 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

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

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 get for typical elastic media.

The eikonal equation includes first-order derivatives of traveltime with respect to position. In order to transform this eikonal equation from the depth to the time coordinate, we need to replace x with $\tilde{x}$.Using the chain rule, $\frac{\partial t}{\partial x}$ in the eikonal equation 4 is given by  

$$\frac{\partial t}{\partial x} = \frac{\partial t}{\partial \tilde{x}} + \frac{\partial t}{\partial \tau} \sigma,$$ (5)

where $\sigma$, extracted from equation (3), is written as  

$$\sigma (x,z)= \frac{\partial \tau}{\partial x} = \int_0^z \... ...ial}{\partial x}\left(\frac{1}{v_v(x,\zeta)}\right)\,\, d\zeta.$$ (6)

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

$$\frac{\partial t}{\partial z} = \frac{2}{v_v} \frac{\partial t}{\partial \tau}.$$ (7)

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

$$J = \left(\matrix{1& \sigma\cr 0& \frac{2}{v_v}\cr}\right)$$ (8)

Substituting equations (5) and (7) into the eikonal equation (4) yields the equation

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

which is indirectly independent of the vertical velocity. However, according to equation (6), $\sigma$ still depends on the vertical P-wave velocity. Rewriting equation (6) in terms of the two-way vertical time (see Appendix A) gives us  

$$\sigma(\tilde{x},\tau)=\frac{-1}{v_v(\tilde{x},\tau)} \int_0... ...ilde{x},\tilde{\tau})}{ \partial \tilde{x}}\,\, d\tilde{\tau},$$ (10)

where $\tilde{x}$ corresponds to the new coordinates ($\tilde{x}$, $\tau$). In the case of $v_v(x,z)=\alpha(z) v(x,z)$, which is a special case of lateral inhomogeneity, referred to here as laterally factorized, equation (6) takes the form  

$$\sigma (x,\tau)= \int_0^{\tau} \frac{\partial}{\partial x}\left(\frac{1}{v}\right)\,\, v d\tilde{\tau},$$ (11)

which is clearly independent of the vertical P-wave velocity. Also, equation (10) becomes  

$$\sigma(\tilde{x},\tau)=\frac{-1}{v(\tilde{x},\tau)} \int_0^{... ...ilde{x},\tilde{\tau})}{ \partial \tilde{x}}\,\, d\tilde{\tau}.$$ (12)

The eikonal equation can be used to compute seismic traveltimes in laterally factorized inhomogeneous media without the need to estimate the vertical P-wave velocity. The departure of the medium from this special condition of laterally factorized media will cause errors in traveltime calculation. We can estimate these errors by evaluating how much $\sigma$ varies between equations (6) and (11). Specifically, if $v_v(x,z)=\alpha(x,z) v(x,z)$ then  

$$\Delta \sigma (x,\tau) = \int_0^{\tau} \frac{\partial}{\partial x}\left(\frac{1}{\alpha}\right)\,\, d\tilde{\tau}.$$ (13)

If the ratio of the vertical to NMO velocity, $\alpha$, does not change laterally, $\Delta \sigma$ is equal to zero, and thus no errors will occur in traveltime calculation. The departure of $\sigma$ from zero affects only the x axis component of the wavefront; according to equations (5) and (7) it is only $\frac{\partial t}{\partial x}$ that depends on $\sigma$. The vertical component of the traveltime remains accurate no matter how much $\alpha$ varies laterally. Also, because the eikonal equation is independent of $\sigma$ for vertically traveling waves ($\frac{\partial t}{\partial \tau}$=0), such waves are error-free. The majority of the errors caused by lateral $\alpha$ variation occurs around 45-degree wave propagation.

In terms of VTI parameters, the NMO velocity is given by Thomsen (1986)

$$v(x,z) = v_v(x,z) \sqrt{1+2\delta(x,z)}.$$

Therefore,

$$\alpha(x,z) = \frac{1}{\sqrt{1+2\delta(x,z)}},$$

and

$$\frac{d \alpha}{dx}= -\frac{1}{(1+2\delta(x,z))^{\frac{3}{2}}}\frac{d \delta}{dx} \approx -\frac{d \delta}{dx}.$$

Then

$$\Delta \sigma = -2 \int_0^{\tau} \frac{d \delta}{dx} d\tilde{\tau}.$$

We can see that the absolute error, resulting from the integral formulation, clearly increases with time.

In addition, when we use the new coordinate system $(x,\tau)$, the transport equation becomes independent of the vertical velocity under the same condition of laterally factorized media (see Appendix B). Bellow, and for simplicity, we will replace $\tilde{x}$ with x to denote the lateral coordinate in the new coordinate system.