Paraxial eikonal equation

Denote by (x_s,z_s) the coordinates of a source point, and by (x,z) the coordinates of a general point in the subsurface. The first arrival traveltime field $\tau(x,z;x_{s},z_{s})$ is the viscosity solution of the eikonal equation

$$\begin{eqnarray} \left(\frac{\partial \tau}{\partial x}\right)^{2}+\left(\frac{\partial \tau}{\partial z}\right)^{2} & = &s^{2}(x,z)\end{eqnarray}$$ (1)

with the initial condition

$$\begin{eqnarray} \lim \left(\frac{\tau(x,z;x_{s},z_{s})}{\sqrt{(x-x_{s})^{2}+(z-z_{s})^{2}}}-\frac{1}{v(x,z)}\right) &=& 0 \nonumber\end{eqnarray}$$

as $(x,z) \rightarrow (x_{s},z_{s})$, where v is the velocity, $s=\frac{1}{v}$ is the slowness Lions (1982).

In some seismic applications, the traveltime field is only needed in regions where

$$\begin{eqnarray} \frac{\partial \tau}{\partial z} &\ge& s\cos \theta_{\max}\gt, \nonumber\end{eqnarray}$$

i.e., along downgoing, first-arriving rays making an angle $\le \theta_{\max}< \frac{\pi}{2}$ with the vertical.

To enforce this condition, we modify the eikonal equation as an evolutionary equation in depth, as suggested by Gray and May 1994:

$$\begin{eqnarray} \frac{\partial \tau}{\partial z}= H(\frac{\partial \tau}{\parti... ...\tau}{\partial x}\right)^{2}, s^{2}\cos^{2}\theta_{\max}\right)}},\end{eqnarray}$$ (2)

where $\rm{smmax}$ is a smoothed $\max$ function:

$$\begin{eqnarray} \rm{smmax}(x,a) &=& \left\{ \begin{array} {lll} \frac{1}{2}a &... ... < a$}, \\ x & \mbox{if $x\ge a$}. \end{array} \right. \nonumber\end{eqnarray}$$

This equation defines a stable nonlinear evolution in z, suitable for explicit finite difference discretization. The solution $\tau$ is identical to the solution of the eikonal equation provided that the ray makes an angle $\le \theta_{\max}< \frac{\pi}{2}$ with the vertical.