3-D ALGORITHM

In 3-D the eikonal equation is

 

u²+v²+w²=s² (8)

where

$$\left \{ \begin{array} {l} u={\partial{t(x,y,z)} \over \part... ...\ \\ w={\partial{t(x,y,z)} \over \partial z}\end{array} \right.$$

For a spherical-coordinates system  

$${w^2+{v^2 \over r^2}+{u^2 \over {r^2{\sin \phi}^2}} }=s^2$$ (9)

where

$$\left \{ \begin{array} {l} w={\partial{t(r,\theta,\phi)} \ov... ...al{t(r,\theta,\phi)} \over \partial \theta} \end{array} \right.$$

The cross derivative equation (4) is transformed into the spherical coordinates system  

$$\left \{ \begin{array} {l} {\partial u \over \partial r}= {\... ...partial r}= {\partial w \over \partial \phi}\end{array} \right.$$ (10)

The finite-difference equivalent of equation (5) is the system  

$$\left \{ \begin{array} {l} u(r+\Delta r,\theta,\phi)=u(r,\th... ...\over \Delta \phi} {\Delta w(r,\theta,\phi)}\end{array} \right.$$ (11)

which is used to advance the stencil for a new radial increment. Once the values of the functions $u(r,\theta,\phi)$ and $v(r,\theta,\phi)$ are known on the spherical front with constant radius ($r+\Delta r$), the third function $w(r,\theta,\phi)$can be calculated using the eikonal equation  

$$w(r+\Delta r,\theta,\phi)={ \sqrt{s^2(r+\Delta r,\theta,\phi... ...Delta r,\theta,\phi) \over {{(r+\Delta r)}^2 \sin ^2 \phi}} }}.$$ (12)

The value of the traveltime is found by integration:

$$t(r+\Delta r,\theta,\phi)=t(r,\theta,\phi)+{\Delta r \over 2} [w(r+\Delta r,\theta,\phi)+w(r,\theta,\phi)].$$

In equation (11), the Engquist-Osher scheme is applied twice, once for calculating the values of $\Delta w$ across three points of consecutive values of $\theta$, and second for calculating the values of $\Delta w$ across three consecutive values of $\phi$.The computational front advances in spherical shells, and on each shell the computations advance a circle at a time. The angle $\theta$ is the horizontal angle while the angle $\phi$ is the vertical angle. The Engquist-Osher scheme is applied along each three consecutive points on the circle with constant vertical angle $\phi$ to determine $\Delta u$ from the equation $\Delta u = {\Delta r \over \Delta \theta} {\Delta w}$.For each circle of constant vertical angle $\phi$ the Engquist-Osher scheme is applied for three points ($\phi - \Delta \phi$), $\phi$ and ($\phi + \Delta \phi$), which are perpendicular on the circle in the $(r,\theta,\phi)$ coordinates. The scheme is completely vectorizable.