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Using the method of characteristics, we can derive
a system of ordinary differential equations that define the
ray trajectories. To do so, we need to
transform equation (9) to the
following form:
| ![\begin{displaymath}
F \left(x,\tau,\frac{\partial t}{\partial x},\frac{\partial t}{\partial \tau} \right)=0,\end{displaymath}](img34.gif) |
(14) |
or
| ![\begin{displaymath}
F \left(x,\tau,p_x,p_{\tau} \right)=0,\end{displaymath}](img35.gif) |
(15) |
where
and
. According to
the classic rules of mathematical physics (Courant, 1966), the solutions of this
kinematic equation can be obtained from the system of ordinary differential equations
| ![\begin{eqnarray}
\frac{d x}{d s} = \frac{1}{2} \frac{\partial F}{\partial p_x} &...
...d p_{\tau}}{d s} = - \frac{1}{2} \frac{\partial F}{\partial \tau},\end{eqnarray}](img38.gif) |
|
| (16) |
where s is a running parameter along the rays, related to the traveltime t as follows:
![\begin{displaymath}
\frac{d t}{d s} = \frac{1}{2} p_{\tau} \frac{\partial F}{\partial p_{\tau}}+ p_x \frac{\partial F}{\partial p_x},\end{displaymath}](img39.gif)
with
| ![\begin{eqnarray}
\frac{d x}{d t} = \frac{d x}{d s} \left/ \frac{d t}{d s}\right....
...\tau}}{d t} = \frac{d p_{\tau}}{d s} \left/\frac{d t}{d s}\right..\end{eqnarray}](img40.gif) |
|
| (17) |
Using equation (9), we obtain
| ![\begin{displaymath}
\frac{d x}{d s} = a\,{v^2}\,\left( 1 + 2\,\eta \,\left( 1 - 4\,{{{p_{\tau }}}^2} \right) \right),\end{displaymath}](img41.gif) |
(18) |
| ![\begin{displaymath}
\frac{d \tau}{d s} = 4\,{p_{\tau }} - a\,{v^2}\,\left( -\sig...
...\,{p_{\tau }} + 8\,\sigma \,{{{p_{\tau }}}^2}
\right) \right),\end{displaymath}](img42.gif) |
(19) |
| ![\begin{eqnarray}
\frac{d p_{x}}{d s} = -a^2\,v\,\left( 1 + 2\,\eta \,\left( 1 - ...
...a - 8\,\eta \,{{{p_{\tau }}}^2} \right) \,
{{\sigma }_x} \right),\end{eqnarray}](img43.gif) |
|
| (20) |
| ![\begin{eqnarray}
\frac{d p_{\tau}}{d s} = -v\,a^2\,\left( 1 + 2\,\eta \,\left( 1...
...\,\eta \,{{{p_{\tau }}}^2} \right) \,{{\sigma }_{\tau }}
\right),\end{eqnarray}](img44.gif) |
|
| (21) |
and
![\begin{displaymath}
\frac{d t}{d s} = 4\,{{{p_{\tau }}}^2} + {a^2}\,{v^2}\,
\le...
... + 2\,\eta \,\left( 1 - 8\,{{{p_{\tau }}}^2} \right)
\right), \end{displaymath}](img45.gif)
where
![\begin{displaymath}
a={p_x} + \sigma \,{p_{\tau }}, \end{displaymath}](img46.gif)
and
and
, and
the same holds for
and
.
To trace rays, we must first
identify the initial values x0,
, px0, and
. The variables x0 and
describe the source position, and px0 and
are extracted from the initial
angle of propagation. Note that, from equation (9),
![\begin{displaymath}
p_{\tau 0}=1- \frac{v^2 p_{x0}^2}{1-2 \eta v^2 p_{x0}^2}, \end{displaymath}](img51.gif)
because
=0 at the source position (z=0).
The raytracing system of equations (18-21)
describes the ray-theoretical aspect of wave propagation in the
-domain, and can
be used as an alternative to the eikonal equation. Numerical solutions of the raytracing equations,
as opposed to the eikonal equation, provide multi-arrival traveltimes and amplitudes.
In the numerical examples,
we use raytracing to highlight some of the features of the
-domain coordinate system.
Next: The X-TAU acoustic wave
Up: VTI processing in inhomogeneous
Previous: Representing depth with VERTICAL
Stanford Exploration Project
9/12/2000