![next](icons/next.png) |
![up](icons/up.png) |
![previous](icons/previous.png) |
![](icons/left.png) | A robust approach to time-to-depth conversion and interval velocity
estimation from time migration in the presence of lateral velocity variations | ![](icons/right.png) |
![[pdf]](icons/pdf.png) |
Next: Appendix D: Analytical expressions
Up: Li & Fomel: Time-to-depth
Previous: Appendix B: The Fréchet
In order to derive the time-to-depth conversion analytically, we first trace image rays in the depth
coordinate for
and
. Then we carry out a direct inversion to find
and
. The Dix velocity can be obtained at last following equations 3 and 4.
Continuing from equation 15, we write the velocity in a coordinate relative to the image ray
![\begin{displaymath}
v (z,x) = v_0 + g_z z + g_x x = \tilde{v}_0 + \mathbf{g} \cdot (\mathbf{x} - \mathbf{x_0})\;,
\end{displaymath}](img163.png) |
(39) |
where
and
. At the starting point, image ray satisfies
![\begin{displaymath}
\left\{ \begin{array}{lcl}
\mathbf{x_0} & = & [0, x_0]^T\;, ...
...ilde{v}_0^{-1}, 0]^T\;, \\
t_0 & = & t\;.
\end{array} \right.
\end{displaymath}](img166.png) |
(40) |
Here we denote ray parameter
and
is the ray parameter at source. The
Hamiltonian for ray tracing reads
.
The corresponding ray tracing system is (Cervený, 2001):
![\begin{displaymath}
\left\{ \begin{array}{lcl}
d \mathbf{x} / d \xi & = & \mathb...
...i = \mathbf{p} \cdot \mathbf{p} v^3 = v\;.
\end{array} \right.
\end{displaymath}](img170.png) |
(41) |
Equation C-1 indicates
, which means
can be
integrated analytically and provides
![\begin{displaymath}
\mathbf{p} = \mathbf{p_0} - \mathbf{g} \xi\;.
\end{displaymath}](img173.png) |
(42) |
From the eikonal equation and considering
and
, we have
![\begin{displaymath}
v = \frac{1}{\sqrt{\mathbf{p} \cdot \mathbf{p}}} =
\left( \...
..._0} \cdot \mathbf{g} \xi + g^2 \xi^2 \right)^{-\frac{1}{2}}\;.
\end{displaymath}](img176.png) |
(43) |
Integrating equation C-5 over
gives
![\begin{displaymath}
t = \frac{1}{g} \mathrm{arccosh}
\left( 1 + \frac{g^2 \xi^2...
...lde{v}_0^{-1}
- \mathbf{p_0} \cdot \mathbf{g} \xi} \right)\;.
\end{displaymath}](img178.png) |
(44) |
Meanwhile, combining equations C-1 and C-3, we find
,
i.e.,
. Suppose
![\begin{displaymath}
\mathbf{x} - \mathbf{x_0} = \alpha \mathbf{p_0} + \beta \mathbf{g}
\end{displaymath}](img181.png) |
(45) |
then
![\begin{displaymath}
\left\{ \begin{array}{lcl}
\mathbf{g} \cdot (\mathbf{x} - \m...
...}_0 \xi + (v - \tilde{v}_0) \xi = v \xi\;.
\end{array} \right.
\end{displaymath}](img182.png) |
(46) |
Solving equation C-8 provides
and
, which after substituting into
equation C-7 leads to
![\begin{displaymath}
\mathbf{x} = \mathbf{x_0} + \frac{(v - \tilde{v}_0) \left[\m...
...xi}
{g^2 - (\mathbf{p_0} \cdot \mathbf{g})^2 \tilde{v}_0^2}\;.
\end{displaymath}](img185.png) |
(47) |
Note equation C-2 states
and thus equations
C-4, C-6 and C-9 can be further simplified.
To connect depth- and time-domain attributes, we first invert equation C-6 such that
is
expressed by
and
![\begin{displaymath}
\xi (t_0,x_0) = \frac{g_z (1 - \cosh (\vert g t_0\vert))
+ g \sinh (g t_0)}{g^2 \tilde{v}_0^2}\;.
\end{displaymath}](img187.png) |
(48) |
Next, we insert equations C-5 and C-10 into C-9 in order to change its
parameterization from
to
. The result is written for the
and
components of
separately, as follows:
![\begin{displaymath}
x (t_0,x_0) = x_0 + \frac{\tilde{v}_0 g_x (1 - \cosh (g t_0))}{g (g \cosh (g t_0) - g_z \sinh (g t_0))}\;,
\end{displaymath}](img190.png) |
(49) |
![\begin{displaymath}
z (t_0,x_0) = \frac{\tilde{v}_0 \left[ g_z (1 - \cosh (g t_0...
... (g t_0) \right]}
{g (g \cosh (g t_0) - g_z \sinh (g t_0))}\;.
\end{displaymath}](img191.png) |
(50) |
Inverting equations C-11 and C-12 results in
![\begin{displaymath}
x_0 (z,x) = x + \frac{\sqrt{(v_0+g_x x)^2 + g_x^2 z^2} - (v_0 + g_x x)}{g_x}\;,
\end{displaymath}](img97.png) |
(51) |
![\begin{displaymath}
t_0 (z,x) = \frac{1}{g} \mathrm{arccosh} \left\{ \frac{g^2 \...
...2 + g_x^2 z^2}
+ g_z z \right] - v g_z^2}{v g_x^2} \right\}\;.
\end{displaymath}](img192.png) |
(52) |
In the last step, we derive the analytical formula for the Dix velocity. Note that from equation
C-13
, i.e., there is no geometrical spreading. The image rays are circles parallel
to each other. Therefore according to equation 3
and is found by combining equations
C-5 and C-10
![\begin{displaymath}
v_d (t_0,x_0) = \frac{(v_0 + g_x x_0) g}{g \cosh (g t_0) - g_z \sinh (g t_0)}\;.
\end{displaymath}](img101.png) |
(53) |
The time-migration velocity
, on the other hand, is
![\begin{displaymath}
v_m (t_0,x_0) = \frac{(v_0 + g_x x_0)^2}{t_0 ( g \coth (g t_0) - g_z )}\;.
\end{displaymath}](img195.png) |
(54) |
![next](icons/next.png) |
![up](icons/up.png) |
![previous](icons/previous.png) |
![](icons/left.png) | A robust approach to time-to-depth conversion and interval velocity
estimation from time migration in the presence of lateral velocity variations | ![](icons/right.png) |
![[pdf]](icons/pdf.png) |
Next: Appendix D: Analytical expressions
Up: Li & Fomel: Time-to-depth
Previous: Appendix B: The Fréchet
2015-03-25