![next](icons/next.png) |
![up](icons/up.png) |
![previous](icons/previous.png) |
![](icons/left.png) | Interferometric imaging condition for wave-equation migration | ![](icons/right.png) |
![[pdf]](icons/pdf.png) |
Next: Appendix B
Up: Appendix A
Previous: Appendix A
Consider a medium whose behavior is completely defined by the acoustic
velocity, i.e. assume that the density
is
constant and the velocity
fluctuates around a homogenized
value
according to the relation
![\begin{displaymath}
\frac{1} {v^2 \left (x,y,z \right)} =
\frac{1 + \sigma m \left (x,y,z \right)}{v_0^2 \left (x,y,z \right)}\;,
\end{displaymath}](img67.png) |
(12) |
where the parameter
characterizes the type of random fluctuations
present in the velocity model, and
denotes their strength.
Consider the covariance orientation vectors
defining a coordinate system of arbitrary orientation in space. Let
be the covariance range parameters in the
directions of
,
and
, respectively.
We define a covariance function
![\begin{displaymath}
\mathrm{cov} \left (x,y,z \right)= \exp \left [-l^{\alpha} \left (x,y,z \right)\right]\;,
\end{displaymath}](img80.png) |
(16) |
where
is a distribution shape parameter and
![\begin{displaymath}
l \left (x,y,z \right)=
\sqrt{\left (\frac{{\bf a}\cdot {\b...
...ight)^2 +
\left (\frac{{\bf c}\cdot {\bf r}}{r_c} \right)^2}
\end{displaymath}](img82.png) |
(17) |
is the distance from a point at coordinates
to the
origin in the coordinate system defined by
.
Given the IID Gaussian noise field
, we obtain the random
noise
according to the relation
![\begin{displaymath}
m \left (x,y,z \right)=
\mathscr{F}^{-1}
\left [\sqrt{\wideh...
...ight)} \;
\widehat{ n} \left (k_x,k_y,k_z \right)
\right]\;,
\end{displaymath}](img87.png) |
(18) |
where
are wavenumbers associated with the spatial
coordinates
, respectively. Here,
are Fourier transforms of the covariance function
and the noise
,
denotes Fourier transform, and
denotes inverse Fourier transform. The
parameter
controls the visual pattern of the field, and
control the size and orientation of a
typical random inhomogeneity.
![next](icons/next.png) |
![up](icons/up.png) |
![previous](icons/previous.png) |
![](icons/left.png) | Interferometric imaging condition for wave-equation migration | ![](icons/right.png) |
![[pdf]](icons/pdf.png) |
Next: Appendix B
Up: Appendix A
Previous: Appendix A
2013-08-29