 |
 |
 |
 | Numeric implementation of
wave-equation migration velocity analysis operators |  |
![[pdf]](icons/pdf.png) |
Next: Summary of operators
Up: Wave-equation migration and velocity
Previous: Survey-sinking migration and velocity
Wavefield reconstruction for multi-offset migration based on the
one-way wave-equation under the shot-record framework is performed by
separate recursive extrapolation of the source and receiver
wavefields,
and
. The wavefield extrapolation progresses
forward in time (causal) for the source wavefield and backward in time
(anti-causal) for the receiver wavefield:
In equations 25-26,
and
represent
the source and receiver acoustic wavefield for a given frequency
at all positions in space
at depth
, and
and
represent the same wavefields extrapolated to
depth
. The phase shift operation uses the depth wavenumber
which is defined by the single square-root (SSR) equation
![\begin{displaymath}
{k_z}= \sqrt{ \left [{ {\omega s} \left ({\bf m}\right)} \right]^2 - \left\vert {{{\bf k}_{\bf m}}} \right\vert^2}
\end{displaymath}](img107.png) |
(27) |
The image is obtained from the extrapolated wavefields by selection of
the zero cross-correlation lags in space of time between the source
and receiver wavefields, an operation which is usually implemented as
summation over frequencies:
 |
(28) |
An alternative imaging condition (Sava and Fomel, 2006) preserves
the space and time cross-correlation lags in the image.
Linearizing the depth wavenumber given by the equation 27
relative to the background slowness
similarly to the case
case of zero-offset migration, we can reconstruct the acoustic
wavefields in the background model using a phase-shift operation
which define the causal
and the
anti-causal
wavefield
extrapolation operators for shot-record migration constructed using
the background slowness
and producing the wavefields
and
at depth
from the
wavefields
and
at depth
, respectively. A
typical implementation of shot-record wave-equation migration follows
the algorithm:
This algorithm is similar to the one used for zero-offset or survey
sinking migration, except that the source and receiver wavefields are
reconstructed separately using wavefield extrapolation. Unlike the
zero-offset extrapolation operator, the shot-record extrapolation
operator uses the background slowness
since the operation
involves sinking of the source and receiver wavefields from the
surface toward the image positions. Wavefield extrapolation is usually
implemented in a mixed domain (space-wavenumber), as briefly
summarized in Appendix A.
Similarly to the derivation of the wavefield perturbation in the
zero-offset migration case, we can write the linearized wavefield
perturbation for shot-record migration as
and
Equations 31-32 define the forward scattering operators
producing the scattered
wavefields
from the slowness perturbation
, based
on the background slowness
and background wavefield
. In this case, the symbol
stands for either
or
, given the appropriate choice of sign in the forward scattering
operator. The image perturbation at depth
is obtained from the
source and receiver scattered wavefields using the relation
 |
(33) |
which corresponds to the frequency-domain zero-lag cross-correlation
of the source and receiver wavefields required by the imaging
condition.
Given an image perturbation
, we can construct the scattered
source and receiver wavefields by the adjoint of the imaging condition
 |
|
|
(34) |
 |
|
|
(35) |
for every frequency
. Then, the slowness perturbations due to the
source and receiver wavefields at depth
under the influence of the
background source and receiver wavefields at the same depth
can be
written as
and
Equations 36-37 define the adjoint scattering operators
, producing the slowness
perturbation
from the scattered wavefield
, based
on the background slowness
and background wavefield
. In this case,
stands for either
or
, given
the appropriate choice of sign in the adjoint scattering operator. A
typical implementation of shot-record forward and adjoint scattering
follows the algorithms:
These algorithms are similar to the one used for zero-offset or survey
sinking migration, except that the source and receiver wavefields are
reconstructed separately using wavefield extrapolation. Unlike the
zero-offset scattering operators, the shot-record scattering operators
use the background slowness
since the operation involves sinking
of the source and receiver wavefields from the surface toward the
image positions.
Both forward and adjoint scattering algorithms require the background
wavefields,
and
, to be precomputed at all depth
levels.
Scattering and wavefield extrapolation are implemented in the mixed
space-wavenumber domain, as briefly explained in Appendix A.
 |
 |
 |
 | Numeric implementation of
wave-equation migration velocity analysis operators |  |
![[pdf]](icons/pdf.png) |
Next: Summary of operators
Up: Wave-equation migration and velocity
Previous: Survey-sinking migration and velocity
2013-08-29