|
|
|
| Lowrank one-step wave extrapolation for reverse-time migration | |
|
Next: Variable velocity and anisotropy
Up: Theory
Previous: Theory
When V is constant, after Fourier transform in space, the wave equation takes the form
|
(2) |
where
is the spatial wavenumber and
is the spatial Fourier transform of
:
|
(3) |
The analytical solution to equation 2 can be expressed as
|
(4) |
where
represents the forward-propagating wavefield, i.e., positive frequencies, and
represents the backward-propagating wavefield, i.e., negative frequencies. The time derivative of
has the following form:
|
(5) |
Zhang and Zhang (2009) used the Hilbert transform to define an additional function:
|
(6) |
where
is the Hilbert transform of
, and
.
Combining equations 4, 5 and 6,
and
can be expressed as
|
|
|
(7) |
|
|
|
(8) |
Equation 2 can be split into a pair of first-order equations and expressed in the following matrix form:
|
(9) |
With the help of the Hilbert transform and equations 7 and 8, a more symmetric expression can be achieved:
|
(10) |
We can further decompose the first matrix on the right-hand side as follows:
|
|
|
(11) |
Substituting equation 11 into equation 10, and using equations 7 and 8, we arrive at:
|
(12) |
In RTM, only one branch of the total wavefield is needed at one time.
The two parts of wave propagation decouple according to
|
(13) |
Modeling seismic wave propagation requires the source function. Letting the source function be
, wave equation 2 can be rewritten in the following form:
|
(14) |
Correspondingly, equation 13 becomes:
The application of operator
can be implemented in either time domain or Fourier domain; it can also be directly incorporated into the definition of source functions. For example, operator
can be regarded as
, which in the time domain corresponds to cascading the Hilbert-transform with the first-order integration.
In constant velocity, the forward-propagating wavefield away from the source at the next time step
can be expressed as:
|
(16) |
|
|
|
| Lowrank one-step wave extrapolation for reverse-time migration | |
|
Next: Variable velocity and anisotropy
Up: Theory
Previous: Theory
2016-11-16