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The theory of least-squares reverse time migration is well established (Symes and Carazzone, 1991; Mulder and Plessix, 2004; Dai et al., 2012).
In this section, I will first review the theory of LSRTM assuming the constant density acoustic wave equation,
|
(14) |
where
is the velocity distribution, and
is the pressure field associated with the source term
. A perturbation in the velocity model
will generate a wavefield
, which obeys the equation
|
(15) |
Expanding the velocity term according to
|
(16) |
and subtracting equation from equation yields the wave equation for the wavefield perturbation
|
(17) |
Neglecting the higher order terms and defining the reflectivity model as
, the above equation becomes
|
(18) |
Equations and will be used to derive the Born modeling operator. Numerically, the calculation of the reflection data
requires two finite-difference simulations: one to solve equation to obtain the wavefield
, and one to solve equation for the reflection data
. The wavefield
will be recorded at the receiver position
to give the shot gather
.
By the adjoint state method (Plessix, 2006), the migration operation of a shot gather
requires two finite-difference simulations, one for the source-side wavefield and one for the receiver-side wavefield:
|
|
|
(19) |
|
|
|
(20) |
where
is the receiver-side wavefield.
Note that the source-side wavefield
propagates forward in time but the receiver-side wavefield
propagates backward in time.
The migration image associated with the shot at
is produced by applying the imaging condition
|
(21) |
To simplify the formulas, matrix-vector notation will be used to represent the Born modeling operator
|
(22) |
where
is the reflection data vector for the
shot,
is a reflectivity model, and
represents the Born modeling operator associated with the
shot.
Similarly, the reverse time migration operator can be expressed as
|
(23) |
with
indicating the migration image for the
shot and
representing the migration operator associated with the
shot.
Subsections
Next: Least-squares Migration
Up: Plane-wave Least-squares Reverse Time
Previous: Introduction
Contents
Wei Dai
2013-07-10