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Migration is, by the standard definition, the operation adjoint to
forward modeling.
It is performed by the downward continuing the recorded wavefield
and imaging at zero time:
 
(3) 
We substitute Equation (1) into
Equation (3), and expand
the integral in Equation (1)
to negative depth, for which the image is zero, by definition:
 
(4) 
We then change the integration variable to k_{z} and obtain:
 
(5) 
The pair of integrals in Equation (5)
describe forward and inverse Fourier transforms,
and thus the effect of chaining modeling and migration
on the image is simply the equivalent of applying
the Jacobian .This result is valid only for real values of k_{z}, which is
what we want, since we are not interested in the wavefield
component for which k_{z} becomes imaginary
(that is, we neglect evanescent waves).
In constant velocity, the frequencywavenumber representation of the
Jacobian is simply a multiplication:
 

 (6) 
The Jacobian weighting is introduced by the imaging step,
therefore, the Jacobian depends on the coordinates used to define
the wavefield during imaging: constant rayparameter
or constant offset wavenumber.
In matrix notation, Equation (1) is written
 
(7) 
where , , and are respectively
the data, image, and reflectivity vectors,
is a diagonal matrix representing the reflection operator,
and is the modeling operator.
With this notation, Equation (5) becomes
 
(8) 
where is a diagonal matrix representing the Jacobian
.
Next: Amplitude correction in variable
Up: Modeling and migration amplitudes
Previous: Modeling and migration amplitudes
Stanford Exploration Project
4/30/2001