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In order to find a conjugate operator,
the forward modeling operator must be formulated explicitly.
The acoustic wave equation in the one dimensional space is
 
(1) 
where P is pressure field, v is the velocity, t and z are
time and depth axis, respectively.
Approximating the derivatives by finite differences, equation (1) becomes
 
(2) 
where i and j represent time and depth indices, respectively.
By rearranging equation (2) for timeextrapolation,
we get the following equation:
 
(3) 
The differencing star in equation (3) covers
the (t,z) space along the zaxis to extrapolate in time
as shown schematically by Figure .
rt1dmdl
Figure 1 Schematic for time extrapolation modeling
for 1D acoustic wave equation.
The input is a source function at t=0 and
the output is a trace at z=0.

 
To make the operator small enough to show on a page,
I choose a small space like (t,z) = (4,5).
The time extrapolation shown in equation (3) can be
formulated algebraically as follows
 
(4) 
where is the model vector at t=1 with length 5,
is the data vector with length 4,
and and represent 5 by 5 identity and zero matrices.
The matrix in equation (4) represents tridiagonal matrix
which looks like:
 
(5) 
where was shown in equation (3).
The matrix samples the wave field on the surface at ith time
and has the following form:
 
(6) 
Now, reversetime migration as the conjugate operator
to the forward timeextrapolation becomes clear.
It is obtained by transposing the operator in equation (4)
and has the following form :
 
(7) 
where and are transpose matrices
of and .It is interesting that the conjugate algorithm follows
the reversetime extrapolation but with an operator which differs
from the forwardtime extrapolation.
Instead of using wave fields from the two previous time steps together
for an extrapolation,
the conjugate operator uses the wave fields of
the two previous time steps separately.
The code for this algorithm, which has passed the dotproduct test
(Claerbout, 1992), is listed in the Appendix.
Next: Example
Up: Ji: Conjugate RTM
Previous: Introduction
Stanford Exploration Project
11/17/1997