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The timedistance relation for a shotreceiver pair is
 
(9) 
where t_{h} is the twoway traveltime of a nonzerooffset shotreceiver pair, h_{x} is the inline component of the halfoffset, and h_{y} is the crossline component of the halfoffset. For simplicity, the connection line of the shot and receiver points is parallel to the xaxis of the Cartesian coordinate system. Therefore, we have the following simple equation which delineates the isochron surface of the prestack migration:
 
(10) 
where ,and a_{z} are the halflengths of the axes of the rotary isochron ellipse in the case of constant velocity.
If , then ; If , then ; If , then . The variable t_{n} is the twoway traveltime after NMO.
Equation () can be rewritten as
 
(11) 
Further, equation () can be changed into
 
(12) 
Defining yields:
 
(13) 
Equation () is in the form of a poststack migration. Therefore, prestack migration can be explained as a poststack migration on a postNMO data set.
We know that
 
(14) 
Therefore, the dispersion relation of equation () is
 
(15) 
Substituting () into the above formula yields
 
(16) 
which can be rewritten as follows:
 
(17) 
This is the dispersion relation of the commonoffset prestack migration equation.
In the time domain, the dispersion relation is
 
(18) 
Therefore, commonoffset prestack time migration (PSTM) can be implemented with the following relation:
 
(19) 
The term in the braces represents the wavefield extrapolation, and the integral at t_{n}=0 serves to extract the imaging values.
Then, the commonoffset inverse PSTM is
 
(20) 
Similarly, the term in the braces represents the wavefield extrapolation, which is an inverse migration. The integral is an inverse Fourier transform.
In the presence of moderate lateral velocity variations, prestack time migration can be expressed as follows:
 

 (21) 
where . W_{1} is the amplitude weight, and is the two way traveltime along the imaging ray.
The inverse PSTM is
 

 (22) 
() give a general theory of data mapping. From here, we will develop some practical approaches for data mapping.
Next: Aliasing and antialiasing
Up: Seismic data preprocessing
Previous: Seismic data preprocessing
Stanford Exploration Project
5/3/2005