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## Common-offset prestack time migration and data regularization

The time-distance relation for a shot-receiver pair is
 (9)
where th is the two-way traveltime of a non-zero-offset shot-receiver pair, hx is the in-line component of the half-offset, and hy is the cross-line component of the half-offset. For simplicity, the connection line of the shot and receiver points is parallel to the x-axis of the Cartesian coordinate system. Therefore, we have the following simple equation which delineates the isochron surface of the prestack migration:
 (10)
where ,and az are the half-lengths of the axes of the rotary isochron ellipse in the case of constant velocity. If , then ; If , then ; If , then . The variable tn is the two-way 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 post-NMO 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 common-offset prestack migration equation. In the time domain, the dispersion relation is
 (18)
Therefore, common-offset prestack time migration (PSTM) can be implemented with the following relation:
 (19)
The term in the braces represents the wave-field extrapolation, and the integral at tn=0 serves to extract the imaging values. Then, the common-offset inverse PSTM is
 (20)
Similarly, the term in the braces represents the wave-field 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 . W1 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 anti-aliasing Up: Seismic data preprocessing Previous: Seismic data preprocessing
Stanford Exploration Project
5/3/2005