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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:
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(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
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(11) |
Further, equation () can be changed into
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(12) |
Defining yields:
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(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
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(14) |
Therefore, the dispersion relation of equation () is
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(15) |
Substituting () into the above formula yields
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(16) |
which can be rewritten as follows:
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(17) |
This is the dispersion relation of the common-offset prestack migration equation.
In the time domain, the dispersion relation is
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(18) |
Therefore, common-offset prestack time migration (PSTM) can be implemented with the following relation:
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(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
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(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