Next: Conclusion Up: Shan and Zhang: Migration Previous: Source-Receiver Migration

# Demonstration of equivalence

Although shot-profile migration and source-receiver migration look totally different, they obtain both the same image and CIG. In this section, we prove that the mono-frequency image and CIG in the shot-profile migration are exactly the same mono-frequency image and CIG in the source-receiver migration, respectively.

We define a new wavefield , which is the cross-correlation between the source wavefield and the receiver wavefield in the shot-profile migration for shot s, that is
 (8)
and the wavefield is the stack of along all the shots,
 (9)
Obviously, from equation (7), is the surface data in source-receiver migration. We will demonstrate that the wavefield satisfies the DSR equation,
 (10)
where . By extension, also satisfies the DSR equation. Thus shot-profile migration and source-receiver migration are two different ways to obtain wavefield Q at the subsurface. In shot-profile migration, source and receiver wavefields are downward continued into the subsurface with the one-way wave equation, and the wavefield is formed by cross-correlating the source wavefields and receiver wavefields and stacking over all shots at all depths. But in source-receiver migration, the wavefield at the surface is obtained by cross-correlating the source wavefield and the receiver wavefield at the surface, and is formed by extrapolating to all depths with the DSR equation.

From the Leibniz rule, we have
 (11)
where and .Since is an up-going wavefield, it satisfies the up-going wave equation(1), so we have
 (12)
is not dependent on xU, so it is constant with respect to the operator , and we have
 (13)
Summarizing equation (12) and (13), we have
 (14)
It is easy to prove that
 (15)
So the second term of equation (11) changes to
 (16)
Since is a down-going wavefield, it satisfies the down-going wave equation (2), and we have
 (17) (18)
Again, does not depend on xD, so we have
 (19) (20)
Summarizing equations (15-20), we have
 (21)
Finally, from equation (11), equation (14) and equation (21), we know Qs satisfies the DSR equation (10). Q is the stack of Qs over all shots, so by extension Q satisfies the DSR equation also.

It is obvious that the image of shot-profile migration in equation (3) is
 (22)
and the corresponding CIG in equation (5) is
 (23)
In traditional source-receiver migration, is the stack of the cross-correlation between the impulse source and the recorded data along shots, which is the CMP gather of the recorded data at the surface. Since both and are obtained by propagating to the subsurface with the DSR equation (10), they are equivalent. If the source in source-receiver migration is not an impulse function, is the stack of the cross-correlation between the source wavefield and the receiver wavefield, and the same conclusion is reached. Thus we have
 (24)
and
 (25)

Next: Conclusion Up: Shan and Zhang: Migration Previous: Source-Receiver Migration
Stanford Exploration Project
7/8/2003