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# Theory

Traditional source-receiver migration assumes that the source is an impulse function and the migration is based on survey sinking. Source-receiver migration downward continues the CMP gather into the subsurface by the Double Square Root equation
 (1)
where x is midpoint, h is half-offset, xs is the source point, xr is the receiver point, and velocity vs=v(xs,z) and vr=v(xr,z). It images by extracting the wavefield at zero subsurface offset and adding along all frequencies. In generalized source-receiver migration, instead of using the CMP gather of the recorded data directly, the cross-correlation between the source wavefield and the receiver wavefield at the surface is extrapolated into the media Shan and Zhang (2003). Namely, the wavefield to be downward continued by the Double Square Root equation is
 (2)
where x=(xU+xD)/2, h=(xU-xD)/2 and s means an areal shot. When the source is an impulse function, the cross-correlation between source and receiver wavefields is exactly the CMP gather of the recorded data and the generalized source-receiver migration algorithm is exactly the same as the traditional source-receiver migration.

Source-receiver migration of multiple reflections is a special case of generalized source-receiver migration, in which the source wavefield is the primary reflection and the receiver wavefield is the corresponding multiple reflection. The wavefield to be downward continued is the cross-correlation between primaries and the corresponding multiples. Since it behaves very similarly to a primary, I call it pseudo-primary data. There are two steps for source-receiver migration of multiples. First, pseudo-primary data are calculated by cross-correlating the primary reflections with the corresponding multiple reflections at the surface. Second, a traditional source-receiver migration is run on the pseudo-primary data. Figure illustrates the principle of source-receiver migration of multiples. The phase of the trace at (x,h) in the pseudo-primary data is exactly the same as a trace at (x,h) from the CMP gather of primary, if we would have put a source at xD and a receiver at xU.

mm
Figure 1
Left: Two traces in originally recorded data. The trace at xD records the primary reflection traveltime t1 of . The trace at xU records the multiple reflection traveltime t1+t2 of .Right: The trace of pseudo-primary data related to trace xD and xU. The trace at (x,h) is the cross-correlation between trace xD and trace xU, where x, h are mid-point and half offset of xD and xU, respectively.

post
Figure 2
Left: One trace in original data. The trace at x has two impulses. The first one records the traveltime of the primary reflection t1: and the second one records the traveltime of the multiple reflection t2: .Right: Zero-offset dataset of pseudo-primary data. The trace in the pseudo-primary data is the cross-correlation of the trace in the left figure with itself. The traveltime of the impulse in the trace is double the traveltime between x and R2.

Zero-offset data is very important in amplitude work, but it is never recorded in a real survey. The zero-offset dataset can be obtained from the pseudo-primary data very easily. In equation (2), letting xU=xD=x, we can get the zero-offset surface dataset
 (3)
Figure illustrates how to generate a zero-offset surface dataset from pseudo-primary data. The phase information of the zero-offset dataset of the pseudo-primary data is exactly the same as the zero-offset would be if we would have put a source and receiver at x.

Next: Synthetic Data Example Up: Shan: Migration of multiples Previous: Introduction
Stanford Exploration Project
7/8/2003