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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, *x*_{s} is the source point,
*x*_{r} is the receiver point, and velocity *v*_{s}=*v*(*x*_{s},*z*) and *v*_{r}=*v*(*x*_{r},*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*=(*x*_{U}+*x*_{D})/2, *h*=(*x*_{U}-*x*_{D})/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 *x*_{D}
and a receiver at *x*_{U}.

**mm
**

Figure 1
Left: Two traces in originally recorded data. The trace at *x*_{D} records the primary
reflection traveltime *t*_{1} of . The trace at
*x*_{U} records the multiple reflection traveltime *t*_{1}+*t*_{2} of
.Right: The trace of pseudo-primary data related to trace *x*_{D} and *x*_{U}. The trace at
(*x*,*h*) is the cross-correlation between trace *x*_{D} and trace *x*_{U}, where
*x*, *h* are mid-point and half offset of *x*_{D} and *x*_{U}, 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 *t*_{1}:
and the second one records the traveltime of the multiple
reflection *t*_{2}: .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 *R*_{2}.

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 *x*_{U}=*x*_{D}=*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