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

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 *x*_{U}, 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 *x*_{D}, so we have
| |
(19) |

| (20) |

Summarizing equations (15-20), we have
| |
(21) |

Finally, from equation (11), equation (14) and equation (21), we know
*Q*_{s} satisfies the DSR equation (10). *Q* is the stack of *Q*_{s} 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