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A

This appendix analytically derives SODCIGs for one shot and well-sampled receiver wavefields in
locally constant-velocity media. Let's consider the SODCIG for a dipping reflector
at image point (*x*_{m}, *z*_{m}), as shown in Figure (). The dipping angle is , while the reflection
angle at (*x*_{m}, *z*_{m}) is (assuming positive sign for angles measured in a clockwise direction).
The red circle shows the source wavefront at some time *t*, which intersects the dipping reflector
at (*x*_{m}, *z*_{m}). If we put receivers
for every location on the surface, i.e. the receiver wavefield is sufficiently densely sampled, when we downward continue or
backward propagate the receiver wavefield, the circular wavefront, which is in blue, can be well reconstructed. It also intersets
the dipping reflector at (*x*_{m}, *z*_{m}) at the same time *t*. Therefore, cross-correlation of the source wavefield and receiver
wavefield will generate the image at (*x*_{m}, *z*_{m}).

dip.full.rec When the receiver wavefield is sufficiently densely sampled,
the actual circular wavefront in blue can be well reconstructed.
Figure 14 |

The equation for the source wavefront in red is

(8) |

(9) |

(10) |

(11) |

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(18) |

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(28) |

B

This appendix analytically derives SODCIGs for one shot and one receiver in locally constant-velocity media.
Let's again consider a dipping reflector with the dip angle , as shown in Figure . The
source is located at (*s*,0), while receiver is located at (*r*,0). Since there is only one receiver, when we downward continue
or backward propagate the receiver wavefield, the receiver wavefront that intersects (*x*_{m}, *z*_{m})
is the green one shown in Figure instead of the blue one shown in Figure .

dip.one.rec In the extreme situation where there is only one receiver,
the receiver wavefront is the one shown here in green, instead of the blue one
shown in Figure .
Figure 15 |

The image point at (*x*_{m}, *z*_{m}) on the dipping reflector certainly satisfies:

(29) | ||

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(31) |

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5/6/2007