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From the discretized data space, the discretized model space, and their relation formula,
| |
(23) |

some causes of aliasing can be clearly seen. The sources of aliasing can be divided into the following three types:
(1) Overly coarse sampling intervals. For example, and/or , and/or are too coarse, where and are the shot-point and receiver-point intervals respectively, and and are the CMP and half-offset intervals, respectively.

(2) Unsuitable modeling or imaging operators, such as the integrated DMO operator (), or the Kirchhoff integral operator (, , , ).

(3) Insufficient output resolution, where the output spatial intervals, such as and are too coarse.
We can analyze the aliasing in the cases of shot-gather migration, receiver-gather migration, common-offset gather migration, and midpoint half-offset domain migration.

Aliasing in shot gather migration is described by the following equation:

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(24) |

where is the conjugate of the extrapolated shot wave-field; is the extrapolated receiver wave-field. *S*_{k} stands for the *K*^{th} common-shot gather. In general, single common-shot-gather imaging presents no aliasing, because the receiver-point interval is easily arranged regularly and small enough. However, imaging an entire 2D line or 3D area will present severe aliasing problems because of the irregular and large shot-point intervals.
Aliasing in receiver-gather migration is described by the following equation:

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(25) |

where is the extrapolated so-called shot wavefield, which corresponds to a specific receiver point; is the extrapolated receiver wavefield, which is sorted from shot gathers. *R*_{k} stands for the *K*^{th}
common-receiver-gather. In general, single common-receiver-gather imaging profiles are susceptible to aliasing.
Aliasing in midpoint half-offset domain migration follows equation ():

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(26) |

where is the extrapolated wavefield with a double-square-root equation. Aliasing will result if and/or are too coarse.
Aliasing in common-offset-gather migration is described by the following equation:

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(27) |

where is the extrapolated wavefield with a double-square-root equation. Aliasing will result if is too coarse.
In the case of rugged topography, the geometry of the acquisition configuration becomes more and more irregular, especially the shot-point coordinates. Therefore, antialiasing processing is necessary in redatuming, seismic data regularization and migration imaging for land-data imaging.

** Next:** Seismic wave illumination analysis
** Up:** Seismic data preprocessing
** Previous:** Common-offset prestack time migration
Stanford Exploration Project

5/3/2005