Numeric implementation of wave-equation migration velocity analysis operators |

All wave-equation migration velocity analysis operators described in the preceding sections are similar in that they relate perturbations of the image with perturbations of the (slowness) model. In all cases, this velocity estimation procedure takes advantage of features of migrated images which indicate incorrect imaging. The imaging inaccuracies can have several causes, i.e. incorrect downward continuation, irregular illumination, limited acquisition aperture etc., but the wave-equation MVA operators translate all inaccuracies in model updates. This feature, however, is a fundamental limitation of all migration velocity analysis techniques and we do not expand on this topic further.

Since the migration velocity analysis operators link image perturbations with slowness perturbations, they are all composed of several common parts, but with implementations that are specific for each imaging configuration. Thus, a wave-equation MVA operator is composed of an extrapolation operator (for wavefield reconstruction from recorded data), an imaging operator (for image construction from reconstructed wavefields) and a scattering operator (for relating wavefield perturbations to slowness perturbations). The following table summarizes the wave-equation MVA operator components in different imaging configurations, as described in detail in the preceding sections.

extrapolation operator |
imaging operator |
scattering operator |
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---|---|---|---|---|---|

equation | velocity | extrapolation | type | equation | |

zero-offset |
SSR | half | backward | zero time | linearized SSR |

equations 1, 2 |
equation 3 |
equations 8, 11 |
|||

survey-sinking |
DSR | full | backward | zero time | linearized DSR |

implicit cross-corr. | |||||

equations 12, 15 |
equation 16 |
equations 20, 23,24 |
|||

shot-record |
SSR | full | forward | zero time | linearized SSR |

backward | explicit cross-corr. | ||||

equations 25,26,27 |
equation 28 |
equations 31,32,36,37 |

Numeric implementation of wave-equation migration velocity analysis operators |

2013-08-29