Numeric implementation of wave-equation migration velocity analysis operators |

Wavefield reconstruction for multi-offset migration based on the
one-way wave-equation under the survey-sinking framework
(Claerbout, 1985) is implemented similarly to the zero-offset
case by recursive phase-shift of prestack wavefields

(13) | |||

(14) |

where and are coordinates of sources and receivers on the acquisition surface. In equation 12, represents the acoustic wavefield for a given frequency at all midpoint positions and half-offsets at depth , and represents the same wavefield extrapolated to depth . The phase shift operation uses the depth wavenumber which is defined by the double square-root (DSR) equation

The image is obtained from this extrapolated wavefield by selection of time , which is usually implemented as summation over frequencies:

Similarly to the derivation done in the zero-offset case, we can
assume the separation of the extrapolation slowness
into a
background component
and an unknown perturbation component
. Then we can construct a wavefield perturbation
at depth and frequency related linearly to the slowness
perturbation
. Linearizing the depth wavenumber given by the
DSR equation 15 relative to the background slowness
,
we obtain

(18) |

Here, represents the spatially-variable

(19) |

This algorithm is similar to the one described in the preceding section for zero-offset migration, except that the wavefield and image are parametrized by midpoint and half-offset coordinates and that the depth wavenumber used in the extrapolation operator is given by the DSR equation using the background slowness . Wavefield extrapolation is usually implemented in a mixed domain (space-wavenumber), as briefly summarized in Appendix A.

Similarly to the derivation of the wavefield perturbation in the
zero-offset migration case, we can write the linearized wavefield
perturbation for survey-sinking migration as

Equation 20 defines the forward scattering operator , producing the scattered wavefield from the slowness perturbation , based on the background slowness and background wavefield . The image perturbation at depth is obtained from the scattered wavefield using the time imaging condition, similar to the procedure used for imaging in the background medium:

(21) |

Given an image perturbation
, we can construct the scattered
wavefield by the adjoint of the imaging condition

(22) |

and

Equations 23-24 define the adjoint scattering operator , producing the slowness perturbation from the scattered wavefield , based on the background slowness and background wavefield . A typical implementation of survey-sinking forward and adjoint scattering follows the algorithms:

These algorithms are similar to the ones described in the preceding section for zero-offset migration, except that the wavefield and image are parametrized by midpoint and half-offset coordinates. Furthermore, the two square-roots corresponding to the DSR equation update the slowness model separately, thus characterizing the source and receiver propagation paths to the image positions. Both forward and adjoint scattering algorithms require the background wavefield, , to be precomputed at all depth levels. Scattering and wavefield extrapolation are implemented in the mixed space-wavenumber domain, as briefly explained in Appendix A.

Numeric implementation of wave-equation migration velocity analysis operators |

2013-08-29