Numeric implementation of wave-equation migration velocity analysis operators |

Wavefield reconstruction for multi-offset migration based on the
one-way wave-equation under the shot-record framework is performed by
separate recursive extrapolation of the source and receiver
wavefields, and . The wavefield extrapolation progresses
forward in time (causal) for the source wavefield and backward in time
(anti-causal) for the receiver wavefield:

In equations 25-26, and represent the source and receiver acoustic wavefield for a given frequency at all positions in space at depth , and and represent the same wavefields extrapolated to depth . The phase shift operation uses the depth wavenumber which is defined by the single square-root (SSR) equation

The image is obtained from the extrapolated wavefields by selection of the zero cross-correlation lags in space of time between the source and receiver wavefields, an operation which is usually implemented as summation over frequencies:

An alternative imaging condition (Sava and Fomel, 2006) preserves the space and time cross-correlation lags in the image.

Linearizing the depth wavenumber given by the equation 27
relative to the background slowness
similarly to the case
case of zero-offset migration, we can reconstruct the acoustic
wavefields in the background model using a phase-shift operation

(29) | |||

(30) |

which define the causal and the anti-causal wavefield extrapolation operators for shot-record migration constructed using the background slowness and producing the wavefields and at depth from the wavefields and at depth , respectively. A typical implementation of shot-record wave-equation migration follows the algorithm:

This algorithm is similar to the one used for zero-offset or survey sinking migration, except that the source and receiver wavefields are reconstructed separately using wavefield extrapolation. Unlike the zero-offset extrapolation operator, the shot-record extrapolation operator uses the background slowness since the operation involves sinking of the source and receiver wavefields from the surface toward the image positions. 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 shot-record migration as

and

Equations 31-32 define the forward scattering operators producing the scattered wavefields from the slowness perturbation , based on the background slowness and background wavefield . In this case, the symbol stands for either or , given the appropriate choice of sign in the forward scattering operator. The image perturbation at depth is obtained from the source and receiver scattered wavefields using the relation

(33) |

Given an image perturbation , we can construct the scattered
source and receiver wavefields by the adjoint of the imaging condition

(34) | |||

(35) |

for every frequency . Then, the slowness perturbations due to the source and receiver wavefields at depth under the influence of the background source and receiver wavefields at the same depth can be written as

and

Equations 36-37 define the adjoint scattering operators , producing the slowness perturbation from the scattered wavefield , based on the background slowness and background wavefield . In this case, stands for either or , given the appropriate choice of sign in the adjoint scattering operator. A typical implementation of shot-record forward and adjoint scattering follows the algorithms:

These algorithms are similar to the one used for zero-offset or survey sinking migration, except that the source and receiver wavefields are reconstructed separately using wavefield extrapolation. Unlike the zero-offset scattering operators, the shot-record scattering operators use the background slowness since the operation involves sinking of the source and receiver wavefields from the surface toward the image positions. Both forward and adjoint scattering algorithms require the background wavefields, and , 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