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If a velocity perturbation is applied at some depth level, a
perturbed wavefield can be derived from the background wavefield
by application of the chain rule to equation 1,
| |
(8) |

where represents the scattered wavefield generated at by the interaction of the velocity model at depth *z*. Field is the accumulated wavefield perturbation corresponding to the
slowness perturbations at all levels above. It is computed by
extrapolating the wavefield perturbations from the level above ,plus the scattered wavefield at this level, .
Equation 8 is also a recursive equation that can be
written in matrix form

| |
(9) |

| (10) |

or in more compact notation as,
| |
(11) |

Operator denotes a perturbation of the extrapolation
operator , while quantity represents a
scattered wavefield and is a function of the medium perturbation given
by the scattering relationship derived in Appendix A. For single
scattering we write,
| |
(12) |

where is the scattering operator, and is slowness
perturbation. This expression yields a recursive relationship that
can be written in matrix form:
| |
(13) |

or in more compact notation
| |
(14) |

where vector denotes slowness perturbations at all depths.
Finally, introducing

| |
(15) |

we can write a simple relationship between slowness and
wavefield perturbations:
| |
(16) |

This expression represents the wavefield scattering caused by the
interaction of the background wavefield with the a medium
perturbation. The total modeled field is defined as,
| |
(17) |

where is the background wavefield modeled by
equation 3.

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Stanford Exploration Project

5/6/2007