(61) |

(62) |

(63) | ||

(64) |

If the downward-continuation operator, , contains
purely a phase-shift, then its adjoint, will
fully describe the inverse process of upward-continuation.
However, for the amplitudes to be treated accurately,
must respect the physics of wave propagation.
Stolt and Benson (1986) show that *v*(*z*) extrapolators based on WKBJ
Green's functions contain an amplitude term as well as a phase term,
and for *v*(*x*,*y*,*z*) earth models this effect is even more pronounced.
So while kinematically-correct extrapolators are
pseudo-unitary, true-amplitude depth extrapolators are not.
In this introductory section, I follow conventional seismic processing
methodology, and treat the depth extrapolator as a unitary operator;
however, in section , I discuss how to model
amplitudes correctly.

The recursion in equations () and () can be rewritten as:

(65) |

(66) |

Equation () encapsulates the idea that we can reconstruct the wavefield at every depth-step in the earth from the wavefield at the surface by inverting matrix, .

As a final step, to produce a migrated image we need to invoke an
imaging condition. For the case of exploding-reflector (zero-offset)
migration [e.g. Claerbout (1995)], we need to extract the image,
, corresponding to *t*=0.
In the temporal frequency domain, we can do this by summing
over frequency. This stacking process is described by the matrix
equation,

or again more simply,

(67) |

The process of imaging by exploding-reflector migration can then be summarized as the chain of composite operators:

(68) |

5/27/2001