(28) |

(29) |

Due to lateral velocity variations, and the desire to avoid spatial
Fourier transforms, approximations to *R* are often applied
in the domain.
Typically *R* is split into a `thin-lens' term that propagates the
wave vertically, and a `diffraction' term that models more complex
wave phenomena. In the domain, the thin-lens term
can be applied as a simple phase-shift, while the diffraction term is
approximated by a small finite-difference filter. The method of
extrapolation determines the nature of the finite-difference
filter.
The mathematical forms of different extrapolators are summarized in
Table 1, and discussed below.

Gazdag: | |

Implicit: | |

Explicit: | |

Helmholtz factorization: |

Implicit extrapolation (discussed in more detail in following chapters) approximates with a rational form, consisting of a convolutional filter, and an inverse filter,

(30) |

Practical 3-D extrapolation is often done with an explicit operator using McClellan transforms. This approach amounts to approximating by with a simple convolutional filter, .Explicit extrapolators, therefore, have the form

(31) |

In contrast to these methods, the minimum-phase factorization of the Helmholtz operator provides a recursive depth extrapolator of a different form:

(32) |

The apparent contradiction that we are approximating the unitary (delay) operator in equation () with the minimum-phase extrapolator in equation () is resolved by examining the impulse response of the operator shown in Figure . The impulse response consists of two `bumps'. The first bump is the response of the impulse at the same depth step as the impulse. Because it looks like a delta function, it leaves that depth step essentially unchanged. The second bump, on the other hand, is the response to the impulse at the following depth step -- it describes the wave propagation in depth. When taken together, the first and second bumps are indeed minimum phase; however, the second bump controls wave propagation in depth and is almost pure delay.

impresp
Amplitude of impulse response of
polynomial division with minimum-phase factorization of the Helmholtz
equation. The top panel shows the location of the impulse. The bottom
panel shows the impulse response. Helical boundary conditions mean
the second bump in the impulse response corresponds to energy
propagating to the next depth step.
Figure 2 |

5/27/2001