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Downward continuation can be done in the Fourier domain as a phase
shift applied to *U*_{z}, the wavefield at depth *z* Claerbout (1985):
| |
(6) |

The vertical wavenumber *k*_{z} depends on the laterally varying
velocity. In order for us to be able to implement the formula, we have
to decompose *k*_{z} into a part not influenced by lateral velocity
variations (*k*_{z}_{o}) and a part influenced by them (*k*_{z}_{x}):
| |
(7) |

The Born approximation is equivalent to a linearization of the
exponential , and therefore
| |
(8) |

In the case of the complexified local Born-Fourier (complexified
pseudo-screen) method, with the notations in equations (12) and (13)
of Sava (2000), we can rewrite it as:
| |
(9) |

where is the background wavefield
downward continuation operator applied in the domain:
| |
(10) |

is the scattering operator, applied in the domain:
| |
(11) |

and where *s* is the slowness at the depth , is the wavenumber across the midpoint direction (scalar for 2D, vector
for 3D), *s*_{o} is the constant background slowness, is the
frequency, and is a small dimensionless quantity introduced for
numerical stability; and must contain a constant. The output of this operator can be seen in Figure
10.
Although with equation (9) we went a step towards
linearity with respect to
the slowness perturbation term, it is not fully linear because the slowness
perturbations compose with themselves.
This is visible if we examine the first
two steps of the downward continuation. At *z*=0, *U*_{z=0} = *Data*
(Ricker wavelet at zero-time in the middle of the x axis). At
,

| |
(12) |

At ,
| |
(13) |

and by plugging in the expression for and because
and do not commute,
| |
(14) |

In order to obtain a downward continuation that is linear in the
slowness perturbations , we have to drop the last term at
each step. Thus, after the *n*^{th} depth step, the wavefield will be:
| |
(15) |

The above formula is equivalent with stating that at each level, we
compute the scattered wavefield only from the background wavefield
from the previous level, then we propagate it down until the last
level with the background operator. The results of this approach are
visible in Figure 5.

** Next:** : APPENDIX C -
** Up:** Vlad: Velocity estimation for
** Previous:** APPENDIX A - Derivation
Stanford Exploration Project

11/11/2002