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Methods for calculating the gradient without explicitly computing the partial derivatives of the data are well established Lailly (1983); Pratt et al. (1998); Pratt and Worthington (1989); Tarantola (1984). The main result in the time-domain inversion literature is that the gradient vector can be computed by a zero-lag correlation of the wavefield propagated forward from the source point, , and the residual wavefield propagated backwards from the farthest receiver location toward the source point past each successive receiver, . The frequency-domain equivalent to this zero-lag correlation is the multiplication of these two wavefields according to
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(6) |

where *Re* indicates the real component of the multiplication result. The summation over sources and receivers is done for each non-linear iteration at each frequency. Following Sirgue and Pratt (2004) in assuming a point source of unit amplitude and zero phase, the forward-propagated wavefield *P*_{f} is given by
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(7) |

while back-propagated wavefield *P*_{b} is defined by
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(8) |

where and represent the monochromatic Green's functions for an excitation at the source and receiver points in the medium, respectively. Hence, the full gradient vector expression is
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(9) |

Note that represents the back-projection of the data residuals and is similar to a "migration with the residual wavefield data" Mora (1987).

** Next:** Conjugate Gradient Definition
** Up:** Review of Frequency-domain waveform
** Previous:** The Inverse Problem
Stanford Exploration Project

1/16/2007