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The goal of waveform inversion is to invert for the optimal set of
velocity perturbations that minimize the difference between
forward-modeled waveforms and acquired data. The first step in setting
up the inverse problem is defining data residuals, ,
| |
(18) |

where is the recorded data. The *L*_{2} residual norm
is used to set up an objective function,
| |
(19) |

that is minimized with respect to slowness perturbations
| |
(20) |

This results in the following least-squares estimate of the slowness
perturbations
| |
(21) |

From here on, the sum over all sources and receivers is implicitly
assumed. Also, we discuss only the gradient vector and the filtering
of the gradient by the inverse Hessian matrix is implicitly assumed.
The adjoint gradient operator is a composite matrix
consisting of a number of chained operators (from
equation 16):

| |
(22) |

where scattering operator or at each extrapolation interval, *S*_{z}
is defined by (see Appendix A),
| |
(23) |

where is considered a filter. This allows us to write
composite operator with scattering and
filter matrices as,
| |
(24) |

Inserting this expression into equation 22 yields,
| |
(25) |

Thus, using the relationship in equations 6 and
7, leads to the following result,
| |
(26) |