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The approach to solving the waveform inversion problem followed here Pratt and Worthington (1989) is based on minimizing the residual misfit at each successive frequency, . Assuming an *L*_{2} norm, the misfit is defined
| |
(3) |

where denotes complex conjugate of wavefield . In this study, I approach the minimization of through computing the negative gradient, or the direction of greatest decrease of the misfit function, with respect to the variation in model parameters. If model parameters are represented by , then the descent direction is defined by
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(4) |

The gradient vector, , is considered an image of the model space and can be used to update the model parameter estimates according to
| |
(5) |

where is the step length discussed below, and subscript *n* indicates the current iteration number.

** Next:** Gradient Vector Definition
** Up:** Review of Frequency-domain waveform
** Previous:** Forward modeling
Stanford Exploration Project

1/16/2007