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After computing the (conjugate) gradient vector, one must calculate the step length, , used to update model parameters (equation 5). This computation is not straightforward because the acoustic wave equation is non-linear in model parameters and the Frechét derivatives are never explicitly calculated. One approach is to use a linear approximation technique based on perturbation methods Mora (1987). This involves calculating an approximate Frechét derivative, , by performing an additional forward modeling using a set of model parameters perturbed by a scaled version of the computed conjugate gradient (i.e. ) and comparing the result with the initial forward modeled data. This is summarized notationally as
| |
(12) |

where and are the perturbed and original wavefields, respectively. Perturbation scaling factor is constrained to be within of the current model parameter values. The step-length is then given by Mora (1987)
| |
(13) |

Again, the step-length in equation 13 is equal to that in Mora (1987) where covariance matrices are represented by identity operators.

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Stanford Exploration Project

1/16/2007