Step-length Definition

After computing the (conjugate) gradient vector, one must calculate the step length, $\gamma_n ({\bf x})$, used to update model parameters (equation 5). This computation is not straightforward because the acoustic wave equation is non-linear in model parameters $m ({\bf x}) = c^{-2}({\bf x})$ 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, $\hat{F}$, by performing an additional forward modeling using a set of model parameters perturbed by a scaled version of the computed conjugate gradient (i.e. $m_n ({\bf x}) + \eta c_n ({\bf x})$) and comparing the result with the initial forward modeled data. This is summarized notationally as  

$$\hat{F}_n = {\bf L}(m_n+\eta c_n) - {\bf L}(m_n)= \Psi_{pert}-\Psi_{orig},$$ (12)

where $\Psi_{pert}$ and $\Psi_{orig}$ are the perturbed and original wavefields, respectively. Perturbation scaling factor $\eta$ is constrained to be within $1\%$ of the current model parameter values. The step-length is then given by Mora (1987)  

$$\gamma_n = \frac{c^{*}_n g_n}{ \frac{1}{\eta^2} \hat{F}^{*}_n \hat{F}_n + c^{*}_n c_n}.$$ (13)

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