ESTIMATING A NEW SLOWNESS MODEL

Now that the data have been migrated, and velocity errors are evident, use the following algorithm to compute an update to the interval slowness model.

Outer loop over interval slowness models ${\bf w}_i$

{

Migrate the data with ${\bf w}_i$

Apply residual NMO+DMO: compute $S(\gamma,\tau)$

Estimate dips on the stacked image

Pick important reflector points

Compute the operator ${\bf G}$

$\Delta {\bf w}_0=\hat \alpha \nabla_{\bf w} Q$; j=1

While $\Vert \Delta {\bf w}\Vert < 20\%$ and $\nabla_{\bf w} Q \gt 0$

{

Compute $\nabla_\gamma Q$

$\nabla_{\bf w} Q = \biggl[G^T_\gamma - {\delta \gamma \over \delta \tau}G^T_\tau \biggr] \nabla_\gamma Q$

line search $\Delta {\bf w}_j+\alpha \nabla_{\bf w} Q$

line search $\Delta {\bf w}_{j-1}+\beta (\Delta {\bf w}_j-\Delta {\bf w}_{j-1}+\hat \alpha \nabla_{\bf w} Q)$

Update $\Delta {\bf w}_{j+1}=\Delta {\bf w}_{j-1}+\hat \beta (\Delta {\bf w}_j-\Delta {\bf w}_{j-1}+\hat \alpha \nabla_{\bf w} Q)$

Update reflector position map $x(\tau)$,$z(\tau)$ , $\delta \gamma / \delta \tau$

j=j+1

}

${\bf w}_{i+1}={\bf w}_i+\Delta {\bf w}_{\rm final}$ ; i=i+1

}

The objective function Q is the sum of the semblance $S(\gamma ({\bf w}),\tau)$over events. ${\bf G}$ is the relation between changes in the interval slowness model ${\bf w}$and changes in $\gamma$ the residual slowness of the chosen reflection events. $\nabla_\gamma Q$ finds the direction and magnitude of increasing semblance versus $\gamma$ for each reflector. $\delta \gamma / \delta \tau$ guides the change in $\gamma$ due to reflector movement. The line searches return the values $\hat \alpha$ and $\hat \beta$ to describe the maxima of the objective function in their respective search directions. The conjugate-gradient algorithm I use is called PARTAN and is described by Luenberger (1984). The algorithm is robust for nonlinear functions and when line searches are inexact. This is important because my objective function is non-quadratic in shape and for values of $\gamma$ intermediate to those computed the objective function has to be computed by interpolation. The bound on the magnitude of the change in interval slowness is taken with respect to the interval slowness model used for migration. $20\%$ is not meant to be taken as a firm figure; the data should be remigrated once residual migration can no longer adequately describe the reflector movement that accumulates as the iterations proceed. Also, the norm measuring relative change in the slowness model can be an ${\bf L}_\infty$ or ${\bf L}_2$ norm.