From $\gamma$ to $\Delta {\bf t}$

The residual moveout velocity analysis, measuring $\gamma$,that was described in the previous section is an indirect way of measuring traveltime errors $\Delta {\bf t}$,which are required to compute the perturbation $\Delta {\bf w}$ in reference slowness as shown in equation ().

The traveltime errors are the differences between the observed traveltimes ${\bf t_m}$ and the computed traveltimes ${\bf t}$ using an assumed slowness model

$$\Delta {\bf t} = {\bf t} - {\bf t_m}.$$ (33)

Both ${\bf t}$ and ${\bf t_m}$ are obtained by integrating slowness along the ray path traced through the assumed reference model,

$$\Delta {\bf t} = L ({\bf w} - {\bf w_m}),$$ (34)

and this is equivalent to an integration of average slowness as follows:  

$$\Delta {\bf t} = L (\bar{\bf w} - \bar{\bf w}_m).$$ (35)

where $\bar{\bf w}$ and $\bar{\bf w}_m$ represent average slownesses of the true model and of the reference model, respectively.

By substituting $\bar{\bf w}=\gamma\bar{\bf w}_m$into equation (), we get

$$\Delta {\bf t} = L ( \gamma - 1 ) \bar{\bf w}_m.$$ (36)

Therefore, we calculate $\Delta {\bf t}$ from the picked $\gamma$ and average reference slowness along the the ray trajectory.