We emphasize the simple relationship between and and differentiating equation (37) with respect to to obtain
(38)
where the second term on the right-hand side disappear due to Fermat's principle (
). Therefore, we have
(39)
Further differentiating equation (39) with respect to leads to
(40)
To evaluate the derivative in equation (40), we need , which can be found from differentiating the Fermat's condition (
) with respect to . This leads to
which relates the second-order total traveltime derivative at the surface (
) to that of the interface below (
). All pertaining derivatives in equation (43) can be found from equation (15) in the main text that include the first-order effects from lateral heterogeneity.
2layer
Figure 15. The ray configurations two- and three-layered media as the basis for relating the second-order traveltime derivatives at different interfaces.
Effects of lateral heterogeneity on time-domain processing parameters