Vertical velocity variation

In vertically inhomogeneous media, traveltimes can be calculated numerically using any standard numerical technique. Alternatively, traveltimes in v(z) media can be approximated using the homogeneous-medium equations [i.e., equation ()] with replacing the medium parameters by their effective values.

First, as usual, the normal-moveout velocity involves a root-mean-squared average of velocities in the previous layers. Specifically  

$$V^2(\tau)=\frac{1}{\tau} \int_0^{\tau} v^2(t) dt,$$ (130)

where all lower-case variables V correspond to interval-velocity values, and all upper-case variables V correspond to RMS averaged values.

From Appendix B, the anisotropy parameter$\eta$ in equations () and (), is replaced by  

$$\eta_{{\rm eff}}(\tau)=\frac{1}{8} \{ \frac{1}{t_0 V^4(\tau)} \int_0^{\tau} v^4(t)[1+8 \eta(t)] dt -1\},$$ (131)

which includes a summation over the fourth power in velocity. These two equations are similar to what Alkhalifah (1997) used for his nonhyperbolic analysis. Thus, now it is safe to say that such averaging holds for dipping events as well since equation () handles dipping events. However, for dipping events the effective values are computed along the zero-offset ray. In v(z) media, the effective values for dipping and horizontal reflectors are the same. The difference appears only when lateral inhomogeneity exists.