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Appendix A: LINEAR VELOCITY MODEL

The linear velocity model is defined by


\begin{displaymath}
V(z) = V_0\,(1+g\,z)\,
\end{displaymath} (38)

where $g$ is the velocity gradient and $V_0$ is velocity at zero depth.

The reflection traveltime can be expressed in an analytical form as a function of offset (Slotnick, 1959)

\begin{displaymath}
t(x) = {\frac{2\,H}{V_0\,(r-1)}}\,
{\mbox{arccosh}{\left[1...
...r-1)^2}{2\,r}\,\left(1+\frac{x^2}{4\,H^2}\right)\right]}}\;,
\end{displaymath} (39)

where $H$ is the depth of the reflector, and $r=V(H)/V_0$ is the ratio of velocity at the bottom and the top of the model. The traveltime parameters are given by
$\displaystyle t_0$ $\textstyle =$ $\displaystyle \frac{2\,H}{V_0}\,\frac{\ln{r}}{r-1}\;,$ (40)
$\displaystyle v^2$ $\textstyle =$ $\displaystyle V_0^2\,\frac{r^2-1}{2\,\ln{r}}\;,$ (41)
$\displaystyle A$ $\textstyle =$ $\displaystyle \frac{1}{2}\,\left(1-\frac{r^2+1}{r^2-1}\,\ln{r}\right)\;.$ (42)

This model has maximum (critical) offset and traveltime that are defined by
$\displaystyle X$ $\textstyle =$ $\displaystyle 2\,H\,\sqrt{\frac{r+1}{r-1}}\;,$ (43)
$\displaystyle T$ $\textstyle =$ $\displaystyle \frac{2\,H}{V_0}\,\frac{\mbox{arccosh}\,{r}}{r-1}\;.$ (44)

Substituting equations A-6 and A-7 into equations 22-23 and also using the expressions for traveltime parameters A-3, A-4, and A-5 results in complicated but analytical expressions for additional parameters $B$ and $C$.


next up previous [pdf]

Next: Appendix B: LINEAR SLOTH Up: Fomel & Stovas: Generalized Previous: ACKNOWLEDGMENTS

2013-03-02