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Appendix B: LINEAR SLOTH MODEL

The linear sloth model is defined by

\begin{displaymath}
{\frac{1}{V^2(z)}} = {\frac{1}{V_0^2}}\,(1+G\,z)\;.
\end{displaymath} (45)

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

The equation for traveltime can be computed analytically, as follows (Cervený, 2001):

\begin{displaymath}
t^2(x) = t_0^2 + \frac{x^2}{v^2} -
\frac{x^4\,(r^2-1)^2\,(2\,Q+1)}{144\,Q^3\,H^2\,v^2\,(Q+1)^2}\;,
\end{displaymath} (46)

where $H$ is the depth of the reflector, $r=V(H)/V_0$ is the ratio of velocity at the bottom and the top of the model and

\begin{displaymath}
Q = \sqrt{1-\frac{x^2\,(r^2-1)^2}{16\,r^2\,H^2}}\;.
\end{displaymath}

The main traveltime parameters are given by
$\displaystyle t_0$ $\textstyle =$ $\displaystyle \frac{4\,H}{3\,V_0}\,\frac{1+r+r^2}{r\,(r+1)}\;,$ (47)
$\displaystyle v^2$ $\textstyle =$ $\displaystyle V_0^2\,\frac{3\,r^2}{1+r+r^2}\;,$ (48)
$\displaystyle A$ $\textstyle =$ $\displaystyle -\frac{(r-1)^2}{6\,r}\;.$ (49)

The maximum offset and traveltime are defined by
$\displaystyle X$ $\textstyle =$ $\displaystyle \frac{4\,H}{\sqrt{r^2-1}}\;,$ (50)
$\displaystyle T$ $\textstyle =$ $\displaystyle \frac{4\,H}{3\,V_0}\,\frac{r^2+2}{r\,\sqrt{r^2-1}}\;.$ (51)

Substituting equations B-6 and B-7 into equations 22-23 and using the expressions for traveltime parameters B-3, B-4, and B-5 results in the following analytical expressions for additional parameters $B$ and $C$:
$\displaystyle B$ $\textstyle =$ $\displaystyle - \frac{(r-1)^2\,(1+r+r^2)}{2\,r\,(r+2)\,(2\,r+1)}\;,$ (52)
$\displaystyle C$ $\textstyle =$ $\displaystyle - \frac{(r-1)^4\,(1+r+r^2)^2}{3\,r\,(r+2)\,(2\,r+1)^2}\;.$ (53)


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Next: Appendix C: REFLECTION FROM Up: Fomel & Stovas: Generalized Previous: Appendix A: LINEAR VELOCITY

2013-03-02