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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

is the velocity gradient and

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

is the depth of the reflector, and

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

and

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

2013-03-02