Traveltime for velocity linear with depth

**Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing**
Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	2
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

{\begin{aligned}t&{\rm {=}}\int _{{\theta }_{0}/pa}^{\theta {\rm {/}}pa}{\frac {ds}{v}}{\rm {=}}{\frac {\rm {1}}{a}}\int _{{\theta }_{0}}^{\theta }{\frac {d\theta }{{\rm {\ sin\ }}\theta }}.\end{aligned}}

(52)

Integration gives the traveltime as

{\begin{aligned}&t{\rm {=}}{\left[{\frac {\rm {l}}{a}}{\rm {\ log\ }}{\frac {{\rm {1}}-{\rm {\ cos\ }}\theta }{{\rm {\ sin\ }}\theta }}\right]}_{{\theta }_{0}}^{\theta }{\rm {=}}{\left[{\frac {\rm {l}}{a}}{\rm {\ log\ \ tan\ }}{\frac {\theta }{\rm {2}}}\right]}_{{\theta }_{0}}^{\theta }{\rm {=}}{\frac {\rm {l}}{a}}{\rm {\ log\ }}{\frac {{\rm {\ tan\ }}\left(\theta {\rm {/2}}\right)}{{\rm {\ tan\ }}\left({\theta }_{0}{\rm {/2}}\right)}},\end{aligned}}

(53)

which gives

{\begin{aligned}at&{\rm {=\ log\ }}{\frac {{\rm {\ tan\ }}\left(\theta {\rm {/2}}\right)}{{\rm {\ tan\ }}\left({\theta }_{0}{\rm {/2}}\right)}}\end{aligned}}

(54)

If we take the exponential of both sides, we obtain the implicit equation for traveltime given by

{\begin{aligned}e^{at}&{\rm {=}}{\frac {{\rm {\ tan\ }}\left(\theta {\rm {/2}}\right)}{{\rm {\ tan\ }}\left({\theta }_{0}{\rm {/2}}\right)}}.\end{aligned}}

(55)

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Traveltime for velocity linear with depth