# Appendix B: Exercises

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

1. Show that (in the case of two dimensions x, y) if the velocity is ${\displaystyle v{\rm {=}}v_{0}{\rm {\ cosh\ }}\left(\tau y/r_{0}\right)}$ then the rays are brought to a common focus (both time and distance) with common emergence distance ${\displaystyle x{\rm {=}}r_{0}}$ and common emergence time ${\displaystyle t_{0}{\rm {=}}r_{0}/v_{0}}$. This exercise represents the traditional case of diving waves.

2. In the case when velocity increases linearly with depth, we first found by integration that ${\displaystyle x{\rm {=}}\rho \left(-{\rm {\ cos\ }}\theta {\rm {+\ cos\ }}{\theta }_{0}\right)}$. Show that depth y can be found without integration by simply using Snell’s law in the velocity equation ${\displaystyle v{\rm {=}}v_{0}{\rm {+}}ay}$.

3. Let the velocity (in the case of two dimensions x,y) be ${\displaystyle v(y)\ {\rm {=}}v_{0}{\rm {/}}{\sqrt {{\rm {1+}}ay}}}$. Show that the equation for the raypath with initial angle ${\displaystyle {\theta }_{0}}$ to the vertical is

{\displaystyle {\begin{aligned}ay{\rm {=}}{\left({\rm {\ cos\ }}{\theta }_{0}{\rm {+}}{\frac {ax}{\rm {2}}}{\rm {\ sin\ }}{\theta }_{0}\right)}^{\rm {2}}-{\rm {cos}}^{\rm {2}}{\theta }_{0},\end{aligned}}}

which is a parabola. Find the vertex of the parabola. Show that the ray is progressively bent toward the normal if a is positive and away from the normal if a is negative.

4. Let velocity (in the case of two dimensions x,y) be the exponential ${\displaystyle v\left(y\right){\rm {=}}v_{\rm {0}}{\rm {\ exp\ }}\left(ay\right)}$. Show that

{\displaystyle {\begin{aligned}ax=2(\ arcsin\left(pv_{0}{\rm {\ exp\ }}\left(ay\right)-{\rm {\ arcsin\ }}\left(pv_{0}\right)\right)\end{aligned}}}

{\displaystyle {\begin{aligned}at{\rm {=2}}p\left({\frac {\sqrt {{\rm {1}}-p^{\rm {2}}v_{0}^{\rm {2}}}}{pv_{0}}}\right).\end{aligned}}}