Appendix B: Exercises
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| 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 |
1. Show that (in the case of two dimensions x, y) if the velocity is $ 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 $ x{\rm {=}}r_{0} $ and common emergence time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): v{\rm =}v_0{\rm +}ay .
3. Let the velocity (in the case of two dimensions x,y) be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): v(y)\ {\rm =}v_0{\rm /}\sqrt{{\rm 1+}ay} . Show that the equation for the raypath with initial angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\theta }_0 to the vertical is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} a y{\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{align}
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): v\left(y\right){\rm =}v_{{\rm 0}}{\rm \ exp\ }\left(ay\right) . Show that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} ax =2(\ arcsin\left(pv_0{\rm \ exp\ }\left(ay\right)-{\rm \ arcsin\ }\left(pv_0\right)\right) \end{align}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} at{\rm =2}p\left(\frac{\sqrt{{\rm 1}-p^{{\rm 2}}v^{{\rm 2}}_0}}{pv_0}\right). \end{align}
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Also in this chapter
- Reflection seismology
- Digital processing
- Signal enhancement
- Migration
- Interpretation
- Rays
- The unit tangent vector
- Traveltime
- The gradient
- The directional derivative
- The principle of least time
- The eikonal equation
- Snell’s law
- Ray equation
- Ray equation for velocity linear with depth
- Raypath for velocity linear with depth
- Traveltime for velocity linear with depth
- Point of maximum depth
- Wavefront for velocity linear with depth
- Two orthogonal sets of circles
- Migration in the case of constant velocity
- Implementation of migration
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