Two orthogonal sets of circles
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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 |
We now must show that the rays and wavefronts always intersect at right angles (Figure 18). The wavefronts are arcs of circles. By the eikonal equation, we know that the wavefront circles are orthogonal to the raypaths. Let H be the point of intersection of a raypath (characterized by 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 }_{{\rm 1}} ) and a wavefront (characterized by traveltime t). The center and radius of the raypath circle are, respectively,
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} E{\rm =}\left(x_E,y_E\right){\rm =}\left(\frac{v_0}{a{\rm \ tan\ }{\theta }_{{\rm l}} },-\frac{v_0}{a}\right)\\ \text{and}\ \lambda {\rm =}\frac{v_0}{a{\rm \ sin\ }{\theta }_{{\rm 1}}}. \end{align} ()
The center and radius of the wavefront circle are, respectively,
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} G{\rm =\ }\left(x_G{\rm \ ,\ }y_G\right){\rm =}\left({\rm 0,\ }\frac{v_0}{a}\left({\rm \ cosh\ }at-{\rm l}\right)\right) \end{align}
and
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} r&=\frac{v_0}{a}{\ sinh\ }\ \textit{at}. \end{align} ()
The distance between point E and point H 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} EH=\lambda=\frac{v_{0}} {a\mathrm{sin}\theta_{1}} \end{align} ()
The distance between point G and point H 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} GH&{\rm =}r=\frac{v_0}{a}\ \mathrm{sinh}\ \textit{at}. \end{align} ()
By the Pythagorean theorem, we have
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} GE^{{\rm 2}} {\rm =}GL^{{\rm 2}}{\rm +}LE^{{\rm 2}}\\ {\rm =}{\left(y_G-y_E\right)}^{{\rm 2}}{\rm +}x^{{\rm 2}}_E\\ {\rm =}{\left[\frac{v_0}{a}\left({\rm \ cosh\ }at-{\rm 1}\right){\rm +}\frac{v_0}{a}\right]}^{{\rm 2}}\\ {\rm +}{\left[\frac{v_0}{a{\rm \ tan\ }{\theta }_{{\rm 1}}}\right]}^{{\rm 2}}\\ {\rm =}\frac{v^{{\rm 2}}_0}{a^{{\rm 2}}}\left[{{\rm cosh}}^{{\rm 2}}{\rm \ }at-{\rm \ }\frac{{\rm 1}}{{{\rm tan}}^{{\rm 2}}{\theta }_{{\rm 1}}}\right]. \end{align} ()
Consider the triangle GHE. The two circles are orthogonal if and only if this triangle is right, with the right angle at H. For this triangle to be a right triangle, the Pythagorean theorem must hold. Thus, we must verify 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/":): GH^{{\rm 2}}{\rm +}EH^{{\rm 2}}{\rm =}\ GE^{{\rm 2}} . By inserting the above values, we obtain
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} &{\left(\frac{v_0}{a}{\rm \ sinh\ }at\right)}^{{\rm 2}} {\rm +}{\left(\frac{v_0}{a{\rm \ sin\ }{\theta }_{{\rm 1}}}\right)}^{{\rm 2}}{\rm =}\frac{v^{{\rm 2}}_0}{a^{{\rm 2}}}\left({{\rm cosh}}^{{\rm 2}}{\rm \ }at-{\rm \ }\frac{{\rm l}}{{{\rm tan}}^{{\rm 2}}{\theta }_{{\rm 1}}}\right) . \end{align} ()
This equation reduces to
$ {\begin{aligned}&\mathrm {sinh} ^{\rm {2}}\ at{\rm {+}}{\frac {\rm {1}}{{\rm {sin}}^{\rm {2}}{\theta }_{\rm {1}}}}{\ =\ }{\rm {cosh}}^{\rm {2}}\ {\textit {at}}-{\frac {\rm {1}}{{\rm {tan}}^{\rm {2}}{\theta }_{\rm {1}}}},\end{aligned}} $ ()
which gives
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} &\frac{{\rm 1}} {{{\rm sin}}^{{\rm 2}}{\theta }_{{\rm 1}}}-\frac{{\rm 1}}{{{\rm tan}}^{{\rm 2}}{\theta }_{{\rm 1}}}{\rm =}{{\rm cosh}}^{{\rm 2}}at-{{\rm sinh}}^{{\rm 2}}\ \textit{at}. \end{align} ()
The right side and the left side of equation 71 are each equal to one; hence, this Pythagorean theorem is indeed satisfied for triangle GHE. Thus, the wavefront circles and the raypath circles are orthogonal.
The slope of the radius of the wavefront must be the same as the slope of the ray. The center 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/":): \left(x_W,y_W\right) of the wavefront circle must lie on the y-axis, because otherwise, wavefronts would intersect. The radius of the wavefront circle has the direction 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/":): \mathbf{u=}\left({\rm \ sin\ }\theta {\rm ,\ cos\ }\theta \right) at point 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 on the ray. The radius r of the wavefront circle is equal to the distance in this direction from the given point H on the ray to the y-axis.
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| Wave Motion | Visualization |
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
- Migration in the case of constant velocity
- Implementation of migration
- Appendix B: Exercises
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