Two orthogonal sets of circles

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

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 ${\theta }_{\rm {1}}$) and a wavefront (characterized by traveltime t). The center and radius of the raypath circle are, respectively,

Figure 18.  Pythagorean confirmation that wavefront circles and raypath circles are orthogonal.

${\begin{aligned}E{\rm {=}}\left(x_{E},y_{E}\right){\rm {=}}\left({\frac {v_{0}}{a{\rm {\ tan\ }}{\theta }_{\rm {l}}}},-{\frac {v_{0}}{a}}\right)\\{\text{and}}\ \lambda {\rm {=}}{\frac {v_{0}}{a{\rm {\ sin\ }}{\theta }_{\rm {1}}}}.\end{aligned}}$

(64)

The center and radius of the wavefront circle are, respectively,

${\begin{aligned}G{\rm {=\ }}\left(x_{G}{\rm {\ ,\ }}y_{G}\right){\rm {=}}\left({\rm {0,\ }}{\frac {v_{0}}{a}}\left({\rm {\ cosh\ }}at-{\rm {l}}\right)\right)\end{aligned}}$

and

${\begin{aligned}r&={\frac {v_{0}}{a}}{\ sinh\ }\ {\textit {at}}.\end{aligned}}$

(65)

The distance between point E and point H is

${\begin{aligned}EH=\lambda ={\frac {v_{0}}{a\mathrm {sin} \theta _{1}}}\end{aligned}}$

(66)

The distance between point G and point H is

${\begin{aligned}GH&{\rm {=}}r={\frac {v_{0}}{a}}\ \mathrm {sinh} \ {\textit {at}}.\end{aligned}}$

(67)

By the Pythagorean theorem, we have

${\begin{aligned}GE^{\rm {2}}{\rm {=}}GL^{\rm {2}}{\rm {+}}LE^{\rm {2}}\\{\rm {=}}{\left(y_{G}-y_{E}\right)}^{\rm {2}}{\rm {+}}x_{E}^{\rm {2}}\\{\rm {=}}{\left[{\frac {v_{0}}{a}}\left({\rm {\ cosh\ }}at-{\rm {1}}\right){\rm {+}}{\frac {v_{0}}{a}}\right]}^{\rm {2}}\\{\rm {+}}{\left[{\frac {v_{0}}{a{\rm {\ tan\ }}{\theta }_{\rm {1}}}}\right]}^{\rm {2}}\\{\rm {=}}{\frac {v_{0}^{\rm {2}}}{a^{\rm {2}}}}\left[{\rm {cosh}}^{\rm {2}}{\rm {\ }}at-{\rm {\ }}{\frac {\rm {1}}{{\rm {tan}}^{\rm {2}}{\theta }_{\rm {1}}}}\right].\end{aligned}}$

(68)

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 $GH^{\rm {2}}{\rm {+}}EH^{\rm {2}}{\rm {=}}\ GE^{\rm {2}}$. By inserting the above values, we obtain

${\begin{aligned}&{\left({\frac {v_{0}}{a}}{\rm {\ sinh\ }}at\right)}^{\rm {2}}{\rm {+}}{\left({\frac {v_{0}}{a{\rm {\ sin\ }}{\theta }_{\rm {1}}}}\right)}^{\rm {2}}{\rm {=}}{\frac {v_{0}^{\rm {2}}}{a^{\rm {2}}}}\left({\rm {cosh}}^{\rm {2}}{\rm {\ }}at-{\rm {\ }}{\frac {\rm {l}}{{\rm {tan}}^{\rm {2}}{\theta }_{\rm {1}}}}\right).\end{aligned}}$

(69)

This equation reduces to

${\begin{aligned}&\mathrm {sinh} ^{\rm {2}}\ at{\rm {+}}{\frac {\rm {1}}{{\rm {sin}}^{\rm {2}}{\theta }_{\rm {1}}}}{\ =\ }{\rm {cosh}}^{\rm {2}}\ {\textit {at}}-{\frac {\rm {1}}{{\rm {tan}}^{\rm {2}}{\theta }_{\rm {1}}}},\end{aligned}}$

(70)

which gives

${\begin{aligned}&{\frac {\rm {1}}{{\rm {sin}}^{\rm {2}}{\theta }_{\rm {1}}}}-{\frac {\rm {1}}{{\rm {tan}}^{\rm {2}}{\theta }_{\rm {1}}}}{\rm {=}}{\rm {cosh}}^{\rm {2}}at-{\rm {sinh}}^{\rm {2}}\ {\textit {at}}.\end{aligned}}$

(71)

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 $\left(x_{W},y_{W}\right)$ of the wavefront circle must lie on the y-axis, because otherwise, wavefronts would intersect. The radius of the wavefront circle has the direction $\mathbf {u=} \left({\rm {\ sin\ }}\theta {\rm {,\ cos\ }}\theta \right)$ at point $\theta$ 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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Table of Contents (book)

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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