Raypath 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

The differentials of horizontal distance and vertical distance 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} &dx{\ =\ sin\ }\theta \ ds{\ =}\ \rho {\rm \ sin\ }\theta\ d\theta {\ ,\ }dy{\ =\ cos\ }\theta \ ds{\ =}\ \rho {\rm \ cos\ }\theta\ d\theta. \end{align}

(42)

Let a particle of energy start at the origin 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{\rm =0,\ }y{\rm =0}\right) , where the path makes at an 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 with the vertical. The particle travels along a circular arc to 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/":): \left(x{\rm ,\ }y\right) , where the path makes an 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 with the vertical. We wish to find the coordinates 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{\rm ,\ }y\right) . The horizontal distance is given by

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} x&{\ =\ }\rho\int^{\theta }_{{\theta }_{{\rm o}} }{{\rm \ sin\ }}\theta\ d\theta {\ =}-\rho{\rm \ cos\ }\theta {\rm +}\rho{\rm \ cos\ }{\theta }_0 \end{align}

(43)

and the vertical distance is given by

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} y&{\rm =}\rho\int^{\theta }_{{\theta }_0}{{\rm \ cos\ }} \theta \ d\theta {\ =\ }\rho{\rm \ sin\ }\theta -\rho{\rm \ sin\ }{\theta }_0. \end{align}

(44)

Equations 43 and 44 are the parametric equations of a circle (Figure 11). The center of this circle is point C, with coordinates

Figure 11.  The arcs of the circles through the origin represent raypaths with different initial angles. The circles with centers on the depth axis represent wavefronts with different traveltimes.

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} C&{\ =\ }\left(x_C{\rm,\ }y_C\right){\rm =}\left(\rho{\rm \ cos\ }{\theta }_0{\rm,\ }-\rho {\rm \ sin\ }{\theta }_0\right){\rm =}\left(\frac{v_0}{a{\rm \ sin\ }{\theta }_0}{\rm \ cos\ }{\theta }_0{\rm \ ,\ }-{\rm \ }\frac{v_0}{a{\rm \ sin\ }{\theta }_0}{\rm \ sin\ }{\theta }_0\right)\\ &{\rm =}\left(\frac{v_0}{a{\rm \ tan\ }{\theta }_0}{\rm ,\ }-\frac{v_0}{a}\right) . \end{align}

(45)

In terms of Snell’s parameter p, the center 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} C&{\rm =}\left(x_C,y_C\right){\rm =}\left(\frac{{\rm 1}} {ap}{\rm \ cos\ }{\theta }_0{\rm ,\ }-\frac{v_0}{a}\right){\rm =}\left(\frac{{\rm 1}}{ap}{\left({\rm 1}-p^{{\rm 2}}v^{{\rm 2}}_0\right)}^{{\rm 1/2}}{\rm ,\ }-\frac{v_0}{a}\right) . \end{align}

(46)

As observed by Slotnick (1959), the raypath is given by the equation of a circle of radius 1/ap, whose center is at the point with coordinates

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} x_c{\rm =} {\left({\rm 1}-p^{{\rm 2}} v^{{\rm 2}}_0\right)}^{{\rm 1/2}} /{ap},\ y_C{\rm =}-v_0\ \textit{la}. \end{align}

(47)

Note that the y-coordinate of the center is independent of the parameter p.

We therefore can write the parametric equations of the circular raypath as

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} x&{\rm =}-\left(\frac{v_0}{a{\rm \ sin\ }{\theta }_{{\rm o}}}\right){\rm \ cos\ }\theta {\rm +}\frac{v_0}{a{\rm \ }{\rm tan\ }{\theta }_0} \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} &y{\rm =}\left(\frac{v_0}{a{\rm \ sin\ }{\theta }_0}\right){\rm \ sin\ }\theta -\frac{v_{{\rm o}} }{a}. \end{align}

(48)

Note that the vertical component 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_0/a does not depend on the 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 . As a result, the family of rays with the given velocity function is made up of the origin and that have their centers on the line 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/":): y{\rm =} -v_0/a. . The circular raypath starts at the origin (which is the shot point). The tangent to the circle at the shot point makes an 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 with the positive y-axis.

Often, we will know only that the ray passes through a certain point, say, $ B{\rm {=}}\left(x{\rm {,\ }}y\right) $, and we must determine the equation of the ray. This problem can be solved as described here and shown in Figure 12. Draw line OB. The midpoint of this line 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/":): A{\rm =}\ {\rm (}x/{\rm 2}, \textit{y/2}{\rm )} . The center of the raypath circle must lie on the perpendicular bisector of the line OB. The center of the raypath circle also must lie on the horizontal line through the 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/":): \left({\rm 0,\ }-v_0/a\right) . The intersection of these two lines of determination gives the center E 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/":): E{\rm =}\left(x_E{\rm ,\ }-v_0/a\right) of the raypath circle. From the geometry, we see that

Figure 12.  Determination of the equation of a ray that passes through a point B. Use the fact that the origin O is also a point on the ray. Connect these two known points with line OB. Use the fact that the ray is a circle. Hence the center lies on the perpendicular bisector AE of line OB. Use the fact that this center also lies on line JE on which the velocity would be zero. Thus the center is at the intersection E of these two lines. The radius is given by line EB. The raypath is the circle with this center and radius.

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} x_E{\rm =}JE{\rm =}JG{\rm +}GE{\rm =}OF{\rm +}GE{\rm =}OF{\rm +}AG{\rm \ tan\ }\alpha\\ {\rm =}\frac{x}{{\rm 2}} {\rm +}\left(\frac{y}{{\rm 2}}{\rm +}\frac{v_0}{a}\right)\frac{y}{x}{\rm =}\frac{x^{{\rm 2}}{\rm +}y^{{\rm 2}}}{{\rm 2}x}{\rm +}\frac{v_0}{a}\frac{y}{x}. \end{align}

(49)

Thus, we have determined the center E. The radius 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} &\rho {\rm =}OE{\rm =}\sqrt{OJ^{{\rm 2}} {\rm +}JE^{{\rm 2}}}{\rm =}\sqrt{{\left(\frac{v_0}{a}\right)}^{{\rm 2}}{\rm +}{\left(\frac{x^{{\rm 2}}{\rm +}y^{{\rm 2}}}{{\rm 2}x}{\rm +}\frac{v_0}{a}\frac{y}{x}\right)}^{{\rm 2}}}. \end{align}

(50)

The initial angle of the raypath 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} {\theta }_0&{\rm =\ arcsin\ }\frac{v_{0}} {a\rho} \end{align}

(51)

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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
• 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
• Appendix B: Exercises

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