Wavefront 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

Our next task is to use the eikonal equation to find the wavefronts. The eikonal equation tells us that the wavefronts are orthogonal to the raypaths. We will use this orthogonal property to construct a wavefront (Figure 14). To this end, it is advantageous to make a one-to-one relationship between raypaths and wavefronts. For each point on the (x,y) plane, a raypath exists whose tangent at that point is horizontal. As we have seen, this is the point of maximum depth. By the eikonal equation, the tangent of the wavefront passing through that point must be vertical. The center C of the circular raypath lies vertically under this point. For the moment, we will assume that the wavefront is also circular. It follows that the center G of this circular wavefront must lie horizontally to the side of this point. But where does the center reside? Because everything is symmetric about the vertical axis, it follows that the center of this circular wavefront must lie on the vertical axis. Thus, the required wavefront is a circle with center and radius given, respectively, by

${\begin{aligned}G&{\rm {=\ }}\left(x_{G}{\rm {\ ,\ }}y_{G}\right){\rm {=}}\left(0,y_{D}\right){\rm {=}}\left({\rm {0,\ }}{\frac {v_{0}}{a}}\left({\rm {\ cosh\ }}at_{D}-{\rm {l}}\right)\right){\rm {\ }}r{\rm {=}}x_{D}{\rm {=}}{\frac {v_{0}}{a}}{\rm {\ sinh\ }}at_{D}.\end{aligned}}$

(62)

Figure 14.  Determination of the center of the wavefront circle through maximum depth point D.

We now can drop the subscript from $t_{D}$. As observed Slotnick (1959), the wavefronts are circles whose centers are along the y-axis at the points $\left({\rm {0,\ }}\left(v_{0}/a\right)\left({\rm {\ cosh\ \ }}at-{\rm {\ 1}}\right)\right)$ and whose radii are $\left(v_{0}/a\right){\rm {\ sinh\ }}$ at where t is the traveltime corresponding to each wavefront.

An arbitrary point $W{\rm {=}}\left(x_{W},\ y_{W}\right)$ on the wavefront circle is given by

${\begin{aligned}x_{W}{\rm {=}}r{\rm {\ cos\ }}\psi {\rm {=}}{\frac {v_{0}}{a}}{\rm {\ sinh\ }}\ {\textit {at}}\ {\rm {\ cos\ }}\varphi \\y_{W}{\rm {=}}y_{G}{\rm {+}}r{\rm {\ sin\ }}\psi \\{\rm {=}}{\frac {v_{0}}{a}}\left({\rm {\ cosh\ }}at-{\rm {1}}\right){\rm {+}}{\frac {v_{0}}{a}}{\rm {\ sinh\ }}\ {\textit {at}}{\rm {\ sin\ }}\varphi .\end{aligned}}$

(63)

All the raypaths are circles with centers on the horizontal line where vertical velocity would be zero (Figure 15). All the wavefronts are circles with centers on the y-axis (Figure 16). The set of raypaths and the set of wavefronts are mutually orthogonal (Figure 17).

Figure 15.  The raypath circles all have centers on the horizontal line where vertical velocity would be zero.

Figure 16.  The wavefront circles all have centers in the vertical axis.

Figure 17.  Both the raypaths of Figure 15 and the wavefronts of Figure 16.

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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
• Two orthogonal sets of circles
• Migration in the case of constant velocity
• Implementation of migration
• Appendix B: Exercises

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