# Raypath for velocity linear with depth

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,

**(**)

Let a particle of energy start at the origin , where the path makes at an initial angle with the vertical. The particle travels along a circular arc to point , where the path makes an angle with the vertical. We wish to find the coordinates . The horizontal distance is given by

**(**)

and the vertical distance is given by

**(**)

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

**(**)

In terms of Snell’s parameter *p*, the center is

**(**)

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

**(**)

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

**(**)

Note that the vertical component does not depend on the angle . 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 . The circular raypath starts at the origin (which is the shot point). The tangent to the circle at the shot point makes an angle with the positive *y*-axis.

Often, we will know only that the ray passes through a certain point, say, , 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 . 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 . The intersection of these two lines of determination gives the center *E* of the raypath circle. From the geometry, we see that

**(**)

Thus, we have determined the center *E*. The radius is

**(**)

The initial angle of the raypath is

**(**)

## References

- ↑ Slotnick. M. M., 1959, Lessons in seismic computing: SEG.

## Continue reading

Previous section | Next section |
---|---|

Ray equation for velocity linear with depth | Traveltime for velocity linear with depth |

Previous chapter | Next chapter |

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
- 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