Ray equation

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 need to understand how a single ray moving along a particular path - say, a seismic ray - can know what is an extremal path in the variational sense. To illustrate the problem, consider a seismic wave traveling through a medium whose slowness n increases in the direction of travel. Because the raypath is parallel to the gradient of the slowness, it undergoes no bending. However, if the contour lines of slowness are at an angle to the ray, then the ray will bend, even though the slowness at each point along the path is identical to what it was before when there was no bending. This shows that a ray’s path cannot be explained solely in terms of the value of the slowness on that path. We also must consider the slowness along neighboring paths - hat is, along paths not taken.

The classical wave explanation, proposed by Huygens, resolves this problem by saying that light does not propagate in the form of a single ray. According to the wave interpretation, light propagates as a wavefront possessing transverse width. If a small section of a propagating wavefront encounters the same value of slowness along its width, then the ray will not bend. If a small section of a propagating wavefront encounters different values of slowness along its width, then the ray will bend. The amount of bending depends on the gradient of the slowness. The wavefront propagates more rapidly on the side where the slowness is low (i.e., where the velocity is high) than it does on the side where the slowness is high. As a result, the wavefront naturally turns in the direction of increasing slowness.

In this section, the position vector r always represents a point on a specific raypath and not any arbitrary point in space. As time increases, r traces out the particular raypath in question. The seismic ray at any given point follows the direction of the gradient of the traveltime field $t\mathbf {(} r)$. As before, let u be the unit vector along the ray. The ray in general will follow a curved path, and the raypath vector $n\mathbf {\ u}$ will be tangent to this curved raypath.

Let us derive the ray equation. This equation tells us how the raypath vector $n\mathbf {u}$ changes along the curved raypath. The equation for the unit tangent vector is

${\begin{aligned}\mathbf {u=} &{\frac {d\mathbf {r} }{ds}}.\end{aligned}}$

(17)

We also know that

${\begin{aligned}{\frac {d\mathbf {r} }{ds}}&{\rm {=}}\left({\frac {dx}{ds}},{\frac {dy}{ds}}\right).\end{aligned}}$

(18)

Thus, the unit tangent vector u is

${\begin{aligned}\mathbf {u=} &\left({\frac {dx}{ds}},{\frac {dy}{ds}}\right).\end{aligned}}$

(19)

The slowness $n\left(x,y\right)$ is a scalar function that depends on coordinates x and y. However, we can hold y constant and consider the curve that gives the variation of slowness with x alone. The slope of this curve $\partial n/\partial x$ is called the partial derivative of slowness with respect to x. The partial derivative $\partial n/\partial y$ is defined similarly. The gradient of the slowness surface $n\left(x,y\right)$ is the vector with these partial derivatives as components; that is,

${\begin{aligned}\mathrm {grad} \ {\textit {n}}&{\ {\rm {=}}}\left({\frac {\partial n}{\partial x}}{\rm {,\ }}{\frac {\partial n}{\partial y}}\right).\end{aligned}}$

(20)

We know that the gradient of the traveltime is

${\begin{aligned}\mathrm {grad} \ {\textit {t}}&{\ {\rm {=}}}\left({\frac {\partial t}{\partial x}},{\frac {\partial t}{\partial y}}\right).\end{aligned}}$

(21)

We recall eikonal equation 14. Now we will do the mathematics. We take the derivative of the eikonal equation with respect to path length s. We obtain

${\begin{aligned}{\frac {d}{ds}}\ \left(n\mathbf {u} \right)&{\rm {=}}{\frac {d}{ds}}{\rm {(}}\mathrm {grad} \ {\textit {t}}{\rm {)}}.\end{aligned}}$

(22)

On the right side, we interchange the two operations to obtain

${\begin{aligned}{\frac {d}{ds}}&{\rm {(}}\mathrm {grad} \ {\textit {t}}{\rm {)=}}\mathrm {grad} \ ({\frac {dt}{ds}}{\rm {)}}.\end{aligned}}$

(23)

We recognize $\partial t{\rm {/}}\partial s$ as the slowness n. Thus, the right side of equation 23 is the gradient of the slowness, so we can write

${\begin{aligned}{\frac {d}{ds}}&{\rm {(}}\mathrm {grad} \ {\textit {t}}{\rm {)=}}\mathrm {grad} \ (n{\rm {)}}.\end{aligned}}$

(24)

If we put together the above equations, we obtain the ray equation

${\begin{aligned}{\frac {d}{ds}}\ \left(n\mathbf {u} \right)&{\rm {=}}\mathrm {grad} \ {\textit {n}}.\end{aligned}}$

(25)

Ray equation 25 says that the rate of change of the raypath vector $n\mathbf {u}$ is equal to the gradient of the slowness. We know that eikonal equation 14 says that the raypath vector is a flow line on the traveltime surface. The ray equation says that the rate of change of the ray-path vector is a flow line on the slowness surface.

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

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