The eikonal equation - book

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

If the given wavefront is at time t and the new wavefront is at time $t{\rm {+}}dt$, then the traveltime along the ray is dt. If s measures path length along the given ray, then the travel distance in time dt is ds. The increments dt and ds are related by the slowness, that is, dt = nds. Thus, The directional derivative in the direction of the raypath is equal to the slowness, that is, $n{\rm {=}}dt/ds$. The directional derivative can be written in terms of its components as

${\begin{aligned}{\frac {dt}{ds}}&{\rm {=}}{\frac {\partial t}{\partial x}}{\frac {dx}{ds}}{\rm {+}}{\frac {\partial t}{\partial y}}{\frac {dy}{ds}}{\ =}\ \mathrm {grad} \ {\textit {t}}\cdot {\frac {d\mathbf {r} }{ds}}.\end{aligned}}$

(13)

Because $d\mathbf {r} /ds=\mathbf {u}$, it follows that equation 13 becomes ${dt/ds}{\ =}\ \mathrm {grad} \ {t}\cdot \mathbf {u}$ or $n{\ =}\mathrm {grad} {\textit {t}}\cdot \mathbf {u.}$ The Fermat requirement that the unit tangent u have the same direction as the vector grad $t\left(x,y\right)$ means that these two vectors are related by the scalar $n\left(x,y\right)$. Thus, the relationship can be written as

${\begin{aligned}\mathrm {grad} \ {\textit {t}}\left(x,y\right){\ =}\ n\left(x,y\right)\mathbf {u} \left(x,y\right).\end{aligned}}$

(14)

This equation is the eikonal equation (in vector form). We call $n\left(x,y\right)\mathbf {u} \left(x,y\right)$ the raypath vector. The eikonal equation says that the raypath vector is equal to the gradient of the traveltime. the gradient grad $t\left(x,y\right)$ gives the flow lines on the traveltime surface. Thus, the eikonal equation asserts that the raypath vector is a flow line on the traveltime surface.

Because u is a unit vector in the same direction as that of the gradient, it follows that $n{\rm {=}}{\rm {|}}\mathrm {grad} {\textit {t}}{\rm {|}}$. In other words, the slowness is equal to the magnitude of the gradient of traveltime. If we take the square of each side, we obtain the eikonal equation (in scalar form)

${\begin{aligned}n^{\rm {2}}&{\rm {=}}{\left({\frac {\partial t}{\partial x}}\right)}^{\rm {2}}{\rm {+}}{\left({\frac {\partial t}{\partial y}}\right)}^{\rm {2}}\end{aligned}}$

(15)

Eikonal equation 15 says that the magnitude of the gradient of the traveltime is equal to the slowness (Robinson and Clark, 2003).

Eikonal equation 14 says that at any point, the gradient of the traveltime is equal to the slowness n times the unit tangent to the ray. Therefore, the gradient and the tangent go in the same direction. Because the gradient is orthogonal to the wavefront and the tangent is along the ray, it follows that the ray is orthogonal to the wavefront:

${\begin{aligned}\left({\frac {\partial t}{\partial x}}{\rm {\ ,\ }}{\frac {\partial t}{\partial y}}\right){\rm {=\ }}n\left({\rm {\ cos\ }}\theta {\rm {,\ sin\ }}\theta \right).\end{aligned}}$

(16)

The left-hand side of equation 16 involves the wavefront; the right-hand side involves the ray. As we have seen, velocity is called swiftness. The reciprocal of velocity is slowness. The connecting link is the slowness. In the above equation, the function $t\left(x,y\right)$ is the traveltime from the source to the point with the coordinates (x,y), and $n\left(x,y\right){\rm {=}}{\rm {1}}/v\left(x,y\right)$ is the slowness (or reciprocal velocity) at that point. The apparent swiftnesses along the coordinate directions are, respectively, $\partial x{\rm {/}}\partial t,\partial y/\partial t$. Thus, the apparent slownesses along the coordinate directions are $\partial t{\rm {/}}\partial x,\partial t/\partial y$. The actual swiftness along the raypath direction is $v{\rm {=}}ds/dt$. Thus the actual slowness along the raypath direction is $n{\rm {=}}dt/ds$.

The eikonal equation describes the traveltime propagation in an isotropic medium. To obtain a well-posed initial-value problem, it is necessary to know the velocity function $v\left(x,y\right)$ at all points. Moreover, as an initial condition, the source or some particular wavefront must be specified. Furthermore, one must choose one of the two branches of the solutions (namely, either the wave going from the source or the wave going to the source). The eikonal equation then yields the traveltime field $t\left(x,y\right)$ in the heterogeneous medium, as required for Migration and other seismic processing needs.

The eikonal equation is a restatement of Fermat’s principle of least time. In other words, a raypath must be a flow line. A flow line is orthogonal to all the wavefronts. The eikonal equation is the fundamental equation that connects the ray (which corresponds to the fuselage of the airplane) to the wavefront (which corresponds to both wings of the airplane). The wings let the fuselage feel the effects of points removed from the path of the fuselage. The eikonal equation makes a traveling wave (as envisaged by Huygens) fundamentally different from a traveling particle (as envisaged by Newton). Hamilton perceived that there is a wave-particle duality, which provides the mathematical foundation of quantum mechanics. Hamilton’s work is based on the principle of least action, which is a more general formulation of the principle of least time (Robinson and Douze, 1985).

Let us summarize. We defined the gradient of the traveltime surface. We also defined the raypath curve. We established that the ray direction is always perpendicular to the traveltime surface. A wave as it travels must follow the path of least time. The wavefronts are like contour lines on a hill. The height of the hill is measured in time. Take a point on a contour line. In what direction will the ray point? Suppose the ray points in the direction of the contour line. In other words, suppose that the raypath lies directly on the wavefront. As the wave travels a certain distance along this ray, it takes time, but all time is the same along the wavefront. Thus, a wave cannot travel along a wavefront. It follows that a ray must point away from a wavefront.

Suppose now that a ray points away from the wavefront. The wave wants to take the least time to travel to the new wavefront. By isotropy, the wave’s velocity is the same in all directions. Because the traveltime is velocity multiplied by distance, the wave wants to take the raypath that goes the shortest distance. The shortest distance is along the path that has no component along the wavefront; that is, the shortest distance is along the normal to the wavefront. In other words, the raypath must be orthogonal (i.e., at right angles) to the wavefront. Thus, the ray’s unit tangent vector u must be orthogonal to the wavefront. By definition, the gradient is a vector that points in the direction orthogonal to the wavefront. Thus the ray’s unit tangent vector u and the gradient grad t of the wavefront must point in the same direction.

The Pythagorean property of the right triangle is fundamental to an understanding of wave propagation in an isotropic medium, in which the wavefront moves along raypaths that are always perpendicular to the wavefront. Thus, the key to wave motion in an isotropic medium is the right angle. Both wavefronts and raypaths reveal the propagation of a traveling wave, and each approach has its own merits.

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

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