Fermat’s principle - 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	1
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

The laws of reflection and refraction and indeed the manner in which waves propagate in general can be understood by means of Fermat’s principle. This principle provides an insightful and highly useful way of appreciating and anticipating the behavior of waves. Heron of Alexandria first set forth a so-called variational principle. In his formulation of the law of reflection, Heron asserted that the path actually taken by light traveling from a point S to a point P via a reflecting surface is the shortest possible one. For more than 1500 years, Heron’s observation stood alone - until in 1657, Fermat propounded his principle of least time, which encompassed both reflection and refraction. Because a beam of light obliquely traversing an interface does not travel in a straight line or assume a minimal spatial path length between a point in the incident medium and a point in the transmitting medium, Fermat reformulated Heron’s statement as follows: The actual path between two points taken by a beam of light is the one that is traversed in the least time (Robinson and Clark, 2006a).

Suppose that we have a stratified medium composed of m layers, each layer having a different wave velocity, as shown in Figure 16. The transit time from A to B is then

${\begin{aligned}t{\rm {=}}&{\frac {s_{\rm {1}}}{v_{\rm {1}}}}{\rm {+}}{\frac {s_{\rm {1}}}{v_{\rm {1}}}}{\rm {+}}\cdots {\rm {+}}{\frac {s_{m}}{v_{m}}},\mathrm {or} t{\rm {=}}\sum _{i{\rm {=1}}}^{m}{\frac {s_{j}}{v_{i}}},\end{aligned}}$

(40)

where $s_{i}$ and $v_{i}$ are the path length and the velocity associated with the ith layer. For an inhomogeneous medium in which v is a function of position, the summation must be changed to an integral:

${\begin{aligned}t{\rm {=}}\int _{\rm {A}}^{B}{\frac {ds}{v\left(s\right)}}.\end{aligned}}$

(41)

Fermat’s principle states that when the raypath travels from point A to point B, it traverses the route with the smallest possible transit time (Robinson and Clark, 1987a).

Figure 16.  A stratified medium consisting of three layers.

The original statement of Fermat’s least-time principle, as given above, needs some modification. Recall that given a function $f\left(x\right)$, we can determine the specific value of the variable x that makes $f\left(x\right)$ stationary if we set $df{\rm {/}}dx{\rm {=0}}$ and solve for x. By a stationary value, we mean a value for which the slope of $f\left(x\right)$ versus x is zero. Equivalently, a stationary value is one for which the function is either at a maximum or a minimum or occurs at an inflection point where the (horizontal) tangent is zero.

In its modern form, Fermat’s principle reads as follows: A ray proceeding from point A to point B must traverse a path for which the transit time is stationary with respect to variations of that path. In other words, the transit time for the true trajectory equals (to a first approximation) the transit times of paths immediately adjacent to it. Thus, there will exist neighboring paths along which the ray takes very nearly the same time to traverse each such path. This latter point makes it possible to begin to understand how a ray manages to be so clever in its meanderings. Suppose that we have a beam of waves advancing through a homogeneous isotropic medium so that a ray passes from point A to point B. Particles within the material are driven by the incident disturbance, and they reradiate in all directions. Quite generally, wavelets originating in the immediate vicinity of a stationary path will arrive at B by routes that differ only slightly, and therefore the wavelets will reinforce each other. Wavelets taking other paths will arrive at P out of phase and therefore tend to cancel each other. That being the case, energy will propagate effectively along the path from A to B that satisfies Fermat’s principle.

To show that the transit time for a ray need not always be a minimum, examine Figure 17, which depicts a segment of a 3D ellipsoidal surface. If the source A and the observer B are at the foci of the ellipsoid, then by definition, the length ACB will be constant regardless of where on the perimeter the point C happens to be. It is also a geometric property of the ellipse that ${\theta }_{i}{\rm {=}}{\theta }_{r}$ for any location of C. Therefore, all transit times from A to B via a reflection point are precisely equal. None is a minimum, and the transit time is clearly stationary with respect to path variations along the ellipse. Rays leaving A and striking the surface will arrive at the focus B. From another viewpoint, we can say that radiant energy emitted by A will be scattered by the ellipsoidal surface so that the wavelets will reinforce each other substantially only at B, where they have traveled the same distance and have the same phase.

Figure 17.  Ellipsoid with foci A and B.

Figure 18.  Illustration of cases of maximum and minimum reflection time.

Let us now examine in more detail the conditions that determine whether a raypath in the process of satisfying Fermat’s principle is a maximum, a minimum, or a stationary time path and whether these conditions have any particular significance with regard to the laws of reflection and refraction.

In Figure 18, the curve CPD is the elliptical reflecting surface (in two dimensions) associated with the two given points A and B and the reflection point P. Curve EPF is a reflecting surface (in two dimensions) tangent to the elliptical surface CPD at P but with greater curvature. If Q is any point on the curve EPF close to the point P, then the hypothetical travel path AQB is obviously shorter than the raypath APB because the path is constant for all points on the elliptical curve CPD. Because Q is any neighboring point on the curve EPF, it follows that the raypath APB is a maximum with respect to all hypothetical neighboring reflections from the surface EPF. Likewise, if GPH is a reflecting surface tangent to the elliptical surface at P but with less curvature, then the raypath APB is a minimum time path with respect to the reflector GPH. It can be concluded, therefore, that if any arbitrary reflecting surface has greater curvature than the elliptical surface at the common reflecting point at which they are tangent, the raypath is a maximum with respect to neighboring hypothetical reflection paths that start at one focus of the elliptical surface and end at the other. If the arbitrary reflecting surface has less curvature than the tangent elliptical surface, the raypath is a minimum. It is evident that the tangent plane SPT always has less curvature than the elliptical surface, so a reflection from a plane always follows a minimum time path.

Again refer to Figure 18. It can be seen that two other points, J and K, which are closer to P but still on the raypath APB, can be found so that their associated elliptical surface (shown by dashes) will have greater curvature than the surface EPF. This means that the raypath JPK is a minimum rather than a maximum, as was the case for the ray APB. Therefore, whether a path is a maximum or a minimum depends not only on the shape of the reflector but also on the points that are selected as the beginning and the end of the path. The question is of no importance as far as the law of reflection is concerned. The law of reflection depends only on the vanishing of the first-order time variation and is independent of whether the time is a maximum, a minimum, or stationary. Fermat’s principle is stated sometimes as a minimum-time principle, but it also must be understood to include the maximum and the stationarity conditions as well as the minimum condition. A well-known seismic example of a traveltime maximum is the reflection from the bottom of a syncline in the seismic “bow-tie” response.

If the curvature of the reflecting surface is greater than that of the tangent elliptical surface, the reflected path ADB traversed by a ray then would be a relative minimum. At the other extreme, if the reflecting surface conforms to a surface lying within the ellipsoid, such as the dashed one shown, that same ray along ADB would negotiate a relative maximum transit time. This is true even though other unused paths (where ${\theta }_{i}\neq {\theta }_{r}$) would be shorter (i.e., apart from inadmissible curved paths). In all cases, the rays travel a path with either a minimum, maximum, or stationary transit time in accordance with Fermat’s principle. Note that because the principle speaks only about the path and not about the direction along it, a ray going from A to B will trace out the same route as one from B to A. This is the very useful principle of reciprocity.

As we have seen, an ellipsoid represents a surface for which all reflecting paths between the two foci A and B have equal traveltimes. In a similar way, an equal-time refracting surface can be defined between two media so that all refraction paths between two points A and B, one in each medium, have equal refraction times. That is, an equal-time refraction surface associated with two points A and B is defined as a surface of separation between two media of velocities $v_{\rm {1}}$ and $v_{\rm {2}}$, such that if A lies in the material of velocity $v_{\rm {1}}$ and B lies in the material of velocity $v_{\rm {2}}$, the traveltime from A to any point P on the equal-time refraction surface at velocity $v_{\rm {1}}$ plus the traveltime from P to B at velocity $v_{\rm {2}}$ will be constant. Any raypath APB for which P is a point on the equal-time refraction surface automatically satisfies Snell’s law of refraction. This follows from a simple application of Fermat’s principle.

In Figure 19, CRPD is an equal-time refraction surface associated with points A and B. Point A is in material of velocity $v_{\rm {1}}$ and B is in material of velocity $v_{\rm {2}}$, with $v_{\rm {2}}{\rm {>}}v_{\rm {1}}$. EPF is an actual refracting interface separating the two media and tangent to the equal-time surface CRPD at P. Choose a point Q on the actual refracting surface close to P and draw the paths AQRB and AQB. Then APB is a true raypath because P lies on both surfaces CRPD and EPF. In addition, $QB{\rm {<}}{QRB}$ because a straight line is the shortest distance between two points. If $t_{QB}$ is the traveltime over the path QB at velocity $v_{\rm {2}}$ and $t_{QRB}$ is the traveltime over the segment QR at velocity $v_{\rm {1}}$ plus the time over the path RB at velocity $v_{\rm {2}}$, then $t_{QB}{\rm {<}}t_{QRB}$ because $v_{\rm {2}}{\rm {>}}v_{\rm {1}}$. However, $t_{AQRB}{\rm {=}}t_{APB}$ because these are rays refracted by the equal-time refraction surface. It follows that $t_{AQB}{\rm {<}}t_{APB}$. Because Q is any neighboring point on the refracting surface, $t_{APB}$ is a maximum traveltime with respect to all hypothetical neighboring paths refracted by the surface EPF.

Figure 19.  Illustration of cases of maximum refraction time.

In conclusion, if the curvature of the actual refracting surface is greater than that of the equal-time refraction surface where they are tangent, then the true raypath is a maximum time path. Likewise, it also can be shown that if the curvature of the actual refracting surface is less than that of the tangent equal-time surface, the raypath is a minimum. A refracting surface that is a plane is always of this latter type and therefore always produces a refracted path of minimum traveltime. These conclusions are identical to those derived above for reflecting surfaces. Furthermore, all results stated above that relate maximum, minimum, and stationary reflection paths to various types of reflecting surfaces are also valid for refracting surfaces.

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Table of Contents (book)

Also in this chapter

• Introduction
• Wavefronts and raypaths
• d’Alembert’s solution
• One-dimensional waves
• Sinusoidal waves
• Phase velocity
• Wave pulses
• Geometric seismology
• The speed of light
• Huygens’ principle
• Reflection and refraction
• Ray theory
• Fermat’s principle and reflection and refraction
• Diffraction
• Analogy
• Appendix A: Exercise

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