Fermat’s principle and reflection and refraction

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
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

According to Fermat’s principle, the propagation of a seismic wave between any two points A and B will take place in such a way that the first-order infinitesimal variation of the traveltime over the raypath between the two points will be zero with respect to all conceivable neighboring paths. The material between the two points might consist of any number of media with different velocities. The raypaths, therefore, might include any number of reflections and refractions. As a consequence of Fermat’s principle, it follows that the traveltime over a true raypath can be a maximum, a minimum, or stationary with respect to all neighboring paths that start and end at the same two points A and B. In most practical cases, the traveltime is a minimum. For instance, when the medium is uniform, the raypath is a straight line, which is the shortest distance between the two points, so the time is a minimum. In addition, the time is always a minimum when the interfaces separating the media of differing velocities are plane surfaces. In other cases, it can be a maximum. Fermat’s principle has been used widely to derive useful properties of raypaths. For instance, it can be used to derive the laws of reflection and refraction from plane surfaces.

Figure 20.  Law of reflection.

First let us give a graphic derivation of the law of reflection using Fermat’s principle. In Figure 20, ST is the trace (in the plane of the paper) of a reflecting interface that is a plane perpendicular to the plane of the paper. Points A and B are any two points in the plane of the paper above the plane ST. Point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A^'} is the image of point A with respect to the plane ST. It is located by drawing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle AA^'} normal to ST and making AD equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle DA^'} . Draw the straight line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A^'CB} , which cuts the line ST at point C. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^'} be any point whatever in the plane ST that is not coincident with C. (Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^'} is not necessarily on the line ST.) Then ACB and ACFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^'} B are two conceivable travel paths from A to the plane ST to B.

In addition, the following relations hold as a consequence of the proposition that a straight line is the shortest distance between two points. Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^'} is any point on the plane ST, it follows that ACB is the shortest path and therefore the minimum time path from A to the plane ST to B. Now because PC is normal to ST and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A^'} is the image of A, both the angle of incidence ACP and the angle of reflection PCB are complements of the angle DCFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A^'} . Hence, the angle of incidence is equal to the angle of reflection. This is the law of reflection.

Next let us derive graphically the law of refraction by using Fermat’s principle. In Figure 21, ST is a plane interface separating two media of velocities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_{{\rm 1}}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_{{\rm 2}};PC} is a normal to this plane. A and B are any two points in the plane normal to the plane ST and containing the normal PC. C is any point on the line ST. D is a neighboring point on the line ST. The time over the path ACB is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t_{ACB}=\frac{AC}{v_{1}} +\frac{CB}{v_{2}} \end{align} } (42)

If CE is normal to AD and DF is normal to CB and if CD is very small, the time over the path ADB can be written as


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t_{ADB}&=\frac{AD}{v_{1}} +\frac{DB}{v_{2}}=\frac{AC+ED}{V_{1}}+\frac{CB-FC}{v_{2}} \end{align} } (43)

so that


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t_{ADB}-t_{ACB}&=\Delta t\frac{ED}{v_{1}} -\frac{FC}{v_{2}}\approx \frac{CDsin\theta_{i}}{v_{1}}-\frac{CDsin\theta_{t}}{v_{2}} \end{align} } (44)

Now if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{ACB}} is such that its first-order variation is zero, we must have


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\Delta t}{CD}=0 \ \mathrm {or} \ \frac{sin\theta_{i}} {v_{1}}=\frac{sin\theta_{t}}{V_{1}}, \end{align} } (45)

and when this condition is satisfied, we can see from the figure that the traveltime will be a minimum and not a maximum. This is Snell’s law of refraction, which already has been derived by using Huygens’ principle.

If point C is displaced to either side and at right angles to the line ST so that it still remains in the plane ST, the traveltime over the displaced path will be greater than that over the undisplaced path. This is true for any arbitrary undisplaced position of point C on the line ST. It follows that the absolute minimum time occurs when C lies on the line ST, and therefore the refracted ray CB lies in the same plane as the incident ray AC and the normal PC.

Figure 21.  Law of refraction.

Fermat’s principle states that a raypath between two points separated by several media of different velocities is a path such that the traveltime is stationary to the first order of differentials with respect to all conceivable neighboring paths. Huygens’ principle can be used to derive the law of reflection and Snell’s law of refraction. These same laws also can be derived by using Fermat’s principle, as we have done in this section.

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