Reflection and refraction laws and Fermat’s principle

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	6
Pages	181 - 220
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 6.3a

Use Fermat’s principle of stationary time to derive the law of reflection.

Background

In the solution of problem 3.1a we showed that the angle of incidence equals the angle of reflection and that, for the angle of refraction $\theta _{2}$, ${\rm {\;sin\;}}\theta _{2}=(V_{2}/V_{1}){\rm {\;sin\;}}\theta _{1}$ [see also equation (3.1a)].

These are the laws of reflection and refraction. Fermat’s principle of least time (more accurately, of stationary time) states that wave travel between any two points is along the path for which the traveltime is either a maximum or a minimum value (i.e., the derivative of the traveltime equals zero) compared with the traveltimes along adjacent paths.

Figure 6.3a.  Deriving Snell’s law.

Solution

In Figure 6.3a, the source $S$ and the receiver $R$ have coordinates $(0,\;h_{1})$ and $(a,\;h_{2})$. The traveltime for a wave from $S$ to $R$ with reflecting point $M(x,\;0)$ is

${\begin{aligned}t=(1/V)\{(x^{2}+h_{1}^{2})^{1/2}+[(a-x)^{2}+h_{2}^{2}]^{1/2}\}.\end{aligned}}$

To find the point $M$ for which the value of $t$ is stationary, we differentiate $t$ with respect to $x$ and set the result equal to zero. Thus,

${\begin{aligned}{\frac {{\rm {d}}t}{{\rm {d}}x}}=(1/V)\left\{{\frac {x}{(x^{2}+h_{1}^{2})^{1/2}}}-{\frac {(a-x)}{[(a-x)^{2}+h_{2}^{2}]^{1/2}}}\right\}=0.\end{aligned}}$

The two terms in the brackets are the sines of the angles $\theta _{1}$ and $\theta _{1}^{'}$; hence, ${\rm {\;sin\;}}\theta _{1}={\rm {\;sin\;}}\theta _{2}^{'}.$

Problem 6.3b

Repeat part (a) for the refracted path SMQ, in Figure 5.3a.

Solution

The traveltime for the path SMQ is

${\begin{aligned}t=(x^{2}+h_{1}^{2})^{1/2}/V_{1}+[(b-x)^{2}+h_{3}^{2}]^{1/2}/V_{2}.\end{aligned}}$

Differentiation gives

${\begin{aligned}{\frac {{\rm {d}}t}{{\rm {d}}x}}={\frac {x}{V_{1}(x^{2}+h_{1}^{2})^{1/2}}}-{\frac {(b-x)}{V_{2}[(b-x)^{2}+h_{3}^{2}]^{1/2}}}=0,\end{aligned}}$

that is, ${\rm {\;sin\;}}\theta _{1}/V_{1}={\rm {\;sin\;}}\theta _{2}/V_{2}$.

Problem 6.3c

Repeat parts (a), (b) for reflected and refracted converted S-waves.

Solution

If we replace the angles $\theta$ with the angles $\delta$ and use the S-wave velocities $\beta$, the foregoing proofs are otherwise unchanged.

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Geometry of seismic waves	Characteristics of seismic events
Table of Contents (book)

Also in this chapter

• Characteristics of different types of events and noise
• Horizontal resolution
• Reflection and refraction laws and Fermat’s principle
• Effect of reflector curvature on a plane wave
• Diffraction traveltime curves
• Amplitude variation with offset for seafloor multiples
• Ghost amplitude and energy
• Directivity of a source plus its ghost
• Directivity of a harmonic source plus ghost
• Differential moveout between primary and multiple
• Suppressing multiples by NMO differences
• Distinguishing horizontal/vertical discontinuities
• Identification of events
• Traveltime curves for various events
• Reflections/diffractions from refractor terminations
• Refractions and refraction multiples
• Destructive and constructive interference for a wedge
• Dependence of resolvable limit on frequency
• Vertical resolution
• Causes of high-frequency losses
• Ricker wavelet relations
• Improvement of signal/noise ratio by stacking

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