huygens snell fermat seismic reflection

Huygens, Snell, And Fermat

With respect to visualizing the seismic method in physical terms, one of the most helpful concepts is that of Huygens' principle.

We consider the position of a wavefront at one instant of time; the wavefront may be plane (as

in Figure 1 (a)), spherical ( Figure 1 (b)), or arbitrary in form.

Huygens tells us to visualize each point on the wavefront as the origin of a new spherical wavelet, propagating at the local velocity; the position of the wave

front at the next instant of time is the envelope of these spherical wavelets.

The concept is easy to accept. Figure 2 ( A seaside illustration of the excitation of a secondary source, in this case circular waves.

Figure 1

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(a) The incident plane wavefront. (b) The resulting circular waves ) makes us comfortable with the idea that any point on a wave front can be regarded as the

source of a new secondary wavelet. And the concepts of linearity and superposition make us comfortable with the idea of obtaining a resultant wave

by adding together many constituent wavelets.

Hugyens' principle is also immensely useful. First, it allows us to take processes usually considered in terms of rays, and to construct their counterparts in terms of wave fronts.

Second, it allows us to visualize processes such as diffraction, by which we hear around corners. Let us set up some examples in each category.

In the first category—rays and wave fronts—we start with the process of reflection. Figure 3

(a) illustrates the basic process in terms of rays, and reminds us of the familiar law that, for "specular" reflection, the angle of incidence is equal to the angle of

reflection.

Figure 2



Figure 3 (b) illustrates the same process in terms of Huygens' principle; clearly, it leads to the same law.

Next, there is the process of refraction. Figure 4 illustrates the basic process in terms

of rays, and reminds us of the familiar law (Snell's law) that the ratio of the sines of the angles is the ratio of the velocities.

Figure 3



Figure 5 illustrates the same process in terms of Huygens' principle; again it is seen to lead to the same law.

Figure 4

Figure 5



In the illustration the lower material has the higher velocity, in which case the refracted ray is bent toward the interface; where the lower material has the lower velocity, the bend is away from the interface.

Next, there is Fermat's principle, which tells us that the path represented by a ray between any

two points (such as A and B in Figure 4 ) is the fastest path between those points. In Figure 5 we can see that the operation of drawing the Huygens wavelets does

guarantee this "least-time" path.

Then there is the generation of head waves. Figure 6 (a)

repeats Figure 4 for the limiting case when the angle of incidence is the critical

angle; then, the refracted ray in the higher-velocity material grazes the interface, and at first sight there is no mechanism for getting it back into the

upper material. As the wave passes any point such as A, however, the particles must be locally compressed; this compression is relieved in part by an upward

bulge of the interface, and the upward bulge becomes, in effect, a secondary source. As shown in Figure 6 (b), this succession of Huygens sources along the

refractor, each generating its spherical Huygens wavelet in the upper material, yields an observable wave—the head wave—in the upper material. This forms the basis of the refraction method of seismic exploration, illustrated in Figure 6

(c). Huygens' construction shows that the geometry of the upcoming ray in Figure 6 (b) is similar to that of the downgoing ray in Figure 5 ; Snell's law

applies to both.

Figure 6



Then there is the principle of reciprocity, of which the above duality is an example. Our

theoretical friends take great care with the exact statement of this principle, in order to minimize the number of situations to which it does not apply. For ourselves, however, and for present purposes, the principle just says that if a certain source at a point P produces a certain wave at a point Q—however complex the path between them—then the same source at point Q would produce the same wave at point P. At least as regards the travel paths between the two

points, we can see that this conclusion follows from the repeated application of Huygens'

principle; thus, in Figure 6 (c), using the constructions of Figures 5 and Figure 6

(b), it is clear that we would obtain the same result (for each source-receiver pair) by interchanging the source and the receiver. Our standard field

techniques assume this type of reciprocity as a matter of course; Figure 7 ( An illustration of reciprocity; (b) is a normal record of one shot into many

geophones, while (a) is the record fabricated from many shots into one geophone group, over the same subsurface. ) is a case in point.

Now let us turn from situations describable in terms of rays to those, such as diffraction, for which rays are inappropriate.

First we might illustrate the difference using the seismic equivalent of a shadow. Figure 8 (a) repeats a simple Huygens construction: the advancing wave may be represented as the combined effect of many spherical wavelets, or by the ray.

Figure 7



The ray implies that the wave at point B is totally defined by the wave at point A. Huygens, however, says that the wave at point B incorporates contributions

from many secondary sources along the wavefront through point A. The difference is highlighted if we move to Figure 8 (b), in which an obstruction is

placed in the path of the wave. The ray is unchanged, but the Huygens construction now has only half the number of secondary sources. Huygens thus

prepares us for the fact of diffraction: the shadow is not sharp. The wave does propagate (although with progressively reduced strength) into the shadow zone behind the obstruction. Just as Huygens would predict, the propagation can be

viewed as that of spherical wavelets centered on the edge of the obstruction. The energy carried by the wave in the shadow zone must come from

somewhere, and the only possibility is that this energy is subtracted from the wave in the open zone; again, just as Huygens would predict, the wave at point

B is one-half of what it would have been without the obstruction. The resulting wave is illustrated schematically in Figure 8 (c).

Next, we can use Huygens to help us understand that the propagation of a seismic wave

involves a large volume of rock. This is simply illustrated by Figure 9 , which represents a

wave (of any sort, plane or spherical) passing across point C and later across point D; somewhere off to the side is a zone of anomalous propagation—

perhaps a zone of different velocity.

Figure 8



Then, Huygens' principle tells us that the wave received at D must be affected by the presence of the anomaly. It also prepares us to accept that the degree of

the effect depends in some way on the relative magnitudes of the separation CD and the offset IJ; the effect must be small if CD is small or if IJ is large.

Developing this, we are led to the concept of the volume of rock contributing meaningfully to the propagation between a source 0 and a receiver G

( Figure 10 .(a)).

Figure 9



Described thus, the "edges" of this volume may appear very indistinct, depending as they do on what we regard as a "meaningful" contribution.

However, we can set some bounds to it if we consider sinusoidal waves, for then it is clear that a path length exceeding the direct path by more than half a

wavelength must actually detract from the composite amplitude of the wave. We are then led to the definition sketched in Figure 10 (b); the edges of the

volume represent path lengths (such as OAG, OBG) that exceed the direct path OG by half a wavelength. We must not think of this boundary as sharp—indeed, the message of Huygens is that in seismics all boundaries are fuzzy. Further,

intuition suggests that the inner part of this volume makes a greater contribution than the outer parts, and we shall find later that this is so. For the

moment, let us carry forward the broad concept that any seismic path between source and receiver involves a volume of rock, broadly ellipsoidal in shape,

whose "breadth" is large at low frequencies and small at high frequencies ( Figure 10 (c)).

We may draw four useful corollaries from this. First, we see that if the physical properties of the earth change within this volume, the received wave represents some sort of center-weighted average of these properties. Second, we see that there is a continuum between the ray view and the wave view; at very high frequencies the breadth of the ellipsoid becomes small, and the propagation is closer to that suggested by a ray. Third, we see that a propagating wave

includes some contributions that do not satisfy Fermat' s principle of least time; strictly,

the principle applies only over path lengths long relative to a wavelength. Fourth, we are braced for the important conclusion that, seismically speaking,

we cannot expect to "see" very small inhomogeneities in the earth; specifically, reflectors of small extent can return only weak reflections.

Figure 10



As our final illustration of the usefulness of Huygens' principle, let us consider again the

difficulty of generating plane waves. Obviously we would prefer to generate plane waves, in that we would avoid the major loss of amplitude associated with spherical spreading, and the major loss of the low frequencies associated with a small source. But Huygens is telling us, clearly, that this cannot be done. True, we could generate a plane wave over perhaps a hundred square meters; that would be easy enough. But after 2 ms of propagation the wave would be

rounded at the edges ( Figure 11 (a)),

and after 2 s of propagation its shape would be indistinguishable from that of a spherical wave ( Figure 11 (b)).

That is not to say that the amplitude of the wave would be the same in all directions. By increasing the extent of the source to a hundred square meters we do not substantially change the sphericity of the wave, but we do increase its amplitude in the downward direction relative

to the sideways direction. This feature is quite distinct from the shape of the wavefront; it requires that the source dimensions be on the order of a wavelength, and arises because of constructive interference in the downward direction and destructive interference in the sideways direction. Thus, there is no contradiction in having a directional source of substantially spherical waves.

We should note that this directivity has nothing to do with the direction of particle motion. Thus, we may drop a flat weight on the ground, and be confident that the initial particle motion

is vertical; Huygens tells us, nevertheless, that the generated wave is effectively spherical, with the particle motion vertical in the vertical direction, but horizontal in the horizontal direction. When we hear the term "a directional source," we must ask in what sense the term is meant; it may mean a source generating a wave in which only some directions of particle motion are represented, or it may mean a source in which all directions of particle motion are represented

but with systematically varying amplitude. Thus, many or most seismic sources used in practice are omnidirectional in one sense, but directional in the other.

Figure 11



Huygens' principle is of enormous value to us; it correctly predicts many features of wave

propagation which are essential to seismic exploration. What is more, it is delightfully simple. When something is at once useful, physically satisfying, and simple, there is a temptation to look on it as a law.

But Huygens' principle is not a physical law. Indeed, it has some major problems.

The first of these is merely a limitation: strictly, Huygens' principle does not work in two dimensions. It requires spherical wavelets, and spherical wavelets exist only in three dimensions.

The other problems are concerned with the wavelets in directions other than the forward direction. In the forward direction, we have agreed, the tangent to the spherical wavelets correctly defines the advancing wavefront, but what of the wavelets in the sideways directions,

and the backward direction? We sometimes read that in these directions the wavelets "cancel,"

but we can see from Figure 12 that this is not directly so; at point Q the

components of motion in the tangential direction do indeed cancel, but the components in the direction of propagation clearly reinforce.

The effect of this would be to add an integrating tail to the propagating wave-

front. Further, it is not immediately obvious how the backward-traveling part of the wavelets would cancel.

A mathematical solution to Huygens' difficulties was found by Kirchhoff. In essence, it requires

that the process of generating the spherical wavelets involves two additional operations not included in Huygens' original concept. For one, the process of generating each spherical wavelet involves a differentiation—a step becomes a spike. For the other, each spherical wavelet—while remaining truly spherical—has directivity; the amplitude is multiplied by a cosine factor that leaves it unchanged in the forward direction but brings it to zero in the backward direction.

Figure 12



While these modifications may make possible the mathematical application of Huygens'

principle, they do nothing but cloud its elemental physical appeal. We cannot see, physically, what it is that imparts directivity to the wavelets. And even though consideration of spherical waves satisfies us physically that any real source of small dimensions involves a differentiation, we cannot see the same mechanism at work in the notional sources of Huygens.

So what do we do? First, we remain totally comfortable with Huygens as a means of telling us whether a wave will be present, what the shape of the wavefront will be, and what the time of arrival will be. Second, we have the mathematical formulation—with its differentiation and

directivity—where we need it. But third, we do not regard Huygens as a means of describing what is actually happening as a wave propagates.

Thus, as regards the shape of the wavefront in space (but not as regards the details of the waveform in time) we are happy to accept that wave propagation is as though it took place according to Huygens. We are not saying it does take place this way, merely that the result is as though it took place this way.

If this sounds unscientific, we can take comfort in the thought that other scientists do it too. The biologists who talk of the selfish gene are not suggesting that the gene is really selfish—

merely that it behaves as though it were selfish.

