Eisner's reciprocity paradox and its resolution

Jon Claerbout and Joe Dellinger

This paper appeared in pages 34 to 37 of the October, 1987, copy of ``The Leading Edge''.

(Click here to get quick access to all the figures and their captions.)

part I: the paradox

The principle of reciprocity says that when a vertical source (a vibrator) and a vertical receiver (a geophone) are interchanged, the same seismogram will be recorded in each case. In a field study, Fenati and Rocca [1984] found that the reciprocal principle also applies surprisingly well even when the required conditions are technically violated, such as when an isotropic source (dynamite) is used with a vertical receiver (again a geophone).

In a split-spread survey (such as in the top part of Figure 1 ) waves can be recorded along the same ray path in both a forward and reversed direction. Since both directions should give the same result, why pay the cost of recording twice? Recording costs money.

Perhaps the reason is the minor sampling advantage that can be gained by using a slightly different split-spread geometry such as that in the bottom part of Figure 1 . This shooting geometry has an advantage over an off-end geometry with the same number of geophones because for the off-end case CDP gather geometries vary with midpoint. This advantage of split spread over off end must be balanced against the additional cost of having twice as much wire out on the ground. Note that in the split-spread geometry shown here the reciprocity principle is implicitly invoked, because traces shot in opposite directions are interleaved in the gather.

``Eisner's paradox'' is a clever attack on the principle of reciprocity from a theoretical point of view, but the paradox can be resolved by elementary (nonmathematical) physics. Understanding this paradox might help you decide whether it makes sense to shoot split spreads.

EISNER'S COUNTER-EXAMPLE TO RECIPROCITY

Eisner's paradox was first presented at the 1980 SEG Friday workshop on the topic of ``Seismic Reciprocity''. Reciprocity is counter to intuition, and just when everyone in the audience was starting to believe in reciprocity, Elmer Eisner (like a magician) presented the following example purporting to show that reciprocity contradicts energy conservation! He presented the paradox in a simple, direct way, much as we shall here, with no mathematics. I (JFC) was there that day and heard much discussion of Elmer's talk but I do not recall anyone able to resolve his paradox, nor could I, so I urged Elmer to publish it. The paper appeared as a short note in Geophysics [1983] . Unlike the talk, the paper included a lot of mathematics --- just the kind of distraction a clever magician might conjure up to distract an audience of theoreticians. Eisner states that half of the Geophysicists and Mathematicians he has presented this paradox to consider it to be a valid counterexample.

Ellipse definition

To construct an ellipse you start with two points on a line. The points are the foci of the ellipse. The line is the axis of the ellipse. Place a tack at each point. Loop a string from one tack to the other, leaving the string loose. Tighten the string by pushing it away from the tacks with the tip of a pencil. Moving the pencil around the tacks to all ``tight-string'' locations draws the ellipse. A ray emitted from one focus reflects from this ellipse and arrives at the other focus having traveled one string length, so all rays arrive at the other focus simultaneously.

Eisner's ``counter-example''

Consider a perfectly reflecting ellipsoid with an impulsive source at one focus and a receiver at the other. Now, remove one end of the ellipsoid (as you might cut the end off a cucumber) by a cut through one focus perpendicular to the axis.

There are two experiments to consider: (A) source at the focus on the enclosed end, and (B) source at the focus on the cut-off end. These two experiments are shown together in Figure 2 , with experiment A on the top and B on the bottom. In each case the source (marked by an S) emits rays evenly in all directions. Each ray can either escape through the hole in the cut-off end, or can reflect from the ellipse and arrive at the receiver (marked by an R).

A. Source at the interior focus (top)

When the source is at the interior focus, almost all the emitted energy reflects from the ellipse and then passes through the cut-end focus. Only a bit of the energy, much less than half, escapes through the hole. Of the rays shown in the figure, for example, only 2 escape, while 34 do not and thus arrive at the receiver.

B. Source at the cut-end focus (bottom)

When the source is at the cut-end focus half of the energy radiates to infinite space, never to return. The other half of the energy passes through the interior focus after one reflection from the ellipsoid.

The paradox

The reciprocal principle says that we should observe the same seismogram if we interchange an isotropic source with an isotropic receiver. The complete seismogram in case A must equal the complete seismogram in case B. The reciprocity principle seems to be violated here, because the primary reflection energy in experiment A is almost twice that in B.

RESOLUTION OF EISNER'S PARADOX

Integrals on a small sphere

F. A. Dahlen and R. I. Odom in their reply [1984] to Eisner's original short note analytically calculate the amplitude at the focus for each case and show that they are the same. To work the problem their way, replace each focus by a small sphere and let the outgoing wave in each case have unit amplitude in all directions. This avoids infinities and allows one to use standard ray theory to calculate the amplitudes on the spheres.

We can now restate the problem. Reciprocity says that the integral of the received wave over the cut sphere in experiment A equals the integral of the received wave over the uncut sphere in experiment B. Observe that the integrand over the uncut sphere is nonzero over almost all angles, whereas the integrand over the cut sphere is nonzero over only half of all angles. This does not necessarily mean that one integral is twice that of the other, because the arriving energy varies with angle. In fact, we know that it is the cut sphere that receives more energy.

Amplitude versus energy

Reciprocity says the two integrals of wave amplitude are equal. Since each ray represents the same amount of energy, the density of rays (shown in Figure 3 ) gives the density of energy arriving as a function of angle.

Each ray carries a local plane wave and the square root of the energy of each plane wave is its amplitude. Graphing the energy and the amplitude versus angle and numerically integrating the areas shows that the amplitudes in the two cases are the same and the energy in one case is indeed almost twice that in the other.

You saw a paradox only if you confused energy integrals with amplitude integrals.

The sophomore paradox

Sharpen up Eisner's paradox again by letting the small spheres collapse to points. If the energies differ by about a factor of two, how can the amplitudes be the same? Have we really resolved the paradox?

Mathematically, it seems that ``A'' squared approximately equals half ``B'' squared, yet simultaneously ``A'' equals ``B''. This happens if A and B are both zero. Eisner suggested perhaps it can also be true if both A and B are infinite. Neither zero nor infinity seems a satisfactory answer. Dahlen and Odom suggested it is wrong to analyze the problem with geometrical optics and that we should use diffraction theory. But that drags in the limitations of diffraction theory too. Now you know everything in the published record. Has the paradox been resolved or has the mystery grown?

We claim the resolution of the paradox requires another ingredient, not yet mentioned. It is a simple fact that you probably learned in sophomore physics, but may have since forgotten. Please study the diagrams and contemplate the paradox a while before you continue on to the resolution .