Inferences about the Deep Interior of the Earth – Rom, 16.11.1990


James Freeman Gilbert

Balzan Preis 1990 für Geophysik (feste Erde)

Für seinen herausragenden Beitrag zu unserer Kenntnis der tiefsten Schichten des Erdinneren.

1.            INTRODUCTION

A flagella terraemotu, libera nos Domine. The plea, from the Litany of All Saints, is very old, and undoubtedly existed long before the advent of Christianity. It reminds us that our ancestors were very fearful of the loss of life and property that can result from an earthquake. We share that fear. Very likely, the ’scourge of the earthquake‘ was one of the first natural hazards that man attempted to ameliorate, an attempt that continues to this day.

The quantitative study of earthquakes is a relatively recent development, and the use of specialized instruments, seismographs, seismometers, etc., dates from the final score of the last century. Considerable progress has been made in understanding the occurrence of earthquakes, but the prediction of earthquakes remains an elusive goal, although there is some promise of the improved forecasting of earthquakes. Recordings of an earthquake are useful not only for the study of the earthquake itself, but also for the study of the medium through which the shocks travel. That medium is the earth. Almost all of our knowledge about the interior of the earth has been inferred from seismological data, data that are becoming more precise and more numerous as we perfect our methods of recording earthquakes. The process of inference continues today, and, as we learn more about the structure of the earth, we anticipate that we shall make progress in understanding the mechanism of earthquakes, thereby providing a social benefit from intellectual effort.


In the past, when the sun never set on the British Empire, it was a relatively simple matter to establish a global, seismographic network, if you were British. John Milne, who was British, did establish the first global network in 1896, and, in a short time, the first tables of travel times of earthquake waves were prepared. Subsequent early discoveries were spectacular.

The earth has a fluid core, there are very deep earthquakes, earthquakes are confined to narrow zones, there is an inner core that is probably solid; these and other discoveries helped to set the stage for the development of the theory of plate tectonics in the 1960s, a development that some historians of science compare with relativity, quantum mechanics and the double helix of DNA. The role of seismology in this development was vital.

In the late 1950s there were developed improved theories of wave motion, in the earth, improved numerical methods for calculating the motion and digital computers that were capable of performing the calculations. Theory was ahead of observation.


The recently developed theoretical and numericall methods were soon tested by the largest earthquake in this century. On May 22, 1960, the nation of Chile experienced an earthquake of great magnitude that was recorded worldwide, and that revealed for the first time that the earth vibrates in its natural modes of oscillation. The oscillations have very grave periods, from hundreds of seconds to nearly one hour. A comparison between the observed periods and those calculated for existing mechanical models of the earth showed fair agreement with room for improvement. In addition, the data showed that the periods of vibration were sensitive to the rotation of the earth and to the equatorial bulge caused by the rotation. It was quickly understood that the new, low frequency data could greatly improve our knowledge of the interior structure of the earth, including the three-dimensional structure.

In the thirty years since the Chilean earthquake there have been several advances in our ability to use data from low frequency seismology to study the earth and earthquakes. At the same time, the improvement in global networks has made it possible to use higher frequency, travel time data to great advantage. In the past few years, it has been possible to combine vibrational data and travel time data to make more accurate inferences about the properties of the earth. It has been my privilege to have participated in this very stimulating experience of exploring of the earth by ‚remote control‘, and to have collaborated with several colleagues in the process.

4.            INVERSE PROBLEMS

How do you know what you know? The question is imprecise and reminds one of the story about the committee of blind men who were asked to describe the form and function of an elephant. Perhaps it was a camel or giraffe, but the point is that we make inferences about the world outside ourselves with inaccurate data. It is a daily activity. The data may also be inadequate, insufficient, biased and otherwise misleading. With poor data we make poor inferences. When are the data poor? Sometimes it is difficult to say. History is full of examples where the wrong decision was made because of faulty data (or faulty thinking). Science is no different than politics in that regard. There are many examples in science of wrong inferences made from faulty data (or faulty thinking).

As an example of the kind of inference that a geophysicist must face, let us consider the problem of a vibrating string, say a violin string. The string may vary in cross section or thickness; its properties may change along its length. It is clamped at both ends and is under tension. We cause the string to vibrate and measure the periods of its vibrations, the fundamental period and the overtone harmonics. Suppose that we may measure as many periods as we like. Can we determine the properties of the string from its periods? We know that if we know the properties of the string its periods can be calculated very accurately. Can the inverse calculation be done? No. There are many different strings, theoretically an infinite number, all having the same set of periods of vibration. However, if we can measure the amplitudes of vibration, depending on where the string is plucked or excited, then we can solve the inverse problem.

In practice, we must be satisfied with certain local averages of the properties of the string. With increasing data the averages can be made more localized. Thus, it is not only the number of data that is important, but also the kind. With a large number of the right kind of data the geophysicist can make useful inferences about the interior of the earth.

The theory of geophysical inverse problems began to be developed in the 1960s in parallel with the development of low frequency seismology and was partially stimulated by that subject. Subsequently, it has found application in other branches of seismology and geophysics as well as other branches of science.


If we have data about travel times and periods of oscillation, these data are not dependent on the size of an earthquake. They are properties of the earth alone. However, if we also need data about amplitudes of oscillations and pulses, then these data depend both on the properties of the earth and the earthquake. Since our models of the earth appear to be fairly good, we can use them to calculate amplitudes for a particular earthquake of a given size, location and orientation. Theoretically, the orientation is known as a tensor quantity, depending on six different numbers, and has the dimensions of force x length, or moment. We say that an earthquake has a certain moment tensor, size and location. Using our models we can calculate the response for a hypothetical moment tensor and location. We use all of the amplitude data to obtain an estimate of the size and of the changes in the moment tensor needed to give a better fit to the data. Finally, we use the remaining information in the amplitude data, and the data about travel times and frequencies, to improve our models of the interior of the earth.

The use of moment tensors to describe the orientation of the source of an earthquake has become standard practice in the past 15 years and is usually included in most earthquake bulletins and catalogs.


The earth is nearly spherically symmetric and its interior properties are dominantly functions only of radius. In qualitative terms the radial structure of modem models of the earth is very similar to the models that were being used 50 years ago. The differences are not great but they are important. The basic features of a solid mantle and a fluid core are preserved. The boundary between the mantle and the core has a radius of 3484 km compared with 6371 km for the outer surface of the earth. The radial distribution of mass, the ‚density‘ of the earth, is now better constrained, but problems of resolution remain. The inner core, discovered in the 1930s, is almost certainly solid, but, partly because of the very high pressures near the center of the earth, it transmits waves of shear much more slowly than waves of compression. The radius of the inner core is not as well determined as that of the outer, fluid core. It is approximately 1223 km.

The earth is not perfectly elastic. Amplitudes of oscillation slowly decay as mechanical motion is turned into heat. The dissipation of elastic energy causes the earth to be more stiff at high frequencies than at low frequencies. The effect is small, a few per mil, but easily measurable.

Problems remain. A subset of the data is sensitive to the very deep structure, the outer, fluid core and the inner, solid core. These data are poorly fit by existing models. The explanation for the poor fit has not been found and is a challenge. Mathematically, our present models may be ‚too far‘ from the truth for our present computational procedures to work effectively. Alternatively, a good model may exist on another ‚manifold‘ from our present models. If the manifold is not known a good model cannot be found. Our ignorance is profound.


The determination of the aspherical structure of the earth requires one or two orders of magnitude more data than the determination of the radial structure. Since the volume of data is not yet available, and since the computational facilities for processing the large volume of data have yet to be developed, it is understandable that progress is modest.

Because of the great pressures in the interior of the earth and the small finite strength of the material that composes the mantle, the earth behaves like a fluid over long time scales. For example, the equatorial bulge is very nearly equal to that calculated on the assumption that the earth behaves like a fluid. It is a good approximation that the interior of the earth is in hydrostatic equilibrium. Deviations from this equilibrium must be very small in the mantle and effectively zero in the fluid, outer core. Such deviations in the mantle may provide insight on steady flows related to convection or on compositional changes.

The periods of oscillation of the earth are organized into groups called multiplets. Each multiplet is analogous to a musical chord that embraces a narrow band of notes. The dominant period of the chord changes with the emphasis given by the player to each note. Similarly, the dominant period in a multiplet of the oscillation of the earth changes as the orientation on the earth changes between earthquake source and seismographic receiver. Remarkably enough, the primary effect on the dominant period is the orientation of the great circle between source and receiver. The dominant period of a given multiplet is nearly the same for all recordings that belong to the same great circular path. The dominant period varies from one great circle to another.

An analysis of thousands of recordings and many tens of multiplets of free oscillation reveals a very simple pattern. First, there is the effect of the equatorial bulge. When this effect, which can be calculated, is removed, there remains a simple pattern that is a strong function of longitude and a less strong function of latitude. The longitudinal pattern has two diametrically opposed positive sectors with negative sectors in between. The cause of this very simple pattern is an aspherical structure in shear strength (about 1,7% in amplitude) located about 500 km deep and spread over a few hundreds of km (say 200 km) in radius. Many more data are required for a more quantitative statement to be made.

Subsequent analysis shows that the boundary between the mantle and core also has a large scale, small amplitude structure, and that the structure in the upper mantle may have more features than the original longitudinal sectors. All of these aspherical structures, preliminary as they are, have recently been used to support the argument for a dynamic, convecting earth. As the time nears for the simultaneous interpretation of the recently acquired oscillation and travel time data from the digital, global seismographic networks, geophysicists are eager to learn what new features of the aspheric structure will be revealed.

Finally, we must honestly state that some of our data defy explanation. About 30 multiplets embrace a narrow band of frequencies that is double that predicted by existing models. These data are quite sensitive to the structure of the fluid, outer core and the solid, inner core. Some of the data are sensitive only to the fluid, outer core and lead us to believe that the cause of the ‚double embrace‘ lies in that region. Such effects as the magnetic field, steady flow and differential rotation can be ruled out, leaving us with few options. To explain these data is another challenge, but challenges of this kind make the life of a scientist all the more interesting.

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