## Equivalence of Tectonic Motions

No one puts new wine into old wineskins.

— Luke 5:37

In 1920, Albert Einstein suggested a famous thought experiment. A man drops an object from his pocket while he is falling from the rooftop of his house. To him the object does not appear to be falling; rather, it remains stationary at his side. Einstein concludes that gravity and acceleration are equivalent, and that “free-fall acceleration is a powerful argument for extending the postulate of relativity to non-uniform motions between coordinate systems.”

This formulation of the equivalence principle can be extended to other realms of physics. In electromagnetic induction, for example, there is equivalence between relative motion of the coil or of the magnet, yet “the theoretical interpretation of the phenomenon in each case is quite different,” as Einstein points out.

Consider the example of earthquakes. Two tectonic plates are moving against each other while the boundary remains locked. Two different interpretations are possible: 1) The stress σ(t) at the boundary is an increasing function of time t while the strength σc at the same boundary remains constant; or 2) the stress σ at the boundary remains constant while the strength σc(t) at the same boundary is a decreasing function of time. In either case the rupture will occur when σ ≥ σc.

Let us call the first case “elastic rebound” and the second “strength degradation.” The question I propose to address is: How can we distinguish between the two cases?

### THE GEODETIC ARGUMENT

Geodetic techniques for measuring displacements at the earth’s surface are developing rapidly. These techniques include the Global Positioning System (GPS) and Interferometric Synthetic Aperture Radar (InSAR) imaging. Geodetic evidence of systematic relative displacements between adjacent plates frequently is cited in support of elastic rebound.

However, the available measurements could equivalently be interpreted as resulting from strength degradation, depending on the assumptions used in converting geodetic displacement rate to stress. In the case of recent GPS observations at the San Andreas fault in California (Parsons 2006; Murray and Langbein 2006), the rock is assumed to behave as a linear viscoelastic Maxwell solid. In this case the deviatoric creep stress σd = σ1 – σ3 may be found from

where  · εc is the effective steady-state creep rate, A is the creep coefficient in MPans–1, n is the creep exponent (usually taken to be 1.9 to 3.5), Q is the activation energy in J/mol, R is the universal gas constant, and T is the temperature (Albert and Phillips 2002).

However, it is found that the creep contribution predicted from equation 1 is unrealistically low, at least for near-surface conditions. Thus,   · εcd 1.9 = 0.11 × 10–26 at T = 273°K, suggesting that tectonic stresses on the order of 10,000 MPa or 100,000 bar should be required to account for the GPS observations by creep alone. Actually, however, transient creep rates in earth materials are orders of magnitude higher than predicted from equation 1.

Two tectonic plates are moving against each other while the boundary remains locked. Two different interpretations are possible.

This is because the Maxwell solid assumed in equation 1 describes steady-state creep only. Transient creep rates are initially much higher than steady-state creep rates and may take more than a decade to converge to equation 1. Lord Kelvin argued as much when he proposed his modified linear viscoelastic model, now known as the Kelvin solid, instead of the Maxwell model (Kinsky 1986). The standard linear model for rocks (Lomnitz 1957) leads to a similar conclusion.

By assuming “the creep term to become vanishingly small” (Parsons 2006), GPS measurements may systematically overestimate the amount of elastic energy stored in rocks by plate motion. The experimental strain rates at the San Andreas fault are below 100 nanostrains per year. Such small strain rates could easily be absorbed by creep deformation at constant tectonic stress if the observed creep rates from GTSM strainmeters were used.

If plate boundaries have capacity to spare for storing platetectonic deformation as creep strain, the argument for elastic strain accumulation at plate boundaries becomes fallacious. In agreement with this conclusion, unexpected large rates of creep strain recovery are being observed on the San Andreas fault after the 2004 Parkfield earthquake (Langbein et al. 2006).

### WEATHER AND CLIMATE: AN APPARENT DIGRESSION

It is often stated that the model of elastic rebound has been confirmed by plate tectonics. This “confirmation” exists in a global kinematic framework; the process itself remains nebulous.

Science is a system for understanding the world. The more a scientific theory generates new predictions, the more it is valued, provided that the predictions can be quickly and easily confirmed or refuted. The proponents of elastic rebound (including Harry F. Reid and Andrew Lawson) expected positive results in earthquake prediction to provide a relevant test of the new model within a short period of time. This expectation was disappointed or repeatedly postponed. A century later, earthquake prediction is notoriously lagging in confirmation. There has been no concerted effort to discover a substitute null hypothesis against which elastic rebound could be tested.

The weather-climate distinction is a basic feature of modern geophysical predictions (Lorenz 1975). Climate is primarily a function of space (e.g., latitude) while weather is mainly a function of time (e.g., seasonal fluctuations). In other words, climate is the response of the system to changes in the boundary conditions when the initial conditions are kept constant, while weather is the time-dependent response of the system to changes in the initial conditions when the boundary conditions are kept constant (Hasselmann 2002).

Large-scale climate fluctuations such as El Niño, or largescale weather variations such as winter storms or thunderstorms, are not known to occur in tectonics as they are in meteorology. This may be due to different time scales. Fluctuations in the atmosphere are exogenous, primarily of solar origin, while those of earthquakes are endogenous. However, this distinction does not necessarily exclude climate and weather effects in plate tectonics.

Lightning is defined as a collective phenomenon, and it makes no sense to identify the largest thunderbolt (the “main bolt”) and the following ones (“afterbolts”).

For example, there are systematic differences in the rates of plate convergence or divergence at different boundaries that cause the seismicity (or seismic climate) of East Asia to be significantly more active than that along the Eastern Pacific boundary. Tectonic weather effects may be present as earthquake swarms, earthquake triggering, or fluctuations in seismicity caused by local depletion of seismic moment or temporary quiescence (“seismic gaps”).

Thus stress accumulation might plausibly be likened to an effect of “earthquake weather.” Thunderstorms are caused by a specific set of weather conditions. The atmosphere is unstable with respect to small perturbations of the initial conditions. This affects predictability: A natural limit of up to 20 days is recognized for purposes of weather prediction (Hasselmann 2002). But weather conditions include a wide range of fluctuations, and predictability is understood not to apply to the fine structure of the phenomenon. In other words, the prediction of a thunderstorm does not extend to predicting the time and location of individual thunderbolts. Lightning is defined as a collective phenomenon, and it makes no sense to identify the largest thunderbolt (the “main bolt”) and the following ones (“afterbolts”). In other words, the fine structure of the process is assumed to be random. Clearly, the analogy with tectonics raises a whole set of new questions.

The model of elastic rebound (Reid 1911) is based on five propositions:

1. Fracture is caused by elastic strains exceeding the strength of the rock.
2. These strains accumulate over a long period of time.
3. Coseismic motion represents the elastic rebound or release of the stored elastic strain.
4. Seismic waves originate at the fracture of the rock.
5. The energy of the earthquake equals the elastic strain energy previously stored in the rock.

These propositions are currently “accepted with only minor modifications” (National Research Council 2003), which is remarkable if one considers the age of the model and the fact that nothing is stated about the origin of strain accumulation. In the atmosphere or in the oceans, no analogous model has been proposed.

What do we know about the dynamics of tectonic processes? At present it is recognized that “earth system processes perform work by degrading sources of free energy, thereby producing entropy” (Kleidon and Lorenz 2005). For example, the work required to drive atmospheric circulation is performed by degrading the temperature difference between the equator and the poles. There is no accumulation of temperature differences anywhere. In elastic rebound, on the other hand, strain energy must accumulate in a bounded region of space. This idea runs counter to cosmic evolution (Chaisson 2005). Converting entropy into free energy is against the Second Law of Thermodynamics. How can long-term local and regional accumulation of energy arise in an open steady-state system? Convection often is invoked at this point. But no energy accumulates anywhere in a convecting system. Convecting systems may be observed in nature, e.g., in lava lakes that are generated in the craters of volcanoes. Lava plates form at the surface through cooling and are recirculated or shored up against the edges of the crater. But there is no local energy accumulation.

Another puzzling feature of the model is the presence of pre-existing faults. Earthquakes never occur in the undamaged host rock. Fault zones are lithospheric features where the rock is damaged as a result of environmental corrosion, usually because of the slow chemical action of water. Faults have a much lower strength than the surrounding rock; therefore fracture occurs at a much lower stress, and the amount of heat produced by friction is much smaller than expected. This fact has been perceived as a paradox (Rice and Cocco 2007; National Research Council 2003, 53–54).

The phenomenon of environmental stress fracture (Fontana 1986) is caused by decaying shear strength at weaknesses in the material under conditions of constant stress. It has been widely studied in metals and other engineering materials. Often there is no warning signal of an impending rupture, because there is no stress accumulation. But environmental stress fracture was unknown in 1911.

A steady-state system is defined as a system in which all state variables have stationary means. Steady state is not the same as equilibrium; for example, there must be a flow through the system. In steady-state convection there may be stress perturbations (an earthquake is such a perturbation), causing the system to generate transient response flows that oppose and eventually cancel the perturbation. The classic example of a steady-state system is a bathtub filling at a constant rate. When the plug is pulled the water will reach a stationary level defined by the inflow equaling the outflow.

In general, complex systems far from equilibrium, which lack fixed boundary conditions, tend to drift toward a steady state in which the entropy production is maximized (Kleidon and Lorenz 2005). In a convection cell the external constraints include the temperature gradient across the cell, but there is an energy balance in the system. The perturbations are short-lived and do not accumulate.

### EVIDENCE FOR ELASTIC REBOUND VERSUS STRENGTH DEGRADATION

A large-scale prediction experiment (1985–2004) was approved and conducted on the San Andreas fault to test prevailing ideas on earthquake processes. Using as input the time series of past earthquakes at Parkfield, California, an event of magnitude 6 was predicted to occur around 1988 plus or minus five years. Arrays of sensitive instruments were placed around the prospective epicenter to detect the expected precursory phenomena caused by strain accumulation.

The predicted earthquake did finally occur in 2004, but it was the wrong earthquake! It was the right magnitude and it occurred near the predicted location, yet it differed in significant details from the earthquake that had been expected. There was no evidence of any long-term or short-term precursory signals. Borehole strain meters recorded no strain accumulation. It looked as if the earth had not known an earthquake was imminent.

Thus it was concluded that the expected “confirmation” by plate tectonics had failed. Many scientists felt that this was negative evidence, which proved nothing. Some proposed that the results be shelved or used to design a better experiment elsewhere. Yet the Parkfield experiment remains the only fullscale test of elastic rebound. It had been carefully designed, and advanced equipment and instrumentation had been used.

Alternate models, such as the model of strength degradation, were not tested. However, the field data of the Parkfield experiment fit strength degradation better than elastic rebound. In terms of internal consistency, the striking absence of precursory signals agreed with the lack of success in earthquake prediction. Observations from borehole strain meters also suggest that the effect of transient creep had been seriously underestimated.

The Parkfield experiment was located at the boundary between two such inertial physiframes of reference. Does this mean that we cannot determine which of these two mechanisms applied in the case of the 2004 earthquake?

### EQUIVALENCE ≠ UNCERTAINTY

The question we set out to answer, about the equivalence between elastic rebound and strength degradation, might now be settled in the affirmative at this point. But are they equivalent in the Einsteinian sense?

Einstein stated that two processes are equivalent when the outcome of a physical experiment in a laboratory that moves in an inertial frame of reference is the same independent of the location and the velocity of the laboratory in space-time. The Parkfield experiment was located at the boundary between two such inertial frames of reference. Does this mean that we cannot determine which of these two mechanisms applied in the case of the 2004 earthquake? Not at all.

Equivalence is not the same as uncertainty. The equivalence principle does not imply that an observer in free fall cannot detect or measure the gravitational field. To use the examples proposed by Einstein himself, equivalence will not prevent an experimenter from deciding whether to move the coil or the magnet in an induction experiment. The outcome may be the same in either case, but extraordinary precautions must be taken if the observer is to remain ignorant of what he is observing. As for the man who falls from the roof, someone might have warned him—before jumping—that he was in a gravity field.

The Parkfield experiment demonstrates unequivocally that there is no detectable strain buildup at plate boundaries before an earthquake. Yet the evidence is not accepted as conclusive, because of the widely accepted interpretation of geodetic observations as “proving” that tectonic stresses are accumulating at the San Andreas fault. The question is: did the Parkfield experiment achieve what it set out to test? If so, a discussion on alternative models of earthquake causation is mandatory. If not, where and how should the hypothesis of elastic rebound be tested?

In the meantime, elastic rebound should be put on hold, as being due for a fundamental revision.

### REFERENCES

Albert, R. A., and R. J. Phillips (2002). Time-dependent effects in elastoviscoplastic models of loaded lithosphere. Geophysical Journal International 151, 612–621.

Chaisson, E. J. (2005). Non-equilibrium thermodynamics in an energy-rich universe. In Non-equilibrium Thermodynamics and the Production of Entropy: Life, Earth, and Beyond, ed. A. Kleidon and R. D. Lorenz, 21–31, Berlin and New York: Springer.

Einstein, A. (1920). Fundamental ideas and methods of relativity (unpublished). Quoted in G. Holton, Einstein’s third paradise, Daedalus (Fall 2002), 26–34, http://www.aip.org/history/einstein/essay-einsteins-third-paradise.htm.

Fontana, M. G. (1986). Corrosion Engineering. New York: McGraw-Hill.

Hasselmann, K. (2002). Is climate predictable? In The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes, eds. A. Bunde, J. Kropp, and H.-J. Schellnhuber, chapter 4, 141–169. Berlin and New York: Springer.

Kinsky, R. (1986). Engineering Mechanics and Strength of Materials. Sydney, Australia: McGraw-Hill.

Kleidon, A., and R. Lorenz (2005). Entropy production by earth system processes. In Non-equilibrium Thermodynamics and the Production of Entropy: Life, Earth, and Beyond, eds. A. Kleidon and R. D. Lorenz, 1–20, Berlin and New York: Springer.

Langbein, J., J. R. Murray, and H. A. Snyder (2006). Coseismic and initial postseismic deformation from the 2004 Parkfield, California, earthquake, observed by Global Positioning System, electronic distance meter, creepmeters, and borehole strainmeters. Bulletin of the Seismological Society of America 96 S304–S320.

Lomnitz, C. (1957). Linear dissipation in solids. Journal of Applied Physics 28, 201–205.

Lomnitz, C. (2007). The Parkfield prediction experiment: Scientific alternatives to elastic rebound (submitted to Bulletin of the Seismological Society of America).

Lorenz, E. N. (1975). Climate predictability: The physical basis of climate and climate modeling. Global Atmosphere Prediction Programme, WMO, Report 16, Geneva.

Murray, J., and J. Langbein (2006). Slip on the San Andreas fault at Parkfield, California, over two earthquake cycles, and the implications for seismic hazards. Bulletin of the Seismological Society of America 96, S283–S303.

National Research Council (2003). Living on an Active Earth: Perspectives on Earthquake Science. Washington, DC: National Academies Press.

Parsons, T. (2006). Tectonic stressing in California modeled from GPS observations. Journal of Geophysical Research 111, B03407, doi:10.1029/2005JB003946.

Reid, H. F. (1911). The elastic-rebound theory of earthquakes. University of California Department of Geology Bulletin 6(19), 413–444.

Rice, J. R., and M. Cocco (2007). Seismic fault rheology and earthquake dynamics. In Tectonic Faults: Agents of Change on a Dynamic Earth [report of the 95th Dahlem workshop on the dynamics of fault zones, Berlin, January 16–21, 2005], ed. M. R. Handy, G. Hirth, and N. Hovius, chapter 5. Cambridge, MA: MIT Press.

Cinna Lomnitz
Email- `cinna [at] prodigy.net.mx`

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Posted: 10 May 2007