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5 Earthquake Physics and Fault-System Science Earthquake research focuses on two primary problems. Basic earthquake science seeks to understand how earthquake complexity arises from the brittle response of the lithosphere to forces generated within the Earth’s interior. Applied earthquake science seeks to predict seismic hazards by forecasting earthquakes and their site-specific effects. Research on the first problem began with attempts to place earthquake occurrence in a global framework, and it contributed to the discovery of plate tectonics; research on the second was driven by the needs of earthquake engineering, and it led to the development of seismic hazard analysis. The historical separation between these two problems, reviewed in Chapter 2, has been narrowed by an increasing emphasis on dynamical explanations of earthquake phenomena. In this context, the term dynamics implies a consideration of the forces (stresses) within the Earth that act to cause fault ruptures and ground displacements during earthquakes. The stress fields responsible for deep-seated earthquake sources cannot be measured directly, but they can be inferred from models of earthquake systems that obey the laws of physics and conform to the relationships between stress and deformation (rheology) observed in the laboratory. This chapter describes how this physics-based approach has transformed the field into an interdisciplinary, system-level science—one in which dynamical system models become the means to explain and integrate the discipline-based observations discussed in Chapter 4. The chapter begins with an essay on the central problems of dynamics and prediction, which is followed by five sections on areas of intense interdisciplinary research: fault systems, fault-zone processes, rupture dynam-
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ics, wave propagation, and seismic hazard analysis. Each of the latter summarizes the current understanding and articulates major goals and key questions for future research. 5.1 EARTHQUAKE DYNAMICS For present purposes, the term “dynamical system” can be understood to mean any set of coupled objects that obeys Newton’s laws of motion—rocks or tectonic plates, for example (1). If one can specify the positions and velocities of each of these objects at any given time and also know exactly what forces act on them, then the state of the system can be determined at a future time, at least in principle. With the advent of large computers, the numerical simulation of system behavior has become an effective method for predicting the behavior of many natural systems, especially in the Earth’s fluid envelopes (e.g., weather, ocean currents, and long-term climate change) (2). However, many difficulties face the application of dynamical systems theory to the analysis of earthquake behavior in the solid Earth. Forces must be represented as tensor-valued stresses (3), and the response of rocks to imposed stresses can be highly nonlinear. The dynamics of the continental lithosphere involves not only the sudden fault slips that cause earthquakes, but also the folding of sedimentary layers near the surface and the ductile motions of the hotter rocks in the lower crust and upper mantle. Moreover, because earthquake source regions are inaccessible and opaque, the state of the lithosphere at seismogenic depths simply cannot be observed by any direct means, despite the conceptual and technological breakthroughs described in Chapter 4. From a geologic perspective, it is entirely plausible that earthquake behavior should be contingent on a myriad of mechanical details, most unobservable, that might arise in different tectonic environments. Yet earthquakes around the world share the common scaling relations, such as those noted by Gutenberg and Richter (Equation 2.5) and Omori (Equation 2.8). The intriguing similarities among the diverse regimes of active faulting make earthquake science an interesting testing ground for concepts emerging from the physics of complex dynamical systems. One consequence of recent interactions between these fields is that theoretical physicists have adopted a family of idealized models of earthquake faults as one of their favorite paradigms for a broad class of nonequilibrium phenomena (4). At the same time, earthquake scientists have become aware that earthquake faults may be intrinsically chaotic, geometrically fractal, and perhaps even self-organizing in some sense. As a result, an entirely new subdiscipline has emerged that is focused around the development and analysis of large-scale numerical simulations of deformation
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dynamics. Combined with insightful physical reasoning and intriguing new laboratory and field data, these investigations promise a better understanding of seismic complexity and predictability. Complexity and the Search for Universality Earthquakes are clearly complex in both the commonsense and the technical meanings of the word. At the largest scales, complexity is manifested by features such as the aperiodic intervals between ruptures, the power-law distribution of event frequency across a wide range of magnitudes, the variable patterns of slip for earthquakes occurring at different times on a single fault, and the richness of aftershock sequences. Individual events are also complex in the disordered propagation of their rupture fronts and the heterogeneous distributions of residual stress that they leave in their wake. At the smallest scales, earthquake initiation appears to be complex, with a slowly evolving nucleation zone preceding a rapid dynamic breakout that sometimes cascades into a big rupture. Among the many open issues in this field are the questions of whether these different kinds of complexity might be related to one another and, if so, how. The most ambitious and optimistic reason for considering the ideas of dynamical systems theory is the hope that one might discover universal features of earthquake-like phenomena. Such features would, of course, be extremely interesting from a fundamental scientific point of view. They might also have great practical value, for example, as a basis for interpreting seismic records or for making long-term hazard assessments. Two thought-provoking, complementary concepts that look as if they might bring some element of universality to earthquake science are fractality and self-organized criticality. The first describes the geometry of fault systems; the second is an intrinsically dynamic hypothesis that pertains to the complex motions of these systems. Although each has provoked its own point of view among earthquake scientists—that seismic complexity is, on the one hand, primarily geometric in origin or, on the other hand, primarily dynamic—it seems likely that both concepts contain some elements of the truth and that neither is a complete description of the behavior of the Earth. There is substantial evidence that fault geometry is fractal, at least in some cases and over some ranges of length scales. Fractality is a special kind of geometric complexity that is characterized by scale invariance (5). That is, images of the same system made with different magnifications are visually similar to one another; there is no intrinsic length scale such as a correlation length or a feature of recognizable size that would enable an observer to determine the magnification simply by looking at the image.
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One result of such a property in the case of fault zones is that there would be a broad, power-law distribution of the lengths of the constituent fault segments (6). If, in the simplest conceivable scenario, the seismic moment of the characteristic earthquake on each segment were proportional to its length, and each segment slipped at random, then the moment distribution would also be a power law. This picture is too simplistic to be a plausible explanation of the Gutenberg-Richter relation, but it may contain some element of the truth. Self-organized criticality refers to the conjecture that a large class of physical systems, when driven persistently away from mechanical equilibrium, will operate naturally near some threshold of instability, and will therefore exhibit something like thermodynamic critical fluctuations (7). Earthquake faults, or arrays of coupled faults, seem to be natural candidates for this kind of behavior; such systems are constantly being driven by tectonic forces toward slipping thresholds (8). If the thermodynamic analogy were valid, then the fluctuations—the slipping events—would be self-similar and scale invariant, and their sizes would obey power-law distributions. More important, systems with this self-organizing property would always be at or near their critical points. Critical behavior, with strong sensitivity to small perturbations and intrinsic unpredictability, would be a universal characteristic of such systems. Elementary Models of Earthquake Dynamics The ideas of fractality and dynamic self-organization have inspired a wide range of theoretical models of seismic systems. These models are almost invariably numerical: that is, they are studied primarily by means of large-scale computation. One class is cellular automata in which highly simplified rules for the behavior of large numbers of coupled components attempt to capture the essential features of complex seismic systems (9). Almost all cellular automata are related in some ways to the original one-dimensional slider-block model of Burridge and Knopoff (10), illustrated in Figure 5.1. Perhaps the most important result to emerge from such studies so far is the discovery that some of the simplest of these models, even the completely uniform Burridge-Knopoff model with a plausible, velocity-weakening dynamic friction law, are deterministically chaotic (11). A chaotic system, by definition, is one in which the accuracy needed to determine its motion over some interval of time grows rapidly, in fact exponentially, with the length of the interval. Two identical systems that are set in motion with almost but not quite the same initial conditions may move in nearly the same way for a while. If these systems are chaotic, however, their motions eventually will differ from each other and, after a
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FIGURE 5.1 Two-dimensional version of the Burridge-Knopoff spring-slider model. Leaf springs (K1) connect a moving plate to an array of smaller sliding blocks. These blocks are in turn connected to their nearest neighbors via coil springs (K2, K3). Sliding blocks also have a frictional contact with a fixed plate. Simulated earthquakes from models of this type display a wide variety of complexity. SOURCE: P. Bak, How Nature Works: The Science of Self-Organized Criticality, Springer-Verlag, New York, 226 pp., 1996. Copyright permission granted by Springer-Verlag. sufficiently long time, will appear to be entirely uncorrelated. The correlation time depends sensitively on the difference in the initial conditions. In the context of predictability, this means that any uncertainty in one’s knowledge of the present state of a deterministically chaotic system produces a theoretical limit on how far into the future one can determine its behavior reliably, a topic explored further below. One theoretical issue that has attracted a lot of attention has come to be known as the question of smooth versus heterogeneous fault models. This issue arose initially as a result of the unexpected success of the uniform Burridge-Knopoff slider-block models in producing very rough but interesting caricatures of complex earthquake-like behavior, which fueled speculation that some of the slip complexity of natural earthquakes might be generated by the nonlinear dynamics of stressing and rupture on essentially smooth and uniform faults. The more conventional and perhaps obvious assumption is that the heterogeneity of fault zones—their geometric disorder and strong variations of lithological properties—plays the dominant role. It appears that earthquake faults, when modeled in any detail, have relevant length and time scales that invalidate simple scaling assumptions. For example, the tectonic loading speed (meters per century) combined with known friction thresholds and elastic moduli of rocks suggests natural characteristic intervals (hundreds of years) between large slipping events. Models that incorporate these features produce event distributions in which the large events fail to be self-similar (12).
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Another example is the thickness of the seismogenic layer, which is less than the rupture scale for larger earthquakes. It, too, seems likely to produce scaling violations both in dynamic behavior and in the geometry of fault systems (see Section 5.2). The existence of relevant length and time scales does not, per se, invalidate dynamical scaling theories; it may merely limit their ranges of validity. In some smooth-fault models, for example, it appears that the small, localized seismic events are self-similar over broad ranges of sizes; however, the large, delocalized events look quite different and are substantially more frequent than would be predicted by extrapolating the scaling distribution for the small events (13), as in the “characteristic earthquake” model discussed in Section 2.6. The picture may change appreciably if one considers large arrays of coupled faults and, especially, if one includes the mechanism for creation of new faults as a part of the dynamical system. It is possible that this global system, in some as yet poorly understood average sense, may come closer to a pure form of self-organized criticality. Chaos and Predictability The theoretical issue of earthquake predictability (as distinct from the practical issue of how to predict specific earthquakes) remains a central, unresolved issue. The wide range of event sizes described by the Gutenberg-Richter law, the obvious irregularities in intervals between large events, the fact that chaotic behavior occurs commonly in very simple earthquake-like models, and many other clues, all argue in favor of chaos and thus for an intrinsic limit to predictability. The interesting question is what bearing this theoretical limit might have on the kinds of earthquake prediction that are discussed elsewhere in this report. If one could measure all the stresses and strains in the neighborhood of a fault with great accuracy, and if one knew with confidence the physical laws that govern the motion of such systems, then the intrinsic time limit for predictability might be some small multiple of the average interval between characteristic large events on the fault. Most of the seismic energy is released in the large events; thus, it seems reasonable to suppose that the system suffers most of its memory loss during those events as well. If this supposition were correct, earthquake prediction on a time scale of months or years—intermediate-term prediction of the sort described in Section 2.6—would, in principle, be possible. The difficulty, of course, is that one cannot measure the state of a fault and its surroundings with great accuracy, and one still knows very little about the underlying physical laws. If these gaps in knowledge could be filled, then predicting earthquakes a few years into the future might be no
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more difficult than predicting the weather a few hours in advance. However, the geological information needed for earthquake prediction is far more complex than the atmospheric information required for weather prediction, and almost all of it is hidden far beneath the surface of the Earth. Thus, the practical limit for predictability may have little to do with the theory of deterministic chaos, but may be fixed simply by the sheer mass of information that is unavailable. Progress Toward Realism Two general goals of research in this field are to understand (1) how rheological properties of the fault-zone material interact with rupture propagation and fault-zone heterogeneity to control earthquake history and event complexity, and (2) to what extent scientists can use this knowledge to predict, if not individual earthquakes, then at least the probabilities of seismic hazards and the engineering consequences of likely seismic events. Finding the answers is an ambitious and difficult task, but there are reasons for optimism. The speeds and capacities of computers continue to grow exponentially; they are now at a point where numerical simulations can be carried out on scales that were hardly imagined just a decade ago. At the same time, the sensitivity and precision of observational techniques are providing new ways to test those simulations. There exists, at present, a substantial theoretical and computational effort in the United States and elsewhere devoted to developing increasingly realistic models of earthquake faults. Given a situation in which such a wide variety of physical ingredients of a problem remain unconstrained by experiment or direct observation, numerical experiments to show which of these ingredients are relevant to the phenomena may be crucial. Consider, for example, the assumptions about friction laws that are at the core of every fault model. For slow slip, the rate- and state-dependent law discussed in Section 4.4 may be reliable, at least in a qualitative sense. On the other hand, for fast slip of the kind that occurs in large events, there is little direct information. It seems likely that dynamic friction in those cases is determined by the behavior of internal degrees of freedom such as fault gouge, pore fluids, and the like. Laboratory experiments on multicomponent lubricated interfaces may provide some insight, but the solution to this problem may have to rely on comparisons between real and simulated earthquakes. There are suggestions that a friction law with enhanced velocity-weakening behavior (i.e., stronger than the logarithmic weakening in the rate and state laws) is needed to produce slip complexity and perhaps also to produce propagating slip pulses in big events (14). This conjecture needs to be tested.
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Friction is not the only constitutive property that may be relevant. The laws governing deformation and fracture may play important roles, especially if the latter processes are effective in arresting large events and/or creating new fault surfaces. Other uncertainties in this category include the geometric structure of faults, the ways in which constitutive properties vary as functions of depth or position along a fault, the statistical distribution of heterogeneities on fault surfaces, and the parameters that govern the interactions between neighboring faults during seismic events. An equally serious issue is whether small-scale physical phenomena are relevant to large-scale behavior. A truly complete description of an earthquake would involve length and time scales ranging from the microscopic ones at which the dynamics of fracture and friction are determined to the hundreds of kilometers over which large events occur. Numerical simulations, especially three-dimensional ones, would be entirely infeasible if they were required to resolve such a huge range of scales. There are, however, examples in other scientific areas where this is precisely what occurs. In dendritic solidification, for example, it is known that a length scale associated with surface tension—a length usually on the order of ångströms—controls the shapes and speeds of macroscopic pattern formation (15). Any direct numerical simulation that fails to resolve this microscopic length scale produces qualitatively incorrect results. There are indications that similar effects occur in some hydrodynamic problems, perhaps even in turbulence (16). At present, it is not known whether any such sensitivities occur in earthquake problems, but there are possibilities. For example, it remains an open question whether simulations of earthquakes must resolve the details of the initial fracture and/or the nucleation process. It is possible that many features of this small-scale behavior are imprinted in important ways on the subsequent large-scale events, but it is also possible that only one or two parameters pertaining to nucleation—perhaps the location and initial stress drop (plus the surrounding stress and strain fields, of course)—have to be specified in order to predict accurately what happens next. Similarly, if the solidification analogy is a guide, then the small-scale, high-frequency behavior of the constitutive laws might be relevant to pulse propagation, interactions between rupture fronts and heterogeneities, and mechanisms of rupture arrest. In order to study large systems on finite computers, investigators frequently study two-dimensional models, often accounting for deformations in the crustal plane perpendicular to the fault (in models of transverse faults) and omitting or drastically oversimplifying variations in the fault plane (i.e., motions that are functions of depth beneath the surface). How relevant is the third dimension? Some investigators have argued
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that it must be crucial because, without a coupling between the top and bottom of the fault, there is no restoring force to limit indefinitely large slip or, equivalently, to couple kinetic energy of slip back into stored elastic energy. It is hard to see how the dynamics of large events, especially rupture arrest and pulse propagation, can be studied sensibly without full three-dimensional analyses. The issues of how to make progress toward realism are theoretical as well as computational. There is an emerging realization among theorists working on earthquake dynamics, and in solid mechanics more generally, that the problems with which they are dealing are far more difficult mathematically than they had originally supposed. One of the reasons that small-scale features can control large-scale behavior, as mentioned above, is that these features enter the mathematical statement of the problem as singular perturbations. For example, the surface tension in the solidification problem and the viscosity in certain shock-front problems enter the equations of motion as coefficients of the highest derivative of the dependent variable. As such, they completely change the answer to questions as basic as whether or not physically acceptable solutions exist and how many parameters or boundary conditions are needed to determine them. A related difficulty that is emerging, especially in problems involving elasticity, is that the equations of motion are often expressed most accurately as singular integral equations. Except for a few famous cases due largely to Muskhelishvili (17), such equations are not analytically solvable. There are not even good methods for determining the existence of solutions, nor are there reliable numerical algorithms for finding solutions when they do exist. In general, the ability to resolve the uncertainties regarding connections between model ingredients and physical phenomena will depend on advances in both mathematics and computer science. These problems are solvable, but they are indeed difficult. 5.2 FAULT SYSTEMS Most theories of earthquake dynamics presume that essentially all major earthquakes occur on thin, preexisting zones of weakness, so that the behavior of the biggest events derives from the slip dynamics of a fault network. There are strongly different conceptions of fault systems, all of which may have merit for some purposes (18). Faults can be modeled as smooth Euclidean surfaces of displacement discontinuity in an otherwise continuous medium; fault systems can be represented as fractal arrays of surfaces; fault segments can be regarded as merely the deforming borders between blocks of a large-scale granular material transmitting stress in a force-chain mode. Representing the crust as a fault system is especially useful on the interseismic time scales relevant to fault interac-
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tions, seismicity distributions, and the long-term aspects of the postseismic response. Fault-system dynamics involves highly nonlinear interactions among a number of mechanical, thermal, and chemical processes—fault friction and rupture, poroelasticity and fluid flow, viscous coupling, et cetera— and sorting out how these different processes govern the cycle of stress accumulation, transfer, and release is a major research goal. Moreover, progress on the problem of seismicity as a cooperative behavior within a network of active faults has the potential to deliver huge practical benefits in the form of improved earthquake forecasting. The latter consideration sets a direction for the long-term research program in earthquake science. Architecture of Fault Systems Thermal convection and chemical differentiation are driving mass motions throughout the planetary interior, but the slip instabilities that cause earthquakes appear to be confined to the relatively cold, brittle boundary layers that constitute the Earth’s lithosphere. With sufficient knowledge of the rheologic properties of the lithosphere and the necessary computational resources, it should be possible to set up simulations of mantle convection that reproduce plate tectonics from first principles, including the localization of deformation into plate boundary zones. However, the nonlinearity of the rheology and its sensitivity to pressure, temperature, and composition (especially the minor but critical constituent of water) make this a difficult problem (19). Tough computational issues are also posed by the wide range of spatial scales that must be represented in numerical models. Strain localization is most intense on plate boundaries that involve the relatively thin oceanic crust, although there are exceptions. One is a region of diffuse though strong seismicity (up to moment magnitude [M] 7.8) in the central Indian Ocean that may represent an incipient plate boundary (20). The study of these juvenile features may shed light on the localization problem. In continents, earthquakes are typically distributed across broad zones in which active faults form geometrically and mechanically complicated networks that accommodate the large-scale plate motions. This diffuse nature is clearly related to the greater thickness and quartz-rich composition of the continental crust, as described in Section 2.4. The structure of continental fault zones is thought to be complicated by variations in frictional behavior with depth, changes in wear mechanisms, and a brittle-ductile transition (Figure 4.30), although the details remain highly uncertain. Interesting issues also arise from attempts to understand how the
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complexities are related to the long geological history of the continents. In the southwestern United States, for example, the fault systems that produce high earthquake hazards have developed over tens of millions of years by tectonic interactions among the heterogeneous ensemble of accreted terrains that constitute the North American continental lithosphere and the oceanic lithosphere of the Farallon and Pacific plates. These interactions have created a zone of deformation a thousand kilometers wide that extends from the continental coastline to the Rocky Mountains. The “master fault” of this plate-boundary zone is the strike-slip San Andreas system, but other types of faults participate in the deformation, from extension in the Basin and Range to contraction in the Transverse Ranges. Likewise, the great thrust faults that mark the subduction zones of the northwestern United States and Alaska are accompanied by secondary faulting distributed for considerable distances landward of the subduction boundary. Within the continental interior far from the present-day plate boundaries, deformation is localized on reactivated, older faults, and some of these structures are capable of generating large earthquakes (see Section 3.2). The geometric complexity of fault systems is fractal in nature, with approximately self-similar roughness, segmentation, and branching over length scales ranging from meters to hundreds of kilometers (Figure 3.2). Fault systems also have mechanical heterogeneities due to litho-logic contrasts, uneven damage, and possibly pressurized compartments within fault zones (21). The understanding of fault system architecture and earthquake generation in such systems is at a rudimentary stage of development. Fault Kinematics and Earthquake Recurrence The subject of fault kinematics pertains to descriptions of earthquake occurrence and slip of individual faults at different time scales, and the partitioning of slip among faults to accommodate regional deformation. An important goal of this characterization is to address the fundamental question of how slow and smoothly distributed regional deformations across fault systems, as seen in geodetic observations, are eventually transformed, principally at the time of earthquakes, into localized slip on particular faults. To build a comprehensive picture of this process requires synthesis of detailed geologic, geophysical, and seismic observations. At present, some regions—particularly portions of California and Japan— have sufficient information to describe the recent history of large earthquakes, to make estimates of the long-term average of slip rates of the principal faults, and to map the surface strain field across fault systems. Though comprehensive descriptions of fault-system kinematics are not
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124. Fault healing was first modeled by R. Madariaga (Dynamics of an expanding circular fault, Bull. Seis. Soc. Am., 66, 639-666, 1976) using a finite-difference approach. Healing of an expanding circular rupture was initiated simultaneously at the edges, and subsequently propagated inward toward the center of the fault at a large fraction of the shear-wave velocity. The fault did not stop slipping at interior points until information that the rupture had stopped propagated inward from the edges. 125. S.M. Day, Three-dimensional finite difference simulation of fault dynamics: Rectangular faults with fixed rupture velocity, Bull. Seis. Soc. Am., 72, 705-727, 1982. 126. T.H. Heaton, Evidence for and implications of self-healing pulses of slip in earthquake rupture, Phys. Earth Planet. Int., 64, 1-20, 1990. 127. Quasi-dynamic models are dynamic models constructed to reproduce the principal features of kinematic models, which were previously obtained by modeling seismic data. Examples of quasi-dynamic models with short rise times include the following: H. Quin, Dynamic stress drop and rupture dynamics of the October 15, 1979 Imperial Valley, California, earthquake, Tectonophysics, 175, 93-117, 1990; T. Miyatake, Dynamic rupture processes of inland earthquakes in Japan; Weak and strong asperities, Geophys. Res. Lett.,19, 1041-1044, 1992; E. Fukuyama and T. Mikumo, Dynamic rupture analysis; Inversion for the source process of the 1990 Izu-Oshima, Japan, earthquake (M = 6.5), J. Geophys. Res., 98, 6529-6542, 1993; G.C. Beroza and T. Mikumo, Short slip duration in dynamic rupture in the presence of heterogeneous fault properties, J. Geophys. Res., 101, 22,449-22,460, 1996. 128. T.H. Heaton, J.F. Hall, D.J. Wald, and M.W. Halling, Response of high-rise and base-isolated buildings to a hypothetical Mw 7.0 blind thrust earthquake, Science, 267, 206-211, 1995. 129. Early references suggesting such behavior include E. Vesanen (On the character interpretation of seismograms, Ann. Acad. Sci. Fenn., AIII, 5, 1942); T. Usami, Quart. J. Seis., 21, 1-13, 1956); S.S. Miyamura, R. Omote, R. Teisseyre, and E. Vesanen, Multiple shocks and earthquake series pattern, Bull. Int. Inst. Seis. Earthquake Eng., 2, 71-92, 1965); and M. Båth (Seis. Bull., Uppsala, February 4, 1965). M. Wyss and J.N. Brune (The Alaska earthquake of 28 March 1964: A complex multiple rupture, Bull. Seis. Soc. Am., 57, 1017-1023, 1967) found that the Good Friday earthquake of 1964 consisted of six subevents of higher than average slip and that, based on the timing of these subevents, rupture propagated primarily to the southwest from the epicenter. Studies of moderate to large earthquakes in the far field using P- and S- wave pulse shapes often depict earthquake rupture with multiple point sources (H. Kanamori and G.S. Stewart, Seismological aspects of the Guatemala earthquake of February 4, 1976, J. Geophys. Res., 83, 3427-3434, 1978), which have been termed asperities (T. Lay and H. Kanamori, An asperity model of great earthquake sequences, in Earthquake Prediction—An International Review, D.W. Simpson and P.G. Richards, eds., American Geophysical Union, Maurice Ewing Series, 4, Washington, D.C., pp. 579-592, 1980). For an isolated point source or even for multiple point sources, it is often sufficient to allow for the finite duration through the moment rate function, without considering the spatial extent of each source. 130. For a thorough review of source tomography in the far field, see the review by L. Ruff, Tomographic imaging of seismic sources, in Seismic Tomography, G. Nolet, ed., D. Reidel, Boston, pp. 339-366, 1987. The pulse shape of far-field body waves can be used to infer the spatial and temporal distribution of slip on the fault; however, only in the case of particularly large earthquakes is it possible to resolve spatial and temporal variation of slip on the fault plane at teleseismic distances. See, for example the study of the 1985 Michoacan, Mexico, earthquakes by C. Mendoza (Coseismic slip of two large Mexican earthquakes from teleseismic body waveforms: Implications for asperity interaction in the Michoacan plate boundary segment, J. Geophys. Res., 98, 8197-8210, 1993). Moreover, it is theoretically impossible to reconstruct the slip distribution from the far-field data alone. In the Fraun-
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hoffer approximation (i.e., to the extent that the only difference in the Green’s function from different parts of the fault is a difference in phase) (K. Aki and P.G. Richards, Quantitative Seismology: Theory and Methods, vol 2., W.H. Freeman, San Francisco, pp. 803-805, 1981), there are a range of wavenumbers that are theoretically impossible to recover. Hence, a complete recovery of the source history is impossible. 131. The first such study by K. Aki (Seismic displacements near a fault, J. Geophys. Res., 73, 5359-5376, 1968) modeled a single near-source record of the 1966 Parkfield earthquake, specifying the depth extent and length of the fault, the rupture velocity, and the average slip. Near-field ground motion data can be supplemented by teleseismically recorded data (S.H. Hartzell and T.H. Heaton, Inversion of strong ground motion and teleseismic waveform data for the fault rupture history of the 1979 Imperial Valley, California earthquake, Bull. Seis. Soc. Am., 73, 1553-1583, 1983) and geodetic data (D.J. Wald, and T.H. Heaton, Spatial and temporal distribution of slip for the 1992 Landers, California earthquake, Bull. Seis. Soc. Am., 84, 668-691, 1994). 132. For example, the 1979 Imperial Valley earthquake (A.H. Olson, and R.J. Apsel, Finite faults and inverse theory with applications to the 1979 Imperial Valley earthquake, Bull. Seis. Soc. Am., 72, 1969-2001, 1982; S.H. Hartzell and T.H. Heaton, Inversion of strong ground motion and teleseismic waveform data for the fault rupture history of the 1979 Imperial Valley, California earthquake, Bull. Seis. Soc. Am., 73, 1553-1583, 1983), the 1989 Loma Prieta earthquake (see the October 1991 special issue of the Bulletin of the Seismological Society of America), the 1994 Northridge earthquake (see the February 1996 special issue of the Bulletin of the Seismological Society of America), the 1995 Hyogo-ken Nanbu earthquake (D.J. Wald, A preliminary dislocation model for the 1995 Kobe (Hyogo-ken Nanbu), Japan, earthquake determined from strong motion and teleseismic waveforms, Seis. Res. Lett.,66, 22-28, 1995); and the 1992 Landers earthquake (D.J. Wald and T.H. Heaton, Spatial and temporal distribution of slip for the 1992 Landers, California earthquake, Bull. Seis. Soc. Am., 84, 668-691, 1994; B.P. Cohee and G.C. Beroza, Slip distribution of the 1992 Landers earthquake and its implications for earthquake source mechanics, Bull. Seis. Soc. Am., 84, 692-712, 1994; F. Cotton and M. Campillo, Frequency domain inversion of strong motions; Application to the 1992 Landers earthquake, J. Geophys. Res., 100, 3961-3975, 1995). 133. Examples of quasi-dynamic rupture models include a study of the 1979 Imperial Valley earthquake (H. Quin, Dynamic stress drop and rupture dynamics of the October 15, 1979 Imperial Valley, California, earthquake, Tectonophysics, 175, 93-117, 1990) and the 1984 Morgan Hill earthquake (G.C. Beroza and T. Mikumo, Short slip duration in dynamic rupture in the presence of heterogeneous fault properties, J. Geophys. Res., 101, 22,449-22,460, 1996). M. Bouchon (The state of stress on some faults of the San Andreas system as inferred from near-field strong motion data, J. Geophys. Res., 102, 11,731-11,744, 1997) studied the stress drop and strength excess for several earthquakes. A quasi-dynamic rupture model was derived for the 1995 Hyogo-ken Nanbu earthquake by S. Ide and M. Takeo (Determination of constitutive relations of fault slip based on seismic wave analysis, J. Geophys. Res., 102, 27,379-27,391, 1997) and for the 1992 Landers earthquake by K.B. Olsen, R. Madariaga, R.J. Archuleta (Three-dimensional dynamic simulation of the 1992 Landers earthquakes, Science, 278, 834-838, 1997) using a slip-weakening constitutive model. Quasi-dynamic models are parameterized in terms such as the strength excess (how close the fault is to failure before the earthquake begins), the dynamic stress drop (difference between the initial stress and the residual sliding friction), and the slip-weakening displacement (amount of slip required to reach the sliding frictional stress). 134. M. Guatteri and P. Spudich, What can strong motion data tell us about slip-weakening friction laws?, Bull. Seis. Soc. Am., 90, 98-116, 2000. 135. As earthquake rupture evolves, it exerts a time-varying stress field on the points around it, including portions of the fault that are slipping. As shown by P.K.P. Spudich (On
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the inference of absolute stress levels from seismic radiation, Tectonophysics, 211, p. 99-106, 1992) under certain assumptions the response of the fault to this load can be used to estimate the absolute stress level if the slip direction at a point on the fault varies as a function of time during rupture. Application of this technique to the 1995 Hyogo-ken Nanbu earthquake has shown promising results that match borehole stress data where available and suggest a low level of absolute stress (P. Spudich, M. Guatteri, I. Otsuki, and J. Minagawa, Use of fault striations and dislocation models to infer tectonic shear stress during the 1995 Hyogo-ken Nanbu (Kobe) earthquake, Bull. Seis. Soc. Am., 88, 413-427, 1998; M. Guatteri and P. Spudich, Coseismic temporal changes of slip direction; The effect of absolute stress on dynamic rupture, Bull. Seis. Soc. Am., 88, 777-789, 1998). 136. Early studies of strong ground motion characterized it as emanating from the source in short impulses (G.W. Housner, Characteristics of strong-motion earthquakes, Bull. Seis. Soc. Am., 37, 19-31, 1947; G.W. Housner, Properties of strong ground motion earthquakes, Bull. Seis. Soc. Am., 45, 197-218, 1955; W.T. Thompson, Spectral aspect of earthquakes, Bull. Seis. Soc. Am., 49, 91-98, 1959). A stochastic model for earthquake slip was formulated by N.A. Haskell (Total energy and energy spectral density of elastic wave radiation from propagating faults, 2, A statistical source model, Bull. Seis. Soc. Am., 56, 125-140, 1966) in a study of the radiated seismic energy. Haskell characterized earthquake slip as a spatial random field using both a correlation length and a correlation time to specify the autocorrelation function. 137. R. Madariaga (High-frequency radiation from crack (stress drop) models of earthquake faulting, Geophys. J. R. Astron. Soc., 51, 625-651, 1977) demonstrated that smooth rupture propagation produces relatively little high-frequency ground motion. 138. As, for example, a model of circular rupture and arrest studied by R. Madariaga (Dynamics of an expanding circular fault, Bull. Seis. Soc. Am., 66, 639-666, 1976) in which rupture was stopped artificially and simultaneously around the entire circumference of the fault. 139. The spectral behavior of ground motion above the corner frequency is often cast in terms of the displacement spectra (K. Aki, Scaling law of seismic spectrum, J. Geophys. Res., 72, 1217-1231, 1967). A spectrum that is flat in acceleration will decay as f–2 in displacement (J.N. Brune, Tectonic stress and the spectra of seismic shear waves from earthquakes, J. Geophys. Res., 75, 4997-5009, 1970). 140. R. Madariaga, High frequency radiation from dynamic earthquake fault models, Ann. Geophys., 1, 17-23, 1983; A.S. Papageorgiou, and K. Aki, A specific barrier model for the quantitative description of inhomogeneous faulting and the prediction of strong ground motion; Part I, Description of the model, Bull. Seis. Soc. Am., 73, 953-978, 1983. 141. Y. Zeng, K. Aki, and T.-L. Teng, Mapping of the high-frequency source radiation for the Loma Prieta earthquake, California, J. Geophys. Res., 98, 11,981-11,993, 1993. 142. See, for example, C.K. Saikia and P.G. Somerville (Simulated hard-rock motions in Saint Louis, Missouri, from large New Madrid earthquake (Mw = 6.5), Bull. Seis. Soc. Am., 87, 123-139, 1997) or Y. Zeng and J.G. Anderson (A composite source modeling of the 1994 Northridge earthquake using Genetic Algorithm, Bull. Seis. Soc. Am., 86, 71-83, 1996). 143. C.H. Scholz, Microfracturing and the inelastic deformation of rock in compression, J. Geophys. Res., 73, 1417-1432, 1968. 144. Shear rupture is naturally limited by the Earth’s surface. There is also evidence that rupture through the shallowest sedimentary layers of the Earth’s crust may be delayed and occur as aseismic afterslip. The decay of afterslip with time is consistent with steady-state velocity strengthening friction on this part of the fault (C. Marone and C. Scholz, The depth of seismic faulting and the upper transition from stable to unstable slip regimes, Geophys. Res. Lett., 15, 621-624, 1988). 145. J.S. Tchalenko, Similarities between shear zones of different magnitudes, Geol. Soc.
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Am. Bull., 81, 1625-1640, 1970; W.L. Power, T.E. Tullis, S. Brown, G.N. Boitnott, and C.H. Scholz, Roughness of natural fault surfaces, Geophys. Res. Lett., 14, 29-32, 1987. 146. For example, fault segmentation was used to estimate the size of future earthquakes in the report by the Working Group on California Earthquake Probabilities (Probabilities of Large Earthquakes Occurring in California on the San Andreas Fault, U.S. Geological Survey Open-File Report 88-398, Reston, Virginia, 62 pp., 1988). 147. For examples of earthquakes that may have been terminated by a fault discontinuity, see the following studies: W.H. Bakun, R.M. Stewart, C.G. Bufe, and S.M. Marks, Implication of seismicity for failure of a section of the San Andreas fault, Bull. Seis. Soc. Am., 70, 185-201, 1980; A.G. Lindh and D.M. Boore, Control of rupture by fault geometry during the 1966 Parkfield earthquake, Bull. Seis. Soc. Am., 71, 95-116, 1981; P. Reasenberg and W.L. Ellsworth, Aftershocks of the Coyote Lake, California, earthquake of August 6, 1979: A detailed study, J. Geophys. Res., 87, 10,637-10,655, 1982; A. Barka and K. Kadinsky-Cade, Strike-slip fault geometry in Turkey and its influence on earthquake activity, Tectonics,7, 663-684, 1988. 148. Examples of earthquakes that ruptured through fault discontinuities include the Borrego Mountain, California (M.M. Clark, Surface rupture along the Coyote Creek Fault, in The Borrego Mountain Earthquake, U.S. Geological Survey Professional Paper 787, Reston, Va., pp. 55-86, 1972), the Erzincan, Turkey (A. Barka and K. Kadinsky-Cade (Strike-slip fault geometry in Turkey and its influence on earthquake activity, Tectonics,7, 663-684, 1988), and the 1966 Parkfield, California, earthquakes (P. Segall and Y. Du, How similar were the 1934 and 1966 Parkfield earthquakes? J. Geophys. Res., 98, 4527-4538, 1993). 149. K.E. Sieh, and 19 others, Near-field investigations of the Landers earthquake sequence, April to July 1992, Science, 260, 171-176, 1993. 150. Working Group on California Earthquake Probabilities, Seismic hazards in southern California: Probable earthquakes, 1994-2024, Bull. Seis. Soc. Am., 85, 379-439, 1995; Working Group on California Earthquake Probabilities, Earthquake Probabilities in the San Francisco Bay Region: 2000 to 2030—A Summary of Findings, U.S. Geological Survey Open File Report 99-517, Reston, Va., 46 pp., 1999. 151. Based on a study of Ms = 6.8 earthquakes on the North Anatolian fault in Turkey. See A. Barka and K. Kadinsky-Cade (Strike-slip fault geometry in Turkey and its influence on earthquake activity, Tectonics,7, 663-684, 1988). 152. R.A. Harris and S.M Day, Dynamics of fault interaction; Parallel strike-slip faults, J. Geophys. Res., 98, 4461-4472, 1993. 153. A possible complicating factor is the effect of pore fluids. If pore fluids are present, they will tend to drop when an extensional jog is stressed. The drop in fluid pressure will tend to lock the fault by counteracting the drop in normal stress across it (R. Sibson, Stopping of earthquake ruptures at dilational fault jogs, Nature, 316, 248-251, 1985). 154. R.A. Harris and S.M. Day, Dynamic three-dimensional simulations of earthquakes on en echelon faults, Geophys. Res. Lett., 26, 2089-2092, 1999. 155. C. Marone and C. Scholz, The depth of seismic faulting and the upper transition from stable to unstable slip regimes, Geophys. Res. Lett., 15, 621-624, 1988. 156. A.J. Michael and D. Eberhart-Phillips (Relations among fault behavior, subsurface geology, and three-dimensional velocity models, Science, 253, 651-654, 1991) studied five large earthquakes in California and found variations in the three-dimensional velocity structure at seismogenic depths that tended to correlate with regions of high mainshock slip. Other studies (A. Michelini and T.V. McEvilly, Seismological studies at Parkfield, I, Simultaneous inversion for velocity structure and hypocenters using cubic B-splines parameterization, Bull. Seis. Soc. Am., 81, 524-552, 1991; C. Nicholson and J.M. Lees, Travel-time tomography in the northern Coachella Valley using aftershocks of the 1986 ML 5.9 North Palm Springs earthquake, Geophys. Res. Lett., 19, 1-4, 1992; W. Foxall, A. Michelini, and T.V.
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McEvilly, Earthquake travel time tomography of the southern Santa Cruz Mountains; Control of fault rupture by lithological heterogeneity of the San Andreas fault zone, J. Geophys. Res., 98, 17,691-17,710, 1993) have noted a similar correlation. 157. D.H. Oppenheimer, W.H. Bakun, and A.G. Lindh (Slip partitioning of the Calaveras Fault, California, and prospects for future earthquakes, J. Geophys. Res., 93, 9007-9026, 1988) found areas on the southern Calaveras fault that failed repeatedly during small to moderate earthquakes. These same areas were devoid of microearthquakes in the interseismic period and hence presumably stuck. Much of the slip on the southern Calaveras fault appears to take place aseismically as indicated by the postseismic transient to the 1984 Morgan Hill, California, earthquake, which was larger than that of the mainshock itself (W.H. Prescott, N.E. King, and G. Guohua, Preseismic, coseismic, and postseismic deformation associated with the 1984 Morgan Hill, California, earthquake, in The 1984 Morgan Hill, California Earthquake, J.H. Bennet and R.W. Sherburne, eds., California Division of Mines and Geology Special Publication 68, Sacramento, pp. 137-148, 1986). 158. D.P. Schaff, B. Shaw, and G.C. Beroza, Postseismic response of repeating aftershocks, Geophys. Res. Lett., 25, 4549-4552, 1998; T.E. Tullis, Perspective—Deep slip rates on the San Andreas fault, Science, 285, 671-672, 1999. 159. M.I. Husseini, D.B. Jovanovich, M.J. Randall, and L.B. Freund, The fracture energy of earthquakes, Geophys. J. R. Astron. Soc., 43, 367-385, 1975. 160. R.L. Wesson and W.L. Ellsworth, Seismicity preceding moderate earthquakes in California, J. Geophys. Res., 78, 8527-8546, 1973. 161. T. Lay and H. Kanamori, An asperity model of great earthquake sequences, in Earthquake Prediction—An International Review, D.W. Simpson and P.G. Richards, eds., American Geophysical Union, Maurice Ewing Series, 4, Washington, D.C., pp. 579-592, 1980. 162. Low prestress levels on the Emerson fault inferred from dynamic rupture modeling (M. Bouchon, M. Campillo, and F. Cotton, Stress field associated with the rupture of the 1992 Landers, California, earthquake and its implications concerning the fault strength at the onset of the earthquake, J. Geophys. Res.,103, 21,091-21,097, 1998) and the lack of post-mainshock shear stress across the Emerson fault based on a stress inversion of the aftershock focal mechanisms (E. Hauksson, State of stress from focal mechanisms before and after the 1992 Landers earthquake sequence, Bull. Seis. Soc. Am., 84, 917-934, 1994) support this hypothesis. Three-dimensional simulations of dynamic rupture across a fault discontinuity indicate that the rupture on the second fault segment will nucleate at shallow depth because the normal stress is lower and because the free surface amplifies the dynamic stress field. Analyses of strong-motion (B.P. Cohee and G.C. Beroza, Slip distribution of the 1992 Landers earthquake and its implications for earthquake source mechanics, Bull. Seis. Soc. Am., 84, 692-712. 1994; D.J. Wald and T.H. Heaton, Spatial and temporal distribution of slip for the 1992 Landers, California earthquake, Bull. Seis. Soc. Am., 84, 668-691, 1994) and geodetic (J. Freymueller, N.E. King, and P. Segall, Co-seismic slip distribution of the 1992 Landers earthquake, Bull. Seis. Soc. Am., 84, 646-659, 1994) data indicate that mainshock slip on the Emerson fault was shallow compared with slip on the other segments. This signature is reflected in the aftershock depths as well, which are found to be shallow over the transition from the Homestead Valley to the Emerson faults (K.R. Felzer and G.C. Beroza, Deep structure of a fault discontinuity, Geophys. Res. Lett., 26, 2121-2124, 1999). If the Emerson fault was far from failure before the Landers mainshock, then it may have terminated at shallow depth with rupture nucleating at shallow depth but not propagating to greater depths or further along the fault. 163. A point made by R.A. Harris and S.M. Day (Dynamic three-dimensional simulations of earthquakes on en echelon faults, Geophys. Res. Lett., 26, 2089-2092, 1999) based on the comparison of the 1934 and 1966 earthquakes reported by P. Segall and Y. Du (How similar were the 1934 and 1966 Parkfield earthquakes?, J. Geophys. Res., 98, 4527-4538, 1993).
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164. Earthquakes below crustal depths in descending slabs can shake the surface hard enough to cause deadly secondary effects. For example, the 1970 intermediate-focus Peru earthquake (M 8.0), which occurred at a focal depth of 64 kilometers, initiated the huge Huascarán slide (Figure 3.3) that killed 60,000 people. Seismic waves propagate efficiently below the asthenosphere (i.e., below about 300-kilometer depth) and within the deeper parts of the thickened cratonic lithosphere; the giant M 8.2 deep-focus earthquake 660 kilometers beneath Bolivia was felt by people as far away as Canada. 165. M. Vassilou and B. Hager, Subduction zone earthquakes and stress in slabs, Pure Appl. Geophys., 128, 547-624, 1988. 166. B. Isacks and P. Molnar, Distribution of stresses in the descending lithosphere from a global survey of focal-mechanism solutions of mantle earthquakes, Rev. Geophys. Space Phys., 9, 103-174, 1971. 167. C. Frohlich, The nature of deep focus earthquakes, Ann. Rev. Earth Planet. Sci., 17, 227-254, 1989; H.W. Green, II and H. Houston, The mechanics of deep earthquakes, Ann. Rev. Earth Planet. Sci., 23, 169-213, 1995. 168. P.B. Stark and C. Frohlich, Depths of the deepest earthquakes, J. Geophys. Res., 90, 1859-1869, 1985. 169. The problem of initiating shear instabilities at very high pressures was broached by D. Griggs and J. Handin (Observations on fracture and a hypothesis of earthquakes, in Rock Deformation, D. Griggs and J. Handin, eds., Geological Society of America Memoir 79, Boulder, Colo., pp. 347-364, 1960). Early experimental and theoretical work focused on the volumetric instabilities in polymorphic phase transitions (e.g., P.W. Bridgman, Polymorphic transitions and geologic phenomena, Am. J. Sci., 243A, 90-97, 1945; F.F. Evison, On the occurrence of volume change at the earthquake source, Bull. Seis. Soc. Am., 57, 9-25, 1967, L. Liu, Phase transformations, earthquakes, and the descending lithosphere, Phys. Earth Planet. Int., 32, 226-240, 1983). The volumetric-instability hypothesis was encouraged by seismological evidence that the great 1970 Columbia deep-focus earthquake radiated energy with a significant isotropic component (A. Dziewonski and J.F. Gilbert, Temporal variation of the seismic moment tensor and the evidence of precursive compression for two deep earthquakes, Nature, 257, 185-188, 1974); however, it is now believed that the isotropic component of deep-focus source mechanisms is small compared to the shear component (D. Russakoff, G. Ekstrom, and J. Tromp, A new analysis of the great 1970 Colombia earthquake and its isotropic component, J. Geophys. Res., 102, 20,423-20,434, 1997). 170. D. Griggs, The sinking lithosphere and the focal mechanism of deep earthquakes, in The Nature of the Solid Earth, E.C. Robertson, ed., McGraw-Hill Inc., New York, pp. 361-384, 1972; M. Ogawa, Shear instability in a viscoelastic material as the cause of deep focus earthquakes, J. Geophys. Res., 92, 13,801-13,810, 1987; B.E. Hobbs and A. Ord, Plastic instabilities: Implications for the origin of intermediate and deep focus earthquakes, J. Geophys. Res., 93, 10,521-10,540, 1988. 171. C.B. Raleigh, Tectonic implications of serpentinite weakening, Geophys. J. R. Astr. Soc., 14, 113-118, 1967; M.S. Paterson, Experimental Rock Deformation—The Brittle Field, Springer, Berlin, 254 pp., 1978. 172. S.H. Kirby, Localized polymorphic phase transitions in high-pressure faults and applications to the physical mechanisms of deep earthquakes, J. Geophys. Res., 93, 13,789-13,800, 1987; C. Meade and R. Jeanloz, Acoustic emissions and shear instabilities during phase transformations in Si and Ge at ultra high pressures, Nature, 339, 616-618, 1989; H.W.I. Green and P.B. Burnley, A new self-organizing mechanism for deep-focus earthquakes, Nature, 341, 733-737, 1989. 173. H.W.I. Green and P.B. Burnley, A new self-organizing mechanism for deep-focus earthquakes, Nature, 341, 733-737, 1989; C. Meade and R. Jeanloz, Deep-focus earthquakes
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and recycling of water into the Earth’s mantle, Science, 252, 68-72, 1991; C. Frohlich, A break in the deep, Nature, 368, 100-101, 1994. 174. S.H. Kirby, Intraslab earthquakes and phase-changes in subducting lithosphere, Rev. Geophys. Suppl., 33, 287-297, 1995. 175. H.W. Green, T.E. Young, D. Wlaker, and C. Scholz, Anti-crack assisted faulting at very high pressure in natural olivine, Nature, 348, 720-722, 1990; H.W. Green, Solving the paradox of deep earthquakes, Sci. Am., 271, 64-71, 1994. 176. H.W. Green II and H. Houston, The mechanics of deep earthquakes, Ann. Rev. Earth Planet. Sci., 23, 169-213, 1995. 177. The March 9, 1994, Tonga earthquake (Mw 7.6, 564 kilometers deep) was recorded by an array of nine instruments deployed directly over the hypocenter (D.A. Wiens, J.J. McGuire, P.J. Shore, M.G. Bevis, K. Draunidao, G. Prasad, and S. Helu, A deep earthquake aftershock sequence and implications for the rupture mechanism of deep earthquakes, Nature, 372, 540-543, 1995). On June 9, 1994, the Mw 8.3 636-kilometer depth Bolivian earthquake was recorded by 26 portable instruments deployed close to the epicenter (P.G. Silver, S.L. Beck, T.C. Wallace, C. Meade, S.C. Myers, S.E. James, and R. Kuehnel, Rupture characteristics of the deep Bolivian earthquake of 9 June 1994 and the mechanism of deep-focus earthquakes, Science, 268, 69-73, 1995). The Bolivian earthquake was the largest deep earthquake ever recorded. 178. In the case of the Bolivian earthquake, the rupture extended about 60 kilometers in the horizontal direction through the thermal boundary layer of the subducting lithosphere (P.G. Silver, S.L. Beck, T.C. Wallace, C. Meade, S.C. Myers, S.E. James, and R. Kuehnel, Rupture characteristics of the deep Bolivian earthquake of 9 June 1994 and the mechanism of deep-focus earthquakes, Science, 268, 69-73, 1995). 179. H. Kanamori, D.L. Anderson, and T.H. Heaton, Frictional melting during the rupture of the 1994 Bolivian earthquake, Science, 279, 839-842, 1998. 180. In an isotropic Earth, where properties do not depend on direction, three independent elastic parameters are needed to describe seismic-wave propagation. These are usually taken to be compressional-wave speed vp, shear-wave speed vs, and mass density ?; alternatives to the first two are the compressional modulus and shear modulus Two anelastic parameters describe isotropic attenuation, such as the inverses of the nondimensional compressional-wave and shear-wave quality factors, and . In most regions of the Earth, the attenuation of pure compression appears to be negligible, which implies that . An anelastic, isotropic Earth model is thus described by the quadruple function of position (?, vp, vs,). 181. K.B. Olsen, Site amplification in the Los Angeles Basin from 3D modeling of ground motions, Bull. Seis. Soc. Am., 90, S77-S94, 2000. 182. A modern standard is the spherically symmetric Preliminary Reference Earth Model (PREM) of A.M. Dziewonski and D.L. Anderson (Preliminary reference Earth model, Phys. Earth Planet. Int., 25, 297-356, 1981). 183. L.C. Pakiser and W.D. Mooney, eds., Geophysical Framework of the Continental United States, Geological Survey of America Memoir 172, Boulder, Colo., 826 pp. 1989. 184. The normal modes can either be standing waves (free oscillations) or traveling waves (surface-wave overtones). An exhaustive discussion of normal mode methods applied to seismology is given by F.A. Dahlen and J. Tromp, Theoretical Global Seismology, Princeton University Press, Princeton, N.J., 1025 pp., 1998. 185. A comprehensive treatment of ray-theoretic methods can be found in K. Aki and P.G. Richards, Quantitative Seismology: Theory and Methods, vols. I & II, W.H. Freeman, San Francisco, 932 pp., 1980. 186. Codes based on the fourth-order staggered velocity-stress formulation, for example, have proven efficient and flexible (e.g., R.W. Graves, Simulating seismic wave propa-
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gation in 3d elastic media using staggered-grid finite differences, Bull. Seis. Soc. Am.,86, 1091-1106, 1996). 187. Features include parallelism (K.B. Olsen, R. Madariaga, and R.J. Archuleta, Three-dimensional dynamic simulation of the 1992 Landers earthquakes, Science, 278, 834-838, 1997; H. Bao, J. Bielak, O. Ghattas, D.R. O’Hallaron, L.F. Kallivokas, J. R. Shewchuk, and J. Xu, Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers, Comput. Meth. Appl. Mech. Eng.,152, 85-102, 1998), realistic anelastic losses (S.M. Day, Efficient simulation of constant Q using coarse-grained memory variables, Bull. Seis. Soc. Am.,88, 1051-1062, 1998; S.M. Day and C.R. Bradley, Memory-efficient simulation of anelastic wave propagation, Bull. Seis. Soc. Am., 91, 520-531, 2001), discontinuous gridding (multiple structured grids coupled at simple interfaces; S. Aoi and H. Fujiwara, 3D finite-difference method using discontinuous grids, Bull. Seis. Soc. Am.,89, 918-930, 1999), un-structured meshing (Bao et al., op. cit.), memory optimization (Graves, op. cit.), surface topography, propagating kinematic earthquake sources, an interface to optional rupture dynamics modules, and links to unified structural representations, with options for user-defined model modifications. 188. D. Komatitsch and J. Tromp, Introduction to the spectral element method for three-dimensional seismic wave propagation, Geophys. J. Int., 139, 806-822, 1999. 189. This steep decay of amplitude with distance from deeper sources was observed for sites within about 40 kilometers of the Loma Prieta earthquake, a relatively deep crustal source. 190. Waves travelling downward from the hypocenter can be critically reflected by layer interfaces below the hypocenter, causing relatively large amplitudes at the surface for certain distance ranges corresponding to critical reflection at these interfaces, such that the reflected waves are larger than the direct waves. One example of this is the SmS phase, which is an S wave critically reflected from the crust-mantle interface (Moho). This SmS phase was one factor responsible for increased ground motions and damage about 90 kilometers away from the Loma Prieta earthquake. The ground motion at these distances was actually greater than at some closer sites with similar geology. Analysis of ground-motion data from aftershocks of the Northridge earthquake demonstrates that reflections from midcrustal interfaces can increase the amplitude of seismic waves from shallow sources at certain distances less than 100 kilometers. 191. Moho reflections were partially responsible for the strong ground motions that damaged San Francisco after the 1989 Loma Prieta earthquake (P.G. Somerville and J. Yoshimura, The influence of critical Moho reflections on strong ground motions recorded in San Francisco and Oakland during the 1989 Loma Prieta earthquake, Geophys. Res. Lett. 17, 1203-1206, 1990). 192. Generally, the shear-wave velocity and density increase with depth in surficial geological materials. As seismic waves propagate to the surface, the decreasing impedance (given by the product of shear-wave velocity and density) gives rise to a corresponding increase in amplitude of seismic waves, so that within the linear range, the less stiff the surface materials, the higher is the ground motion. Current building codes (1997 NEHRP Provisions and 1997 UBC) include ground-motion amplification factors in two period ranges that are related to the shear-wave velocity averaged over the top 30 meters (the typical depth of geotechnical borings). These amplification factors also account for nonlinear effects, which generally reduce the elastic ground motions at short periods. 193. H. Kawase, The cause of the damage belt in Kobe: “The basin-edge effect,” constructive interference of the direct S-wave with the basin-induced diffracted/Rayleigh waves, Seis. Res. Lett., 67, 25-34, 1996; A. Pitarka, K. Irikura, T. Iwata, and H. Sekiguchi, Three-dimensional simulation of the near-fault ground motion for the 1995 Hyogo-ken Nanbu (Kobe), Japan, earthquake, Bull. Seis. Soc. Am., 88, 428-440, 1998.
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194. D.J. Wald and R.W. Graves, The seismic response of the Los Angeles basin, California, Bull. Seis. Soc. Am., 88, 337-356, 1998. 195. A. Frankel, Three-dimensional simulations of ground motions in the San Bernardino Valley, California for hypothetical earthquakes on the San Andreas fault, Bull. Seis. Soc. Am., 83, 1020-1041, 1993; K.B. Olsen, R.J. Archuleta, and J.R. Matarese, Magnitude 7.75 earthquake on the San Andreas fault; Three-dimensional ground motion in Los Angeles, Science, 270, 1628-1632, 1995; K.B. Olsen and R.J. Archuleta, Three-dimensional simulation of earthquakes on the Los Angeles fault system, Bull. Seis. Soc. Am., 86, 575-596, 1996; R.W. Graves, Three-dimensional finite-difference modeling of the San Andreas fault: Source parameterization and ground motion levels, Bull. Seis. Soc. Am., 88, 881-897, 1998; T. Sato, R.W. Graves, and P.G. Somerville, 3-D finite difference simulations of long period strong motions in the Tokyo metropolitan area during the 1990 Odawara earthquake (Mj 5.1) and the great Kanto earthquake (Ms 8.2) in Japan, Bull. Seis. Soc. Am., 89, 579-607, 1999. 196. P.N. Sahay, T.J.T. Spanos, and V. de la Cruz, Seismic wave propagation in inhomogeneous and anisotropic porous media, Geophys. J. Int., 145, 209-223, 2001. 197. P.G. Somerville, N.F. Smith, R.W. Graves, and N.A. Abrahamson, Modification of empirical strong ground motion attenuation relations to include the amplitude and duration effects of rupture directivity, Seis. Res. Lett., 68, 199-222, 1997. 198. Present research addresses how the Q values and their frequency dependence relate to the physical state of the lithosphere and to the physical mechanisms that cause attenuation. It has been proposed that the regional differences in crustal Q are caused by differences in crustal temperature, crack properties, continuity of geologic structures, and/ or scatterer properties. See H.M. Benz, A. Frankel, and D.M. Boore, Regional Lg attenuation for the continental United States, Bull. Seis. Soc. Am., 87, 606-619, 1997. 199. H.B. Seed and I.M. Idriss, Analyses of ground motions at Union Bay, Seattle during earthquakes and distant nuclear blasts, Bull. Seis. Soc. Am.,60, 125-136, 1970; M. Zeghal and A.-W. Elgamal, Analysis of site liquefaction using earthquake records, J. Geotech. Engr.,120, 996-1017, 1994; E.H. Field, P.A. Johnson, I.A. Beresnev, and Y.H. Zeng, Nonlinear ground-motion amplification by sediments during the 1994 Northridge earthquake, Nature,390, 599-602, 1997; J. Aguirre and K. Irikura, Nonlinearity, liquefaction and velocity variation, of soft soil layers in Port Island, Kobe, during the Hyogo-ken Nanbu earthquake, Bull. Seis. Soc. Am., 87, 1244-1258, 1997. 200. K.-L. Wen, Nonlinear soil response in ground motions, Earthquake Eng. Struct. Dynamics,23, 599-608, 1994; I.A. Beresnev, K.-L. Wen, and Y.T. Yeh, Nonlinear soil amplification—Its corroboration in Taiwan, Bull. Seis. Soc. Am., 85, 496-515, 1995; J. Aguirre and K. Irikura, Nonlinearity, liquefaction and velocity variation of soft soil layers in Port Island, Kobe, during the Hyogo-ken Nanbu earthquake, Bull. Seis. Soc. Am., 87, 1244-1258, 1997. 201. In general, stiff soils have higher thresholds and can exhibit quasi-linear response; soft, cohesionless soils have lower nonlinear thresholds. The frequency of vibration is also a critical parameter. At low frequencies, response spectral ratios are unaffected by strain amplitudes. At intermediate frequencies, including those usually most prominent in large-earthquake spectra, damping dominates and strong motion-soil amplification is less than it is for weak motions. At high frequencies, reduction in the shear modulus dominates and soil amplification increases with increased dynamic strain levels. The transition frequencies will vary depending on soil type and thickness and the input seismic spectrum. 202. K. Arulanandan and R.F. Scott, Verification of Numerical Procedures for the Analysis of Soil Liquefaction Problems, 2 vols, Balkema, Rotterdam, 1801 pp., 1994. 203. Housner reported in the 1940s that strong-motion records could be described as white noise. He attributed the complex character of the records to inhomogeneity on the fault. In many respects, these basic concepts have withstood the test of time and the accumulation of more data. However, as described above, deterministic features of strong-
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ground-motion waveforms, including some produced by source processes such as rupture directivity and others produced by seismic-wave propagation phenomena, such as the trapping of waves in basins, have since been recognized and quantified. 204. The f–2 falloff in the displacement amplitude spectrum was recognized in spectra from regional and teleseismic events by K. Aki (Scaling law of seismic spectrum, J. Geophys. Res., 72, 1217-1231, 1967). The low-frequency spectral level is proportional to the seismic moment. Greater stress drops produce a larger high-frequency spectral level for a given seismic moment. It is generally observed that stress drops are approximately uniform over a wide range of earthquake sizes (see Section 2.5). 205. It has been shown that the falloff can be explained by attenuation in the near-surface material beneath a site. fmax is observed to correlate with the site geology, with soil sites having lower fmax values than rock sites. The falloff above fmax can be described using a frequency-independent Q. Typically, this is parameterized by k, which is related to the slope of the spectral falloff on a log-linear plot. Although it was initially proposed that fmax is produced by a characteristic length scale of the rupture process, fmax is observed to increase for seismometers located in boreholes, indicating that it is at least in part an artifact of near-surface attenuation. For small earthquakes (M lower than 4), the source corner frequency is often obscured by the effects of near-surface attenuation. 206. J.N. Brune (Tectonic stress and the spectra of seismic shear waves from earthquakes, J. Geophys. Res., 75, 4997-5009, 1970) derived this behavior from a earthquake model in which a stress pulse propagated along the fault. He was the first to relate the corner frequency to the radius of rupture and derived the relation between stress drop, seismic moment, and corner frequency. Madariaga (Dynamics of an expanding circular fault, Bull. Seis. Soc. Am., 66, 639-666, 1976) showed that a simple dynamic model of crack nucleation could not produce enough high frequency for an f–2 falloff and suggested that the f–2 falloff was caused by the stopping phase of an earthquake rupture. More recently, the hypothesis that the high-frequency level of the acceleration spectrum is a manifestation of the complexity of the rupture process has been explored. In this view, smaller-scale variations of stress along the fault plane produce higher frequencies of radiated ground motion. Recently, several investigators have proposed fractal models of fault stress heterogeneity. In these models the stress on the fault is a random, self-similar variable with a fluctuation spectrum whose spectral amplitude is proportional to the wavelength raised to some power. Asperities on the fault that produce subevents are described with a power-law distribution of sizes. In separate studies, T. Hanks (b values and ?-? seismic source models; Implications for tectonic stress variations along active crustal fault zones and the estimation of high-frequency strong ground motion, J. Geophys. Res., 84, 2235-2242, 1979), D.J. Andrews (A stochastic fault model, 2. Time-independent case, J. Geophys. Res., 86, 10,821-10,834, 1981), and A. Frankel (High frequency spectral falloff of earthquakes, fractal dimension of complex rupture, b value, and the scaling strength on fault, J. Geophys. Res., 96, 6291-6302, 1991) showed that a flat acceleration spectrum could be explained by a variation of stress drop that was independent of length scale on a fault. Such a scale-independent stress drop is consistent with observations of stress drop being independent of seismic moment. Interestingly, this same variation in stress drop on the fault produced a population of subevents with b values of 1. This b value is similar to those reported for earthquakes in most regions, suggesting that the population statistics of earthquakes may be related to the same stress drop variation responsible for the generation of high-frequency ground motion. 207. For example, N.A. Abrahamson, J.F. Schneider, and J.C. Stepp, Empirical coherency functions for applications to soil-structure interaction, Earthquake Spectra, 7, 1-27, 1992; M.I. Todorovska and M.D. Trifunac, Amplitudes, polarity and time of peaks of strong ground motion during the 1994 Northridge, California, earthquake, Soil Dyn. Earthquake Engr., 16, 235-258, 1997; P. Bodin, S.K. Singh, M. Santoyo, and J. Gomberg, Dynamic defor-
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mations and shallow sediments in the Valley of Mexico, I: Three-dimensional strains and rotations recorded on a seismic array, Bull. Seis. Soc. Am., 87, 540-550, 1997. 208. Such short-range incoherence may be attributed to spatially varying site conditions, which cause seismic waves to be distorted from the plane waves. In the case of Pinyon Flat, F. Vernon, J. Fletcher, L. Carroll, A. Chave, and E. Sembera (Coherence of seismic body waves from local events as measured by a small-aperture array, J. Geophys. Res., 96, 11,981-11,996, 1991) concluded that coherence was reduced due to slight irregularities in depth of the weathered layer of granodiorite at the site. 209. For example, see Working Group on California Earthquake Probabilities, Seismic hazards in southern California: Probable earthquakes, 1994-2024, Bull. Seis. Soc. Am., 85, 379-439, 1995. 210. The Northridge earthquake had more than 20,000 recorded aftershocks. 211. Y. Okada, Internal deformation due to shear and tensile faults in a half-space, Bull. Seis. Soc. Am., 82, 1018-1040, 1992. 212. Several different probability distributions can be used to characterize earthquake recurrence, although the most common for time-dependent hazard analysis is the log-normal distribution. The most critical parameter in these calculations is the ratio of the standard deviation (s) of the interoccurrence times and the average of these times (Tave), or the coefficient of variation s/Tave. For low values of s/Tave, earthquake occurrence is almost periodic; high values indicate large variability in recurrence times. Present research is focused on determining the magnitude and physical origin of this coefficient for different regions of the world. 213. See Working Group on California Earthquake Probabilities, Seismic hazards in southern California: Probable earthquakes, 1994-2024, Bull. Seis. Soc. Am., 85, 379-439, 1995. 214. For a summary of the contents of this issue, see the introductory paper by N.A. Abrahamson and K.M. Shedlock, Overview of ground motion attenuation models, Seis. Res. Lett., 68, 9-23, 1997. 215. E.H. Field and the SCEC Phase III Working Group, Accounting for site effects in probabilistic seismic hazard analyses of southern California: Overview of the SCEC Phase III report, Bull. Seis. Soc. Am., 90, S1-S31, 2000. 216. Senior Seismic Hazard Analysis Committee, Recommendations for Probabilistic Seismic Hazard Analysis: Guidance on Uncertainty and Use of Experts, U.S. Nuclear Regulatory Commission, NUREG/CR-6372, Washington, D.C., 1997. 217. International Conference of Building Officials, Uniform Building Code, Whittier, Calif., 3 volumes, 1997; Building Seismic Safety Council, NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures, Part 1—Provisions, FEMA 302, Washington, D.C., 336 pp., 1997; Building Seismic Safety Council, The 2000 NEHRP Recommended Provisions for New Buildings and Other Structures, FEMA 368, Washington, D.C., 374 pp., 2000; International Building Council, International Building Code, Falls Church, Va., 756 pp., 2000. 218. At present, there is debate among proponents of this stochastic method as to whether the source spectrum is best represented by a Brune spectrum with a single corner frequency or by a model having two corner frequencies. See D.M. Boore, Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra, Bull. Seis. Soc. Am., 73, 1865-1894, 1983; G.M. Atkinson and D.M. Boore, Evaluation of models for earthquake source spectra in eastern North America, Bull. Seis. Soc. Am., 88, 917-934, 1998. 219. For example, see R.W. Graves (Simulating seismic wave propagation in 3D elastic media using staggered-grid finite-differences, Bull. Seis. Soc. Am., 86, 1091-1106, 1996) and K.B. Olsen (Site amplification in the Los Angeles Basin from 3D modeling of ground motions, Bull. Seis. Soc. Am., 90, S77-S94, 2000).
Representative terms from entire chapter: