Geochemistry, Geophysics, Geosystems

Nonlinear attenuation and rock damage during strong seismic ground motions



Strong seismic waves cause nonlinear behavior in the shallow subsurface in fractured rocks. Seismologists use low-amplitude signals from small repeating earthquakes to measure S wave velocity decrease after strong motion. The 2004 Parkfield, California, earthquake provides examples of such velocity changes in fractured sandstone with an S wave velocity of ∼300 m s−1. This nonlinear behavior occurred around the wave number depth of the incident waves, ∼30 m, for the ∼10 s−1 dominant angular frequency on a velocity seismogram. The low-amplitude S wave velocity gradually recovered with the logarithm of time. The attenuation of strong waves in general depends nonlinearly on their amplitude. High dynamic stress triggered small, very shallow earthquakes, at sites including Parkfield. The theoretical frictional behavior of a fractured medium with heterogeneous prestress relates these phenomena. Failure occurs in the highly prestressed domains causing small earthquakes and opening-mode cracks. The energy to dilate the cracks dissipates a significant fraction of the incoming seismic energy. The local high-porosity domains close with the logarithm of time, as expected from the aging law of rate and state friction, increasing the S wave velocity. The domain model indicates that nonlinear effects increase gradually over a range of dynamic Coulomb stresses as observed and as included in the widely used Masing rules. The Linker and Dieterich (1992) relationship provides the maximum sustainable dynamic coefficient of friction needed to utilize the Masing rules. This parameter is the coefficient of friction at a laboratory normal traction plus a constant ∼0.15 times the logarithm of the ratio of field normal traction to the laboratory normal traction. It is helpful to relate S wave velocity to starting frictional strength, as coefficient of friction near the quarter-wavelength depth determines nonlinear behavior. Then the dynamic coefficient of friction and equivalently the maximum sustainable acceleration at the dominant frequency depend weakly on S wave velocity. For example, the coefficient of friction at an angular frequency of 10 s−1 is less than 1.5 for rocks ranging from Parkfield sandstone to intact granite.

1. Introduction

Strong ground motions pose major hazards to structures and human life. Prediction of these motions helps society prepare for this eventuality. Seismology provides quantification of the motions. It is easiest to represent the Earth as an elastic though highly heterogeneous material. Then one can use records for many small events as Green's functions to mathematically construct the shaking for a large event. In addition, one can apply knowledge of the seismic velocity structure of the Earth to compute the shaking from kinematic and dynamic models of rupture.

Yet the behavior of the Earth is clearly not fully elastic during strong shaking. Comparison of the seismograms from weak and intense shaking indicates that attenuation becomes nonlinear at high amplitudes. Recent works include those by Frankel et al. [2002], Beresnev [2002], Hartzell et al. [2002, 2004], Bonilla et al. [2005], and Tsuda et al. [2006a]. Comparison of borehole and surface records indicates that the nonlinear response occurs in the shallow subsurface [e.g., Beresnev, 2002; Bonilla et al., 2005; Tsuda et al., 2006b, Lee et al., 2006].

Other numerous observations indicate that the S wave velocity of the shallow subsurface decreases in the aftermath of large nearby events [e.g., Rubinstein and Beroza, 2004a, 2004b, 2005; Peng and Ben-Zion, 2006; Sawazaki et al., 2006]. These studies utilize low-amplitude signals from small repeating earthquakes occurring before and after brief episodes of strong shaking. They hence measure changes in the low-amplitude (linear regime) shear modulus and seismic velocity. This concept differs from the change in the instantaneous dynamic shear modulus during strong shaking. Comparison of borehole and surface seismograms indicates that the seismic velocity changes are limited to a depth of ∼100 m [Rubinstein and Beroza, 2005; Sawazaki et al., 2006]. The process responsible for this decrease in velocity shares aspects of rock damage observed during laboratory frictional sliding experiments and predicted by rate and state dependent friction. Rapid shearing in the laboratory dilates gouge, which should decrease its low-amplitude seismic velocity. The low-amplitude S wave velocity decrease recovers to its prequake value with the logarithm of time after the event. Sheared laboratory gouge similarly compacts after sliding with the logarithm of time. (In contrast, material affected by liquefaction quickly (in hours) recovers its initial properties [Aguirre and Irikura, 1997].) The damage from one strong event seems to make the ground more easily damaged by subsequent events [Rubinstein and Beroza, 2004a]. Analogously the “starting” coefficient of friction of recently sheared gouge is less than that of gouge that has healed over a long time interval.

More limited data show that strong seismic waves trigger small high-frequency events in the shallow subsurface [Fischer et al., 2008a, 2008b]. Fischer et al. [2008b] point out that these events are likely to be both a form of nonlinear attenuation and a process that damages the shallow subsurface. Large events of this type may produce brief pulses of extreme acceleration [Sleep and Ma, 2008]. This stick-slip behavior and the analogous behavior of low-amplitude S wave velocity changes in laboratory gouge warrant application of the rate and state friction formalism to the behavior of the shallow subsurface during strong shaking.

We utilize field measurements of changes in low-amplitude S wave velocity to constrain the energy budget during strong shaking. We can estimate only the total diminution of energy as we have only the net change in the condition of the rock from before to after the seismic event. We strive for factor of 100.5 (or equivalently π) accuracy in our quantitative estimates. Specifically, the observation that strong seismic waves damage the ground implies that these waves lost energy via work against lithostatic stress and frictional heat as inferred above from direct observation of the attenuation of shaking. The purpose of this paper is to relate S wave velocity anomalies to nonlinear attenuation and to consider the physics of the process. A second goal is to predict physical limits on the maximum shaking from extreme seismic waves at (or near) the Earth's surface in terms of sustained acceleration.

We consider shallow materials that can be meaningfully represented as cracked rocks that fail in a macroscopically brittle manner. These include crystalline “hard” rocks and sandstones as in the Parkfield area. We do not consider soils or “soft” rocks that fail by macroscopically ductile creep or comminution. Liquefaction [e.g., Bonilla et al., 2005] is beyond the scope of this paper except for a brief caveats. Rock damage from trapped waves during fault zone rupture is a related phenomenon [e.g., Li et al., 2006], and is also beyond the scope of the paper.

We begin in section 2 by considering the kinetic and elastic strain energy carried by a propagating shear wave in terms of to describe dynamic stresses and strains. We use the ratio of dynamic stress to lithostatic stress to calculate the work done to open crack space and use this to establish a relationship between attenuation and rock damage. We then derive a relationship between changes in damage and changes in low-amplitude velocity, which provides a means to monitor the evolution of damage after strong ground motion.

In section 3, we test the validity of our analysis using the 2004 Parkfield earthquake as a case study and show that it is feasible to link nonlinear attenuation of strong ground motion and damage using the energy-based relationships developed in section 2. We also demonstrate that the predicted loss of energy at generic “hard” rock and “sandstone” sites is compatible with direct observations of nonlinear attenuation.

Having established relationships between attenuation, damage, and velocity, we then develop a physical basis for the observed phenomena by applying rate and state friction theory. In section 4, we derive equations that relate S wave velocity and damage to the evolution of the state parameter and nonlinear attenuation to dynamic stress changes. We conclude the section by showing that the dynamic stress-strain response predicted by rate and state friction is compatible with response described by the Masing rules for soil deformation.

In sections 5, we attempt to predict the field behavior described in section 3 using the rate and state formalism developed in section 4. While rate and state theory provides a physical basis for the observed relationships among nonlinear attenuation, rock damage, and velocity changes associated with strong ground motion, we had insufficient laboratory and field data to calibrate the equations. We conclude this section with a discussion of the types of data needed to fully calibrate and test the rate and state formalism.

2. Seismic Waves Near the Earth's Surface

The actual shaking in earthquakes is complicated. Starting with a full three-dimensional numerical model would serve only to obscure simple dimensional relationships. To delineate the physics and to define notation, we present basic well-known equations for seismic waves. We use Cartesian horizontal coordinates, x and y. The vertical coordinate z is depth.

2.1. Propagating Wave

Earthquake waves start out below the free surface of the Earth. For simplicity, we represent shaking with a monochromatic S wave moving vertically upward in the negative z direction,

equation image

where u is displacement polarized in the x direction, u0 is the maximum displacement, ω is angular frequency, t is time, and k is wave number. The S wave velocity is c = ω/k. The particle velocity is

equation image

where U0 is the maximum particle velocity. The particle acceleration is

equation image

where A0 is the maximum acceleration. The shear stress is

equation image

where G is the shear modulus. It is well known from (2) and (4) that shear stress scales with particle velocity.

One needs to keep track of energy to represent its attenuation. The strain energy per volume equals the kinetic energy per volume at every point in space and time,

equation image

where the equation for seismic velocity c = equation image where ρ is density confirms the equality [e.g., Timoshenko and Goodier, 1970, p. 491]. The kinetic plus strain energy per volume averaged over a wavelength is

equation image

The energy flux per area in the direction of propagation is thus cE.

2.2. Free-Surface Standing Wave

As shown in the remainder of this section, much of the rock damage occurs at shallow depths where the boundary condition at the Earth's surface has significant effects. As a practical matter, many structures and many seismographs reside at the surface. A standing wave provides a simple approximation for this part of the Earth,

equation image

Seismograms measure the velocity

equation image

The stress at depth causes inelastic deformation and hence attenuation

equation image

At shallow depths where kz < 1, the Taylor series expression yields dimensional approximations,

equation image

where we retain the magnitude of the dimensional quantities but not the phase. The depth 1/k is a natural basis for additional scaling relationships as stresses are closest to satisfying frictional failure criteria τ0 = μ0P = μ0ρgz (where P is confining pressure strictly on a horizontal plane, μ0 is the first order coefficient of friction, and g is the acceleration of gravity) in that region [e.g., Hartzell et al., 2004, p. 1614]. That is, the energy of a reflecting wave is kinetic energy near the free surface (that does not cause dissipation) and shear strain energy around the quarter wavelength depth π/2k. The dynamic stress reaches its maximum at the quarter wavelength depth and is a factor of 1/sin(1) = 1.19 greater than its value at the scale depth. Lithostatic stress continues to increase below the quarter wavelength depth while the dynamic stress is bounded by its value at that depth. The combination of significant shear-strain energy and dynamic stresses approaching the confining pressure imply that dissipation should occur around the scale depth if the rock is not fully elastic. We do not distinguish the quarter wavelength depth π/2k from the scale depth 1/k in dimensional results.

2.3. Energy Budget for Opening Cracks

Seismologists measure changes in the P-S delay from small repeating earthquakes. Damage to the rock decreases both P wave and S wave velocities. We use the measured change in P-S to represent to total change in S wave traveltime ΔtS. This procedure underestimates the actual change in S wave traveltime by a modest amount.

We estimate the change in rock properties from before and after the strong shaking and constrain the energy budget of this process. Before the shaking, the rock has material properties shear modulus Gbef, density ρbef, and porosity fbef and a S wave velocity of cbef. The properties measured some time after the strong shaking are Gaft, ρaft, faft, and caft, respectively. We express formulas in terms of the S wave velocity c, which is approximately known from site surveys, and the increase in porosity from strong shaking in porosity Δffaftfbef, which is needed to constrain energy.

Damage requires energy to open cracks against lithostatic stress and to slide cracks against friction. We explicitly consider the former effect in this section. The work per volume in both cases scales as strain times lithostatic stress at the scale depth 1/k,

equation image

where ɛ is strain. One obtains the porosity change (volumetric strain) Δf from traveltime variations as shown in this subsection. Thus it is convenient to represent work per volume in this way

equation image

where λ ≥ 1 is a dimensionless constant and Δf is dilatational strain.

Seismic velocity is a function of crack porosity f. We use a linear expression to obtain trial dimensional results. The approach applies without loss of generality as a Taylor series approximation over a limited range of porosities. See section 5.3 for application of percolation theory to represent shear modulus as a function of porosity in the full range from intact hard to highly fractured rock.

We expand the Taylor series about the conditions present at the site before shaking. Formally, the shear modulus is G = Gbef + Δf(∂G/∂f)bef. We compact notation with the linear expression,

equation image

where the shear modulus extrapolates to G0 at porosity 0 and to 0 at porosity γ. One can guess G0 from the shear modulus of uncracked rock at depth and the parameter γ at ∼20–30% from the porosity of highly cracked rock. Note that the expression applies only near the actual conditions in the rock and should not be interpreted that the shear modulus goes to zero. The traveltime variations measure the volume-averaged change in properties of the rock rather than the local damage within highly strained “domains” in the rock that actually fail in stick-slip.

We differentiate to obtain the change in low-amplitude seismic velocity with respect to porosity is

equation image

where the density of intact rock is ρ0. As the maximum porosity of rock that fails by cracking is much less than 1, γ ≪ 1, the second term involving the change in density can be ignored in a dimensional calculation.

The low-amplitude S wave traveltime across the easily damaged region scales with the dimensional depth 1/k. The traveltime across this region is

equation image

where DW is a dimensionless factor. It is π/2, for example, when the thickness of the damaged region is the quarter–wavelength depth. The change in traveltime that occurs over the fixed depth range of Dw/k is

equation image

where we use (13) to obtain the final equality and evaluate derivatives at ambient conditions before shaking. That is, the change in traveltime is proportional to the product of the original traveltime in (15), the porosity change, and a term that relates the change of shear modulus to porosity. Equation (16) has the testable feature that low frequency strong motions (at a given acceleration) cause damage over a great depth range and thus larger traveltime anomalies.

To return to a field situation, the seismologist measures the traveltime anomaly and the frequency of the strong motion. The near-surface seismic velocity and hence the shear modulus Gbef are known from site surveys. The computed value of Δf from (16) is thus proportional to γfbef. The parameters in (13) yield the observed near-surface shear modulus and hence constrain (γfbef)/γ if G0 is obtained from uncracked rock.

In general, γfbef is small for rocks having low S wave velocities and large ∼γ for nearly intact rocks. Thus, a small given porosity change implies large traveltime changes at stations underlain by low-velocity highly cracked rocks and slight traveltime anomalies at stations underlain by intact hard rocks γfbef.

Using the relationship in (12) between porosity change and work per volume yields that the traveltime change is

equation image

where k/ω = 1/c. This quantity is independent of the frequency of the seismic wave in a homogeneous half-space. However, a low-frequency, long-wavelength seismic wave in a real location tends to sample higher velocities at depth than a high-frequency wave.

Finally, we consider the energy arriving in the strong seismic wave. The length of the wave train scales with 1/k and which we express as DE/k where the dimensionless parameter DE represents the length of the wave train. It is 2π, for example, for a sinusoidal wave consisting of positive and negative pulses. Thus the energy in the arriving seismic wave is obtained by combining (5) and (17), and yields the fraction of the wave energy that goes into work to open cracks

equation image

in terms of measured quantities. It is thus unnecessary to know the depth Dw/k range over which damages occurs. We did need to specify, however, that the damage was concentrated near the scale depth 1/k.

Finally, the tendency of a material to fail in friction or dilation depends on the ratio of dynamic stresses to lithostatic stresses. This Coulomb ratio at (and above) the scale depth is

equation image

which is the measured ratio of particle acceleration to gravity and independent of material properties. This mechanical definition of strong motion in (19) is equivalent to the conventional definition that accelerations are on the order of that of gravity. This useful relationship is entrenched in work on the nonlinear attenuation of seismic waves. For example, Beresnev [2002] compiled amplification ratio as a function of peak ground acceleration.

Data from borehole seismometers at the Large Scale Seismic Test array in Taiwan provide appraisal of our inference that nonlinear attenuation occurs at depths scaling to 1/k. Lee et al. [2006] studied the strong motions from several earthquakes. They found that nonlinear attenuation occurred between 6 and 17 m depth. The S wave velocity increases gradually from 115 m s−1 to 234 m s−1 at 17 m depth. The dominant frequency of the signals is ∼10 s−1, so 1/k is within the interval where attenuation was observed.

Similarly, data of high-frequency energy bursts triggered by the ground motion of strong seismic waves from the Parkfield event are consistent with a shallow source. Events cannot be correlated between stations as close as 50 m [Fischer et al., 2008b].

2.4. Pulse-Like Arrival

Subtleties, however, arise in the application of (19). We derived this expression for a standing monochromatic sine wave. We now discuss the behavior with more realistic pulses using well-known results involving kinematic waves. We will show that pulse-like waves and sine waves produce similar dynamic stress to lithostatic stress ratios at the 1/k depth, which supports our use of the simplifying assumption of a sine wave in the previous analysis.

In general, a function for particle velocity satisfies the wave equation and the free slip boundary condition at the surface if it has the kinematic wave form,

equation image

where F is an arbitrary well-behaved function and the first term represents upcoming energy and the second downgoing energy. The dynamic stress is proportional to

equation image

where it is evident that the stress is zero at the surface z = 0.

The Brune [1970] velocity pulse illustrates one implication of (21). The particle velocity suddenly rises to its maximum value at the start of the pulse

equation image

where time is normalized so that the zero crossing is at t = 1. Despite the infinite acceleration at zero time, the pulse conserves the energy supplied by fault rupture.

Figure 1 compares the Brune pulse with a sinusoidal pulse with the same zero crossing and maximum amplitude. The peak dynamic stress from the sinuoid increases gradually with depth as expected from (9) and (10) (Figure 2). The dynamic stress from the Brune pulse is half that of the maximum from the sinusoidal pulse. This happens because the start of (22) is a step function. The maximum velocity from the upcoming wave reaches a point while the reflected downgoing wave is still zero. The gradient of dynamic stress with depth at the free surface from the Brune pulse is infinite but stress remains bounded. Equation (19) applies only at infinitesimal depth.

Figure 1.

The particle velocity of the Brune pulse and a sinusoidal pulse are normalized to have the maximum amplitude of 1 and a first zero crossing at the time 1.

Figure 2.

The dynamic shear traction is normalized so that the maximum for the sine wave in Figure 1 is 1. Depth is normalized to the half wavelength of the sine wave. The maximum is at the quarter wavelength depth.

We continue with the implications of more realistic pulses with brief high accelerations. Figure 3 shows a velocity pulse based loosely on the high acceleration record at station FZ16 from the Parkfield Earthquake [Shakal et al., 2006b] in Figure 4. The pulse is sinusoidal except for a brief period of high accelerations of 2.5 times that of the sinusoid near the zero crossing at t = 2 (Figure 3). The gradient of peak stress near the surface is 2.5 times that of the sinusoid as expected from (19) (Figure 5). The peak stress approaches that of the sinusoid at modest depths. Overall, high accelerations of brief duration do produce proportionally high ratios of dynamic stress to lithostatic stress, but only in the very shallow subsurface. The cause of such brief high accelerations is beyond the scope of this paper.

Figure 3.

Normalized velocity as a function of normalized time. The sine wave is modified to have high acceleration near the normalized time of 2.

Figure 4.

Northward particle (top) velocity and (bottom) acceleration from station FZ16 for the Parkfield earthquake, redrawn from the work of Shakal et al. [2006b]. The maximum acceleration (X) occurs between velocity trough C and velocity peak D. The period between peaks B and D is 0.43 s. The maximum amplitude range in velocity is 1.4 m s−1.

Figure 5.

The normalized dynamic shear traction for the sine and high-acceleration pulses in Figure 3 are plotted as a function of normalized depth. The high-acceleration pulse implies a high ratio of dynamic stress to confining pressure in the shallow subsurface.

3. Observed Damage and Strong Motions

The most intense ground motion of some moderate-sized events, like the 2004 Parkfield event [Shakal et al., 2006a], consists of a single pulse (Figures 6 and 7). In contrast, sedimentary basins, such as Palm Springs and Los Angeles, California, trap seismic waves. Ground motion involves repeated reverberations (Figure 8). A single pulse provides a simpler situation, but trapped waves are important hazards to structures. We discuss the effects of shallow structure and repeated reverberations in sedimentary basins in section 3.2.

Figure 6.

Eastward displacement as a function of time for the Parkfield earthquake recorded at station Cholame 2 West. Data from Shakal et al. [2006a]. Zero time is arbitrary.

Figure 7.

Particle velocity as a function of time for the Parkfield earthquake recorded at station Cholame 2 West. Motion is predominately horizontal. Data from Shakal et al. [2006a].

Figure 8.

Particle velocity as a function of time for a generic event recorded in a southern California sedimentary basin. The dominant period is ∼4.5 s and motion is mainly horizontal. Synthetic seismogram from Terashake.

3.1. Attenuation by a Wave Pulse and Its Energy Balance

We construct a generic example of a pulse with parameters appropriate for the 2004 Parkfield event [Rubinstein and Beroza, 2005] to show that a link between nonlinear attenuation of strong ground motions and damage is plausible with regard to energy. A precise treatment is intractable because seismologists currently measure low-amplitude S wave delays and strong motions at different sites. The seismic motion at station Cholame 2 West [Shakal et al., 2006a] is a brief dominantly S wave pulse (Figures 6 and 7). In round numbers typical of soft rock sites in the region, the dominant angular frequency ω is 10 s−1 and the maximum particle velocity of the seismometer is 0.5 m s−1. The maximum displacement is 0.05 m. The S wave velocity in the upper 30 m is ∼300 m s−1 at many sites. The scale depth k is 30 m, implying that VS30 in seismic station tables is an appropriate estimate of seismic velocity in the region where damage is likely.

The strong motion data set and traveltime data set of Rubinstein and Beroza [2005] come from different groups of stations. We use a typical S wave traveltime change of 0.007 s for generic numerical examples. We obtain the porosity from (13) with reasonable parameters. With forethought, we let the extrapolated failure porosity be γ = 0.24, the intact shear modulus be G0 = 20 GPa, and the density ρ = 2000 kg m−3. This yields that the porosity is 0.24–0.0216 = 0.23784. The low-amplitude velocity change ∼7% implies a porosity increase of 0.003 from (16). This would produce an uplift scaling to Δf/k of 0.009 m, which would be marginally observable.

We have enough information to obtain the diminution of energy from (18). The energy per volume in the wave from (5) is 250 J m2. The parameter DE = 2π as the wave train consists of positive and negative pulses (Figures 6 and 7). We do not know the parameter λ, the ratio of total strain to volumetric strain. We assume 2 to obtain trial results. The diminution of the upcoming energy is 0.23/(1 + 0.23) = 0.19. We use this quantity as a stable measure of nonlinear attenuation.

We repeat the calculation with the actual seismic velocity of 185 m s−1 at Cholame 2 West to appraise the sensitivity of the result to local seismic velocity. This yields γfbef = 0.00082, Δf = 0.000115, and a fractional energy change of 0.05. The diminution of the upcoming energy is 0.05/(1 + 0.05) = 0.05.

This computed range of fractional energy loss is compatible with direct observations of nonlinear attenuation. For example, Beresnev [2002] infers that the acceleration amplitude of strong waves to ∼0.6 g was reduced by a factor of 1.7 to 2 in amplitude at “soil” sites. One needs the area under a velocity-squared curve to estimate diminution of energy. However, the diminution of acceleration amplitude gives an upper limit of the diminution of velocity amplitude. This limit implies that remaining energy is 1/3 to 1/4 of the original energy and the energy diminution is 66–75%.

We also show that the additional porosity associated with damage is hard to detect at a hypothetical hard rock persists all the way to the surface. To show this, let the energy diminution in (18)DWW/DEE be 0.6. We let the shear wave velocity be 2000 m s−1 and retain the maximum particle velocity and the density. The porosity from (13) is 0.144. The traveltime delay from (18) is 0.00006 s, which is unobservable with real repeating earthquakes. Hence the lack of traveltime anomalies at hard rock stations does not imply that nonlinear attenuation does not happen beneath those sites.

3.2. Effects of Shallow Structure and Reverberations

The hypothetical basin event (Figure 8) is more complicated than the Parkfield event (Figures 6 and 7). Low-velocity sedimentary rocks in the basin trap seismic waves. In detail, the shaking is complicated and one cannot pigeonhole wave types in a three-dimensional medium. In terms of a laterally homogeneous medium, the waves are analogous to Rayleigh and Love waves, which produce deviatoric stresses at the free surface. The long-period energy propagates nearly horizontally within high-velocity rock at great depth. The raypaths refract in the low-velocity shallow subsurface. Hence, the shaking may be represented with nearly vertically propagating S waves that reflect repeatedly off the free surface and the particle velocity near the surface is horizontal. The latter wave type continues to provide a simple model. The dominant period is about 4.5 s, implying that ω = 1.4 s−1. The scale length 1/k is 7 times that for the Parkfield event.

Only the seismic energy that makes it into the shallow subsurface (around the scale depth 1/k) is subject to significant nonlinear attenuation in our model. Low seismic velocities near the surface reflect some of the upcoming energy back to depth sparing this reflected energy from attenuation. We consider two end-member cases to illustrate the effect: (1) The S wave velocity varies gradually so that all the upcoming energy arrives at the surface [e.g., Hartzell et al., 2004, Figure 1; Lee et al., 2006, Table 2]. The compiled average velocities of typical California rocks increase gradually with depth [Brocher, 2008, Figure 10], so we expect this situation is common at sites where there is no shallow discontinuity between different rock types. (2) There is a sharp discontinuity between shallow compliant rocks and a deep hard rock half-space.

Equation (6) provides the amplitude of the constant energy flux wave,

equation image

where the subscript 0 refers to the uppermost material and the subscript 1 refers to conditions in underlying hard rock. Solving, the shallow to deep amplitude ratio is

equation image

where ρc is the seismic impedance. The transmitted amplitude through a stepwise increase in velocity is

equation image

which is the free-surface result 2 in the limit of a very slow upper layer. Applying (23) and (24) gives the fraction of transmitted energy.

For an example, we assume a strong velocity contrast between 185 and 2000 m s−1. (Lee et al. [2006] use 2000 m s−1 and 1980 kg m−3 for their model half-space.) We keep density constant, as it does not vary a lot. The amplitudes of the constant energy flux wave and the wave through the interface are 3.29 and 1.83 of the incident wave respectively. The energy flux through the interface is 30% of the incident energy flux. We conclude that the detailed surface structure does affect the fraction of the seismic energy exposed to attenuation.

A second effect is that nonlinear attenuation reflects upcoming energy sparing some of it from further attenuation at shallower depth. Qualitatively, the nonlinear attenuation occurs over a depth range scaling to 1/k. This is between the abrupt interface limit where much of the energy reflects and the gradual-change limit where little energy reflects, implying a modest amount of reflection.

In practice, the site response of stations for small events constrains the fraction of deep upcoming energy that impinges the surface in the linear regime. Complications [e.g., Beresnev, 2002] based on surface shaking include both the effects of both linear and nonlinear reflection. Sorting out linear amplification effects due to velocity structure from nonlinear attenuation is complicated but manageable [e.g., Beresnev, 2002; Tsuda et al., 2006a, 2006b, Lee et al., 2006]. One can mathematically model nonlinear attenuation for a single pulse [e.g., Hartzell et al., 2004; Bonilla et al., 2005; Lee et al., 2006]. Numerically, one may represent a depth range with transmission, reflection, and attenuation within a stack of layers [e.g., Kramer, 1996, p. 176; Hartzell et al., 2004; Lee et al., 2006]. We do not attempt this onerous calculation.

In contrast, the downgoing wave is relevant to energy trapped within basins. It includes the energy reflected by nonlinear attenuation and energy reflected from the surface that attenuated on the way up and on the way down. The energy arriving at a site has been repeatedly exposed to nonlinear attenuation each time it reflected off the surface at various remote locations.

4. Application of Rate and State Friction

We estimated using (18) in section 3.1 that opening cracks dissipated a significant fraction of the incoming seismic wave energy at stations where S wave delay changes were observed. This reasoning is consistent with the inference that the energy to open cracks caused nonlinear attenuation.

We apply the rate and state friction formalism in this section to quantify the dissipation on the scale of cracks. Sites, like those around Parkfield that lie near a major seismogenic fault, have experienced strong shaking and hence local failure on cracks numerous times in the past. Stress concentrations thus exist near some crack tips and there are conversely stress shadows. The dynamic stresses from strong seismic waves cause local failure on crack surfaces with favorably oriented prestress, analogous to earthquake triggering. Failure tends to occur near the peak stress, causing small earthquakes in the shallow subsurface [Fischer et al., 2008a, 2008b].

These processes cannot be usefully modeled as linear attenuation that preferentially damps high frequencies. The purpose of this section is to quantify nonlinear attenuation as a form of triggering within the formalism of rate and state friction. We explicitly consider the key feature of rate and state friction that shear failure dilates cracks.

4.1. Rate and State Equations

The formalism for rate and state friction evolved to represent time-dependent friction in laboratory experiments. One measures (or controls) the macroscopic quantities of normal traction P (in general effective pressure, but we stick to drained conditions for simplicity), shear traction τ, and sliding velocity V (using the notation of Sleep et al. [2000]). We obtain testable hypotheses by applying the formalism to the ensemble of stress states in the shallow subsurface during strong ground motions.

Ruina [1980] first represented frictional creep in terms of the mesoscopic (averaged in time and in space over enough grains that a continuum is meaningful) parameter strain rate. That is, ɛ′ ≡ V/wf, where wf is the width of the sliding zone. This allows one to model strain rate localization. Here, it lets us consider a myriad of potential failure “domains” in a rock mass, motivated by the observed triggering of small earthquakes by strong seismic waves in the shallow subsurface [Fischer et al., 2008a, 2008b].

We summarize a unified theory of rate and state friction compiled by Sleep [1997, 2005, 2006a] and Sleep et al. [2000] to illustrate its implications and to define friction parameters. First, the coefficient of friction μτ/P does not vary much during steady sliding from a constant value for a given surface (Amonton's law). There are two second-order effects that arise in nonlinear attenuation. An instantaneous increase in the shear traction increases the sliding velocity. The coefficient of friction depends on the surface's history because sliding causes damage, which decreases the coefficient. The surface also “heals” allowing a steady state balance between damage and healing at constant sliding velocity. The state variable ψ represents these historical effects; its inverse is a measure of damage on a sliding surface. The unified rate and state friction equation retains these aspects of the traditional theory, while explicitly including the effects of varying normal traction; the shear traction is

equation image

where μ0 is the coefficient of friction at reference conditions, a and b are small dimensionless constants, the reference strain rate is ɛ0′ ≡ V0/wf, where V0 is a reference velocity, and ψnorm is a normalizing value for the state variable. We discuss evolution laws that represent the change of the state variable ψ over time and the relationship of ψnorm to changes in normal traction in Appendix A.

We express the state variable in terms of a directly observable property as we wish to model its change during strong ground motion. The state variable is empirically

equation image

where ϕ is a reference porosity and Cɛ is a dimensionless material property [Segall and Rice, 1995]. The aging evolution law (A1) then becomes

equation image

This expression has simple kinematic implications. The first term represents that the increase of porosity from dilatancy is proportional to the shear strain rate. That is, the dilatancy coefficient βCɛint. The second term represents power law compaction creep during holds where the exponent N = α/b.

It is also helpful to write the friction equation (26) as a flow law

equation image

The exponential dependence arises because creep at real asperities is thermally activated and scales exponentially with the real stress at contacts [e.g., Berthoud et al., 1999; Baumberger et al., 1999; Rice et al., 2001, Nakatani, 2001, Nakatani and Scholz, 2004]. Opening-mode cracks are analogous and fail by exponential creep at the crack tips [Beeler, 2004].

4.2. History of Single Domain and Its Ensemble

We combine (28) and (29) to represent the porosity history of a single domain with a given prestress (Figure 9) in a more or less intact rock,

equation image

where the second compaction term in (28) is negligible over the brief duration of seismic shaking. We show that the intuitive approximation of an ensemble of domains each with its own yield stress is reasonable. During the initial part of the strong motion the state variable is still near its prequake value. We explicitly treat a vertically propagating S wave on a horizontal surface so that normal traction stays constant. The rate and state equations are singular for negative normal traction, but the property that the initial creep on a contact under tension depends exponentially on this stress carries through as in the work of Beeler [2004].

Figure 9.

Schematic diagram of the interaction of a vertically propagating S wave with the free surface. (a) The stress has opposite polarity in the two parts of the incident pulse. (b) The stress in the reflected pulse and (c) the second part of the incident pulse add to produce (d) the maximum stress at the quarter wavelength depth. Local domains along preexisting and prestressed cracks fail around that time causing damage and nonlinear attenuation. The first half of the incident pulse is not subject to strong nonlinear attenuation. Modified after Sleep and Ma [2008].

With the inference that the shallow subsurface in a quake-prone region has experienced numerous episodes of strong shaking and crack failure, we invoke self-organized criticality where parts of the rock mass are near failure to constrain prestress. Each domain has a preexisting stress σ that scales to the lithostatic stress and a local value of the state variable. Retaining only the dominant terms with scalars, (30) becomes

equation image

where f0′ collects the constants and σ is the component of (prestress) traction acting in the direction of the dynamic stress τ on the domain surface. We consider the effect of heterogeneous state variable in later in this subsection.

As τ/P ≈ 1 near failure and a ≈ 0.01, small dynamic stresses have big effects. We represent the variation of dynamic stress with a single-frequency sinusoid for simplicity. We use a Taylor series to consider behavior near the time of the peak stress when significant creep occurs

equation image

Without loss of generality, time is defined so that this peak stress is reached at time 0. The total porosity change in the local porosity during a wave cycle is

equation image

where the limits of the quickly converging integral are extended to infinity to obtain a closed-form solution. The square root term accounts for the effect that peak stresses (within a factor of e of the maximum) occur over only a small part of the wave period. The exponential term in (33) dominates. The predicted dilatation is either tiny or astronomically large except over a small range of dynamic stress. Large slips and dilations cannot occur, however, as modest inelastic strains relax the dynamic stress within a domain and transfer it to other domains in the cracked rock. Once finite slip begins, the state variable decreases, further weakening the contact, but macroscopically, the rock cannot attenuate more energy than that supplied by the wave.

We consider the ensemble of domains with a distribution of prestress. Without loss of generality, we define the leading constant f0′ in (33) so that failure of a domain occurs at a failure stress

equation image

The failure stress depends on the preexisting state variable distribution in the rock.

Overall the failure process is complicated. Elastic modulus and prestress heterogeneities are likely causes of localized absolute tensile tractions from macroscopic dynamic stresses, leading to opening mode failure of some domains (Figure 10). To illustrate this effect, we represent an imaginary plane in the ground as a large number of springs (domains) acting in parallel between parallel surfaces. The total force on the domains divided by the area of the plane is the macroscopic stress. This normal traction is approximately the lithostaic stress as the ground behaves as a deformable fluid over long periods of time. A passing SH wave causes extension moving 45° dipping surfaces apart. The macroscopic normal traction is still less than the lithostatic stress. The local normal traction becomes somewhat more tensile on the compliant domains and reaches absolute tension on the least compliant ones. These regions of local tension may then fail transferring their stress to other domains. Prestress increases the effect. Some domains may have low initial compressive stress. The least compliant of these domains are the first to reach absolute tension. The weakest of this subset of domains fail first.

Figure 10.

Springs between two rigid plates schematically represent stresses in shallow rocks. (a) Lithostatic pressure loads the sample. All the springs are in compression and support the load. Dynamic stresses reduce the stress between the plates and the plates move about. The stiff spring comes under tension and will fail if its tensile strength is exceeded.

We obtain tractable expressions for attenuation assuming a fractal distribution of the prestress σ in Appendix B. Using (B3) for the inelastic strain energy yields that fractional attenuation is dimensionally

equation image

where there is no nonlinear attenuation at the limit of zero dynamic stress. We dimensionally denote attenuation as Q−1, the familiar form of the inverse of the quality factor.

The diminution of local seismic energy in the arriving seismic wave W/E in (33) is the physical quantity of interest. The fractal cumulative distribution (στ0)−1.5 is unbounded as its argument approaches zero. Thus (35) is unbounded as the driving stress τ0 approaches the failure stress σf. However, the number of domains (that is, the volume) of a region is finite. Attenuation cannot dissipate more energy than is present in the original wave.

We thus define a dynamic stress τ1 where the available energy in a seismic wave would be dissipated if this shaking with this maximum stress continued over a full wave cycle. It is approximately the dynamic stress above which a further increase in the amplitude of the upcoming wave produces little increase in the amplitude observed at the surface. It is not evident how to be more rigorous in a generic situation. In general, strong nonlinear attenuation decreases the amplitude of the remaining wave sparing it from further strong attenuation. As already noted in section 3.2, nonlinear attenuation causes dynamic stresses to differ from those obtained by considering a nonlinear medium.

The parameter ratio τ1/σf gives a simple representation of the variation of nonlinear attenuation with peak dynamic stress. If τ1/σf is close to 1 significant attenuation occurs over a small relative stress range (Figure 11). If τ1/σf is sufficiently less than 1, attenuation occurs over a broad range of stress. The latter case has the desirable aspect that attenuation Q−1 is a weak function of dynamic stress over a large range (Figure 11). This feature is compatible with the observation that seismologists frequently detect S wave velocity changes after strong motion. In contrast, failure at a single prestress approaching σf would leave either no damage or pulverized rock in its wake.

Figure 11.

Fractional attenuation of incoming seismic energy is a function of the normalized dynamic stress τ0/τ1. The label on the curves is ratio τ1/σf, where σf is the failure stress, and τ1 is the dynamic stress at which the available energy is fully attenuated. Attenuation increases over a range dynamic stresses if this parameter somewhat less than 1.

4.3. Comparison With Data and Soils-Engineering Formalism

Attenuation over broad ranges of stress is compatible with direct observations of strong motions. Beresnev [2002] obtained detectable attenuation (diminution of acceleration amplitude) at ∼0.2 g and strong attenuation to ∼0.7 energy loss at 0.6 g at soil sites. Lee et al. [2006] detected nonlinear attenuation beginning between 0.04 to 0.0075 g. In practice, any smooth curve for nonlinear attenuation that approaches zero with low to zero slope at zero stress and increases rapidly at high stresses serves to capture the phenomenon. Beresnev's [2002, Figure 4] use of a logarithmic plot implies an exponential dependence.

Soil engineers often model nonlinear attenuation with extended Masing rules [e.g., Hartzell et al., 2004; Lee et al., 2006]. The Masing rules empirically describe macroscopic stress-strain behavior similar to that predicted by the rate and state formalism discussed in sections 4.1 and 4.2. One plots stress τ versus strain ɛ. In the limit of low stresses and strains the material is elastic, τ = Gmaxɛ. The stress reaches a maximum τmax at high strains (Figure 12). On loading, the stress-strain curve is

equation image

where the function Mback is known as the backbone curve [e.g., Kramer, 1996]. Lee et al. [2006] use an empirical curve with these limits from the work of Chang et al. [1990]. Loading continues until reaching a strain rate reversal at τr, ɛr. Upon unloading, the stress strain trajectory is given by

equation image

which implies that the instantaneous shear modulus at the start of unloading is the initial elastic value,

equation image

That is, the stress concentrations that fail on loading are different than those that fail on reverse loading. Additional rules represent multiple cycles. The stress-strain curve follows the initial backbone curve after it intersects it and it follows any previous curve with the same sense of strain rate that it intersects. Dissipation of energy is proportional to the area of the hysteresis loop on a stress-strain plot. The fraction of the supplied energy that is diminuted is, however, sensitive to how the backbone curve approaches the maximum stress. The hyperbolic curve implies that the available energy is dissipated in the limit of very high strains. The empirical curve from the work of Chang et al. [1990] approaches ∼22% damping in the limit of high strains [Lee et al., 2006].

Figure 12.

Stress-strain curves illustrate Masing rules. The sample starts at the origin. Initial elastic deformation is parallel to the thin line. Loading follows the (dark) backbone curve to point A where unloading begins. Initial deformation is elastic and parallel to the thin line. The direction of stress reverses and the stress-strain curve intersects the backbone curve near point B where the sense of deformation again reverses. The attenuation turning the hysteresis loop from B to A and back to B is proportional to the green shaded area. Stress is normalized to τmax and strain is normalized to τmax/Gmax.

One can use laboratory and field data to select the maximum stress τmax. Lee et al. [2006] use τmax = Gmax/2000, which accounts for the inference that more elastically compliant material should be weaker. Hartzell et al. [2004] use a Coulomb failure criterion.

Overall, the predictions of rate and state friction and Coulomb-based Masing rules are similar especially with regard to the net diminution of energy. The Masing rules, unlike rate and state friction, are not time-dependent. This matters little for strong shaking as we ignored the frequency dependent term in estimating energy loss using rate and state friction in (33). The dynamic stress τ1 where attenuation consumes the available energy and the maximum stress that the rock can sustain σf have implications similar to the Masing parameter τmax.

The Masing rules, however, do not bear on the slow healing of fractured rock following strong shaking. In particular, one needs to consider evolution laws in Appendix A to represent time-dependent changes of low-amplitude S wave delays after strong shaking and the effect of previous strong shaking ∼1 day to years ago on current nonlinear attenuation of seismic waves.

4.4. Application of the State Variable to Transient Traveltime Anomalies

Rate and state formalism lets one represent the time dependence of traveltime anomalies and the failure stress in terms of observable quantities. The former bears on our assumption of domains.

We assumed in subsection 4.2 that the shallows subsurface fails locally within the domains during strong shaking. The rest of the rock remains intact. The porosity change Δf in (16) obtained from low-amplitude S wave traveltime anomaly is the macroscopic average. The local porosity change within failed domains ΔfL in (33) is relevant to their healing.

We apply the aging law (28) and (A1) as they apply in the laboratory to sandstones similar to those at the Parkfield sites [Hagin et al., 2007]. (The slip law gives the inapplicable null result of no compaction and no seismic velocity change.) Referring to the evolution law for state (A1), the second term on the right-hand side is zero when the surface is not sliding (analogous to a laboratory hold) ɛ′ = 0 during an intraseismic period. The first term on the right hand side is a constant if the normal traction P remains constant. We compactly represent this term as tN, as it has dimensions for time. The state variable ψ is then dimensionless and is ψafttaft/tN (where taft has dimensions of time and is estimated in the subsection) immediately after strong shaking. The state variable increases to ψposttaft/tN + tpost/tN in the postseismic time tpost.

We apply the feature of rate and state friction that the strain in a failed region is greater than the intrinsic strain ɛint in (A1) to quantify the state ψaft. The state variable approaches equilibrium ψss = ɛint/ɛ′tNtseis/tN where tseis is the duration of seismic slip. This duration is comparable to the duration of peak stresses ∼0.1 s at Parkfield. Its precise value is unimportant as it is much less than the postseismic time tpost where seismologists study the low-amplitude arrivals from small repeating earthquakes. These events are unobservable during the coda of strong shaking, which continues at least for several seconds.

These assumptions yield an expression for the post-seismic change in local porosity,

equation image

where the reference porosity ϕ is set (without loss of generality) to the immediate postseismic value immediately after the shaking tpost = 0, and the porosity in the failed domain decreases by ΔfL in the postseismic period. The result is independent of the parameter tN. The macroscopic porosity decrease is the sum of the decreases over the failed domains. From (16), the traveltime change thus scales to the logarithm of time as observed.

As already noted, failed domains slide extensively over a time scaling with the period of the wave, ∼0.1 s for the Parkfield generic example. The interval between strong shakings is ∼30 years or 109 s. The ratio of these times is 1010. Most of the change occurs quickly after the earthquake. That is, one fourth of the change occurs in the first 101.5 s (∼30 s), half of the change occurs in the first 104 s or (∼3 h), and three fourths occurs in 106.5 s (∼1 month). Studies of postseismic S wave delays [e.g., Rubinstein and Beroza, 2004a, 2004b, 2005; Peng and Ben-Zion, 2006; Sawazaki et al., 2006] thus underestimate immediate postshaking properties (the S wave velocity change, the porosity change, and the work to open cracks) by a factor of ∼2. One can crudely account for this effect by extrapolating S wave delays back to the duration of shaking.

The damaged domains remain weak for a considerable time after the strong shaking. For illustration, we define strength from (26) as the shear traction needed to drive slip at a strain rate inverse to the duration of strong shaking. With this definition from (26) and (27), the increase in strength in a failed asperity is proportional to the porosity decrease

equation image

Rubinstein and Beroza [2004a] detected S wave delays from the 1990 Chittenden earthquake series in 1990 in California. These events followed the nearby Loma Prieta event by ∼6 months (107.2 s). Using the porosity changes from the previous paragraph, 18% of the original strength reduction remained at that time.

Equation (39) provides an estimate of the porosity change within failed domains. The rate and state parameter Cɛ was 0.0020 in an analysis of sandstones by Hagin et al. [2007]. (Sleep et al. [2000] obtained 0.0034 for gouge.) Using the sandstone parameter, state variable increases by a factor of 1010 over the postseismic period and the porosity decreases by 0.046.

In contrast, we do not obtain a logarithmic decrease in seismic velocity and porosity with time if we assume that the change in porosity uniformly distributed within the rock. For example, the macroscopic change in porosity inferred from low-amplitude S wave traveltime variations is 0.0003 in one of our examples. Assuming that Cɛ is ∼0.0020 produces a fractional change in the state variable equal to 15%. In terms of (39), the postseismic state variable taft is only slightly less than its typical ambient value tpost. The logarithmic dependence of porosity on time would not become evident until times exceeding the interseismic interval ∼30 years and thus is not observable in the immediate aftermath of strong shaking. Note that small seismic events triggered by dynamic stresses [Fischer et al., 2008b] cause local rather than uniform rock damage.

5. Porosity, Percolation Theory, and Failure

We return to rock physics to constrain the maximum expected sustained accelerations from seismic waves and to discuss the difficulties in obtaining an improved relationship between S wave velocity and porosity change. From the work of Beresnev [2002], it is evident that nonlinear attenuation increases from too small to detect to removing significant energy of the incident wave over a factor of 3 of peak ground acceleration, in his case from 0.2 to 0.6 g. Our parameterization has this feature. Taken at face value, we cannot expect to see sustained ground accelerations far above the threshold for nonlinear attenuation, whether measured directly or inferred from later S wave delays.

It is also evident that our micromechanical model implies that attenuation depends on heterogeneous material properties and on heterogeneous prestress. Some calibration is thus necessary, both in the field and the laboratory. Neither studies of strong motions or later S wave delays are ideal, as they require both a strong earthquake and a seismic station at the point of interest before action can be taken. This difficulty is especially true for waves that reverberate everywhere from the surface of basins. In section 5.1, we obtain direct inferences from rate and state friction. Then we present a method that uses near-surface S wave velocities.

5.1. Failure Stress

For engineering applications, it is both convenient and practical to concentrate on the maximum dynamic stress that can be sustained in the shallow subsurface. From (19), this yields the maximum sustained acceleration. The Masing rules parameter τmax in (36) has this property if a Coulomb criterion is assumed [e.g., Hartzell et al., 2004]. So does the flow law equation (29) for shear creep. Our treatment based on an ensemble of domains represents this stress with σ1 in Figure 11. We have already noted in section 4.2 that a significantly high dynamic stress will bring most of the domains in a rock into shear failure.

The evolution equation (A1) and the friction equation (26) bear on frequently damaged soft rock. We select properties appropriate to sandstone under lithostatic stress [Hagin et al., 2007] to provide examples relevant to the Parkfield, California, region. We extrapolate from the laboratory compressive stress range of 30–100 MPa to the field range of 0.1 to 1 MPa (5–50 m depth). Without loss of generality, we here let the reference strain rate be fast enough that failure actually occurs on the timescale of 1/ω or ∼0.1 s. The reference coefficient of friction is a typical laboratory value of 0.7. We let b = 0.015 and α = 0.15. We presume the neutral stability criterion approximately holds so that ab.

These parameters provide enough information to obtain the strength of ambient rock. From (39), the reference state variable at the time of failure scales with 1/ω and the state variable in the damaged domains increases to the intersesimic time, ∼30 years. The ratio (109 s/0.1 s) is ∼1010. Letting b = 0.015, the coefficient of friction in (26) in the damaged domains increases by 0.345 from 0.7 to 1.045. This value is not sensitive to the interseismic time. For example, the coefficient of friction in domains damaged 3000 years ago (1011 s) is 1.114. The results are also not sensitive to the strain rate that we define as failure. For example, changing the failure strain rate by a factor of 100 changes the coefficient of friction in (26) by 0.069.

In general, strong shaking changes the normal traction P as well as the shear traction τ. The parameter α accounts for this effect from sudden but not instantaneous changes. The flow law for shear strain (29) implies that the rate of frictional sliding remains constant when Δτ = (μ0αP rather than the simple frictional result Δτ = μ0ΔP. This result, however, does not apply for a sufficiently rapid change in normal traction. The dynamic equations of rupture between two dissimilar materials are well posed only if normal traction is continuous in time [Ranjith and Rice, 2001; Perfettini et al., 2001]. That is, αμ0 at the limit of an instantaneous change of ΔP ≠ 0 (implying Δτ → 0). See Hong and Marone [2005] for friction experiments on oscillating normal traction. We assume constant normal traction to obtain an upper limit on strength. Coulomb-favored planes with low normal traction will from the discussion in this paragraph be somewhat weaker than constant normal traction planes.

The effects in the previous paragraphs have effects similar to those of cohesion C where τ = μ0P + C. (1) The initial coefficient of friction is somewhat above that for steady sliding in the laboratory. (2) Rapidly decreasing normal traction does not immediately greatly decrease the shear traction during sliding.

5.2. Calibration Using Experiments on Gouge and Failed Rock

One would like to infer the maximum dynamic stress at a site from laboratory or field measurements that can be made in advance of strong ground motions. Our rate and state treatment uses porosity as a measurable parameter.

From (26), the shear traction from friction is not strongly dependent on strain rate. As in (40), we assume a reference strain rate appropriate for rapid failure in the laboratory and in the shallow surface during strong shaking. For the field, one would like to obtain the failure properties of a rock at a given depth and ambient lithostatic stress PL. One can measure the ambient porosity fbef. From (26), (A4), and (27), gouge sliding in steady state at the reference velocity would have this porosity at the normal traction Pss, defined by

equation image

where the steady state porosity is ϕ at normal traction Pss. The shear traction at normal traction PL is

equation image

The “starting” coefficient of friction of near zero porosity intact rock provides an upper limit for the strength of rocks with finite porosity. (Intuitively, cracked rock is weaker than intact rock.) This stress if sustained would pulverize a rock. We apply (42) and laboratory data on intact rock to obtain an upper limit for near-surface ground strength (Figure 13). The strength of intact rock approaches that of gouge at Pint = ∼1500 MPa [Lockner, 1995; Sleep, 1999], which we let be Pss in (41) and (42). We assume a density of 2500 kg m−3 and shear modulus of 40 GPa to evaluate (42). The initial coefficient of friction of hard rock at 0.1 MPa (4 m depth), 1 MPa (40 m depth), 10 MPa (400 m depth), and 100 MPa (4 km depth) from (42) is 2.14, 1.80, 1.45, and 1.11, respectively. The shear tractions are 0.214, 1.80, 14.5, and 111 MPa, respectively. The S wave velocity is thus 4000 m s−1. The coefficient of friction at scale depth 1/k (400 m) with ω = 10 s−1 is relevant to nonlinear attenuation in the hypothetical case that the hard rock extents all the way to the surface.

Figure 13.

(top) The initial coefficient of friction of intact rock increases as normal traction decreases. The point of intersection (square) of the coefficients of friction of intact rock (red) and sliding gouge (green) determines the shape of the curves. The coefficient of friction of broken rock is expected to lie between that of intact rock and gouge (shaded area). (middle) Fractured surface rock has the porosity of sliding gouge at 20 MPa. Rocks with higher porosity have lower initial coefficients of friction and fall in the shaded area. (bottom) The initial shear traction as a function of normal tractions for intact rock and rock with the porosity of the surface rock in Figure 13(middle). Tangents to the curves project to finite shear traction at zero normal traction.

One gets a better limit on strength by extrapolating from experiments on the rock present at the site of interest (Figure 13). This requires inferring the steady state porosity of gouge as a function of normal traction to calibrate (41). The normal traction Pss of this gouge is less than Pint for intact rock but is greater than the ambient confining pressure, because as already noted, the near surface has experienced lithostatic stress. As noted in section 5.1, the “starting” coefficient of friction is at least ∼1.0 for rock that was damaged in the previous strong seismic event and healed for tens to thousands of years. We return to the topic with actual samples in section 5.4.

The mathematic singularity in (42) does not imply a physical singularity. The shear traction is (42) goes to zero as the normal traction PL → 0 (note equation image [PL ln(PL)] = 0). However, real data taken at finite normal traction extrapolates to positive shear traction at zero normal traction (Figure 13). That is, it gives the appearance of cohesion.

5.3. Calibration Using S Wave Velocity

Shear velocity provides a means for determining the strength of both intact rock and gouge that can be measured remotely from the surface. The shallow S wave velocity parameter VS30 is widely tabulated which we use to obtain the scale depth 1/k. This exercise provides an illustrative application of our theoretical framework.

Using S wave velocity is attractive. Creep and failure properties of rock in rate and state friction in (27) and (40) and the shear modulus in (13) both depend on porosity. The state variable has the properties of a nonlinear ductile stiffness given by ψ for compaction in (28) and ψb/a for shear in (29). Percolation theory provides a formalism to represent the effect of porosity on the state variable over the range of porosity from intact rock to gouge [Sleep, 1999]. The goal is to represent the shear modulus G(f) from intact rock f = 0 to failure with a single expression. In contrast, the elastic shear modulus in (13) is adequately expressed as a linear function of porosity only over a limited range of that parameter.

Incorporating nonlinearity into the relationship G(f) between shear modulus and porosity illustrates the analogy with the state-porosity relationship ψ(f) and is more realistic for strong ground motion. Intuitively, stiffness should vary linearly with porosity near zero and have power law dependence as a critical porosity is approached:

equation image

the stiffness of intact rock with modulus GI extrapolates to zero at f = ϕl, and the stiffness approaches the (power law) percolation theory limit in the approximate equality (on the right-hand side) at the critical porosity ϕc (Day et al. [1992], typo corrected). The term in the brackets needs to be positive for the equation to have physical meaning. This occurs for f < ϕc for ϕ1 > ϕc/2M. The formula is merely a convenient heuristic mathematical form with the limiting behavior and should not be regarded a rigorous [Krajcinovic, 1997]. In general the behavior near the critical porosity cannot be obtained from the low-porosity limit [e.g., Jusiak et al., 1994].

We confine discussion to the case where grain-grain contacts bear lithostatic stress so that the shear modulus is finite. Equation (43) then suffices as it is intended to provide numerical examples in this paper. It may well be inapplicable for obtaining the shear modulus during liquefaction (beyond critical porosity) and it certainty should not be applied to fragmented rock that has been ballistically ejected from the surface.

Continuing percolation theory, the exponent M in theory depends on the exponent N = α/b of the rheology of the material. Krajcinovic [1993] provides the expression

equation image

where χ depends on the geometry of the cracks. It is 0.85 for a three-dimensional lattice. The power for linear elasticity is N = 1, implying M = 3.70. The parameter ϕl is somewhat less than the critical porosity [Jusiak et al., 1994]. The computed stiffness does not depend strongly on this parameter (Figure 14).

Figure 14.

(a) The shear modulus normalized to that of intact rock is a function of porosity normalized to the critical porosity. The curves are labeled with the ratio ϕ1c and depend modestly on that parameter. (b) The green-shaded area in Figure 14a is enlarged for visibility. The curves approach the axis at the critical porosity (normalized porosity of 1) with zero slope. However, the slopes of the curves for finite shear moduli are a factor of ∼7 less than that in the linear model. They extrapolate to normalized shear moduli of ∼1/7 at zero porosity.

We discuss behavior of rock with the slow S wave velocity of 185 m s−1 (with an intact seismic velocity of 4000 m s−1) to bring the linearization (13) into context. The ratio of stiffness to intact stiffness is (∼(185/4000)2 = 0.002). The porosity extrapolates to 0 giving the linear parameter γP at slightly less than the critical porosity. The stiffness extrapolated 0 porosity GP is a factor of ∼7 less than the intact stiffness or 40/7 = 5.7 GPa.

One needs the linear parameter λfbef to evaluate traveltime changes in (17) and the attenuation in (18). We estimate the uncertainty that arises from our lack of knowledge of this parameter. In practice, we will know the seismic velocity of the rock and hence its stiffness Gbef at in situ conditions. Both the linear parameters that we used to evaluate (13) and the extrapolations from percolation theory must yield this stiffness.

equation image

We also know that the porosity, as a function of stiffness, extrapolates to zero in the linear model γ, and the percolation model result is only modestly greater than the observed porosity. The denominators in (45) are thus approximately equal. We assumed that G0 = 20 GPa in section 3.1. Using the percolation theory value of GP = 5.7 GPa for G0 in (17) and (18) implies that we replace γfbef with the parameter γPfP. From (45) the ratio (γPfP)/(γfbef) is 20/5.7 = 3.5. We underestimated the amount of energy diminution by that factor in section 3.1 if percolation theory in fact gives a better representation of the dependence of shear modulus on porosity than the simple linear formula (13). Overall, (13) is not likely to give gross errors in the observed range of seismic velocities for shallow rocks.

With regard to friction, the measured power law exponent N = α/b for sandstone in compaction is ∼10 [Hagin et al., 2007]. The percolation exponent M is thus 11.35. The computed “state” modulus is quite small over a wide range of porosity (Figure 15). It is approximately linear on a semilog plot over a broad range of porosity (Figure 15). Equation (27) is the linear representation, which holds for sandstone in the laboratory [Hagin et al., 2007]. The state variable then does not go exactly to zero at a critical porosity.

Figure 15.

The normalized shear modulus in percolation theory represents the state variable. The curves are labeled with the ratio ϕ1c and depend quite weakly on that parameter. The relationship is log linear over a large range of porosity. The linear extrapolation may represent the state variable in rock and gouge.

As a practical matter, we need both shear modulus and state in terms of porosity to obtain the work per volume involved in increasing pore space. We do not need to explicitly consider porosity if we only want to obtain failure properties from S wave velocity including the dependence of strength or porosity in (40). The number density of tabular cracks is a more fundamental parameter that provides this relationship [Krajcinovic, 1993; Mallick et al., 1993]. It is Γ = f/Ω, where Ω is the short/long aspect ratio of the cracks. Equation (13) then becomes

equation image

where the shear modulus becomes zero at Γ0. The percolation theory formula (43) may be similarly modified with the critical crack density approximately equal to 2. The porosity state equation becomes

equation image

where ΓR is a reference crack density and CΓ = Cɛ/Ω. One can eliminate the crack density the shear modulus equation (46) and the state equation (47) to obtain an expression of the form

equation image

and hence use S wave velocity to predict instantaneous strength.

5.4. Example Calculations With Intact Rock

In principle, one can use the porosity-shear modulus relationship in (43) and the rate and state equations (41) and (42) to obtain the failure stress at the scale depth 1/k using the intact rock limit where Pss = Pint and ϕ = 0. As noted in section 5.3, this procedure should yield essentially the same results as one using the crack number density Γ.

The percolation formula for shear modulus (43) has five independent parameters. The independent parameters ρ0 = 2500 kg m−3 and GI = 40 GPa are reasonably well constrained. We let the percolation exponent M have its theoretical value of 3.70. This leaves the percolation coefficients ϕc (which is measurable on samples) and ϕl (which is poorly constrained unless we have data on low-porosity hard rock). The shear wave velocity is then c = equation image, where ρ0 is the density of zero porosity rock. The lithostatic stress at the scale depth 1/k is

equation image

Thus the weak dependence of density on 1 − f affects both calculation of pressure and seismic velocity.

There are three independent parameters in the rate and state friction equations (41) and (42). The normal traction Pint = 1500 MPa where intact rock behaves like gouge is reasonably well constrained, especially since the computed shear traction depends logarithmically on that parameter. We let the parameter Cɛα/b = 0.002. Hagin et al. [2007] measured Cɛ for sandstone and the ratio a/b is ∼1 in laboratory experiments. The main restriction is that this measurable parameter and the assumed critical porosity ϕc = 19.23% should lead to failure in the porosity range at normal tractions relevant to the shallow subsurface. We let α have its measured value for sandstone 0.15 [Hagin et al., 2007]; in another calculation, we assume its measured value for hard rock 0.21 [Sleep, 1999].

This procedure yields the “starting” (similar to the intact) coefficient of friction of rock at the scale depth 1/k and the shear wave velocity as parametric functions of porosity (Figure 16). From (19), these curves provide an estimate of the maximum acceleration that can be sustained without greatly damaging the rock as the upper limit for Coulomb failure. They presume vertical SH waves acting on horizontal planes. As already noted, the optimal failure plane for all waves is a surface with reduced dynamic normal traction.

Figure 16.

The starting coefficient of friction at the scale depth 1/k from percolation theory is a function of S wave velocity. The curves depend on the (labeled) percolation parameter ϕ1. Three curves have α = 0.15 and one curve has α = 0.21. The curves satisfy the minimum limit of starting friction below (μ0 = 0.7) for gouge.

The curves are smooth and do not vary a lot, except for unphysically small values of the shear wave velocity. The weak dependence arises from (42). The parameter Pss approaches Pint at high S wave velocity and the lithostatic pressure PL in (49) is evaluated at a large depth scaling to the high S wave velocity. A low S wave velocity decreases both Pss and PL. The logarithm of the ratio Pss/Pint in (42) does not change much. This smooth variation provides motivation for continuing with the approach.

However, the curves do depend significantly on the rate and state parameter α and the percolation porosity ϕl. In particular, one can obtain the unphysical result that the rock at lithostatic pressure is weaker than gouge (the coefficient of friction < μ0) with seemingly reasonable parameters (Figure 16).

5.5. Application Using Laboratory Data

Our use of intact rock at ∼1500 MPa in section 5.4 as a standard state involves major extrapolation to the shallow subsurface. It is thus convenient to use laboratory experiments on the failure of rocks at lower normal tractions. We spot check and illustrate our method with data on the Darley Dale sandstone [Ayling et al., 1995; Read et al., 1995]. We also check our inference that the strength of a rock depends on its S wave velocity. We select this sandstone because data are available and because similar rock types underlie several stations in the Parkfield area.

That is, we apply (42) where shear traction at failure τlab is measured at normal traction Plab, which is near the ambient normal traction of field interest. The coefficient of friction at failure at other normal tractions is expected to be

equation image

which has the advantages that the logarithmic term is small if the two normal tractions are comparable and the parameter α can be measured from other experiments.

Ayling et al. [1995] and Read et al. [1995] measured the stress at failure of the Darley Dale sandstone under uniaxial compression. This rock is a “hard” sandstone with some equidimensional porosity. The rock failed in shear for normal tractions less than 100 MPa and by collapse of pore space at higher normal tractions. We use only the data where shear failure occurred. We plot the true coefficient of friction and normal traction on the failure surface in Figure 17. The laboratory value of α = 0.15 [Hagin et al., 2007] provides a good fit to the data.

Figure 17.

The starting coefficient of friction for the Darley Dale sandstone as a function of confining pressure [Ayling et al., 1995; Read et al., 1995]. The Linker and Dieterich [1992] relationship (50) allows explanation to lower confining pressure. The coefficient of friction near the scale depth (for an angular frequency of 10 s−1) is relevant to the transmission of extreme seismic waves and approximately maximum sustainable acceleration. The starting coefficient of friction of more cracked (and lower S wave velocity) rock lies in the shaded region below the curve.

Ayling et al. [1995] and Read et al. [1995] also measured the S wave velocity as failure commenced (Figure 18). Their results confirm our inference that instantaneous coefficient of friction depends on S wave velocity. There is too much scatter to determine the form of the velocity coefficient of friction curve. A linear relationship would suffice within the actual range of data.

Figure 18.

The coefficient of friction for the Darley Dale sandstone during failure as a function of the S wave velocity normalized to the S wave velocity of intact rock [Ayling et al., 1995; Read et al., 1995]. An eyeball line is shown for reference. It extrapolates to a reasonable coefficient of friction for intact rock (normalized velocity equals 1) but extrapolates to coefficients of friction below that of gouge at moderately reduced S wave velocities. These data are probably affected by strain localization where the measured velocity includes both damaged and intact domains.

The data may be affected by strain localization. That is, the measured S wave velocity is an average of weakly and strongly damaged parts of the rock. Data on wellbore collapse would not suffer from this difficulty. One can measure the seismic velocity of pristine rock away from the borehole. One can constrain the ambient stress state. This exercise is beyond the scope of this paper.

6. Discussion on Dilatancy

From section 3, rock failure that produces S wave traveltime anomalies occurs in the shallow subsurface when dynamic stresses are comparable to lithostatic stress as is assumed in some dynamic models [e.g., Hartzell et al., 2004]. From (28), the energy per volume to open cracks is the product of the shear strain, the dilatancy coefficient β, and the confining pressure, ɛβP. The energy per volume dissipation in shear is the product of shear strain, the coefficient of friction, and confining pressure ɛμ0P. The coefficient λ = β/μ0 in (18) is the ratio of these energies.

The laboratory dilatancy coefficient for mature gouge in the aging law (28) is ∼4% [Sleep et al., 2000; Sleep, 2006b], implying that λ = 17.5. We assumed that λ = 2 with some forethought to estimate the fractional energy dissipation in section 3.1. We obtained 0.23 and 0.05 in two generic examples. The percolation theory discussion in section 5.3 indicates that these ratios should be increased by a factor of ∼3.5 to 0.81 and 0.18. This range indicates significant but reasonable dissipation of the incoming energy.

In contrast, we obtain unreasonable estimates of energy dissipation if we use the laboratory value of λ = 17.5 as we need to increase the above estimates by a factor of 17.5/2. This result indicates that the laboratory value of the dilatancy coefficient is inappropriate for cracking in the shallow subsurface.

We briefly discuss the physics of dilatancy to put this result into context. The rate and state observation that rock dilates when sheared at constant normal traction may seem counterintuitive to some readers.

The common observation that if one breaks an object it does not fit back together applies. On all scales, material deforms inelastically in a ductile manner before it fails in a brittle manner. Local brittle and ductile failure leave residual stresses and strains once stresses (here from strong seismic motion) relax. The residual strains prevent the object from fitting back together. The residual stresses do work to expand porosity against the confining pressure. Sleep [2006b] discussed residual stresses in gouge and showed that the dilatancy coefficient is approximately half the ratio of the real stress yield stress to the stress modulus of the grains.

Bonilla et al. [2005] discussed kinematic dilatancy of rolling grains in a granular material. For example, rolling of a square block initially dilates a region. The stress and dilatational strains are comparable so that λ ≈ 2. However, there is no net dilatation if the block rolls many times as would occur in a gouge [Sleep, 2006b].

Opening-mode cracks have the attractive property that they do not produce shear strain so that λ ≈ 1 in (17). The derivation in section 4.2 for shear cracks carries through for opening-mode cracks in a heterogeneous material with heterogeneous prestress. Crack tips fail in exponential creep [e.g., Beeler, 2004]. As in (33), the predicted amount of strain in a domain at a given stress is either infinitesimal or huge. This feature gives rise to the useful approximation of a stress intensity factor at crack tips analogous to the failure stress in (34).

However, opening-mode fractures differ from shear fractures in that the ground is continually subjected to gravity and hence to lithostatic stress. The prestress of compaction thus everywhere opposes dynamic tension (Figure 10). For example, a vertically propagating P wave produces a vertical acceleration at the surface. A loose object on the surface does not get thrown into the air unless the acceleration exceeds the ambient gravity 1 g. Damage as observed by S wave arrival time variations and nonlinear attenuation, however, occurs at well below 1-g accelerations and with predominately horizontal motion.

Rayleigh waves and nonvertically propagating body waves do produce absolute tensile traction on vertical planes near the free surface. This effect, however, is unlikely to be the source of damage in low-velocity material as incident vibrations from depth refract into nearly vertically propagating waves. In any case, seismic stations do not detect damage right at the free surface as their piers extend downward to finite depths.

We continue with dilatational processes that might be observed on an outcrop to thin section scale. Opening-mode cracks occur at local stress concentrations associated with shear failure. This process is equivalent to the concept that residual strains associated with local shear failure prevent the rock from fitting back together. With regard to the occurrence of small, triggered events [Fischer et al., 2008a, 2008b], tail cracks dilate at the end of shear cracks. Conversely, small “transform” shear stepovers may unlock opening-mode cracks.

For completeness, crushing of pore space by high compressive stress during the passage of the seismic wave is also an unattractive failure mechanism. As noted in section 4.4, porous rocks compact under lithostatic stress. The ambient compaction rate in (28) is less than the compaction rate by a factor of ψ which scales to the ratio the interseismic time to the shaking time 1/ω, that is ∼1010. As N = α/b in (28) is ∼10 for sandstone [Hagin et al., 2007], the dynamic compressional stress would have to be ∼10 times the lithostatic stress. From (19), this requires an acceleration of ∼10 g. Conversely with this gross extrapolation, compressional collapse provides an upper limit on the amplitude of seismic waves that can propagate through porous rock, but not a useful one for earthquakes.

7. Conclusions

We began with two nonlinear processes that accompany strong seismic shaking. (1) The attenuation of strong seismic waves increases nonlinearly with their amplitude. (2) Strong shaking damages the shallow subsurface decreasing the low-amplitude S wave velocity. This damage heals gradually over time. The observation that strong seismic waves trigger small earthquakes in the shallow subsurface links these processes and permits construction of a quantitative physical model. Ground failure occurs within domains rather than uniformly throughout the rock. We usefully represent this failure as a Coulomb frictional process and the rheology of the rock with rate and state friction.

The well-known equations of seismic waves reflecting off the free surface indicate that the nonlinear processes occur at depths scaling to the inverse of the wave number 1/k of the dominant frequency of the seismic wave or equivalently with the quarter wavelength depth. The length 1/k provides a scale for a continuum. That is, weak or prestressed domains that are of a common scale govern nonlinear phenomena. These include places with little tensile strength and/or cohesion. The scale 1/k indicates the likely depth of shallow earthquakes driven by dynamic stress. Data are compatible with this prediction but do not mandate it [Fischer et al., 2008a, 2008b].

The ratio of dynamic stress to lithostatic pressure scales with the sustained dynamic acceleration at the dominant frequency on a velocity seismogram. It is already standard practice to use this parameter as a measure of strong motion.

One would like to include the effects of nonlinear attenuation in predictions of shaking by large earthquakes. We discussed methods that apply to site response and others to reverberating seismic waves that repeatedly pass through the shallow subsurface. Methods can attempt to extrapolate upward from modest nonlinear effects observed in moderately strong shaking or downward from the limiting strength of the subsurface.

The domain model indicates that nonlinear effects increase gradually over a range of dynamic accelerations as observed and as included in the Masing rules. For example, Beresnev [2002] showed an increase from too small to detect to removing significant energy of the incident wave from 0.2 to 0.6 g. Thus detection of S wave velocity changes or triggered shallow seismic events indicates nonlinear attenuation occurred and that somewhat stronger seismic waves would have been strongly attenuated. Triggered events unlike seismic velocity changes are potentially observable with hard rocks.

One can use the in situ properties of rocks to estimate their Coulomb failure properties during dynamic stress. This exercise yields a maximum stress that would pulverize the rock. We represented this quantity as σf in our model of domain failure and as τmax in the Masing rules. Strong attenuation occurs well below this dynamic stress within weak and prestressed domains.

We obtained an estimate of ∼1 for the starting coefficient of friction of domains that failed in the last earthquake from rate and state friction. We used rate and state friction to extrapolate the starting coefficient of friction of hard rock and sandstone to the in situ lithostatic stress in the shallow subsurface. We obtained ∼1.5 and ∼1.3, respectively.

Both the strength from the state variable and the S wave velocity depend on the amount of crack-like porosity within the rock. It appears promising to estimate state and hence the starting coefficient of friction from field measurements of shallow S wave velocity. Limited laboratory data did not suffice for this task.

Overall, the feature that the ratio of dynamic stress to lithostatic stress scales with sustained dynamic acceleration provides simple intuition for site response. Frequently shaken “soil” sites experience strong attenuation below an acceleration of 0.7 g, representing typical steady state coefficient of friction of 0.7. Sustained accelerations of over 1 g do occur, but nonlinear attenuation makes sustained accelerations of 2 g highly unlikely.

The calculation of nonlinear attenuation for reverberating waves is not simple. The energy passes through the surface producing its highest ratios of dynamic stress to lithostatic stress. In addition at a given time, much of the energy may reside at moderate depths where the ratio of dynamic stress to lithostatic stress is around the threshold for nonlinear attenuation, ∼0.2. It is unclear whether strong shallow attenuation or deeper modest attenuation dominates. We do not attempt relevant sophisticated calculations.

Appendix A:: Summary of Rate and State Friction Evolution Laws

We need to represent both damage during strong motion and postseismic healing. The aging or Dieterich [1979] evolution law represents these effects explicitly with separate terms

equation image

where the first term represents healing and the second represents damage. The variable t is time, the intrinsic strain is ɛintDc/W, where Dc is the critical displacement to significantly change the properties of the sliding surface, α is a dimensionless parameter that represents the behavior of the surface after a change in normal traction from Linker and Dieterich [1992], and P0 is a reference normal traction.

The slip or Ruina [1983] evolution law gives the combines the effects of healing and damage in one term

equation image

This law predicts no healing as ɛ′ → 0. It cannot represent postseismic S velocity changes. Physically, this relationship presumes that the stresses at real contacts scale to the macroscopic ones and that the material deforms mainly in shear [Sleep, 2005, 2006a]. The latter assumption does not apply to the shallow subsurface deforming under lithostatic stress. (A. Ruina and J. Dieterich (personal communications, 2006) prefer the descriptive terms “slip” and “aging,” for clarity.)

The state variable reaches a steady state value at constant strain rate in both evolution laws.

equation image

If the steady state coefficient of friction is independent of normal traction,

equation image

Equations (A4) and (1) represent the effects of the sudden (but not instantaneous) changes of normal traction on friction [Linker and Dieterich, 1992].

Appendix B:: Fractal Stress Distribution

We apply a fractal theory for prestress in fractured rock in the shallow subsurface developed for an analogous purpose. Marsan [2005] obtained that the cumulative “number” N of domains with stress greater than σ scales as σ−1.5 in a medium with seismically active faults. We modify this relationship to account for the inference that stresses greater than the failure stress σf are unlikely, as they would have already caused failure during the intersesimic interval between strong ground motions. The cumulative number of domains is hence proportional to

equation image

The cumulative number of domains that fail for dynamic stress τ0 is

equation image

Each domain does work scaling to the acting stress τ0 and produces an inelastic strain scaling to that quantity. The work per volume is thus proportional to

equation image

This relationship has one physical parameter σf and is not sensitive to the fractal exponent −1.5.


This research was in part supported by NSF grant EAR–0406658. This research was supported by the Southern California Earthquake Center. SCEC is funded by NSF Cooperative agreement EAR-0106924 and USGS Cooperative agreement 02HQAG0008. The SCEC contribution number for this paper is 1217. Adam Fischer and Charles Sammis provided preprints of their work and helpful discussion at the 2007 SCEC meeting and e-mail correspondence. We thank an anonymous reviewer and Ralph Archuleta for critically reading the paper.