The S-wave velocity in the shallow subsurface within seismically active regions self-organizes so that typical strong dynamic shear stresses marginally exceed the Coulomb elastic limit. The dynamic velocity from major strike-slip faults yields simple dimensional relations. The near-field velocity pulse is essentially a Love wave. The dynamic shear strain is the ratio of the measured particle velocity over the deep S-wave velocity. The shallow dynamic shear stress is this quantity times the local shear modulus. The dynamic shear traction on fault parallel vertical planes is finite at the free surface. Coulomb failure occurs on favorably oriented fractures and internally in intact rock. I obtain the equilibrium shear modulus by starting a sequence of earthquakes with intact stiff rock extending all the way to the surface. The imposed dynamic shear strain in stiff rock causes Coulomb failure at shallow depths and leaves cracks in it wake. Cracked rock is more compliant than the original intact rock. Cracked rock is also weaker in friction, but shear modulus changes have a larger effect. Each subsequent event causes additional shallow cracking until the rock becomes compliant enough that it just reaches Coulomb failure over a shallow depth range of tens to hundreds of meters. Further events maintain the material at the shear modulus as a function where it just fails. The formalism provided in the paper yields reasonable representation of the S-wave velocity in exhumed sediments near Cajon Pass and the San Fernando Valley of California. A general conclusion is that shallow rocks in seismically active areas just become nonlinear during typical shaking. This process causes transient changes in S-wave velocity, but not strong nonlinear attenuation of seismic waves. Wave amplitudes significantly larger than typical ones would strongly attenuate and strongly damage the rock.
 Construction engineers need to know the probability that a given level of shaking will be exceeded during the expected lifetime of a structure. The likelihood of infrequent extreme ground motion is essential in the design of structures, including nuclear waste depositories, which must persist safely over long time frames [e.g., Hanks et al., 2006; Andrews et al., 2007]. An empirical approach is to examine the levels of past shaking. Instrumental records are typically inadequate, as they cover much less than an earthquake cycle in most areas. There are two related geological approaches: (1) past failure of a “fragile” geological structure indicates that shaking exceeded a certain level at least once; and (2) the survival of the structure indicates that this threshold of shaking did not occur over the duration of its existence. Both classes of observations provide information on the statistical recurrence of shaking at a given location, provided that the geological duration of vulnerability can be constrained.
 The purpose of this paper is to treat the uppermost tens to few hundreds of meters of surface rock as a fragile geological feature. Individual masses of rock remain near the surface with the potential for damage for long periods of time particularly in desert regions [Brune, 2001; Hanks et al., 2006; Anderson et al., 2008; Sleep, 2010a; Girty et al., 2008; Sleep, 2011; Replogle, 2011]. Seismic failure leaves cracks in its wake (Figure 1). This damage makes the rock elastically compliant, reducing the dynamic shear stress in strain-driven deformation. Over time the shallow seismic velocity evolves so that regolith barely becomes nonlinear during typical strong shaking. The results of work reported here constrain past peak ground velocity (PGV) and may be compared with the precariously balanced rocks studies ofAnderson et al.  and are applicable where such rocks do not exist.
2. Stresses Within Seismically Damaged Regolith
 Observations indicate that seismically damaged regolith exists near major faults. Brune . Wechsler et al.  and Dor et al.  studied the tendency of repeatedly damaged rock near faults to easily erode more easily by rain and runoff than distal intact rock. McCalpin and Hart  note that ridge tops in the San Gabriel Mountains of California resemble rock fractured by blasting. Girty et al.  speculated that large volumetric strains within saprock near the Elsinore fault were largely due to dilation produced by ground shaking. Notably Replogle showed that such strains appear to decrease relative to near-fault areas within the band of precariously balanced rocks studied byBrune et al. .
 I consider locations where the effect of ground shaking where strain boundary conditions at depth provide a reasonable representation of the net effect of impinging seismic waves on the shallow subsurface. Specifically, finite dynamic shear stresses occur all the way to the free surface for Love and Rayleigh waves. Near-field velocity pulses are more complicated, but the formalism for Love and Rayleigh waves provides guidance. In contrast, the shear traction approaches zero at the free surface for vertically ascending S-waves. The Coulomb ratio of shear traction to lithostatic normal traction scales with the dynamic acceleration.
 With forethought to section 5, I begin with long-wavelength Love wave followingBullen and Bolt . Cartesian coordinates are x1 in the horizontal direction of propagation, x2 horizontally in the direction of propagation, and x3 ≡ −Z vertically upward, where depth Z is measured from the free surface throughout this paper. A monochromatic Love wave produces a displacement of
where UL is the peak dynamic displacement, k is the wave number, t is time, and ω is the angular frequency. The peak ground velocity is VL = ωUL. Horizontal displacement and velocity vary continuously with depth. At shallow depths Z ≪ 1/k, the peak dynamic displacement, the peak dynamic velocity, and the dynamic shear strain ∂U2/ ∂x1 have approximately their values at the free surface.
 The peak dynamic shear stress depends on the shear modulus at depth G = ρβ2 where the (interval) shear wave velocity β and the bulk density ρ may vary with depth. In terms of convenient parameters,
where cL = ω/k is the phase velocity of the Love wave. The analogous expression for peak horizontal dynamic tension τ11 from a Rayleigh wave is
where the second Lamé constant λ may vary with depth, VR is the peak dynamic horizontal velocity, and cR is the phase velocity of the Rayleigh wave.
2.2. Shallow Near-Field Dynamic Shear Stress
Equation (2)with minor modification is applicable to mainly horizontal velocity pulses in the near field of strike-slip faults. Shallow damaged regolith may grade into flower structures of near-fault and fault zone damage [Ma, 2008; Finzi et al., 2009]. I concentrate on regolith damage for brevity.
Makris and Black  provided an applicable construct. Their Type A records have a single velocity pulse that has one polarity, and that leaves a static displacement in its wake. Makris and Black  cite station LUC near the 1992 Landers California earthquake with PGV of ∼1.5 m s−1. Pump House 10 for the Denali earthquake is similar with PGV of ∼1.8 m s−1 [Ellsworth et al., 2004]. High dynamic velocities persisted for well over a second.
 The phase velocity of the Love wave is then approximately the S-wave velocity of the underlying hard rock half-spaceβH [e.g., Bullen and Bolt, 1985, p. 115]. The peak dynamic shear strain is εD = VNF/βH, where VNis near field PGV velocity. The dynamic shear stress in the underlying half-space isτH = ρβHVN. The dynamic shear stress is the regolith is
where the subscript reg indicates regolith properties.
 I use a simplified Coulomb ratio between dynamic shear stress and lithostatic pressure at a given depth Z. This criterion makes it unnecessary to include geometrical terms for the orientation of cracks relative to dynamic stress. This criterion reduces to the Drucker and Prager  criterion of Love waves that do not change the mean compressive stress. Failure occurs when this Coulomb ratio exceeds the coefficient of friction
where g is the acceleration of gravity, and μreg is the local coefficient of internal friction for the regolith. The second equality assumes that density does not vary much with depth. If pore fluid is present, ρ is replaced with the difference in between bulk and fluid density ρ − ρf in the right hand side and the factor ρ/(ρ − ρf) appears in the middle term. I omit these factors to compact dimensional relationships.
 Failure in terms of given PGV occurs at depths where S-wave velocity is sufficiently great to satisfy
where the second bracket is the inverse of the dynamic shear strain. The singularity at zero dynamic shear strain implies that regolith remains linear for feeble shaking and reasonable S-wave velocities. S-wave velocity at failure depends on the square root of the coefficient of friction.
Anderson et al. considered an alternative failure criterion involving absolute tension in regolith. For horizontal S-waves, the maximum dynamic tension is equal to the dynamic shear stress and 45° from the plane of shear. The failure criterion is
where C is the tensile strength at zero pressure. I use this formulation in models, but do not algebraically expand equations derived from it in the text.
3. Failure Within Seismically Damaged Regolith
 The failure criteria as in (2), (3), (5), (6), and (7)imply self-organization at sites where strong shaking is frequent and evolution starts with exhumed intact rock(6), the imposed dynamic shear strain εdyncauses failure within the stiff hard rock down to considerable depth. Each failure event increases the density of cracks in the rock, reducing its shear wave velocity. Eventually, the S-wave velocity becomes low enough that the dynamic shear stress barely reaches the failure criterion. Little additional damage occurs in subsequent events and the shear modulus remains at its critical value, which increases with depth in(6). Macroscopic cracks and volumetric strain allow groundwater to percolate through the rock mass increasing chemical weathering [Replogle, 2011]. This effect increases that rate that S-wave velocity evolves in the regolith. I do not specifically analyze it.
3.1. Rate and State Friction Rheology
 To apply (6), it is necessary to have frictional failure criteria for both intact and cracked rock. I apply the well-known construct of rate and state friction and consider three simple cases: (1) Intact rock failure criterion gives an upper limit on rock strength. Cracks reduce the S-wave velocity but do not change the failure criterion. (2) Crack friction controls failure, giving a lower limit of strength. The density of cracks affects seismic velocity but not strength. (3) I then consider the intermediate case of pulverized rock, where the local porosity controls both frictional failure and seismic velocity. I relegate it to Appendix A, as I do not have suitable field measurements of S-wave velocity. I do not consider the failure of granular material by crushing as would occur from strongly compressive P waves. CAP and critical state theory would be applicable [e.g.,Karner et al., 2008].
Sleep [1997, 2006] and Sleep et al.  compiled the rate and state theory in terms of strain rate. The instantaneous shear traction τ as a function of normal traction P is then
where the dominant term μ0 ≈ τ/P represents the approximation that the coefficient of friction has a constant value for a given surface (Amonton's law), a and b are small ∼0.01 dimensionless constants, ε′ ≡ V/W is the frictional strain rate, V is sliding velocity, W is the width of the sliding zone, ε′ref is a reference strain rate, and ψ is the state variable that includes the effects of healing and damage, and the ψnorm is the normalizing value of the state variable which represents transient variations of normal traction and the behavior of intact rocks.
 The aging evolution law represents the state variable ψ in terms of the past history of the fault. This law has kinematically explicit terms [Dieterich, 1979]
where the first term represents healing and the second damage. The variable t is time, εint is the strain 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 and is from the work of Linker and Dieterich , and Pref is a reference normal traction that may be selected for convenience. The steady state value of the state variable in (9) is
If the coefficient of friction at a given strain rate is independent of normal traction, the normalized value of the state variable is
3.2. Intact Rock Failure
 Experiments provide calibration for the failure of hard rock (Figure 2) and sandstone (Figure 3). I follow Sleep [2009, 2010a] and Sleep and Hagin . The stress at failure is not strongly dependent on the frictional strain rate in (8) so it is unnecessary to know the precise frictional strain rate value. Furthermore, the term μ0 − aln (ε′ref) is a constant in (8). It thus can set the reference strain rate to the actual frictional strain rate at failure to eliminate the a term and implicitly adjust μ0 without loss of generality. The state variable is ψhard, a value appropriate for intact rock, again without having to know its precise value. Then (5) and (8) become
where the coefficient of internal friction at the reference normal traction is μref. Application of this Linker and Dieterich  equation involves measurement of the starting coefficient of internal friction over a range of normal tractions. The parameter α is ∼0.30 for granite (Figure 2) and ∼0.15 for hard sandstone (Figure 3). In both cases, it is necessary to extrapolate from experimental conditions to low normal tractions present within shallow rock. The coefficient of friction in Figure 2 for granite extrapolates to 2.894 (retaining extra digits for comparison) at 1 MPa at 40 m depth for a density of 2500 kg m−3. Its value for hard sandstone in Figure 3 extrapolates to 1.491 at 1 MPa at 50 m depth for a density of 2000 kg m−3. The coefficient of friction at 10 MPa normal traction is 2.203 and 1.452, respectively.
3.4. Cracked Rock Failure
 I use the evolution equation (9) to simply calibrate the starting friction of cracked rocks in the field. Strong seismic shaking with crack damage occurs episodically. Interseismically, the strain rate in the second term of (9) is ∼0. Furthermore, the steady state value of the state variable in (10) for cracks that fail is quite small just after shaking. Thus, the state variable at the start of renewed shaking in (9) is proportional to the ts since shaking
Laboratory gouge in Figure 2 provides a proxy for rapidly slipping cracks. I again let the strain rate at rapid slip be the reference strain rate without having to know its precise value. The steady state coefficient of friction is ∼0.8 at low normal tractions. Equation (8) becomes
The duration of strong shaking time of seconds or less provides a constraint on the ta ≡ εint/ε′ref time of significant slip. The seismic interval ts is decades or longer. It is not critical to know the precise ratio of the times. I assume 1010-1013 and a = 0.005–0.015 for an example calculation to obtain 0.961–1.249, again retaining extra digits, and use the rounded value of 1 in later calculations. As expected, these values are less than the computed starting coefficient of intact hard rock and intact sandstone at shallow depths.
 Some care is needed in defining the effective normal traction and stress in pervasively fractured regolith. Centimeter scale contact occurs only on a small fraction of the surface area of joints, cracks, and bedding planes. The ratio of normal to shear traction scales with the macroscopic value, but the centimeter-scale stresses are much greater than the macroscopic ones and hence conveniently for purposes of this paper nearer the values where friction is typically measured in the laboratory as inFigures 2 and 3.
4. Cajon Pass Field Example
 I use sandstone near Cajon Pass as an example in this section, as data are available and as near-field velocity pulses are likely to produce strong shaking. Insection 5, I analyze shallow damage from Love waves in sedimentary basins as another well-characterized example. I note that shallow stiff rock occurs in seismically active areas. For example, andesite with S-wave velocity of ∼1700 m s−1 occurs beneath 7 m depth near Tehran [Shafiee and Azadi, 2007]; S-wave velocities in hard Paleozoic sediments over 3000 m s−1 occur at a few 10 m depth near St. Louis [Williams et al., 2007]. However, the type of seismic waves that might produce the strongest shaking is less obvious in these regions than in the Californian localities that I model.
 Due diligence observation of sandstone near Cajon Pass by the author in September 2011 indicated that macroscopic cracks are in fact absent as reported by Anderson et al. . There are a few (∼1-cm wide) mineralized joints. Subvertical probably tectonic faults occur on the road cut approximately perpendicular to bedding on the north side of SR 138. The largest had apparent offset of ∼4 cm and the smallest ended upward at a bedding plane. With regard to another class of fragile geological structure, there are hard rock clasts to over 30 cm in diameter that might act as stress concentrations. I found no macroscopic cracks in the surrounding sandstone nucleated on these features.
Anderson et al.  measured the interval seismic velocity of sandstone near Cajon Pass, using ambient noise (Figure 4). The S-wave velocity has a nominal precision of 10%. The analysis method yielded piecewise constant interval velocity with depth. The velocity steps may not represent real interfaces, so it is not of great concern that failure criteria predict smooth velocity variation with depth. Lines L101 and L102 (34.31454 N 117.46626 W) overlap. Line L103 (34.31861 N 117.50412 W) is nearby. The overall data represent the general increase of velocity with depth.
 The nearby San Andreas fault has repeatedly shaken this rock. That is, the rock has experienced thousands of near-field velocity pulses. I apply my formalism for intact rock to these data, as the rock appears massive in outcrop (Figure 4), to get a maximum estimate of its strength from the sandstone behavior in Figure 3. I use intact rock S-wave velocity of 2000 m s−1 and density 2000 kg m−3 from Anderson et al.  for calculations. For comparison, I represent Cajon Pass sandstone as a cracked rock with a constant of coefficient of friction of 1 (Figure 5) and the tensile failure from (4) (Figure 6).
 All three similar failure criteria provide reasonable representations of the increase of seismic velocity with depth. Following Anderson et al. , high S-wave velocities indicate the survival of fragile material over geological time scales. Conversely, processes including weathering unrelated to earthquakes can decrease seismic velocity. In agreement withAnderson et al.  the rock has not experienced particle velocities in excess of 2 m s−1 and may not have even experienced 1 m s−1near-field pulses. Overall, the data support the concept that S-wave velocity self organizes so that regolith barely becomes nonlinear, but cannot distinguish between model failure criteria.
 The models with zero cohesion in Figure 4 and 5 do not fit the finite measured seismic velocity at very shallow depths. Equation (7) reduces to τR = ρcR2ε = Cat the free surface. The predicted cohesion for observed surface S-wave velocity of ∼500 m s−1 is 0.25, 0.5, and 0.75 for MPa dynamic shear strains of 5, 10, 15 × 10−4, respectively. The model with 1 MPa of cohesion in Figure 6a predicts excessive velocity at very shallow depths. The cohesion of 0.5 MPa gives in Figure 6b a better fit.
 Alternatively, damage cannot decrease the seismic velocity of sandstone below that of loose sand in the presence of gravity and hence not all the way to 0. Winds would quickly remove loose sand from protruding outcrops at Cajon Pass. Anderson et al.  placed their instruments on intact outcrop. Intact parts of the outcrop likely have the cohesion measured by Anderson et al.  on samples that remained intact all the way to the laboratory. Flaws within the outcrop may well fail at very low normal tractions. There are locally thin 1–2 cm layers of weathered saprock at outcrops, but they would not affect the seismic experiment or friction at depth.
 It is thus unclear whether very shallow apparent cohesion indicates real cohesion at depths of tens of meters. In any case, modest cohesion at depth has a mild effect of computed shear velocity at failure. Compare Figures 5, 6a, and 6b that share the same coefficient of friction 1, but have different cohesions.
5. Love Waves Within Deep Sedimentary Basins
 Deep sedimentary basins, including Los Angeles and Palm Springs, trap surface waves from large earthquakes on the nearby San Andreas fault. Numerical simulations predict PGV of ∼1.5 m s−1 [Graves et al., 2008; Olsen et al., 2009] and ∼2.5 m s−1 [Graves et al., 2011]. It is feasible to include a well-posed representation of nonlinear attenuation in these codes [Graves, 1996; Graves et al., 2008, 2011]. I proceed by presenting data and then discuss the behavior of accumulating alluvium.
5.1. Basin Data From California
 Generic modeling of rock damage in California basins using (2) to obtain stress and strain is feasible, as strongest shaking is essentially Love waves [Joyner, 2000; Olsen et al., 2006]. I follow Sleep [2010b] using the underlying velocity structure near Whittier Narrows from the SCEC Community Modeling Environment [Graves et al., 2008] and obtain Love wave phase velocities using code by Herrmann . The Love wave phase velocity for 3 s and 4 s period waves is 1190 and 1532 m s−1, respectively. I use the rounded value of 1500 m s−1for example calculations. Phase velocity at these long periods depends weakly on the detailed S-wave velocity of the shallow subsurface.
 The exhumed sediments are relatively stiff. The observed compiled interval S-wave velocity corresponds to failure at a coefficient of friction of 1 and PGV of 1.5 m s−1 (Figure 7). O'Connell and Turner  present data on stiff sediments exhumed by the Mission Hills anticline (Figure 8). Interval S-wave velocity corresponds to failure at a coefficient of friction of 1 and PGV of 0.75 m s−1.
 Accumulating alluvium is less stiff than exhumed sediments. A failure curve for the San Fernando Valley corresponds to PGV of 3 m s−1 for a coefficient of friction of 1 without cohesion (Figure 8). The failure curve for the Santa Clara Valley near San Jose corresponds to PGV of 3 m s−1 for a coefficient of friction of 1 with 0.25 MPa of cohesion (Figure 9).
 In all the cases presented, the S-wave velocity as a function of depth corresponds to a frictional failure curve at constant dynamic shear strain. This property does not depend of the assumed coefficient of friction. Cohesion may have a mild effect. The data indicate that the exhumed sediments have not been strongly shaken while they resided at shallow depths. The accumulating alluvium does not appear to have experienced extreme shaking with PGV above 3 m s−1. At a minimum, material over a substantial shallow depth range becomes strongly nonlinear at the same dynamic shear strain. A Love wave producing more than this dynamic shear strain would experience strong nonlinear attenuation. The results thus yield a limit on the amplitude of Love waves that can propagate across a basin.
5.2. Physics of Compacting Sediments
 Sites with high S-wave at shallow depths are likely to be better indicators of maximum past PGV than sites with low seismic velocities. It is not reasonable to assume that extreme waves somehow avoid areas of exhumed sediment. Processes other than shaking can reduce seismic velocity. Chemical weathering is an obvious example that is likely coupled to damage from shaking [Replogle, 2011]. In particular, compaction may have never increased the S-wave velocity in accumulating alluvium to values where it could fail during past strong shaking.
 From (6), the S-wave velocity in accumulating alluvium needs to increase with the square root of depth andGlinearly with depth to fail at a constant coefficient of friction and strain. I discuss how this property might arise in the absence of strong shaking following the treatment of pulverized rock in Appendix A on the inference that compaction, state, and seismic velocity depend on grain-scale properties rather than macroscopic cracks. I make no attempt to develop compaction theory from CAP or critical state formalism [Karner et al., 2008]. In the absence of frictional slip, the state in (9) increases linearly with accumulation or burial time tA = ts in (13). The state and hence the S-wave velocity increase with normal tractionP and hence depth. The ratio in the stress equation (8) and hence the coefficient of friction,
is independent of pressure. Deeper sediments have been buried longer than shallow ones, but the logarithm of burial time from (15) in (8) does not vary much in a given active basin.
 The state variable in (A1) and shear modulus in (A2)both increase with decreasing porosity. Deeper sediments are expected to be stiffer than shallow ones, qualitatively in agreement with observations. Some caveats are in order. First, the observed S-wave velocity does not go all the way to zero at the free surface (Figures 7 and 9). Rather, it varies by a factor of ∼3 in the uppermost ∼150 m with considerable scatter. Any process that linearly increases stiffness with depth, perhaps chemical cementation, would qualitatively produce the observed curved trend of S-wave velocity versus depth. Cohesion provides another free parameter in the absence of laboratory data, where the theoretical curves need not extrapolate to zero at the free surface. Modeling using(A2)is likely to be unproductive as there are additional free parameters. That is, the process controlling S-wave velocity in accumulating alluvium is not obvious, but frequent strong seismic shaking with PGV of ∼3 m s−1 would be needed to control compaction.
 Near-field velocity pulses from major strike-slip faults and Love waves within sedimentary basins impose strain boundary conditions on shallow regolith that scale with peak ground velocity. Coulomb failure occurs at a given PGV when the regolith is sufficiently stiff or equivalently has a significantly high S-wave velocity. Failure adds cracks to the regolith reducing its S-wave velocity to a critical value where typical subsequent dynamic shear strain just reaches nonlinearity. Survival of stiff rock at shallow depths provides very widespread exceedance information.
 S-wave velocities in sandstone near Cajon Pass are compatible with the self-organization of S-wave velocity and the inference ofAnderson et al.  that these rocks have not experience 2 m s−1 PGV. Exhumed sediments in the San Fernando Valley provide somewhat local lower limits of 0.75 and 1.5 m s−1. The S-wave velocity at failure in(6) depends on the square root of the coefficient of friction. It is not productive to attempt to resolve detailed failure criteria from the available data.
 Overall, intact rock and regolith have desirable properties of fragile geological features. They are widespread and have experienced vast numbers of earthquakes, which can be inferred where exhumation rates are known [e.g., DiBiase et al., 2010]. Shallow seismic velocity is measurable. Damaged rock may be examined in the laboratory and in the field. In particular, macroscopic cracks, bedding planes, and joints that preclude significant cohesion are apparent. Numerical modeling of specific seismic waves from putative events in observed cracked near-surface rock is feasible.
 The tendency for shallow compliant rocks to become weakly nonlinear in typical strong motions may be a general feature whenever dynamic motion in stiff deep rocks imposes recurring essentially strain boundary conditions. This process is plausible whenever lithostatic pressure is low enough that compliant cracked rock can persist. This concept (if applicable) is frustrating in that nonlinear shallow rock failure does not sap typical strong waves and that detailed knowledge of shallow seismic velocity is necessary in nonlinear dynamic models of shaking. It is pleasing that more extreme ground motions are likely to be strongly nonlinear at shallow depths in active areas and not propagate effectively. It is convenient in that linear calculations based on linear elasticity are grossly accurate for typical events.
 I relate the state variable ψ in the friction equation (8)to the S-wave velocity, which is measured in the field. Porosity is a convenient intermediate as it is normally measured in the field and in the laboratory. The number density Γ =f/Ω of tabular cracks (where f is porosity and Ω is the long/short aspect ratio of cracks) is a more fundamental parameter [Krajcinovic, 1993; Mallick et al., 1993]. The relationship between state and S-wave velocity obtained using porosity as an intermediate proxy parameter for number density thus may be more accurate than relationships of porosity to state and porosity to S-wave velocity.
 Percolation theory provides formalism for the shear modulus of intact and highly fractured rock
where the modulus at low porosities GI extrapolates to zero at f = ϕl, and the stiffness approaches the (power law) percolation theory limit proportional to (ϕc − f)M at the critical porosity ϕc [Day et al., 1992]. The formula applies below the critical porosity. Further, it is physically necessary that the shear modulus is positive. Thus ϕl > ϕc/2M. This requirement is a result of the form of (A2)and does not imply a physical relationship between low-porosity and high-porosity behavior [Krajcinovic, 1997]. In general, one cannot deduce critical porosity behavior from the low porosity limit [e.g., Jasiuk et al., 1994]. The exponent M is theoretically 3.70 [Krajcinovic, 1993]. A further practical matter is that the approximation in (A2)does not hold in extremely porous gouge that deforms at high strain rates under very low normal tractions. S-wave velocity is not measurable in this case (the material is not elastic on a short time scale) and the gouge porosity may exceed the critical porosity obtained from studying shear modulus of rock. An exponential form analogous to(A1) at high porosities is attractive, but is not necessary the effective starting friction of in situ rock with measurable elastic parameters. A formulation for high porous very shallow uniform regolith may be useful. For example (A1) may be replaced with a form having a percolation limit,
where ψP → 0 at the limiting porosity fmax and ψP ≈ ψ for f ≈ ϕ. Then ϕ has a typical porosity of laboratory gouge. I have no data to calibrate this equation.
 With regard to mildly damaged intact rocks, using the low porosity limit of (A2) provides considerable algebraic simplification. I assume that the dynamic shear strain εdyn and the ambient normal traction P are given. Without loss of generality, I take intact rock with ϕ = f = 0 as my reference in (A1). I take the reference strain rate at a convenient value where co-seismic failure occurs to eliminate the strain rate term in(8) without having to know the precise value. Failure then occurs when the frictional stress in (8) equals the stress predicted from the shear modulus in (A2)
where the coefficient of friction of intact rock and gouge at ∼0 porosity are both μint at normal traction Pint. Any point on the empirical curve for intact granite in Figure 2 serves to calibrate it mathematically. I use the right end at 0.7 and 1500 MPa.
Equation (A4) may be solved for porosity. The rate and state porosity coefficient Pb/Ceps on the left hand side is small at low normal tractions. I use a generic range of b between 0.007 and 0.015. Sleep et al.  obtained Ceps = 3.4 × 10−3 for laboratory gouge. Hagin and Sleep obtained 2 × 10−3 for sandstone. The rate and state porosity coefficient for 1 MPa normal traction then ranges between 2.06 and 7.5 MPa. With forethought, I let the dynamic shear strain be 3.3 × 10−4 and use an intact granite shear modulus of 40 GPa. I let the linear coefficient ϕl range between 0.02 and 0.06. The coefficient multiplying f in (14) is between 220 and 660 MPa. Thus, increased porosity makes nearly intact rock more elastically compliant while only moderately decreasing the starting coefficient of friction. It is thus not critical to know rate and state properties.
 I present the results of a series of numerical calculations using (8) and (A2) to illustrate the applicability of the implication of the linearization in (A4). I obtain normal traction by integrating bulk density ρ = (1 − f)ρint, where the ρint density of intact rock is 2500 kg m−3. The intact shear modulus is 40 GPa and its S-wave velocity is 4000 m s−1. The critical porosity is 0.01, b = 0.007, and Ceps = 3.3 × 10−3. I vary the parameter α and the elastic linear coefficient ϕl.
 The computed S-wave velocity as expected from(A4) depends somewhat on the parameter α and significantly on the dynamic shear strain (Figure A1). It depends very weakly on the linear coefficient ϕl because the shear modulus on the right hand side of (A3) must vary to satisfy the equation with only a slight change in density in β = .The computed coefficient of friction depends mainly on the parameter α (Figure A2).
 For comparison, I adjust parameters to make the frictional term in (A4) more important. I let Ceps = 2 × 10−3, ϕl = 0.06, ϕc = 0.20, α= 0.30. The coefficient of friction and the S-wave velocity at failure are strongly dependent on the dynamic shear strain (Figure A3). The predicted coefficient of friction becomes relatively constant with depth.
 The 2010 Workshop on Applications of Precarious Rocks and Related Fragile Geological Features to U.S. National Hazard Maps was informative. John Anderson, Ileana Tibuleac, and Jim Brune quickly responded to my questions on precarious rocks on dip slopes seen on the field trip at this conference. John Anderson supplied the Cajon Pass report. Jerry Treiman pointed out the existence of California Geological Survey Open-File Report 1. Eric Dunham, Jesse Lawrence, and Gregory Beroza answered numerous near-field seismological questions. This work benefited from reviews by Gary Girty and Stephen Karner. This research was in part supported by NSF grant EAR-0909319. This grant is funded under the American Recovery and Reinvestment Act of 2009 (ARRA) (Public Law 111–5). 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 515.