3.1 Frictional Creep
 The well-known construct of rate and state friction provides quantification for frictional creep within a pervasively cracked material. The macroscopic strain is the sum of deformation on a large number of fractures. I consider the fate of a typical fracture to obtain the change in shear stress over the seismic cycle and its history over time.
 I apply the strain rate form of a unified theory of rate and state friction compiled by Sleep [1997, 2006] and Sleep et al. , as I am interested in macroscopic distributed strain. That is, failure is macroscopically internal deformation and internal friction but slip on random cracks when viewed in detail. The instantaneous shear traction τ on a crack as a function of normal traction PN is then
where the dominant term as in (3) μ0 ≈ τ/PN represents the approximation that the coefficient of friction has a constant value for a given surface (Amontons’ law), a and b are small ~0.01 dimensionless constants, ε′ ≡ V/W is the strain rate, V is sliding velocity, W is the width of the sliding gouge 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 arises if normal traction changes and with regard to apparent cohesion at low normal tractions (Appendix A). For now, I consider constant normal traction on a given fracture and set ψnorm = 1 without further loss of generality.
 Implications of (11) become evident when it is rewritten as a scalar flow law
 Note that ε′ref exp(−μ0/a) is effectively a multiplicative constant. Thus, one may set ε′ref to a physically relevant value that fixes μ0 using experimental data. Furthermore, the change of shear stress from the aftermath of the earthquake to the start of the next earthquake is relevant here. One does not need to know μ0 precisely for this purpose. Also, note that the logarithmic singularity at zero strain rate in equation (11) does not arise in equation (12); the predicted strain rate is unobservably small at very low shear stresses.
 In general, one needs to know the change in the state variable ψ over the interseismic interval to evaluate equation (12). I use the “aging” evolution law to represent the state variable ψ in terms of the past history of the fault to illustrate well-known implications of equations (11) and (12) [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 from the work of Linker and Dieterich , and Pref is a reference normal traction. Equation (13) implies that the state variable increases strengthening the surface when sliding is not occurring ε′ = 0. Experiments where P-wave transmissivity directly constrained the logarithm of state and showed that surfaces do strengthen in the predicted manner [Nagata et al., 2012].
 Equation (2) empirically represents the tendency of highly elastically compliant material to rapidly anelastically compact at depth and thus not typically be present within the Earth. In terms of equation (13), the state variable increases with time when sliding is not occurring, and the state variable increases, decreasing porosity [Segall and Rice, 1995]. Compaction and closing of numerous cracks increases the shear modulus (Appendix A of Sleep ).
 I continue with the history of postseismic stress and strain. I set Pref = PN to compact notation, as (by the assumption of a strike-slip fault) normal traction does not change with a given place in the rock and relegate discussion of this term to Appendix A. The steady-state value of the state variable in equation (13) is then
 I consider a simple example to obtain the basic form of the evolution of shear stress and strain to illustrate general features of the seismic cycle. I assume that coseismic slip and shaking do not reset the state variable along many of the fractures in the lid. That is, the state variable is typically large and does not change much in equation (13) over the seismic cycle. The earthquake suddenly increases shear stress by ΔτF. Thereafter, shear stress evolves as
 High shear stresses are very quickly relaxed by the rapid strain rates in equation (15). The widely used logarithmic decay of stresses and elastic strains arises as an approximation from equation (15) at large times tpost after the earthquake
where the shear traction is τyield at time tyield. The measurable relative change in stress is independent of ambient normal traction, provided that coseismic strain brings the material to yield.
 Alternatively, coseismic damage resets the state variable for some fractures in the lid to a low value ψlow. At long times after the event, the state variable in equation (13) is
when the state variable is still well below its steady-state value in equation (14). The strain rate in equation (12) decreases, as ψ increases in addition to the direct effect from stress relaxation.
 Physically, resetting of the state variable to low values by cracking on some fractures is likely to be unimportant to total postseismic strain. Slip on the main fault imposes the geometry of macroscopic shear strain within the lid. Local failure on a crack relaxes the shear stress on that crack. Failure of neighboring cracks over numerous cycles is required to reload the local stress so that the crack can fail again. Thus, the state variable from equation (13) is typically at a high value for the cracks that actually creep. Hence, equation (16) gives a maximum and likely estimate of the observable relative shear stress change over the seismic cycle for frictional rheology.
 I obtain the shear modulus and seismic wave velocity at failure in terms of equation (16). These quantities are observable within the lid at quasi-steady state. Coseismic displacement leaves material at frictional failure at μyieldPN, the preseismic stress is PN(μyield − a ln(tcycle/tyield)) ≡ PNμpre, where tcycle is the length of the seismic cycle. The criterion in equation (4) becomes
 The S-wave velocity for failure in equation (5) becomes
where the material is compliant enough that observable postseismic slip accommodates all the tectonic deformation. The seismic velocity predicted by equation (19) is much less than that predicted by equation (5) where shear stress fully relaxes in the interseismic interval. The shear modulus in equation (18) cannot decrease below the value where lithostatic pressure closes cracks in equation (2).
 It is not necessarily to precisely know physical parameters to obtain the implications of equations (16), (18), and (19). In addition, equation (18) applies for both frictional and ductile relaxation of stress when μpre is defined more generally to give pre-event shear stress, which remains positive. If μpre and hence the preseismic stress in the lid are small, a small coseismic strain does not bring the lid to frictional failure. The coseismic deformation of the lid is elastic, and the interseismic deformation is ductile. Conversely, a large coseismic strain brings the lid to frictional failure even if the preseismic stress is ~0.
 With regard to geodetic methods, comparison of immediate postseismic data with preseismic data provides coseismic slip and the shallow slip deficit compared with the deep fault. One then extrapolates observed postseismic creep measured over the remaining seismic cycle. The definition of postseismic, however, depends on available observations. In an optimal case, seismologists might have an array of strong-motion instruments around the fault. Off-fault deformations might become observable (defining tyield) after the passage of the crack tip in a fraction of a second. More likely, the duration of slip over the entire fault ~30 s for Landers [Wald and Heaton, 1994] is relevant if there are nearby continuous geodetic instruments. I let the seismic cycle time tcycle = 1000 a (3.15 × 1010 s), for an example. The argument of the logarithm in equation (16) is then between 1010 and 1012. Much of the eventual displacement occurs in the first year 3 × 107 s, 6/10 and 2/3 of the total, respectively. The eventual displacement is well constrained if continuous geodetic measurement are available, as more than half of it occurs within the first year. The computed final postseismic slip is often insufficient to balance slip on the underlying fault [e.g., Kaneko and Fialko, 2011].
 Overall, neither μyield nor a are precisely known from actual field examples, but this information is not necessary to obtain the essence of frictional behavior. The apparent coefficient of friction of 0.6 observed within boreholes in stable regions is a possibility [Townend and Zoback, 2000]. It is not likely to greatly exceed 1 for cracked rock. I use generic parameters of a = 0.01 and μyield = 1 as an example; 23% and 28% of the peak stress relaxes before the next event, respectively for my two time ratios.
3.2 Ductile Creep
 I contrast the time history implied by ductile creep with that implied by frictional creep in section 3.1. Ductile rheology is attractive for exhumed quartz-rich clay-bearing sedimentary rocks. I begin with the scalar linear Maxwell viscoelastic rheology, which leads to gradual relaxation of postseismic shear stress. The strain rate is the sum of the effects elasticity and viscous creep
where η is viscosity.
 The shear stress at constant strain is approximately applicable with a very elastically compliant damaged lid (Figure 1) and evolves as, τ = τpost[1 − exp(−Gt/η)], where the postseismic shear stress is τpost. The well-known Maxwell time tMax = η/G illustrates the effect of viscous creep. If this time is much longer that the earthquake cycle time, negligible viscous creep occurs; computation proceeds with elastic and frictional formalism. If the Maxwell time is much shorter than the cycle time, shear stresses quickly approach zero, and the net effect as assumed in equations (4) and (5) is that the lid is not under shear stress at the time of the next earthquake.
 The main difference with frictional rheology in section 3.1 is that the strain rate decreases slowly over the interseismic interval rather than very rapidly at first. I give two examples for a cycle time of 1000 a with the intact S-wave velocity of 2500 m s-1 and the cracked rock velocity of 600 m s-1. I retain the sediment density of 2200 kg m-3. The computed viscosity with this Maxwell time is 4.3 × 1020 and 2.5 × 1019 Pa s, respectively.
 The long-term anelastic strain over the geological time tgeol to form a tectonic feature, tgeolτ/η, provides field constraints on the order of magnitude of viscosity. I give a generic example applicable to lids of blind faults in California, Algeria, Haïti, and Iran. A strain of ~1 occurs over ~1 million years. The normal traction at ~1 km depth is 22 MPa, where faults and ductile deformation coexist, providing a scaling for stress. (A similar stress is needed to lift a protruding 1-km anticline against gravity.) The computed viscosity is 6.9 × 1020 Pa s, similar to that obtained from considering Maxwell times for stiff rock.
 Conversely, very low sediment viscosities cannot extend to depths of even a few 100 m within regions with significant topographic relief. Otherwise, the relief would have already spread from gravitational forces. I provide an example for shale outcropping on the Niagara escarpment [Barlow, 2002]. This rock has been exposed at the base of a cliff for ~104 a; the inelastic strain again is ~1. The shale is buried about 100 m deep, so the shear stress is ~1/10 that in the previous example. So the long-term viscosity is ~6.9 × 1017 Pa s, which would imply a brief Maxwell time even in compliant rocks. This material would have a strain of 1000 for the stress of 22 MPa over 1 Ma in the previous paragraph compared with observed strain of ~1 involved with producing observed topographic relief. Material with this viscosity would have spread laterally rather than forming an anticlinal hill. I discuss more sophisticated rheologies for interbedded clay-bearing sand-rich rocks in Appendix A.
3.3 Compactional Creep and Postseismic Subsidence
 Postseismic subsidence is a way to detect distributed co-seismic brittle failure on cracks. Brittle fracture dilates material reducing the value of the state variable on the failed cracks. This dilation reduces the postseismic low-amplitude shear wave velocity. It also produces co-seismic uplift and postseismic subsidence. Fielding et al.  studied the latter near Bam Iran.
 I quantify my analysis by considering an empirical relationship between state and porosity f,
where ϕ is a reference porosity where ψ = 1 and Cε is a dimensionless coefficient [Segall and Rice, 1995]. I assume that cracking reduces the state variable within numerous fractures to small values immediately after the earthquake. The state variable in the rest of the rock mass does not change significantly. Combining equations (17) and (21) yields that porosity within failed cracks decreases with the logarithm of time
where t is time after the earthquake and porosity change is measured after t0. The dilatation occurs over a depth scaling with the lid thickness ZF. So the initial uplift and subsequence subsidence with quasi-steady cycles are approximately this porosity change times the lid thickness, ΔU ≈ ΔfZF.
 The construct dilatancy angle (the ratio of dilatational strain to anelastic shear strain Φ) in engineering geology [e.g., Ribacchi, 2000; Alejano and Alonso, 2005; Zhou and Cai, 2010] allows estimation of the coseismic change in porosity. The observed Φ ratio in jointed sedimentary rocks at normal tractions relevant to lids is a few tenths. That, is Δf ≈ ΦεF and ΔU ≈ ΦZFεF.
 In practice, geodetic studies measure the postseismic subsidence ΔUobs for a finite interval after the earthquake, and one must infer the total subsidence for the earthquake cycle ΔU. I base an example calculation on the work of Fielding et al.  for Bam, Iran. The coseismic rupture was complicated enough that they could study postseismic subsidence but not directly extract the coseismic uplift related to dilatation. Their first measurement was 12 days after the earthquake, and they continued to measure for ~3.5 years. Subsidence of ~0.02 m apparently following equation (22) occurred over that time. The unmeasured subsidence before and after the event does not sensitively depend on those times. I assume 30 s for the beginning and 1000 years for the end. These times imply unrecorded early subsidence of 0.046 and 0.024 m of future subsidence for a total of 0.09 m. The brittle anelastic shear displacement associated with this subsidence is 0.09 m divided by the dilatancy coefficient. For example, Φ = 0.3 implies ~0.3 m of anelastic shear displacement. Taking this result at face value, the anelastic slip was a significant but not dominant part of the slip per cycle of ~1 m. Ductile shear deformation could well accommodate the remaining slip over the next ~1000 years.
 This behavior differs from that of ductile compaction and poroelastic compaction. The difference between lithostatic and pore pressure drives both ductile compaction and brittle compaction. However, the ductile creep rate depends on this pressure linearly or to a small power. As with ductile shear creep, deformation spreads out over the interseismic interval and is not particularly fast just after the earthquake. Ignoring ductile creep is thus justified in my analysis and the study by Fielding et al. .
 Some caveats are in order with regard to poroelastic effects. Fielding et al.  did not have data until 12 days after the event to fully resolve that equation (22) applies. Pore pressure declines exponentially over a time scale related to permeability, pore compliance, and the length scale of pressure variation. In that case, the observed subsidence of 0.02 m could well represent most of the total subsidence per cycle and not require significant brittle dilatation. Unmodeled poroelastic effects would reduce our computed brittle subsidence per cycle of 0.09 m and the coseismic brittle shear deformation of 0.3 m. This would increase the shear deformation available for ductile creep in the lid.