A slower fault may produce a smaller preseismic moment rate: Non-1/tf acceleration of moment rate during nucleation and dependency on the background slip rate
Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, Yokohama, Japan
Corresponding author: H. Noda, Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, 3173–25 Showa-machi, Kanazawa-ku, Yokohama, Kanagawa 236–0001, Japan. (firstname.lastname@example.org)
 A recent global study has revealed that seismicity near the hypocenter prior to large earthquakes, which could be a proxy for preseismic moment rate, accelerates before interplate earthquakes, while it rarely does before intraplate earthquakes. Understanding the amplitude of preseismic deformation is important in assessing the possibility of its detection. For a class of rate-state friction laws without a characteristic speed-related parameter (e.g., aging law and slip law), a dimensional analysis has shown that if the moment rate increases more mildly than 1/tf where tf is the time-to-failure, then the amplitude of preseismic moment rate is smaller for a smaller quasistatic slip rate. Three-dimensional numerical simulations have revealed that the aging law yields 1/tf acceleration, while the slip law causes milder acceleration. If the latter is the case, faults with very low long-term slip rates (e.g., intraplate faults) may have very small preseismic moment rates.
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 Under what circumstances a large earthquake has a large preseismic deformation is an important problem. Bouchon et al.  have revealed that seismicity rate near the hypocenter detectably accelerates prior to large earthquakes for interplate faults, while it rarely does for intraplate faults. One possibility is that the interplate faults may have different mechanical properties from the intraplate ones in terms of, for example, the frictional parameters and the shape of the friction law possibly because of difference in fault structures and physicochemical conditions. Alternatively, nucleation processes may be affected by boundary conditions such as the long-term slip rate. In the present study, we investigate the latter possibility for a class of friction laws theoretically and numerically.
Noda et al.  conducted earthquake sequence simulations with the aging law (AG), one of the most commonly used version of the rate- and state-dependent friction laws (RSFs) [Ruina, 1983], and showed that the preseismic moment release rate in the seismogenic patch dM/dt increases as 1/tf where tf is the time-to-failure. The 1/tf acceleration was explained by a model of spontaneously accelerating circular patch with a fixed size [Dieterich, 1992; Noda et al., 2013] equal to the nucleation size [Rubin and Ampuero, 2005; Chen and Lapusta, 2009] dictated by the RSF parameters. Here spontaneous acceleration means that loading to the nucleation patch due to slip in the surrounding region does not play a role; even if we glued the surrounding region of ongoing nucleation, it would accelerate by itself. In this sense, it does not seem straightforward to ascribe the observation by Bouchon et al.  to the differences in the loading rate.
 In the present paper, we shall show by a dimensional analysis that if acceleration of dM/dt is milder than 1/tf, the amplitude of dM/dt positively depends on the scale of quasistatic slip rate Vqs for a certain class of the RSFs. Further, we shall demonstrate that it is the case for the nucleation with the slip law (SL).
2 Dimensional Analysis
 In the present study, we choose speed, length, and stress as three independent dimensions. RSFs [e.g., Dieterich, 1979; Ruina, 1983] are frequently used in modeling earthquakes and their sequences [e.g., Tse and Rice, 1986], and the simplest and most frequently used formulations (e.g., AG and SL) do not have a parameter related to a dimension of speed. The focus here is such a dimensionally simple class of the friction laws. Most commonly, the shear stress on the fault τ is given by
where V is the slip rate, τ0 is the steady state frictional resistance at a reference slip rate V0,A and B represent the amounts of the direct effect and the evolution effect, respectively, and L is the state-evolution distance. The evolution equation for the state variable θ is written as
in AG and
in SL. V0 can be taken arbitrarily without loss of generality and is not a characteristic slip rate of the friction law.
 Let us consider nucleation on a fault embedded in a linearly elastic medium. Elastodynamics is characterized by the shear wave speed cs, the shear modulus μ, and the Poisson's ratio ν. From the boundary and initial conditions, we have the length scale R of the system size, the scales of the slip rate, and the shear stress in a quasistatic situation before nucleation Vqs and τqs. Since the reference slip rate in the friction law V0 can be selected arbitrarily, we shall use Vqs for V0. In general, the initial condition prior to nucleation is heterogeneous on the fault. We regard such heterogeneity as distributions of nondimensionalized slip rate and shear stress by Vqs and τqs. Later, we shall consider the dependency of the solution on Vqs by a dimensional analysis. That is, we shall discuss how the solution changes when the scale of the quasistatic slip rate changes while keeping the nondimensional distribution. Under equation (1), only two out of V, τ, and θ are independent variables. Although θ is useful in understanding the concrete mechanism of nucleation [Rubin and Ampuero, 2009], we choose V and τ as the independent variables as they are more directly related to geodetic and seismic observations.
 In the quasistatic limit, cs does not matter by definition, and it is known that V diverges at a finite time [e.g., Tse and Rice, 1986] which we shall call a failure time in this section. Actual solutions of fully dynamic (or quasidynamic) problems deviate from quasistatic solutions when inertial effects become significant. In later sections where numerical results are discussed, the failure time is defined by setting a threshold slip rate.
 Let us consider the behavior of quasistatic solutions as the time t approaches to the failure time or as the time-to-failure tf decreases to zero. The moment rate dM/dt is the product of the averaged slip rate Vav over a seismogenic patch, the patch area, and the shear modulus μ. Since only Vav changes with time, we consider how Vav evolves as tf decreases. The variables and parameters are listed in Table 1. The list of speed-related variables and parameters which matters in quasistatic problems consists of Vav (speed), Vqs (speed), and tf (length/speed).
Table 1. List of Parameters and Their Dimensions Used in the Simulations
where π3, π4, …, πn are the nondimensional parameters independent of Vav, Vqs, and tf. If we assume the existence of a nonzero implicit function, we obtain
where G is the reciprocal of the implicit function. Therefore, if Vav does not depend on Vqs, then Vav is inversely proportional to tf and vice versa. The difference in the initial value of θ (with keeping Vqs) implicitly affects some of π3 to πn through τqs.
 Equation (5) also means if Vav depends on Vqs, which is quite possible in general, then Vav is not inversely proportional to tf. If the divergence behavior of Vav is characterized by a power law in terms of tf, it must take a form:
where r is a real number. If r is not 1, the amplitude of the preseismic moment rate should be different for faults with different Vqs, even if the mechanical properties are the same.
 If we choose Vav/Vqs as one of the nondimensional parameters instead of Vavtf/L, an expression corresponding to equation (5) becomes
 In this form, it is visible that the nondimensional moment rate Vav/Vqs = G′ depends on nondimensional time Vqstf/L, and its functional form does not depend on Vqs because π3, π4, …, πn are independent of the speed-related parameters. The same is true for the distribution of V since the system of equations for quasistatic problems becomes identical when nondimensionalized by Vqs.
3 Numerical Experiments
3.1 Problem Settings
 An example problem is illustrated in Figure 1. For the parameter values, please see Table 1. Problem setting is similar to, and numerical methodology is identical to what is used by Noda et al. . For detailed numerical methodology, please also see Noda and Lapusta . A planar fault is embedded in a linearly elastic infinite medium. The x1, x2, and x3 axes are set in the overall slip direction, normal to the fault, and in the other orthogonal direction, respectively. We set a rate-weakening circular patch of radius R = 4 km at the center surrounded by a rate-strengthening region. The fault is loaded by prescribing steady sliding near periodic boundaries by Vqs in the x1 direction. In order to see the dependency of Vav(tf) on Vqs, we have used various Vqs ranging from 10−12 to 10−6 m/s. The simulated earthquakes are about Mw 6. The origin of the Cartesian coordinate is at the center of the rate-weakening patch.
 The L is uniformly distributed on the fault. We choose L for AG and SL differently so that resultant nucleation patches under respective friction law have convenient size to track over the tested range of Vqs. The present paper only concerns Vqs dependence within each friction law; nucleation size between SL and AG should not be compared because whatever size can be obtained by setting an appropriate L for each friction law. The value of L in nature is totally an open question [e.g., Scholz, 2002].
 For AG, fully dynamic simulations have been conducted with L = 2.72 mm which corresponds to a nucleation size [e.g., Chen and Lapusta, 2009; Noda et al., 2013] one fifth of the rate-weakening patch in length. For SL, nucleation is numerically much more difficult to resolve than AG [Rubin and Ampuero, 2009]. Then we have adopted a quasidynamic approximation [Rice, 1993] with a longer state-evolution distance L = 17.0 mm so that a critical wavelength for impossibility of steady state sliding (λcr in Rice et al. ) is equal to R. The periodic boundaries are set every 25.6 km, and a squared region defined by them is discretized by 1,5362 grid points for AG and by 12,8002 grid points for SL. We also conducted fully dynamic simulations for SL with the same resolution as AG. The results (not shown) were essentially the same, though some oscillations due to poor resolution occurred.
 At t < 0, the fault is assumed to be sliding steadily by V1 = Vqs and V3 = 0. At t = 0, we break the steady state and the symmetry by adding a spatially smooth slip perturbation of a 1 cm amplitude in a circular region of 1.5 km radius centered at (x1, x3) = (1, 1 km). The simulations have been conducted until the first earthquake, and the preseismic processes are compared in order to illuminate the effect of Vqs.
3.2 Non-1/tf Acceleration and Initial Condition Dependency With the Slip Law
 Figure 2 represents snapshots of the slip rate distribution in the cases with Vqs = 10−12 m/s with AG (left) and SL (right). With AG, an aseismic transient (Figure 2a) hosts an elliptic nucleation which accelerates to a coseismic slip rate level (Figures 2c, 2e, and 2g). The overall acceleration of the nucleation patch follows 1/tf expected from a classical analytical approximation [e.g., Dieterich, 1992; Noda et al., 2013], with some spatial complexity and temporal fluctuation [Rubin and Ampuero, 2005, Figure 9] (also see supporting information, Movie 1).
 On the other hand with SL, pulse-like nucleation takes place, with the slip rate taking its spatially maximum value Vmax always near the propagating front (Figures 2b, 2d, 2f, and 2h). The curvature of the front increases with propagation because a part of the front having larger slip rate causes faster propagation [Ampuero and Rubin, 2008]. The region of relatively high slip rate becomes smaller and smaller as the transient propagates (Figures 2d, 2f, and 2h and supporting information, Movie 2).
 Figures 3a and 3b represent nondimensional preseismic moment rate in the rate-weakening patch Vav/Vqs for AG and SL, respectively. The failure time is defined by a threshold 0.1 × 2csA/μ = 0.032 m/s for Vmax at which the radiation damping term [Rice, 1993] is significant compared with the direct effect. As expected from equation (7), the solutions for different Vqs trace each other until inertial effects become significant as indicated by the leveling off of each curve. With AG, Vav or the moment rate roughly follows the estimate by Noda et al. . The fluctuation relative to a straight line for the 1/tf dependency reflects the complexity in the nucleation patch. For SL, however, the moment rate increases more mildly than 1/tf. On the other hand, Vmax in the simulations with SL follows 1/tf dependency (supporting information Movie 2). Thus, the shrink of the high slip rate region (Figure 2) is probably responsible for the milder acceleration of Vav.
Rubin and Ampuero  showed in 2-D cases that Vmax increases as 1/tf if the maximum slip of a nucleation pulse δ is proportional to ln(Vmaxθbg/L)1/2 where θbg is θ ahead of the pulse, which is also the case in numerical simulation by Ampuero and Rubin . However, the length scale of the pulse in such a case does not shrink as the pulse propagates. In the present 3-D cases, the high slip rate region in Figures 2d, 2f, and 2h shrinks in the direction parallel to the front as well as normal to it. Therefore, a 3-D effect may be important.
 The significance of 1/tf acceleration is illuminated if we select a different scale. In Figures 3c and 3d, the preseismic moment rate dM/dt (= πR2Vavμ) is plotted against tf in the “real” scale. When the scale is changed from Figures 3a and 3b to Figures 3c and 3d, the plotted solutions shift parallel to 1/tf dependency by the amounts corresponding to Vqs. Therefore, the common 1/tf acceleration recognized in Figure 3a is preserved in Figure 3c, which is a characteristic dM/dt as a function of tf dictated by the frictional parameters and not affected by Vqs. However, for SL, the quasistatic solution in Figure 3b does not show 1/tf acceleration, and thus, dM/dt as a function of tf depends on Vqs (Figure 3d). Since the power exponent is smaller than 1, dM/dt is smaller for smaller Vqs at a certain tf. For example, at tf = 1,000 s, dM/dt increases by about two orders of magnitude for an increase of Vqs by six orders of magnitude (the vertical dotted arrow in Figure 3d). Dependence at further smaller tf may be of interest, however, for a small tf, relative positioning between curves for different Vqs is significantly affected by the arbitrary definition of failure time.
4 Discussion and Conclusions
 The present simulations with SL have shown that the amplitude of preseismic moment rate dM/dt depends positively on Vqs (Figure 3d). From a dimensional analysis (section 2), this is a necessary and sufficient condition for acceleration milder than 1/tf (equation (6)), which was the case in the simulations (Figure 3d). In the present problem setting, Vqs is the slip rate of the creeping region that loads up the asperity, suggesting a possibility that the loading rate affects the amplitude of preseismic moment release. This could be a reason for the recent observation by Bouchon et al. , although there are other possibilities such as differences in the frictional properties between intra- and interplate faults.
 As mentioned earlier, however, the nucleation we are looking at is a self-accelerating process at the last stage of a seismic cycle, so the loading rate itself should not affect the nucleation. We rather speculate that Vqs affects nucleation via its control on the background slip rate Vbg and hence on the state θbg just ahead of the nucleation patch because θbg dictates the strength drop. As shown by Rubin and Ampuero , effect of θbg should be especially strong in the pulse-like propagation under SL because nucleation keeps migrating into a fresh region still slipping at Vbg. As seen from Figure 2, Vbg ~ Vqs in the present simulations.
 Strictly speaking, it is desirable to confirm the Vqs dependence in sequence simulations whereby Vbg and θbg are naturally set by interseismic processes following earlier earthquakes. Although sequence simulations under AG show that nucleation tends to start in the region where V has been raised not far below the slip rate of the creeping region outside the rate-weakening patch [e.g., Lapusta and Liu, 2009; Noda et al., 2013], fully dynamic simulations for SL required for this purpose demands too much numerical resources.
 We have used AG and SL as demonstrative examples of the general implications of equation (6) that Vqs affects the amplitude of nucleation if the growth is non-1/tf. Although it may be arguable that SL is more important to nucleation because it better describes the strength drop behavior in the laboratory [e.g., Nakatani 2001; Ampuero and Rubin, 2008], we do not intend to imply that SL is preferred. We rather emphasize the general existence of a nonstraightforward effect of Vqs, which is suppressed under AG which happens to cause nucleation growth following 1/tf. There are other examples of the dimensionally simple RSFs such as the Nagata's law [Nagata et al., 2012] and the Perrin-Rice-Zheng law [Perrin et al., 1995] and the preseismic acceleration with those laws may deserve further study.
 Other friction laws with characteristic speed-related parameters are sometimes used. For example, flash heating has the weakening slip rate, and thermal pressurization of pore fluid has diffusivities [e.g., Rice, 2006]. Those realistic friction laws probably make a difference when the corresponding physical process (e.g., frictional heating) becomes significant [e.g., Schmitt et al., 2011]. But the notion derived here from the dimensionally simple friction laws may be the case in other stages in nature and useful in discussing numerical and experimental results and in comparing simulations with observations.
 The comments by A. Rubin and an anonymous reviewer were helpful in improving the manuscript. The Earth Simulator was used for all simulations. This study is supported by the Observation and Research Program for the Prediction of Earthquakes and Volcanic Eruptions of the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
 The Editor thanks Allan Rubin and an anonymous reviewer for their assistance in evaluating this paper.