Volume 126, Issue 4 e2020JB020430
Research Article
Free Access

Precursory Slow Slip and Foreshocks on Rough Faults

Camilla Cattania

Corresponding Author

Camilla Cattania

Department of Geophysics, Stanford University, Stanford, CA, USA

Now at Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

Correspondence to:

C. Cattania,

[email protected]

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Paul Segall

Paul Segall

Department of Geophysics, Stanford University, Stanford, CA, USA

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First published: 18 March 2021
Citations: 59

Abstract

Foreshocks are not uncommon prior to large earthquakes, but their physical mechanism remains controversial. Two interpretations have been advanced: (1) foreshocks are driven by aseismic nucleation and (2) foreshocks are cascades, with each event triggered by earlier ones. Here, we study seismic cycles on faults with fractal roughness at wavelengths exceeding the nucleation length. We perform 2-D quasi-dynamic, elastic simulations of frictionally uniform rate-state faults. Roughness leads to a range of slip behavior between system-size ruptures, including widespread creep, localized slow slip, and microseismicity. These processes are explained by spatial variations in normal stress (σ) caused by roughness: regions with low σ tend to creep, while high σ regions remain locked until they break seismically. Foreshocks and mainshocks both initiate from the rupture of locked asperities, but mainshocks preferentially start on stronger asperities. The preseismic phase is characterized by feedback between creep and foreshocks: episodic seismic bursts break groups of nearby asperities, causing creep to accelerate, which in turns loads other asperities leading to further foreshocks. A simple analytical treatment of this mutual stress transfer, confirmed by simulations, predicts slip velocities and seismicity rates increase as 1/t, where t is the time to the mainshock. The model reproduces the observed migration of foreshocks toward the mainshock hypocenter, foreshock locations consistent with static stress changes, and the 1/t acceleration in stacked catalogs. Instead of interpreting foreshocks as either driven by coseismic stress changes or by creep, we propose that earthquake nucleation on rough faults is driven by the feedback between the two.

Key Points

  • Rough fault simulations exhibit simultaneous foreshocks and creep caused by heterogeneity in normal stress induced by roughness

  • Stress transfer between foreshocks and creep produces a positive feedback and 1/t acceleration prior to the mainshock

  • The precursory phase is characterized by migratory seismicity and creep over an extended region

Plain Language Summary

Understanding premonitory seismicity leading up to large earthquakes has been a central problem in seismology for several decades. In spite of constantly improving observational networks and data analysis tools, we are still grappling with the fundamental question: what causes foreshocks? Do they represent a chain of isolated events, or are they driven by slow slip over a large fault area, gradually accelerating before the mainshock? In this study, we tackle this question with numerical simulations of slip on a fault with a realistic (fractal) geometry. This geometrical complexity causes spatial variations in stress: compression or extension occur as irregularities on opposite sides of the fault are pressed closer together or pulled apart. This spatial heterogeneity modulates slip stability across the fault, causing simultaneous occurrence of slow slip and foreshocks. The two processes are linked by a positive feedback, since each increases stress at the location of the other; under certain conditions, this can culminate in a large earthquake. Our model reproduces a number of observed foreshock characteristics, and offers new insights on the physical mechanism driving them.

1 Introduction

Foreshocks have been observed before many moderate and large earthquakes (Abercrombie & Mori, 1996; Ende & Ampuero, 2020; Jones & Molnar, 1976; Trugman & Ross, 2019), and even though modern seismic networks and analysis techniques have imaged foreshock sequences in unprecedented detail (Ellsworth & Bulut, 2018; Tape et al., 2018), the physical mechanisms driving them remains debated (Gomberg, 2018; Mignan, 2014). One interpretation is that foreshocks represent failures of seismic sources (asperities) driven by an otherwise aseismic nucleation process (Abercrombie & Mori, 1996; Bouchon et al., 20112013; Kato, Fukuda, Nagakawa & Obara, 2016; McGuire et al., 2005; Ruiz et al., 2014; Sugan et al., 2014; Tape et al., 2018). Aseismic acceleration prior to instability is predicted by theory (Ampuero & Rubin, 2008; Dieterich, 1992; Rubin & Ampuero, 2005; Ruina, 1983) and has been observed in laboratory experiments (Dieterich & Kilgore, 1996; McLaskey, 2019; McLaskey, 2013; McLaskey & Lockner, 2014) and numerical simulations (e.g., Dieterich, 1992; Lapusta, 2003; Lapusta et al., 2000). On the other hand, foreshocks have also been interpreted as a cascade of events triggered by one another, not mediated by an aseismic process (Hardebeck et al., 2008; Helmstetter & Sornette, 2003; Schurr et al., 2014). Recent studies have shown that the relative locations of foreshocks are in fact consistent with static stress triggering (Ellsworth & Bulut, 2018; Yoon et al., 2019), and the lack of detectable aseismic slip preceding most moderate to large earthquakes supports the view of a triggering cascade.

The occurrence of foreshocks implies fault heterogeneity: if they are driven aseismically, heterogeneity leads to simultaneous occurrence of seismic and slow slip; in the cascade model, it is required to explain why foreshocks remain small, while the mainshock evolves into a large rupture. Previous modeling studies of foreshocks have considered various sources of heterogeneity: velocity-weakening asperities in a velocity strengthening fault (Dublanchet, 2018; Yabe & Ide, 2018); spatial variations in nucleation length on a velocity-weakening fault caused by heterogeneous state evolution distance (Noda et al., 2013) or effective normal stress (Schaal & Lapusta, 2019). In these studies, aseismic slip can take place around the asperity due to either velocity strengthening behavior or frictional properties that lead to large nucleation dimensions; however, the presence of asperities with a small nucleation dimension can nevertheless lead to a cascading sequence (Noda et al., 2013).

Perhaps the most ubiquitous and best characterized source of heterogeneity is geometrical roughness: faults are fractal surfaces (Brodsky et al., 2016; Candela et al., 20092012; Power & Tullis, 1991; Power et al., 19871988; Sagy et al., 2007). Numerical and theoretical studies have shown that fault roughness has a first-order effect on rupture nucleation (Tal et al., 2018), as well as propagation and arrest (Dunham et al., 2011; Fang & Dunham, 2013; Heimisson, 2020; Ozawa et al., 2019).

Here, we focus on the effect of long wavelength roughness (exceeding the nucleation length) on the nucleation phase and precursory seismicity leading up to a mainshock. We perform quasi-dynamic simulations of rough but otherwise uniform velocity-weakening faults embedded in a linear elastic medium. Numerical simulations show that a rich slip behavior ranging from slow slip to seismic ruptures arises as a consequence of normal stress heterogeneity induced by fault roughness, which causes spatial variations in strength and fault stability. Early in the cycle, low normal stress regions start to creep stably while high normal stress regions (from now on referred to as “asperities”) remain locked. The mainshock nucleation phase is characterized by an interplay between accelerating creep and episodic foreshocks: creep loads asperities, until they fail seismically; foreshocks increase stress on nearby asperities and creeping areas, causing the latter to accelerate in turn triggering subsequent foreshocks; asperities do not fully relock after failure, gradually unpinning the fault and increasing the creeping area and velocities. We introduce a simple analytical model based on these interactions, which predicts acceleration in seismicity rate and creep as 1/t, where t is the time to the mainshock. Simulated sequences reproduce a number of observations, such as the relative location of foreshocks, their migration toward the mainshock hypocenter and the power-law acceleration of foreshocks in stacked catalogs.

2 Numerical Model

We run 2-D plane strain simulations with the quasi-dynamic boundary element code FDRA (Segall & Bradley, 2012). The following equation of motion governs fault slip:
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0001(1)
where μ is the shear modulus, τf is the frictional resistance, and τel is the shear stress due to remote loading and stress interactions between elements. The stress from each element is computed from dislocation solutions (e.g., Segall, 2010), accounting for variable element orientation. The right-hand side is the radiation damping term, which represents stress change due to radiation of plane S-waves (Rice, 1993), with cs the shear wave speed.
Frictional resistance evolves according to rate-state friction (Dieterich, 1978):
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0002(2)
where a and b are the constitutive parameters; dc is the state evolution distance; σ is the effective normal stress; v* is a reference slip velocity; f0 is the steady-state friction coefficient at v = v*, and θ is a state-variable. Model parameters are listed in Table 1. We employ the aging law (Ruina, 1983) for state evolution:
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0003(3)
such that steady-state friction at sliding velocity v is
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0004(4)
Table 1. Model Parameters
Parameter Value
a 0.015
b 0.02
dc 10−4 m
f0 0.6
σ0 10 MPa
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0005 0.004 Pa s−1
μ 30 GPa
ν 0.25
Lmin 100 m
Lf 5.2 km
Ch 0.013 m2(1−H)
H 0.7
  • Note. Lf is the total fault length and urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0006 is the spatially averaged stressing rate. Other parameters are described in the text.
We apply remote loading such that the stress-rate tensor is pure shear
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0007(5)
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0008(6)
where σ1,3 are the principal stresses and σD is the differential stress. Resolving these on to the fault yields shear and normal stressing rates
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0009(7)
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0010(8)
where Ψ is the average fault angle with respect to σ1 and θ(x) is the local slope. In general, both shear and normal stress vary in time; here, we take Ψ = 45°, so that the spatially averaged effective normal stress is constant and equal to a uniform value σ0 = 10 MPa. In addition to the remote loading, slip on a rough fault also causes normal stress changes, which in our case dominate the effect of spatially variable loading rate described by Equations 7 and 8. In Appendix A, we show how perturbations in normal stress depend on fault roughness and slip. Normal stresses can locally become tensile and induce opening if a purely elastic response is assumed. In contrast, tensile stresses are reduced or entirely inhibited in an elastoplastic medium with a Drucker-Prager yield surface (e.g., Dunham et al., 2011). We approximate this behavior by setting a minimum value σmin for normal stress, σmin = 1 kPa ≪ σ0. Earthquakes are defined as times when the slip velocity anywhere on the fault exceeds the threshold velocity Vdyn = 2aσcs/μ (Rubin & Ampuero, 2005), here ∼4 cm/s.
The fault profile is fractal, characterized by power spectral density
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0011(9)
with β = 2H + 1, where H is the Hurst exponent. For natural faults, this is typically between 0.4 and 0.8 (Renard & Candela, 2017); here, we set H = 0.7. For computational reasons, we only include wavelengths greater than Lmin = 100 m, close to the nominal nucleation length defined below, unless otherwise specified.

2.1 Model Resolution

To correctly describe rupture behavior, both the nucleation length and the cohesive zone Λ0 need to be well resolved (e.g., Lapusta et al., 2000; Perfettini & Ampuero, 2008). Erickson et al. (2020) found that a suite of planar fault models, including FDRA, produced well resolved simulations with Λ0x ≥ 3, with Λ0 = μdc/ (Rubin, 2008), in agreement with previous studies (Day et al., 2005). A resolution of Λ0x ≈ 1.7 produced similar temporal patterns, but slight differences in the frequency-magnitude distribution of simulated events. On a rough fault, normal stresses change with time and can locally be higher than the average, requiring a finer resolution. Moreover, we found that rough fault simulations are less forgiving than may be expected from the results above. For instance, a simulation resolving the nominal cohesive zone size with four grid points and a small fraction (10–15%) of the fault with Λ0x ≈ 1–2 produced abundant microseismicity and no full ruptures, while doubling the number of grid points generated full ruptures. Since earthquakes tend to arrest where σ is high and the cohesive zone is small, a few under-resolved regions can determine the event size statistics. Here, we specify a uniform resolution with nominal Λ0x ≈ 8, and for the foreshock sequence discussed through most of the paper Λ0x > 2 everywhere. We tested a few individual foreshocks and verified that their rupture length does not change when doubling the resolution.

3 Summary of Simulation Results

The first-order effect of fault roughness during the interseismic phase is a decrease in fault locking: as seen in Figure 1a, and previously noted by Tal et al. (2018), the maximum slip velocity on the fault during the interseismic period is several orders of magnitude larger for a rough fault than for its planar counterpart. Figure 1b shows that this is due to patches of higher velocity between locked patches. For simplicity, in the remainder of the paper, we refer to these slowly slipping regions as “creeping,” even though their slip velocity (estimated in Section 4) can be several orders of magnitude lower than typically measurable fault creep.

Details are in the caption following the image

(a) Maximum slip velocity over multiple cycles on a rough (black) and comparable planar (gray) fault. The dotted line is the threshold velocity used to define earthquakes. (b) Slip velocity across the entire fault during one cycle showing alternating creeping and locked patches. The lower panel shows the slip velocity on a planar fault during the same time period (only a small region is shown, since velocity is effectively uniform).

During most of the interseismic phase the average slip velocity slowly increases, as creeping patches widen; this process is entirely aseismic, even though brief slow slip episodes with velocities up to about 10−6–10−5 m/s occur as creep fronts coalesce and break asperities (Figure 1, 6–8 years into the cycle). Only in the final part of the cycle do asperities rupture in seismic events while creep rates increase (Figure 2). During the acceleration leading up to the mainshock slip velocity on the fault does not increase gradually but in abrupt steps, associated with bursts of microseismicity. This pattern repeats at increasingly short temporal scales as the background slip velocity increases.

Details are in the caption following the image

Average slip velocity on the fault leading up to the mainshock, showing a similar pattern across multiple temporal scales. Earthquakes are marked with crosses, and each gray box indicates the extent of the next panel.

Foreshocks only occur once sufficient slip has accumulated on the fault, and the first few sequences consist of events spanning the entire domain (system-size ruptures). This is due to an increase in the amplitude of normal stress perturbations with total slip, quantified in Appendix A: microseismicity starts when the root-mean-square normal stress perturbation Δσrms is of the order of the background normal stress σ0. In the rest of the paper, we will focus on one of the first sequences with foreshocks (Δσrms/σ0 = 1.1), since later sequences, with more net slip, may not be well resolved (as discussed in Section 2). Other sequences are qualitatively similar (Figure S1).

4 Relationship Between Fault Roughness and Interseismic Locking

Previous studies have shown that slip on a rough surface leads to perturbations in normal stress (Chester & Chester, 2000; Dunham et al., 2011; Sagy & Lyakhovsky, 2019). In Appendix A, we summarize these findings and derive a simple expression for normal stress perturbations as a function of cumulative slip and fault topography. Normal stress perturbations on a fractal fault with uniform slip S have a Gaussian distribution; for a fractal fault with Hurst exponent H, its standard deviation is given by
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0012(10)
where μ′ = μ/(1 − ν) with ν the Poisson’s ratio, and α is the roughness, which quantifies the amplitude of topography such that the root-mean-square elevation measured over a length l is given by yrms = αlH (Section A1). These variations in normal stress are responsible for the occurrence of alternating creeping and locked regions, as shown in Figure 4: creep takes place where roughness decreases the normal stress, while regions with increased σ remain locked.
A simple model illustrating the heterogeneous response of a rough fault loaded at uniform shear stressing rate is shown in Figure 3. After a system-wide rupture, all points on the fault are at steady-state friction fco = fss(Vco), given by Equation 4, with Vco the seismic slip velocity (this applies if fault healing occurs on a much longer timescale than the earthquake itself, as in the case of the aging rate-state friction). As the fault is loaded at a uniform stressing rate, points with low σ reach “static” strength sooner than those at high σ (Figure 3a). A creeping patch may then become unstable if it exceeds a critical elasto-frictional length, or creep at constant stress otherwise. The steady-state velocity is urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0013, where κ is the stiffness, which for a region of size L is of the order of μ′/L so that urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0014. The critical length for instability (nucleation length) was first estimated from a spring-slider linear stability analysis (Ruina, 1983); later, Rubin and Ampuero (2005) used energy balance arguments to derive expressions for aging rate-state faults. In general, this critical length has the form
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0015(11)
where f(a, b) is a function of rate-state parameters a, b; for rate-state friction with the aging law and a/b = 0.75 (as in our case), f(a, b) = b/[π(b − a)2] and the nucleation length is denoted by L (Rubin & Ampuero, 2005). Expressions for nucleation length derived for a homogeneous fault cannot directly be applied to a heterogeneous one. However, linear fracture mechanics can be used to derive alternative expressions for these cases, as done by Tal et al. (2018) for rough faults with small scale (sub-Lc) roughness, and Dublanchet (2018) for heterogeneous friction. With these caveats in mind, here we appeal to the concept of a heterogeneous nucleation length as an intuitive way to relate spatial variations in normal stress to slip behavior.
Details are in the caption following the image

Conceptual model and simulation results for the evolution of stress on the fault. (a) Expected state of stress after the entire fault has ruptured (orange) and later in the cycle: points at low σ reach the end of their cycle first and start creeping (green), while asperities are still locked (blue). σ0 is the unperturbed normal stress, and a Gaussian distribution of σ due to slip (as derived in Appendix A) is shown in black. The gray lines indicate the static and dynamic strength (i.e., the steady-state strength at interseismic and coseismic slip velocities, respectively). (b–d) Shear and normal stresses from the simulation, right after an earthquake (b); during the aseismic phase of the cycle (c); toward the end of the nucleation phase (d).

Due to the inverse proportionality between Lc and σ, the first patches to reach static strength are the most stable ones (large Lc), thus favoring stable creep. During this phase we expect the average slip velocity on the fault to increase for several reasons: (1) The area of creeping patches increases as more points reach static strength, since the time to failure is given by urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0016, where Δτ = [fss(Vco) − fss(Vcr)]σ is the difference between the dynamic and static strength (Figure 3). (2) Creep on low σ patches redistribute stresses onto locked patches, contributing to the acceleration by causing points to be closer to failure than predicted from tectonic loading in Figure 3c. (3) The steady-state slip velocity on each patch increases as it widens, since the average slip velocity in the creeping regions Vcr ∼ Lcr where Lcr is the dimension of creeping patches. This leads to the interseismic acceleration seen in Figure 1. As creep occurring in low σ regions penetrates into asperities, it can cause them to fail in localized slow slip or earthquakes (velocity peaks in Figure 1). Microseismicity occurs late in the cycle since the most locked patches, where the nucleation length is small enough to allow seismic rupture, are the last to reach failure.

5 Seismicity on Strong Patches

Foreshocks occur in subclusters at multiple temporal scales: Figures 2 and 4 show three events occurring a few days before the mainshock, followed by quiescence and a second cluster about a day later; more clusters occur a few hours and a few minutes before the mainshock. Each burst represents the rupture of a group of nearby asperities (Figure 4 and Figure S2), and the relative location of each event is consistent with static stress transfer from previous ones. This gives rise to migration (e.g., events 1–8, 9–14), which can also reverse due to repeated rupture of the same asperity (e.g., events no. 1, 13, 14 and 2, 12, 14 among others). Seismic clusters are bounded by stronger or wider asperities, which typically fail in later bursts: the increase in shear stress imparted by earthquakes on surrounding low σ patches leads to a sudden creep acceleration, which in turn loads nearby asperities until they fail (see, e.g., accelerated creep at the edge of previous foreshocks leading up to events 6, 11, and 14 in Figure 4). Similarly, the mainshock initiates at the edge of the previous events and the creeping region. The asperity on which it nucleates has a higher normal stress than nearby asperities and previous foreshocks.

Details are in the caption following the image

Creep acceleration and seismicity leading up to a mainshock. (top) Slip velocity on the fault vs. time to the end of the mainshock, with red bars marking the rupture length and triangles marking the nucleation point (midpoint of the region where v > Vdyn during the first earthquake time step). The inset on the left shows normal stress at the beginning of this cycle. Note the sudden acceleration in nearby creeping patches and the widening of the fast-slipping region with each successive seismic burst. The instantaneous localized accelerations seen at ∼3–5 km, just before the arrival of the mainshock front are a consequence of the model assuming instantaneous stress changes. (bottom) Subset of the top panel, with events numbered by occurrence time. Small black dots and lines indicate the location of maximum slip velocity at each time step, showing accelerated creep at the edges of each burst, where the subsequent ones initiate. Gray panels show close ups of a few clustered foreshocks.

In spite of the elevated normal stress on asperities, foreshocks do not have particularly high stress drops (0.1–2 MPa): in agreement with Schaal and Lapusta (2019), who observed a similar behavior in 3D simulations, we find that foreshocks are not confined to asperities, but propagate into the surrounding low σ regions, thus lowering the average stress drop. The presence of such low stress-drop regions is also responsible for the partial overlap between consecutive events, even though in some cases asperities themselves rerupture (Figure 4).

5.1 Feedback Between Creep and Foreshocks

The average slip velocity during the foreshock sequence increases in sudden steps after failure of one more asperities (Figures 4 and 5a). The acceleration occurs even at large distances from the foreshocks compared to their rupture dimension, so that foreshocks contribute to widening the fast-creeping area. Average slip velocities on the fault increase approximately with the inverse of time to mainshock (Figure 5), similar to studies of velocity-weakening asperities embedded in a velocity strengthening (creeping) fault (Dublanchet, 2018; Yabe & Ide, 2018). However, neither asperities nor creeping patches follow this trend individually (Figure 5b).

Details are in the caption following the image

Slip velocity and seismicity rates during the foreshock sequence shown in Figure 4. (left) Black solid line: average slip velocity in the mainshock nucleation region (between 2.2 and 4.5 km) versus time to the end of the mainshock. Red circles: seismicity rates estimated by the inverse of intervent times, plotted at the midpoint between each pair of events. The y axes are scaled with respect to one another according to eq. C1. The theoretical evolution of slip velocity (Equation 14) is indicated by the dotted line (for the median value of foreshock stress drop) and gray band (for the entire range of stress drops). (right) Slip velocity vs. time for the asperity (A) and a nearby creeping patch (B), which are identified in Figure 4. The horizontal line indicates the threshold velocity Vdyn used to define earthquakes.

To understand the effect of a seismic rupture on weak patches, consider the change in velocity caused by an instantaneous shear stress perturbation Δτ through the direct effect
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0017(12)
where V0 is the starting velocity. For a given stress change, areas at low normal stress are particularly susceptible to stress increases due to foreshocks, even if they are several rupture lengths away. As an example, Figure 5b shows slip velocities on the asperity which ruptured in a foreshock (event no. 8 in Figure 4) and a nearby creeping patch, marked in Figure 4. After the earthquake, the asperity does not fully relock, but continues slipping about four orders of magnitude faster than it did before. This behavior can be explained by the faster loading rate from the nearby creeping patches, which prevents the asperity from fully relocking. We can gain some intuition into this by treating the asperity as a spring-slider driven at a constant stressing rate, which in turn depends on the creep rate around it. The solution for velocity evolution derived in Appendix B predicts that the minimum slip speed right after an earthquake grows with stressing rate urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0018:
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0019(13)

After a mainshock, urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0020 (the background loading rate); during the nucleation phase, creep velocities adjacent to the asperities increase (in this case, Vcr ∼ 1 × 10−8 m/s; see Figure 5), giving a stressing rate on the asperity of the order of urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0021 = 4 Pa/s, here about 103 times larger than the background loading rate urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0022. Plugging these numbers in the expression above, we expect Vlock after the foreshock to be about ∼104 times larger than its minimum value early in the cycle, consistent with the simulation (Figure 5). The creeping patches and asperities subsequently decelerate, but the asperity slip velocity remains several orders of magnitude larger than before rupture (Figures 4 and 5).

The positive feedback between creep rates and seismicity rates leads to an overall acceleration and expansion of the creeping region. In Appendix C, we derive a simple analytical expression based on the observations described above. It relies on the following assumptions: (1) seismicity rate is proportional to average creep rate; (2) creep rates increase by a constant factor after each foreshock (derived from Equation 12), and do not change otherwise. This simple model predicts that the average slip velocity evolves as
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0023(14)
where L is the dimension of the nucleation region, Δτ is the foreshock stress drop and β is a factor quantifying the increase in creep velocity after each foreshock; t is time since the first foreshocks and t0 is the time to instability, given by
urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0024(15)
where <V0> is the average slip velocity before the first foreshock. We estimated β by applying Equation 12 to the creep patches in the nucleation region, and treating foreshocks as uniform stress drop cracks of fixed size, and we obtained values between 1.1 and 1.3 (the range is given by variability in foreshock stress drops). Overall, the average slip velocity in the nucleation region increases approximately as predicted by this expression (Figure 5a).

5.2 Stacked Foreshock and Aftershock Catalogs

The prediction of 1/t acceleration in creep rates and seismicity rates does not account for temporal clustering due to elastic interactions between asperities, visible in Figure 5. Therefore, the 1/t acceleration in seismicity rates may not be readily visible in individual catalogs. To better capture temporal patterns, we stack the catalogs from all cycles. All foreshocks-aftershock sequences are shifted so the mainshock occurs at t = 0, and then combined in a single catalog. As shown in Figure 6a, the rate of foreshocks increases with the inverse time to the mainshock, as observed for stacked catalogs of natural sequences (Jones & Molnar, 1979; Ogata et al., 1995).

Details are in the caption following the image

Moment per unit length and interevent times in the stacked catalog. (a) Seismicity rates estimated as the inverse of interevent time showing power-law acceleration. The dotted line is proportional to 1/t. (b) Moment per unit length as a function of time to mainshock. Open circles indicate mainshocks.

5.3 Onset of Foreshocks and Mainshock

The occurrence of foreshocks in the vicinity of the mainshock hypocenter raises the following question: why do some ruptures arrest, while others in the same region grow into large events? Figure 4 shows that the mainshock, like most foreshocks, nucleates at the edge of a fast-creeping region, on an asperity which arrested the previous event. The mainshock nucleation asperity has the highest normal stress on the entire fault. To verify whether other mainshocks also nucleate on high σ asperities, we compare normal stresses in the nucleation region of mainshocks and nearby foreshocks. Figure 7 shows that mainshocks tend to nucleate on stronger asperities than most of their foreshocks. This may not be surprising in light of the simple model shown in Figure 3, since patches with higher normal stress take longer to reach static strength. Once a strong asperity breaks, its stress drop is high and leads to a more pronounced stress concentration at its edge, allowing it to grow further than earlier events. This also explains why larger foreshocks tend to occur later in the cycle (Figure 6b).

Details are in the caption following the image

Difference between average normal stress in the nucleation region of foreshocks and their respective mainshocks. Nucleation is defined as the region between points exceeding a velocity threshold at the beginning of an earthquake (see Section 2). We consider mainshocks as events with a rupture length exceeding 2 km, and only select foreshocks within the mainshock rupture area. Numbers indicate Δσrms/σ0 for each sequence.

Rupture arrest is also determined by the strength of asperities ahead of the rupture tip, which act as barriers. We consider all asperities which are either within or adjacent to a rupture, and as expected we find that stronger (higher normal stress) asperities are more likely to arrest ruptures. We also find that a rupture nucleating at normal stress σnuc has a 62% probability of breaking an asperity with normal stress exceeding σnuc, and a 77% chance of breaking an asperity with normal stress lower than σnuc. A selection bias could originate when grouping asperities according to this criterion: on average, asperities with σasp > σnuc for a given earthquake are stronger than those with σasp < σnuc. However, we find that a difference remains when comparing asperities with approximately the same normal stress, indicating that σnuc also affects rupture arrest.

6 Discussion

The results presented above show that the preseismic phase on a velocity-weakening fault with fractal roughness is characterized by a complex interplay between slow slip and foreshocks. Most of the period between mainshocks is devoid of seismicity, and characterized by localized patches of slow slip; late in the cycle, strong asperities start failing in short bursts, each of them in turn accelerating creep in its neighborhood. This process leads to acceleration over an extended region (here about 20 times larger than the nominal nucleation dimension), with migration of seismicity toward the mainshock hypocenter.

6.1 Model Limitations

The central result of this study is the coexistence and interaction of slow slip and foreshocks during nucleation on a rough fault. The primary control on this mixed behavior are normal stress perturbations due to roughness, and their effect on fault stability and slip patterns (Section 4). These findings are not specific to rate-state (aging law) friction, and likely apply for other frictional laws and weakening mechanism. On the other hand, certain simplifications in our study may be more consequential and deserve further investigation. The quasi-dynamic approximation can affect rupture velocity, rupture arrest and lengths, even though based on previous planar fault studies (Lapusta et al., 2000; Thomas et al., 2014) we do not expect the qualitative pattern to change dramatically with aging-law rate-state friction. Considering the three-dimensional nature of fault surfaces can modify certain aspects of fault dynamics, such as the ability of an asperity to arrest rupture or the migration patterns caused by stress redistribution. In particular, static stress changes extend further in 2D than 3D. Another significant assumption in our study is the purely elastic response: inelastic processes would limit the amplitude of stress perturbations, in particular at the smallest length scales (e.g., Dunham et al., 2011).

6.2 Conditions for Foreshock Occurrence

The dimension of asperities relative to characteristic elasto-frictional length scales is expected to affect foreshock behavior. Previous numerical studies of foreshocks on heterogeneous faults found that foreshocks only occur in a particular regime (Dublanchet, 2018; Schaal & Lapusta, 2019): asperities must be larger than the local nucleation dimension for seismic slip to occur, but smaller than a critical dimension (such as the nucleation dimension outside the asperity) to arrest without generating system-size ruptures. Here, the amplitude of spatial variations in σ controls the range of local nucleation lengths Lc. As more slip accrues and normal stress perturbations grow, the nucleation length shrinks on the asperities and grows around them: therefore, microseismicity only appears for sufficiently large normal stress perturbations (here Δσrms ≈ σ0).

A similar transition from few large ruptures to many smaller ones was found by Heimisson (2020) when increasing kmax; since the amplitude of normal stress perturbations grows with kmax (Equation 10), this is consistent with our findings. Similarly, we expect that increasing fault roughness would have the same effect, since Δσrms increases with the product of roughness and accrued slip. In our simulations, we chose kmax ∼ 2π/L, for computational efficiency. To verify the effect of smaller wavelengths, we also ran simulations for a smaller domain and kmax up to 4 times higher (Figure S2). We find that the presence of sub-L asperities leads to more frequent aseismic ruptures (similar to those in Figure 1). Both seismic and aseismic failures contribute to a gradual unpinning of the fault, as described above. The temporal evolution of slip velocities, with an abrupt increase during bursts and an overall 1/t trend, is similar to the nominal case.

6.3 Preslip vs. Nucleation on Rough Faults

In the “preslip” model, aseismic slip is generally understood to occur at the location of the mainshock hypocenter, reflecting the notion that seismic instabilities develop over a region of finite size, as predicted by spring-slider and continuum models (e.g., Rubin & Ampuero, 2005; Ruina, 1983). It is conceivable that heterogeneity within the nucleation region could lead to foreshocks driven by accelerating slip (e.g., Noda et al., 2013); however, our results favor a different interpretation. Here, the large scale precursory accelerating slip is not mainshock nucleation in the classical sense: since slow slip occurs in stable low σ patches which do not accelerate when subject to slow loading, it does not directly evolve into a seismic rupture. Instead, slow slip triggers smaller scale nucleation on locked asperities, which can remain small or grow into a mainshock.

A similar relationship between preslip and mainshock initiation in presence of heterogeneity has been has been inferred in laboratory experiments. McLaskey and Lockner (2014) observed acoustic emissions (analogous to foreshocks) and slow slip leading up to failure in a centimeter-scale laboratory sample, and noted that system-size ruptures begin as acoustic emissions, with local strength variations perhaps controlling whether they evolve into larger ruptures. Similarly, meter-scale experiments by McLaskey (2019) show evidence of abrupt earthquake initiation caused by creep penetration from weak regions into a locked patches, “igniting” large ruptures.

The migratory behavior of microseismicity, and the earthquake hypocenter on the edge of the creeping region, also indicate a different mechanism than self-nucleation. Recent observations of precursory slip leading up to glacial earthquakes by Barcheck et al. (2021) are similar to our results: slow slip and microseismicity migrate toward the mainshock hypocenter. Similar seismicity migration has also been observed prior to several events (Tohoku, 2011, Kato et al., 2012; Iquique, Brodsky & van der Elst, 2014; l’Aquila, Sugan et al., 2014), and it is sometimes interpreted as evidence for aseismic slip.

On the other hand, migratory behavior can also be interpreted as evidence for direct triggering between foreshocks: seismicity prior to the 1999 Izmit (Ellsworth & Bulut, 2018) and 1999 Hector Mine (Yoon et al., 2019) exhibit a cascade behavior similar to that observed here (Figure 4): successive failure of neighboring asperities, with each event nucleating at the edge of the previous ones, and rare rerupture of the same asperity (as in Figure 4). Here, we find that the migration is in some cases caused by direct stress triggering (leading to rapid failure of nearby asperities in a short burst), but it can also be mediated by accelerated creep between asperities. Note that direct stress transfer between asperities would be less effective in 3D, and aseismic slip is therefore likely to play a more important role in this geometry (see also Lui & Lapusta, 2016).

Stacked earthquake catalogs exhibit a gradual power-law acceleration (Bouchon et al., 2013; Jones & Molnar, 1979; Ogata et al., 1995), analogous to Figure 6. However, individual sequences typically display more irregular patterns: Chen and Shearer (2013) observed burst-like behavior for foreshock sequences in California, and Kato et al. (2016) documented abrupt changes in seismicity and aseismic slip prior to the 2014 Iquique earthquake. A gradual 1/t acceleration is predicted by spring-slider models of nucleation on rate-state faults (Dieterich, 1992; Rubin & Ampuero, 2005); on the other hand, Helmstetter and Sornette (2003) derived the same results from earthquake triggering with foreshocks producing offspring at a rate given by the Omori-Utsu law, without requiring aseismic slip, and Felzer et al. (2015) invoked the same mechanism to explain the apparent acceleration prior to large interplate earthquakes. Here, we suggest that both processes are at play, and demonstrate that a 1/t acceleration can arise from the interaction of seismic failure on asperities and the surrounding creeping regions. Unlike seismicity driven by gradually accelerating slow slip, in this case both earthquake rates and slip velocities increase in abrupt steps, so that the power-law behavior is visible for stacked catalogs but not for individual sequences.

An intriguing observation is the occurrence of earthquakes in the vicinity of a future mainshock hypocenter. The 2004 Mw6 Parkfield and the Mw9 Tohoku earthquakes were both preceded by moderate events within a few years of the mainshock, a much shorter timescale than the respective earthquake cycles. Based on our results, which should be further verified with fully dynamic simulations, we suggest that local strength variations between potential nucleation patches within a small region may determine which earthquakes evolve into destructive events.

7 Conclusions

We find that fault roughness can lead to simultaneous occurrence of aseismic slip and foreshocks in the precursory phase of mainshocks, modulated by normal stress variations caused by fault geometry. The precursory phase can be described as a gradual unpinning of the fault by episodic asperity failure, mediated by aseismic slip. The creeping area widens and accelerates through each seismic burst, leading to migration of seismicity toward the eventual mainshock hypocenter. A simple model for the positive feedback between creep and seismicity predicts that slip accelerates as 1/t, as confirmed by the simulations.

This process results in precursory slip on a larger scale than, and spatially distinct from, classical rate-state nucleation on flat faults. Our results provide a physical interpretation for laboratory and field evidence of migratory preslip and foreshocks in the vicinity of a future mainshock hypocenter.

Acknowledgments

We would like to thank Aitaro Kato, John Rudnicki, and the Associate Editor whose constructive reviews helped improve and clarify this manuscript. C. Cattania was funded by SCEC award 18166 and NSF award 1620496.

    Appendix A.: Normal Stress Variations

    Here we derive the spatial distribution of normal stresses due to slip on a rough fault with small perturbations in elevation y(x) (i.e., distance from the mean fault position). Fang and Dunham (2013) derived the following expression for normal stress perturbations due to uniform unit slip
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0025(A1)
    where μ′ = μ/(1 − ν) and compressive stresses are positive. The elevation profile can be written as
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0026(A2)
    Taking the second derivative and inserting into Equation A1 gives
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0027
    where u = ξ − x. We use the following results:
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0028
    Thus, the inner integral takes the value of and
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0029(A3)
    where we have scaled the stress by the total slip S. The integral has a form similar to the second derivative of the topography, but with a phase shift of π/2 in each Fourier component. This result is consistent with the findings of Romanet et al. (2020), who demonstrated that normal stress perturbations on a curved fault are proportional to the local curvature (which to first order is equal to the second derivative of the slope). The phase shift can be intuitively understood by considering a sinusoidal profile: a phase shift of π/2 places maximum compressive and tensile stresses at the inflection point of restraining and releasing bends (see Figure A1). Since stress perturbations depend on the second derivative of the elevation profile, they are dominated by the shortest wavelengths. We emphasize that in the simulations normal stress perturbations are not prescribed or computed by the above expressions. Rather they arise in the boundary element calculations from stress transfer between elements with variable orientation (Section 2). Nevertheless, the analytical result may prove useful to approximate the effect of roughness by imposing normal stress perturbations on a planar fault (e.g., Schaal & Lapusta, 2019), even though this method would not account for the increase in perturbations with slip.

    Roughness also affects shear stresses on the fault. The two are related by Equation 1, which during most of the cycle can be approximated as τel ≈  (and since with rate-state friction fractional changes in f are small compared to σ, τel ≈ f0σ). Equilibrium is achieved by a heterogeneous slip distribution modulating stresses. On a fault with small deviations from planarity, slip gradients efficiently modify shear stresses, but have little effect on normal stresses (this can be understood intuitively from the fact that slip on a planar fault has no effect on σ; for a more general derivation, see Romanet et al., 2020). Therefore, roughness induced variations in τ are accommodated by slip gradients, while the normal stress distribution remains virtual unchanged between large earthquakes and determines the shear stress profiles.

    Details are in the caption following the image

    (top) Normal stresses from BEM calculations used in FDRA (blue) and Equation A3 (dotted yellow), as a function of normalized position, with unit slip, normalized by μ′/2. Black: fault profile rescaled by a factor of 500. (bottom) Zoomed in (inset in top figure), with fault profile scaled by 4,000, showing normal stress perturbations corresponding to releasing and restraining bends.

    A1 Self-Similar Roughness

    Consider a fault with a profile y characterized by power spectral density
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0030(A4)
    between kmin = 2π/L and kmax, with β = 2H + 1 and H is the Hurst exponent. Using Parseval’s theorem, it can be shown that the root-mean-square elevation in the limit kmax ≫ kmin is
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0031(A5)
    where α is the surface roughness. Similarly, by applying Parseval’s theorem to the second derivative of y we obtain the root-mean-square value
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0032(A6)

    Here, we used fractal surfaces with random phases, resulting in a Gaussian distribution in y(x); y′′(x) is also Gaussian with standard deviation urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0033 (e.g., Persson et al., 2005). Combining this result with Equation A3, we find that normal stress perturbations are Gaussian distributed with zero mean and standard deviation urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0034, where S is the accrued slip.

    Appendix B: Spring-Slider

    To obtain the interseismic evolution of slip velocity, we consider a spring-slider with stiffness κ driven at constant rate urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0035:
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0036(B1)
    where δ is slip and τ0 is shear stress at time t = 0 (see also Rubin & Ampuero, 2005, Equation A12). Since we are interested in the velocity during the interseismic phase, the radiation damping term is not included. Time is measured since the last earthquake, and τ0 is the residual stress after rupture. More specifically, we define t = 0 as the moment when the system last crossed steady-state, and
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0037(B2)
    with Vdyn as defined in Section 2. Inserting Equation B2 into Equation B1 and solving for V gives
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0038(B3)
    further assuming that the fault is locked (/ ≪ 1) and far below steady-state (θ ∼ t), velocity evolves as
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0039(B4)
    The minimum velocity occurs at urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0040 and is given by
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0041(B5)

    Appendix C: Preseismic Acceleration

    As discussed in Section 5.1, the acceleration leading up to the mainshock is controlled by a feedback between creep in low normal stress patches and foreshocks on asperities. Here we develop a simple model of these interactions and the temporal evolution of acceleration.

    Seismicity rate is controlled by the surrounding creep rate, which for simplicity we take as uniform. The interevent time on a single asperity is of the order of urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0042, where Δτ is the stress drop. Note that this expression does not apply if some interseismic slip takes place within the rupture area; however, Cattania and Segall (2019) obtained a similar expression, within a factor of order one, allowing for creep to penetrate the asperity. The overall seismicity rate on the fault is therefore urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0043, where N ≈ L/Lmin is the number of asperities in a nucleation region or length L. During nucleation we can neglect tectonic loading, so urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0044, with κ ∼ μ′/2Lmin so that the seismicity rate is
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0045(C1)
    where n is the cumulative number of foreshocks, and 〈V〉 denotes average slip velocity. We further assume that each earthquake increases the average creep rate by a constant factor β, derived below, and we neglect self-acceleration of creeping patches. Slip velocities are then given by
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0046(C2)
    where <V0> is the average slip velocity before the first foreshock. Differentiating Equation C2 and combining with Equation C1 results in
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0047(C3)
    which has solution
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0048(C4)
    where t is time since the first foreshocks and t0 the time to instability, given by
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0049(C5)
    Note that we assumed that the creep velocity remains high after each foreshock. For a creep patch of fixed dimension (stiffness) subject to a sudden stress increase, we would instead expect velocity to decay to the steady-state value determined by the background loading rate; however, simulations show that creep velocities remain high after each step (Figures 4 and 5), possibly due to the reduction in stiffness after each foreshock described in Section 5.1.

    The functional form of Equations C1 and C2 is not expected to change in 3D (even though β and the prefactor in Equation C1 will differ). Therefore, we expect the main result of this analysis, which is the growth of velocity as the inverse of time to instability, to remain valid.

    Appendix C1. Estimating

    To obtain a rough estimate of β, the fractional change in creep rate due to a foreshock, we consider a simple model of periodic locked asperities alternating creeping patches (Figure C1). We assume that asperities break in events with uniform stress drop, confined to a single asperity and the creeping patch on each side, with the next asperity acting as barrier. Since the response to stress changes is dominated by regions with low σ, we consider the change in velocity in creeping patches only.

    Details are in the caption following the image

    Simple model used to estimate changes in creep rate after a foreshock. (top) Schematic spatial distribution of normal stress. (middle) Shear stress change caused by a constant stress drop crack normalized by stress drop. (bottom) Foreshock slip distribution. Dotted lines and circles indicate the center of creeping patches and locations at which stress changes are calculated.

    The stress field outside a constant stress drop crack of length 2l and stress drop Δτ is (Bonafede et al., 1985)
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0050(C6)
    where x is the distance from the crack center on the crack plane. Since the system is symmetric around x = 0, in what follows we consider x > 0. We approximate the stress change within each creeping patch by the value at its center; as shown in Figure C1, creeping patches are centered at positions x = 2l, (2 + 4/3)l, (2 + 8/3)l, …. The stress change at position x = nl is given by
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0051(C7)
    The local velocity after a stress step given by the direct effect is
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0052(C8)
    where Vpre is the velocity before the stress step and σ is the normal stress in creeping patches. Assuming the same initial velocity Vpre in all creeping patches, the new average velocity is the sum of the velocity change in each patch divided by the total number of creeping patches Np
    urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0053(C9)
    where ni = 2 + 4i/3. The fractional change in slip velocity is simply β = 〈V〉/Vo. At the onset of the foreshock sequence considered in the main text, slip velocities in creeping patches are of the order of 10−11 m/s (as expected from urn:x-wiley:21699313:media:jgrb54795:jgrb54795-math-0054), and their average normal stress is about 5 MPa. Foreshocks have stress drops between 0.1 and 2 MPa, with a median value of 0.5 MPa. Considering the nucleation region between 1.7 and 4.7 km (Figure 4), the number of creeping patches is ≈3 km/Lmin = 30; and since the analysis above only considers one side of the fault, Np = 15. Plugging these values into Equation C9 gives β between 1.1 and 1.3, depending on the stress drop.

    Data Availability Statement

    No data were used in this study.