# Earthquake Nucleation

Earthquakes nucleate as accelerating slip over a region of finite size.  Understanding what controls the length and time scales of this process is important for assessing earthquake early-warning systems and for interpreting data obtained by monitoring earthquake-generating faults within tens to hundreds of meters of the source (such as at the 2.5-km-deep San Andreas Fault Observatory at Depth, or SAFOD; see Figure 1).

Theoretical models of earthquake nucleation require coupling the equations of elasticity with a constitutive law for the evolving strength of the fault surface.  For more than 2 decades the most complete constitutive laws have been various incarnations of “rate- and state dependent friction”, meaning that the frictional strength depends upon the micromechanical state of the fault surface as well as the current slip speed.  Despite this lengthy history, it is most likely that no numerical simulation has ever employed the “correct” constitutive law.  This is because no proposed law fits even all the available experimental data, let alone the conditions of temperature, pressure, and fluid chemistry that might be appropriate in situ.  To make matters worse, the strongly nonlinear nature of friction has made it very difficult to obtain an intuitive understanding of the differences between the underlying equations that people use.  This combination of poorly-constrained constitutive laws and opaque equations is a significant impediment to extrapolating from numerical simulations to fault slip in the Earth.

Over the last several years, the work I have done with post-doc Jean-Paul Ampuero (now an assistant professor at Caltech) has gone a log way toward developing this intuitive understanding.  Using standard methods of fracture mechanics, we now have analytic expressions for the length and time scales of nucleation under the most commonly-used law for the evolution of state (the “aging” law; see Figure 2).  Under this law, nucleation zones can grow to be so large that they might often be detectable from the Earth’s surface.  However, our analytical solutions also let one see immediately that the properties of the aging law that generate these large nucleation zones are directly contradicted by lab experiments. This has led to a collaboration with Chris Marone (Penn State), where we recently showed that lab data relevant to earthquake nucleation are much more consistent with the “slip” evolution law (Figure 3).

 Figure 2. Snapshots of slip speed, from 10-11 to 1 m/s, from a numerical simulation of nucleation under the “aging” evolution law.  Linf is our analytical prediction of the nucleation length for these parameters. The normalizing length scale Lb could be anywhere from centimeters to many meters. Figure 3. Measured change in friction (black curve) as a function of slip, for step velocity increases and decreases of first 1 and then 2 orders of magnitude, on simulated quartz gouge (glass beads). Data were obtained in Chris Marone’s lab at Pen State. The symmetric response to velocity increases and decreases, and the similar decay distance for the different magnitudes of velocity step, are consistent with the slip law (red curve) but not the aging law.

Our work shows that if lab results can be safely extrapolated to the Earth (a very big if!), the “slip” law is the proper law for studies of earthquake nucleation. Nucleation under the “aging” and “slip” laws is entirely different, taking the form of an expanding crack in the first case and a unidirectional slip pulse in the second (Figure 4). The extent of this difference is surprising in that both laws have been advertised as being adequate at some level. However, this difference is understandable in terms of the much larger fracture energy at the edge of the expanding nucleation zone implied by the “aging” law. Nucleation under the “slip” law is much less likely to be observable by surface instruments.

## Related publications:

75 Publications
Journal Article
High-resolution three-dimensional discrete element method (DEM) simulations of sandbox-scale models of accretionary wedges suggest thrusts follow a variety of propagation processes and orientations depending on a number of factors. These include the stage of development of the wedge (precritical vs. critical), basal friction, and type of thrust …
Journal Article
Nearly all frictional interfaces strengthen as the logarithm of time when sliding at ultra-low speeds. Observations of also logarithmic-in-time growth of interfacial contact area under such conditions have led to constitutive models that assume that this frictional strengthening results from purely time-dependent, and slip-insensitive, contact…
Journal Article
During rock friction experiments at large displacement, room temperature and humidity, and following a hold test, the fracture energy increases approximately as the square of the logarithm of hold duration. While it s been long known that failure strength increases with log hold time, here the slip weakening distance, dh, also increases. The…
Journal Article
The empirical constitutive modeling framework of rate- and state-dependent friction (RSF) is commonly used to describe the time-dependent frictional response of fault gouge to perturbations from steady sliding. In a previous study (Ferdowsi & Rubin, 2020), we found that a granular-physics-based model of a fault shear zone, with time-independent…
Journal Article
Although tremor is believed to consist of myriad low-frequency earthquakes (LFEs), it also contains longer-period signals of unknown origin. We investigate the source of some of the longer-period signals by locating tremor windows independently in relatively high-frequency (“HF”, 1.25–6.5 Hz, containing typical LFEs) and low-frequency (“LF”, 0.5–1…
Journal Article

Rate‐ and state‐dependent friction (RSF) equations are commonly used to describe the time‐dependent frictional response of fault gouge to perturbations in sliding velocity. Among the better‐known versions are the Aging and Slip laws for the evolution of state. Although the Slip law is more successful, neither can predict all the robust features…

Journal Article

Abstract In this study, we analyze high-resolution tremor catalogs from northern Cascadia, Guerrero, and northern Kii Peninsula. We find that tremor often occurs in short bursts that repeatedly occupy the same source area within a slow slip event. We hypothesize that these bursts are driven by loading from slow slip in areas surrounding the…

ISBN
0094-8276
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Journal Article

The popular constitutive formulations of rate-and-state friction offer two end-member views on whether friction evolves only with slip (Slip law) or with time even without slip (Aging law). While rate stepping experiments show support for the Slip law, laboratory-observed frictional behavior near zero slip rates has traditionally been inferred…

ISBN
2169-9313
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Journal Article

Abstract Whether rate- and state-dependent friction evolution is primarily slip dependent or time dependent is not well resolved. Although slide-hold-slide experiments are traditionally interpreted as supporting the aging law, implying time-dependent evolution, recent studies show that this evidence is equivocal. In contrast, the slip law…

ISBN
2169-9313
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Journal Article

Abstract Slow slip events exhibit significant complexity in slip evolution and variations in recurrence intervals. Behavior that varies systematically with recurrence interval is likely to reflect different extents of fault healing between these events. Here we use high-resolution tremor catalogs beneath Guerrero, Mexico, to investigate the…

ISBN
0094-8276
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Journal Article

Abstract We have recently suggested that the nearly constant duration of low-frequency earthquakes (LFEs) (and, equivalently, the band limitation of tectonic tremor) manifests a moment-duration scaling that is fundamentally different from regular earthquakes and is most easily explained as rupture on asperities of roughly constant dimension. In…

ISBN
2169-9313
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Journal Article

Eruptions on the ice moon Enceladus provide access to materials from Enceladus’ ocean. The mechanism that drives and sustains the eruptions is unclear, and it is also not known what sets the rate of volcanism. We found that a simple model in which the erupting fissures are underlain by slots that connect the surface to the ocean can explain the…

Journal Article

Abstract Episodic tremor and slip (ETS) in subduction zones is generally interpreted as the manifestation of shear slip near the base of earthquake-generating portion of the plate interface. Here we devise a new method of cross-correlating stacked Array of Arrays seismic data that provides greatly improved tremor locations, a proxy for the…

ISBN
1525-2027
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Journal Article
We develop a cross-station method to detect and locate tremor and low-frequency earthquakes (LFEs), based on the original work of Armbruster et al. (2014) that compares waveforms from the same time window at stations separated by roughly 10 km. To improve the signal-to-noise ratio, we first rotate the horizontal components into the empirical shear…
Journal Article

Abstract We employ 130 low-frequency earthquake (LFE) templates representing tremor sources on the plate boundary below southern Vancouver Island to examine LFE magnitudes. Each template is assembled from hundreds to thousands of individual LFEs, representing over 269,000 independent detections from major episodic-tremor-and-slip (ETS) events…

ISBN
2169-9313
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Journal Article

Abstract The variations in the response of different state evolution laws to large velocity increases can dramatically alter the style of earthquake nucleation in numerical simulations. But most velocity step friction experiments do not drive the sliding surface far enough above steady state to probe this relevant portion of the parameter space…

ISBN
2169-9313
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Journal Article

Nonvolcanic tremor is an important component of the slow slip processes which load faults from below, but accurately locating tremor has proven difficult because tremor rarely contains clear P or S wave arrivals. Here we report the observation of coherence in the shear and compressional waves of tremor at widely separated stations which allows…

ISBN
2169-9313
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Journal Article

Earthquake ruptures on the San Andreas Fault are affected by the material contrast across the fault. Previous observations of microearthquakes at the northern end of the creeping section have found strong signals of asymmetry in both rupture directivity (preferential propagation to the SE), and aftershock asymmetry (many more to the NW, on…

ISBN
0956-540X
Journal Article

A new state evolution law has recently been proposed by Nagata et al. (2012) that includes a dependence upon stressing rate through a laboratory derived proportionality constant c. It has been claimed that this law, while retaining the time-dependent healing of the Dieterich (or Aging) law, can also match the symmetric response of the Ruina (or…

ISBN
2169-9313
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Journal Article

We apply a new method to obtain accurate locations of tremor sources beneath southern Vancouver Island. Unlike more standard "cross-time" methods, which compare waveforms from different time windows at the same station, this "cross-station" method compares waveforms from the same time window at widely separated stations. It performs well,…

ISBN
1525-2027
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Journal Article

Zhou et al. (2012) proposed that longitudinal dunes in the Qaidam Basin, China, formed like yardangs: by erosion into sediment that was not deposited by those dunes. Because erosion occurs on the upwind fl anks of most migrating dunes (Rubin and Hunter, 1982, 1985), the key to demonstrating a yardang-like origin is to show that the dunes did…

ISBN
0016-7606
Journal Article

We use borehole strain and seismic data to show that slow slip and tremor in central Cascadia are correlated on a range of time scales shorter than 1 day. The recorded strain rate is our proxy for the slow slip moment rate, and the seismic amplitude is our proxy for the tremor amplitude. We find that, on average, the strain rate is higher when…

ISBN
2169-9313
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Journal Article

We investigate the behavior of simulated slow slip events using a rate and state friction model that is steady state velocity weakening at low slip speeds but velocity strengthening at high slip speeds. Our simulations are on a one-dimensional (line) fault, but we modify the elastic interactions to mimic the elongate geometry frequently…

ISBN
2169-9313
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Journal Article

We examine tidal modulation and back-propagating fronts in simulated slow slip events using a rate and state friction law that is steady state velocity weakening at low slip rates and velocity strengthening at high slip rates. Tidal forcing causes a quasi-sinusoidal modulation of the slip rate during the events, with the maximum moment rate…

ISBN
2169-9313
Keywords
Journal Article

Since the discovery of slow slip events along subduction zone interfaces worldwide, dense geodetic and seismic networks have illuminated detailed characteristics of these events and associated tremor. High-resolution observations of tremor, where the spatial-temporal evolution is presumed to reflect that of the underlying slow slip events, show…

ISBN
0094-8276
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Journal Article

Earthquakes often occur on faults separating materials with different elastic properties. On theoretical grounds, it is expected that earthquakes on such bimaterial interfaces might have a preferred rupture propagation direction, that being the direction of motion of the more compliant material. The goal of this paper is to determine whether a…

ISBN
0956-540X
Journal Article

A striking observation from both Cascadia and Japan is that the tremor associated with slow slip often migrates along strike at speeds close to 10 km/d but updip and downdip at speeds approaching 100 km/h. In this paper I adopt the view that the friction law appropriate for these regions is unknown, and I ask what constraints the observed…

ISBN
1525-2027
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Journal Article

Several studies have shown that the seismic tremor in episodic tremor and slip is tidally modulated, suggesting a sensitivity to the rather small tidal stresses. We address whether the slip rate in slow slip events is also tidally modulated by examining data from six borehole strainmeters in northwest Washington and southern Vancouver Island…

ISBN
0148-0227
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Journal Article

The mechanics of slow slip events (SSE) in subduction zones remain unresolved. We suggest that SSE nucleate in areas of unstable friction under drained conditions, but as slip accelerates dilatancy reduces pore pressure p quenching instability. Competition between dilatant strengthening and thermal pressurization may control whether slip is…

ISBN
0148-0227
Keywords
Journal Article

In the vicinity of episodic aseismic transients in several subduction zones, the presence of interstitial fluids and near-lithostatic pore pressure has been proposed to interpret seismic observations of high P to S wave speed ratio and high Poisson's ratio. Under such conditions, fault stabilization by dilatancy-induced suction during increased…

ISBN
0148-0227
Keywords
Journal Article

For a wide range of conditions, earthquake nucleation zones on rate- and state-dependent faults that obey either of the popular state evolution laws expand as they accelerate. Under the "slip" evolution law, which experiments show to be the more relevant law for nucleation, this expansion takes the form of a unidirectional slip pulse. In…

ISBN
0148-0227
Keywords
Journal Article

There are several ways of generating episodic slow slip events in models of rate-and-state friction. Here I explore the possibility that they arise on velocity-weakening faults whose length is "tuned" in some sense. Unlike spring-block sliders, which have a unique critical stiffness for instability, elastically deformable faults have…

ISBN
0148-0227
Keywords
Journal Article

We compare 2-D, quasi-static earthquake nucleation on rate-and-state faults under both "aging" and "slip" versions of the state evolution law. For both versions mature nucleation zones exhibit 2 primary regimes of growth: Well above and slightly above steady state, corresponding respectively to larger and smaller fault weakening rates. Well…

ISBN
0148-0227
Keywords
Journal Article
To better understand the asymmetric distribution of microearthquake aftershocks along the central San Andreas fault, we study dynamic models of slip-weakening ruptures on an interface separating differing elastic half-spaces. Subshear ruptures grow as slightly asymmetric bilateral cracks, with larger propagation velocities, slip velocities, and…
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Journal Article

Taiwan's 1999 Mw 7.6 earthquake generated over 85 km surface rupture along the Chelungpu thrust fault. Paleoseismic studies at the Shi-Jia site near Nantou city, reveal folding as the predominant form of deformation. Stratigraphic relations across the 1999 fold scarp show the style and degree of deformation caused by the penultimate event is…

ISBN
0148-0227
Keywords
Journal Article

Recent relocation and focal mechanism analyses of deep earthquakes beneath Kilauea volcano, Hawaii indicate that seismicity is concentrated on a horizontal fault zone at a depth of 30 km, with seaward slip of the upper block on a low-angle plane. We discuss whether the observed localization of the earthquakes can be explained primarily by…

ISBN
0956-540X
Journal Article

Correction to “Aftershock asymmetry on a bimaterial surface”

Journal Article

[1] To better understand the asymmetric distribution of microearthquake aftershocks along the central San Andreas fault, we study dynamic models of slip‐weakening ruptures on an interface separating differing elastic half‐spaces. Subshear ruptures grow as slightly asymmetric bilateral cracks, with larger propagation velocities, slip…

Journal Article

We obtain quasi-static, two-dimensional solutions for earthquake nucleation on faults obeying Dieterich's ?aging? version of the rate and state friction equations. Two distinct nucleation regimes are found, separated by roughly a/b ? 0.5, where a and b are the constitutive parameters relating changes in slip rate V and state ? to frictional…

ISBN
0148-0227
Keywords
Journal Article

[1] The recurrence intervals for 194 repeating clusters on the Calaveras fault follow a power‐law decay relation with elapsed time after the 1984 M6.2 Morgan Hill, California, mainshock. The decay rates of repeating aftershocks in the immediate vicinity of a high‐slip patch that failed during the mainshock systematically exceed those that…

Journal Article

We use relocated catalogs of microearthquakes to investigate earthquake interaction along sections of the Sargent, Calaveras, and San Andreas faults in California. We examine the stress dependence of seismicity rate change along the three fault segments and find that the seismicity rate following a mainshock decays approximately as 1/time, the…

ISBN
0037-1106
Journal Article

Dieterich [1994] modeled the response to a stress step of a population of faults governed by rate- and state-dependent friction. This model assumes that aftershocks nucleate over areas on the fault that at the time of the main shock are already accelerating toward failure and disregards the effect of interactions among aftershocks. The main…

ISBN
0148-0227
Keywords
Journal Article

Using a waveform cross-correlation technique, Rubin and Gillard [2000] obtained precise relative locations for 4300 0.5 < M < 3.5 earthquakes occurring along 50 km of the San Andreas fault. This study adds to that another 5000 earthquakes distributed along 10 km of the San Andreas fault and 20 km of the Calaveras fault. Errors in relative…

ISBN
0148-0227
Keywords
Journal Article

We use seismic waveform cross correlation to determine the relative positions of 2747 microearthquakes near Mount Lewis, California, that have waveforms recorded from 1984 to 1999. These earthquakes include the aftershock sequence of the 1986 ML5.7 Mount Lewis earthquake. Approximately 90% of these aftershocks are located beyond the tips of the…

ISBN
0148-0227
Keywords
Journal Article

Waveform cross-correlation allows one to measure the relative arrival times of similar microearthquakes with errors of less than 1/10 of 1 sample. Location algorithms based on these measurements have greatly improved images of earthquake distribution. For the Northern California Seismic Network catalog, however, the relative location errors…

ISBN
0037-1106
Journal Article

Using a waveform cross-correlation technique, we have obtained precise relative locations for nearly 75% of the Northern California Seismic Network catalog (4300 earthquakes) occurring between 1984 and 1997 along 50 km of the San Andreas fault. Errors in relative location are meters to tens of meters for events separated by tens to hundreds of…

ISBN
0148-0227
Journal Article

The main objective of this study is to see if a lower threshold for earthquake triggering exists. Resolving this issue is important for the understanding of earthquake mechanics and for the purpose of hazard analysis. We compute the cumulative static stress changes imposed on 63 M ≥ 4.5 earthquakes in central California between 1969 and 1998,…

ISBN
0148-0227
Journal Article

We investigate the ability of magma to propagate along preexisting fractures oblique to the least compressive stress. Relaxation of the preexisting shear stress to zero over the portion of the fracture dilated by magma (the dike) results in slip for some distance along the closed portion of the fracture ahead of the dike tip and a stress…

ISBN
0148-0227
Journal Article

Crustal faults that produce most of their slip aseismically typically generate large numbers of small earthquakes. These events have generally been interpreted as coming from localized patches of the fault that undergo unstable (stick–slip) sliding, surrounded by larger regions of stable sliding (creep). In published catalogues the…

ISBN
1476-4687