Asperity formations and their relationship to seismicity...

submitted to Geophys. J. Int.

Asperity formations and their relationship to seismicity on a

planar fault in the laboratory

P. A. Selvadurai1⋆ & S. D. Glaser1

1 Civil and Environmental Engineering, University of California, Berkeley, 94720, USA

Received 2015 September XX; in original form 201X

SUMMARY

Earthquake faults, and all frictional surfaces, are in contact through touching asperities. We

offer an explanation for why certain asperities are more prone to foreshock, and why others

would be more likely to slide aseismically. This increased knowledge of how asperities form

will provide a better understanding of the manner in which they communicate with each other

(i.e. static or dynamic stress transfer) during the foreshock failure sequences leading to the

larger main shock. We present results of experiments where a pressure sensitive film was used

to map, size, and measure the normal stresses of the asperities along the laboratory fault. As

the surface roughness plays an important role in how asperities are formed, two Hurst expo-

nents were measured to characterize the interface: (1) long length scale estimate (H ∼ 0.45)

and (2) short length scale estimate that was present over asperity junction points (H ∼ 0.8 to

1.2). Macroscopically, the number of asperities, real contact area, and the mean asperity size

increased with additional applied normal force while the mean normal stress remained con-

stant. Larger asperities carried higher levels of normal stress on average and displayed higher

levels normal stress heterogeneity than smaller ones. The critical slip distance on foreshocking

asperities was estimated to be d0 ∼ 0.65 µm using frictional stability theory. d0 was ∼ 2.5

to 1.9% of the premonitory slip needed to initiate gross fault rupture of the interface, rang-

ing from between 20 to 40 µm. The overall slip necessary to initiate gross fault rupture was

2 P. A. Selvadurai & S. D. Glaser

between 77 to 87.5 % of average asperity radius which follows micromechanical theories in

previous laboratory studies.

Key words: Instability analysis – Earthquake interaction, forecasting, and prediction – Me-

chanics, theory, and modelling – Dynamics and mechanics of faulting – Rheology: crust and

lithosphere

1 INTRODUCTION

Laboratory experiments (e.g., Dieterich, 1978, 1994) have been used to improve our understanding

of earthquakes and faulting (Scholz, 2002). Laboratory models have provided a phenomenolog-

ical understanding of the dynamics surrounding the transition of friction from static to kinetic.

These mechanistic models allow for estimates of the changing stress states on fault in nature in-

ferred using standard seismological techniques. Over the last two decades, the level of detail from

the field measurements has increased due to improved seismometer array coverage, broadband

seismometers, down borehole seismometer and more accurate global positioning systems (GPS)

among other factors.

The mechanical model describing the transition of frictional sliding from slow (stable) to rapid

(unstable) is described as a shear rupture where stress is broken down by the accumulation of

slip (e.g., Ida, 1972; Andrews, 1976). Both laboratory (e.g., Dieterich, 1978; Ohnaka & Shen,

1999; McLaskey & Kilgore, 2013; Selvadurai & Glaser, 2015b) and numerical studies (e.g., La-

pusta et al., 2000; Chen & Lapusta, 2009; Ampuero & Rubin, 2008; Kaneko & Ampuero, 2011)

have shown that a premonitory (nucleation) phase exists where a region of slow aseismic accel-

erating fault slip precedes a large earthquake. Detection of the premonitory phase is extremely

difficult using current geodetic measurements (Tullis, 1996; Roeloffs, 2006) due to array cover-

age and location constraints, and the small size and limited timing of the accelerating aseismic

region. With recent advances in broadband seismometers, recorded events are becoming smaller

and were previously undetected. This seismicity is generated at the fringe of the nucleation re-

⋆ Corresponding Author

Foreshocks seismicity in relation to asperity formations 3

gion. Seismologists have begun looking at this seismic activity as an indicator of the otherwise

aseismic preparatory phase (Chiaraluce et al., 2011; Marsan & Enescu, 2012; Kato et al., 2014;

Yagi et al., 2014; Meng et al., 2015). In nature, localized and rapid failure of asperities within a

growing nucleation zone, if large enough, are referred to as foreshocks and have been observed to

precede larger earthquakes. With better seismological instruments, clusters and swarms of small

earthquakes and/or foreshocks are now being detected weeks to minutes before and in close spa-

tial proximity the eventual main shocks (e.g., Mogi, 1963; Jones & Molnar, 1979; Ohnaka, 1993;

Nadeau et al., 1994; Dodge et al., 1995, 1996; Chiaraluce et al., 2011; Govoni et al., 2013; Tape

et al., 2013). Recent studies show evidence that at least 50% of major interplate earthquakes have

foreshocks (Brodsky & Lay, 2014), whereas Bouchon et al. (2013) believe it to be more in the

range of approximately 70%. Foreshock sequences and slow slip processes prior to recent, well-

monitored large earthquakes, i.e. the 2011 Tohoku-Oki, Japan, (Kato et al., 2012) and the 2014

Iquique earthquakes (Brodsky & Lay, 2014; Yagi et al., 2014) are forcing scientists to reconsider

original estimates of spatial and temporal scales over which preparatory phases occur.

A priori detection of foreshocks is currently difficult (if not impossible) to distinguish from

non-premonitory seismic activity – much of this is the result of our lack of understanding of

the mechanical processes controlling premonitory seismicity. Stress changes along faults due to

preparatory seismicity is not well understood; in certain cases, earthquake swarms do not cul-

minate in a major event (Holtkamp et al., 2011). The exact relation between foreshocks and the

aseismic preparatory phase is also debatable (Mignan, 2014). Two concurrent conceptual models

exist that explain the presence of preparatory seismicity: (i) the pre-slip and (ii) the earthquake

cascade models (Ellsworth & Beroza, 1995; Beroza & Ellsworth, 1996; Vidale et al., 2001). In

the pre-slip model, slow aseismic slip is responsible for foreshocks and other precursory seismic

activity. If the slow slip is below the detection threshold of modern geodetic networks, the seis-

micity produced is a means to monitor the extent and growth rate of the slow slip region. The size

of this region determines the frictional stability of the shear rupture – once a nucleation region

reaches a critical size, unstable rupture will ensue. Better estimates of the size and growth rate of

the nucleation zone could lead to better estimates and the possible prediction of the larger impend-


ing earthquakes. Conversely, the earthquake cascade model does not require an initial region of

slow slip. Here, earthquakes trigger the subsequent event. Each event is a ‘foreshock’ to the pre-

vious event – the main shock thus has only one foreshock (not a sequence) and triggering is due

to dynamic or after-slip stress perturbations. Epidemic-Type Aftershock Sequence models (ETAS)

(Ogata, 1988) use empirically based laws (i.e. Gutenberg-Richter and Omori time diffusion law) to

study after-slip triggering. This model implicitly assumes that any transient change in the overall

seismicity is due to the triggering of an earthquake by another one. Prediction becomes futile for

the breakaway earthquake model; there is a lack of understanding of the fundamental mechanics,

which is unlike that describing the nucleation theory for the pre-slip model.

The behavior of a fault is controlled by the nature of the contact surfaces. In this study, we

perform laboratory tests to study interface contacts (i.e. asperities) formed between two rough

surfaces that are direct byproducts of the global stress states and physical properties along the

frictional fault. Surface roughness has influence on many facets of natural phenomena such as

adhesion, friction, wear and electrical/thermal transmission (e.g., Popov, 2010; Pohrt & Popov,

2012), and geophysical problems such as rock friction leading to earthquakes(Scholz, 2002). As-

perities represent strength heterogeneity caused by the coming together of the rough surfaces and

produce local stress fluctuations that dictates how slip accrues. We look to the laboratory to un-

derstand how asperities interact during the premonitory slip phase (Tullis, 1996) and the initial

prestress states of the fault. We relate the findings presented here to results presented from the

concerted study by Selvadurai & Glaser (2015b) in which foreshocks occurred within a nucleation

zone of a frictional fault – similar to how they appear on natural faults.

1.1 Asperity formation in terms of contact mechanics

Understanding how asperities are formed, in terms of contact mechanics, is in large part due to

seminal research by Hertz (1882). He calculated the contact area and displacements formed be-

tween interacting quadratic surfaces in elastic contact, most notably two spheres pressed together

under normal loads. Real surfaces exhibit sparse sets of dilute contacts over a range of length

scales (e.g. Persson, 2006). We call an isolated contact patch an asperity and the sum of all as-


perities along the interface equates to the real contact area (Ar). We limit ourselves to length scale

variations of 20 µm due to a limitation of our experimental facilities discussed later. Bowden

& Tabor (2001) first noted that the real contact formed along an interface were typically orders

of magnitude lower than the nominal area (A0) of the interface. Idealized models (e.g. Archard,

1957, 1961; Greenwood & Williamson, 1966) examined the contacts formed between rough elas-

tic surfaces in terms of statistically prescribed smooth spherical bumps with a range of radii. These

models take advantage of the foundations provided by the Hertzian contact solutions in superpo-

sition (Timoskenko & Goodier, 1970). Geophysical studies (e.g., Brown & Scholz, 1985; Power

& Tullis, 1991) build on these idealized contact models and apply it to rough interacting rock

surfaces. More recent studies describe self-affine roughness characterized by the Hurst exponent

H (Bouchaud, 1997; Candela et al., 2009, 2011; Kirkpatrick & Brodsky, 2014; Brodsky et al.,

2016). Using the Greenwood-Williamson model (GW), interaction of these self-affine surfaces

cause scale-dependent heterogeneous normal stress fields (Hansen et al., 2000; Batrouni et al.,

2002; Persson, 2006) when a purely elastic constitutive relation was employed. Bowden & Tabor

(2001) realized that asperities are local stress concentrations and can grow while maintaining a

constant pressure (Pm), equal to the materials indentation hardness, due to local yielding. More-

over, as the normal force is increased on a single smooth contact, the material undergoes a tran-

sition through an elastic to elastic-plastic, to fully plastic regime (∼ 3 · Pm) (see p. 178 Johnson,

1985).

The GW model can be used to model fully plastic deformation. During fully plastic defor-

mation, three conditions must be satisfied (Baumberger & Caroli, 2006): (1) the real contact area

must be proportional to the normal load and independent of the nominal faulting area, (2) the aver-

age contact radius ā is load independent and (3) asperity radii are of micrometric order. Dieterich

(1994) directly confirmed that contacts were indeed experiencing plastic deformations via opti-

cal experiments performed on roughened Lucite-Lucite interfaces. This model is commonly used

in geophysical studies. We present laboratory findings that show both conditions (2) and (3) are

not well satisfied on a fault that exhibited interesting frictional features: i.e., a growing nucleation

region that exhibited localized foreshocks (Selvadurai & Glaser, 2015b).


Foreshocks are believed to be the sudden failure of an asperity cause by nearby stress per-

turbations. Their presence (or lack thereof) may hold information regarding the impending larger

earthquake and provide information about the breakdown zone and overall growth of the nucleat-

ing shear rupture. More recent laboratory results (McLaskey & Kilgore, 2013; McLaskey et al.,

2014) and the study directly linked to the results presented here (Selvadurai & Glaser, 2015b)

showed that local asperities within the breakdown region can fail dynamically as the shear rupture

expands and prior to gross fault rupture (main shock).

1.2 Nucleation of rapid slip

Asperity interactions may have important features when trying to understand nucleation processes

along a fault modeled in terms of an outwardly growing shear crack (e.g., Andrews, 1976; Ohnaka,

1992). Shear stress ahead of the crack-tip τi is broken down to a residual level τr by accruing slip.

The amount of slip necessary to reach breakdown shear stress is referred to as the critical slip dis-

tance Dc and occurs over a finite breakdown/cohesive zone of size Xc. Once the crack has grown to

a sufficient size h∗, it begins to propagate unstably and accelerates to speeds closer to the Rayleigh

wave velocity of the material. This produces seismic waves we interpret as earthquakes. One factor

controlling h∗ is the effective normal stress field σ̄ (Rice & Ruina, 1983; Dieterich, 1992; Rice,

1993). Effective normal stress is also a parameter controlling asperities formations that may cause

local fluctuations in strength that inhibit the growth of the nucleation region. Consequently, asper-

ity formations and interactions may shed light on how stress breakdown occurs within the cohesive

zone (Ida, 1972) at the tip of the shear crack as it attempts to grows large enough to generate an

earthquake.

A major focus of this study is to examine the asperities within a breakdown region in more

detail. A novel pressure sensitive film was used to characterize the unique features of the asper-

ities: their locations, size, and local normal stresses distributions. Our goal is to provide a better

understanding of two factors within the breakdown region: (i) how asperities communicate with

each other based on their location and size and (ii) what measured features (e.g. size and normal


stress) may promote a foreshock on an asperity; i.e. what constitutes a seismic versus aseismic

asperity.

2 EXPERIMENTAL METHODS

2.1 General

We examine the mature faulting surface after a suite of experiments described by Selvadurai

& Glaser (2015b). General experimental configuration, fault geometry and sample dimensions

are shown in Fig. 1(a) and surface preparation techniques are described by Selvadurai & Glaser

(2015b). The mature fault (i.e. one exhibiting negligible wear) in this study had previously accu-

mulated approximately 58.4 mm of slip over 32 stick-slip events (SS) at a range of applied normal

loads (Fn = 2000 to 4400 N). During the premonitory phase (i.e. the time prior to SS) a total of

68 foreshocks occurred at asperity length scale determined acoustically by Selvadurai & Glaser

(2015b).

The polymethyl methacrylate (PMMA)-PMMA interface was held under simple normal load

Fn for thold = 900 s then sheared by driving the rigid loading platen at a constant velocity (VLP =

0.010 mm/s) until gross fault rupture (SS) occurred (Fig. 1(b)). Slip sensors (NC1-NC7) captured

the slow accumulation of slider block slip along the fault, which accumulated non-uniformly in

the loading (x-) direction. Typical observations of normal force Fn, shear force Fs and slip δ for

a stick-slip event is shown in Fig. 1(b). Tests were performed at four levels of applied force: 4400

N, 3400 N, 2700 N and 2000 N. Macroscopic nucleation of gross rupture was found to occur in

a repeatable manner. A slowly propagating shear rupture (between 1 to 9.5 mm/s) moved from

the trailing (loaded) edge of the sample into a relatively locked region and its propagation speed

was normal stress dependent. Foreshocks seismicity is generated by localized, rapid sliding as an

asperity failed. This was captured by an array of absolutely calibrated Glaser-type piezoelectric

acoustic emission sensors (PZ1-PZ15) (McLaskey & Glaser, 2012). Foreshocks emanated from a

seismogenic region extending from x = 150 mm to 300 mm. The entire catalog of foreshocks from

the seismogenic region (recorded by Selvadurai & Glaser (2015b)) is shown in Fig. 1(c). The event

magnitudes are given as the average fault-normal peak ground displacements (PGD) over the three


AE sensors nearest to the seismic hypocenter. An example of seismic data recorded from a single

foreshock is shown in Fig. 1(d). Foreshocks occurred on length-scales of 0.4 to 2 mm estimated

acoustically (Selvadurai & Glaser, 2015b).

2.2 Microscopy of asperity surfaces

Microscopy was performed on the surface of the slider block from within the seismogenic region

shown in Fig. 1(c). Full asperity surfaces were initially visualized using light microscopy (dark

field illumination). To reconstruct the asperity, multiple images were stitched together manually

in an image processing software. Two different optical profilometers (using white light interfer-

ometry) were used to characterize the surface roughness at different length scales. For longer

wavelengths (millimetric), a Nanovea PS50 Profilomter with a 3000 µm optical pen was used.

Scans sizes were 25 mm x 13 mm with 10 µm spatial resolution and 0.5 µm height resolution. For

shorter wavelengths, higher magnification (micron length scales) roughness measurements were

required. Non-contact interferometry measurements were performed using the ADE MicroXAM-

100 Optical Profilometer. The non-contact profilometry provided height measurements (0.01 µm

resolution) over a scanning area of 256 µm x 196 µm (0.95 µm spatial resolution) in the x- and y

-directions, respectively.

3 RESULTS

3.1 Size and location of asperities

The size and location of the asperities formed during the initial contact between the slider block

and base plate were measured using Fujifilm Prescale medium range (12-50 MPa) pressure-sensitive

film (Selvadurai & Glaser, 2012, 2015b,a). The pressure film was independently calibrated inde-

pendently by Selvadurai & Glaser (2015a). Detailed calibration performed in our laboratory show

that the spatial resolution of the pressure film was about 20 to 30 µm and, for this study, the film

was digitized using a pixel size of 20 µm x 20 µm (Apixel = 400 µm2 = 4 x 10−4 mm2) using a

HD scanner (MUSTEK SE A3 USB 2400 Pro). The digitized image was converted to stress (pro-

portional to the coloration of the film) and a lower threshold was used to determine real contact


versus film noise. As a check on the sum measured force, we calculated the total reactive normal

force Fr supported across the asperities in the seismogenic region (x = 150 to 300 mm). The total

reactive normal force Fr balancing the normal force in the seismogenic region was calculated by

summing the reactive force on each ith asperity,

Fr =n∑

i=1

σ̄iAi. (1)

Regions without contact were prescribed null normal stress conditions (Persson, 2006).

Fig. 2 shows a section of the interface (x = 270 to 290 mm, y = -2 to 6 mm) for two different

confining normal forces Fn = 4400 N (top) and 2700 N (bottom). The inset of each image shows an

enlarged view of a few asperities at the two different load levels and the manner in which asperities

grew with the application of normal force. Highlighted in Fig. 2 is the (i) full normal stress field

σi, (ii) average of the normal stress field σ̄i, (iii) the asperity area Ai and (iv) the coefficient of

variation CVi = std (σi/σ̄i) which is a measure of dispersion in the stress field.

3.2 Macroscopic contact properties for various levels of Fn

Using the pressure film we examined the number of asperities n formed in the seismogenic zone

under various far-field normal loads Fn.

Fig. 3(a) shows the relationship between reactive normal force Fr and real contact (Ar orange

stars) and the number of asperities (n blue circles). We found that real contact area in the seismo-

genic region Ar increased at 0.059 mm2 per Newton of reactive force (R2 =0.996). The number of

asperities did have an increasing trend with reactive normal force but did not observe the amount

of linearity seen in real contact area. Variations in the numbers of asperities could be due to the

contact processes (e.g., Archard, 1957, 1961; Greenwood & Williamson, 1966; Nayak, 1973) and

also our inability to recreate the fault along very small length scales (Selvadurai & Glaser, 2012).

In Fig. 3(b) we examine the relationship between mean asperity area (〈A〉 blue squares) and

mean asperity normal stress (〈σ〉 orange triangles) with the normal reactive force Fr. From this

we can also calculate upper and lower limits of the mean asperity radius (assuming a circular rep-

resentation) which ranges from 〈a〉 = 26 to 36 µm for lower and higher normal reactive force, re-

spectively. We note that average asperity radius was not insensitive to normal force which should


be the case for plastic asperity deformations. The average asperity normal stress remained rela-

tively constant 〈σ〉 = Fr/Ar = 15.42 MPa and insensitive to the normal force. Converse to the

previous observation regarding 〈A〉, this supports the theory that asperities formed plastically at a

flow pressure (Archard, 1957). The counteracting observations suggests that the random processes

models (Nayak, 1973) for the rough surfaces of PMMA may be more complex than previously

anticipated using standard contact theories.

3.3 Local variations asperity properties at various levels of Fn

We can measure the normal stress field σi from each asperity allowing the calculation of the scalar

mean of the normal stress field σ̄i for all asperities within the seismogenic region. Fig. 4(a) shows

the probability density functions (PDF) of σ̄i for various levels of Fn. Catalogs were created at

the four levels of applied load Fn and load levels are indicated in the legend and is applicable

throughout Fig. 4(a). The distributions have not been normalized since the number of n asperities

varied as the normal force was increased (Fig. 3(a)). We see that the distributions of asperities

supporting lower σ̄i is relatively uniform for all Fn. However, for asperities with σ̄i > 16.2 MPa

we seen that an increase amount of asperities support higher σ̄i for increased Fn as seen by the

decreasing slope ξ in Fig. 4(a).

It is evident that no single value of mean normal stress (e.g. 〈σ〉 from Fig. 3(b)) is necessary and

sufficient to characterize the distributions of σ̄i on all asperities a various normal loads. A single

value would be the case for the plastic formation of asperities and we denote 〈σ〉 (dashed lines in

Fig. 3) to show how the distributions deviate. The heterogeneity described by these distributions

again suggests that formation of asperities is more complex than typical models for our fault.

In Fig. 4(b) we plot relationship between σ̄i and the individual asperity area Ai. We see that

as the asperities increase in size Ai they also tend to support more mean normal stress σ̄i, which

was the case for all levels of applied normal force Fn. The only variation in the distributions in

Fig. 4(b) was that more large asperities were present at higher normal loads Fn (inferred from the

results in In Fig. 4(a)).

A similar clustering of asperity sizes versus the coefficient of variation CVi is observed in Fig.


4(c). Asperities with lower values of CVi indicate a more uniform distribution, whereas higher

values indicate a larger degree of heterogeneity in the normal stress field σi. This parameter may

have implications on the seismicity generated from a foreshock and will be discussed later.

Fig. 4(c) shows that asperities of all sizes show some dispersion in the normal stress field

(CVi = 0.061 = 6.1%). As the normal force was increased the dispersion increased along asper-

ities that were smaller and continued to increase as asperities became larger (dashed line in Fig.

4(c)). This might been due to the increased complexity in as asperities shape and normal stress

grew as the normal force was increased as shown directly in Fig.2.

Fig. 4(d) plots the relationship between the mean normal stress σi and the coefficient of varia-

tion CVi for individual asperities. We see that as the average normal stress increases along the fault

so does the level of dispersion. We estimate that dispersion in the normal stress field σi increased

by 4.1% per 1 MPa rise in σ̄i along individual asperities.

3.4 Microscopy of asperity surfaces

After the sequence of tests was performed, the slider block sample was examined using microscopy

techniques. Fig. 5(a) shows a typical asperity located at x ≈ 235 mm and y ≈ 5.1 mm on the slider

sample. The asperity surfaces appear flat on top, most likely due to wear processes that occurred

during the surface preparation techniques. We noticed interesting features reminiscent of asperities

on rock faults, such as visual striations that align with the direction of slip and what appeared

to be slickensides-like features over portions of the asperity (Candela et al., 2011; Kirkpatrick &

Brodsky, 2014; Brodsky et al., 2016). Two small section on the asperity was analyzed using optical

profilometery and the results are shown in Fig. 5(b) and (c). The colorbars are indicative of the

surfaces height but we have also normalized this by the scan length L = 0.2571 mm. We see that

the height is ranges from ∼ 0.0014·L to 0.0007·L which was the case in the scans taken on 33

different locations using the ADE MicroXAM-100 optical profilometer.

Transects of the roughness profile along slip (blue) and normal to slip (magenta) are shown in Fig.

5(d) and (e) for the optical scans in (b) and (c), respectively. Below (in Fig. 5(f) and (g)) are Fourier

power spectrum estimates of the slip profiles in Fig. 5(d) and (e), respectively, in the sliding (blue)


and normal to sliding (magenta) directions. The Hurst exponent H was calculated using the spatial

power spectrum. In this study, we calculate Hpar (along slip) and Hper (normal to slip) using

P (k) ∝ k−(1+2H), (2)

where k is the wavenumber. The Hurst exponent was measured for longer and shorter wavelengths

about the wavenumber k = 3 · 102 mm−1. The longer wavelength (k ≤ 3 · 102 mm−1) Hurst

exponents, averaged over 33 surface scans similar to those shown in Fig. 5(b) and (c), was Hpar

= 0.4179 (along slip) and Hper = 0.5033 (normal to slip). Shorter wavelength (k > 3 · 102 mm−1)

Hurst exponents was Hpar = 0.8334 (along slip) and Hper = 1.2036 (normal to slip).

Fig. 6(a) gives measurements from the Nanovea PS50 Profilomter of a 25 mm x 12 mm section

of the fault within the seismogenic region. Transects A and B are drawn through the lower (y =

-4 mm) and upper (y = 4 mm) section of the fault, respectively. Fig. 6(b) shows the probability

density function (PDF) for the surface height for the entire surface shown in Fig. 6(a). We note

that the PDF is not gaussian and seems to be display some skew, which we have attributed to the

wearing process; this possible anisotropy of the PDF needs to be investigated in more detail in

future studies. Fig. 6(c) shows the height profile for the two transects A and B in Fig. 6(a). Along

the y = 4 mm Transect B the surface has wider, flatter surfaces that in the lower y = -4 mm Transect

A. Fig. 6(c) shows ‘valleys’ (with height h) and ‘flat-topped’ sections (with width w) which were

prominent features of the fault. The ratio of h/w ∼ 0.06, similar to faulting outcrops in nature

(Kirkpatrick & Brodsky, 2014). We also show the height profile normalized by the length of the

scan (L = 25 mm) for a relative height of 100 µm, which corresponds to h = 0.004 · L that is also

similar to natural faults (Brodsky et al., 2016).

The Hurst exponent was again calculated but for length scales using the measurements ob-

tained from the Nanovea PS50 optical profilometer. that spanned the ‘valleys’ observed between

the ‘flat-topped’ sections of the surface. In Fig. 6(d), we analyzed the Fourier power spectrum

(FPS) between length scales of 60 µm to 1000 µm for height profiles in the direction (black) and

normal (red) to the direction sliding. Hurst exponent estimates was Hpar = 0.4123 (along slip) and

Hper = 0.4621 (normal to slip). This matched the longer wavelength (k ≤ 3 · 102 mm−1) estimates

made using the ADE MicroXAM-100 optical profilometer. The Fourier power spectrum for scans


from the ADE MicroXAM-100 optical profilometer is shown in Figure 6(d) for comparison. The

larger sampling region displayed less anisotropy in Hurst exponents between the direction and nor-

mal to the direction of sliding. It is possible that wearing processes affected shorter wavelengths

(


separation distances dsep to this problem from a range (1 to 10,000) of equally sized asperities

(Ngrid) occupying a unit square. Fig. 7(c) show an example schematic of the separation distance

dsep calculated from the pressure film measurements.

Using the metrics for each cell in the seismogenic zone, we calculated the asperity separation

distances at normal forces of Fn = 4400 N. The average number of asperities over all 120 cells was

Ngrid = 117 (ranging from 43 to 251 asperities). The average equivalent radius of the asperities

in all cells was agrid = 0.0553 mm = 53 µm (ranging from 31 µm to 120 µm) mm. Using these

measurements and the tabular packing data (Specht, 2015) we found on average dsep/agrid = 9.33

(ranging from 3.95 to 19.75). At lower normal forces Fn =2000 N, we found on average dsep/agrid

= 14.66 (ranging from 4.97 to 76.90). As expected, the ratio of separation distance to average

asperity radius increased as the fault was unloaded.

4 DISCUSSION

4.1 General contact observations

Fig. 3 shows that the real contact area increased with increase of normal force Fn increased. This

is a typical observation in contact mechanics studies where Ar


plastically) and Papp is the apparent pressure (Baumberger & Caroli, 2006). Assuming that the

flow pressure Pm = 15.42 MPa, we can evaluate both sides of eq. 3, within the seismogenic region

independently using the results from Fig. 3. We use the apparent area Aapp = 1800 mm2 to estimate

the apparent pressure Papp = Fr/Aapp. For Fr ≃ 1600 N, the apparent pressure becomes Papp =

0.88 MPa and the ratio of Papp/Pm ≈ 0.057. At the same reactive normal force, we measure

the real contact area Ar ≈ 93 mm2 and the ratio Ar/A0 ≈ 0.051. This calculation promotes

the idea that contact deformation was plastic as interpreted from the film. However, plastic (and

elastic) deformation on nominally flat rough surfaces predicts that the average size of asperities

should remain constant (Greenwood & Williamson, 1966; Johnson, 1985) – a feature that was

not the case for our interface (see Fig. 3(b)). The GW model has limiting features that seem to

be violated: (1) asperities must deform independently of each other and (2) the roughness profile

must be isotropic gaussian. Separation distances were small within the seismogenic region (see

Fig. 7) and the distribution of height on the slider block were not gaussian (see Fig. 6(b)) both of

which violates this standard model. These atypical features, describing the frictional fault reported

here, may lead to increased complexity at the microscopic level.

4.2 Asperity separation relating to nucleation processes

One question which remains unanswered is the impact of a failing (foreshocking) asperity on the

manner in which the stress is transferred to its surroundings. Foreshock sequences have been stud-

ied as precursory phenomena to a larger main shock (e.g., Yagi et al., 2014; Kato et al., 2012;

Brodsky & Lay, 2014). Two predominant theories surrounding foreshock sequences have been

postulated: the cascade and preslip models. A better understanding of stress states along asperi-

ties and the range of their elastic interactions may help scientists understand the implications of

foreshock(s) and how they relate to the larger main shock.

Whether asperities on the fault communicate elastically or are isolated will affect the local

stiffness of the fault and may be important when attempting to understand breakdown processes

in the cohesive zone of the shear rupture. The interactions between asperities through the off-fault

material becomes an issue at small separation distances. This elastic interaction probably affects


local variations in fault stiffness (Campaña et al., 2011; Ciavarella et al., 2008; Yastrebov et al.,

2012). In-plane asperities separation distances in the seismogenic zone, even at the lower normal

loads, were too small (results relating to Fig. 7) to neglect the elastic interactions (Baumberger &

Caroli, 2006). Tribological studies of the local interactions between closely packed asperities are

now being more often investigated (e.g., Ciavarella et al., 2008; Afferrante et al., 2012). These

interactions may affect sliding dynamics during the premonitory phase and therefore cannot be

ignored, as is the case for the standard GW model. The work pioneered by Bowden & Tabor

(2001) and Archard (1953) suggest that typical separation distances for multicontact interfaces are

much larger, approximately 100 radii – which has been confirmed by direct observations (Dieterich

& Kilgore, 1994, 1996b). Discrepancies between these studies and that presented here may be

influenced by the surface roughness; surfaces in the aforementioned studies are much smoother

(∼ 0.1 µm RMS) than the long length scale roughness (∼ 5 µm RMS roughness) seen in this

study (see Fig. 6).

4.3 What constitutes a foreshock susceptible asperity?

Asperity interactions at longer length-scales, i.e. the length of the seismogenic region ∼ 150 mm,

may account for the large amounts of macroscopic slip (from ∼ 20 to 40 µm) observed experi-

mentally before gross fault rupture (see figure 5 from Selvadurai & Glaser, 2015b). Here we use

frictional stability theory to estimate features of asperities that may have allow localized fore-

shocks.

We know that foreshocks observed in this study are the sudden localized failure of an indi-

vidual asperities that form at length-scales from tens of microns to a few millimeters. This was

determined by Selvadurai & Glaser (2015b) using the acoustic emission signals and the standard

Brune relationship (Brune, 1970) for the sudden failure of a circular asperity. McLaskey & Kil-

gore (2013) suggested that foreshocks in their laboratory experiments arose as local slip rates

increased in close proximity to a local strength heterogeneity (i.e. asperity). They attributed this

strength variation to the geometric interaction of the two faulting surfaces and in our experiments

we quantified these interactions using the pressure film (see Fig. 4). It is possible for an asperity to


be seismogenic, independent of the rate at which it slides or how fast sliding occurs nearby. The

potential for seismicity here depends on the contact size and normal stress measurements along

asperities and takes advantage of frictional stability theory (Rice & Ruina, 1983; Ruina, 1983;

Baumberger et al., 1994).

We first assume frictional stability theory scales to the length-scales at which asperities form.

This assumption implies that along an asperity a small nucleation region grows outwards from a

relatively weak section along the asperity. The asperity exhibits an initial shear stress distribution

that is broken down to residual levels by allowing characteristic amount of slip Dc to accrue across

the asperity junction. A critical nucleation half-length h∗ describes the maximum size at which the

asperity-level nucleation region can achieve before frictional instability occurs (i.e. a foreshock).

For each ith asperity, we examine the theoretical foreshock nucleation size using two formulations

(Kaneko & Ampuero, 2011):

hRR∗i =π

4

G∗

σ̄i

Dc(b− a)

, (4)

hRA∗i =1

π

G∗

σ̄i

Dc · b

(b− a)2, (5)

where the Poissons ratio ν 0.32, G∗ is the shear modulus (G∗ = G/(1 − ν)) for in-plane sliding,

G is the shear modulus of PMMA (1410 MPa (Saltiel et al., 2016)), Dc is the characteristic slip

distance over which the state parameter evolves, σ̄i is the average normal stress on an individual

asperity (measured using the pressure sensitive film) and a and b are the rate and state constitutive

parameter for steady-state velocity weakening friction capable of forming an instability (i.e. b−a >

0). The estimate hRR∗i was derived from the linear stability analysis of steady sliding by Rice &

Ruina (1983), while hRA∗i was obtained for (a/b) >= 0.5 by Rubin & Ampuero (2005) on the

basis of energy balance for a quasi-statically expanding crack governed by the aging law.

We used values of a = 0.01 and b = 0.0108 in conjunction with a numerical study Kaneko

& Ampuero (2011) that attempts to explain nucleation processes in a frictional laboratory study

between two blocks of Columbia Resin (polymer) (Nielsen et al., 2010). In Fig. 8(a) we plot eqs

4 (solid line) and 5 (dashed line) for a range of normal stresses assuming the constants mentioned

on the graph. The cartoon in Fig. 8(a) describes the relationship between asperity radius ri (shaded


region) to the critical nucleation size h∗ (green circle) and how to interpret these curves when the

experimental data provided from the pressure sensitive film is introduced (blue dots). From Fig.

4 we obtain the average normal stress σ̄i and radius ri (assuming a circular representation). For

each asperity we can compare whether it lies above or below the h∗ estimates provided by eqs

4 or 5. If an asperity radius is smaller than critical nucleation size, i.e. ri < h∗, we expect these

asperities to be aseismic – they would be incapable of developing foreshocks since linear stability

is not violated (within the lower limits of our instruments). Asperities with radii greater than the

critical nucleation length, i.e. ri > h∗, could potentially generate a local foreshock according to

nucleation theory.

Selvadurai & Glaser (2015b) estimated source length scales from foreshocking asperities to

range from R0 ∼ 0.21 to 1.09 mm. In Fig. 8(b) we enlarge the experimental pressure film data

and include the range of foreshock sizes R0 (shaded gray region). We see that using the parameters

shown on Fig. 8(a), eq. 5 predicts that more asperities should be susceptible to foreshock seismicity

based on the size and normal stress on asperities measured experimentally. They also seem to

bound the lower estimates of foreshock sizes (i.e. R0 ∼ 0.21 mm) determined experimentally.

Typical laboratory values Dc have been reported between 24 µm (calculated from curves in

Dieterich & Kilgore, 1996a) to 100 µm (Marone, 1998) for a range of materials sometimes with

the presence of gouge. This is significantly higher than the low critical slip distance predicted

here Dc = 0.65 µm and may be explained by variations in the surface roughness. Laboratory

tests have shown that critical slip-weakening distance Dc decreases as the interacting surfaces

become smoothed (Dieterich, 1994; Yoshioka & Iwasa, 1996; Yoshioka, 1997; Ohnaka & Shen,

1999). Ohnaka (1992) suggested that strength inhomogeneities (i.e. asperities) are local barriers to

an expanding nucleation front that can have lower characteristic slip displacements (d0) than the

critical displacement D∞ describing the spatially larger breakdown processes along the fault.

Each laboratory experiment studying nucleation processes on pre-existing faults is unique.

Distinct foreshock sequences observed in our experimental study may have been caused by the

unique surface roughness that varied at shorter (Fig. 5(f) and (g)) and longer wavelengths (Fig.

6(d)). In Fig. 6(d), we see that the mature faulting surface can be characterized using a Hurst


exponent at smaller and longer wavelength. Smaller estimates of Dc = 0.65 µm (Fig. 8) promotes

the idea that foreshocks from on smoother junctions of the fault (see Fig. 5) if lower values of Dc

are characteristic of the interaction between smoother surfaces (Yoshioka & Iwasa, 1996).

Selvadurai & Glaser (2015b) measured approximately 20 to 40 µm of slow premonitory slip

leading up to gross fault rupture. A standard interpretation of the critical slip distance Dc has been

that of the slip necessary to renew surface contacts (Dieterich, 1979; Marone, 1998) and usually is

on the order of the mean contact diameter (Dieterich, 1994; Yoshioka & Iwasa, 1996). From Fig.

3(b) we see that mean contact radius ranges from 26 to 35 µm. The overall premonitory slip was

approximately 77 to 87.5 % of the average asperity radius in the seismogenic region. Comparing

Dc = 0.65 µm (Fig. 8) to the mean asperity radii, we obtain a critical slip distance that is 2.5 to 1.9%

of the mean asperity radius. This supports Ohnakas theory that the two length scales of critical slip

distances may be affecting the nucleation processes along fault. Foreshocks may be attributed to

smoother localized sections of the fault, i.e. asperity top, that exhibit a lower critical slip distance

(d0) than the overall critical slip distance (D∞) defined by a longer length scale roughness. While

this mechanism could explain the presence of foreshock, more numerical studies may be necessary

to determine the effect of rate and state parameters (a and b), locally varying normal stress states

and the ratio of asperity-level to overall critical slip distance (d0/D∞) during nucleation processes

(Higgins & Lapusta, 2014, 2015).

4.4 Local variations in normal stress on individual asperity junctions

Looking at the image of a smoother section of the fault (Fig. 5(a)) we see features (slickensides,

striations) that would suggest that normal stress variations would vary along a given asperity. This

was confirmed using the pressure film visually (Fig. 2) and quantified using the coefficient of varia-

tion CVi (Figs 4(b) and (c)). The normal stress field along an individual asperities (which fluctuates

due to the asperities on asperity concept (Archard, 1957, 1961; Persson, 2006)) is then indicative

of local variations in shear strength along the individual asperity. Variation seismicity can be di-

rectly indicative of the strength fluctuations along the fault (Bouchon et al., 1998; Boatwrigth &

Quin, 1986; Mai & Beroza, 2002; Ripperger et al., 2007). Reasons for the increased dispersion in


the normal stress field (i.e. the increased CVi shown in Fig. 4(c) and (d)) on asperities could be re-

lated to the local contact processes of asperities formed between the rough-rough interfaces (e.g.,

Nayak, 1973). Using the techniques here it may be possible to study characteristics of foreshocks

seismicity (e.g. the high-frequency content of the elastodyamic signals) and its relationship to how

asperities form and their local stiffness along the natural fault. Using seismicity (foreshocks or

swarms) to help develop a better understanding of the local strength heterogeneity prior to gross

fault rupture (main shock) could help better estimate seismic hazard but would require increased

laboratory efforts.

4.5 Effects of multiscale roughness on contact formation

Surfaces forming the asperities on our fault appeared to be flat-topped islands amongst a large flat

sea, as seen in Fig. 6, and were characterized over longer length scales with a Hurst exponent of

∼ 0.41 to 0.45. Along the tops of individual asperities themselves, such as the images seen in Fig.

5(a), the surfaces were smoother at smaller length scales.

The two varying levels of roughness may be important during earthquake nucleation: (i) longer

length scale roughness may define the inter-asperity spacing (dsep) and the manner in which asper-

ities interact during the slow slip process and (ii) the shorter length scale roughness may defines

how/if foreshock(s) will be present and may dictate the elastodynamic radiation patterns released

upon failure. The larger asperities, which were more likely to form a foreshock according to the

model proposed in Fig. 8, appear to contain non-uniform normal stress distributions and are irreg-

ularly shaped (containing perforations on the periphery) – perhaps more appropriately described

contact processes involving two roughness profiles (Nayak, 1973).

5 CONCLUSIONS

Selvadurai & Glaser (2015b) observe a number of foreshocks occurring as premonitory slip accu-

mulated and transitioned from slow to rapid sliding (Selvadurai & Glaser, 2015b). Foreshocks in

that study were constrained to a ‘seimsogenic section of the fault. In this study, we investigated (in

more detail) the measurements from a pressure sensitive film that mapped the location of asperities


and their local normal stress. Macroscopically the contacts formed in the seismogenic region in

a standard manner; maintaining an average normal stress of Pm = 15.42 MPa for different levels

of applied nominal normal force (Fn). Closer inspection showed that asperities formed in a more

complicated manner than standard geophysical contact models. Larger asperities accrued higher

normal stress levels than smaller asperities. The variability in normal stress along the asperities

also varied with both size Ai and average normal stress σ̄i they supported.

Separation distances between asperities within the seismogenic region, appeared to be small.

Traditional models (e.g. the GW model) assume that the formation of asperities are independent of

each other and the long range elastic interactions between asperities (via the material substrate) can

be neglected. This may be an oversimplification and future investigations into frictional breakdown

may need to include asperity interactions in the cohesive breakdown zone of a slowly growing

shear rupture.

Upon examining the surface roughness, we found that asperity formation was likely controlled

by two levels of surface roughness. On asperity length scales (micron to sub-millimeter length

scales) a smoother surface was measured (Hpar = 0.8 (along slip) and Hper = 1.2 (normal to slip))

than at longer length scale variations (tens of millimeter) where the fault was characterized by

a Hurst exponent of H = 0.45. We proposed that asperities forming at the smaller length scales

are responsible for localized foreshocks. Frictional stability theory was used in conjunction with

measurements of asperity radius and normal stress to determine a critical slip distance d0 required

to generate a foreshock on the smoother asperity surface itself. We found a d0 = 0.65 µm and

was 2.5 to 1.9% of the mean asperity radius along the interface. The global level of slip need to

induce gross fault rupture was closer to the mean asperity radius (77 to 87.5 %), a value that agrees

with traditional laboratory estimates (Marone, 1998). From these findings, we proposed that two

critical slip distance may be controlling nucleation processes on our frictional fault – small d0 –

is responsible for the generation of foreshocks (that represent localized barriers resistant to shear

rupture) and large (D∞) that control the general growth of the nucleation zone. The variations in

critical slip distance can be related to the smoother and rougher estimates of surface roughness at

smaller and longer length scales, respectively.


ACKNOWLEDGMENTS

All data sets required to reproduce the results presented here is freely available to the reader upon

request. Please contact the corresponding author for access. The National Science Foundation

Grant CMMI-1131582 provided funding for this research, P.A. Selvadurai received funding from

the Jane Lewis Fellowship (University of California, Berkeley) and the National Science and En-

gineering Research Council of Canada (PGSD3-391943-2010).

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0 50 100 150Time (s)

0

0.5

FN

(kN

)S

lip,

(m

)δ

μ

a)

z

x-direction (mm)

Piezoelectric sensors(PZ1 - PZ15)

LE TE

PMMA slider block

PMMA base plate

Loading platen

FS(k

N)

0

0.1

Stick-slip event (SS)

0 mμ

250

b)F

S

FN

/ 2FN

/ 2

Seismogenic region[ ]Selvadurai and Glaser, 2015

Slip sensors (NC1 - NC7)

150 200 250 300

-5

0

5

0 406 mm

y-d

irection (

mm

)

c)PZ12

PZ4 Rela

tive d

ispla

cem

ent (n

m)

0

20 nm

0 100 sμ

PZ5

PZ6

PZ7

PZ8PZ9

PZ10

PZ11d)PGD (nm) µ M

100

10-1

10-2

10-3

0 kN

0

4.4 kN

Figure 1. Experimental configuration of the direct shear friction apparatus used for the experiments on

a PMMA-PMMA interface. (a) Side view schematic depiction of the interface created by compressing the

slider block into the base sample through the rigid loading platen. The normal force, Fn, was applied through

the rigid loading platen and driven at a constant velocity VLP using the shear actuator. The shear actuator

was located at the trailing edge (TE) of the slider block. (b) Measurements from a typical stick-slip event

(SS) during a single loading cycle. Slip (δ) accumulated slowly in the interseismic phase and transitioned to

rapid sliding during the seismic phase of the loading cycle. (c) Total catalog of foreshock seismicity from

Selvadurai & Glaser (2015b). A total of 68 foreshocks were observed over 8 stick-slip events and the size of

the event was determined from the peak ground displacements (PGD) averaged over local sensors. (d) An

individual seismic event (black event from (c)) as measured by the sensors in the acoustic emission array.


Figure 2. Normal stress distributions recorded from the pressure sensitive film for a small section (x = 270

to 290 mm, y = 6 to -2 mm) of the interface at Fn = 4400 N (top) and 2700 N (bottom). The inset shows how

the asperities vary with changing normal force Fn is changed over the mature faulting surface. For each ith

asperity, we measure the normal stress field σi (MPa), the mean normal stress σ̄i (MPa), the asperity area

Ai (mm2) and the coefficient of variation CVi.

600 800 1000 1200 1400 16002

2.5

3

3.5

4× 10-3

13

14

15

16

600 800 1000 1200 1400 16001.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3× 104

20

30

40

50

60

70

80

90

100

Mean a

sperity

norm

al

str

ess,

(MP

a)

σ

15.42 MPa

Mean a

sperity

siz

em

m)

A(

2

Num

ber

of asperities,

n

Real conta

ct are

a,

(mm

)A

r

2

Reactive normal force, (N)Fr

Reactive normal force, (N)Fr

A Fr r= 0.059·

R = 0.9962

b)a)

Figure 3. (a) Changes in the amount of asperities n (blue circles) and real contact area Ar (orange stars)

as the reactive normal force Fr increased within the seismogenic region (x = 150 to 300 mm). Reactive

normal force was calculated as the summation of forces on all i asperities within the seismogenic region.

(b) Changes in the mean asperity area 〈A〉 (blue squares) and mean asperity normal stress 〈σ〉 (orange

triangles) as the reactive normal force Fr in the seismogenic region was increased.


15 16 17 18 19 20 21100

105

15 16 17 18 19 20 2110-4

10-2

100

102

Fn = 4400 N

Fn = 3400 N

Fn = 2700 N

Fn = 2000 N

σ

Mean asperity normal stress, (MPa)is Mean asperity normal stress, (MPa)is

Asperity

are

a(m

m)

Ai

2

σ

Counts

a) b)

ξ

15 16 17 18 19 20 210

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.2510-4

10-2

100

102

Coefficient of Variation Mean asperity normal stress, (MPa)is

c) d)

Asperity

are

a(m

m)

Ai

2

Coeffic

ient of V

ariation

6.1%

4.1%/MPa

Figure 4. Normal stress measurements from individual asperities within the seismogenic region of the

fault (x = 150 mm to 300 mm) compiled into a catalog for various levels of far-field normal force Fn.

(a) The probability distribution function (PDF) for the distribution of the mean asperity normal stress σ̄i.

Relationships between (b) asperity mean normal stress σ̄i and contact area Ai, (c) asperity coefficient of

variation CVi and contact area Ai and (d) asperity mean normal stress σ̄i and coefficient of variation CVi.


Figure 5. (a) Optical microscopy images of an asperity, on the slider block surface, located in the highly

seismogenic region of the fault (approximately x = 235 mm and y = 5.1 mm) after the suite of experiments

were performed. Stitching four sub-images created the image. We see evidence of features similar to geo-

logical features on natural faulting surfaces (Candela et al., 2011), such as, striations and slickensides. (b

and c) show separate 2D surface profiles using the ADE MicroXAM-100 optical profilometer. (d) and (e)

shows height profiles along transects from (b) and (c), respectively, in the direction of slip (blue) and normal

to slip (magenta). (f and g) Fourier power spectrum P (k) of the 1D profiles from (d and e), respectively,

in the direction and normal to slip on the interface. The Hurst estimate H (eq. 2) was used to measure the

self-affinity of the surface.


Figure 6. (a) A large section (25 mm x 12 mm) of the seismogenic region is shown. (b) The Power Density

Function (PDF) of surface heights is shown for (a). (c) Surface profiles from Transect A (y= -4 mm) and

Transect B (y= 4 mm) are shown for the section shown in (a). Transect B has more and larger ‘flat-topped’

surfaces than the lower region. A small section of Transect B was examined to better see the ‘flat-topped’

feature. We see that this feature has a width w ∼ 0.7 mm and height h ∼ 0.03 mm (h/w = 0.04). (d) Fourier

power spectrum P (k) of the height profiles using the Nanovea PS50 optical profilometer are shown in the

direction (red) and normal to slip (black) on the interface. These are compared with the smaller and more

spatially resolved measurements taken using the ADE MicroXAM-100 and shown for comparison (blue

and magenta lines).


5 m

m

50 mm

Fn 4400 N≈a)

dsep

Lgrid

A , Ngrid grid

b)

Lgrid

A , Ngrid grid

agrid

c)

Figure 7. Average asperity separation distance calculation. (a) Asperity contact measurements from the

pressure sensitive film are examined in the seismogenic zone (gray region). This is discretized into 40 x 3

square cells with side length Lgrid. (b) One cell contains Ngrid asperities with a total real contact area of

Agrid. From this we can calculate the average asperity radius āgrid. (c) Using the cellular packing solutions

from Specht (2015) we can calculate the optimal separation distance dsep for each cell.

Figure 8. (a) Asperity radius ri versus the mean asperity normal stress σ̄i measured at an applied normal

force Fn = 4400 N determined using the pressure sensitive film (red dots). Eqs 4 and 5 calculate the critical

nucleation length required for foreshock seismicity. Parameters used are shown in the legend. Asperities

with radii greater than the critical nucleation length ri > h∗ are susceptible to foreshocks whereas ri < h∗

were not. (b) Detailed view of experimental results. The gray shaded region express the limits to the radius

of the foreshocks source determined by Selvadurai and Glaser (Selvadurai & Glaser, 2015b).

Asperity formations and their relationship to seismicity...

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