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From laboratory friction to numerical models of fault dynamics
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From laboratory friction to numerical models of fault dynamics
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FROM LABORATORY FRICTION TO NUMERICAL MODELS OF FAULT DYNAMICS
by
Shiying Nie
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(GEOLOGICAL SCIENCES)
May 2024
Copyright 2024 Shiying Nie
Dedication
To my children, Charlotte and Vincent, with all my love.
ii
Acknowledgements
My PhD journey was deeply affected by the global impact of COVID-19. During this time, facing challenges
like limited childcare, mixed-up work-life schedules, and difficult communication, we all learned to be
resilient and adapt. Without the tremendous support from my family, friends, and academic community,
I wouldn’t have been able to manage through this period, which has also transformed this journey into a
deeply meaningful part of my life.
First, I want to thank my family—Charlotte, Vincent, and Dr. Yongfei Wang—for their constant love,
encouragement, and both emotional and physical support during this time. Professors Steve Day and
Jennifer FitzGerald, who lifted me when I was down, guided me when I felt lost, and cheered me up when
I doubted myself. Thank you all for being the guiding light in the dark moments of my journey.
I am thankful to my thesis advisor, Professor Sylvain Barbot, for his guidance, patience, and valuable
feedback that played a crucial role in shaping my thesis. His expertise not only influenced my work but
also molded my thinking. I’m also grateful to the faculty members, especially my committee members,
Professors John Vidale, John Platt, Heidi Houston, Tom Jordan, James Dolan, Steve Nutt, and many others, for their insightful courses, constructive feedback, and valuable discussions, all of which significantly
contributed to my academic growth. I am thankful to Professors John Platt and Frank Corsetti for their
support when I faced progress challenges. I’m thankful to my peers and colleagues—Dr. Sharadha Sathiakumar, Dr. Xiaofeng Meng, and others—for sharing their knowledge, ideas, and experiences, and for the
friendships we’ve formed, helping me grow academically and personally.
iii
Regarding the contributions to my dissertation, Chapter 2 is a full reprint of the material previously
published in Earth and Planetary Science Letters 2022 (Nie, Shiying, and Sylvain Barbot. "Rupture styles
linked to recurrence patterns in seismic cycles with a compliant fault zone"), with me as the first author.
Similarly, Chapter 3 is a full reprint of the material previously published in Earth and Planetary Science
Letters (Nie, Shiying, and Sylvain Barbot. "Seismogenic and tremorgenic slow slip near the stability transition of frictional sliding"). Chapter 4 has been submitted for publication, showcasing the material as it
may appear in Geophysical Research Letters (Nie, Shiying, and Sylvain Barbot. "Velocity- and temperaturedependence of steady-state friction of natural gouge controlled by competing healing mechanisms"), with
me as the first author.
iv
Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Physical-based friction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The implications of frictional behaviors on fault dynamics . . . . . . . . . . . . . . . . . . 4
1.3 Motivation and thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 2: Rupture styles linked to recurrence patterns in seismic cycles with a compliant fault zone 12
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Physical assumptions and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Rupture styles and recurrence patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 The effect of compliant fault zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 3: Seismogenic and tremorgenic slow slip near the stability transition of frictional sliding 41
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Physical assumptions and modeling method . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Seismogenic slow-slip events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Tremorgenic slow-slip events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Chapter 4: Velocity- and temperature-dependence of steady-state friction of natural gouge
controlled by competing healing mechanisms . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Plain Language Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Constitutive framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
v
4.5 Constitutive properties of San Andreas Fault gouge . . . . . . . . . . . . . . . . . . . . . . 69
4.6 Application to other fault gouges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Governing equations and non-dimensional parameters . . . . . . . . . . . . . . . . . . . . . . . 111
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Dieterich-Ruina-Rice number for a compliant fault zone . . . . . . . . . . . . . . . . . . . . . . . 115
Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Formulations based on the age of contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
vi
List of Tables
2.1 Physical parameters of fixed fault zone thickness simulations (Figure 2.2 to 2.8) . . . . . . 30
2.2 Physical parameters of fault zone thickness experiments (Figure 2.8 to 2.10). . . . . . . . . 30
3.1 Physical parameters for aseismic and seismogenic slow-slip events and earthquakes
simulations (Figure 3.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Physical parameters of slow-slip and tremors simulations (Figure 3.3, 3.4, 3.5). . . . . . . . 56
4.1 Constitutive and physics parameters of the simulated velocity-step experiments on the San
Andreas Fault Observatory at Depth SDZ sample shown in Figure 4.9. The uncertainties
correspond to plus or minus a standard deviation. The parameters d0 = 1 µm, V0 = 1 µm/s,
and T0 = 25◦C represent scaling factors, not constitutive parameters per se. The reference
friction coefficient µ0 is a material property corresponding to the ratio of plowing to
indentation hardness. The gouge thickness h = 1 mm is a laboratory setting. . . . . . . . . 75
4.2 Constitutive parameters of natural gouge constrained by velocity-step experiments
(Figures 4.9 and 4.10). The activation energies H1 and H2 are in kJ/mol. The activation
temperatures T1 and T2 are in degrees Celsius. The ratio T1/T2 is better constrained than
the absolute value of T1 and T2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Constitutive parameters using RSFit3000. The velocity steps are imposed sequentially
from V1 to V2, expressed in µm/s. Experiments are conducted at different temperatures
expressed in ◦C. The characteristic weakening distance L is expressed in µm. The
parameters a and b and their uncertainties are expressed in ‰. . . . . . . . . . . . . . . . . 76
vii
List of Figures
1.1 Schematic of the contact of micro-asperities and the healing mechanisms, using
compaction creep and pressure-solution and precipitation as examples. . . . . . . . . . . . 2
1.2 Behavior of a spring-slider system under constant loading velocity: (a) Evolution of the
friction coefficient over time. (b) Displacement of the system over time. The black and red
lines represent the stick-slip and stable sliding behaviors, respectively. . . . . . . . . . . . 5
1.3 The schematic diagram of the evolution of friction coefficient during a velocity-jump
experiment. The velocity-weakening (a − b < 0) and velocity-strengthening (a − b > 0)
conditions are represented in black and red, respectively. . . . . . . . . . . . . . . . . . . . 7
1.4 Schematic of the distribution of earthquakes and slow earthquakes on the strike-slip fault
(a) and thrust fault (b). (modified from Burgmann, 2018 [54]) . . . . . . . . . . . . . . . . . 8
1.5 Velocity and temperature dependence of friction of crustal faults from laboratory
experiments. (a) shows the measurement of a-b under different loading velocities at room
temperature. (b) depicts the measurement of a-b under different temperature conditions.
The velocity jump is labeled next to the points, and the unit is µm/s. The offset of
the points within one study indicates the dataset from different samples for (a) or the
decrease/increase of velocity in (b). (c) and (d) display the interpolated a-b as a function of
both velocity and temperature using data from the Alpine Fault and Central San Andreas
Fault, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Schematic representation of the simplified compliant fault zone model. The rigidity of the
bedrock and the fault zone are G and Gcz, respectively. The fault zone thickness is T. The
central unstable patch of width W is surrounded by a velocity-strengthening fault. . . . . 31
2.2 Seismic cycle simulations under variable characteristic weakening distance and fault zone
compliance level. a) Sub-domains for creep, slow slip, bilateral and unilateral periodic
ruptures, crack-like ruptures and pulse-like ruptures. Contours are the Ru numbers of the
models. The symbols represent the rupture styles of the models used to produce this phase
diagram. The background color indicates the peak slip velocity and the shading represents
the crack-like to pulse-like rupture transition. b to i) Examples of different rupture styles,
represented by the slip velocity. The x- and y-axes represent down-dip distance on the
fault and time steps, respectively. The area between dash lines is the velocity-weakening
domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
viii
2.3 Representative behavior of numerical simulations for a range of compliant zone rigidity,
characteristic weakening distance. Each rectangle represents the evolution of fault slip
velocity as a function of numerical time steps. All the simulations showed here include a
fault zone thickness of 2 km. R = Gcz/G is the compliant ratio and L is the characteristic
weakening distance in units of millimeters. The peak slip velocity, proportion of cracklike versus pulse-like ruptures, proportion of full and partial ruptures, and number of
aftershocks per mainshock shown in Figures 2.2, 2.4, 2.5 and 2.6 is extracted from these
simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Crack-like to pulse-like rupture transitions under variable characteristic weakening
distance and compliance level of the fault zone. a) The proportion of crack-like versus
pulse-like rupture style. Contours are for constant Ru numbers. b) A typical crack-like
rupture propagation for a simulation with G/Gcz = 2 and L = 2mm. The upper panel is
the slip velocity versus time steps; middle panel shows the rupture history with 1-second
interval contour; the lower panel is the source time function, the sampling locations are
shown as the blue line in upper panel, close to the hypocenter. c) and d) pulse-like rupture
models differing from b) by increasing the compliance level or reducing the characteristic
weakening distance, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 The relationship between rupture styles and temporal recurrence patterns. a) Phase
diagram of coefficient of variation (CoV, standard deviation divided by mean of the
recurrence time of major earthquakes) of the major events (events that rupture at least
half of the fault). The contours are the Ru numbers of the models. b) to e) The left plots
show the earthquake timelines and moments for four parametric regimes labeled in a).
The stars indicate the major events used to calculate the CoV in a), while the triangles
indicates the minor events. The right plots show the corresponding slip deficit. The
upper and lower dash lines in each figure show the approximated limit of slip- and timepredictable, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Relationship between rupture styles and spatial recurrence patterns. a) Phase diagram
of proportion of full-ruptures, excluding the events that rupture less than 20% of the
seismogenic zone. The threshold for a full rupture is 90% of the seismogenic zone. The
contours are the Ru numbers of the models. b) to e) The rupture area of the earthquakes
from four parametric regimes labeled in a). The stars indicate the hypocenters of major
events used to calculate CoV, while the triangles indicates the hypocenter of minor events.
The dark blue lines represent rupture areas of full ruptures mentioned in b). The light
blue lines are counted as partial ruptures. The black ones are ignored for this analysis. . . 36
2.7 The statistics of aftershocks under variable characteristic weakening distance and
compliance level of the fault zone. a) Number of aftershocks per mainshock. The contour
lines are the Ru number of the simulations. The inset shows the number of aftershock
as a function of Ru for all the 525 simulations. b) Velocity vs time step and terminology,
the parameters are shown on a). c), e), and g) display the time history and moment of the
mainshocks and aftershock sequences. The shaded areas are the postseismic periods. d),
f), and h) is the corresponding peak slip velocity used to define the postseismic period. i)
Seismicity rate of aftershocks as a function of time from the mainshock. In all of b-i), the
stars represent the mainshocks. The asterisk and diamonds represent shallow and deep
aftershocks, respectively. The triangles are the remaining events. . . . . . . . . . . . . . . 37
ix
2.8 Efficiency of the Ru number to describe the rupture style. a) Rupture style for varying
compliance level and characteristic weakening distance for a fixed compliant zone
thickness of T = 2, 000 m. b) Rupture style for varying fault zone thickness, characteristic
weakening distance, and compliance ratio G/Gcz. The corresponding parameters are in
Table 2.1 and Table 2.2. The dash lines are the approximation of Ru thresholds of rupture
styles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.9 Representative behavior of numerical simulations for a range of compliant zone rigidity,
compliant zone thickness, and characteristic weakening distance. Each rectangle
represents the evolution of fault slip velocity as a function of numerical time steps.
R = Gcz/G is the compliant ratio, L is the characteristic weakening distance in
millimeters and Th represents compliant zone thickness in units of meters. The analysis
of Figure 2.8 is constructed from these simulations. . . . . . . . . . . . . . . . . . . . . . . 39
2.10 Equivalent sets of fault zone thickness, compliance, and characteristic weakening distance
for the same Ru number. a) to d) Relationship among fault zone thickness, compliance,
and characteristic weakening distance for Ru = 5.33, 11, 28.5 and 98, respectively. The
contours are the required characteristic weakening distance, in units of millimetres. The
red dash lines indicate the thickness T = 2h
∗
. For each Ru number, the rupture history
of three models with different physical units is plotted below each phase diagram. The
x- and y-axes of the color plots are time steps and distance, respectively. The white text
represents their actual Ru number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 Concurrence of slow-slip events and slow earthquakes at major plate boundaries. A) Slowslip events [206] and tremors (NIED catalogue) in 2012, and rupture area (thick dashed
line) of historical large earthquakes [210] at the Nankai Trough, Japan. B) Collocated
tremors [129] and slow-slip [262] at the Cascadia subduction zone. The seismogenic
zone is situated above the region of high geodetic coupling [190]. The thin dash lines in
A) and B) correspond to the USGS Slab2 model [116]. C) Distribution of low-frequency
earthquakes along the San Andreas Fault [268]. D) Low-frequency earthquakes and
correlation with surface geodetic measurements [246]. The seismogenic region [17]
surrounds the hypocenters of the 1966 (green star) and 2004 (red star) earthquakes. . . . . 57
3.2 Seismic cycle simulations under variable Rb and Ru. A) Sub-domains for creep, creep
waves, earthquakes, aseismic and seismogenic slow-slip events. The background color
indicates the peak slip velocity. The white curves show the time series of peak slip
velocities for a one or more events, with varying time scales. B to E) Examples of waves
of partial coupling, aseismic slow-slip from low Ru velocity-weakening regime, aseismic
slow-slip from medium Ru velocity-neutral regime, and earthquakes. F to I) Seismogenic
slow-slip events from high Ru in the near-neutral velocity-weakening regime. The Ru
number is controlled by the characteristic weakening distance. The velocity weakening
area is between the two dashed lines. The segmentation lines in A) are only conceptual. . . 58
3.3 Simulated collocated fast and slow-slip events with a 2D heterogeneous model. A) Peak
velocity of the slip cycles. B) Synthetic slip cycles along the entire fault length. The
velocity-weakening area is located between the two vertical black dashed lines. The
hypocenters of the seismic events (circles with size scaled by seismic moment) are plotted
on top of slip velocity of the rupture cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
x
3.4 Space-time distribution of seismic hypocenters for the entire simulation period. A)
Histograms of daily slow earthquakes. B) The time distribution of slow (grey bands) and
fast (circles scaled by seismic moment and colored by peak velocity) events. The slow-slip
events are detected by the average velocity (10−6
as the start of a event and 5 × 10−9
as the end) with a duration longer than 60 days. Slow-slip events generally last longer
than the seismogenic period, and most seismicity occurs during an underlying slow-slip
episode. C) to E) enlarge the three seismogenic periods shown by the dashed lines in
figure B). The background color indicates the slip velocity. . . . . . . . . . . . . . . . . . . 60
3.5 Geodetic and seismic signatures of seismogenic slow slip. A) Location of the geodetic and
seismic receivers relative to the source fault. The source is simplified as a point located
in the middle of the velocity-weakening area (orange star). The red triangle indicates
the primary station used to simulate geodetic and seismic data for the time series in B).
The yellow triangles represent a seismic array with 16 receivers along the same azimuth.
B) Synthetic geodetic displacement and velocity (upper panel) and clipped seismograms
showing the ground velocity (bottom panel) for the entire sequence. C) Seismograms
generated by ruptures highlighted in Figure 3.4D and recorded by the orange seismic
array in A). The start time is labeled by dashed red line in B). The P and S wave arrivals
are marked by the blue and red lines, respectively. D) is the seismograms highlighted by
blue rectangle in C) with a smaller amplitude range. . . . . . . . . . . . . . . . . . . . . . . 61
3.6 Schematic distribution of frictional properties on subduction megathrust and continental
transform faults. A) Subduction megathrust. The seismogenic zone spans the unstable
weakening fault region. Seismogenic slow-slip events take place above and below, in the
near-neutral weakening region. B) Continental transform. The seismogenic zone and
the seismogenic slow-slip area are associated with unstable weakening and near-neutral
weakening friction properties, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1 Velocity and temperature dependence of friction of natural gouge from laboratory
experiments. a) Schematic of friction evolution during a velocity-jump experiment with
possible velocity-strengthening (a − b > 0) and velocity-weakening (a − b < 0) at
steady-state. b) Measured (diamonds) and interpolated (background color) value of a − b
as a function of velocity and temperature for the SDZ sample cored from San Andreas
Fault Observatory at Depth in the Central San Andreas Fault (CSAF). c) Laboratory
measurements of a − b from natural gouge for different loading velocities at room
temperature. d) Variation of a − b under various temperature slip-rate conditions. . . . . . 77
4.2 Mechanisms of deformation and healing enabling seismic cycles across a frictional
interface. a) Case of solid-solid or bare contact with contact rejuvenation by dilatant
shear during sliding and compaction creep accommodated by viscoelasticity or pressuresolution creep during relocking. b) Case of solid-gouge-solid contact where granular flow,
fracturing, and subcritical crack growth accommodate dilatant shear and comminution.
The closure/cementation of cracks enables healing of the interface. Each mechanism
acting on different minerals is associated with specific constitutive properties. . . . . . . . 78
xi
4.3 Friction coefficient as a function of velocity and displacement from simulated velocity
jump experiment for the constitutive law with two mechanisms. a) The friction coefficient
under the temperature of 25, 150, and 300◦C as a function of velocity. The black solid
and dashed red lines represent the experiments utilizing Equation (3) and Equation (4),
respectively. b) The friction coefficient as a function of displacement, conducted by
step-up and step-down velocity jump experiments. Equations (3) and (4) correspond to
the aging-law and slip-law end-members, respectively. . . . . . . . . . . . . . . . . . . . . 78
4.4 Micro-asperity size and frictional velocity dependence as a function of sliding velocity
and temperature. The dashed lines show the evolution of one single mechanism. The first
mechanism, represented by red lines leads to velocity-strengthening behavior while the
second mechanism (blue lines) tends to velocity-weakening. The solid lines are the results
of two competing mechanisms that are dominant in different velocity and temperature
ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.5 Effective frictional parameters obtained from regression of velocity step experiments
using the SDZ sample. a) and b) Direct effect parameter a as a function of velocity
and temperature, respectively. c) and d) Evolution parameter b as a function of velocity
and temperature, respectively. e) and f) Same for the characteristic weakening distance
L. g) and h) Same for the stiffness k. The up-pointing and down-pointing triangles
represent data from step-up and step-down experiments, respectively. Colors represent
the temperature and loading rate of the experiments. The parameters are summarized in
Table 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Evolution of the friction coefficient during simulated velocity jumps experiments. a) Effect
of different temperatures. b) Effect of loading rate. c) and d) Effect of stiffness. . . . . . . . 81
4.7 Constraints on the friction coefficient µ for the SDZ sample. a) Root-mean-square error
(RMSE) of the observed and simulated steady-state friction coefficient µ as a function
of activation energy Q and reference friction coefficient µ0. b), c) and d) Comparison
between the simulated and the regressed steady-state friction coefficient µ as a function of
loading velocity and temperature. The simulated data is the background color for (b) and
the solid lines for (c) and (d), and the regressed data is label colors for (b). The colors in
(c) and (d) label temperature and velocity, respectively. The circles and triangles represent
the value of steady-state µ before and after the velocity changes. . . . . . . . . . . . . . . . 82
4.8 Optimization of constitutive parameters for the SDZ sample. a), b), c) and d) show the
root-mean-square deviation (RMSE) of synthetic and lab-observed a − b measurement as
a function of exponents n/m, evolution exponents ratio p1/p2, activation enthalpy ratio
H1/H2, and activation temperature ratio T1/T2. The parameters used to generate the
preferred model shown in Figure 3 (red dot) are labeled. . . . . . . . . . . . . . . . . . . . . 83
xii
4.9 Comparison between simulated and measured frictional data for samples from the SDZ
of the San Andreas Fault Observatory at Depth. a) Comparison between simulated
(background color) and measured (colored triangles) steady-state velocity-dependence
parameter a − b. b) Simulated (lines) and regressed (triangles) a − b parameters as a
function of velocity colored by temperature. c) Simulated (lines) and regressed (triangles)
a − b parameters as a function of temperature colored by velocity. d) Comparison of
the synthetic (lines) and raw friction coefficient measurements (gray dots). The red
solid lines and black dashed lines denote simulations conducted using Equation (4.3) and
Equation (4.4), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.10 Comparison between simulated (lines) and laboratory-derived (dots) velocity dependence
parameter (a − b) of natural samples from a) South Alpine fault [27], b) and c) CDZ and
SDZ of central San Andreas Fault, respectively [58], d) Zuccale Fault [69], e) lizardite-rich
serpentinite sample [140], f) Alpine DFDP-1 sample [48], g) South Alpine outcrop [47], h)
Nankai (simulated) [112], and i) Central San Andreas Fault CDZ [195]. Colors represent
experiments under different temperatures. The dashed line is for a − b = 0. . . . . . . . . 85
xiii
Abstract
Faults display a broad spectrum of behaviors in terms of their rupture styles and recurrence patterns,
varying from aseismic creep to chaotic earthquake sequences. However, the underlying mechanisms and
potential links behind these phenomena are poorly understood. In this study, I consider the frictional properties as essential ingredients to understand the variation in seismic observations. I utilize a physics-based
friction constitutive law that considers the thermally activated healing mechanisms of the micro-asperities.
I start by exploring the relationship between rupture styles and their recurrence patterns within the seismogenic zone. I use quasi-dynamic simulations of faults embedded in a compliant zone and explore a broad
spectrum of friction properties under isothermal conditions, assuming a single healing mechanism. The
research demonstrates that as the fault zone rigidity or characteristic weakening distance decreases, the
rupture style shifts from crack-like to pulse-like. Concurrently, the recurrence behavior transitions from
slip- and time-predictable sequences to chaotic sequences characterized by varying sizes of ruptures and
aftershock sequences. In this research, I also suggest that the effect of the fault zone can be represented
by a single non-dimensional number, allowing for comparable outcomes in quasi-dynamic simulations regardless of whether the fault zone is considered. I then extend the research to slow-slip events and tremors
that take place downdip of the seismogenic zones, also using the quasi-dynamic approach coupled with a
similar physical-based friction law. This study considers both the characteristic weakening distance and
the friction dependence conditions to create a more comprehensive parametric space of friction parameters and associated rupture behaviors. I find that the simultaneous and co-located occurrence of slow- and
xiv
fast-events is a natural outcome under a velocity-neutral condition with a small characteristic nucleation
size. This condition aligns with the frictional conditions of the fault beneath the seismogenic zone, which
acts as a transition from velocity-weakening to velocity-strengthening. Moreover, tremors are a burst of
slow earthquakes initiated when slow-slip events interact with small-scale structural asperities. In the final part of the study, I focus on the findings that frictional parameters vary with temperature and velocity,
which goes beyond the scope of the condition of isothermal and a single healing mechanism. I utilize the
physical-based friction constitutive law that incorporates the competition of healing mechanisms dominant in various temperature and velocity regimes. This model explains the friction properties of a range of
natural gouges from the San Andreas Fault, Alpine Fault, and the Nankai Trough. The friction framework
has the potential for further understanding a broader range of fault dynamics in future studies.
xv
Chapter 1
Introduction
1.1 Physical-based friction model
The evolution of friction strength is believed to be determined by the deformation processes of the microasperities that bear the load and create the real contact between two surfaces [79]. These processes are
activated under different hydrothermal conditions, and typically associated with factors such as mineral
types, surface roughness, fluid presence, and other fault conditions. For example, compaction creep in
micro-asperities exhibits a wide range of activation energies (50–700 kJ/mol), while fluid-related processes
like pressure-solution and precipitation operate at relatively lower activation energies (<130 kJ/mol), and
the ranges vary depending on the rocks types [18]. In reality, these processes collaborate and compete,
leading to complex dynamics in frictional sliding. Laboratory studies further illustrate this complexity
by showing the significant variation in frictional parameters under different hydrothermal conditions,
loading velocities, pore-fluid pressures, and confining pressures, even for the same lithology. Therefore,
it is crucial to move beyond the empirical friction laws and develop a physical-based friction framework
that incorporates micro-mechanisms in order to capture these laboratory findings.
Many models have been proposed to interpret the results from experimental studies. For example, Sleep
(1997) linked the evolution of the state variable to the creation of porosity by frictional dilatancy and the
1
Mineral 1
Compaction
Creep
Mineral 2
σ
Precipi
-tation
Dissolution
Micro-asperity
d + δd
d
Figure 1.1: Schematic of the contact of micro-asperities and the healing mechanisms, using compaction
creep and pressure-solution and precipitation as examples.
closure of porosity by compaction [286]. Niemeijer and Spiers (2007) developed a model based on the competition between intergranular dilatation and pressure solution to explain the strong velocity-weakening
behavior at rapid slip rates in halite-muscovite samples [203, 62]. The dynamic weakening effect that occurs at the speed of earthquake rupture is also included in seismic cycle simulations [208]. The physical
friction law utilized in this study is based on the evolution of the real area of contact, which is controlled
by the healing of the contacted micro-asperities, subsequently changing the friction strength [11]. This
model unifies the physical concepts at the microscopic scale such as the healing of grain, real area of
contact, gouge thickness, and activation energy, and has been proven to explain a number of laboratory
observations [18].
The constitutive law of friction strength is expressed as:
µ = µ0
V
V0
1
n
d
d0
m
n
exp
Q
nR
1
T
−
1
T0
, (1.1)
This flow law describes the relationship between the friction coefficient µ, the slip rate V , and the temperature T. The asperity size d represents the local radius of curvature at the contact junctions (Figure
1.1). The parameters n and m are the stress and micro-asperity-size power exponents, respectively. The
constants µ0, V0, d0, and T0 are reference values for the friction coefficient, velocity, size of asperities, and
2
temperature, respectively. The exponential term accounts for the Arrhenius activation with the activation
energy Q. R is the universal gas constant. The evolution of the asperity size is governed by:
˙d
d
=
X
N
k=1
Gk
pkd
pk
−
λV
2h
, (1.2)
In this formulation, the first term represents the healing process, which includes N different micro-processes.
The kth healing mechanism is governed by the size power exponent pk and the reference growth rate Gk.
The second term corresponds to asperity reduction mechanisms, such as fracturing or contact rejuvenation
during shear. The healing rate Gk captures the thermal activation of the kth process:
Gk(T) = G
0
k
exp
−
Hk
R
1
T
−
1
Tk
, (1.3)
In which G0
k
represent the reference growth rate, Hk is the activation enthalpy, and Tk is the activation
temperature.
Assuming one healing process is involved (N = 1) under isothermal conditions, Equation 1.1 and 1.2
becomes:
µ = µ0
V
V0
1
n
d
d0
m
n
, (1.4)
˙d
d
=
G
pdp
−
λV
2h
, (1.5)
in which, the age of micro-asperities θ corresponds to the size of the micro-asperity d:
θ =
1
G
d
p
, (1.6)
3
and the characteristic weakening distance L is related to the gouge thickness h:
L =
2h
λp , (1.7)
After the substitution based on equation 1.6 and 1.7, the consititutive law becomes:
τ = µ0σ¯
V
V0
1
n
θV0
L
m
pn
, (1.8)
and the age of contact follows the aging law [253]:
˙θ = 1 −
V θ
L
, (1.9)
This multiplicative formulation resembles the classic rate-and-state friction law [78, 252]. It suggests that
the classical logarithmic version is, in fact, the first-order approximation obtained by Taylor expanding
equation 1.8, which corresponds to equation 1.1, under the assumption of a single healing process and
isothermal conditions. This constitutive law not only offers a physical explanation for the empirical form
of the rate- and state-law but also enhances it by considering multiple thermally activated deformation
mechanisms of asperities within a micro-physical framework.
1.2 The implications of frictional behaviors on fault dynamics
During friction experiments on a spring-slider system, the interaction between two contacted surfaces in
relative motion exhibits two dominant behaviors. The first is characterized by the contacted surfaces staying relatively stationary for an extended period, followed by a sudden slip associated with stress release.
These slip events occur recursively with ongoing loading and are commonly referred to as "stick-slip" behavior. The second behavior involves continuous and steady motion of the contact surfaces without abrupt
4
Time
Load
Locking phase
Rupture phase
Steady sliding
Creep
Stick slip
Slip
Displacement Friction coefficient (μ) A
B
Figure 1.2: Behavior of a spring-slider system under constant loading velocity: (a) Evolution of the friction
coefficient over time. (b) Displacement of the system over time. The black and red lines represent the
stick-slip and stable sliding behaviors, respectively.
changes in velocity, identified as "stable-sliding" (Figure 1.2). These commonly observed phenomena have
been proposed to describe the mechanisms of seismic events. The recurrent "stick-slip" behavior is believed
to be linked to earthquake sequences [51], while "stable-sliding" is used to explain aseismic creep found
in down-dip or up-dip regions of seismogenic zones. Thus, understanding friction behaviors is crucial in
unraveling the fundamentals of earthquakes and other related phenomena.
The velocity-jump experiments conducted on rock samples typically exhibit a specific pattern in the
friction coefficient. Following a sudden increase in loading velocity, there is a rapid increase in the friction
coefficient by an order of magnitude, commonly referred to as the "direct effect." Subsequently, the friction
coefficient gradually decreases, stabilizing at a steady-state value after a specific sliding distance. This
observed trend can be explained by the constitutive friction law mentioned in the previous section. To
5
align the classic parameters in rate-and-state friction with the physics-based constitutive laws, we further
assume:
a =
µ0
n
, (1.10)
b =
mµ0
np
, (1.11)
Here, the parameters a and b represent the "direct" and "delayed" effects observed in velocity-jump experiments. Additionally, the characteristic weakening distance L governs the distance required for the friction
coefficient to stabilize at the steady-state. These parameters are essential for predicting frictional-related
rupture behaviors. Specifically, the difference between a and b reflects the velocity dependence of friction.
When a−b is positive, indicating an increase in the friction coefficient with higher velocity, this condition
leads to stable sliding and is typically described as the velocity-strengthening condition. Conversely, if
a − b is negative, it allows the stick-slip instability to occur, known as the velocity-weakening condition
(Figure 1.3). In fault dynamics, the velocity-strengthening condition is used to characterize the creep and
afterslip of earthquakes occurring both above and below the seismogenic zone. On the other hand, the
velocity-weakening condition enables the nucleation and propagation of rupture events, thereby permitting the occurrence of earthquakes. Additionally, within a region characterized by velocity-weakening
conditions, the initiation of events is restricted to cases where the asperity size exceeds the characteristic
nucleation size. This size is determined by a combination of parameters including a, b, effective normal
stress, material rigidity (G), and the characteristic distance (L), which is explained in Chapter 2 and Appendix A.
In reality, the complexity of fault dynamics observed in nature extends far beyond ’stick-slip’ and
’stable-sliding.’ Although the simple spring-slider model suggests that earthquakes with the same rupture
area should occur periodically as the strain from the previous one accumulates [237] (Figure 1.2), the spatial
and temporal recurrence of major earthquakes deviates significantly from this periodic and characteristic
6
Displacement
v1 v2>v1
Friction coefficient (μ) μss(v1)
μss(v2)
μss(v2)
a - b >0
V-strengthening
V-weakening
a - b < 0
Figure 1.3: The schematic diagram of the evolution of friction coefficient during a velocity-jump experiment. The velocity-weakening (a − b < 0) and velocity-strengthening (a − b > 0) conditions are represented in black and red, respectively.
pattern. Additionally, the rupture styles of earthquakes are complex, including a variety of styles, such as
unilateral or bilateral ruptures[245], crack-like or pulse-like ruptures [311], or even sub-Rayleigh or supershear waves. While these phenomena are increasingly documented by seismic or paleoseismic records, the
underlying mechanisms are still not well understood and remain a mystery to researchers.
Furthermore, improvements in various seismographic and geodetic networks, have enabled the discovery of phenomena other than earthquakes that accommodate tectonic motion. These phenomena include
“slow earthquakes", which include a range of processes such as slow slip events, tremors, low-frequency
earthquakes (LFEs), and very low-frequency earthquakes (VLFEs). Such phenomena have been observed
below the seismogenic zone of major subduction zones and strike-slip faults, such as the Cascadia, Nankai
Trough, and San Andreas Faults (Figure 1.4). These findings enrich the diversity of fault rupture styles
while simultaneously presenting new challenges to understanding fault mechanics.
1.3 Motivation and thesis structure
To establish a link between laboratory experiments and the well-recognized fault behaviors discussed in
the previous section, and thus provide a comprehensive explanation for a wide range of rupture dynamics
observed in nature, two non-dimensional numbers derived from the combination of frictional parameters,
7
A
B
Figure 1.4: Schematic of the distribution of earthquakes and slow earthquakes on the strike-slip fault (a)
and thrust fault (b). (modified from Burgmann, 2018 [54])
8
Ru and Rb, are introduced [12]. The Ru number, defined as the ratio between asperity and critical nucleation size (W/h∗
), governs various fault slip styles from slow-slip to aperiodic pulse-like fast ruptures.
Meanwhile, the Rb number, which is the ratio of velocity dependence ((b − a)/b), not only indicates the
possibility of instability but also indicates the occurrence of concurrent slow-slip events and slow earthquakes when combined with Ru. In Chapters 2 and 3, I demonstrate that the parametric space defined
by these non-dimensional numbers can explain a wide range of seismic observations using seismic cycle
simulations. The friction constitutive models are described in Equations 1.8 and 1.9.
In Chapter 4, I focus on an observation that the velocity dependence a − b, which was believed to be
constant with the same lithology, is observed to change under different velocity and temperature regimes.
Several studies show that a decrease in velocity and/or an increase in temperature can lead to a transition
from velocity-strengthening to velocity-weakening (Figure 1.5). This transition has the potential to impact
fault behavior since the velocity dependence determines the possibility of event nucleation. I utilize the
physics-based constitutive law to account for thermal activation and the competition between two different
healing mechanisms, and discuss how it captures these features observed in laboratory experiments. The
friction constitutive law used in this study are described in Equations 1.1 and 1.2.
My dissertation is structured as follows: In Chapter 2, I examine how the frictional properties control
the rupture styles and recurrence patterns of earthquake sequences. I explore the correlation between rupture styles and the recurrence behavior of seismic events using a two-dimensional fault model embedded
in a fault zone. This study also explores a broad spectrum of frictional and fault zone properties. Additionally, I conflate the effects of the fault zone into a single non-dimensional number, allowing quasi-dynamic
simulations to replicate similar seismic cycles with or without the application of a fault zone. In Chapter
3, I expand my focus from the seismogenic zone to the downdip of seismogenic regions, investigating the
well-documented behavior of slow-slip events and slow earthquakes. I discover that concurrent slow- and
fast-slip behavior can originate on homogeneous faults with velocity-neutral conditions, aligning with the
9
CSAF (SDZ) Moore et al.,2016
CSAF (SDZ) Moore et al.,2016
CSAF (SDZ) Carpenter et al., 2015
CSAF (CDZ) Carpenter et al, 2015
CSAF (CDZ) Moore et al.,2016
CSAF (CDZ) Moore et al.,2016
South Alpine Boulton et al., 2018
South Alpine Boulton et al., 2018
Zuccale Colletini et al., 2011
0 0.004
a - b a - b
0.008 0.012
Northern Apennines Tesei et al., 2014 Nankai den Hartog et al., 2011
South Alpine Barth et al., 2013
Alpine Boulton et al., 2014 Alpine Boulton et al., 2014
-0.01 0 0.01
0.01~0.001
0.001~0.01
0.01~0.001
0.001~0.01
0.1~1.0
1.0~10.0
10.0~1.0
1.0~0.1
0.1
100
200
300
B
(o
c) 102
100
10-2
A
μm/s
Velocity-dependence of a-b Temperature-dependence of a-b
0.01 0.1 1
50
100
Temperature (o
c)
Velocity (μm/s)
150
200
50
100
150
200
250
0.01
0
-0.01
C a-b
0.01 0.1 1
Velocity (μm/s)
D
0.005
0
a-b
Alpine Boulton et al., 2014 CSAF (SDZ) Moore et al.,2016
velocity weakening
velocity strengthening
velocity strengthening
velocity
weakening
Figure 1.5: Velocity and temperature dependence of friction of crustal faults from laboratory experiments.
(a) shows the measurement of a-b under different loading velocities at room temperature. (b) depicts the
measurement of a-b under different temperature conditions. The velocity jump is labeled next to the points,
and the unit is µm/s. The offset of the points within one study indicates the dataset from different samples
for (a) or the decrease/increase of velocity in (b). (c) and (d) display the interpolated a-b as a function of both
velocity and temperature using data from the Alpine Fault and Central San Andreas Fault, respectively.
10
frictional condition of the downdip of the seismogenic zone. This study also suggests that tremors are
triggered when a slow-slip rupture bypasses small-scale asperities. In Chapter 4, I use the physical-based
friction constitutive model to clarify the extensively observed frictional properties that depend on loading
rate and temperature in the laboratory. The transition from velocity-weakening to velocity-strengthening
behavior is associated with the competing healing mechanisms dominant in different temperature/velocity
regimes. This model explains the friction properties of gouge from the San Andreas Fault, Alpine Fault, and
Nankai Trough across a wide range of temperatures and slip rates. In Appendix A, I describe the governing
equations of the friction laws and the associated non-dimensional parameters Ru and Rb used in Chapters
2 and 3. Appendix B explains the representation of the Dieterich-Ruina-Rice number Ru incorporating
the effect of the fault zone. Appendix C describes the relationship between the empirical friction law and
the physical-based friction law used in my study, explaining how the empirical laws can be modified to
incorporate non-stationary velocity dependence found in laboratory observations.
11
Chapter 2
Rupture styles linked to recurrence patterns in seismic cycles with a
compliant fault zone
2.1 Abstract
Seismic cycles emerge in a broad range of rupture styles, from slow-slip events to pulse-like earthquake
sequences. Meanwhile, large earthquakes in paleoseismic and instrumental catalogues exhibit various recurrence patterns going from periodic to chaotic cycles with characteristic or dissimilar ruptures. The
potential connection between these observations is still poorly known. Here, we investigate the link between rupture styles and recurrence patterns in quasi-dynamic models of seismicity in two-dimensional
faults embedded in a compliant zone, exploring a wide range of frictional and fault zone properties. The
recurrence patterns evolve from purely periodic to multiple-periodic time- and slip-predictable cycles, to
chaotic sequences of super-cycles with full and partial ruptures with an increasing number of aftershocks.
This transition is accompanied by changes of rupture styles from crack-like to pulse-like ruptures. These
behaviors can be obtained either by a more compliant fault zone or a reduced characteristic weakening
distance of friction. The effects of the compliant zone and other physical characteristics can be conflated
into a single non-dimensional number, such that seismic cycles with similar behaviors can be obtained
with or without a compliant fault zone in quasi-dynamic simulations. The connection between recurrence
12
patterns and rupture styles implies that the paleoseismic record can bring useful constraints to rupture
characteristics and fault zone properties.
2.2 Introduction
The seismic cycle is maintained by the large-scale motion of tectonic plates that slowly reload faults after
a rupture [237]. The recurrence pattern of large earthquakes is controlled by many factors, including
the structural fabric [317, 316, 80], frictional properties [159, 304], and tectonic assembly [98] of the host
fault, modulated by static and remote triggering [146, 93, 102]. Earthquake sequences on a given fault
segment often approach the time-predictable and slip-predictable models based on a fixed yield strength
or a coseismic basal friction, respectively [53, 277, 321]. However, paleoseismic and instrumental records
indicate considerable deviation from these end-members [315, 64], highlighting the inherent complexity
of the faulting process.
The space-time recurrence pattern of earthquakes is documented at major plate boundaries worldwide
along continental transform faults [280, 315, 242, 261, 2, 175, 235], continental collisions [258, 46], and subduction zones [191, 260, 100, 199, 220, 251]. The paleoseismic record indicates substantial variations of the
time intervals and the spatial distributions of coseismic slip of consecutive events [281, 288]. Although
some faults rupture fairly regularly [34] and others seem to produce characteristic coseismic slip distributions [148], a majority of faults exhibit more temporal clustering and more dissimilar slip distributions. For
example, seismic super-cycles showcase smaller-magnitude earthquakes between much larger events [279,
101, 228, 300, 207, 227, 10].
While increasingly well documented, the physical mechanisms underpinning the spatial and temporal variability of the seismic cycle remains poorly understood, but there is more. Seismicity embodies
a wide range of rupture styles varying from slow-slip events and tremors to slow earthquakes and fast
ruptures [225, 209, 205]. Large earthquakes exhibit much diversity, with unilateral and bilateral rupture
13
propagation [245], crack-like ruptures and self-healing sustained, decaying, or growing pulses [118, 311],
and sub-Rayleigh and super-shear ruptures [85, 9]. Although fundamentally linked by fault mechanics,
the relation between recurrence patterns and rupture styles is poorly understood. Here, we use numerical simulations that consistently resolve all phases of the seismic cycles in space and time to describe the
possible connection between rupture styles and recurrence histories assuming a simple structural setting
with a single fault embedded in a compliant fault zone.
The control of frictional properties on the rupture style in a homogeneous medium is well documented [e.g., 156, 318, 141, 305, 23, 200], but the surrounding compliant zone may have a strong influence
on rupture propagation, peak slip velocity, ground motion [111, 82, 126], and on the emergence of pulselike ruptures [138, 1, 130, 291]. Compliant zones develop from the successive damage accumulation during
seismic ruptures that causes measurable elastic moduli reduction [89, 15, 16]. A combination of petrologic,
seismological, and geodetic studies indicate a wide range of compliance and fault zone thickness [e.g. 66,
67, 162, 306, 185, 63, 88, 109, 68, 266]. The effect of a compliant fault zone within the quasi-dynamic or
quasi-static approximations has been documented for one or more physical parameters within a limited
range. The goal of this study is to jointly explore the recurrence patterns, slip distributions, and source
characteristics of the ruptures that consistently unfold for a wider range of attributes of the compliant
zone and frictional properties of the fault interface.
The content of this manuscript is arranged as follows. In Section 2.3, we describe the physical assumptions and the numerical model setup. In Section 2.4, we characterize the rupture styles, from creep and
slow-slip events to complex pulse-like rupture sequences, and their associated recurrence patterns in the
parametric space of the characteristic weakening distance and the fault zone rigidity. In Section 2.5, we
explore the effect of the fault zone properties on rupture cycles. Our simulations indicate a wide range of
recurrence patterns, from single- to multiple-periodic, to apparently chaotic, culminating with mainshockaftershock sequences, with ruptures going from characteristic to more dissimilar. These variations seem
14
to go hand in hand with specific rupture styles, from crack-like to pulse-like ruptures, associated with a
specific range of structural and frictional properties that can be described by a single non-dimensional
parameter representative of the velocity-weakening region and its surrounding compliant zone.
2.3 Physical assumptions and methods
Our goal is to determine the possible relationship between the style and recurrence pattern of ruptures that
emerge spontaneously in seismic cycles under the rate- and state-dependent frictional framework [78, 253,
240]. We consider a simplified mechanical system consisting of a single fault embedded in a compliant zone
as a numerical analog for the seismogenic zone. We ignore the viscoplastic deformation of the compliant
zone [189, 87] and the resulting incremental damage accumulation during each seismic rupture [173]. We
assume that the evolution of fault slip is controlled by the frictional resistance along the fault plane and
that all aspects of fault dynamics are determined by the combination of frictional, physical, and structural
properties of the system. We consider a two-dimensional approximation in condition of anti-plane strain
relevant to long strike-slip faults. While the two-dimensional approximation is sufficient to capture many
ruptures styles and recurrence patterns, some important end-members, such as period-multiplying cycles
of both slow and fast ruptures, only occur in three-dimensional models. For example, successive slow
and fast rupture cycles emerge near the stability transition in finite faults due to coexisting mode II and
mode III ruptures, which have slightly differing stability conditions [305, 23, 304].
We use the multiplicative form of the rate- and state-dependent friction law without thermal effects.
The friction law can be derived from first principles assuming that the real area of contact density formed
at contact junctions follows [11]
A =
µ0σ¯
χ
θV0
L
b
µ0
(2.1)
15
where µ0 is the reference friction coefficient at reference sliding velocity V0, σ¯ is the effective normal stress
including the Terzaghi pore-pressure effect, and χ is the indentation hardness of the surface. In addition
to a linear dependence on effective normal stress, the real area of contact density is modulated by a state
variable θ representing the age of contact with the reference age L/V0 and the power exponent b ≪ 1,
allowing contact growth during locked periods and contact erosion at finite sliding velocity. In isothermal
condition, the sliding velocity follows the power-law constitutive relationship [11]
V = V0
τ
χA
µ0
a
, (2.2)
where V is the local sliding velocity, the product χA represents a time- and state-dependent yield strength,
and a ≪ 1 is a power exponent. Inverting the relationship, we obtain the multiplicative form of rate- and
state-dependent friction
τ = µ0σ¯
V
V0
a
µ0
θV0
L
b
µ0
(2.3)
independent of the indentation hardness of the material, explaining why hard and soft rocks have the same
coefficient of friction [55]. As the logarithmic form of rate- and state-friction represents the linear terms of
the Taylor series expansion of the multiplicative form, similar behaviors are expected under common conditions. However, the multiplicative form supersedes the logarithmic form for vanishing velocity and truly
stationary contact. Further, we assume that contact aging and erosion follow the aging law in isothermal
conditions [253]
˙θ = 1 −
V θ
L
, (2.4)
allowing healing at truly stationary contact. The aging law captures the time-dependent flattening and
rejuvenation of contact junctions that form the real area of contact at the fault interface with the characteristic slip distance L. The fault is situated along the center of a compliant zone with a uniform thickness
16
T and rigidity Gcz, potentially differing from the country rocks with rigidity G in the surrounding two
half-spaces. The resulting computational domain is a full space — without a free surface — centered on the
compliant fault zone (Figure 2.1).
To investigate a wide range of fault zone and frictional properties efficiently, we use the spectral
boundary-integral method. We simulate quasi-dynamic seismic cycles within a whole space with the radiation damping approximation. The Green’s functions of stress interactions are derived in the frequency
domain and the numerical efficiency is obtained with the fast Fourier transform, fifth-order accurate adaptive time steps, and shared memory parallelism [19]. To incorporate the compliant zone surrounding the
fault, we use the corresponding stress interaction kernel, which has a closed-form expression in the Fourier
domain [130]. For all the models considered, the effective normal traction is σ¯ = 100 MPa, the reference
coefficient is µ0 = 0.6, and the reference velocity is V0 = 10−6 m/s. The power exponent for the velocity
dependence is uniform along the fault with a = 10−2
. The long-term loading rate that drives the seismic cycle is VL = 10−9 m/s or about 32 mm/yr. To represent the seismogenic zone, we consider a single
velocity-weakening area surrounded by a velocity-strengthening region. The fault extends from -8.2 to
8.2 km and the velocity-weakening region spreads from -2.5 to 2.5 km from the origin at the center of the
computational domain. The power exponent for the state-dependence is piece-wise homogeneous, with
b = 1.4 × 10−2
in the central velocity-weakening area and b = 0.6 × 10−2
in the surrounding region.
We consider the quasi-static process zone Λ0 at the rupture speed 0
+, which describes the area near the
rupture front during the breakdown of the fault resistance, as the critical length scale in the system [71]
Λ0 =
GczL
bσ¯
. (2.5)
The finest models hold 2
15 surface elements of 50 cm width. For all the models presented herein, the grids
are finer than Λ0/4, which we found sufficient to attain numerical convergence.
17
Within our physical assumptions, the rupture style and recurrence pattern are controlled by a small
set of independent non-dimensional parameters (Appendix A). The Dieterich-Ruina-Rice number
Ru =
W
h
∗
(2.6)
corresponds to the ratio of the asperity size W to a characteristic nucleation size h
∗
that depends on the
rigidity structure and controls the dynamics of ruptures. The Rb number
Rb =
b − a
b
(2.7)
controls the state evolutionary effects and the emergence of complex slow-slip events. The range of rupture
styles that spontaneously emerge as a function of these parameters in a homogeneous crust is well understood [318, 23, 200]. However, the presence of a compliant zone introduces two other non-dimensional
parameters, namely the ratio of country-rock to fault zone rigidity G/Gcz and the width of the compliant
zone relative to the seismogenic zone T /W. A key question that has long animated the community is
whether a compliant zone generates unique rupture styles and recurrence patterns that cannot take place
in the simpler structural setting of a homogeneous medium [126, 127, 221, 125, 1, 130, 291]. To address
this question for quasi-dynamic seismic cycles simulations, we explore a range of characteristic weakening
distance, compliance ratio, and fault zone thickness compatible with laboratory and field observations of
fault zones.
2.4 Rupture styles and recurrence patterns
We examine the rupture styles and recurrence patterns that spontaneously evolve in numerical models of
the seismic cycle under the rate- and state-dependent friction framework in the presence of a pre-existing
compliant zone. We explore a wide range of characteristic weakening distance from 0.5 to 125 mm covering
18
representative values inferred from the laboratory [178, 188, 187] and geophysical observations [86, 94]
within the practical constraints of numerical calculations. Varying the characteristic weakening distance is
an efficient way to control the rupture styles, from slow-slip events to fast earthquakes. We consider a fixed
ratio of power exponents a and b because varying the Rb number is only critical to control the seismogenic
potential of slow-slip events [200], which is not the focus of the current study. Additionally, we explore a
wide range of compliance levels from homogeneous with the country rocks to a factor of five of rigidity
reduction within the fault zone, compatible with seismo-geodetic observations for various faults [89, 108,
16, 68]. In a preliminary step, we consider a fixed compliant zone thickness of 2 km. This forms a wide twodimensional parameter space documenting the effects of frictional and structural properties of fault zones.
We explore the effect of the thickness of compliant zones in the next section. We sample the characteristic
weakening distance with 25 values nearly uniformly distributed in logarithmic space. For each case, we
explore 21 compliance levels, uniformly distributed between G/Gcz = 1 to 5. The resulting 525 simulations
document the evolution of possible rupture style and recurrence patterns in the two-dimensional space of
structural and frictional parameters. The simulations run for up to 107
iterations, but the simulated time
varies from less than a year to thousands of years, depending on the parameters.
The simulations feature seismic cycles with distinct rupture styles including steady sliding, waves of
partial coupling, periodic slow-slip events, bilateral and unilateral ruptures, crack-like ruptures with aftershock and foreshock sequences, and chaotic cycles of pulse-like ruptures (Figures 2.2 and 2.3). Each style
of seismic cycle occurs within a well delineated range of compliance level and characteristic weakening
distance. The transition from any rupture style to another can be obtained by varying either the characteristic weakening distance or the compliance level. At first order, the domain boundaries of all distinct
rupture styles can be predicted by contours of constant Dieterich-Ruina-Rice number if the characteristic
19
nucleation size incorporates the effect of elastic layering. As the effective rigidity near the fault zone depends on the wavelength of fault slip, the estimate of nucleation size solves the transcendental equation
(Appendix B) [138, 130]
1
Ru
tanh
λ
T
W
Ru + arctanhGcz
G
=
Gcz
(b − a)¯σ
L
W
, (2.8)
where the coefficient λ = π/4 is introduced for the anti-plane setting [138] and to fit the observed boundaries of rupture styles in our simulations. The solution to Equation (2.8) provides a smooth transition
between the two end-member cases whereby either the fault zone or country rock rigidity fully controls
the elastic interactions, in which case the domain is effectively homogeneous. The transitions from creep
to waves of partial coupling, to slow-slip events, to bilateral ruptures and unilateral ruptures, to crack-like
and pulse-like ruptures with aftershock sequences occur at Ru = 2, 3, 7, 18, 56, and 88, respectively. In all
of our simulations, we do not observe any rupture style or recurrence pattern with a compliant zone that
does not also emerge in a homogeneous medium with a smaller characteristic weakening distance.
The exploration of a vast parameter space shows that the transition from crack-like to pulse-like rupture propagation is promoted by increasingly compliant fault zones, consistent with previous findings [126,
221], but also by a smaller characteristic weakening distance (Figure 2.4). We examine every rupture in the
simulations to determine the rupture style, looking to identify growing, steady, and decaying pulse-like,
as well as crack-like rupture propagation by visual inspection. Additionally, we calculate the ratio of fast
rupture area (velocity larger than 10−2 m/s) over the total rupture area in space and time as a proxy of
pulse-like or crack-like modes. The average among all events is used as a representative value for phase
diagram in Figure 2.4a. For a relatively large characteristic weakening distance or a mild rigidity contrast,
the ruptures propagate like cracks, e.g., the case in Figure 2.4b for L = 2 mm and G/Gcz = 2. The rupture stops when it reaches the velocity-strengthening area and the rise time is commensurate with the
20
rupture time. The simulated events with either a lower fault zone rigidity, e.g., the case in Figure 2.4c for
L = 2 mm and G/Gcz = 5, or shorter characteristic slip weakening distance, e.g., case in Figure 2.4d
with L = 0.5 mm and G/Gcz = 2, tend to propagate like a pulse albeit with multiple sub-events and
back-propagating rupture fronts. The source-time functions include several peaks, leading to a less reliable rise-time determination. Despite the crack-like back-propagation at the end of the rupture, a rise time
of about 20% of the rupture duration for a point located near the hypocenter makes it a firmly pulse-like
rupture. Among all the simulations considered, some ruptures exhibit steady pulse-like propagation, e.g.,
Figure 2.4c. Other simulations produce decaying pulses, e.g., Figure 2.4d, but we could not identify any
growing pulses. The transition from crack-like to pulse-like ruptures occurs gradually for Ru numbers
between about 85 and 100.
The sequences with distinct rupture styles show different temporal and spatial recurrence patterns
(Figure 2.5 and 2.6). Following a common practice in paleoseismic studies, we describe the variability of
temporal recurrence patterns with the coefficient of variation (CoV), i.e., the standard deviation divided by
the average recurrence time. We use the proportion of the number of full ruptures over total number of
events to describe the spatial recurrence behavior. The bilateral-rupture family exhibits periodic sequences
with minimum coefficient of variation (Figure 2.5b) and purely characteristic ruptures (Figure 2.6b). The
cycle of unilateral full and partial ruptures showcase another periodic recurrence pattern including two
major and two minor events within one cycle (Figure 2.5c and Figure 2.6c), which we refer to as periodicn sequences, here with n = 4. Both periodic and periodic-4 sequences are considered time- and slippredictable based on their slip deficit evolution (right panels of Figure 2.5b and 2.5c). The recurrence
pattern becomes less regular for larger Ru number for which aftershock sequences occur (panels d and e in
Figure 2.5 and 2.6). The crack-like sequences can still be considered quasi-periodic, having a CoV less than
1 and 75% of the earthquakes rupturing a similar area. For even larger Ru numbers, the pulse-like ruptures
occur more randomly, associated with a high CoV and less characteristic ruptures. The complex recurrence
21
patterns of the cycles at high Ru number is manifest by their slip-deficit history that falls neither in the slippredictable nor time-predictable end-members (right panels of Figure 2.5d and e), and the cycle of pulselike ruptures exhibiting apparently larger deviations. For Ru > 125, the period separating consecutive full
ruptures is a super-cycle that includes multiple smaller ruptures. The CoV cannot be used to differentiate
all the cases in our simulations. For instance, the periodic-n cycles display a high CoV and low proportion
of full-ruptures (Figures 2.5a and 2.6a) because the large earthquakes are systematically followed by two
partial ruptures, reducing the mean and enlarging the standard deviation of the recurrence intervals. The
pulse-like sequences at high Ru numbers have variable CoV and proportion of full ruptures, primarily due
to fewer simulated events in each simulation due to numerical limitation, but the recurrence behavior of
these models is visibly complicated.
As the cycles increase in complexity with a transition from crack-like to pulse-like ruptures with a
lower proportion of full ruptures, aftershocks also appear at higher Ru numbers (Figure 2.7). For this analysis, an aftershock is defined as an earthquake that happens during the postseismic phase of a mainshock,
characterized by fast afterslip in the velocity-strengthening region. In our simulations, we define a mainshock as a event that ruptures more than half of the velocity-weakening patch. We determine the end of
a postseismic period when the peak slip rate drops below a threshold of twice the background loading
rate or another mainshock occurs. The number of aftershocks increases for more compliant fault zones
and for smaller characteristic weakening distance, which can be captured by boundaries of constant Ru
numbers. The aftershocks start to occur for Ru > 30, but only one aftershock occurs per rupture tip for
30 < Ru < 150 (Figure 2.7c and d). For the models with Ru between 150 to 400, 2 aftershocks occur
for each rupture tip, resulting in an average amount of 4 aftershocks per mainshock (Figure 2.7e and f).
The number of aftershocks can be more than 10 for extremely large Ru cases, which occurs during nested
aftershock sequences — when large aftershocks trigger their own aftershock sequence. We use the sequences with at least two aftershocks at each rupture tip to analyze the time-dependent seismicity rate,
22
and we treat the aftershocks at the top and bottom of the seismogenic zone separately (Figure 2.7i). The
seismicity rate is calculated as the reciprocal of the time gap between two subsequent aftershocks. The
seismicity rate decays following approximately t
−1 with some scatter associated with nested aftershock
sequences.
The rate of natural aftershocks follows the Omori-Utsu law [217, 301]
N˙ (t) = K
(c + t)
p
, (2.9)
where K, c and p are site-dependent constants and tis the time since the mainshock. Our aftershock dataset
can be explained by N˙ (t) = 1/t, which conforms to the Omori law with K = 1, p = 1 and c = 0. In our
models, the aftershocks occur in the velocity-weakening region as a result of a temporary acceleration of
the local loading rate that accompanies afterslip in the nearby velocity-strengthening region. As afterslip
decays as 1/t [192], it drives the aftershock sequence with a similar decay rate [226]. The value of p and c
of our models is in the observational range. However, the productivity parameter K is much smaller than
in nature due to the limited structural and rheological complexity in the model.
Our findings suggest that the spatial and temporal recurrence behaviors are controlled by the Ru
number of the fault zone. At high Ru number, the characteristic nucleation size is small enough that partial
ruptures develop near the boundaries of the velocity-weakening region, where the loading rate is higher.
These partial ruptures delay the wall-to-wall ruptures and leave behind stress concentrations within the
velocity-weakening region that affects the rupture propagation of subsequent events. These two effects
contribute to the breakdown of the time-predicable model. Furthermore, partial ruptures produce different
slip from the full ruptures despite a similar stress drop, due to the variation of the rupture length. This
effect breaks down the slip-predictable model. As a result, the seismic cycle at high Ru number follows
neither the time- nor the slip-predictable models. Our results also suggest that the change of recurrence
23
behavior is deeply linked to associated changes in rupture styles. Models with high Ru numbers tend
to feature complex recurrence patterns in time and space, manifesting as pulse-like events with random
recurrence patterns, culminating in mainshock-aftershock sequences. In contrast, the low Ru cases are
relatively simple, with crack-like ruptures and more periodically occurring large events.
2.5 The effect of compliant fault zones
In this study, we explore the frictional and compliant zone properties to disentangle their respective effects.
The Dieterich-Ruina-Rice number provides a good estimate of the boundaries of the parameter domains
producing similar rupture styles. This is highlighted by laying out the rupture characteristics found in the
previous section (Figure 2.2) based on the corresponding Ru number instead of the physical units used.
With a few exceptions, the rupture styles are associated with a clear range of Ru numbers (Figure 2.8a). In
order to draw a more complete picture of the impact of fault zone structure on seismic cycles, we explore
additional models with a compliant zone thickness ranging from 20 meters to 200 kilometers with the
characteristic weakening distance ranging from 0.6 mm to 63 mm, and the fault zone rigidity contrast also
increasing from 1.2 to 4.6 (Figure 2.9). Without exception, the boundaries between different rupture styles
still occur at fixed Ru number (Figure 2.8b).
For the purpose of predicting rupture styles and recurrence patterns, the parameter space can be reduced from the three-dimensional parameter volume of L, Gcz, and T to a one-dimensional line of varying
Ru numbers. This is further confirmed by comparing simulations for varying compliance, thickness, and
characteristic weakening distance with a similar Ru number for the cases of Ru = 5.33, 11, 28.5, 98 (Figure 2.10). The sequences with similar Ru show a similar behavior. The simulations for varied fault zone
thickness combined with insights from the definition of Equation (2.8) of the Ru number indicate that the
effect of a finite fault zone thickness is of consequence only within a narrow range (Figure 2.10). A fault
zone thickness T > 2h
∗
, with h
∗ = W/Ru, is large enough to form an effectively uniform domain and its
24
rigidity controls the characteristic nucleation size. Similarly, if the thickness is smaller than this threshold
by an order of magnitude or more, the fault dynamics is controlled by the rigidity of the country rocks.
The rupture behaviors documented in previous studies can be obtained for various combinations of
frictional and structural parameters or any change of Ru number (Figures 2.2 and 2.8). For example, for
the large characteristic weakening distance regime of L = 5 cm, increasing the compliance level brings
the behavior from creep, to slow-slip events, and ultimately to earthquakes, thus increasing the peak slip
velocity significantly. In a moderate characteristic weakening distance regime, such as for L = 16 mm,
the ruptures transition from bilateral to unilateral, resulting in the alternation of hypocenter locations.
The cases with small characteristic weakening distance, e.g., L = 1 mm, show a transition from crack-like
to pulse-like rupture propagation, with the apparition of back-propagating rupture sub-fronts. At first
order, these phenomena are mainly associated with varying the Ru number, and can be achieved by either
modifying the characteristic weakening distance or other fault zone properties.
2.6 Discussion
In this study, we describe the quasi-dynamics of a two-dimensional planar fault embedded in a compliant
zone. The numerical simulations document the link between the rupture styles and the recurrence patterns
of major earthquakes. The rupture styles range from slow-slip events, waves of partial coupling, unilateral
and bilateral ruptures, full and partial ruptures, culminating with mainshock-aftershock sequences with
either crack-like or pulse-like rupture propagation. The recurrence patterns for bilateral and unilateral
ruptures follow the slip- and time-predictable models. For increasingly unstable faults, as the sequence
changes from strictly periodic to multiple periodic, i.e., a repeating sequence that involves multiple events,
aftershocks emerge. In the parameter regime that produces more pulse-like ruptures and an increasing
amount of aftershocks, the slip evolution deviates wildly from the slip- or time-predictable models. The
variability of recurrence time is fundamentally linked to the emergence of partial ruptures. The models
25
with the most unstable faults produce coefficients of variation greater than unity, a telltale sign of deterministic chaos.
The strong relationship between rupture style and recurrence patterns may shed new light on paleoseismic records. For example, the occurrence of characteristic ruptures at the Fuyun Fault in China [148]
is a telltale sign of a large characteristic nucleation size, which is associated with crack-like rupture propagation in our models. The otherwise scarcity of characteristic ruptures implies that most active faults
have a small characteristic nucleation size compared to their seismogenic length, which also implies recurrence patterns that tend to deviate from slip- and time-predictable models. A well-documented example
is at Wrightwood, California [315], where the evolution of fault slip shows a substantial deviation from
the time- and slip-predictable models, compatible with a parameter regime leading to a high DieterichRuina-Rice number. This implies that the Mojave section of the San Andreas Fault is greatly unstable, its
along-strike length being much greater than the characteristic nucleation size of earthquakes. This can
be accomplished by a substantial reduction of rigidity in the surrounding compliant zone and/or a small
characteristic weakening distance. Even though our geometrically simple model can produce complex recurrence patterns that deviate from time- and slip-predictable behavior, other aspects may also contribute
in nature, such as geometrical and frictional heterogeneity.
The chaotic sequence of full and partial ruptures may explain the so-called seismic super-cycles observed at the Sumatra [279, 228, 227], Northeast Japan [300], Ecuador [207] and Cascadia [101] subduction
zones. The model implies a strong deviation from the time- and slip-predictable models during supercycles, as observed [279], presumably caused by numerous smaller ruptures, aftershocks and creep within
the velocity-weakening region that redistribute stress and yet cannot be detected in the geologic record.
The succession of several partial ruptures before a through-going rupture is a hallmark of super-cycles,
explaining why the short paleoseismic records at the Japan trench was insufficient to herald an imminent
giant earthquake [10]. This behavior is, in fact, the natural response of even a simple model with only
26
a single unstable patch. As seismic super-cycles seem common at subduction zones, it implies that subduction megathrusts have a small characteristic nucleation size compared to the down-dip width of the
seismogenic zone.
Understanding the rupture propagation style of earthquakes has important implications on the radiated
energy and ground motion [153]. Some finite-fault inversions suggest that large earthquakes are preferentially pulse-like [119]. Several mechanics have been proposed to explain pulse-like ruptures, such as
strong weakening during flash heating and thermal pressurization [311, 153], heterogeneity of stress [33],
and specific geometry of the seismogenic zone [72, 312]. Our results corroborate others based on a quasidynamic approach [23, 130] and suggest that a large Ru number, whether due to the presence of a compliant
fault zone or any other relevant physical characteristics, can also yield pulse-like ruptures. In this parameter regime, stress heterogeneities spontaneously develop as multiple partial ruptures leave behind residual
stress concentration at the rupture tip. This heterogeneous initial stress at the onset of ruptures creates the
condition for a pulse-like rupture propagation. In our models, pulse-like ruptures occur in the regime that
also produces complex recurrence patterns, non-characteristic slip distributions, and substantial deviation
from the time- and slip-predictable models.
Our results also help characterize a physical range for the coefficient of variation in paleoseismic
records. For example, the longest continuous record of paleo-earthquakes at Hokuri Creek in the Alpine
Fault, New Zealand [34] shows remarkable consistency, with a coefficient of variation of approximately
0.37. Our models for a wide range of parameters produce coefficients of variation between zero for strictly
periodic earthquakes and above 1 for multi-periodic cycles and random sequences. In this context, the
variation of recurrence times at Hokuri Creek is notable, compatible with a parameter regime that exhibits large deviations from time- and slip-predictable recurrence patterns. In turn, such paleo-seismic
record can place constraints on the frictional regime of the fault. Although Hokuri Creek is situated in
a relatively isolated mature fault of South Island, New Zealand, the recurrence pattern in other tectonic
27
contexts may be strongly influenced by step-overs, neighboring faults, and interactions with the ductile
layers. More complex frictional behavior may also contribute to more variable recurrence patterns, including strong-weakening mechanisms during seismic ruptures, and long-term evolution of the fault zone
associated with damage evolution, cataclasis, and metamorphism. Simultaneously, the paleo-seismic observations have uncertainties and limitations, making detailed explanations of long-term seismic behavior
exceptionally challenging.
Our study shows that the effect of a compliant fault zone on the style of ruptures and recurrence patterns in quasi-dynamic seismic cycles can be approximated by a homogeneous model with physical properties leading to the same Ru number. In contrast, incorporating a compliant zone in fully dynamic simulations leads to a perturbation of the seismic cycle with increasing variability of the hypocenter location [e.g.,
126, 127, 291]. This difference occurs because radiation damping does not capture all the wave-mediated
stress transfer. Fault zones create several seismic phases such as trapped waves and head waves [185,
160] interacting with the rupture that affect its style during the coseismic period. Previous studies did not
show evidence that trapped and head waves affect the rupture styles or earthquake statistics drastically.
However, these studies have been limited to low Ru numbers. The development of off-fault plasticity also
affects earthquake recurrence as inelastic deformation dissipates the energy and redistributes the stress
field, resulting in more partial ruptures and temporal clustering of earthquakes [189].
Conclusion
We show that the mechanics of faulting implies a strong relationship between the rupture style and recurrence patterns of earthquakes. Fault dynamics depends on the parameter regime, which is determined
by the physical properties of the fault zone in and around the seismogenic zone. The presence of a compliant zone reduces the characteristic nucleation size of the seismogenic zone, making it more unstable.
However, a similar effect can be obtained in a number of ways in a homogeneous medium, by reducing
28
the characteristic weakening distance of friction, or increasing the effective normal stress. Importantly,
there is no particular style of rupture or recurrence pattern that emerges within a compliant zone that is
not also found in a homogeneous elastic medium in our models. Nevertheless, considerable complexity
can be found in various parameter regimes linked to specific rupture styles and recurrence patterns. For
example, in our models, pulse-like rupture propagation is always associated with large deviations from
the time- and slip-predictable models, with partial ruptures of the seismogenic zones, and with the emergence of aftershocks. Earthquake sequences that appear quasi-periodic in the paleo-seismic record are
compatible with seismic cycles with full and partial ruptures of the seismogenic zone with crack-like or
pulse-like ruptures. We conclude that long-term paleo-seismic records inform earthquake physics beyond
seismic hazards and fault activity, also providing fundamental constraints into the fault properties and the
frictional regime.
29
Table 2.1: Physical parameters of fixed fault zone thickness simulations (Figure 2.2 to 2.8)
Parameter Symbol Value
Static friction coefficient µ0 0.6
Reference slip rate V0 10−6 m/s
Loading rate VL 10−9 m/s
Fault width WF 16.4 km
Width of the velocity-weakening zone W 5 km
Effective normal stress σ¯ 100 MPa
Rigidity of the country rock G 30 GPa
Velocity power exponent a 0.01
State-dependence (vel-weakening) b 0.014
State-dependence (vel-strengthening) b 0.006
Characteristic weakening distance L 0.5 ∼ 125 mm
Fault zone compliance level G/Gcz 1 ∼ 5
Fault zone thickness T 2 km
Mesh size 0.5, 2, 8, 32 m
Table 2.2: Physical parameters of fault zone thickness experiments (Figure 2.8 to 2.10).
Parameter Symbol Value
Static friction coefficient µ0 0.6
Reference slip rate V0 10−6 m/s
Loading rate VL 10−9 m/s
Fault width WF 16.4 km
Width of the velocity-weakening zone W 5 km
Effective normal stress σ¯ 100 MPa
Rigidity of the country rock G 30 GPa
Velocity power exponent a 0.01
State-dependence (vel-weakening) b 0.014
State-dependence (vel-strengthening) b 0.006
Characteristic weakening distance L 0.6 ∼ 63 mm
Fault zone compliance level G/Gcz 1.2 ∼ 4.6
Fault zone thickness T 20 m ∼ 200 km
Mesh size 0.5, 2, 8, 32 m
30
Country
rocks
Country
rocks
T
W
Unstable
asperity
Compliant
fault zone
G
Rigidity Rigidity
Antiplane
simple shear
Gcz
Figure 2.1: Schematic representation of the simplified compliant fault zone model. The rigidity of the
bedrock and the fault zone are G and Gcz, respectively. The fault zone thickness is T. The central unstable
patch of width W is surrounded by a velocity-strengthening fault.
31
Distance (km)
-2.5
Distance (km)
2.5 -2.5
Distance (km)
2.5 -2.5
Distance (km)
2.5
Time steps (integer) Time steps (integer)
5.0
4.5
4.0
3.5
3.0
2.5
2.0
Compliance level (G/Gcz) 1.5
1.0
125100 80 63 50 40 30 25 20 16 12 10 8 6 5 4 3 2.5 2 1.5 1.3 1 0.8 0.6 0.5
Characteristic weakening distance L (mm)
Bilateral
ruptures
E
G/Gcz=1.6
L=20 mm
Waves of
partial coupling
C
G/Gcz=2.0
L=50 mm
Slow-slip
events
D
G/Gcz=2.8
L=63 mm
Full and
partial ruptures
G
G/Gcz=2.0
L=4 mm
Cracks
aftershocks
H
G/Gcz=2.6
L=2.5 mm
Pulses
Multi-pulses
Aftershocks
I
G/Gcz=3.8
L=1.5 mm
-2.5 2.5
Unilateral
ruptures
F
G/Gcz=3.0
L=16 mm
B Creep
G/Gcz=1.4
L=80 mm
Slip velocity (m/s)
10-9
10-6
10-3
1
A
Creep
Bilateral
ruptures
Unilateral
ruptures
Full and
partial
ruptures
Pulse-like ruptures,
multi-pulses
and aftershocks
with
aftershocks
Crack-like
ruptures
Slow-slip
events and slow
earthquakes
I
H
G
F
E
D
C
B
2
2
2
3
3
3
7.5
7.5
7.5
18.35
Ru =18.35
56.4
56.4
56.4
88
88
88
10
Crack
Pulse
10-4
10-9
Slip
velocity(m/s)
Figure 2.2: Seismic cycle simulations under variable characteristic weakening distance and fault zone compliance level. a) Sub-domains for creep, slow slip, bilateral and unilateral periodic ruptures, crack-like
ruptures and pulse-like ruptures. Contours are the Ru numbers of the models. The symbols represent the
rupture styles of the models used to produce this phase diagram. The background color indicates the peak
slip velocity and the shading represents the crack-like to pulse-like rupture transition. b to i) Examples of
different rupture styles, represented by the slip velocity. The x- and y-axes represent down-dip distance
on the fault and time steps, respectively. The area between dash lines is the velocity-weakening domain.
32
Figure 2.3: Representative behavior of numerical simulations for a range of compliant zone rigidity, characteristic weakening distance. Each rectangle represents the evolution of fault slip velocity as a function of
numerical time steps. All the simulations showed here include a fault zone thickness of 2 km. R = Gcz/G
is the compliant ratio and L is the characteristic weakening distance in units of millimeters. The peak
slip velocity, proportion of crack-like versus pulse-like ruptures, proportion of full and partial ruptures,
and number of aftershocks per mainshock shown in Figures 2.2, 2.4, 2.5 and 2.6 is extracted from these
simulations.
33
125 80 50 30 20 12 8 5 3 2 1.3 0.8 0.5
Characteristic weakening distance L (mm)
5
4
3
2
Compliance level (G/Gcz) 1
Healing phase
-3 0 3 -3 0 3 -3 0 3
-3 0 3 -3 0 3
Slip velocity (m/s)
10-9 10-6 10-3 1
Rupture time (s)
0 10 20
Distance (km) Distance (km) Distance (km)
1
Slip (m) Timesteps (integer)
-3 0 3
0
1
0
0.6
0
0.8
0.8
1.25
1.25
1.25
2
2
2
2
3
3
3
3
5
5
5
5
8
8
8
8
12.5
Ru =
12.5
12.5
12.5
20
20
20
20
30
30
30
30
50
50
50
50
80
80
80
80
125
125
125
125
200
200
200
300
300
500
A Crack-like Pulse-like
B
C
D
0
0 1 0 1 2
2
Time/rupture duration Time/rupture duration
0 1 2
Time/rupture duration
Velocity (m/s)
B C D
Back-propagation
Steady pulse
Crack-like back-propagating crack
Decaying pulse
back-propagating crack
Figure 2.4: Crack-like to pulse-like rupture transitions under variable characteristic weakening distance
and compliance level of the fault zone. a) The proportion of crack-like versus pulse-like rupture style.
Contours are for constant Ru numbers. b) A typical crack-like rupture propagation for a simulation with
G/Gcz = 2 and L = 2mm. The upper panel is the slip velocity versus time steps; middle panel shows the
rupture history with 1-second interval contour; the lower panel is the source time function, the sampling
locations are shown as the blue line in upper panel, close to the hypocenter. c) and d) pulse-like rupture models differing from b) by increasing the compliance level or reducing the characteristic weakening
distance, respectively.
34
5
4
3
2
Compliance level (G/Gcz)
1
125 80 50 30 20 12 8 5 3 2 1.3 0.8 0.5
Characteristic weakening distance L (mm)
B
C
D
E
0.8
0.8
1.25
1.25
1.25
2
2
2
2
3
3
3
3
5
5
5
5
8
8
8
8
12.5
Ru = 12.5
12.5
12.5
20
20
20
20
30
30
30
30
50
50
50
50
80
80
80
80
125
125
125
125
200
200
200
300
300
500
Coefficient of variation (CoV)
A
1
0
0.5
2000 2200 2400 2600 2800 3000 2000 2200 2400 2600 2800 3000
2000 2200 2400 2600 2800 3000
2000 2200 2400 2600 2800 3000
2000 2200 2400 2600 2800 3000
1
0
-1
-2
-2
-3
0
-1
-4
-6
-4
-8
-10
-8
-12
-14
2000 2200 2400 2600 2800 3000
2000 2200 2400 2600 2800 3000
2000 2200 2400 2600 2800 3000
108
10Moment (N)
4
Slip deficit (m) Slip deficit (m) Slip deficit (m) Slip deficit (m)
108
104 Moment (N)
108
104 Moment (N)
108
104 Moment (N)
Time (years) Time (years)
Periodic
sequences
Period-4
sequences
Quasi-periodic
cycles
Random
sequences
B
C
D
E
Periodic sequences
Period-4 sequences
Quasi-periodic cycles
Random sequences
Figure 2.5: The relationship between rupture styles and temporal recurrence patterns. a) Phase diagram
of coefficient of variation (CoV, standard deviation divided by mean of the recurrence time of major earthquakes) of the major events (events that rupture at least half of the fault). The contours are the Ru numbers of the models. b) to e) The left plots show the earthquake timelines and moments for four parametric
regimes labeled in a). The stars indicate the major events used to calculate the CoV in a), while the triangles
indicates the minor events. The right plots show the corresponding slip deficit. The upper and lower dash
lines in each figure show the approximated limit of slip- and time- predictable, respectively.
35
3
0
-3
3
0
-3
3
0
-3
3
0
-3
Distance (km) Distance (km)
Time (years) Time (years)
2000 2200 2400 2600 2800 3000 2000 2200 2400 2600 2800 3000
2000 2200 2400 2600 2800 3000 2000 2200 2400 2600 2800 3000
125 80 50 30 20 12 8 5 3 2 1.3 0.8 0.5
5
4
3
2
Compliance level (G/Gcz) 1
Characteristic weakening distance L (mm)
B
C
D
E
0.8
0.8
1.25
1.25
1.25
2
2
2
2
3
3
3
3
5
5
5
5
8
8
8
8
12.5
Ru =
12.5
12.5
12.5
20
20
20
20
30
30
30
30
50
50
50
50
80
80
80
80
125
125
125
125
200
200
200
300
300
500
1
0.2
0.6
Proportion of full-ruptures
A
B C
D E
CoV: 0.91
Full: 47.4%
CoV: 1.22
Full: 65.2%
CoV: <0.01
Full: 100%
CoV: 0.70
Full: 75%
Periodic
sequences
Period-4
sequences
Quasi-periodic
sequences
Random sequences
Supercycles
Figure 2.6: Relationship between rupture styles and spatial recurrence patterns. a) Phase diagram of proportion of full-ruptures, excluding the events that rupture less than 20% of the seismogenic zone. The
threshold for a full rupture is 90% of the seismogenic zone. The contours are the Ru numbers of the models. b) to e) The rupture area of the earthquakes from four parametric regimes labeled in a). The stars
indicate the hypocenters of major events used to calculate CoV, while the triangles indicates the hypocenter of minor events. The dark blue lines represent rupture areas of full ruptures mentioned in b). The light
blue lines are counted as partial ruptures. The black ones are ignored for this analysis.
36
200 240 280 320 360 400 1
0.2 0.4
Time (years)
0.6 0.8 1
2 3
108
104
102
10-10
10-3
102 104 106 108
10-9
10-6
102
10-10 Moment (N) Vmax (m/s)
Time after mainshock (s)
Seismicity rate (s-1)
Vmax (m/s)
108
10
Moment (N)
4
No
aftershocks
1
per
rupture tip
2 per
rupture tip
0.8
0.8
1.25
1.25
1.25
2
2
2
2
3
3
3
3
5
5
5
5
8
8
8
8
12.5
Ru = 12.5
12.5
12.5
20
20
20
20
30
30
30
30
50
Ru = 50
50
50
80
80
80
80
125
125
125
125
200
200
200
300
300
500
125 80 50 30 20 12 8 5 3 2 1.3 0.8 0.5
Characteristic weakening distance L (mm)
5
4
3
2
Compliance level (G/Gcz) 1
-2.5
2.5
0
Distance (km)
Shallow aftershocks
Deep aftershocks
Afterslip
Nested aftershock
sequence
Nested aftershock
sequence
Two aftershocks
for each boundary
Nested aftershock
sequence
Mainshock
C E
F
I
D
G
H
Mainshocks
Shallow aftershocks
Deep aftershocks
Other events
Ṅ(t) = 1/t
C, D
E, F
B, G, H
B
Nested
aftershocks
Number of aftershocks
A
6
3
0
10-1
0
2
4
6
8
10
12
100 101
Ru
102 103
No aftershock
1 per rupture tip
2 per rupture tip
Nested aftershock
sequences
Figure 2.7: The statistics of aftershocks under variable characteristic weakening distance and compliance
level of the fault zone. a) Number of aftershocks per mainshock. The contour lines are the Ru number of
the simulations. The inset shows the number of aftershock as a function of Ru for all the 525 simulations.
b) Velocity vs time step and terminology, the parameters are shown on a). c), e), and g) display the time
history and moment of the mainshocks and aftershock sequences. The shaded areas are the postseismic
periods. d), f), and h) is the corresponding peak slip velocity used to define the postseismic period. i)
Seismicity rate of aftershocks as a function of time from the mainshock. In all of b-i), the stars represent
the mainshocks. The asterisk and diamonds represent shallow and deep aftershocks, respectively. The
triangles are the remaining events.
37
1
2
3
4
5
1 10 100 1000
Compliance level (G/Gcz)
Dieterich-Ruina-Rice number (Ru)
1 10 100 1000
0.01
0.1
1
10
Relative thickness (Thickness/Width)
Creep
Bilateral ruptures
Unilateral ruptures
Slow slip and
slow earthquakes
Full and partial ruptures
Pulse-like ruptures,
multi-pulses
Crack-like ruptures
with aftershocks
A
B
Figure 2.8: Efficiency of the Ru number to describe the rupture style. a) Rupture style for varying compliance level and characteristic weakening distance for a fixed compliant zone thickness of T = 2, 000 m.
b) Rupture style for varying fault zone thickness, characteristic weakening distance, and compliance ratio
G/Gcz. The corresponding parameters are in Table 2.1 and Table 2.2. The dash lines are the approximation
of Ru thresholds of rupture styles.
38
63 L
R
Th
50 40 30 25 20 16 12 10 8 6 6 5 5 4 3 2.5 2 1.5 1.3 0.8 0.6
1.2
20
100
200
400
103
104
2×
103
4×
103
1.6×
104
2×
104
2×
105
1.2 1.4 1.6 1.6 1.6 1.8 2.0 2.2 2.2 2.4 2.6 2.8 3.0 3.4 3.6 3.8 4.0 4.2 4.4 4.4 4.6
Figure 2.9: Representative behavior of numerical simulations for a range of compliant zone rigidity, compliant zone thickness, and characteristic weakening distance. Each rectangle represents the evolution of
fault slip velocity as a function of numerical time steps. R = Gcz/G is the compliant ratio, L is the characteristic weakening distance in millimeters and Th represents compliant zone thickness in units of meters.
The analysis of Figure 2.8 is constructed from these simulations.
39
13 13 13
16
16
16
20
20
20
20
L = 25 mm
25
25
25
30
30
30
30
40
40
50
50
Ru = 5.33
1.0
2.0
3.0
4.0
5.0
Compliance level (G/Gcz)
102 103 104
Fault zone thickness (m)
I
II III
Ru5_I
5.71 5.34 5.34
Ru5_II Ru_III
Ru28_I
28.33 28.9 28.9
Ru28_II Ru28_III Ru98_I
93.33 101.34 97.82
Ru98_II Ru98_III
Ru11_1
10.56 10.88 11.12
Ru11_2 Ru11_3
A
C
I
II III
B
102 103 104
Fault zone thickness (m)
Ru = 11
8
8 8
10
10
10 10
L = 12 mm
12
12
12
16
16
16
20
20
25
25
30
II III
I
Ru = 28.5
1.0
2.0
3.0
4.0
5.0
Compliance level (G/Gcz)
102 103 104
Fault zone thickness (m)
2.5 2.5 2.5
3
3 3
4
4
4 4
5
5
5 5
6
6
6
8
8 L = 8 mm
10
10
III
II
I
102 103 104
Fault zone thickness (m)
Ru = 98
0.8 0.8 0.8
1
1 1 1
1.3
1.3
1.3 1.3
1.5
1.5
1.5 1.5
2
2
L = 2 mm 2
2.5
2.5 2.5
3
3 3
D
Figure 2.10: Equivalent sets of fault zone thickness, compliance, and characteristic weakening distance
for the same Ru number. a) to d) Relationship among fault zone thickness, compliance, and characteristic
weakening distance for Ru = 5.33, 11, 28.5 and 98, respectively. The contours are the required characteristic weakening distance, in units of millimetres. The red dash lines indicate the thickness T = 2h
∗
. For
each Ru number, the rupture history of three models with different physical units is plotted below each
phase diagram. The x- and y-axes of the color plots are time steps and distance, respectively. The white
text represents their actual Ru number.
40
Chapter 3
Seismogenic and tremorgenic slow slip near the stability transition of
frictional sliding
3.1 Abstract
Slow-slip events and tremors occur below the seismogenic zone of major plate boundaries. While the
physics of aseismic slow-slip events is relatively well understood, the mechanics of seismogenic slow slip
remains elusive because the conditions leading to slow or fast ruptures are thought to be mutually exclusive. Here, we explore fault dynamics in the parametric space of frictional conditions to show that seismogenic slow-slip events are the natural behavior of homogeneous faults, as long as the velocity dependence
approaches velocity neutral with a small characteristic nucleation size. Tremors can originate from rapid
bursts of slow earthquakes that are triggered as the slow-slip rupture spreads over small-scale asperities.
The near velocity-neutral conditions explain the underlying mechanics of collocated slow and fast slip
of seismogenic slow-slip events commonly found below the seismogenic zone. The presence of material
heterogeneity may explain the spatio-temporal clustering and migration features of tremor activity.
41
3.2 Introduction
Fault dynamics involve a wide range of rupture styles from slow-slip events to large earthquakes. However,
slow-slip events are often associated with so-called slow earthquakes. The slow earthquake family includes
several types of events based on their frequency and characteristics. Events identified in a frequency band
higher than 1 Hz are called low-frequency earthquakes, and those found in the 0.01-0.10 Hz bandwidth are
called very-low-frequency earthquakes [181]. Furthermore, tremors are distinguished from low-frequency
earthquakes based on their seismic wave phases. While low-frequency earthquakes have identifiable P or
S wave arrivals, tremors have no distinct body wave arrivals. Tremors are sometimes interpreted as a burst
of low-frequency earthquakes [184, 271].
Concurrent slow-slip events and slow earthquakes have been found in several subduction zones (Figure
3.1), including Cascadia [81, 243, 222], the Aleutian [247], the Nankai Trough in Japan [272, 320, 278], the
Mid-American Trench in Mexico [150, 92], and Hikurangi in New Zealand [145, 295]. Concurrent slow-slip
events and tremors are also found at major strike-slip faults, for example at the San Andreas Fault near
Parkfield [196, 269, 246] and at the Alpine Fault in New Zealand [61, 313].
Slow-slip events and slow earthquakes represent widely different rupture behaviors. Slow earthquakes
last minutes to hours and radiate broadband seismic signals up to 10 Hz. In contrast, slow-slip events have
a slip velocity only slightly higher than background relative plate motion and last from days to months.
Despite their different characteristics, the two phenomena are mechanically coupled, as most slow-slip
events are accompanied by slow earthquakes with temporal and spatial correlation [e.g., 32]. The intensity
of tremor activity varies among slow-slip events. For example, the gap between the seismogenic zone and
short-term tremorgenic slow-slip area at the Nankai [289, 210, 97] and Mexico [323, 223] subduction zones
are filled with long-term, less seismically active slow-slip events.
Dominantly aseismic slow-slip events are a relatively well-understood phenomenon through numerical
simulations. Slow-slip events may occur due to conditional stability when rupture nucleation is limited
42
by fault width [166], or through enhanced stability by dilatant strengthening [265], by restrengthening
at high slip speed [276, 183, 131] or by spontaneous thermal instabilities arising from a positive feedback
between shear heating and temperature-weakening friction [310]. Slow-slip events may also emerge near
the brittle-ductile transition due to the stabilizing effect of viscoelastic flow [103, 38, 296]. The circulation
of fluids along the fault may also trigger slow slip [31, 70, 37].
In contrast, the physics underlying coincident slow slip and slow earthquakes is unresolved because
the conditions leading to slow or fast slip are thought — incorrectly — to be mutually exclusive. Sustained
sequences of slow and fast ruptures on the same asperity have been obtained under specific conditions [e.g.,
303, 244, 304], but the two rupture styles do not occur within the same event. Recent studies cover the joint
occurrence of slow and fast ruptures [23, 274], but this phenomenon has only been described briefly. Other
works explore the emergence of seismic events during creep or slow slip invoking a heterogeneous rock
matrix in the fault zone [158] or lateral variations of frictional properties along the fault [84]. Elucidating
the physics of concurrent tremor and slow slip still constitutes a major challenge in tectonophysics [134].
In this study, we argue that coincident slow slip and slow earthquakes are the natural response of faults
near the transition between velocity-weakening and velocity-strengthening properties. This condition is
universally found below the seismogenic zone and can be accomplished by a wide range of frictional properties. We present numerical simulations of the seismic cycle that produce recurring slow-slip events while
resolving the simultaneous nucleation, rupture propagation, and arrest of triggered slow earthquakes in
a continuum. The model provides a source mechanism for slow slip and slow earthquakes and explains
the occurrence of this phenomenon below the seismogenic zone at many subduction zones and in a continental transform setting. Further, we explain the generation of tremors by the presence of small-scale
asperities that are triggered during the passage of a slow-slip rupture front. The model helps us understand
the clustering, stationarity, and rapid migration of tremors.
43
In the next section, we describe the physical assumptions and the numerical method. We then describe
the parametric domains leading to aseismic or seismogenic slow-slip events using a homogeneous asperity
model. Finally, we present seismic cycle simulations of tremorgenic slow-slip events in the presence of
small-scale frictional heterogeneities. We produce synthetic geodetic time series and seismic waveforms
that bear resemblance with observed slow-slip events, low-frequency earthquakes, and tremors for natural
faults. Our results help understand why seismogenic slow-slip events occur in a wide range of conditions,
the variability of tremorgenic potential being controlled by the degree of material heterogeneity of the
host fault.
3.3 Physical assumptions and modeling method
Concurrent slow-slip events and tremors occur in various tectonic settings, including subduction zones and
continental transforms. Since slow-slip events often occupy a fault area with a large aspect ratio, we conduct numerical simulations of the seismic cycle using a two-dimensional approximation. The anti-plane
strain and in-plane strain approximations are most relevant for strike-slip and thrust faulting, respectively,
but fault dynamics in these conditions are similar, only differing by the stiffness of asperities, which depends on Poisson’s ratio for mode II cracks. We proceed with the anti-plane strain approximation, but
the results remain relevant for in-plane strain given a systematic change of parameters that affects the
nucleation size.
We conduct numerical simulations of fault dynamics based on a physics-based rate- and state-dependent
friction law, building on a long legacy of studies that describe a wide range of faulting behaviors during
seismic cycles, including slow-slip events, fast ruptures, and aperiodic seismic cycles [e.g., 299, 168, 155,
17, 318, 232, 218, 10]. We consider the multiplicative form of a friction constitutive law [22] in isothermal
conditions given by
τ = µ0σ¯
V
V0
a
µ0
θV0
L
b
µ0
, (3.1)
44
where µ0 represents the static friction coefficient, V0 is a reference velocity, and a and b are power exponents for the rate and state dependence of the frictional resistance, respectively. Velocity-weakening at
steady state is obtained for a − b < 0 and velocity-strengthening for a − b > 0. The effective normal
stress σ¯ is affected by the pore-fluid pressure as σ¯ = σ − pf , where σ is the normal traction and pf is the
pore-fluid pressure in the fault zone. The frictional resistance is modulated by the age of contact following
the aging law [253] in isothermal conditions, given by
˙θ = 1 −
V θ
L
, (3.2)
where L represents a characteristic weakening distance and θ represents the age of contact.
Slow-slip events are often associated with elevated pore-fluid pressure [e.g., 97], revealed by an elevated
ratio of compressional to shear wave speeds in major plate boundaries, including Cascadia [8], the Nankai
Trough [149], Hikurangi [30], Mexico [91], and the San Andreas Fault [219]. The direct effect of high
pore-fluid pressure in the fault zone is to decrease the effective normal stress [287], compatible with the
exceedingly small stress drop of slow-slip events [198]. We incorporate this effect by considering nearlithostatic pore-fluid pressure, resulting in an effective normal stress of 20 MPa for all models considered.
The wide spectrum of rupture styles that develop during seismic cycles is controlled by the frictional
and geometrical properties of a fault. These parameters can be combined into dominantly two nondimensional parameters that, within realistic bounds, exert a strong control on rupture dynamics [23].
The Dieterich-Ruina-Rice number
Ru =
(b − a)¯σ
G
W
L
, (3.3)
45
incorporates the asperity size W, the rate dependence at steady state (a−b), the characteristic weakening
distance, the rigidity G of the host rocks, and the effective normal stress and controls the complexity of
fast rupture cycles [59]. The other number
Rb =
b − a
b
(3.4)
controls the ratio of dynamic to static stress drops [96] and reflects the relative importance of the evolutionary effects. The development of instabilities within the framework of rate- and state-dependent friction has
been typically associated with a critical stiffness or a ratio of nucleation size to asperity size [253, 238, 239].
Accordingly, numerical models of slow-slip events are often obtained using a small ratio of the seismogenic
width to a characteristic nucleation size, i.e., Ru ∼ 1 [e.g., 166, 168, 249, 167, 314], unless different mechanical behaviors are assumed, but the role of the Rb number has been largely overlooked. Slow-slip events
can in fact occur for a wide range of frictional properties as Rb approaches zero from above [249, 318, 23].
In addition, consideration of nonlinear stability analysis [308, 307] indicates the apparition of new Hopf
bifurcations during rupture nucleation when the velocity-weakening friction properties converge towards
velocity neutral, i.e., as Rb approaches zero.
We develop quasi-dynamic simulations of seismic cycles with adaptive time steps to explore a broad
spectrum of seismic and aseismic activity while maintaining high numerical accuracy. Adaptive timestepping is key to resolve the enormous range of time scales relevant to slow slip and slow earthquakes.
We use the spectral boundary-integral method with shared memory parallelism to resolve the stress interactions during rupture dynamics with great numerical efficiency [20]. For all simulations, we set the fault
width as 15 km and the velocity-weakening region extends from 5 to 10 km. We load the fault at a rate of
46
83 mm/yr or 2.6 × 10−9 m/s, which is typical for a subduction zone setting. For all models considered, the
numerical grid size is chosen to resolve the cohesion length
Lb =
GL
bσ¯
. (3.5)
Following a convergence test, we choose a grid size smaller than Lb/4 for the homogeneous models of
Section 3 and of Lb/15 = 15 cm for the heterogeneous model of Section 4.
3.4 Seismogenic slow-slip events
Episodic tremor and slow-slip events often take place near the spatial transition between the seismogenic
zone and regions of long-term stable sliding [e.g. 210, 97]. In a continental setting, the depth extent of
the seismogenic zone is controlled by the stability of wet granite or quartz-rich gouge [40, 41, 42] and
the top and bottom boundaries of the seismogenic zone correspond to isotherms [263]. This is perhaps
best evidenced by the correlation of heat flow and depth of background seismicity in California [174, 114].
In a subduction zone setting, the stability transition may also be controlled by regional metamorphism
due to the fluid-rich, low-temperature conditions near the subducting slab. Slow-slip events can occur at
greater depths than for a continental transform because of the presence of antigorite-rich serpentinite,
which is velocity-weakening at and above 450◦C [214, 213]. Mantle rocks are also velocity-weakening at
greater temperatures than granitic rocks [45]. Nevertheless, in both continental and subduction settings,
the slow-slip events and tremors phenomenon seems to be associated with a stability transition.
Motivated by these observations, we explore the dynamics of a single velocity-weakening asperity in
conditions of near-neutral velocity-weakening friction at steady state. We explore the parametric space of
frictional parameters to determine the various styles of slow-slip events that may emerge in various physical conditions near the stability transition. While keeping the width of the velocity-weakening region
47
and effective normal stress fixed, we vary the state-dependence power exponent and the characteristic
weakening distance to explore the two-dimensional space of non-dimensional parameters Ru and Rb (Figure 3.2). We simulate seismic-cycles with Rb ranging from 0.02 to 0.5 and Ru ranging from 0.6 to 220. We
show that slow slip within the rate- and state-dependent friction framework can be classified into three
parametric sub-domains: creep, aseismic slow-slip events, and seismogenic slow-slip events.
Aseismic slow-slip events emerge when the rupture is limited to the nucleation phase by the boundaries
of the velocity-weakening region. They occur for a narrow range of Ru number that depends on the Rb
number. For strongly velocity-weakening asperities with 0.2 ≤ Rb ≤ 1, aseismic slow-slip events take
place for 1 < Ru < 3. For asperities in near velocity-neutral conditions with 0 < Rb < 0.1, aseismic
slow-slip events take place for the range 5 < Ru < 10. Waves of partial coupling (Figure 3.2b) may
propagate for decades with a slip velocity only slightly above and below the background long-term fault
slip-rate. This behavior occurs for intermediate values of Ru and Rb and may be important to understand
decadal-scale variations in fault sliding velocity.
Seismogenic slow-slip events (Figure 3.2f-i) occur in the conditions predicted by Viesca (2016a, 2016b) [308,
307] for Rb ∼ 0 and Ru ≫ 1. The chaotic nucleation regime manifests itself in seismic cycles by numerous fast ruptures embedded in longer and larger slow-slip events. The nucleation of seismogenic slow slip
events may be caused by the coalescence of slow-slip rupture fronts, which is consistent with observations from the laboratory [137, 95] and from the Cascadia subduction zone [44]. For this type of event to
emerge spontaneously requires near-neutral velocity-weakening friction and a small enough characteristic nucleation size. We limit our simulations to Ru = 220 due to the challenges associated with numerical
convergence, but the Ru number has virtually no upper bound. We expect that slow-slip events continue
to grow in complexity for increasing Ru numbers. In other words, the characteristic weakening distance
may be arbitrarily small and still allow seismogenic slow-slip events.
48
Seismogenic slow-slip events may occur for a wide range of parameters, including virtually arbitrarily
small weakening distance, in conditions of near-neutral velocity-weakening properties. These findings
help explain the presence of slow-slip events below the seismogenic zone of continental transforms and
subduction zones at the transition between stick-slip and stable sliding. In addition, the near velocityneutral regime helps explain the typical association of slow-slip events with slow earthquakes. The remaining unknown is the mechanics of tremor generation.
3.5 Tremorgenic slow-slip events
Seismogenic slow-slip events represent complex slow ruptures that incorporate slow earthquakes. However, slow-slip events in nature are frequently associated with tremors, a more specific type of seismicity. As tremors do not seem to form spontaneously in homogeneous velocity-weakening asperities, we
introduce small-scale asperities within the velocity-weakening region. The presence of small-scale asperities changes the macroscopic rupture behavior, for example, by generating foreshocks and aftershocks in
the seismogenic zone [e.g., 83, 319] or by affecting the macroscopic stability of the fault [283, 171]. The
rupture dynamics depend on the density of small-scale asperities and their physical properties. After exploring various configurations and considering the phase diagram of Figure 3.2, we present the details of
a single model that exhibits the characteristic features of tremorgenic slow-slip events in nature, such as
event duration of the order of months, recurrence times of the order of years, and typical seismo-geodetic
signatures.
We simulate seismogenic slow-slip cycles with heterogeneous frictional properties consisting of a matrix of 50 m-wide asperities with Rb = 0.2 embedded in a 5 km-wide near-neutral velocity-weakening fault
with Rb = 0.02 (Figure 3.3). The model produces a non-characteristic sequence of seismogenic slow-slip
events. Fast sub-events are triggered at the velocity-weakening asperities by the passage of the slow-slip
rupture fronts. We identify the fast events using the slip velocity threshold of 0.01 m/s, but these fast events
49
vary in both peak slip velocity and seismic moment. Rapid failure at the asperities disturbs the propagation of the underlying slow-slip events, resulting in multiple secondary slip fronts. Because the location
of small-scale asperities is stationary, the triggered slow earthquakes form repeaters within a single slowslip event and provide markers of the rupture front. The up-dip and down-dip propagation of the rupture
front causes reversal migration of slow earthquakes. In addition, most of the fast sub-events occur within
slow-slip events and cluster in time (Figure 3.4).
We seek to determine the nature of the slow-slip cycle from the point of view of seismo-geodetic
observations. We therefore compute geodetic and seismic recordings to reveal whether the fast sub-events
represent earthquakes or tremors and whether the deformation is nominally seismic or aseismic. We first
compute the geodetic displacement time series for 5.4 years using elasto-static Green’s functions for a
homogeneous half space with two-dimensional line sources [212]. For these calculations, we assume a
fault dip angle of 7.1◦
(Figure 3.5a). We remove the long-term displacement accumulation represented by a
linear trend to accentuate the dynamic range from the slow-slip events. Long-term deformation associated
with slow slip is detected in synthetic geodetic time series (Figure 3.5b), with a rapid first phase followed
by a gradual recovery to background strain accumulation due to propagation of slow slip into the velocitystrengthening region. The cumulative surface displacement per event is of the order of several millimeters,
which is typical for geodetic observations of slow-slip events [e.g., 243].
We then compute the seismic waveform at the same location of the geodetic station. We assume a
homogeneous full elastic space using a closed-form analytical solution [230]. The near-, intermediate-, and
far-field components of the seismic waveforms each correspond to various different linear combinations
of the moment-rate and its integrals, with weights depending on the distance and azimuths from source
to receiver and arrival times corresponding to S- and P-wave speeds. We assume a rigidity of 30 MPa, a
Poisson’s ratio of ν = 0.25, and a density of 2800 kg/m3
. We ignore seismic attenuation. We first produce
the displacement waveforms. Then, we evaluate a time derivative to obtain the velocity waveforms. In our
50
two-dimensional model, the moment-rate per unit length is a simulation outcome. To propagate the seismic
wave in a three-dimensional space, we assume a constant width of 5 km for the source. These assumptions
affect the amplitude of the waveforms, but not the phase nor the arrival times. We resample the momentrate function on the fault at 1 Hz for the entire time series (5.4 years) to reduce the computational cost.
This frequency is high enough to sample tremor-like events, avoid severe aliasing, and capture the seismic
activity from impulsive signals. The station location to the east of the fault results in only shear waves to
be detected.
Each slow-slip events includes seismic activity, and most of the fast events are within slow-slip events
(Figure 3.5b). Some of these fast events can be distinguished as isolated events, but others are concentrated
and difficult to distinguish separately, reflecting the temporal and spatial clustering of the seismic source
(Figure 3.4). The slow-slip events that consist of multiple acceleration phases in the geodetic time series
are associated with more seismic activity. We compute the seismic waveforms at 16 stations aligned along
a fixed azimuthal angle (Figure 3.5a) to test whether the body-wave arrivals can be identified. We focus on
a single event and we sample the synthetic seismograms at 10,000 Hz to further eliminate aliasing. Unlike
the seismo-geodetic station that receives only S waves, these additional seismic stations have an azimuthal
angle of 30 degrees, so that both P and S waves can be detected. The arrival of both P and S waves and the
moveout pattern is apparent for some of the early phases (Figure 3.5c). However, the body-wave arrivals
of individual events within rapid bursts of seismicity are intermingled.
The concurrency of seismic and aseismic slip and the clustering of individual fast slip events along
slow-slip rupture fronts in our simulations indicate that slow slip triggers small earthquakes and that
tremor-like seismic waveforms can be obtained by superposition of the seismic waves radiated by short
bursts of small earthquakes. Our simulations imply that seismic and slow-slip events are deeply coupled
51
by the presence of heterogeneous frictional properties within the parametric regime of seismogenic slowslip. Complex seismogenic slow-slip events emerge spontaneously with a succession of either isolated
slow earthquakes or tremors associated with rapid bursts of seismic sources.
3.6 Discussion
Slow-slip events emerge spontaneously in the framework of rate- and state-dependent friction when the
rupture is limited to the nucleation region [142, 154, 166, 168, 249]. However, this explanation for slowslip events in nature is unsatisfactory for two key reasons. First, the parameter space that produces this
behavior is so limited that it is unlikely to explain the slow-slip phenomenon in diverse tectonic settings.
This realization pushed several investigators to consider coupling with other mechanisms or different
friction laws [e.g., 164, 248]. Second, the slow-slip events produced in this parameter range are aseismic,
failing to explain the strong coupling between slow slip and slow earthquakes observed in nature [e.g., 107,
128, 28]. In this study, we show that seismogenic slow-slip events are also a spontaneous behavior of rateand state-dependent friction faults and that the parameter space producing this behavior is essentially
unbounded, allowing an arbitrarily small characteristic weakening distance or, equivalently, an arbitrarily
small characteristic nucleation size, as long as the rate dependence approaches velocity neutral.
The physical conditions amenable to seismogenic slow-slip events are those found below the seismogenic zone, near the transition between velocity-weakening and velocity-strengthening rate-dependence
of steady state friction. We propose that concurrent slow-slip events and slow earthquakes can be explained
by a region with nearly velocity-neutral properties, also characterized by a relatively short characteristic
nucleation size. The model is appealing because the transition to a velocity neutral rate-dependence is a
necessary condition at the boundary of the seismogenic zone, regardless of tectonic setting (Figure 3.6).
The model explains the underlying mechanism and location of seismogenic slow-slip events up-dip and
down-dip of the seismogenic zones at subduction megathrusts. However, the model requires that the
52
boundary spread over several kilometers. In a subduction setting, the down-dip boundary may be accommodated by fluid-assisted regional metamorphism that forms rocks with weakly velocity-weakening
properties at high temperature, such as antigorite-rich serpentinite [274]. The up-dip boundary may correspond to poorly confined clay-rich sediments that also exhibit weakly unstable friction [10]. That the
Sunda and the Japan trenches do not exhibit deep slow-slip events or tremors may be due to an underdeveloped, i.e., sharp, stability transition or recent giant earthquakes shutting down the slow-slip cycles
for a few decades [274, 10]. Slow-slip and tremors are less frequently observed at continental transforms,
and have overall smaller dimensions. However, mixtures of talc and serpentinite in quartz-rich gouge at
mid- and lower-crustal depths may provide the conditions for seismogenic slow slip in this context [194,
193].
We highlight the importance of the aseismic and seismogenic end-members of slow-slip events associated with different frictional parameter regimes. Aseismic slow-slip events recur in a more characteristic
manner whereas the seismogenic slow-slip events tend to follow a more chaotic cycle. In the seismogenic
slow-slip parameter regime, the slow-slip cycle is aperiodic, with large variability of event size, duration,
and recurrence time. The ruptures are also more complex, with up-dip and down-dip migrations of the
rupture front, marked by similar propagation patterns of seismic sources in the presence of material heterogeneities. The aseismic and seismogenic slow-slip end-members may be used to explain the difference
between so-called long-term and short-term slow-slip events in the Nankai Trough, the later being more
tremorgenic than the former [274]. The tremorgenic potential of slow-slip events may be controlled by
the concentration of small-scale heterogeneities.
Other mechanisms have been proposed to explain tremors, for example, small-scale geometrical heterogeneity [297] or stress perturbation by fluid migration and metamorphism [229]. Given the ample
evidence for complexity of exhumed fault zones [e.g., 151, 26, 147], the role of heterogeneity is important. Recent studies feature fast sub-events by including small scale heterogeneities explicitly [e.g., 171,
53
172], but the detail of the fast ruptures and how they affect the slow-slip propagation is not fully resolved
numerically. As seismogenic slow slip may readily be obtained in a homogeneous region with properties
systematically found below the seismogenic zone, resolving the effects of heterogeneities may be key to
explain the characteristics of tremors, such as the stationary location of tremor hotspots [250] and rapid
tremor migration in the up-dip and down-dip directions [224, 223, 270, 123].
3.7 Conclusions
Our study shows that the frictional conditions found near the boundary of the seismogenic zone naturally
promote the emergence of slow-slip events for a wide range of frictional properties. Slow-slip events
can be categorized by the aseismic and seismogenic end-members depending on the frictional regime,
perhaps best exemplified in nature by the long-term and short-term slow-slip events at the Nankai Trough.
Aseismic slow-slip events occur when the rupture propagation is limited by the overall size of the asperity
and feature a more characteristic cycle. Seismogenic slow-slip events occur in conditions of near-neutral
velocity-weakening friction at steady state and feature a more complex, aperiodic sequence, due to the
strong coupling between slow slip and embedded fast ruptures. The range of frictional properties for
seismogenic slow slip is conceptually unbounded, allowing an arbitrarily small characteristic nucleation
size, as long as the rate-dependence approaches velocity-neutral.
The velocity-neutral rate-dependence of friction at steady state occurs systematically near the boundary of the seismogenic zone. The development of slow slip and slow earthquakes depends on the width of
this transition and the characteristic nucleation size: Aseismic and seismogenic slow-slip events occur for
large and small nucleation sizes, respectively. The width of the stability transition is likely controlled in
nature by the down-dip stratification of metamorphic rocks and the properties of the fault zone.
Tremorgenic slow-slip events can occur when small-scale asperities are embedded within a nearneutral velocity-weakening fault. Tremors can result from the rapid succession of clustered seismic sources
54
when front of slow-slip ruptures pass through these asperities. Rapid tremor reversals [e.g., 99, 124, 115]
follow from the up-dip and down-dip migrations of the slow-slip rupture front. These results explain
the ubiquity of tremorgenic slow-slip events below the seismogenic zone in various tectonic settings. The
range of frictional parameters leading to both seismogenic or and tremorgenic slow ruptures is much wider
than for aseismic slow-slip events, explaining the predominance of episodic tremor and slip as a slow-slip
rupture style in nature.
55
Table 3.1: Physical parameters for aseismic and seismogenic slow-slip events and earthquakes simulations
(Figure 3.2)
Parameter Symbol Value
Static friction coefficient µ0 0.6
Reference slip rate V0 10−6 m/s
Loading rate VL 2.6 × 10−9 m/s
Fault width WF 20 km
Width of the velocity-weakening zone W 5 km
Effective normal stress σ¯ 20 MPa
Background rigidity G 30 GPa
Velocity power exponent a 0.05
Dieterich-Ruina-Rice number Ru 0.6, 0.8, 1.0, 1.2, 1.5,
2.1, 2.25, 2.5, 3.0, 4.0,
4.5, 5.0, 7.5, 10, 20, 34.5,
50, 70, 90, 110, 160, 220
Rb 0.02, 0.03, 0.04, 0.07,
0.2, 0.3, 0.5
State-dependence (vel-weakening) b a/(1 − Rb)
State-dependence (vel-strengthening) a × 0.6
Characteristic weakening distance L (b − a)¯σW/RuG
Mesh size 0.15, 0.6, 3, 15 m
Table 3.2: Physical parameters of slow-slip and tremors simulations (Figure 3.3, 3.4, 3.5).
Parameter Symbol Value
Static friction coefficient µ0 0.6
Reference slip rate V0 10−6 m/s
Loading rate VL 2.6 × 10−9 m/s
Fault width WF 20 km
Width of the velocity-weakening zone W 5 km
Effective normal stress σ¯ 20 MPa
Background rigidity G 30 GPa
Velocity power exponent a 0.05
Characteristic weakening distance L 0.1 mm
Mesh size 0.15 m
Parameters of near-neutral weakening background
Dieterich-Ruina-Rice number Ru 34.5
Rb 0.02
Rate-and State friction parameter b 0.0521 (Vel-Weakening)
0.03 (Vel-Strengthening)
Parameters of small-scale asperities
Width w 50 m
Dieterich-Ruina-Rice number Ru 4.2
Rb 0.2
Rate-and State friction parameter b 0.0625
56
35.5˚
36.0˚
36.5˚
132º 133º 134º 135º 136º 137º 138º
-127º -126º -125º -124º -123º -122º -121º Distance from Parkfield (km)
-121.5º -121º -120.5º -120º
San Andreas Fault
Parkfield
0
10
20
-60 -50
Depth (km)
-40 -30 -20 -10 0 10
2004
1966 D
0 1.5
Correlation (%)
San Andreas
Fault
Co
u
pling
=
0
2.
Cascadia
trench
45˚
46˚
47˚
48˚
49˚
50˚
0 6 Slip (cm)
Cascadia
Nankai
1946
Tonankai
1944
Tokai
1854
33˚
34˚
35˚
36˚
Nankai trough
10 50 Slip (cm)
Nankai
A
B C
Figure 3.1: Concurrence of slow-slip events and slow earthquakes at major plate boundaries. A) Slowslip events [206] and tremors (NIED catalogue) in 2012, and rupture area (thick dashed line) of historical
large earthquakes [210] at the Nankai Trough, Japan. B) Collocated tremors [129] and slow-slip [262] at
the Cascadia subduction zone. The seismogenic zone is situated above the region of high geodetic coupling [190]. The thin dash lines in A) and B) correspond to the USGS Slab2 model [116]. C) Distribution of
low-frequency earthquakes along the San Andreas Fault [268]. D) Low-frequency earthquakes and correlation with surface geodetic measurements [246]. The seismogenic region [17] surrounds the hypocenters
of the 1966 (green star) and 2004 (red star) earthquakes. 57
Rb = (b-a)/b
Dieterich-Ruina-Rice number Ru
1 10
0.02
0.1
0.5
100
Creep
B D F G
C E
H I
Aseismic
slow slip
Seismogenic
slow slip
Earthquakes
Time steps (integer)
Time steps (integer)
-2.5
Distance (km)
2.5
Ru=10
Rb=0.02
Ru=70
Rb=0.3
Ru=1
Rb=0.3
Ru=5
Rb=0.02
Ru=110
Rb=0.02
Ru=160
Rb=0.02
Ru=90
Rb=0.02
Ru=50
Rb=0.02
B C D
A
E
F G H I
Peak slip velocity (m/s)
10-9 10-6 10-3 1
Waves of Long-term
slow-slip event
Seismogenic
slow-slip event
Complex seismogenic
slow-slip event
Complex seismogenic
slow-slip event
Complex seismogenic
slow-slip event
Short-term
slow-slip event
Aftershocks
Earthquake
Creep waves
partial coupling
Figure 3.2: Seismic cycle simulations under variable Rb and Ru. A) Sub-domains for creep, creep waves,
earthquakes, aseismic and seismogenic slow-slip events. The background color indicates the peak slip
velocity. The white curves show the time series of peak slip velocities for a one or more events, with
varying time scales. B to E) Examples of waves of partial coupling, aseismic slow-slip from low Ru velocityweakening regime, aseismic slow-slip from medium Ru velocity-neutral regime, and earthquakes. F to
I) Seismogenic slow-slip events from high Ru in the near-neutral velocity-weakening regime. The Ru
number is controlled by the characteristic weakening distance. The velocity weakening area is between
the two dashed lines. The segmentation lines in A) are only conceptual. 58
0 2 4 6
Peak Velocity (m/s)
-4 -3 -2 -1 0 1 2 3 4
0.5
1.0
1.5
2.0
2.5
3.0
Time steps (integer x106
)
Distance (km)
10-9
10-7
10-3
1
Slip velocity
(m/s)
1/2 year
Primary front
107
101
103
105
Moment
(N)
Velocity strengthening
(Rb=-0.67)
Near-neutral weakening
(Rb=0.02)
Velocity weakening
(Rb=0.2)
Small asperity
(50 m)
-4 -3 -2 -1 0 1 2 3 4
A B
Secondary front
Figure 3.3: Simulated collocated fast and slow-slip events with a 2D heterogeneous model. A) Peak velocity
of the slip cycles. B) Synthetic slip cycles along the entire fault length. The velocity-weakening area is
located between the two vertical black dashed lines. The hypocenters of the seismic events (circles with
size scaled by seismic moment) are plotted on top of slip velocity of the rupture cycle.
59
157.6 min
Panel C Panel D Panel E
-4
-2
0
2
3
-3
10.5 min
Number of events
0
20
40
60
80
Distance (km)
0
Distance (km)
3
-3
0
Distance (km)
Time (years)
Time (days)
E
A
36.7 min
B
Moment (N)
107 102
Peak slip velocity of fast events (m/s)
101 100 10-1
10-11 10-8 10-4 1
Slip velocity (m/s)
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
200 400 600 800 1000 1200 1400 1600 1800 2000
C D
Figure 3.4: Space-time distribution of seismic hypocenters for the entire simulation period. A) Histograms
of daily slow earthquakes. B) The time distribution of slow (grey bands) and fast (circles scaled by seismic
moment and colored by peak velocity) events. The slow-slip events are detected by the average velocity
(10−6
as the start of a event and 5 × 10−9
as the end) with a duration longer than 60 days. Slow-slip
events generally last longer than the seismogenic period, and most seismicity occurs during an underlying
slow-slip episode. C) to E) enlarge the three seismogenic periods shown by the dashed lines in figure B).
The background color indicates the slip velocity.
60
30°
5 km
7.1°
5 km 5 km
35 km
N
E 5.8 km
19.6 km
5
0
10Displacement (mm)
Seismo-geodetic station
GPS Panel C & D
Clipped
Seismic
Seismic station set
Velocity strengthening fault
Velocity neutral/weakening
A
B
-1
0
1
10-9
10-8
10-7Velocity (10-11m/s) Velocity (m/s)
Years
1 2 3 4 5
0 50 100 150 200 250 0 10 20 30 40 50
Time (s) Time (s)
C D
Panel C & D
Figure 3.5: Geodetic and seismic signatures of seismogenic slow slip. A) Location of the geodetic and
seismic receivers relative to the source fault. The source is simplified as a point located in the middle of
the velocity-weakening area (orange star). The red triangle indicates the primary station used to simulate
geodetic and seismic data for the time series in B). The yellow triangles represent a seismic array with
16 receivers along the same azimuth. B) Synthetic geodetic displacement and velocity (upper panel) and
clipped seismograms showing the ground velocity (bottom panel) for the entire sequence. C) Seismograms
generated by ruptures highlighted in Figure 3.4D and recorded by the orange seismic array in A). The start
time is labeled by dashed red line in B). The P and S wave arrivals are marked by the blue and red lines,
respectively. D) is the seismograms highlighted by blue rectangle in C) with a smaller amplitude range.
61
Subduction megathrust
B Continental transform
A
Figure 3.6: Schematic distribution of frictional properties on subduction megathrust and continental transform faults. A) Subduction megathrust. The seismogenic zone spans the unstable weakening fault region.
Seismogenic slow-slip events take place above and below, in the near-neutral weakening region. B) Continental transform. The seismogenic zone and the seismogenic slow-slip area are associated with unstable
weakening and near-neutral weakening friction properties, respectively.
62
Chapter 4
Velocity- and temperature-dependence of steady-state friction of
natural gouge controlled by competing healing mechanisms
4.1 Abstract
The empirical rate- and state-dependent friction law is widely used to explain the frictional resistance of
rocks. However, the parameters vary with temperature and sliding velocity, preventing extrapolation of
laboratory results to natural conditions. Here, we explain the frictional properties of natural gouge from
the San Andreas Fault, Alpine Fault, and the Nankai Trough from room temperature to ∼ 300◦C for a
wide range of slip-rates with constant constitutive parameters by invoking the competition between two
healing mechanisms with different thermodynamic properties. The transition from velocity-strengthening
to velocity-weakening at steady-state can be attained either by decreasing the slip-rate or by increasing
temperature. Our study provides a framework to understand the physics underlying the slip-rate and state
dependence of friction and the dependence of frictional properties on ambient physical conditions.
4.2 Plain Language Summary
The physics of friction is crucial to understanding fault mechanics, impacting virtually every aspect of
earthquake initiation, propagation, and associated hazards. The mechanics of active fault zones exhibit a
63
complex dependence on temperature and sliding velocity among other factors. The frictional resistance
of natural gouge can be explained by empirical rate- and state-dependent friction laws for a limited range
of conditions. However, explaining the non-stationary frictional behavior of gouge friction and extrapolation of laboratory constraints to natural conditions remains challenging. In this study, we describe a
constitutive law that predicts the velocity of sliding of natural gouge based on applied shear stress, effective
confining pressure, and the ambient temperature of the fault. The transition from stable to unstable sliding
is controlled by the competition between micro-mechanisms of deformation within the gouge that operate
in distinct ranges of temperature and slip-rate. Once calibrated to mechanical data for a specific lithology
and confining pressure, the model explains the temperature and slip-rate control on fault stability, allowing
extrapolation of laboratory data to natural conditions.
4.3 Introduction
The slip-rate and state dependence of friction is a key feature of fault mechanics that enables runaway
instabilities and the recurrence of earthquakes [77, 76]. The unstable nature of rock friction is widely
recognized as the origin of stick-slip instabilities in natural faults [51, 56, 263, 211], allowing laboratory
analogs, down-scaled versions of earthquakes [157, 186]. The frictional behavior of rocks may be described
by constitutive laws calibrated with laboratory observations that predict the slip-rate based on shear stress
and one or more state variables representing the evolving texture of fault gouge [74, 75, 73, 252, 239, 106].
Empirical friction laws capture the direct velocity dependence of friction and the transient phase that
follows perturbations of shear stress, normal stress, or temperature [74, 163, 65]. A key aspect of rate- and
state-dependent friction laws is the steady-state velocity dependence parameter, characterized by
a − b =
∂µss
∂ ln V
, (4.1)
64
where µss is the steady-state friction coefficient, defined as the ratio of the shear and effective normal
stresses, and V is the slip-rate across the fault. The sign of a − b controls frictional stability, i.e., the
potential of a fault to generate dynamic ruptures (Figure 4.1a). Steady-state velocity-weakening friction,
which occurs for a − b < 0, allows the nucleation and propagation of earthquakes and slow-slip events.
In contrast, velocity-strengthening condition, with a − b > 0, promotes stable sliding, manifested as fault
creep [110] or afterslip following mainshock ruptures [179].
After calibration, empirical friction laws reproduce experimental data for many lithologies in a wide
range of hydrothermal settings [e.g., 39, 257, 117, 202, 322, 113, 216]. In addition, friction laws successfully
explain a variety of fault behaviors, including creep, slow-slip events, and crack-like or pulse-like seismic
ruptures [e.g. 156, 12, 60, 200, 201, 309], allowing numerical modeling of natural faults [e.g., 298, 17, 232,
305, 10, 165, 259, 273, 136].
The frictional parameters vary widely depending on hydrothermal conditions, slip-rate, pore-fluid and
confining pressure, and other factors in laboratory experiments, even for a given lithology. For example,
phyllosilicate-rich natural gouge from the Alpine Fault, Nankai Trough, and the Central San Andreas
Fault [112, 48, 47, 58, 195, 290, 202] consistently show a drastic reduction of the strengthening effect
or even a transition from velocity-strengthening to velocity-weakening at steady-state with increasing
temperature and/or decreasing slip-rate (Figure 4.1). The velocity dependence of frictional stability can
be captured using a cut-off velocity in the friction law [e.g., 215, 182, 275], but this approach does not
incorporate the temperature dependence. Despite the importance of these effects on fault dynamics, the
origin of the different regimes of stability is still poorly understood, hindering ongoing efforts to build
physical models of the seismic cycle consistent with rock mechanics.
In this study, we use a physics-based constitutive model of rate- and state-dependent friction based on
the competition between multiple healing mechanisms [18] to capture the variations of frictional properties of natural gouge with temperature and slip-rate under constant coefficients. In the next section,
65
we describe the details of the constitutive framework. We then calibrate the model to experimental data
from gouge samples cored from the San Andreas Fault Observatory at Depth that document the frictional
behavior from room temperature to 250◦C and within three orders of magnitude of slip-rates from a few
nanometers to micrometers per second [195]. Finally, we show that the constitutive model applies to a
variety of natural rock samples under varying parametric configurations, including gouge from the South
Alpine Fault in New Zealand [27], the Alpine Fault Deep Fault Drilling Project [48], the Zuccale Fault
in Italy [69], and the Nankai Trough [112] from room temperature to about 300◦C and slip-rate ranging
from nanometers to millimeters per second. The model provides more comprehensive predictions of the
slip rate and temperature regimes of fault stability than empirical friction laws, allowing extrapolation of
laboratory data to natural fault conditions and more realistic simulations of the seismic cycle.
4.4 Constitutive framework
To explain the velocity and temperature control of the steady-state velocity dependence of friction of
natural fault gouge, we consider a thermally activated constitutive law based on the real area of contact
and the healing of micro-asperities [11, 18, 21]. The constitutive law for cataclastic flow describes the
frictional strength in isobaric conditions as
µ = µ0
V
V0
1
n
d
d0
m
n
exp
Q
nR
1
T
−
1
T0
, (4.2)
where µ is the frictional resistance controlled by the instantaneous slip-rate V and the temperature T. The
representative size of micro-asperities d corresponds to the local radius of curvature at contact junctions
and constitutes a state variable for the evolving texture of the gouge layer. The constants V0 = 1 µm/s, d0 =
1 µm, and T0 = 25◦C represent reference values of slip-rate, micro-asperity size and ambient temperature,
respectively. The reference friction coefficient µ0 is a material property corresponding to the ratio of
66
plowing to indentation hardness [49, 50]. The coefficients n and m are the stress and microasperity-size
power exponents, respectively. The exponential term corresponds to an Arrhenius activation with the
energy and temperature of activation Q and T0, respectively, involving the universal gas constant R. As
the laboratory data considered in this study is limited to up to 300◦C, Equation (4.2) incorporates only a
single mechanism of deformation and, therefore, does not capture the brittle-to-ductile transition observed
at higher temperatures and lower slip-rates [39, 112, 202, 216]. Capturing the brittle-to-flow transition
requires multiple deformation mechanisms [21, 25].
The evolution of the effective asperity size incorporates time-dependent healing by multiple microphysical mechanisms and slip-dependent comminution by grain fracturing and contact rejuvenation [18].
The corresponding evolution law can be cast in additive form, following
˙d
d
=
X
N
k=1
Gk
pkd
pk
−
λV
2h
, (4.3)
where the first term on the right-hand side corresponds to healing with the micro-asperity size power exponent pk and the reference growth rate Gk for the healing mechanism k = 1 to N. The fastest mechanism
dominates the healing process and we ignore possible coupling effects between healing mechanisms. The
second term on the right-hand side of Equation (4.3) represents the reduction of asperity size during shear
over the gouge thickness h with the characteristic strain 1/λ due to rejuvenation of the contact population
(Figure 4.2). Alternatively, the evolution law can be expressed in multiplicative form, following
˙d
d
=
λV
2h
ln "X
N
k=1
Gk
pkd
pk
λV
2h
#
, (4.4)
involving the difference in the logarithm of the healing and weakening terms. The evolution laws of
Equations (4.3) and (4.4) may produce different evolutionary effects, for example, varying slip weakening
67
distance (Figure 4.2 and 4.3), but produce the same response at steady-state [29, 197, 3, 236, 35, 36]. In both
formulations, the healing rate Gk is thermally activated following an Arrhenius formulation, as in
Gk(T) = G
0
k
exp
−
Hk
R
1
T
−
1
Tk
, (4.5)
with the reference growth rate G0
k = (1 µm)pk /s, the activation enthalpy Hk, and the activation temperature Tk. The ratio Hk/Tk represents the change of entropy of healing. The evolution of asperity size
leads to a change in the real area of contact that modulates the frictional strength. Equivalent evolution
laws based on the age of contact that fall in the aging-law and slip-law end-members are discussed in
Appendix A.
The healing mechanisms are defined by their distinct thermo-mechanical parameters affected by grain
shape, composition, and deformation process [284, 285, 18]. Healing associated with compaction creep
may occur by subcritical crack growth [7], pressure-solution creep [105], intra-granular deformation [121],
and closure of cracks at different rates depending on mineralogy, texture, and temperature (Figure 4.2).
Additional healing may occur without significant fault-perpendicular shortening by crack healing, crack
sealing, and cementation of the pore space [5, 104, 231].
Considering two healing mechanisms, i.e., using N = 2 in Equation (4.3) or Equation (4.4), we obtain
velocity-weakening and velocity-strengthening friction at steady-state in different velocity and temperature regimes controlled by the growth rates and thermodynamic properties of each mechanism. The
dominant healing mechanism is associated with the largest grain size at steady-state, which depends on
temperature and slip-rate (Figure 4.4). When one mechanism dominates, the associated strain-rate can be
orders of magnitude higher than the other. However, the transition between healing mechanisms leads to
a gradual change of steady-state velocity dependence within specific ranges of temperatures and slip-rates.
68
4.5 Constitutive properties of San Andreas Fault gouge
We first focus the analysis on laboratory experiments on gouge samples from the San Andreas Fault. The
San Andreas Fault Observatory at Depth offered direct access to the local creeping segment, allowing the
study of the mechanical, compositional, and frictional properties of fault gouge [292, 170, 58]. The borehole
crossed two creeping strands of the Central San Andreas Fault at about 3 km depth [324, 325]. Samples
from the Southern Deforming Zone (SDZ) and Central Deforming Zone (CDZ) are rich in phyllosilicates,
including saponite, corrensite, and serpentinite, and exhibit steady-state velocity-strengthening at room
temperature. The slightly stronger SDZ strand, with a lower phyllosilicate and higher quartz and feldspar
content, becomes velocity-weakening under elevated temperature or sufficiently low velocity [195].
We utilize the mechanical data from velocity-step experiments on the SDZ samples for three orders of
magnitude of slip-rates, from nanometers to micrometers per second, from room temperature (25◦C) to
250◦C in isobaric conditions, with a constant effective normal stress of σ¯ = 100 MPa [195]. The loading
rate of the slowest experiments is close to the long-term slip-rate of the San Andreas Fault [14, 13], approaching the conditions of rupture nucleation. We first use the RSFit3000 methodology [282] to derive
the empirical rate- and state-dependent parameters, including the reference friction coefficient µ0, a, b, the
characteristic weakening distance L, and the system stiffness k (Figure 4.5). We calculate the steady-state
parameter a−b using the regressed a and b values. The constitutive parameters are scattered with a small
dynamic range, consistent with a single deformation mechanism. However, there is a clear dependence
of the state dependence parameter b on temperature and velocity, indicating changes in the underpinning
evolutionary process (Figures 4.5). The corresponding trend of a − b shows a clear sensitivity to slip-rate
and temperature, delineating two distinct stability regimes (Figure 4.9). The velocity-weakening regime
operates at high temperatures and low slip rates.
69
To explain the experimental data, we conduct numerical simulations of the velocity jump experiments
with a spring-slider assembly with the constitutive laws of Equations (4.2) and (4.3) and separately with
Equations (4.2) and (4.4). The conservation of linear momentum in isobaric condition implies
µ˙σ¯ = −k(V − VL)
(4.6)
where k is the stiffness of the system and VL represents the imposed loading rate, which varies from V1
to V2 across a velocity step (Figures 4.6). As the loading rate barely exceeds micrometers per second and
the gouge is conditionally stable in the experimental conditions, we ignore the radiation of seismic waves.
We estimate the empirical parameter a−b using a numerical approximation of Equation (4.1). Conducting
such experiments for various temperatures and slip rates allows us to calculate the residuals with the
laboratory observations. We optimize 10 constitutive parameters (µ0, n, m, and Q for the direct effect and
p1, p2, H1, H2, T1, and T2 for the evolutionary effects) by grid search to minimize the root mean square of
the residuals with about 56 measurements of frictional resistance at steady-state and 33 measurements of
a − b (Figures 4.7 and 4.8).
The frictional parameters for the SDZ gouge are best explained by the power exponents n = 24 ± 5,
m = 1.64 ± 0.2 in the flow law of Equation (4.2). The reference friction coefficient µ0 = 0.21 and the
activation energy Q = 85 ± 15 kJ/mol are constrained by the average frictional resistance at steady-state
(Figure 4.7). The distinct regimes of stability are controlled by the competition of healing mechanisms with
activation enthalpies H1 = 80±15 kJ/mol and H2 = 195±45 kJ/mol. The absolute value of the activation
temperatures cannot be determined with the data available, but the ratio T1/T2 = 1.51 is well constrained.
The size power exponents for healing p1 = 1.1 and p2 = 3.6 trade off with each other, but fall within the
expectation of p1 < m for steady-state velocity-weakening and p2 > m for velocity strengthening. The
best-fitting parameters are summarized in Table 4.1.
70
The modeled evolution of the friction coefficient and the steady-state velocity dependence parameter a − b explain the laboratory measurements well (Figure 4.9). As observed, the transition between
velocity-weakening and velocity-strengthening at steady-state is controlled by slip-rate and temperature.
In the higher velocity and/or lower temperature range, the frictional behavior is controlled by the velocitystrengthening healing mechanism with p1 > m. In the complementary range with a velocity-weakening
behavior, the second healing mechanism with p2 < m dominates (Figure 4.9a). These results are corroborated by the evolution of the friction coefficient, whereby only the velocity step from 10−2
to 10−3 µm/s
at 200◦C and 250◦C exhibits velocity-weakening (Figure 4.9d).
The thermodynamic properties of the SDZ samples provide new insights into the dominant healing
mechanisms. The activation energy H1 = 80 kJ/mol is compatible with a variety of healing mechanisms
operating in wet conditions, including subcritical crack growth, pressure solution, and crack healing or
crack sealing [241, 6, 204, 52, 180, 18]. With the activation energy H2 = 195 kJ/mol, the second healing
mechanism may involve viscoelastic collapse of a weak mineral phase within the gouge [152, 256, 120, 18].
4.6 Application to other fault gouges
We now consider the velocity-step experiments conducted on natural samples from the South Alpine
Fault [27, 48, 47], Central San Andreas Fault [58], Zuccale Fault [69, 140], and Nankai Trough [112] at
varying slip rates, temperatures, or both. The majority of these samples exhibit a predominantly velocitystrengthening behavior in the experimental conditions, consistent with the presence of hydrous minerals
and clays in all samples (e.g., smectite, chlorite, illite, saponite). However, a few cases exhibit velocityweakening at steady-state for sufficiently low velocity (e.g., Figure 4.10ehi). Regardless, all samples exhibit
increasing stability with increasing slip-rate or decreasing temperature.
Using the procedure described in the previous section, we obtain constitutive parameters that explain
the evolution of the steady-state velocity dependence parameters a − b with temperature and slip-rate
71
(Figure 4.10), as summarized in Table 4.2. As these experiments document a limited range of slip rates
and temperatures, a wide range of thermodynamic properties can explain the mechanical data equally
well. The activation enthalpy varies in the range 50 − 80 kJ/mol and 80 − 150 kJ/mol for the low- and
high-temperature healing mechanisms, respectively, which is typical in the brittle regime. For example,
hornblende [169], pyroxene [293], and natural gouges [4, 302, 113] in wet conditions feature an activation
enthalpy within 20−65 kJ/mol [18]. Similarly, wet Westerly granite [39], basalt [216], and cataclasite gouge
from the Alpine Fault [202] exhibit an activation enthalpy for healing in the range 30−55 kJ/mol [21]. The
presence of mechanisms characterized by activation energy greater than 100 kJ/mol suggests the activation of compaction creep of anhydrous minerals. For instance, the sample from the Zuccale Fault contains
calcite [69], which shows activation energies of 145-250 kJ/mol for viscoelastic flow [122]. Samples from
the San Andreas Fault Observatory at Depth include quartz and feldspar [58, 195] that showcase activation energies for viscoelastic flow of 150-600 kJ/mol [256] and 135-240 kJ/mol [254, 255], respectively.
Conversely, the low activation energy of healing under specific conditions suggests the deformation of
phyllosilicates or other weak minerals [267, 176, 177, 18].
4.7 Discussion
As evidenced by an abundance of laboratory experiments, the frictional behavior of natural gouge is complex, involving different stability regimes based on ambient temperature and instantaneous slip rate. The
mechanics of gouge friction cannot be explained using empirical friction laws at constant parameters,
at least not as previously defined (see Appendix C). In contrast, the constitutive framework described
in Section 2 captures the frictional behavior of natural gouge for a wide range of rocks upon parametric adjustments based on lithology and confining or pore-fluid pressures. The constitutive model applies
within a nominal range of temperatures and slip rates that exclude the brittle-ductile transition. These
results facilitate the scaling of laboratory observations up to natural fault conditions within the relatively
72
low temperatures of the middle and upper crust or the thick sedimentary layers of accretionary prisms at
subduction zones.
The non-stationary properties of gouge friction documented in the laboratory and captured in the
constitutive model imply complex dynamics of natural faults. The unstable friction of phyllosilicates at a
sufficiently low slip rate and increased stability at intermediate velocity is particularly relevant to creeping
faults. Although the central San Andreas Fault exhibits primarily aseismic slip [13, 264], creep is spatiotemporally variable and episodic [294, 143, 144]. This phenomenon is compatible with phyllosilicate gouge
friction, which is often found velocity-weakening at low slip speed, allowing nucleation of instabilities,
but velocity-strengthening at higher velocity, inhibiting the transition to seismic rupture propagation.
The slip-rate dependence of gouge stability may also explain shallow slow-slip events on the San Andreas
Fault [314] or the North Anatolian Fault [139].
The complex frictional behavior of natural gouge may explain the non-stationary creeping behavior
of other faults, such as the Laohushan segment of the Haiyuan fault [133, 135, 161], and the Pütürge and
Palu segments of the East Anatolian Fault [43, 57, 234]. The temperature dependence of gouge friction also
implies variations in fault behavior with depth, including the prevalence of a shallow slip deficit during
large earthquakes [90, 233, 24] and variations of interseismic coupling as a function of depth [e.g., 132].
4.8 Conclusion
We describe a constitutive framework for gouge friction that explains the widely observed velocity and
temperature dependence of effective mechanical properties. The competition between thermally activated
healing mechanisms and weakening by contact rejuvenation leads to distinct stability regimes with a tendency for increased stability with increasing slip rate or decreasing temperature. The model applies to
a range of temperatures and slip-rates within the brittle field for phyllosilicate-rich gouge. Considering
73
additional deformation mechanisms is required to capture the brittle-ductile transition [21, 25]. The constitutive model explains the frictional behavior of natural gouge from the San Andreas Fault, Zuccale Fault,
Alpine Fault, and the Nankai Trough under varying parametric configurations, allowing the extrapolation
of laboratory data to natural fault conditions within the middle and top crust. If the effects of temperature
and slip rate on the frictional resistance are now better understood, the remaining controls of varying
normal stress and pore-fluid pressure remain elusive. Further experimental work is needed to describe the
properties of a wide range of rocks and calibrate the model for different tectonic contexts and hydrothermal
conditions.
74
Table 4.1: Constitutive and physics parameters of the simulated velocity-step experiments on the San Andreas Fault Observatory at Depth SDZ sample shown in Figure 4.9. The uncertainties correspond to plus
or minus a standard deviation. The parameters d0 = 1 µm, V0 = 1 µm/s, and T0 = 25◦C represent scaling
factors, not constitutive parameters per se. The reference friction coefficient µ0 is a material property corresponding to the ratio of plowing to indentation hardness. The gouge thickness h = 1 mm is a laboratory
setting.
Direct effect
Parameter Symbol Value
Reference friction coefficient µ0 0.21
Reference asperity size d0 1 µm
Reference velocity V0 1 µm/s
Effective normal stress σ¯ 100 MPa
Stress power exponent n 24 ± 5
Asperity-size power exponent m 1.64 ± 0.2
Activation energy Q 85 ± 15 kJ/mol
Reference temperature T0 25◦C
Evolutionary effects
Parameter Symbol Value
Size-sensitivity exponent p1 1.1
p2 3.6
Activation enthalpy H1 80 ± 15 kJ/mol
H2 195 ± 45 kJ/mol
Activation temperature T1 286◦C
T2 106◦C
Reference strain 1/λ 0.1
Gouge thickness h 1 mm
Table 4.2: Constitutive parameters of natural gouge constrained by velocity-step experiments (Figures 4.9
and 4.10). The activation energies H1 and H2 are in kJ/mol. The activation temperatures T1 and T2 are in
degrees Celsius. The ratio T1/T2 is better constrained than the absolute value of T1 and T2.
Fault gouge Mineral assembly n m p1 p2 H1 H2 T1 T2 Reference
Alpine Fault Chlorite, illite 29 0.8 1.0 2.5 80 150 217 338 [27]
San Andreas (CDZ) Saponite 28 1.0 1.2 2.7 80 150 217 325 [58]
San Andreas (SDZ) Corrensite, serpentine 26 1.0 1.2 2.7 80 150 217 275 [58]
Zuccale Fault Calcite, smectite 59 0.6 1.1 2.5 80 150 217 308 [69]
Lizardite-rich serpentinite 15 1.1 1.0 2.7 80 150 165 308 [140]
Alpine Fault Smectite, quartz, K-feldspar 17 1.9 1.3 2.7 60 80 200 150 [48]
Alpine Fault Saponite, serpentine 53 0.9 1.0 3.1 50 180 206 106 [47]
Nankai Trough Smectite, illite, chlorite 29 1.6 1.0 4.1 80 100 277 278 [112]
San Andreas (CDZ) Saponite, serpentine 62 0.9 1.0 2.8 50 180 206 110 [195]
75
Table 4.3: Constitutive parameters using RSFit3000. The velocity steps are imposed sequentially from
V1 to
V2, expressed in
µm/s. Experiments
are conducted at different temperatures expressed in
◦C. The characteristic weakening distance
L is expressed in
µm. The parameters
a and b and
their uncertainties are expressed in
‰.
Step T
V1 V2 a
σa b
σb L σ
L k σk µ0 σ
µ0 µss
1 25 0.08 0.011 6.79 6.21E-2 0.006 1.59E-1 100 7.30E-01 8.18E-03 2.91E-04 0.173 5.68E-02 0.167
2 25 0.01 0.001 6.09 4.42E-2 1.69 4.81E-1 417 2.25E+01 6.08E-03 1.87E-04 0.151 2.69E-02 0.147
3 25 0.001 0.01 4.40 7.71E-2 0.58 1.10E2 101 1.94E+01 8.75E-03 3.35E-04 0.144 2.38E-02 0.148
4 25 0.01 0.10 5.85 5.51E-2 -0.2 1.48E-2 53 7.31E-02 1.39E-02 1.15E-03 0.153 2.26E-05 0.159
5 25 0.06 3.0 3.86 1.58E-1 -1.90 7.49E1 100 4.04E+00 3.11E-03 4.56E-04 0.168 6.86E-05 0.178
6 25 3.0 0.08 4.79 2.99E-1 0.11 7.98E-2 100 1.37E+00 2.43E-03 3.53E-04 0.187 1.41E-04 0.180
1 100 0.3 0.01 3.14 2.59E-1 0.67 2.22E-1 100 2.46E-01 4.90E-03 9.57E-04 0.153 9.50E-05 0.149
2 100 0.01 0.001 4.48 5.15E-2 3.47 3.26E-2 81 5.41E-03 1.31E-03 4.32E-05 0.143 3.17E-05 0.142
3 100 0.001 0.01 3.51 9.07E-2 1.30 2.77E1 100 2.23E+00 2.78E-02 3.01E-03 0.141 4.40E-05 0.143
4 100 0.01 0.08 5.95 1.22E-1 2.03 7.46 150 6.14E+02 1.91E-02 1.76E-03 0.148 3.19E-05 0.151
5 100 0.08 3.0 4.21 1.46E-1 -0.10 5.58E1 100 5.62E+01 2.21E-03 2.20E-04 0.156 5.96E-05 0.163
6 100 2.96 0.08 4.30 1.34E-1 0.11 8.01E-2 100 6.08E-01 4.88E-03 6.60E-04 0.172 2.34E-04 0.166
1 150 0.09 0.01 4.90 6.74E-2 3.17 1.77E-1 88 1.44E-02 1.52E-02 1.29E-03 0.170 8.27E-05 0.168
2 150 0.01 0.001 4.81 1.01E-1 5.00 5.47E-1 200 5.39E-02 1.61E-03 6.88E-05 0.150 5.47E-05 0.150
3 150 0.001 0.009 3.45 1.64E-1 2.41 2.31E2 100 9.82E+00 8.51E-04 2.37E-05 0.146 2.44E-05 0.147
4 150 0.01 0.1 4.36 4.59E-2 0.71 2.51E-1 52 3.02E-02 1.31E-02 1.29E-03 0.148 1.97E-05 0.152
5 150 0.04 3.0 3.15 1.03E-1 -0.024 1.99E2 150 1.23E+03 2.81E-03 2.89E-04 0.153 7.75E-05 0.159
6 150 3.0 0.09 4.42 1.26E-1 0.364 2.91E-1 420 1.40E+00 2.47E-03 3.35E-04 0.168 2.49E-04 0.162
1 200 0.1 0.01 4.44 6.16E-2 2.41 6.86E-2 53.4 2.14E-03 1.73E-03 1.39E-04 0.172 5.31E-05 0.170
2 200 0.01 0.001 3.55 4.49E-2 7.82 8.09E-2 159 6.82E-03 1.79E-03 1.02E-04 0.162 4.35E-05 0.166
3 200 0.001 0.01 2.54 4.24E-2 4.64 4.51E-2 75 1.71E-03 1.15E-02 1.61E-03 0.171 4.95E-05 0.169
4 200 0.01 0.1 3.37 1.42E-1 3.00 1.17 100 5.68E-02 5.96E-03 8.64E-04 0.171 3.36E-05 0.171
5 200 0.13 3.0 4.11 1.18E-1 0.97 4.98E1 200 1.06E+01 8.03E-04 5.07E-05 0.172 3.55E-05 0.176
6 200 3.0 0.1 4.95 6.33E-2 0.72 4.39E-2 316 1.34E-01 3.21E-03 2.49E-04 0.185 1.28E-04 0.179
1 250 0.1 0.01 4.74 1.20E-1 6.81 9.90E-2 92.3 7.11E-03 3.12E-03 3.45E-04 0.168 1.48E-04 0.170
2 250 0.01 0.001 7.71 7.72E-2 14.2 2.90E-1 447 2.57E-02 1.09E-03 4.06E-05 0.162 6.07E-05 0.168
3 250 0.001 0.01 2.70 6.68E-2 5.89 1.33E-1 32 1.54E-03 4.24E-02 9.28E-03 0.165 5.22E-05 0.162
4 250 0.01 0.1 2.02 9.16E-2 1.79 1.43E1 100 8.78E-01 6.81E-03 2.25E-03 0.160 4.46E-05 0.160
5 250 0.05 3.0 2.17 1.10E-1 -1.08 7.55E2 200 1.41E+02 2.02E-03 3.23E-04 0.161 5.98E-05 0.167
6 250 3.0 25 8.20 1.05 0.39 1.80E-01 100 4.94E+01 5.48E-01 3.37E-05 0.170 7.07E-05 0.177
7 250 25 3.0 5.91 1.59 0.440 1.12E-03 100 1.60E+03 3.20E+00 4.84E-03 0.181 1.07E-03 0.176
8 250 3.0 0.08 3.66 1.04E-1 0.429 5.38E-2 130 2.00E-01 3.45E-03 3.83E-04 0.169 1.44E-04 0.164
9 250 0.06 0.01 5.71 9.57E-2 0.005 2.14 1,000 6.15E+02 1.09E-02 9.74E-04 0.155 1.30E-04 0.151
76
Friction coefficient Velocity
Displacement
Velocity
steps
Veloc
V
ity-strengthening
elocity-weakening
Fast
Slow
a-b>0
a-b<0
A
-2
0
2
4
6
8
10
12
14
Steady-state velocity dependence (x10-3)
Steady-state velocity dependence (x10-3)
Velocity (μm/s)
0 50 100 150 200 250 300
-15
-10
-5
0
5
10
CSAF (SDZ) Moore et al. 2016
CSAF (CDZ) Moore et al. 2016
CSAF (SDZ) Carpenter et al. 2015
CSAF (CDZ) Carpenter et al. 2015
South Alpine Boulton et al. 2018
Northern Apennines Teset et al. 2014
Southen Alpine Barth et al. 2013
Zuccale Colletini et al. 2011
Nankai den Hartog et al. 2012
Alpine Boulton et al. 2014
C D
Temperature (o
C)
0.01 0.1 1 10 100
10-3-10-2 μm/s
10-1-100 μm/s
10-3-10-2 μm/s
10-2-10-1 μm/s
100
-101 μm/s
Deformation
at room
temperature
Temperature (o
C)
Velocity (μm/s)
-2
0
0
2
2
2
4
4
4
4
6
6
6
8
8
10
50
100
150
200
250
Velocity-strengthening
Velocity
weakening
CSAF (SDZ) Moore et al. 2016Steady-state velocity dependence (x10-3)
0
5
B
0.01 0.1 1
Figure 4.1: Velocity and temperature dependence of friction of natural gouge from laboratory experiments.
a) Schematic of friction evolution during a velocity-jump experiment with possible velocity-strengthening
(a − b > 0) and velocity-weakening (a − b < 0) at steady-state. b) Measured (diamonds) and interpolated
(background color) value of a − b as a function of velocity and temperature for the SDZ sample cored
from San Andreas Fault Observatory at Depth in the Central San Andreas Fault (CSAF). c) Laboratory
measurements of a−b from natural gouge for different loading velocities at room temperature. d) Variation
of a − b under various temperature slip-rate conditions.
77
Seismic cycle
Compaction
creep
Contact rejuvenation
Pressure-solution
creep
Dissolution Precipitation
Healing and compaction
Shear and dilatancy
Bare contact
Fluid
Healing
Crack
closure
Cementation
Fracturing
Granular flow
Fracturing
Communition
Subcritical
crack growth
Sealing
A B Gouge
Healing and compaction
Shear and dilatancy
Seismic cycle
Figure 4.2: Mechanisms of deformation and healing enabling seismic cycles across a frictional interface.
a) Case of solid-solid or bare contact with contact rejuvenation by dilatant shear during sliding and compaction creep accommodated by viscoelasticity or pressure-solution creep during relocking. b) Case of
solid-gouge-solid contact where granular flow, fracturing, and subcritical crack growth accommodate dilatant shear and comminution. The closure/cementation of cracks enables healing of the interface. Each
mechanism acting on different minerals is associated with specific constitutive properties.
10-4
0.15
0.2
10-2 100 0.2 0.3 0.4 0.5
Velocity (μm/s) Displacement (cm)
Friction coefficient (
µ)
0.14
0.16
0.18
25 o
C
Velocity-weakening
Velocity-strengthening
150 o
C
300 o
C
Slip
Aging
A B
10-4 µm/s 10-3 µm/s 10-4 µm/s
Figure 4.3: Friction coefficient as a function of velocity and displacement from simulated velocity jump
experiment for the constitutive law with two mechanisms. a) The friction coefficient under the temperature of 25, 150, and 300◦C as a function of velocity. The black solid and dashed red lines represent the
experiments utilizing Equation (3) and Equation (4), respectively. b) The friction coefficient as a function
of displacement, conducted by step-up and step-down velocity jump experiments. Equations (3) and (4)
correspond to the aging-law and slip-law end-members, respectively.
78
Mechanism 1
Mechanism 2
2 Mechanisms
Temperature = 250 °C Temperature = 250 °C
Velocity = 10-2 μm/s Velocity = 10-2 μm/s
10-4 10-3 10-2 10-1
10-7
10-5
10-3
10-1
1 101
Velocity (μm/s)
10-4 10-3 10-2 10-1 1 101
Velocity (μm/s)
Micro-asperity size (mm)
a-b ( x10-3 ) -2
0
2
4
A
C
B
0 100 200 300 400
Temperature (°C)
10-14
10-10
10-6
10-2
Micro-asperity size (mm)
0 100 200 300 400
Temperature (°C)
a-b ( x10-3 ) -2
0
2
4
D
Figure 4.4: Micro-asperity size and frictional velocity dependence as a function of sliding velocity and
temperature. The dashed lines show the evolution of one single mechanism. The first mechanism, represented by red lines leads to velocity-strengthening behavior while the second mechanism (blue lines) tends
to velocity-weakening. The solid lines are the results of two competing mechanisms that are dominant in
different velocity and temperature ranges.
79
250°C
up down
200°C
150°C
100°C
25°C
up down
0.001-0.01
0.01-0.1
0.1-1
1-10
2
3
4
5
6
7
8
a (x10-3)
A
b (x10-3) 0
5
10
C
L (mm)
102.0
101.5
102.5
E
10-2.5
10-3.5
10-3.0
10-2.0
10-1.5
10-2 10-1 100 101
Velocity (μm/s) Temperature (o
C)
G
2
3
4
5
6
7
8 B
0
5
10
15 D
1.4
1.6
1.8
2.0
2.2
2.4
2.6
F
0 50 100 150 200 250
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0 H
Stiffness (MPa/mm)
Figure 4.5: Effective frictional parameters obtained from regression of velocity step experiments using the
SDZ sample. a) and b) Direct effect parameter a as a function of velocity and temperature, respectively. c)
and d) Evolution parameter b as a function of velocity and temperature, respectively. e) and f) Same for
the characteristic weakening distance L. g) and h) Same for the stiffness k. The up-pointing and downpointing triangles represent data from step-up and step-down experiments, respectively. Colors represent
the temperature and loading rate of the experiments. The parameters are summarized in Table 4.3. 80
0.15
0.16
0.17
0.18
0.19
Friction coefficient μ Friction coefficient μ Friction coefficient μ Friction coefficient μ
0.15
0.16
0.17
0.18
0.19
0.15
0.16
0.17
0.18
0.19
k = 0.03 MPa/μm
4 4.5 5 5.5 6 6.5 7
Displacement (mm)
0.15
0.16
0.17
0.18
0.19
k = 0.03 MPa/μm
0.001 -> 0.01
k = 1 MPa/μm
25 OC
250 OC
250 OC
k = 1 MPa/μm
k = 1 MPa/μm
k = 1 MPa/μm
250 OC
0.001 μm/s -> 0.01 μm/s
25 OC
0.001 μm/s -> 0.01 μm/s
1 μm/s -> 10 μm/s
0.001 μm/s -> 0.01 μm/s
A
B
C
D
Figure 4.6: Evolution of the friction coefficient during simulated velocity jumps experiments. a) Effect of
different temperatures. b) Effect of loading rate. c) and d) Effect of stiffness.
81
Q (kJ/mol)
Reference friction coefficient
µ0
-2
-2
-1.9
-1.9
-1.9
-1.8
-1.8
-1.8
-1.8
-1.7
-1.7
-1.7
-1.7
-1.7
-1.6
-1.6
-1.6
-1.6
-1.6
-1.6
-1.6
0.16
0.18
0.20
0.22
0.24
0.26
0.28
40 60 80 100 120 140 160
RMSE
10-2.0
10-1.9
10-1.8
10-1.7
10-1.6
A
Temperature (oC)
pre-step
post-step
10-4 10-2 100 102
Velocity (μm/s)
0
50
100
150
200
250
B
Temperature (o
C)
10-4 10-2 100 102
Velocity (μm/s)
C D
0.12
0.14
0.16
0.18
0.20
Steady-state
μss
250°C
200°C
150°C
100°C
25°C
μ
0.14
0.15
0.16
0.17
0.18
0 50 100 150 200 250 300
100
101
10-1
10-2
10-3
Figure 4.7: Constraints on the friction coefficient µ for the SDZ sample. a) Root-mean-square error (RMSE)
of the observed and simulated steady-state friction coefficient µ as a function of activation energy Q and
reference friction coefficient µ0. b), c) and d) Comparison between the simulated and the regressed steadystate friction coefficient µ as a function of loading velocity and temperature. The simulated data is the
background color for (b) and the solid lines for (c) and (d), and the regressed data is label colors for (b).
The colors in (c) and (d) label temperature and velocity, respectively. The circles and triangles represent
the value of steady-state µ before and after the velocity changes.
82
-2.7
-2.7
-2.7
-2.6
-2.6
-2.6
-2.6
-2.5
-2.6
-2.6
-2.5
-2.5
-2.4
-2.4
-2.3
-2.2
15 20 25 30 35 40 45 50
0.8
1
1.2
1.4
1.6
1.8
2
0.8 0.9 1 1.1 1.2 1.3 1.4
-2.7
2.8
3
3.2
3.4
3.6
3.8
4
-2.75
-2.75
-2.7
-2.7
-2.7
-2.7
-2.65
-2.65
-2.65
-2.65
-2.65
-2.6
-2.6
-2.6
-2.6
-2.55
-2.55
-2.55
-2.55
-2.5
-2.5
-2.5
-2.45 -2.45
-2.45
-2.4 -2.4
-2.4
-2.35 -2.35 -2.35
50 60 70 80 90 100 110 120
120
130
140
150
160
170
180
190
200
210
220
-2.75
-2.75
-2.75
-2.75
-2.75
-2.75
-2.7
-2.7
-2.7
-2.7
-2.7
-2.7
-2.7
-2.65
-2.65
-2.65
-2.65
-2.65
-2.6
-2.6
-2.6
-2.6
-2.6
-2.55
-2.55
-2.55
-2.5
-2.5
-2.5
-2.45
-2.4
260 270 280 290 300 310 320 330
90
100
110
120
130
140
150
-2.7
-2.7
-2.7
-2.7
-2.6
-2.6
-2.6
-2.5
-2.5
-2.5
-2.2
-2.3
-2.4
-2.4
-2.5
-2.5
-2.6
-2.6
-2.6
-2.7
n = 24
m = 1.64
p1
= 1.1
p2
= 3.6
H1
= 80
H2
= 195
T1
= 286
T2
= 106
Asperity size exponent n
Asperity size exponent m
A
Asperity size evolution exponent p1
Asperity size evolution exponent p2
B
Activation Enthalpy H1 (kJ/mol)
Activation Enthalpy H2 (kJ/mol)
Activation Temperature T1 (
o
C)
Activation Temperature T2 (oC)
C D
10-2.8
10-2.7
10-2.6
10-2.5
10-2.4
10-2.3
RSME
Figure 4.8: Optimization of constitutive parameters for the SDZ sample. a), b), c) and d) show the rootmean-square deviation (RMSE) of synthetic and lab-observed a−b measurement as a function of exponents
n/m, evolution exponents ratio p1/p2, activation enthalpy ratio H1/H2, and activation temperature ratio
T1/T2. The parameters used to generate the preferred model shown in Figure 3 (red dot) are labeled.
83
Velocity-strengthening
Velocity-weakening
Steady-state velocity
dependence (x10-3)
0
5
A
4
4
4
2
2
-2
-4
-2
0
0
100
-101
10-1-100
10-2-10-1
10-3-10-2 -5 μm/s
0
5
a-b (x10-2)
0 50 100 150 200 250
Temperature (°C)
C
0.001 0.01 0.1 1 10
Velocity (μm/s)
-5
0
5
a-b (x10-2)
250°C
200°C
150°C
100°C
25°C
B
0.01 0.1 1 10
100
150
200
250
50
0
Temperature (°C)
Velocity (μm/s)
Velocity-strengthening Velocity-strengthening
Velocity-strengthening
Velocity-strengthening
0
0.01 0.001μm/s
Slip (μm)
50 100 150 200 0
0.1 0.01μm/s
50 100 150 200
Slip (μm) Slip (μm)
0
1.0 0.1μm/s
50 100 150 200
D
Velocity-weakening
Velocity-neutral
Velocity-neutral
Velocity-strengthening
Velocity-strengthening
Velocity-neutral
Velocity-strengthening
Velocity-strengthening
Velocity-strengthening Velocity-weakening
Velocity-strengthening
25°C 25°C 25°C
100°C 100°C 100°C
150°C 150°C 150°C
200°C 200°C 200°C
250°C 250°C 250°C
Aging-law end member Slip-law end member
Figure 4.9: Comparison between simulated and measured frictional data for samples from the SDZ of
the San Andreas Fault Observatory at Depth. a) Comparison between simulated (background color) and
measured (colored triangles) steady-state velocity-dependence parameter a − b. b) Simulated (lines) and
regressed (triangles) a−b parameters as a function of velocity colored by temperature. c) Simulated (lines)
and regressed (triangles) a−b parameters as a function of temperature colored by velocity. d) Comparison
of the synthetic (lines) and raw friction coefficient measurements (gray dots). The red solid lines and black
dashed lines denote simulations conducted using Equation (4.3) and Equation (4.4), respectively.
84
1 10 100 1000
4
6
8
10
12
South Alpine
Barth et al. 2013
CSAF(CDZ)
Carpenter et al. 2015
saponite, serpentine
CSAF(CDZ)
Moore et al. 2016
CSAF(SDZ)
Carpenter et al. 2015
calcite, smectite
Zuccale
Collettini et al. 2013
saponite
South Alpine
Boulton et al. 2018
smectite, illite
Nankai
den Hartog et al. 2011
serpentinite
Kaproth & Marone 2013
1 10 100
2
3
4
5
6
7
8
9
1 10 100
5
6
7
8
9
10
11
10 100 1000
-5
0
5
10
15
20
0.01 0.1 1 10
-20
-15
-10
-5
0
5
10
15
0.01 0.1 1
0
2
4
6
8
1 10 100
0
10
20
-10
a - b (x10-3) a - b (x10-3) a - b (x10-3)
Velocity (μm/s) Velocity (μm/s) Velocity (μm/s)
smectite, quartz
chlorite, illite saponite corrensite, serpentine
Alpine DFDP-1
Boulton et al. 2018
room temp.
70o
C
140o
C
175o
C
210o
C
100o
C
150o
C
200o
C
250o
C
300o
C
A B C
D E F
G H I
0.001 0.01 0.1 1 10
1
2
3
4
5
6
7
1 10 100 1000
4
5
6
7
Figure 4.10: Comparison between simulated (lines) and laboratory-derived (dots) velocity dependence parameter (a − b) of natural samples from a) South Alpine fault [27], b) and c) CDZ and SDZ of central
San Andreas Fault, respectively [58], d) Zuccale Fault [69], e) lizardite-rich serpentinite sample [140], f)
Alpine DFDP-1 sample [48], g) South Alpine outcrop [47], h) Nankai (simulated) [112], and i) Central San
Andreas Fault CDZ [195]. Colors represent experiments under different temperatures. The dashed line is
for a − b = 0.
85
Bibliography
[1] Mohamed Abdelmeguid, Xiao Ma, and Ahmed Elbanna. “A novel hybrid finite element-spectral
boundary integral scheme for modeling earthquake cycles: Application to rate and state faults
with low-velocity zones”. In: J. Geophys. Res. 124.12 (2019), pp. 12854–12881.
[2] Amotz Agnon. “Pre-instrumental earthquakes along the Dead Sea rift”. In: Dead Sea transform
fault system: reviews. Springer, 2014, pp. 207–261.
[3] J.-P. Ampuero and A. M. Rubin. “Earthquake nucleation on rate and state faults - aging and slip
laws”. In: J. Geophys. Res. 113.B01302 (2008). doi: 10.1029/2007JB005082.
[4] Mengke An, Fengshou Zhang, Derek Elsworth, Zhengyu Xu, Zhaowei Chen, and
Lianyang Zhang. “Friction of Longmaxi shale gouges and implications for seismicity during
hydraulic fracturing”. In: J. Geophys. Res. 125.8 (2020), e2020JB019885. doi: 10.1029/2020JB019885.
[5] Muriel Andreani, Alain Baronnet, Anne-Marie Boullier, and Jean-Pierre Gratier. “A
microstructural study of a “crack-seal” type serpentine vein using SEM and TEM techniques”. In:
European Journal of Mineralogy 16.4 (2004), pp. 585–595.
[6] Barry Kean Atkinson. “Subcritical crack growth in geological materials”. In: J. Geophys. Res. 89.B6
(1984), pp. 4077–4114.
[7] HV Atkinson. “Overview no. 65: Theories of normal grain growth in pure single phase systems”.
In: Acta Metallurgica 36.3 (1988), pp. 469–491. doi: 10.1016/0001-6160(88)90079-X.
[8] Pascal Audet, Michael G Bostock, Nikolas I Christensen, and Simon M Peacock. “Seismic evidence
for overpressured subducted oceanic crust and megathrust fault sealing”. In: Nature 457.7225
(2009), pp. 76–78.
[9] Han Bao, Jean-Paul Ampuero, Lingsen Meng, Eric J Fielding, Cunren Liang,
Christopher WD Milliner, Tian Feng, and Hui Huang. “Early and persistent supershear rupture of
the 2018 magnitude 7.5 Palu earthquake”. In: Nature Geoscience 12.3 (2019), pp. 200–205. doi:
10.1038/s41561-018-0297-z.
[10] S. Barbot. “Frictional and structural controls of seismic super-cycles at the Japan trench”. In: Earth
Planets Space 72.63 (2020). doi: 10.1186/s40623-020-01185-3.
86
[11] S. Barbot. “Modulation of fault strength during the seismic cycle by grain-size evolution around
contact junctions”. In: Tectonophysics 765 (2019), pp. 129–145. doi: 10.1016/j.tecto.2019.05.004.
[12] S. Barbot. “Slow-slip, slow earthquakes, period-two cycles, full and partial ruptures, and
deterministic chaos in a single asperity fault”. In: Tectonophysics 768 (2019), p. 228171. doi:
10.1016/j.tecto.2019.228171.
[13] S. Barbot, P. Agram, and M. De Michele. “Change of Apparent Segmentation of the San Andreas
Fault Around Parkfield from Space Geodetic Observations Across Multiple Periods”. In: J.
Geophys. Res. 118.12 (2013), pp. 6311–6327. doi: 10.1002/2013JB010442.
[14] S. Barbot, Y. Fialko, and Y. Bock. “Postseismic Deformation due to the Mw 6.0 2004 Parkfield
Earthquake: Stress-Driven Creep on a Fault with Spatially Variable Rate-and-State Friction
Parameters”. In: J. Geophys. Res. 114.B07405 (2009). doi: 10.1029/2008JB005748.
[15] S. Barbot, Y. Fialko, and D. Sandwell. “Effect of a Compliant Fault Zone on the Inferred
Earthquake Slip Distribution”. In: J. Geophys. Res. 113.B6 (June 2008). doi: 10.1029/2007JB005256.
[16] S. Barbot, Y. Fialko, and D. Sandwell. “Three-Dimensional Models of Elasto-Static Deformation in
Heterogeneous Media, with Applications to the Eastern California Shear Zone”. In: Geophys. J.
Int. 179.1 (2009), pp. 500–520. doi: 10.1111/j.1365-246X.2009.04194.x.
[17] S. Barbot, N. Lapusta, and J. P. Avouac. “Under the Hood of the Earthquake Machine: Towards
Predictive Modeling of the Seismic Cycle”. In: Science 336.6082 (2012), pp. 707–710. doi:
10.1126/science.1218796.
[18] Sylvain Barbot. “A rate-, state-, and temperature-dependent friction law with competing healing
mechanisms”. In: J. Geophys. Res. 127 (2022), e2022JB025106. doi: 10.1029/2022JB025106.
[19] Sylvain Barbot. “A Spectral Boundary-Integral Method for Quasi-Dynamic Ruptures of Multiple
Parallel Faults”. In: Bull. Seism. Soc. Am. 111.3 (2021), pp. 1614–1630.
[20] Sylvain Barbot. “A spectral boundary-integral method for quasi-dynamic ruptures of multiple
parallel faults”. In: Bull. Seism. Soc. Am. (2021). doi: 10.1785/0120210004.
[21] Sylvain Barbot. “Constitutive behavior of rocks during the seismic cycle”. In: AGU Advances
(2023). doi: 10.1029/2023AV000972.
[22] Sylvain Barbot. “Modulation of fault strength during the seismic cycle by grain-size evolution
around contact junctions”. In: Tectonophysics 765 (2019), pp. 129–145.
[23] Sylvain Barbot. “Slow-slip, slow earthquakes, period-two cycles, full and partial ruptures, and
deterministic chaos in a single asperity fault”. In: Tectonophysics 768 (2019), p. 228171. doi:
10.1016/j.tecto.2019.228171.
[24] Sylvain Barbot, Heng Luo, Teng Wang, Yariv Hamiel, Oksana Piatibratova,
Muhammad Tahir Javed, Carla Braitenberg, and Gokhan Gurbuz. “Slip distribution of the
February 6, 2023 Mw 7.8 and Mw 7.6, Kahramanmaraş, Turkey earthquake sequence in the East
Anatolian fault zone”. In: Seismica 2.3 (2023). doi: 10.26443/seismica.v2i3.502.
87
[25] Sylvain Barbot and Lei Zhang. “Constitutive Behavior of Olivine Gouge Across the Brittle-Ductile
Transition”. In: Geophys. Res. Lett. 50.24 (2023). doi: 10.1029/2023GL105916.
[26] Philip M Barnes, Laura M Wallace, Demian M Saffer, Rebecca E Bell, Michael B Underwood,
Ake Fagereng, Francesca Meneghini, Heather M Savage, Hannah S Rabinowitz, Julia K Morgan,
et al. “Slow slip source characterized by lithological and geometric heterogeneity”. In: Science
Advances 6.13 (2020), eaay3314. doi: 10.1126/sciadv.aay3314.
[27] NC Barth, C Boulton, BM Carpenter, GE Batt, and VG Toy. “Slip localization on the southern
Alpine fault, New Zealand”. In: Tectonics 32.3 (2013), pp. 620–640. doi: 10.1002/tect.20041.
[28] Noel M Bartlow. “A Long-Term View of Episodic Tremor and Slip in Cascadia”. In: Geophys. Res.
Lett. 47.3 (2020), e2019GL085303. doi: 10.1029/2019GL085303.
[29] N. M. Beeler, T. E. Tullis, and J. D. Weeks. “The roles of time and displacement in the evolution
effect in rock friction”. In: Geophys. Res. Lett. 21 (1994), pp. 1987–1990. doi: 10.1029/94GL01599.
[30] Rebecca Bell, Rupert Sutherland, Daniel HN Barker, Stuart Henrys, Stephen Bannister,
Laura Wallace, and John Beavan. “Seismic reflection character of the Hikurangi subduction
interface, New Zealand, in the region of repeated Gisborne slow slip events”. In: Geophysical
Journal International 180.1 (2010), pp. 34–48.
[31] Maxime Bernaudin and Frédéric Gueydan. “Episodic tremor and slip explained by fluid-enhanced
microfracturing and sealing”. In: Geophys. Res. Lett. 45.8 (2018), pp. 3471–3480. doi:
10.1029/2018GL077586.
[32] Gregory C Beroza and Satoshi Ide. “Slow earthquakes and nonvolcanic tremor”. In: Annual review
of Earth and planetary sciences 39 (2011), pp. 271–296.
[33] Gregory C Beroza and Takeshi Mikumo. “Short slip duration in dynamic rupture in the presence
of heterogeneous fault properties”. In: J. Geophys. Res. 101.B10 (1996), pp. 22449–22460.
[34] Kelvin R Berryman, Ursula A Cochran, Kate J Clark, Glenn P Biasi, Robert M Langridge, and
Pilar Villamor. “Major earthquakes occur regularly on an isolated plate boundary fault”. In:
Science 336.6089 (2012), pp. 1690–1693.
[35] Pathikrit Bhattacharya, Allan M Rubin, and Nicholas M Beeler. “Does fault strengthening in
laboratory rock friction experiments really depend primarily upon time and not slip?” In: J.
Geophys. Res. 122.8 (2017), pp. 6389–6430. doi: 10.1002/2017JB013936.
[36] Pathikrit Bhattacharya, Allan M Rubin, Terry E Tullis, Nicholas M Beeler, and Keishi Okazaki.
“The evolution of rock friction is more sensitive to slip than elapsed time, even at near-zero slip
rates”. In: Proc. Nat. Acad. Sci. 119.30 (2022), e2119462119. doi: 10.1073/pnas.2119462119.
[37] Pathikrit Bhattacharya and Robert C Viesca. “Fluid-induced aseismic fault slip outpaces pore-fluid
migration”. In: Science 364.6439 (2019), pp. 464–468. doi: 10.1126/science.aaw7354.
[38] James Biemiller and Luc Lavier. “Earthquake supercycles as part of a spectrum of normal fault
slip styles”. In: Journal of Geophysical Research: Solid Earth 122.4 (2017), pp. 3221–3240.
88
[39] M. L. Blanpied, D. A. Lockner, and J. D. Byerlee. “Frictional slip of granite at hydrothermal
conditions”. In: J. Geophys. Res. 100.B7 (1995), pp. 13045–13064. doi: 10.1029/95JB00862.
[40] Michael L Blanpied, DA Lockner, and JD Byerlee. “Fault stability inferred from granite sliding
experiments at hydrothermal conditions”. In: Geophysical Research Letters 18.4 (1991), pp. 609–612.
[41] Michael L Blanpied, David A Lockner, and James D Byerlee. “Frictional slip of granite at
hydrothermal conditions”. In: Journal of Geophysical Research: Solid Earth 100.B7 (1995),
pp. 13045–13064.
[42] Michael L Blanpied, Terry E Tullis, and John D Weeks. “Effects of slip, slip rate, and shear heating
on the friction of granite”. In: Journal of Geophysical Research: Solid Earth 103.B1 (1998),
pp. 489–511.
[43] Quentin Bletery, Olivier Cavalié, Jean-Mathieu Nocquet, and Théa Ragon. “Distribution of
interseismic coupling along the North and East Anatolian Faults inferred from InSAR and GPS
data”. In: Geophys. Res. Lett. 47.16 (2020), e2020GL087775. doi: 10.1029/2020GL087775.
[44] Quentin Bletery and Jean-Mathieu Nocquet. “Slip bursts during coalescence of slow slip events in
Cascadia”. In: Nature Communications 11.1 (2020), pp. 1–6.
[45] Margaret S Boettcher, Greg Hirth, and Brian Evans. “Olivine friction at the base of oceanic
seismogenic zones”. In: J. Geophys. Res. 112.B1 (2007), p. 13. doi: 10.1029/2006JB004301.
[46] L. Bollinger, Y. Klinger, P. Tapponnier, Y. Gaudemer, and D. Tiwari. “Estimating the return times
of great Himalayan earthquakes in eastern Nepal: Evidence from the Patu and Bardibas strands of
the Main Frontal Thrust”. In: Nature Geosci. 6 (2013), pp. 71–76.
[47] Carolyn Boulton, Nicolas C Barth, Diane E Moore, David A Lockner, John Townend, and
Daniel R Faulkner. “Frictional properties and 3-D stress analysis of the southern Alpine Fault,
New Zealand”. In: Journal of Structural Geology 114 (2018), pp. 43–54. doi:
10.1016/j.jsg.2018.06.003.
[48] Carolyn Boulton, Diane E Moore, David A Lockner, Virginia G Toy, John Townend, and
Rupert Sutherland. “Frictional properties of exhumed fault gouges in DFDP-1 cores, Alpine Fault,
New Zealand”. In: Geophys. Res. Lett. 41.2 (2014), pp. 356–362. doi: 10.1002/2013GL058236.
[49] F. P. Bowden and D. Tabor. The friction and lubrication of Solids, Part I. Oxford: Clarendon Press,
1950. isbn: 9780198507772, 0198507771.
[50] F. P. Bowden and D. Tabor. The friction and lubrication of Solids, Part II. Oxford: Clarendon Press,
1964.
[51] WF Brace and JD Byerlee. “Stick-slip as a mechanism for earthquakes”. In: Science 153.3739
(1966), pp. 990–992. doi: 10.1126/science.153.3739.990.
[52] Susan Louise Brantley, James David Kubicki, and Art F White, eds. Kinetics of water-rock
interaction. Springer, 2008. doi: 10.1007/978-0-387-73563-4.
89
[53] Charles G Bufe, Philip W Harsh, and Robert O Burford. “Steady-state seismic slip–A precise
recurrence model”. In: Geophys. Res. Lett. 4.2 (1977), pp. 91–94.
[54] Roland Bürgmann. “The geophysics, geology and mechanics of slow fault slip”. In: Earth and
Planetary Science Letters 495 (2018), pp. 112–134.
[55] J.D. Byerlee. “Friction of rock”. In: Pure Appl. Geophys. 116 (1978), pp. 615–626.
[56] James D Byerlee and WF Brace. “Stick slip, stable sliding, and earthquakes—effect of rock type,
pressure, strain rate, and stiffness”. In: J. Geophys. Res. 73.18 (1968), pp. 6031–6037. doi:
10.1029/JB073i018p06031.
[57] Ziyadin Cakir, Uğur Doğan, Ahmet M Akoğlu, Semih Ergintav, Seda Özarpacı, Alpay Özdemir,
Tohid Nozadkhalil, Nurdan Çakir, Cengiz Zabcı, M. Hilmi Erkoç Erkoç, Mehran Basmenji,
Mehmet Köküm, and Roger Bilham. “Arrest of the Mw 6.8 January 24, 2020 Elaziğ (Turkey)
earthquake by shallow fault creep”. In: Earth Planet. Sci. Lett. 608 (2023), p. 118085. doi:
10.1016/j.epsl.2023.118085.
[58] BM Carpenter, DM Saffer, and C Marone. “Frictional properties of the active San Andreas Fault at
SAFOD: Implications for fault strength and slip behavior”. In: J. Geophys. Res. 120.7 (2015),
pp. 5273–5289. doi: 10.1002/2015JB011963.
[59] C. Cattania. “Complex earthquake sequences on simple faults”. In: Geophys. Res. Lett. 46.17-18
(2019), pp. 10384–10393. doi: 10.1029/2019GL083628.
[60] Camilla Cattania and Paul Segall. “Crack models of repeating earthquakes predict observed
moment-recurrence scaling”. In: J. Geophys. Res. 124.1 (2019), pp. 476–503. doi:
10.1029/2018JB016056.
[61] Calum J Chamberlain, David R Shelly, John Townend, and Tim A Stern. “Low-frequency
earthquakes reveal punctuated slow slip on the deep extent of the Alpine fault, New Zealand”. In:
Geochemistry, Geophysics, Geosystems 15.7 (2014), pp. 2984–2999.
[62] Jianquan Chen, Chang Liu, Hang Zhang, and Yaolin Shi. “Seismic Activities and Hazards of the
Seismic Gaps in the Haiyuan Fault in Northeastern Tibet: Insights From Numerical Modeling
Earthquake Interaction”. In: Earth and Space Science 9.11 (2022), e2022EA002536.
[63] Qizhi Chen and Jeffrey T Freymueller. “Geodetic evidence for a near-fault compliant zone along
the San Andreas fault in the San Francisco Bay area”. In: Bull. Seism. Soc. Am. 92.2 (2002),
pp. 656–671.
[64] Yuxuan Chen, Mian Liu, and Gang Luo. “Complex temporal patterns of large earthquakes: Devil’s
staircases”. In: Bull. Seism. Soc. Am. 110.3 (2020), pp. 1064–1076. doi: 10.1785/0120190148.
[65] FM Chester. “Effects of temperature on friction: Constitutive equations and experiments with
fault gouge”. In: J. Geophys. Res. 99.B4 (1994), pp. 7247–7261. doi: 10.1029/93JB03110.
90
[66] FM Chester and John M Logan. “Implications for mechanical properties of brittle faults from
observations of the Punchbowl fault zone, California”. In: Pure and applied geophysics 124.1
(1986), pp. 79–106.
[67] Frederick M Chester, James P Evans, and Ronald L Biegel. “Internal structure and weakening
mechanisms of the San Andreas fault”. In: J. Geophys. Res. 98.B1 (1993), pp. 771–786.
[68] Elizabeth S Cochran, Yong-Gang Li, Peter M Shearer, Sylvain Barbot, Yuri Fialko, and
John E Vidale. “Seismic and geodetic evidence for extensive, long-lived fault damage zones”. In:
Geology 37.4 (2009), pp. 315–318.
[69] Cristiano Collettini, André Niemeijer, Cecilia Viti, Steven AF Smith, and Chris Marone. “Fault
structure, frictional properties and mixed-mode fault slip behavior”. In: Earth Planet. Sci. Lett.
311.3 (2011), pp. 316–327. doi: 10.1016/j.epsl.2011.09.020.
[70] Víctor M Cruz-Atienza, Carlos Villafuerte, and Harsha S Bhat. “Rapid tremor migration and
pore-pressure waves in subduction zones”. In: Nature communications 9.1 (2018), pp. 1–13.
[71] S. M. Day, L. A. Dalguer, N. Lapusta, and Y. Liu. “Comparison of finite difference and boundary
integral solutions to three-dimensional spontaneous rupture”. In: J. Geophys. Res. 110.B12307
(2005).
[72] Steven M Day. “Three-dimensional finite difference simulation of fault dynamics: rectangular
faults with fixed rupture velocity”. In: Bull. Seism. Soc. Am. 72.3 (1982), pp. 705–727.
[73] J. H. Dieterich. “Constitutive properties of faults with simulated gouge”. In: Monograph 24:
Mechanical Behavior of Crustal Rocks, ed. by J. M. Logan N. L. Carter M. Friedman and
D. W. Stearns. Washington, DC: AGU, 1981, pp. 103–120. doi: 10.1029/GM024p0103.
[74] J. H. Dieterich. “Modeling of Rock Friction 1. Experimental Results and Constitutive Equations”.
In: J. Geophys. Res. 84.B5 (1979), pp. 2161–2168. doi: 10.1029/JB084iB05p02161.
[75] J. H. Dieterich. “Modeling of Rock Friction 2. Simulation of preseismic slip”. In: J. Geophys. Res.
84.B5 (1979), pp. 2169–2175. doi: 10.1029/JB084iB05p02169.
[76] J. H. Dieterich. “Time-dependent friction and the mechanics of stick-slip”. In: Pure Appl. Geophys.
116.4-5 (1978), pp. 790–806. doi: 10.1007/978-3-0348-7182-2\_15.
[77] J. H. Dieterich. “Time-dependent friction in rocks”. In: J. Geophys. Res. 77 (1972), pp. 3690–3697.
doi: 10.1029/JB077i020p03690.
[78] James H Dieterich. “Modeling of rock friction: 1. Experimental results and constitutive
equations”. In: Journal of Geophysical Research: Solid Earth 84.B5 (1979), pp. 2161–2168.
[79] James H Dieterich and Brian D Kilgore. “Direct observation of frictional contacts: New insights
for state-dependent properties”. In: Pure and applied geophysics 143 (1994), pp. 283–302.
91
[80] J F Dolan, Lee J McAuliffe, Edward J Rhodes, Sally F McGill, and Robert Zinke. “Extreme
multi-millennial slip rate variations on the Garlock fault, California: Strain super-cycles,
potentially time-variable fault strength, and implications for system-level earthquake
occurrence”. In: Earth Planet. Sci. Lett. 446 (2016), pp. 123–136. doi: 10.1016/j.epsl.2016.04.011.
[81] Herb Dragert, Kelin Wang, and Thomas S James. “A silent slip event on the deeper Cascadia
subduction interface”. In: Science 292.5521 (2001), pp. 1525–1528.
[82] Benchun Duan. “Effects of low-velocity fault zones on dynamic ruptures with nonelastic off-fault
response”. In: Geophys. Res. Lett. 35.4 (2008).
[83] Pierre Dublanchet. “The dynamics of earthquake precursors controlled by effective friction”. In:
Geophysical Journal International 212.2 (2017), pp. 853–871.
[84] Pierre Dublanchet, Pascal Bernard, and Pascal Favreau. “Interactions and triggering in a 3-D
rate-and-state asperity model”. In: J. Geophys. Res. 118.5 (2013), pp. 2225–2245.
[85] Eric M Dunham and Ralph J Archuleta. “Evidence for a supershear transient during the 2002
Denali fault earthquake”. In: Bull. Seism. Soc. Am. 94.6B (2004), S256–S268. doi:
10.1785/0120040616.
[86] WL Ellsworth and GC Beroza. “Seismic evidence for an earthquake nucleation phase”. In: Science
268.5212 (1995), pp. 851–855. doi: 10.1126/science.268.5212.851.
[87] Brittany A Erickson, Eric M Dunham, and Arash Khosravifar. “A finite difference method for
off-fault plasticity throughout the earthquake cycle”. In: Journal of the Mechanics and Physics of
Solids 109 (2017), pp. 50–77.
[88] Y Fialko. “Space Geodetic Constraints on the Structure and Properties of Compliant Damage
Zones Around Major Crustal Faults”. In: AGU Fall Meeting Abstracts. Vol. 2004. 2004, S32B–02.
[89] Y. Fialko, D. Sandwell, D. Agnew, M. Simons, P. Shearer, and B. Minster. “Deformation on nearby
faults induced by the 1999 Hector Mine earthquake”. In: Science 297 (2002), pp. 1858–1862. doi:
10.1126/science.1074671.
[90] Y. Fialko, D. Sandwell, M. Simons, and P. Rosen. “Three-dimensional deformation caused by the
Bam, Iran, earthquake and the origin of shallow slip deficit”. In: Nature 435 (May 2005),
pp. 295–299. doi: 10.1038/nature03425.
[91] William B Frank, Nikolaı M Shapiro, Allen L Husker, Vladimir Kostoglodov, Harsha S Bhat, and
Michel Campillo. “Along-fault pore-pressure evolution during a slow-slip event in Guerrero,
Mexico”. In: Earth and Planetary Science Letters 413 (2015), pp. 135–143.
[92] William B Frank, Nikolaı M Shapiro, Vladimir Kostoglodov, Allen L Husker, Michel Campillo,
Juan S Payero, and Germán A Prieto. “Low-frequency earthquakes in the Mexican Sweet Spot”.
In: Geophysical Research Letters 40.11 (2013), pp. 2661–2666. doi: 10.1002/grl.50561.
[93] A. M. Freed and J. Lin. “Delayed triggering of the 1999 Hector Mine earthquake by viscoelastic
stress transfer”. In: Nature 441 (May 2001), pp. 180–183.
92
[94] Eiichi Fukuyama and Takeshi Mikumo. “Slip-weakening distance estimated at near-fault
stations”. In: Geophys. Res. Lett. 34.9 (2007). doi: 10.1029/2006GL029203.
[95] Eiichi Fukuyama, Kotoyo Tsuchida, Hironori Kawakata, Futoshi Yamashita, Kazuo Mizoguchi,
and Shiqing Xu. “Spatiotemporal complexity of 2-D rupture nucleation process observed by direct
monitoring during large-scale biaxial rock friction experiments”. In: Tectonophysics 733 (2018),
pp. 182–192.
[96] A-A Gabriel, J-P Ampuero, Luis A Dalguer, and Paul Martin Mai. “The transition of dynamic
rupture styles in elastic media under velocity-weakening friction”. In: J. Geophys. Res. 117.B9
(2012).
[97] Xiang Gao and Kelin Wang. “Rheological separation of the megathrust seismogenic zone and
episodic tremor and slip”. In: Nature 543.7645 (2017), pp. 416–419.
[98] Judith Gauriau and James F Dolan. “Relative Structural Complexity of Plate-Boundary Fault
Systems Controls Incremental Slip-Rate Behavior of Major Strike-Slip Faults”. In: Geochemistry,
Geophysics, Geosystems 22.11 (2021), e2021GC009938. doi: 10.1029/2021GC009938.
[99] Abhijit Ghosh, John E Vidale, Justin R Sweet, Kenneth C Creager, Aaron G Wech, Heidi Houston,
and Emily E Brodsky. “Rapid, continuous streaking of tremor in Cascadia”. In: Geochemistry,
Geophysics, Geosystems 11.12 (2010).
[100] C. Goldfinger, C H. Nelson, J. E Johnson, and Shipboard Scientific Party. “Holocene earthquake
records from the Cascadia subduction zone and northern San Andreas fault based on precise
dating of offshore turbidites”. In: Ann. Rev. Earth Planet. Sci. 31.1 (2003), pp. 555–577.
[101] Chris Goldfinger, Yasutaka Ikeda, Robert S Yeats, and Junjie Ren. “Superquakes and supercycles”.
In: Seism. Res. Lett. 84.1 (2013), pp. 24–32. doi: 10.1785/0220110135.
[102] Joan Gomberg and Paul Johnson. “Dynamic triggering of earthquakes”. In: Nature 437.7060
(2005), pp. 830–830. doi: 10.1038/437830a.
[103] Arjun Goswami and Sylvain Barbot. “Slow-slip events in semi-brittle serpentinite fault zones”. In:
Scientific reports 8.1 (2018), pp. 1–11.
[104] Jean-Pierre Gratier, Pascal Favreau, and François Renard. “Modeling fluid transfer along
California faults when integrating pressure solution crack sealing and compaction processes”. In:
Journal of Geophysical Research: Solid Earth 108.B2 (2003).
[105] Jean-Pierre Gratier, Robert Guiguet, François Renard, Liliane Jenatton, and Dominique Bernard.
“A pressure solution creep law for quartz from indentation experiments”. In: J. Geophys. Res.
114.B3 (2009). doi: 10.1029/2008JB005652.
[106] J. Gu, J. R. Rice, A. L. Ruina, and S. T. Tse. “Slip motion and stability of a single degree of freedom
elastic system rate and state dependent friction”. In: J. Mech. Phys. Sol. 32 (1984), pp. 167–196.
93
[107] K Hall, D Schmidt, and H Houston. “Peak tremor rates lead peak slip rates during propagation of
two large slow earthquakes in Cascadia”. In: Geochemistry, Geophysics, Geosystems 20.11 (2019),
pp. 4665–4675. doi: 10.1029/2019GC008510.
[108] Y. Hamiel and Y. Fialko. “Structure and mechanical properties of faults in the North Anatolian
Fault system from InSAR observations of coseismic deformation due to the 1999 Izmit (Turkey)
earthquake”. In: J. Geophys. Res. 112.B07412 (2007). doi: 10.1029/2006JB004777.
[109] Yariv Hamiel and Yuri Fialko. “Structure and mechanical properties of faults in the North
Anatolian Fault system from InSAR observations of coseismic deformation due to the 1999 Izmit
(Turkey) earthquake”. In: J. Geophys. Res. 112.B7 (2007).
[110] Ruth A Harris. “Large earthquakes and creeping faults”. In: Reviews of Geophysics 55.1 (2017),
pp. 169–198.
[111] Ruth A Harris and Steven M Day. “Effects of a low-velocity zone on a dynamic rupture”. In: Bull.
Seism. Soc. Am. 87.5 (1997), pp. 1267–1280.
[112] SAM den Hartog, Colin J Peach, DA Matthijs de Winter, Christopher J Spiers, and
Toshihiko Shimamoto. “Frictional properties of megathrust fault gouges at low sliding velocities:
New data on effects of normal stress and temperature”. In: Journal of Structural Geology 38 (2012),
pp. 156–171. doi: 10.1016/j.jsg.2011.12.001.
[113] SAM den Hartog, Marion Y Thomas, and DR Faulkner. “How do laboratory friction parameters
compare with observed fault slip and geodetically derived friction parameters? Insights from the
Longitudinal Valley Fault, Taiwan”. In: J. Geophys. Res. 126.10 (2021), e2021JB022390. doi:
10.1029/2021JB022390.
[114] Egill Hauksson. “Crustal geophysics and seismicity in southern California”. In: Geophys. J. Int.
186.1 (2011), pp. 82–98. doi: 10.1111/j.1365-246X.2011.05042.x.
[115] Jessica C Hawthorne, Michael G Bostock, Alexandra A Royer, and Amanda M Thomas.
“Variations in slow slip moment rate associated with rapid tremor reversals in Cascadia”. In:
Geochemistry, Geophysics, Geosystems 17.12 (2016), pp. 4899–4919. doi: 10.1002/2016GC006489.
[116] Gavin P Hayes, Ginevra L Moore, Daniel E Portner, Mike Hearne, Hanna Flamme, Maria Furtney,
and Gregory M Smoczyk. “Slab2, a comprehensive subduction zone geometry model”. In: Science
362.6410 (2018), pp. 58–61. doi: 10.1126/science.aat4723.
[117] Changrong He, Zeli Wang, and Wenming Yao. “Frictional sliding of gabbro gouge under
hydrothermal conditions”. In: Tectonophysics 445.3-4 (2007), pp. 353–362. doi:
10.1016/j.tecto.2007.09.008.
[118] Thomas H Heaton. “Evidence for and implications of self-healing pulses of slip in earthquake
rupture”. In: Phys. Earth Planet. Inter. 64.1 (1990), pp. 1–20. doi: 10.1016/0031-9201(90)90002-F.
[119] Thomas H Heaton. “Evidence for and implications of self-healing pulses of slip in earthquake
rupture”. In: Physics of the Earth and Planetary Interiors 64.1 (1990), pp. 1–20.
94
[120] G. Hirth, C. Tessier, and W. J. Dunlap. “An evaluation of quartzite flow laws based on
comparisons between experimentally and naturally deformed rocks”. In: Int. J. Earth Sci. 90
(2001), pp. 77–87. doi: 10.1007/s005310000152.
[121] Greg Hirth and Jan Tullis. “Dislocation creep regimes in quartz aggregates”. In: Journal of
Structural Geology 14.2 (1992), pp. 145–159.
[122] Caleb W Holyoke III, Andreas K Kronenberg, and Julie Newman. “Dislocation creep of
polycrystalline dolomite”. In: Tectonophysics 590 (2013), pp. 72–82. doi:
10.1016/j.tecto.2013.01.011.
[123] Heidi Houston, Brent G Delbridge, Aaron G Wech, and Kenneth C Creager. “Rapid tremor
reversals in Cascadia generated by a weakened plate interface”. In: Nature Geoscience 4.6 (2011),
pp. 404–409.
[124] Heidi Houston, Brent G Delbridge, Aaron G Wech, and Kenneth C Creager. “Rapid tremor
reversals in Cascadia generated by a weakened plate interface”. In: Nature Geoscience 4.6 (2011),
pp. 404–409. doi: 10.1038/ngeo1157.
[125] Yihe Huang. “Earthquake rupture in fault zones with along-strike material heterogeneity”. In: J.
Geophys. Res. 123.11 (2018), pp. 9884–9898. doi: 10.1029/2018JB016354.
[126] Yihe Huang and Jean-Paul Ampuero. “Pulse-like ruptures induced by low-velocity fault zones”.
In: J. Geophys. Res. 116.B12 (2011). doi: 10.1029/2011JB008684.
[127] Yihe Huang, Jean-Paul Ampuero, and Don V Helmberger. “Earthquake ruptures modulated by
waves in damaged fault zones”. In: J. Geophys. Res. 119.4 (2014), pp. 3133–3154.
[128] Alexandra A Hutchison. “Inter-episodic Tremor and Slip Event Episodes of
Quasi-spatiotemporally Discrete Tremor and Very Low Frequency Earthquakes in Cascadia
Suggestive of a Connective Underlying, Heterogeneous Process”. In: Geophys. Res. Lett. 47.3
(2020), e2019GL086798. doi: 10.1029/2019GL086798.
[129] Koki Idehara, Suguru Yabe, and Satoshi Ide. “Regional and global variations in the temporal
clustering of tectonic tremor activity”. In: Earth Planets Space 66.1 (2014), p. 66. doi:
10.1186/1880-5981-66-66.
[130] B Idini and J-P Ampuero. “Fault-zone damage promotes pulse-like rupture and back-propagating
fronts via quasi-static effects”. In: Geophys. Res. Lett. 47.23 (2020), e2020GL090736. doi:
10.1029/2020GL090736.
[131] Kyungjae Im, Demian Saffer, Chris Marone, and Jean-Philippe Avouac. “Slip-rate-dependent
friction as a universal mechanism for slow slip events”. In: Nature Geoscience 13.10 (2020),
pp. 705–710. doi: 10.1038/s41561-020-0627-9.
[132] R Jolivet, M Simons, PS Agram, Z Duputel, and Z-K Shen. “Aseismic slip and seismogenic
coupling along the central San Andreas Fault”. In: Geophysical Research Letters 42.2 (2015),
pp. 297–306.
95
[133] R. Jolivet, C. Lasserre, M.-P. Doin, S. Guillaso, G. Peltzer, R. Dailu, J. Sun, Z.-K. Shen, and X. Xu.
“Shallow creep on the Haiyuan Fault (Gansu, China) revealed by SAR Interferometry”. In: J.
Geophys. Res. 117.B06401 (2012), p. 18. doi: 10.1029/2011JB008732.
[134] Romain Jolivet and WB Frank. “The transient and intermittent nature of slow slip”. In: AGU
Advances 1.1 (2020), e2019AV000126. doi: 10.1029/2019AV000126.
[135] Romain Jolivet, Cécile Lasserre, M-P Doin, G Peltzer, J-P Avouac, Jianbao Sun, and R Dailu.
“Spatio-temporal evolution of aseismic slip along the Haiyuan fault, China: Implications for fault
frictional properties”. In: Earth Planet. Sci. Lett. 377 (2013), pp. 23–33. doi:
10.1016/j.epsl.2013.07.020.
[136] Joaquín Julve, Sylvain Barbot, Marcos Moreno, Andrés Tassara, Rodolfo Araya, Nicole Catalán,
Jorge GF Crempien, and Valeria Becerra-Carreño. “Recurrence time and size of Chilean
earthquakes influenced by geological structure”. In: Nature Geoscience (2023), pp. 1–9. doi:
10.1038/s41561-023-01327-8.
[137] Y Kaneko and J-P Ampuero. “A mechanism for preseismic steady rupture fronts observed in
laboratory experiments”. In: Geophys. Res. Lett. 38.21 (2011).
[138] Y Kaneko, J-P Ampuero, and N Lapusta. “Spectral-element simulations of long-term fault slip:
Effect of low-rigidity layers on earthquake-cycle dynamics”. In: J. Geophys. Res. 116.B10 (2011).
[139] Y Kaneko, Y Fialko, David T Sandwell, Xiaopeng Tong, and M Furuya. “Interseismic deformation
and creep along the central section of the North Anatolian fault (Turkey): InSAR observations
and implications for rate-and-state friction properties”. In: J. Geophys. Res. 118.1 (2013),
pp. 316–331. doi: 10.1029/2012JB009661.
[140] Bryan M Kaproth and C Marone. “Slow earthquakes, preseismic velocity changes, and the origin
of slow frictional stick-slip”. In: Science 341.6151 (2013), pp. 1229–1232. doi:
10.1126/science.1239577.
[141] Naoyuki Kato. “Deterministic chaos in a simulated sequence of slip events on a single isolated
asperity”. In: Geophys. J. Int. 198.2 (2014), pp. 727–736.
[142] Naoyuki Kato. “Repeating slip events at a circular asperity: Numerical simulation with a rate-and
state-dependent friction law”. In: Bull. Earthq. Res. Inst. Univ. Tokyo 78 (2003), pp. 151–166.
[143] M Khoshmanesh, M Shirzaei, and RM Nadeau. “Time-dependent model of aseismic slip on the
central San Andreas Fault from InSAR time series and repeating earthquakes”. In: J. Geophys. Res.
120.9 (2015), pp. 6658–6679. doi: 10.1002/2015JB012039.
[144] Mostafa Khoshmanesh and Manoochehr Shirzaei. “Episodic creep events on the San Andreas
Fault caused by pore pressure variations”. In: Nature geoscience 11.8 (2018), p. 610. doi:
10.1038/s41561-018-0160-2.
[145] Marina J Kim, Susan Y Schwartz, and Stephen Bannister. “Non-volcanic tremor associated with
the March 2010 Gisborne slow slip event at the Hikurangi subduction margin, New Zealand”. In:
Geophys. Res. Lett. 38.14 (2011). doi: 10.1029/2011GL048400.
96
[146] G. C. P. King, R. S. Stein, and J. Lin. “Static stress changes and the triggering of earthquakes”. In:
Bull. Seism. Soc. Am. 84.3 (1994), pp. 935–953.
[147] James D Kirkpatrick, Åke Fagereng, and David R Shelly. “Geological constraints on the
mechanisms of slow earthquakes”. In: Nature Reviews Earth & Environment (2021), pp. 1–17. doi:
10.1038/s43017-021-00148-w.
[148] Y Klinger, M Etchebes, P Tapponnier, and C Narteau. “Characteristic slip for five great
earthquakes along the Fuyun fault in China”. In: Nature Geoscience 4.6 (2011), pp. 389–392.
[149] Shuichi Kodaira, Takashi Iidaka, Aitaro Kato, Jin-Oh Park, Takaya Iwasaki, and
Yoshiyuki Kaneda. “High pore fluid pressure may cause silent slip in the Nankai Trough”. In:
Science 304.5675 (2004), pp. 1295–1298.
[150] Vladimir Kostoglodov, Shri Krishna Singh, Jose Antonio Santiago, Sara Ivonne Franco,
Kristine M Larson, Anthony R Lowry, and Roger Bilham. “A large silent earthquake in the
Guerrero seismic gap, Mexico”. In: Geophysical Research Letters 30.15 (2003). doi:
10.1029/2003GL017219.
[151] Alissa J Kotowski and Whitney M Behr. “Length scales and types of heterogeneities along the
deep subduction interface: Insights from exhumed rocks on Syros Island, Greece”. In: Geosphere
15.4 (2019), pp. 1038–1065. doi: 10.1130/GES02037.1.
[152] Andreas K Kronenberg, Stephen H Kirby, and John Pinkston. “Basal slip and mechanical
anisotropy of biotite”. In: J. Geophys. Res. 95.B12 (1990), pp. 19257–19278. doi:
0.1029/JB095iB12p19257.
[153] Valère Lambert, Nadia Lapusta, and Stephen Perry. “Propagation of large earthquakes as
self-healing pulses or mild cracks”. In: Nature 591.7849 (2021), pp. 252–258.
[154] N. Lapusta and J. R. Rice. “Nucleation and early seismic propagation of small and large events in a
crustal earthquake model”. In: J. Geophys. Res. 108.B4, 2205 (2003). doi: 10.1029/2001JB000793.
[155] N. Lapusta, J. R. Rice, Y. BenZion, and G. Zheng. “Elastodynamics analysis for slow tectonic
loading with spontaneous rupture episodes on faults with rate- and state-dependent friction”. In:
J. Geophys. Res. 105.B10 (Oct. 2000), pp. 23765–23789.
[156] Nadia Lapusta and James R Rice. “Nucleation and early seismic propagation of small and large
events in a crustal earthquake model”. In: J. Geophys. Res. 108.B4 (2003).
[157] S Latour, A Schubnel, S Nielsen, R Madariaga, and S Vinciguerra. “Characterization of nucleation
during laboratory earthquakes”. In: Geophys. Res. Lett. 40.19 (2013), pp. 5064–5069. doi:
10.1002/grl.50974.
[158] Luc L Lavier, Xinyue Tong, and James Biemiller. “The Mechanics of Creep, Slow Slip Events and
Earthquakes in Mixed Brittle-Ductile Fault Zones”. In: Journal of Geophysical Research: Solid Earth
(2020), e2020JB020325.
97
[159] JR Leeman, DM Saffer, MM Scuderi, and C Marone. “Laboratory observations of slow earthquakes
and the spectrum of tectonic fault slip modes”. In: Nature Communications 7 (2016), p. 11104. doi:
10.1038/ncomms11104.
[160] Michael A Lewis and Yehuda Ben-Zion. “Diversity of fault zone damage and trapping structures
in the Parkfield section of the San Andreas Fault from comprehensive analysis of near fault
seismograms”. In: Geophys. J. Int. 183.3 (2010), pp. 1579–1595.
[161] Yanchuan Li, Jean-Mathieu Nocquet, Xinjian Shan, and Xiaogang Song. “Geodetic observations of
shallow creep on the Laohushan-Haiyuan fault, northeastern Tibet”. In: J. Geophys. Res. 126.6
(2021), e2020JB021576. doi: 10.1029/2020JB021576.
[162] Yong-Gang Li, Keiiti Aki, David Adams, Akiko Hasemi, and William HK Lee. “Seismic guided
waves trapped in the fault zone of the Landers, California, earthquake of 1992”. In: J. Geophys.
Res. 99.B6 (1994), pp. 11705–11722.
[163] M. H. Linker and J. H. Dieterich. “Effects of variable normal stress on rock friction: Observations
and constitutive relations”. In: J. Geophys. Res. 97 (1992), pp. 4923–4940. doi: 10.1029/92JB00017.
[164] Y. Liu and A. M. Rubin. “Role of fault gouge dilatancy on aseismic deformation transients”. In: J.
Geophys. Res. 115.B10414 (2010), 10.1029/2010JB007522.
[165] Yajing Liu, Jeffrey J McGuire, and Mark D Behn. “Aseismic transient slip on the Gofar transform
fault, East Pacific Rise”. In: Proc. Nat. Acad. Sci. 117.19 (2020), pp. 10188–10194. doi:
10.1073/pnas.1913625117.
[166] Yajing Liu and James R Rice. “Aseismic slip transients emerge spontaneously in
three-dimensional rate and state modeling of subduction earthquake sequences”. In: Journal of
Geophysical Research: Solid Earth 110.B8 (2005).
[167] Yajing Liu and James R Rice. “Slow slip predictions based on granite and gabbro friction data
compared to GPS measurements in northern Cascadia”. In: Journal of Geophysical Research: Solid
Earth 114.B9 (2009).
[168] Yajing Liu and James R Rice. “Spontaneous and triggered aseismic deformation transients in a
subduction fault model”. In: J. Geophys. Res. 112.B9 (2007).
[169] Yang Liu and Changrong He. “Friction properties of hornblende and implications for slow-slip
events in subduction zones”. In: Tectonophysics 796 (2020), p. 228644. doi:
10.1016/j.tecto.2020.228644.
[170] D. A. Lockner, C. Morrow, D. Moore, and S. Hickman. “Low strength of deep San Andreas fault
gouge from SAFOD core”. In: Nature 472 (Apr. 2011), pp. 82–86. doi: 10.1038/nature09927.
[171] Yingdi Luo and Jean-Paul Ampuero. “Stability of faults with heterogeneous friction properties
and effective normal stress”. In: Tectonophysics 733 (2018), pp. 257–272. doi:
10.1016/j.tecto.2017.11.006.
98
[172] Yingdi Luo and Zhen Liu. “Rate-and-state model casts new insight into episodic tremor and
slow-slip variability in Cascadia”. In: Geophys. Res. Lett. 46.12 (2019), pp. 6352–6362. doi:
10.1029/2019GL082694.
[173] V. Lyakhovsky, Y. Ben-Zion, and A. Agnon. “Earthquake cycle, fault zones, and seismicity
patterns in a rheologically layered lithosphere”. In: J. Geophys. Res. 106 (2001), pp. 4103–4120.
[174] Harold Magistrale. “Relative contributions of crustal temperature and composition to controlling
the depth of earthquakes in southern California”. In: Geophys. Res. Lett. 29.10 (2002), pp. 87–1.
doi: 10.1029/2001GL014375.
[175] Shmuel Marco and Yann Klinger. “Review of on-fault palaeoseismic studies along the Dead Sea
Fault”. In: Dead Sea transform fault system: reviews. Springer, 2014, pp. 183–205.
[176] Vanadis Marina Mares and AK Kronenberg. “Experimental deformation of muscovite”. In: Journal
of Structural Geology 15.9-10 (1993), pp. 1061–1075. doi: 10.1016/0191-8141(93)90156-5.
[177] Elisabetta Mariani, Katharine H Brodie, and Ernest H Rutter. “Experimental deformation of
muscovite shear zones at high temperatures under hydrothermal conditions and the strength of
phyllosilicate-bearing faults in nature”. In: Journal of Structural Geology 28.9 (2006),
pp. 1569–1587. doi: 10.1016/j.jsg.2006.06.009.
[178] C. Marone and B. Kilgore. “Scaling of the critical slip distance for seismic faulting with shear
strain in fault zones”. In: Nature 362 (1993), pp. 618–620. doi: 10.1038/362618a0.
[179] C. Marone, C. H. Scholz, and R. Bilham. “On the mechanics of earthquake afterslip”. In: J.
Geophys. Res. 96 (1991), pp. 8441–8452. doi: 10.1029/91JB00275.
[180] Nicolas CM Marty, Francis Claret, Arnault Lassin, Joachim Tremosa, Philippe Blanc, Benoit Madé,
Eric Giffaut, Benoit Cochepin, and Christophe Tournassat. “A database of dissolution and
precipitation rates for clay-rocks minerals”. In: Applied Geochemistry 55 (2015), pp. 108–118. doi:
10.1016/j.apgeochem.2014.10.012.
[181] Koki Masuda, Satoshi Ide, Kazuaki Ohta, and Takanori Matsuzawa. “Bridging the gap between
low-frequency and very-low-frequency earthquakes”. In: Earth, Planets and Space 72 (2020),
pp. 1–9.
[182] T. Matsuzawa, H. Hirose, B. Shibazaki, and K. Obara. “Modeling short- and long-term slow slip
events in the seismic cycles of large subduction earthquakes”. In: J. Geophys. Res. 115.B12301
(2010). doi: 10.1029/2010JB007566.
[183] Takanori Matsuzawa, Hitoshi Hirose, Bunichiro Shibazaki, and Kazushige Obara. “Modeling
short-and long-term slow slip events in the seismic cycles of large subduction earthquakes”. In:
Journal of Geophysical Research: Solid Earth 115.B12 (2010).
[184] Wendy McCausland, Steve Malone, and Dan Johnson. “Temporal and spatial occurrence of deep
non-volcanic tremor: From Washington to northern California”. In: Geophysical Research Letters
32.24 (2005).
99
[185] Jeff McGuire and Yehuda Ben-Zion. “High-resolution imaging of the Bear Valley section of the
San Andreas Fault at seismogenic depths with fault-zone head waves and relocated seismicity”.
In: Geophys. J. Int. 163.1 (2005), pp. 152–164.
[186] Gregory C McLaskey. “Earthquake initiation from laboratory observations and implications for
foreshocks”. In: J. Geophys. Res. 124.12 (2019), pp. 12882–12904. doi: 10.1029/2019JB018363.
[187] Cheng Mei, Sylvain Barbot, Yunzhong Jia, and Wei Wu. “Experimental evidence for multiple
controls on fault stability and rupture dynamics”. In: Earth Planet. Sci. Lett. (2021).
[188] Cheng Mei, Sylvain Barbot, and Wei Wu. “Period-multiplying cycles at the transition between
stick-slip and stable sliding and implications for the Parkfield period-doubling tremors”. In:
Geophys. Res. Lett. 48.7 (2021), e2020GL091807. doi: 10.1029/2020gl091807.
[189] Md Shumon Mia, Mohamed Abdelmeguid, and Ahmed E Elbanna. “Spatio-Temporal Clustering of
Seismicity Enabled by Off-Fault Plasticity”. In: Geophysical Research Letters (2022),
e2021GL097601.
[190] Sylvain Michel, Adriano Gualandi, and Jean-Philippe Avouac. “Interseismic coupling and slow
slip events on the Cascadia megathrust”. In: Pure Appl. Geophys. 176.9 (2019), pp. 3867–3891. doi:
10.1007/s00024-018-1991-x.
[191] Jasper Moernaut, Maarten Van Daele, Karen Fontijn, Katrien Heirman, Philippe Kempf,
Mario Pino, G Valdebenito, Roberto Urrutia, Michael Strasser, and Marc De Batist. “Larger
earthquakes recur more periodically: New insights in the megathrust earthquake cycle from
lacustrine turbidite records in south-central Chile”. In: Earth and Planetary Science Letters 481
(2018), pp. 9–19.
[192] Laurent GJ Montési. “Controls of shear zone rheology and tectonic loading on postseismic creep”.
In: Journal of Geophysical Research: Solid Earth 109.B10 (2004).
[193] Diane E Moore and David A Lockner. “Frictional strengths of talc-serpentine and talc-quartz
mixtures”. In: J. Geophys. Res. 116.B1 (2011). doi: 10.1029/2010JB007881.
[194] Diane E Moore and David A Lockner. “Talc friction in the temperature range 25◦
-400◦C:
Relevance for fault-zone weakening”. In: Tectonophysics 449.1-4 (2008), pp. 120–132. doi:
10.1016/j.tecto.2007.11.039.
[195] Diane E Moore, David A Lockner, and Stephen Hickman. “Hydrothermal frictional strengths of
rock and mineral samples relevant to the creeping section of the San Andreas Fault”. In: Journal
of Structural Geology 89 (2016), pp. 153–167. doi: 10.1016/j.jsg.2016.06.005.
[196] Robert M Nadeau and David Dolenc. “Nonvolcanic tremors deep beneath the San Andreas Fault”.
In: Science 307.5708 (2005), pp. 389–389.
[197] M. Nakatani. “Conceptual and physical clarification of rate and state friction: Frictional sliding as
a thermally activated rheology”. In: J. Geophys. Res. 106.B7 (2001), pp. 13347–13380. doi:
10.1029/2000JB900453.
100
[198] Priyamvada Nanjundiah, Sylvain Barbot, and Shengji Wei. “Static source properties of slow and
fast earthquakes”. In: J. Geophys. Res. 125.12 (2020), e2019JB019028. doi: 10.1029/2019JB019028.
[199] Alan R Nelson, Harvey M Kelsey, and Robert C Witter. “Great earthquakes of variable magnitude
at the Cascadia subduction zone”. In: Quaternary Research 65.3 (2006), pp. 354–365.
[200] S. Nie and S. Barbot. “Seismogenic and tremorgenic slow slip near the stability transition of
frictional sliding”. In: Earth Planet. Sci. Lett. 569 (2021), p. 117037. doi: 10.1016/j.epsl.2021.117037.
[201] Shiying Nie and Sylvain Barbot. “Rupture styles linked to recurrence patterns in seismic cycles
with a compliant fault zone”. In: Earth Planet. Sci. Lett. 591 (2022), p. 117593. doi:
10.1016/j.epsl.2022.117593.
[202] AR Niemeijer, Carolyn Boulton, VG Toy, John Townend, and Rupert Sutherland.
“Large-displacement, hydrothermal frictional properties of DFDP-1 fault rocks, Alpine Fault, New
Zealand: Implications for deep rupture propagation”. In: J. Geophys. Res. 121.2 (2016), pp. 624–647.
doi: 10.1002/2015JB012593.
[203] AR Niemeijer and CJ Spiers. “A microphysical model for strong velocity weakening in
phyllosilicate-bearing fault gouges”. In: Journal of Geophysical Research: Solid Earth 112.B10
(2007).
[204] AR Niemeijer, CJ Spiers, and B Bos. “Compaction creep of quartz sand at 400–600 C:
Experimental evidence for dissolution-controlled pressure solution”. In: Earth Planet. Sci. Lett.
195.3-4 (2002), pp. 261–275. doi: 10.1016/S0012-821X(01)00593-3.
[205] T Nishikawa, T Matsuzawa, K Ohta, N Uchida, T Nishimura, and S Ide. “The slow earthquake
spectrum in the Japan Trench illuminated by the S-net seafloor observatories”. In: Science
365.6455 (2019), pp. 808–813. doi: 10.1126/science.aax5618.
[206] Takuya Nishimura, Takanori Matsuzawa, and Kazushige Obara. “Detection of short-term slow
slip events along the Nankai Trough, southwest Japan, using GNSS data”. In: J. Geophys. Res.
118.6 (2013), pp. 3112–3125. doi: 10.1002/jgrb.50222.
[207] J-M Nocquet, P Jarrin, M Vallée, PA Mothes, R Grandin, Frédérique Rolandone, B Delouis,
H Yepes, Yvonne Font, D Fuentes, et al. “Supercycle at the Ecuadorian subduction zone revealed
after the 2016 Pedernales earthquake”. In: Nature Geoscience 10.2 (2017), p. 145. doi:
10.1038/ngeo2864.
[208] Hiroyuki Noda and Nadia Lapusta. “Stable creeping fault segments can become destructive as a
result of dynamic weakening”. In: Nature 493.7433 (2013), pp. 518–521.
[209] Kazushige Obara and Aitaro Kato. “Connecting slow earthquakes to huge earthquakes”. In:
Science 353.6296 (2016), pp. 253–257. doi: 10.1126/science.aaf1512.
[210] Kazushige Obara and Aitaro Kato. “Connecting slow earthquakes to huge earthquakes”. In:
Science 353.6296 (2016), pp. 253–257.
101
[211] Mitiyasu Ohnaka and Lin-feng Shen. “Scaling of the shear rupture process from nucleation to
dynamic propagation: Implications of geometric irregularity of the rupturing surfaces”. In:
Journal of Geophysical Research: Solid Earth 104.B1 (1999), pp. 817–844.
[212] Y. Okada. “Surface deformation due to shear and tensile faults in a half-space”. In: Bull. Seism. Soc.
Am. 75.4 (Aug. 1985), pp. 1135–1154.
[213] Keishi Okazaki and Ikuo Katayama. “Slow stick slip of antigorite serpentinite under hydrothermal
conditions as a possible mechanism for slow earthquakes”. In: Geophys. Res. Lett. 42.4 (2015),
pp. 1099–1104. doi: 10.1002/2014GL062735.
[214] Keishi Okazaki, Ikuo Katayama, and Miki Takahashi. “Effect of pore fluid pressure on the
frictional strength of antigorite serpentinite”. In: Tectonophysics 583 (2013), pp. 49–53.
[215] P. G. Okubo. “Dynamic rupture modeling with laboratory-derived constitutive relations”. In: J.
Geophys. Res. 94 (1989), pp. 12321–12335. doi: 10.1029/JB094iB09p12321.
[216] Hanaya Okuda, André R Niemeijer, Miki Takahashi, Asuka Yamaguchi, and Christopher J Spiers.
“Hydrothermal friction experiments on simulated basaltic fault gouge and implications for
megathrust earthquakes”. In: J. Geophys. Res. 128.1 (2023), e2022JB025072. doi:
10.1029/2022JB025072.
[217] Fusakichi Omori. “On the after-shocks of earthquakes”. In: J. Coll. Sci., Imp. Univ., Japan 7 (1894),
pp. 111–200.
[218] S. Q. Ong, Sylvain Barbot, and Judith Hubbard. “Physics-based scenario of earthquake cycles on
the Ventura Thrust system, California: the effect of variable friction and fault geometry”. In: Pure
Appl. Geophys. (2019). doi: 10.1007/s00024-019-02111-9.
[219] A Arda Ozacar and George Zandt. “Crustal structure and seismic anisotropy near the San Andreas
Fault at Parkfield, California”. In: Geophysical Journal International 178.2 (2009), pp. 1098–1104.
[220] Jason R Patton, Chris Goldfinger, Ann E Morey, Ken Ikehara, Chris Romsos, Joseph Stoner,
Yusuf Djadjadihardja, Sri Ardhyastuti, Eddy Zulkarnaen Gaffar, and Alexis Vizcaino. “A 6600 year
earthquake history in the region of the 2004 Sumatra-Andaman subduction zone earthquake”. In:
Geosphere 11.6 (2015), pp. 2067–2129. doi: 10.1130/GES01066.1.
[221] Christian Pelties, Yihe Huang, and Jean-Paul Ampuero. “Pulse-like rupture induced by
three-dimensional fault zone flower structures”. In: Pure Appl. Geophys. 172.5 (2015),
pp. 1229–1241. doi: 10.1007/s00024-014-0881-0.
[222] Yajun Peng and Allan M Rubin. “High-resolution images of tremor migrations beneath the
Olympic Peninsula from stacked array of arrays seismic data”. In: Geochemistry, Geophysics,
Geosystems 17.2 (2016), pp. 587–601. doi: 10.1002/2015GC006141.
[223] Yajun Peng and Allan M Rubin. “Intermittent tremor migrations beneath Guerrero, Mexico, and
implications for fault healing within the slow slip zone”. In: Geophys. Res. Lett. 44.2 (2017),
pp. 760–770. doi: 10.1002/2016GL071614.
102
[224] Yajun Peng, Allan M Rubin, Michael G Bostock, and John G Armbruster. “High-resolution
imaging of rapid tremor migrations beneath southern Vancouver Island using cross-station cross
correlations”. In: J. Geophys. Res. 120.6 (2015), pp. 4317–4332.
[225] Zhigang Peng and Joan Gomberg. “An integrated perspective of the continuum between
earthquakes and slow-slip phenomena”. In: Nature Geoscience 3.9 (2010), p. 599. doi:
10.1038/ngeo940.
[226] H Perfettini and J-P Avouac. “Postseismic relaxation driven by brittle creep: A possible
mechanism to reconcile geodetic measurements and the decay rate of aftershocks, application to
the Chi-Chi earthquake, Taiwan”. In: Journal of Geophysical Research: Solid Earth 109.B2 (2004).
[227] Belle Philibosian and Aron J Meltzner. “Segmentation and supercycles: A catalog of earthquake
rupture patterns from the Sumatran Sunda Megathrust and other well-studied faults worldwide”.
In: Quaternary Science Reviews 241 (2020), p. 106390. doi: 10.1016/j.quascirev.2020.106390.
[228] Belle Philibosian, Kerry Sieh, Jean-Philippe Avouac, Danny H Natawidjaja, Hong-Wei Chiang,
Chung-Che Wu, Chuan-Chou Shen, Mudrik R Daryono, Hugo Perfettini, Bambang W Suwargadi,
et al. “Earthquake supercycles on the Mentawai segment of the Sunda megathrust in the
seventeenth century and earlier”. In: J. Geophys. Res. 122.1 (2017), pp. 642–676. doi:
10.1002/2016JB013560.
[229] John P Platt, Haoran Xia, and William Lamborn Schmidt. “Rheology and stress in subduction
zones around the aseismic/seismic transition”. In: Progress in Earth and Planetary Science 5.1
(2018), p. 24.
[230] Jose Pujol. Elastic wave propagation and generation in seismology. Cambridge University Press,
2003.
[231] A Putnis and G Mauthe. “The effect of pore size on cementation in porous rocks”. In: Geofluids 1.1
(2001), pp. 37–41.
[232] Q. Qiu, E. M Hill, S. Barbot, J. Hubbard, W. Feng, E. O Lindsey, L. Feng, K. Dai, S. V Samsonov,
and P. Tapponnier. “The mechanism of partial rupture of a locked megathrust: The role of fault
morphology”. In: Geology 44.10 (2016), pp. 875–878. doi: 10.1130/G38178.1.
[233] Qiang Qiu, S. Barbot, Teng Wang, and Shengji Wei. “Slip complementarity and triggering
between the foreshock, mainshock, and afterslip of the 2019 Ridgecrest rupture sequence”. In:
Bull. Seism. Soc. Am. (2020). doi: 10.1785/0120200037.
[234] Théa Ragon, Mark Simons, Quentin Bletery, Olivier Cavalié, and Eric Fielding. “A stochastic view
of the 2020 Elazığ Mw 6.8 earthquake (Turkey)”. In: Geophys. Res. Lett. 48.3 (2021),
e2020GL090704. doi: 10.1029/2020GL090704.
[235] Yongkang Ran, Xiwei Xu, Hu Wang, Wenshan Chen, Lichun Chen, Mingjian Liang, Huili Yang,
Yanbao Li, and Huaguo Liu. “Evidence of characteristic earthquakes on thrust faults from
paleo-rupture behavior along the Longmenshan fault system”. In: Tectonics 38.7 (2019),
pp. 2401–2410.
103
[236] Andrew P Rathbun and Chris Marone. “Symmetry and the critical slip distance in rate and state
friction laws”. In: J. Geophys. Res. 118.7 (2013), pp. 3728–3741. doi: 10.1002/jgrb.50224.
[237] Harry Fielding Reid. “The mechanics of the earthquake”. In: The California Earthquake of April 18,
1906, Report of the State Earthquake Investigation Commission (1910).
[238] J. R. Rice. “Constitutive relations for fault slip and earthquake instabilities”. In: Pure Appl.
Geophys. 21 (1983), pp. 443–475. doi: 10.1007/978-3-0348-6608-8\_7.
[239] J. R. Rice and A. L. Ruina. “Stability of steady frictional slipping”. In: J. Appl. Mech. 50 (1983),
pp. 343–349. doi: 10.1115/1.3167042.
[240] James R Rice and Andy L Ruina. “Stability of steady frictional slipping”. In: J. Appl. Mech. 50
(1983), pp. 343–349. doi: 10.1115/1.3167042.
[241] James Donald Rimstidt and HL Barnes. “The kinetics of silica-water reactions”. In: Geochimica et
Cosmochimica Acta 44.11 (1980), pp. 1683–1699. doi: 10.1016/0016-7037(80)90220-3.
[242] Thomas K Rockwell, Timothy E Dawson, Jeri Young Ben-Horin, and Gordon Seitz. “A 21-event,
4,000-year history of surface ruptures in the Anza seismic gap, San Jacinto Fault, and implications
for long-term earthquake production on a major plate boundary fault”. In: Pure and Applied
Geophysics 172.5 (2015), pp. 1143–1165.
[243] Garry Rogers and Herb Dragert. “Episodic tremor and slip on the Cascadia subduction zone: The
chatter of silent slip”. In: Science 300.5627 (2003), pp. 1942–1943.
[244] Pierre Romanet, Harsha S Bhat, Romain Jolivet, and Raúl Madariaga. “Fast and slow slip events
emerge due to fault geometrical complexity”. In: Geophysical Research Letters 45.10 (2018),
pp. 4809–4819.
[245] Zachary E Ross, Daniel T Trugman, Kamyar Azizzadenesheli, and Anima Anandkumar.
“Directivity modes of earthquake populations with unsupervised learning”. In: J. Geophys. Res.
125.2 (2020), e2019JB018299. doi: 10.1029/2019JB018299.
[246] Baptiste Rousset, Roland Bürgmann, and Michel Campillo. “Slow slip events in the roots of the
San Andreas fault”. In: Science advances 5.2 (2019), eaav3274.
[247] Baptiste Rousset, Yuning Fu, Noel Bartlow, and Roland Bürgmann. “Week-long and year-long
slow slip and tectonic tremor episodes on the South Central Alaska Megathrust”. In: J. Geophys.
Res. 122 (2019), pp. 13–392. doi: 10.1029/2019JB018724.
[248] A. M. Rubin. “Designer friction laws for bimodal slip propagation speeds”. In: Geochemistry,
Geophysics, Geosystems 12.Q04007 (2011). doi: 10.1029/2010GC003386.
[249] A. M. Rubin. “Episodic slow slip events and rate-and-state friction”. In: J. Geophys. Res.
113.B11414 (2008). doi: 10.1029/2008JB005642.
104
[250] Allan M Rubin and John G Armbruster. “Imaging slow slip fronts in Cascadia with high precision
cross-station tremor locations”. In: Geochemistry, Geophysics, Geosystems 14.12 (2013),
pp. 5371–5392. doi: 10.1002/2013GC005031.
[251] Charles M Rubin, Benjamin P Horton, Kerry Sieh, Jessica E Pilarczyk, Patrick Daly, Nazli Ismail,
and Andrew C Parnell. “Highly variable recurrence of tsunamis in the 7,400 years before the 2004
Indian Ocean tsunami”. In: Nature Communications 8 (2017), p. 16019.
[252] A. Ruina. “Slip instability and state variable friction laws”. In: J. Geophys. Res. 88 (1983), pp. 10,
359–10, 370. doi: 10.1029/JB088iB12p10359.
[253] Andy Ruina. “Slip instability and state variable friction laws”. In: J. Geophys. Res. 88.B12 (1983),
pp. 10359–10370.
[254] EH Rutter and KH Brodie. “Experimental grain size-sensitive flow of hot-pressed Brazilian quartz
aggregates”. In: Journal of Structural Geology 26.11 (2004), pp. 2011–2023. doi:
10.1016/j.jsg.2004.04.006.
[255] EH Rutter and KH Brodie. “Experimental intracrystalline plastic flow in hot-pressed synthetic
quartzite prepared from Brazilian quartz crystals”. In: Journal of Structural Geology 26.2 (2004),
pp. 259–270. doi: 10.1016/S0191-8141(03)00096-8.
[256] E Rybacki and G Dresen. “Dislocation and diffusion creep of synthetic anorthite aggregates”. In: J.
Geophys. Res. 105.B11 (2000), pp. 26017–26036.
[257] Demian M Saffer and Chris Marone. “Comparison of smectite-and illite-rich gouge frictional
properties: application to the updip limit of the seismogenic zone along subduction megathrusts”.
In: Earth Planet. Sci. Lett. 215.1-2 (2003), pp. 219–235. doi: 10.1016/S0012-821X(03)00424-2.
[258] SN Sapkota, L Bollinger, Y Klinger, P Tapponnier, Y Gaudemer, and D Tiwari. “Primary surface
ruptures of the great Himalayan earthquakes in 1934 and 1255”. In: Nature Geoscience 6.1 (2013),
pp. 71–76.
[259] S. Sathiakumar and S. Barbot. “The stop-start control of seismicity by fault bends along the Main
Himalayan Thrust”. In: Communications Earth & Environment 2.1 (2021), pp. 1–11. doi:
10.1038/s43247-021-00153-3.
[260] Yuki Sawai. “Subduction zone paleoseismology along the Pacific coast of northeast
Japan—progress and remaining problems”. In: Earth-Science Reviews (2020), p. 103261. doi:
10.1016/j.earscirev.2020.103261.
[261] Katherine M Scharer and Doug Yule. “A maximum rupture model for the Southern San Andreas
and San Jacinto faults, California, derived from paleoseismic earthquake ages: Observations and
limitations”. In: Geophys. Res. Lett. 47.15 (2020), e2020GL088532.
[262] DA Schmidt and H Gao. “Source parameters and time-dependent slip distributions of slow slip
events on the Cascadia subduction zone from 1998 to 2008”. In: J. Geophys. Res. 115.B4 (2010). doi:
10.1029/2008JB006045.
105
[263] C. H. Scholz. “Earthquakes and friction laws”. In: Nature 391 (Jan. 1998), pp. 37–42. doi:
10.1038/34097.
[264] Chelsea Scott, Michael Bunds, Manoochehr Shirzaei, and Nathan Toke. “Creep along the Central
San Andreas Fault from surface fractures, topographic differencing, and InSAR”. In: Journal of
Geophysical Research: Solid Earth 125.10 (2020), e2020JB019762.
[265] Paul Segall, Allan M Rubin, Andrew M Bradley, and James R Rice. “Dilatant strengthening as a
mechanism for slow slip events”. In: Journal of Geophysical Research: Solid Earth 115.B12 (2010).
[266] Nikolai M Shapiro, Michel Campillo, Laurent Stehly, and Michael H Ritzwoller. “High-resolution
surface-wave tomography from ambient seismic noise”. In: Science 307.5715 (2005), pp. 1615–1618.
[267] William T Shea and Andreas K Kronenberg. “Rheology and deformation mechanisms of an
isotropic mica schist”. In: J. Geophys. Res. 97.B11 (1992), pp. 15201–15237. doi: 10.1029/92JB00620.
[268] D. R. Shelly and J. L. Hardebeck. “Precise tremor source locations and amplitude variations along
the lower-crustal central San Andreas Fault”. In: Geophys. Res. Lett. 37.L14301 (2010), p. 5. doi:
10.1029/2010GL043672.
[269] David R Shelly. “Migrating tremors illuminate complex deformation beneath the seismogenic San
Andreas fault”. In: Nature 463.7281 (2010), pp. 648–652.
[270] David R Shelly, Gregory C Beroza, and Satoshi Ide. “Complex evolution of transient slip derived
from precise tremor locations in western Shikoku, Japan”. In: Geochemistry, Geophysics,
Geosystems 8.10 (2007).
[271] David R Shelly, Gregory C Beroza, and Satoshi Ide. “Non-volcanic tremor and low-frequency
earthquake swarms”. In: Nature 446.7133 (2007), pp. 305–307.
[272] David R Shelly, Gregory C Beroza, Satoshi Ide, and Sho Nakamula. “Low-frequency earthquakes
in Shikoku, Japan, and their relationship to episodic tremor and slip”. In: Nature 442.7099 (2006),
pp. 188–191. doi: 10.1038/nature04931.
[273] Pengcheng Shi, Meng Wei, and Sylvain Barbot. “Contribution of viscoelastic stress to the
synchronization of earthquake cycles on oceanic transform faults”. In: J. Geophys. Res. 127.8
(2022), e2022JB024069. doi: 10.1029/2022JB024069.
[274] Qibin Shi, Sylvain Barbot, Shengji Wei, Paul Tapponnier, Takanori Matsuzawa, and
Bunichiro Shibazaki. “Structural control and system-level behavior of the seismic cycle at the
Nankai Trough”. In: Earth Planets Space 72.1 (2020), pp. 1–31. doi: 10.1186/s40623-020-1145-0.
[275] Bunichiro Shibazaki, Shuhui Bu, Takanori Matsuzawa, and Hitoshi Hirose. “Modeling the activity
of short-term slow slip events along deep subduction interfaces beneath Shikoku, southwest
Japan”. In: J. Geophys. Res. 115.B4 (2010). doi: 10.1029/2008JB006057.
[276] Bunichiro Shibazaki and Toshihiko Shimamoto. “Modelling of short-interval silent slip events in
deeper subduction interfaces considering the frictional properties at the unstable—Stable
transition regime”. In: Geophys. J. Int. 171.1 (2007), pp. 191–205.
106
[277] Kunihiko Shimazaki and Takashi Nakata. “Time-predictable recurrence model for large
earthquakes”. In: Geophys. Res. Lett. 7.4 (1980), pp. 279–282.
[278] Kazuya Shiraishi, Yasuhiro Yamada, Masaru Nakano, Masataka Kinoshita, and Gaku Kimura.
“Three-dimensional topographic relief of the oceanic crust may control the occurrence of shallow
very-low-frequency earthquakes in the Nankai Trough off Kumano”. In: Earth, Planets and Space
72 (2020), pp. 1–14.
[279] K. Sieh, D. H. Natawidjaja, A. J. Meltzner, C.-C. Shen, H. Cheng, K.-S. Li, B. W. Suwargadi,
J. Galetzka, B. Philibosian, and R. L. Edwards. “Earthquake supercycles inferred from sea-level
changes recorded in the corals of West Sumatra”. In: Science 322 (2008), pp. 1674–1678. doi:
10.1126/science.1163589.
[280] K. E. Sieh and R. H. Jahns. “Holocene activity of the San Andreas Fault at Wallace Creek,
California”. In: Geology 95.8 (1984), pp. 883–896. doi:
10.1130/0016-7606(1984)95<883:HAOTSA>2.0.CO;2.
[281] Kerry Sieh. “The repetition of large-earthquake ruptures”. In: Proceedings of the National Academy
of Sciences 93.9 (1996), pp. 3764–3771. doi: 10.1073/pnas.93.9.3764.
[282] Rob M Skarbek and Heather M Savage. “RSFit3000: A MATLAB GUI-based program for
determining rate and state frictional parameters from experimental data”. In: Geosphere 15.5
(2019), pp. 1665–1676.
[283] Robert M Skarbek, Alan W Rempel, and David A Schmidt. “Geologic heterogeneity can produce
aseismic slip transients”. In: Geophys. Res. Lett. 39.21 (2012). doi: 10.1029/2012GL053762.
[284] N H Sleep. “Grain size and chemical controls on the ductile properties of mostly frictional faults
at low-temperature hydrothermal conditions”. In: Pure Appl. Geophys. 143.1-3 (1994), pp. 41–60.
doi: 10.1007/BF00874323.
[285] N H Sleep. “Real contacts and evolution laws for rate and state friction”. In: Geochemistry,
Geophysics, Geosystems 7.8 (2006). doi: 10.1029/2005GC001187.
[286] Norman H Sleep. “Application of a unified rate and state friction theory to the mechanics of fault
zones with strain localization”. In: Journal of Geophysical Research: Solid Earth 102.B2 (1997),
pp. 2875–2895.
[287] Teh-Ru Alex Song, Donald V Helmberger, Michael R Brudzinski, Robert W Clayton, Paul Davis,
Xyoli Pérez-Campos, and Shri K Singh. “Subducting slab ultra-slow velocity layer coincident with
silent earthquakes in southern Mexico”. In: Science 324.5926 (2009), pp. 502–506.
[288] Lynn R Sykes and William Menke. “Repeat times of large earthquakes: Implications for
earthquake mechanics and long-term prediction”. In: Bull. Seism. Soc. Am. 96.5 (2006),
pp. 1569–1596. doi: 10.1785/0120050083.
[289] Ryota Takagi, Kazushige Obara, and Takuto Maeda. “Slow slip event within a gap between tremor
and locked zones in the Nankai subduction zone”. In: Geophys. Res. Lett. 43.3 (2016),
pp. 1066–1074. doi: 10.1002/2015GL066987.
107
[290] Telemaco Tesei, Cristiano Collettini, Massimiliano R Barchi, Brett M Carpenter, and
Giuseppe Di Stefano. “Heterogeneous strength and fault zone complexity of carbonate-bearing
thrusts with possible implications for seismicity”. In: Earth Planet. Sci. Lett. 408 (2014),
pp. 307–318. doi: 10.1016/j.epsl.2014.10.021.
[291] Prithvi Thakur, Yihe Huang, and Yoshihiro Kaneko. “Effects of low-velocity fault damage zones
on long-term earthquake behaviors on mature strike-slip faults”. In: J. Geophys. Res. 125.8 (2020),
e2020JB019587. doi: 10.1029/2020JB019587.
[292] C. Thurber, S. Roecker, H. Zhang, S. Baher, and W. Ellsworth. “Fine-scale structure of the San
Andreas fault zone and location of the SAFOD target earthquakes”. In: J. Geophys. Res. 31.L12S02
(2004), p. 4.
[293] Ping Tian and Changrong He. “Velocity weakening of simulated augite gouge at hydrothermal
conditions: Implications for frictional slip of pyroxene-bearing mafic lower crust”. In: J. Geophys.
Res. 124.7 (2019), pp. 6428–6451. doi: 10.1029/2018JB016456.
[294] Sarah J Titus, Charles DeMets, and Basil Tikoff. “Thirty-five-year creep rates for the creeping
segment of the San Andreas fault and the effects of the 2004 Parkfield earthquake: Constraints
from alignment arrays, continuous global positioning system, and creepmeters”. In: Bulletin of the
Seismological Society of America 96.4B (2006), S250–S268.
[295] Erin K Todd, Susan Y Schwartz, Kimihiro Mochizuki, Laura M Wallace, Anne F Sheehan,
Spahr C Webb, Charles A Williams, Jenny Nakai, Jefferson Yarce, Bill Fry, et al. “Earthquakes and
tremor linked to seamount subduction during shallow slow slip at the Hikurangi margin, New
Zealand”. In: J. Geophys. Res. 123.8 (2018), pp. 6769–6783. doi: 10.1029/2018JB016136.
[296] Xinyue Tong and Luc L Lavier. “Simulation of slip transients and earthquakes in finite thickness
shear zones with a plastic formulation”. In: Nature communications 9.1 (2018), pp. 1–8.
[297] Victor C Tsai and Greg Hirth. “Elastic impact consequences for high-frequency earthquake
ground motion”. In: Geophys. Res. Lett. 47.5 (2020), e2019GL086302. doi: 10.1029/2019GL086302.
[298] S. T. Tse and J. R. Rice. “Crustal earthquake instability in relation to the depth variation of
frictional slip properties”. In: J. Geophys. Res. 91.B9 (1986), pp. 9452–9472. doi:
10.1029/JB091iB09p09452.
[299] Simon T Tse and James R Rice. “Crustal earthquake instability in relation to the depth variation of
frictional slip properties”. In: J. Geophys. Res. 91.B9 (1986), pp. 9452–9472.
[300] Kazuko Usami, Ken Ikehara, Toshiya Kanamatsu, and Cecilia M McHugh. “Supercycle in great
earthquake recurrence along the Japan Trench over the last 4000 years”. In: Geoscience Letters 5.1
(2018), p. 11. doi: 10.1186/s40562-018-0110-2.
[301] Tokuji Utsu. “A statistical study on the occurrence of aftershocks”. In: Geophys. Mag. 30 (1961),
pp. 521–605.
108
[302] RD Valdez II, H Kitajima, and DM Saffer. “Effects of temperature on the frictional behavior of
material from the Alpine Fault Zone, New Zealand”. In: Tectonophysics 762 (2019), pp. 17–27. doi:
10.1016/j.tecto.2019.04.022.
[303] Deepa Mele Veedu and Sylvain Barbot. “The Parkfield tremors reveal slow and fast ruptures on
the same asperity”. In: Nature 532.7599 (2016), pp. 361–365.
[304] Deepa Mele Veedu, Carolina Giorgetti, Marco Scuderi, Sylvain Barbot, Chris Marone, and
Cristiano Collettini. “Bifurcations at the Stability Transition of Earthquake Faulting”. In: Geophys.
Res. Lett. 47.19 (2020), e2020GL087985. doi: 10.1029/2020GL087985.
[305] DM. Veedu and S. Barbot. “The Parkfield tremors reveal slow and fast ruptures on the same
asperity”. In: Nature 532.7599 (2016), pp. 361–365. doi: 10.1038/nature17190.
[306] John E Vidale and Yong-Gang Li. “Damage to the shallow Landers fault from the nearby Hector
Mine earthquake”. In: Nature 421.6922 (2003), pp. 524–526.
[307] Robert C Viesca. “Self-similar slip instability on interfaces with rate-and state-dependent
friction”. In: Proc. R. Soc. A 472.2192 (2016), p. 20160254.
[308] Robert C Viesca. “Stable and unstable development of an interfacial sliding instability”. In:
Physical Review E 93.6 (2016), p. 060202.
[309] Binhao Wang and Sylvain Barbot. “Pulse-like ruptures, seismic swarms, and tremorgenic
slow-slip events with thermally activated friction”. In: Earth Planet. Sci. Lett. 603 (2023), p. 117983.
doi: 10.1016/j.epsl.2022.117983.
[310] L Wang and S Barbot. “Excitation of San Andreas tremors by thermal instabilities below the
seismogenic zone”. In: Science Advances eabb2057 (2020). doi: 10.1126/sciadv.abb2057.
[311] Yongfei Wang and Steven M Day. “Seismic source spectral properties of crack-like and pulse-like
modes of dynamic rupture”. In: J. Geophys. Res. 122.8 (2017), pp. 6657–6684.
[312] Yongfei Wang, Steven M Day, and Marine A Denolle. “Geometric controls on pulse-like rupture in
a dynamic model of the 2015 Gorkha earthquake”. In: J. Geophys. Res. 124.2 (2019), pp. 1544–1568.
[313] AG Wech, CM Boese, TA Stern, and J Townend. “Tectonic tremor and deep slow slip on the
Alpine Fault”. In: Geophysical Research Letters 39.10 (2012).
[314] M. Wei, Y. Kaneko, Y. Liu, and J. J McGuire. “Episodic fault creep events in California controlled
by shallow frictional heterogeneity”. In: Nature geoscience 6.7 (2013), p. 566.
[315] Ray Weldon, Katherine Scharer, Thomas Fumal, and Glenn Biasi. “Wrightwood and the
earthquake cycle: What a long recurrence record tells us about how faults work”. In: GSA today
14.9 (2004), pp. 4–10. doi: 10.1130/1052-5173(2004)014<4:WATECW>2.0.CO;2.
[316] Steven G Wesnousky. “Predicting the endpoints of earthquake ruptures”. In: Nature 444.7117
(2006), pp. 358–360. doi: 10.1038/nature05275.
109
[317] Steven G Wesnousky. “Seismological and structural evolution of strike-slip faults”. In: Nature
335.6188 (1988), pp. 340–343. doi: 10.1038/335340a0.
[318] Yan Wu and Xiaofei Chen. “The scale-dependent slip pattern for a uniform fault model obeying
the rate-and state-dependent friction law”. In: J. Geophys. Res. 119.6 (2014), pp. 4890–4906. doi:
10.1002/2013JB010779.
[319] Suguru Yabe and Satoshi Ide. “Variations in precursory slip behavior resulting from frictional
heterogeneity”. In: Progress in Earth and Planetary Science 5.1 (2018), pp. 1–11. doi:
10.1186/s40645-018-0201-x.
[320] Yusuke Yokota, Tadashi Ishikawa, Shun-ichi Watanabe, Toshiharu Tashiro, and Akira Asada.
“Seafloor geodetic constraints on interplate coupling of the Nankai Trough megathrust zone”. In:
Nature 534.7607 (2016), pp. 374–377.
[321] Robert R Youngs and Kevin J Coppersmith. “Implications of fault slip rates and earthquake
recurrence models to probabilistic seismic hazard estimates”. In: Bull. Seism. Soc. Am. 75.4 (1985),
pp. 939–964.
[322] Lei Zhang, Changrong He, Yajing Liu, and Jian Lin. “Frictional properties of the South China Sea
oceanic basalt and implications for strength of the Manila subduction seismogenic zone”. In:
Marine Geology 394 (2017), pp. 16–29. doi: 10.1016/j.margeo.2017.05.006.
[323] Dimitri Zigone, Diane Rivet, Mathilde Radiguet, Michel Campillo, Christophe Voisin,
Nathalie Cotte, Andrea Walpersdorf, Nikolai M Shapiro, Glenn Cougoulat, Philippe Roux, et al.
“Triggering of tremors and slow slip event in Guerrero, Mexico, by the 2010 Mw 8.8 Maule, Chile,
earthquake”. In: J. Geophys. Res. 117.B9 (2012). doi: 10.1029/2012JB009160.
[324] Mark Zoback, Stephen Hickman, and William Ellsworth. “Scientific drilling into the San Andreas
fault zone”. In: Eos, Transactions American Geophysical Union 91.22 (2010), pp. 197–199.
[325] Mark Zoback, Stephen Hickman, William Ellsworth, and the SAFOD Science Team. “Scientific
drilling into the San Andreas fault zone–an overview of SAFOD’s first five years”. In: Scientific
Drilling 11 (2011), pp. 14–28. doi: 10.2204/iodp.sd.11.02.2011.
110
Appendix A
Governing equations and non-dimensional parameters
The equations that govern the quasi-dynamic evolution of slip on the fault come about from the balance
between shear stress and frictional resistance. The stress evolution satisfies the constitutive relationship
of Equation (2.3), providing
τ˙
τ
=
a
µ0
V˙
V
+
b
µ0
˙θ
θ
, (A1)
where the terms are defined in the main text. The elastic interactions must conserve momentum, leading
to
τ˙ =
Z ∞
−∞
K(x3; y3) (V (y3) − VL) dy3 − Gcz
V˙
2Vs
, (A2)
where VL is the loading rate. For convenience, we operate the change of variable Ω = V θ/L, so that the
stress evolution satisfies
τ˙
τ
=
a − b
µ0
V˙
V
+
b
µ0
Ω˙
Ω
, (A3)
making the steady-state velocity dependence on a − b appearing clearly. This gives rise to the governing
equation
(a − b)¯σ
µ
µ0
V˙
V
+ bσ¯
µ
µ0
Ω˙
Ω
= K∗(V − VL) −
G
2Vs
V˙
(A4)
111
where ∗ is the convolution operator, coupled to the evolution law
Ω =˙
V
L
(1 − Ω) + V˙
V
Ω . (A5)
The stress interaction kernel K(x3, y3) describes the shear stress on the fault plane at depth x3 due to a
line dislocation at depth y3 in the presence of a compliant fault zone of total thickness T and rigidity Gcz
surrounded by country rocks of rigidity G, and is given in conditions of anti-plane strain by the infinite
series
K(x3; y3; G, Gcz, T) =
Gcz
2π
1
(x3 − y3)
2
− 2
X∞
m=1
G − Gcz
G + Gcz m (x3 − y3)
2 − (mT)
2
[(x3 − y3)
2 + (mT)
2]
2
.
(A6)
The non-dimensional parameters that control the rupture style emerge from a dimensional analysis of the
governing equations (A4) to (A6). We scale the coordinate system by the width of the seismogenic zone
and the time scale with the characteristic time of weakening, giving rise to
x3 = W x′
3
; y3 = W y′
3
V = VLV
′
;
∂
∂t =
VL
L
∂
∂t′
.
(A7)
The stress kernel has dimensions scaling with rigidity over square-meters, so we define the dimensionless
stress kernel as
K = −
Gcz
W2 K′
, (A8)
112
where the minus sign indicates that fault slip is associated with a local reduction of stress. This leads to
the following expression for the dimensionless stress interaction kernel
K′
x
′
3
, y′
3
;
G
Gcz
,
T
W
=
1
2π
1
(x
′
3 − y
′
3
)
2
− 2
X∞
m=1
G/Gcz − 1
G/Gcz + 1m (x
′
3 − y
′
3
)
2 − (mT /W)
2
[(x
′
3 − y
′
3
)
2 + (mT /W)
2]
2
.
(A9)
With this change of dynamic variables, the governing equation becomes
V˙ ′
V ′
=
1
Rh
u
Gcz
G
µ0
µ
K′
∗(V
′ − 1) + 1
Rb
Ω˙
Ω
+
µ0
µ
V˙ ′
V
′
th
(A10)
coupled to an evolution law that contains no physical parameters,
Ω =˙ V
′
(1 − Ω) + V˙ ′
V ′
Ω , (A11)
where the time derivative is with respect to non-dimensional time t
′
. This derivation highlights the controlling non-dimensional parameters. The Dieterich-Ruina-Rice number for a homogeneous medium is
given by
R
h
u =
W
L
(b − a)¯σ
G
, (A12)
and controls the elastic interactions. The other non-dimensional number
Rb =
b − a
b
, (A13)
controls the degree of weakening. The threshold velocity for radiation damping is given by
Vth = VLV
′
th = 2Vs
(b − a)¯σ
G
. (A14)
113
The non-dimensional parameters of Equations (A12-A14) appear for a homogeneous elastic domain [23].
The presence of a compliant zone adds two more non-dimensional parameters: the ratio of rigidity G/Gcz
and the aspect ratio T /W, both affecting some aspects of the elastic interactions. This justifies our exploration of these ratios in Figures 2.2-10. As we show in the following Appendix, the effect of the compliance
ratio and the fault zone thickness can be incorporated into a modified Dieterich-Ruina-Rice number that
accounts for the fault zone structure, reducing the number of controlling variables accordingly.
114
Appendix B
Dieterich-Ruina-Rice number for a compliant fault zone
We seek a representation of the Dieterich-Ruina-Rice number that incorporates the effect of the fault zone
structure. As the compliant zone thickness and rigidity intervene in the elastic interactions through a
convolution, it is useful to consider the corresponding term in the Fourier domain. Using the convolution
theorem of the Fourier transform, the stress evolution in the Fourier domain is given by
˙τˆ = Kˆ (Vˆ − Vˆ
L) − Gcz
˙
Vˆ
2Vs
, (A1)
where a hat represents the Fourier transform of the respective field. The stress interaction kernel has the
closed-form solution [130]
Kˆ (k3) = −Gczπk3 coth
π T k3 + arctanhGcz
G
(A2)
for a compliant zone of total thickness T and rigidity Gcz surrounded by country rocks of rigidity G, where
k3 is the wavenumber associated with the vertical axis. Comparing with the analytic solution for the case
of a homogeneous medium
Hˆ (k3) = −Gπk3 (A3)
115
underlines the wavelength dependence of the effective rigidity
Gˆ(k3) = Gcz coth
π T k3 + arctanh(Gcz/G)
, (A4)
such that Kˆ (k3) = −Gˆ(k3)πk3. The effective rigidity varies smoothly between the rigidity of the compliant zone and of the country rocks
lim
k3→0
Gˆ = G
lim
k3→∞
Gˆ = Gcz
(A5)
as long-wavelength and short-wavelength asymptotic solutions, respectively. The Dieterich-Ruina-Rice
number is defined as the size of the velocity-weakening region scaled by a characteristic nucleation size
as Ru = W/h∗
. We then evaluate the effective rigidity at the wavelength of nucleation [138]
Gˆ∗ = Gˆ(1/h∗
)
= Gcz coth
πT /h∗ + arctanhGcz
G
= Gcz coth
πRu
T
W
+ arctanhGcz
G
(A6)
Finally, we redefine the Dieterich-Ruina-Rice number with the effective rigidity
Ru =
W
L
(b − a)¯σ
G∗
. (A7)
This provides the transcendental equation for the Ru number used in Equation (2.8)
Ru coth
πRu
T
W
+ arctanhGcz
G
=
W
L
(b − a)¯σ
Gcz
, (A8)
116
which is affected by the compliance ratio and the relative thickness of the compliant zone. To better explain
our numerical findings in Figure 2.2, we substitute the constant π in Equation (A8) for a parameter λ = π/4
that better captures the observed nucleation size of earthquake ruptures.
117
Appendix C
Formulations based on the age of contact
In this Appendix, we describe how the empirical evolution laws based on the age of contact can be modified
to enable a stability transition with increasing slip-rate. For simplicity, we discuss the constitutive behavior
in isobaric condition. The temperature effects are captured in the temperature dependence of Gi The
evolutionary effects of rate- and state-dependent friction can be described using the age contact [74, 75,
252]. The relationship between size and age of contact is well-defined when a single healing mechanism
operates [11]. However, with the competition of multiple healing mechanisms, the apparent age depends
on the healing rate of reference [18].
Taking arbitrarily the first healing mechanism as a reference and assuming two distinct healing mechanisms, we define the apparent age of contact as [11]
θ =
d
p1
G1
. (A1)
Defining the characteristic weakening distance as a fraction of the gouge thickness [11]
L =
2h
λp1
, (A2)
118
the evolution law of Equations (4.3) and (4.4) can be written as a function of the age of contact. Specifically,
Equation (4.3) becomes
˙θ = 1 + αθβ −
V θ
L
, (A3)
and Equation (4.4) becomes
˙θ = −
V θ
L
ln
V θ
L
1 + αθβ
. (A4)
The competition of two healing mechanisms introduces a new term associated with the parameters
α =
G2
rGr
1
(A5)
and
β = 1 − r (A6)
that depend on the ratio of the micro-asperity size power exponents r = p2/p1. If the second healing
mechanism can be neglected, with G2 = 0, then α = 0 and Equations (A3) and (A4) reduce to the aging
law and the slip law defined by [252], respectively. The correspondence between the apparent age and
size of contact indicates how the empirical evolution laws can be modified to capture the temperature
and velocity dependence of the stability regime of gouge friction. Although the formulations based on
the age of contact are mathematically adequate, they are physically inconsistent due to the ambiguity of
the reference healing rate. When multiple healing mechanisms operate, using the size of contact is more
appropriate.
119
Abstract (if available)
Abstract
Faults display a broad spectrum of behaviors in terms of their rupture styles and recurrence patterns, varying from aseismic creep to chaotic earthquake sequences. However, the underlying mechanisms and potential links behind these phenomena are poorly understood. In this study, I consider the frictional properties as essential ingredients to understand the variation in seismic observations. I utilize a physics-based friction constitutive law that considers the thermally activated healing mechanisms of the micro-asperities. I start by exploring the relationship between rupture styles and their recurrence patterns within the seismogenic zone. I use quasi-dynamic simulations of faults embedded in a compliant zone and explore a broad spectrum of friction properties under isothermal conditions, assuming a single healing mechanism. The research demonstrates that as the fault zone rigidity or characteristic weakening distance decreases, the rupture style shifts from crack-like to pulse-like. Concurrently, the recurrence behavior transitions from slip- and time-predictable sequences to chaotic sequences characterized by varying sizes of ruptures and aftershock sequences. In this research, I also suggest that the effect of the fault zone can be represented by a single non-dimensional number, allowing for comparable outcomes in quasi-dynamic simulations regardless of whether the fault zone is considered. I then extend the research to slow-slip events and tremors that take place downdip of the seismogenic zones, also using the quasi-dynamic approach coupled with a similar physical-based friction law. This study considers both the characteristic weakening distance and the friction dependence conditions to create a more comprehensive parametric space of friction parameters and associated rupture behaviors. I find that the simultaneous and co-located occurrence of slow- and fast-events is a natural outcome under a velocity-neutral condition with a small characteristic nucleation size. This condition aligns with the frictional conditions of the fault beneath the seismogenic zone, which acts as a transition from velocity-weakening to velocity-strengthening. Moreover, tremors are a burst of slow earthquakes initiated when slow-slip events interact with small-scale structural asperities. In the final part of the study, I focus on the findings that frictional parameters vary with temperature and velocity, which goes beyond the scope of the condition of isothermal and a single healing mechanism. I utilize the physical-based friction constitutive law that incorporates the competition of healing mechanisms dominant in various temperature and velocity regimes. This model explains the friction properties of a range of natural gouges from the San Andreas Fault, Alpine Fault, and the Nankai Trough. The friction framework has the potential for further understanding a broader range of fault dynamics in future studies.
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From laboratory friction to numerical models of fault dynamics
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