On the Choice and Implications of Rheologies That Maintain Kinematic and Dynamic Consistency Over the Entire Earthquake Cycle

Abstract

Viscoelastic processes in the upper mantle redistribute seismically generated stresses and modulate crustal deformation throughout the earthquake cycle. Geodetic observations of these motions at the surface of the crust‐mantle system offer the possibility of constraining the rheology of the upper mantle. Parsimonious representations of viscoelastically modulated deformation through the aseismic phase of the earthquake cycle should simultaneously explain geodetic observations of (a) rapid postseismic deformation, (b) late in the earthquake cycle near‐fault strain localization. To understand how rheological formulations affect kinematics, we compare predictions from time‐dependent forward models of deformation over the entire earthquake cycle for an idealized vertical strike‐slip fault in a homogeneous elastic crust underlain by a homogeneous viscoelastic upper‐mantle. We explore three different rheologies as inferred from laboratory experiments: (a) linear Maxwell, (b) linear Burgers, (c) power‐law. The linear Burgers and power‐law rheologies are consistent with fast and slow deformation phenomenology over the entire earthquake cycle, while the single‐layer linear Maxwell model is not. The kinematic similarity of linear Burgers and power‐law models suggests that geodetic observations alone may be insufficient to distinguish between them, but indicate that one may serve as an effective proxy for the other. However, the power‐law rheology model displays a postseismic response that is non‐linearly dependent on earthquake magnitude, which may offer a partial explanation for observations of limited postseismic deformation near some magnitude 6.5–7.0 earthquakes. We discuss the role of mechanical coupling between frictional slip and viscous creep in controlling the time‐dependence of regional stress transfer following large earthquakes and how this may affect the seismic hazard and risk to communities living close to fault networks.

Full text

1. Introduction
Inferring the constitutive relations that describe how the macroscopic stress state of the lithosphere-asthenosphere
system evolves as a function of strain rate, total strain, and intensive system variables (temperature, pressure,
composition, etc.) remains a grand challenge in the geosciences (NSF,2020). Constraining these constitutive
relations, or rheology, is fundamental to our understanding of the dynamics of the solid Earth. From the occur-
rence of earthquakes and their effects at any point within the Earth, to the construction of the geological struc-
ture that surrounds us and the sustenance of plate tectonics itself, the rheology and strength of Earth materials
Abstract Viscoelastic processes in the upper mantle redistribute seismically generated stresses and
modulate crustal deformation throughout the earthquake cycle. Geodetic observations of these motions at
the surface of the crust-mantle system offer the possibility of constraining the rheology of the upper mantle.
Parsimonious representations of viscoelastically modulated deformation through the aseismic phase of the
earthquake cycle should simultaneously explain geodetic observations of (a) rapid postseismic deformation,
(b) late in the earthquake cycle near-fault strain localization. To understand how rheological formulations
affect kinematics, we compare predictions from time-dependent forward models of deformation over the
entire earthquake cycle for an idealized vertical strike-slip fault in a homogeneous elastic crust underlain by
a homogeneous viscoelastic upper-mantle. We explore three different rheologies as inferred from laboratory
experiments: (a) linear Maxwell, (b) linear Burgers, (c) power-law. The linear Burgers and power-law rheologies
are consistent with fast and slow deformation phenomenology over the entire earthquake cycle, while the
single-layer linear Maxwell model is not. The kinematic similarity of linear Burgers and power-law models
suggests that geodetic observations alone may be insufficient to distinguish between them, but indicate that one
may serve as an effective proxy for the other. However, the power-law rheology model displays a postseismic
response that is non-linearly dependent onearthquake magnitude, which may offer a partial explanation for
observations of limited postseismic deformation near some magnitude 6.5–7.0 earthquakes. We discuss the role
of mechanical coupling between frictional slip and viscous creep in controlling the time-dependence of regional
stress transfer following large earthquakes and how this may affect the seismic hazard and risk to communities
living close to fault networks.
Plain Language Summary The solid Earth is a viscoelastic material that displays both solid
and fluid-like behaviors depending on the observational time window and the applied stress. We develop
numerical simulations of how the uppermost solid Earth responds to a sequence of periodic earthquakes and
the earthquake cycle. Our simulations test a range of proposed viscoelastic materials. The predicted surface
displacements from each model are compared with observational features extracted from geodetic datasets
compiled over the past few decades. All existing viscoelastic material descriptions can satisfactorily explain
observational features in the first few years following an earthquake; significant differences between the
viscoelastic models emerge 10–100years following a large earthquake. Identifying the most appropriate
viscoelastic description requires the integration of geodetic data that constrains the velocity evolution from
a sequence of earthquakes (as opposed to a single event) with observations from rock physics laboratory
experiments. A unified description of visceolasticity in the uppermost solid earth has important implications for
understanding stress evolution in fault networks, and improving models of seismic hazard.
MALLICK ETAL.
© 2022. American Geophysical Union.
All Rights Reserved.
On the Choice and Implications of Rheologies That Maintain
Kinematic and Dynamic Consistency Over the Entire
Earthquake Cycle
Rishav Mallick1,2 , Valère Lambert3 , and Brendan Meade4
1Seismological Laboratory, California Institute of Technology, Pasadena, CA, USA, 2Earth Observatory of Singapore,
Nanyang Technological University, Singapore, Singapore, 3Seismological Laboratory, University of California, Santa Cruz,
NC, USA, 4Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA, USA
Key Points:
• We develop a kinematically consistent
periodic earthquake cycle model
with mechanical coupling between
frictional slip and viscous creep
• Steady-state power-law and linear
Burgers rheologies are consistent with
earthquake cycle observations while
linear Maxwell is not
• Distributed viscous creep alters the
(stress) loading rate of a large volume
around the fault and is important for
multifault interactions
Correspondence to:
R. Mallick,
rmallick@caltech.edu
Citation:
Mallick, R., Lambert, V., & Meade, B.
(2022). On the choice and implications
of rheologies that maintain kinematic
and dynamic consistency over the
entire earthquake cycle. Journal of
Geophysical Research: Solid Earth,
127, e2022JB024683. https://doi.
org/10.1029/2022JB024683
Received 28 APR 2022
Accepted 14 SEP 2022
Author Contributions:
Conceptualization: Rishav Mallick,
Valère Lambert, Brendan Meade
Methodology: Rishav Mallick, Valère
Lambert
Software: Rishav Mallick, Valère
Lambert
Validation: Rishav Mallick, Brendan
Meade
Writing – original draft: Rishav
Mallick, Valère Lambert, Brendan Meade
Writing – review & editing: Rishav
Mallick, Valère Lambert, Brendan Meade
10.1029/2022JB024683
RESEARCH ARTICLE
1 of 25

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plays a crucial role in defining these processes (Bürgmann and Dresen,2008; Mulyukova and Bercovici,2019).
However, inferring these constitutive relations at the kilometer scale of geological processes is a difficult task as
aspects of rock failure are shown to be scale-dependent (e.g., Lambert etal.,2021; Yamashita etal.,2015) and
there are limited opportunities to conduct experiments at the crustal or lithospheric scale. Our goal in this article
is to demonstrate that the earthquake cycle, in the vicinity of a mature strike-slip fault, may provide us with the
necessary experimental conditions to probe the rheology of the lithosphere.
While there exist a number of studies that have sought to infer rheological properties of Earth's
lithosphere-asthenosphere system using observations from the earthquake cycle (Bürgmann and Dresen,2008,
and references therein), the interpretation of results from different methodologies for extracting rheologi-
cal parameters can be limited or challenged by three key assumptions. First, a common approach to modeling
geophysical systems is to prescribe a functional form of the rheological model a priori and then estimate the
associated best-fitting set of rheological parameters for that selected model, potentially with limited consideration
of alternative rheological models that may be equally or better supported by the observations. Second, studies are
often limited to a specific observational time window, such as a few years following an earthquake, from which
the aforementioned best-fit model parameters are estimated. As such, inferred parameters are tied to the obser-
vational window that is probed, which may in part explain vastly different rheological estimates determined for
studies of the lithosphere over different observational windows (e.g., Henriquet etal.,2019; Hussain etal.,2018;
Kaufmann & Amelung,2000; Larsen etal., 2005; Milne etal.,2001; Pollitz,2005, 2019; Ryder etal., 2007;
Tamisiea etal.,2007). Finally, a common assumption when processing observed time series is that the signal can
be well-separated into a set of linearly superimposed functions, thereby neglecting nonlinear interactions among
the associated physical processes.
In this work, we seek to develop a framework that overcomes some of these limitations and can reconcile rheo-
logical inferences from different observational windows. As a starting point, we focus on major observational
features in geodetic time series obtained from mature strike-slip fault settings globally, from immediately follow-
ing earthquakes (postseismic period) to late in the earthquake cycle (interseismic period). We do not attempt to
directly optimize the fit to data, rather we consider the generality and descriptive power of popular rheological
models of the lithosphere and study where each model can explain major observational features or is insufficient
(Tarantola,2006). To assist the reader with appreciating the task at hand, we begin by providing some back-
ground on common rheological models that are used to describe lithospheric deformation, general observational
constraints available from geodesy and prevalent modeling strategies in the literature.
1.1. Elasticity, Friction and Viscous Creep
The rheology of the lithosphere does not appear to follow a single simple description at all timescales. For exam-
ple, observations of the passage of seismic waves and the static deformation of the Earth's lithosphere, in response
to an earthquake, allow us to describe the lithosphere as an elastic body over timescales ranging from seconds to a
day. However, the entire lithosphere cannot be elastic since an earthquake source is a frictional rupture restricted
to a narrow shear band (Kanamori & Brodsky,2004). At timescales longer than a day, time-dependent defor-
mation patterns of the solid Earth's surface following large earthquakes reveal the nonelastic nature of the litho-
sphere, that is, deformation that continues well after the initial source of deformation has ceased, and is thought to
result from a combination of two different processes: (a) time-dependent frictional slip on fault planes (afterslip)
while the surrounding medium is elastic (Marone etal.,1991), and (b) time-dependent distributed deformation
of the entire medium itself. This is commonly modeled as a viscoelastic process where the short timescale stress
perturbations are accommodated by the elasticity of the medium (ɛ∝σ), while relaxation following instantane-
ous stress steps or long timescale observations highlight the viscous properties of the medium, that is,
 𝐴 ∝
(ɛ—strain,
 𝐴
—strain rate, n—power exponent, σ—stress) (Hirth & Kohlstedt,2003). Laboratory experiments also
suggest that viscous flow laws exhibit unsteady or transient deformation, that is, the relationship between σ and
 𝐴
is unique once steady state is achieved, which requires a finite amount of strain or time (Post,1977). This style of
deformation is often modeled using a Burgers rheology (e.g., Hetland & Hager,2005; Müller,1986).
1.2. Geodetic Observations
In this study, we focus on mature strike-slip faults and simplify them to a two-dimensional geometry and describe
the characteristics of the interseismic and postseismic period as imaged by the past few decades of geodetic

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observational techniques; these are (a) the interseismic locking depth, (b) the postseismic relaxation time, and (c)
cumulative postseismic deformation over a given time window. These are key features that numerical models of
earthquake cycles along such faults attempt to explain.
1.2.1. Interseismic Observations
In between earthquakes, geodetic time series from most mature strike-slip fault settings appear nearly linear
in time, at least over available observational timescales (1–2decades), and the estimated velocities follow an
S-shaped function in space (Figure1), commonly modeled using the functional form

∞
tan−1
(


 )
(Savage &
Burford,1973) where v
∞ is the estimated long-term slip rate on the fault and Dlock is the depth to which the fault is
locked. The estimated locking depth from this kind of modeling is on the order of 10–20km, which is comparable
to the thickness of the lithosphere over which frictional processes are thought to be dominant (Vernant,2015).
Deviations from this expected behavior do appear in the data, such as nonlinearities in the time series and devia-
tions from the tan
−1 shape function, however, these differences are mostly due to localized creep episodes (in time
and space) or time-invariant creep on some sections of the fault (e.g., Khoshmanesh & Shirzaei,2018; Mallick
etal.,2021; Burgmann,2018).
1.2.2. Postseismic Observations
Following large earthquakes, time-dependent deformation occurs in the near-field as well as far away from
the fault. This time-dependent signal is typically decomposed into a linear term and a decaying curvature term
(Figure1). The linear term is assumed to represent background loading due to the motion of tectonic plates, as
discussed above. The curvature in the time series is typically fit with functional forms such as ∼log(t) and ∼e
−t,
motivated by spring-slider models of afterslip and creep of a linear viscoelastic material, respectively (Perfettini
& Avouac,2004). Poroelastic deformation can also contribute to postseismic deformation (Jónsson etal.,2003;
Peltzer etal.,1998), however, we ignore this process as we are limited to a two-dimensional anti-plane geometry
where no volumetric strains occur.
Figure 1. Schematic displacement and velocity evolution recorded at the Earth's surface over the entire earthquake cycle.
We show both (a) the spatial pattern (in colors varying from blue—early postseismic, to pink—late postseismic) and (b) the
temporal evolution at a chosen location (black lines). The geodetic predictions from steady rigid block motion is shown in
red-dashed lines, and deviations from this motion arise due to effects of the earthquake cycle.

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1.3. The Underlying Physical and Computational Problem
The goal is to infer the rheology of the fault and surrounding medium from the spatio-temporal pattern of surface
deformation that contain the features described in the previous section. Two predominant modeling strategies
are used for such studies—kinematic modeling of the deformation field and parameter estimation using dynamic
models.
1.3.1. Kinematic Models
Kinematic models use principles of linear elasticity to develop an impulse-response type relationship between
unit inelastic shear and displacements at the Earth's surface (e.g., Barbot,2018; Segall,2010). This set of linear
relationships is then used to construct a set of normal equations to estimate slip or strain distributions within the
discretized domain to explain the data. The results of such an exercise are estimates of the inelastic source defor-
mation (fault slip—Δs(t) and distributed strain—Δɛ(t)), which then may be combined with elastic stress compu-
tations to estimate the relationship between stress change and incremental slip/strain and other derived quantities.
1.3.2. Dynamic Models
Dynamic models typically perform physics-based simulations to solve for the stress (σ) and strain-rate evolution
( )
within the Earth's lithosphere consistent with quasistatic equilibrium:
∇
⋅
( )+f

=0
. fb is the equivalent
body force applied to the system, which could arise from gravity or imposed slip and tractions as a bound-
ary condition (e.g., Segall, 2010). To obtain a unique solution for each simulation, boundary conditions and
initial conditions need to be specified. Most simulations apply mixed boundary conditions along the edge of the
domain (e.g., Figure2a). However, the choice of initial conditions remains a difficult task. Many studies treat the
pre-earthquake strain rate as a free parameter that is also estimated as part of the inverse problem. The end goal is
to determine the coefficients relating σ and
 𝐴
; to do that, an optimization is performed such that the misfit between
predicted deformation and the observed deformation time series at sites on the Earth's surface is minimized.
1.3.3. Decomposing the Time Series
To simplify the inverse problem, many kinematic and dynamic modeling studies decompose the observed tectonic
deformation time series into additive contributions arising from (a) a constant in time but spatially variable
velocity field and (b) residual terms that are supposed to correspond to time-dependent postseismic deformation
(Figure 1). This simplification helps split the spatial domain of the problem into a computationally conven-
ient framework—by neglecting the spatially variable velocity field, post-earthquake relaxation studies need only
model inelastic deformation sources that satisfy a zero-displacement boundary condition; a condition that is
satisfied trivially for a finite deformation source. A point to note is that this linear decomposition of the time
series holds exactly for linear dynamical systems, but can be a source of error and bias if the rheology is nonlinear.
1.3.4. Viscoelastic Earthquake Cycle Models
To circumvent issues related to far-field boundary and initial conditions, as well as data decomposition, numer-
ical studies can focus on periodic earthquake cycles. These class of models have been developed in an effort to
predict and explain time-dependent earthquake cycle deformation consistent with not only a single earthquake,
but also the cumulative effects of periodic earthquake sequences integrated over time (across 10's or 100's of
earthquakes) to reach an approximately cycle invariant state.
Analytic and semi-analytic interseismic velocity models have been developed assuming linear viscoelastic rheol-
ogies in both the cases of a finite thickness faulted elastic layer over an unbounded viscoelastic region (Cohen
& Kramer,1984; Hetland & Hager,2005, 2006; Savage & Prescott,1978), depth-averaged rheology models
(Lehner & Li,1982; Li & Rice,1987; Spence & Turcotte,1979), as well as a thin viscoelastic channel (Cohen
& Kramer,1984). These models use linear Maxwell or Burger's rheologies (Hetland & Hager,2005) to describe
the viscoelastic medium and assume that earthquakes rupture the entire elastic layer. More recent studies account
for the mechanical coupling between afterslip and viscoelastic deformation. Since these models involve linear
rheologies, the effect of velocity boundary conditions is weak, and the inverse exercise simply involves fitting the
curvature in the data with an optimum value of the viscosity (or viscosities for a Burger's body) of the system.
An alternative approach is to incorporate rheological parameterizations based on laboratory experiments when
solving for equilibrium conditions. These laboratory-derived rheological models are typically determined from
studies of single crystal or polycrystal assemblages of minerals thought to be the dominant deforming phase

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in the crust (quartz) and mantle (olivine) (Hirth, 2002; Hirth & Kohlstedt,2003). These flow laws are then
evaluated at values determined from geological estimates of compositional and thermal variations within the
lithosphere to derive rock rheologies at the kilometer scale (Lyzenga etal.,1991; Reches etal.,1994; Takeuchi
& Fialko,2012, 2013). Recent numerical studies have incorporated viscoelastic deformation in simulations of
earthquake sequences along a strike-slip fault setting, providing a self-consistent framework that can reproduce
all aspects of the earthquake cycle, including spontaneous earthquake nucleation, propagation, and arrest (Allison
& Dunham,2017,2021; Lambert & Barbot,2016).
Figure 2. (a) Geometry of the numerical experiments. The domain of the stress calculations are separated into an elasto-frictional domain from 0 to 20km depth and a
viscoelastic domain from 20 to 50km depth. Shear resistance in the frictional domain is given by rate-state friction, while the viscoelastic domain is governed by either
a Maxwell rheology (the dashpot can be linear or power-law) or a linear Burgers rheology. (b) These rheologies are shown schematically. ηM—Maxwell viscosity, ηk—
Kelvin viscosity. (c) Long-term viscous strain rate

 ∞
12

2+

 ∞
13

2

as a function of the power exponent n.

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Both classes of numerical simulations pose their own challenges. Linear viscoelastic models are borne out of
computational simplicity and are able to fit many aspects of postseismic deformation, however, they predict late
interseismic locking depths that are significantly deeper than the brittle-ductile transition and are limited in their
ability to match observations (e.g., Takeuchi & Fialko,2012). Numerical simulations that make use of more
sophisticated laboratory-derived flow laws are more numerically challenging and computationally expensive
(e.g., Lambert & Barbot,2016). While they are able to better explain observations over the entire period between
earthquakes, their relatively high computational expense poses a challenge for coupling them into an obser-
vational data-driven optimization problem, limiting their current utility for exploring and identifying effective
constitutive relations of the lithosphere. Thus, there is need for a class of simulations that both satisfies the plate
motion-derived kinematic boundary conditions and enables efficient exploration of various rheological parame-
terizations in order to evaluate what constraints may be afforded from surface deformation data on the effective
rheology of the lithosphere.
1.4. Aim of This Study
In this article, we examine the use of earthquakes and the related cycle of loading and stress release, in an ideal-
ized two-dimensional strike-slip fault geometry, to study the rheological properties of the lithosphere. We develop
numerical models of periodic earthquake cycles that can handle all popularly employed rheological models,
satisfy the applied boundary conditions in the long-term (integrated over many earthquake cycles) as well as
mechanical equilibrium throughout the earthquake cycle, and still remain computationally inexpensive.
We qualitatively compare the predictions from our simulations with observations from strain-rate regimes that
are orders of magnitude apart, that is, the interseismic period
( 𝜀  ∞)
and the postseismic period
(
≥
10  ∞)
,
where
 𝐴 ∞
refers to the steady-state strain rate of the system or the strain rate averaged over geological timescales
(∼1Ma). We do not attempt to solve for a best-fit rheological description like one would in an inverse problem
sense. Instead, we show that linear viscoelastic rheologies need different parameters to explain the interseis-
mic and postseismic periods of the earthquake cycle, as can be modeled by a Burgers rheology (e.g., Hearn &
Thatcher,2015), while steady state power-law rheologies with power exponent n≥3 are able to simultaneously
explain the observed localization of strain preceding great earthquakes on mature faults, as well as the typical
curvature observed in postseismic deformation time series. Discriminating between Burgers and steady state
power-law rheological models using a single earthquake cycle may not be possible using available geodetic time
series. However, we discuss how this task may become significantly more feasible if we include observations over
sequences of earthquakes, particularly of different earthquake size.
2. Methods
Our numerical model is developed in an anti-plane geometry, that is, displacements are only in the out-of-plane
x1 direction, while displacement gradients exist in the x2×x3 plane. We consider a faulted elastic plate supported
by a visco-elastic substrate subject to imposed boundary conditions. The thickness of the elastic plate is DF, while
the viscous substrate extends from (DF, DF+DV). The elastic plate extends infinitely in the x2 direction, and the
viscous domain is chosen to be large enough to approximate this infinite x2 extent (Figure2a).
We first solve the viscous boundary-value problem for the long-term simulation and obtain the inelastic strain rate
and slip rate of the viscous medium and fault, respectively. We combine these long-term rates with an elasticity
kernel to formulate a set of Boundary Integral Equations to simulate the earthquake cycle (Mallick etal.,2021).
2.1. Long-Term Viscous Strain Rate
The governing equation for the viscous boundary-value problem is posed in terms of the scalar velocity field
v(x2, x3),
∇
2𝑣(𝑥2,𝑥
3)=−
(𝜕log 𝜂
𝜕𝑥2
𝜕𝑣 (𝑥
2
,𝑥
3
)
𝜕𝑥2
+
𝜕log 𝜂
𝜕𝑥3
𝜕𝑣 (𝑥
2
,𝑥
3
)
𝜕𝑥3)
(1)
where rheology of the substrate is described as follows,
1

=
(√
2
12 +2
13
)−1
;1= 𝜂 1=
(
1
2

)
(2)

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A is a rheological constant, n is the power in the power-law relation
 𝐴 =
, η is the viscosity, and the individual
stress components are σ1i.
2.1.1. Boundary Conditions and Solution
The boundary conditions on this system are as follows: traction-free at the base
(13 (3=F+V)=0
)
; lateral
edges are subject to anti-symmetric Dirichlet boundaries
(
(2→±∞)=±
∞
2)
; the entire fault slips uniformly
at v
∞ resulting in rigid block-like motion of the elastic layer
(0
≤
3
≤


)
.
There exist analytical solutions to this system, at least for spatially uniform values of A, n (Moore & Parsons,2015).
The viscous strain rates for a choice of power-law rheology only depend on n (Figure2c) and weakly depend
on the dimensions of the system. We present these solutions in terms of rescaled dimensions

′
2,
′
3
, where

′
3=

3
−
F

V
and

′
2=

2

V
. The domain for the solutions are
0
≤
′
3
≤
1,−
≤
′
2
≤

. We choose the aspect ratio
ω=10, which is sufficiently large such that there are negligible effects due to the location of the boundary on the
strain-rate tensor (Moore & Parsons,2015).

∞
12

∞=1
2+1

⎛
⎜
⎜
⎜
⎜
⎜
⎝
∞
∑
=1
cosh 
(
1−′
3
)
√
cosh 
√
cos ′
2
√
⎞
⎟
⎟
⎟
⎟
⎟
⎠

∞
13

∞=− 1
√
⎛
⎜
⎜
⎜
⎜
⎜
⎝
∞
∑
=1
sinh  (1−′
3)
√
cosh 
√
sin ′
2
√
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(3)
We remind the reader that
 𝐴
refers exclusively to the viscous component of the strain rate. The total strain rate,
which is a sum of the viscous and elastic components, is denoted as
 𝐴 total
=
𝐴
+
𝐴 elastic
.
2.2. Periodic Earthquake Cycle Simulations
The steady-state solutions for long-term viscous creep rate (Equation3, Figure2c) can be used to compute an
equivalent background stressing rate to load earthquake cycle simulations (Mallick etal., 2021). We note that
without the long-term strain rates, one would have to assign a spatially variable long-term slip rate and strain rate
to drive the earthquake cycle simulations (e.g., Lambert & Barbot,2016), but this would not necessarily satisfy
the boundary conditions of the system.
Using a background stressing rate that is kinematically and dynamically consistent with the long-term boundary
conditions, we transform the time-dependent partial differential equations for quasi-static equilibrium to a set of
coupled ordinary differential equations (e.g., Lambert & Barbot,2016; Mallick etal.,2021). Here we discuss the
procedure in brief; we discretize the nonelastically deforming part of the domain using constant-slip boundary
elements for faults and constant-strain boundary elements for viscous shear. These boundary elements along with
Equation3 can be used to compute the long-term loading rate of the system as follows,
⎡
⎢
⎢
⎢
⎢
⎣
 ∞

 ∞
12
 ∞
13
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
 𝐹 𝐹12 𝐹13
12𝐹 12𝐹12 12𝐹13
13𝐹 13𝐹12 13𝐹13
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
−∞
− ∞
12
− ∞
13
⎤
⎥
⎥
⎥
⎥
⎦
(4)
Ka,b is a stress-interaction kernel or the boundary-element approximation of the Green's function tensor that
describes the elastic stress transfer to any given element a in response to inelastic shear (slip on faults and strain
in shear zones) on the considered element b (Barbot,2018).
Deviations from the long-term loading rate (Equation4) drive frictional slip and viscous shear within the compu-
tational domain over the earthquake cycle. The set of coupled ordinary differential equations we need to solve is
therefore the instantaneous momentum balance for each boundary element (e.g., Mallick etal.,2021). To do this,

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we account for the full elastic interaction between each point on the fault and
in the viscous shear zones using the above described stress interaction kernel.
⎡
⎢
⎢
⎢
⎢
⎣
 𝐹 𝐹12 𝐹13
12𝐹 12𝐹12 12𝐹13
13𝐹 13𝐹12 13𝐹13
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
−∞
 12 − ∞
12
 13 − ∞
13
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
dfriction
d
 12 d
d+d 12
d
 13 d
d
+d 13
d
⎤
⎥
⎥
⎥
⎥
⎦
(5)
The left-hand side of this set of equations is the stressing rate in the system
arising from elasticity while the right-hand side is the time derivative of the
shear resistance provided by the rheology of the fault zone and viscoelastic
medium. Details about the chosen rheologies are provided in the following
section.
2.2.1. Friction and Viscous Laws
Resistive strength evolution on the fault (Equation 5) is described by
rate-dependent friction (Marone etal.,1991), that is, the resistive strength of
the fault is given by fσn where f is the friction coefficient and σn is the effec-
tive normal strength on the fault, and reference values f0, v0.

friction =()=
(
0+(−)log 
0)


(6)
The values for each parameter is shown in Table1, and are only applicable
to the regions where postseismic creep can occur, that is, between 0–2 and
15–20km on the fault (Figure2a).
The rheological models we test in the viscoelastic domain are the linear
Maxwell, linear Burgers, and power-law rheologies (Figure2b). The total
strain rate in these rheologies are of the form,

total =
+

+ 
⎧
⎪
⎨
⎪
⎩
−

,
Burgers body
0,otherwise
(7)
where
 𝐴 
is the Kelvin strain only present for a Burgers body, ηM is the viscosity of the Maxwell element (for
power-law rheologies, ηM in turn is a function of
 𝐴
, i.e.,
d
d
≠
0
in Equation5) and G is the elastic shear modulus
of the system.
To study the role of viscous rheology in modulating the stress state in this system, and the associated displace-
ment and velocity field at the free surface, we vary the two parameters used to describe the rheology in the
viscous shear layer for the spring-dashpot bodies (linear Maxwell and power-law): A, n; while we vary the Kelvin
and Maxwell viscosities for the Burgers material: ηk, η. We also vary the recurrence time for the earthquake to
see how relaxation in the lithosphere is related to the magnitude of coseismic stress perturbation. We list model
parameters we varied for these simulations in Table1.
2.2.2. Initial Conditions From Coseismic Slip
The set of ordinary differential equations we need to solve is Equation5 in terms of the variables
[ ,  12,  13 ]
,
subject to the rheologies in Equations6 and7. To guarantee a unique solution for this system, we need to deter-
mine the initial condition for
[ ,  12,  13 ]
. This is done by using the stress change due to prescribed coseismic slip
on the fault to instantaneously change values of
[ ,  12,  13 ]
subject to their rheological properties.
We prescribe coseismic slip as a uniform value of u
∞=v
∞Teq within the locked domain (2km≤xco≤15km), and
tapered in the surrounding section of fault (0km≤xas≤2km ∪ 15km≤xas≤20km) such that the stress increase
does not exceed 3MPa and slip within this domain is minimized (Figure2a). The stress change calculations only
require the previously computed Green's function tensor for elastic stress interactions K,
Parameter Range
Fault width 20km
x3 scale 30km
x2 scale 200–500km
Shear modulus (G) 30GPa
Teq 50, 100, 200years
v
∞10
−9m/s
Viscous layer (linear Maxwell, power-law)
ΔxVariable mesh size
n1, 2, 3, 4, 5, 6
A
−1 10
18, 3×10
18, 7×10
18, 10
19, 5×10
19, 10
20
Viscous layer (linear Burgers)
ΔxVariable mesh size
ηM (Pa-s) 10
18, 5×10
18, 10
19, 5×10
19, 10
20
ηk (Pa-s) 5×10
17, 10
18, 5×10
18
Fault parameters
Δx3500m
a−b0.005
σn50MPa
f0, v00.6, 10
−6m/s
Table 1
Model Parameters for Earthquake Cycle Simulations

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
, 
∞
()
+
,
Δ

∞
()≤
3MPa
Δ
∞
()∼0
(8)
where u
∞(xco) is uniform slip applied within the locked domain and Δu
∞(xas) is the tapered slip within the
rate-strengthening frictional domain. This is a linear inequality constrained optimization for Δu
∞(xas) that is done
using the MATLAB function lsqlin. The resulting shape of this profile on the fault is shown in Figure2a.
With the initial conditions determined from coseismic slip, we integrate the system of equations using MATLAB's
Runge-Kutta fourth order solver ode45 to obtain the time history of fault velocity (within the rate-strengthening
domain) and viscous strain rates
[ ,  12,  13 ]
over the entire domain. Since the coseismic slip derived initial condi-
tions only provide a change in the integrable variables, we need to run these earthquake cycles a number of
times until we obtain cycle invariant results (e.g., Hetland & Hager,2005; Takeuchi & Fialko,2012). In that
case, the coseismic slip-derived stress change is imposed every Teq years. We find that, depending on rheology,
10–20 cycles is sufficient to obtain cycle invariant results given the rheological parameters and timescales we
have chosen.
2.3. Parameters That Can Be Estimated Geodetically
The earthquake cycle simulations give us the time history of fault slip rate and viscous strain rates within the
viscoelastic medium, which we then combine with displacement Green's functions to predict displacement time
series at the free surface (Barbot,2018). Since our focus in this article is the behavior of the viscoelastic domain,
we neglect the fault slip rate evolution in the predicted surface deformation time series. We consider two main
parameters that can be inferred geodetically that are generally used to describe the period following and leading
up to large plate boundary earthquakes. In the postseismic period, we estimate the effective relaxation time of the
system, tR; we describe the interseismic signal using an effective locking depth, Dlock.
For postseismic relaxation, we consider only the deviation from steady state behavior, that is, we remove displace-
ments associated with the long-term motion of the plate boundary or the steady-state strain rates
(

∞
12
, 
∞
13)
. We
characterize the transient surface displacements during the first 2years following the earthquake using a two-step
procedure. First, we use singular-value decomposition on the displacement time series and extract the temporal
component associated with the most dominant singular value. We fit this with the following functional form,

()=
(
1−exp
(
−

)).
(9)
β, tR are estimated using a MATLAB-based nonlinear least squares routine, and tR gives the best-fit relaxation
time of the system over the observational window.
Later in the earthquake cycle, we consider the interseismic period as the time period when the maximum surface
velocity is smaller than the relative plate velocity, that is,
|
(2)|≤
∞
2
. The resulting velocity field can then be
fit to an arc-tangent function (Savage & Burford,1973),

(2)=∞
tan−1
(
2

).
(10)
The estimated locking depth controls the effective width of the surface that is experiencing interseismic strain,
and is thus a physically motivated representation of the spatial pattern of the signal.
3. Results
We describe the surface deformation observations predicted at geodetic sites over the entire earthquake cycle,
as well as the corresponding strain rate evolution within the viscoelastic domain from our numerical exper-
iments (Figures3 and 4). Since we are interested in cycle invariant behavior, we only present results from
the last earthquake cycle; the previous cycles are necessary only for spin up.The results are discussed sepa-
rately for linear Maxwell, linear Burgers, and power-law rheologies in terms of interseismic locking depths
(Figure5), cumulative postseismic displacements (Figure6), and effective relaxation timescales (Figure7).
We note that our simulations allow for the mechanical coupling between frictional afterslip on the fault and

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viscous creep in the viscoelastic domain. However, since our focus is on the behavior of the viscoelastic
medium, the surface displacement and velocities that we discuss in subsequent sections do not contain contri-
butions from afterslip.
3.1. Linear Maxwell
For linear Maxwell rheologies, both the amplitude and effective relaxation timescale of the postseismic response
directly depend on the viscosity (ηM). As ηM increases, the timescale for stress relaxation following the coseismic
perturbation (tR) increases, while the magnitude of the initial jump in strain rate
(Δ (Δ= 0))
decreases.
𝑡𝑅=
𝜂𝑀
𝐺
Δ
𝜀 𝑀(Δ𝑡=0)=Δ𝜏𝑐𝑜
𝜂𝑀
≈𝐾(𝑣∞𝑇𝑒𝑞
)
𝜂𝑀
(11)
Figure 4. Surface predictions of postseismic displacements and interseismic velocities for different rheologies for a periodic earthquake cycle of Teq=100years. The
rheologies are chosen such that the cumulative postseismic after 1year is nearly identical for all three models. (a)–(c) Cumulative postseismic displacements normalized
by the coseismic slip amount
(
∞=∞
eq)
for times varying from 1day to 1year. (d)–(f) Interseismic velocities compared to the steady interseismic expectation (black
line).
Figure 3. Surface velocity and internal viscous strain rate evolution over the earthquake cycle for different rheologies for a periodic earthquake cycle of Teq=50years.
The rheologies were chosen such that the early postseismic surface velocity field is nearly identical. (a) Linear Maxwell body (ηM=3×10
18 Pa-s), (b) Linear Burgers
body (ηk=3×10
18 Pa-s, ηM=5×10
19 Pa-s), and (c) Power law rheology (A
−1=3×10
18, n=3). The linear rheologies allow accelerated viscous deformation
of significantly larger volume of material compared to the power-law rheology, which promotes localization of strain. This effect is noticeable in all the strain rate
snapshots.

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A dominant feature from simulations incorporating a linear Maxwell rheology is that they show strain rates
that are diffusive is space and in time (Figure3a). The initial strain rate following the earthquake decays
in space as expected from the stress change Δτco. In time, the elevated strain rate is damped as it diffuses
outwards. At the end of the earthquake cycle (Δt/Teq → 1), nearly the entire viscoelastic medium is at a
uniform strain rate level and the resulting surface velocity field appears to have a near constant spatial gradient
(Figure4d).
Many aspects of the evolution of this viscoelastic system can be explained by a single dimensionless variable,

=


2

=

2

(Savage,2000; Savage & Prescott,1978). Models with αM≫1, in our simulations this mainly
arises from low ηM, generate relatively large magnitude postseismic deformation early in the earthquake cycle
(Figure6) and predict relatively small near fault velocity gradients late in the earthquake cycle (FiguresA3,5d).
Conversely, if αM≪1 or ηM is large, the system response approaches the elastic limit where there is negligible
viscous response and the predicted surface velocities vary only moderately around the steady state elastic expec-
tation throughout the earthquake cycle (FiguresA3,5a,5b).
Figure 5. Compilation of late interseismic locking depths for various rheological choices and two different Teq. Locking depth (assuming an arc-tangent functional fit—
∞

tan−1
2

) for (a) Linear Maxwell and power-law materials with n varying from 1 to 6 for Teq=50years. (b) Same as (a) for Teq=200years. (c) Locking depths for a
linear Burgers rheology for a constant ηk and varying ηM and Teq. Late interseismic locking depths show no dependence on ηk. (d) The estimated locking depth varying
in time over the interseismic period for different rheologies. Both the power-law body and linear Burgers (with large ηM) show nearly time invariant late-interseismic
locking depth.

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3.2. Linear Burgers
The linear Burgers rheology is characterized by two separate timescales: a short-term anelastic timescale



controlled by the viscosity of the Kelvin element, and a long-term Maxwell timescale


(assuming ηk<ηM)
(Hetland & Hager,2005; Müller, 1986). Only the creep associated with the Maxwell element is recorded as
permanent strain, the anelastic term is significant for geodetic observations but does not leave a record in the
long-term.
Similar to the linear Maxwell case, the linear Burgers body also exhibits a tendency to diffuse strain rate away
from the fault with time (Figure3c). This pattern depends on three variables—Teq, and the two relaxation times
associated with ηk and ηM. Large values of ηk, ηM and small values of Teq, lead to small stress perturbations and
hence minimal deviation from a time-invariant steady-state model. Small values of ηk and ηM, or large values of
Teq lead to more pronounced earthquake cycle effects.
Figure 6. Postseismic relaxation times for linear Maxwell and power law bodies estimated over a 2-year period following the earthquake for a recurrence interval of (a)
50years and (b) 200years. Increased Teq leads to larger coseismic slip
(
∞=∞
)
, and hence larger stress change to drive postseismic creep.Linear Maxwell bodies
follow a stress-independent relaxation time given by

≈


. The relaxation time of power law bodies show a significant reduction for larger coseismic slip.We do not
show the results for Burgers bodies, since their relaxation times over the given time window are exactly as predicted by the viscosity of the Kelvin element

≈


.
Figure 7. Magnitude dependent postseismic motions for power law bodies. (a) Cumulative postseismic displacement (steady state component removed) normalized
by the coseismic slip amount (u
∞=v
∞Teq) over 2years for the same rheology. By increasing the earthquake recurrence interval, we increase the coseismic slip amount.
Only power law materials show increasing cumulative deformation with increase in the recurrence interval. (b) As the cumulative deformation increases, the relaxation
timescale decreases, that is, the postseismic deformation becomes faster and larger.

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3.2.1. Interseismic Locking Depth
Late interseismic locking depths show systematic dependence on only two parameters—ηM and Teq (Figure5c).
This indicates that for the parameter range we explore here, the late interseismic behavior is effectively
controlled by the dimensionless variable, αM which we discussed in the linear Maxwell case. Thus, small
αM values (large ηM) result in locking depth estimates comparable to purely frictional-elastic simulations
(Figure5d).
3.2.2. Postseismic Creep and Relaxation Time
Linear Burgers bodies do not have a single relaxation timescale, and thus our estimates of tR depend on the time
window that is considered. We consider a 2-year time window, which is a typical observational window used
in geodetic studies, in order to estimate the relaxation time and effective viscosity of the system. In most of our
simulations, this estimated relaxation time corresponds to sampling the viscous relaxation controlled by ηk (see
caption in Figure6).
3.3. Power-Law
Our numerical experiments governed by power-law rheologies are characterized by two main features—(a) the
interseismic locking depths appear to be a constant in time and only weakly sensitive to the parameters we varied
(Figures5a and5b), and (b) the postseismic relaxation timescale and amplitude appear to depend on the coseis-
mic slip amplitude and conform poorly to the
exp (−∕

)
functional form we chose to fit it with (Figures4c
and7b), that is, the curvature in the timeseries is closer to a logarithmic decay than the exponential function we
chose (e.g., Montési,2004).
3.3.1. Localized Deformation and Interseismic Locking Depth
For our simulations with power-law rheologies, deformation throughout the entirety of the earthquake cycle
is significantly more localized in space than as observed for the linear viscoelastic rheologies discussed above
(Figure3c). The extent of localization depends on the power exponent n as well as the rheological parameter A.
We contrast this with the fact that the solution to the long-term viscous boundary value problem does not depend
on A (Equation 3). Thus, our simulation results suggest that both A and n may be inferred from geodetic data
collected over the entire earthquake cycle.
Larger stress exponents n favor increased localization while large coefficients A reduce the impact of stress
perturbations from coseismic slip, similar to how the magnitude of the viscosity of linear rheologies controls the
change in strain rates in Equation11. While the degree of strain localization depends on the power law stress
exponent, for the parameter space explored, we find that models with power law exponents n≥3 exhibit nearly
identical late interseismic locking depths (Figures5a and5d), and are generally comparable to simple back slip
models of interseismically locked faults.
3.3.2. Postseismic Creep and Relaxation Time
The postseismic deformation time series is not expected to conform to the exponential functional form we used
to fit the time series. This is because the exponential function is a solution to the linear viscoelastic problem, and
the outputs of a power-law rheology correspond to an effective viscosity that systematically increases in time
(Montési,2004). However, since we consider time windows on the order of 1–2years, the relaxation timescale
can be fit using a linear viscoelastic approximation to estimate an average relaxation time over that window.
These relaxation timescales are not only dependent on rheological parameters A, n but also are a function of the
earthquake size, parameterized here in terms of coseismic slip (Figure6).
For a given set of rheological parameters A, n (for n>1), the cumulative postseismic deformation over a given
time window (in this case Δt= 2years), even when normalized by the coseismic slip amount, increases with
earthquake size (Figure7a). The normalized postseismic deformation following small earthquakes in our simula-
tions (u
∞∼ 1.5m) amounts to about 30% of the normalized postseismic deformation following the largest earth-
quakes (u
∞∼ 12m). On the other hand, the estimated relaxation timescale decreases with increasing earthquake
size (Figure7b).

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4. Discussion
We have developed numerical earthquake cycle experiments in order to test how well popular rheological models
are able to qualitatively reproduce different observational features in geodetic studies over the entire interseis-
mic period. Our simulation results illustrate the nonuniqueness of rheological models, and their parameters, in
explaining postseismic data alone (Figures5–7).
Our simulations demonstrate that the nonuniqueness in interpreting postseismic data can to some extent be miti-
gated by incorporating data corresponding to strain accumulation in the late interseismic period (Figure5). We
find that steady-state power-law rheologies with n≥ 3 as well as linear Burgers rheology with ηM≈10
20 Pa-s
and ηk≈10
18 Pa-s are able to explain early postseismic relaxation as well as the strain localization observed near
strike-slip faults late in the interseismic period. While we do not show it explicitly, nonlinear Burgers rheologies
with n ≥1 (with relevant A values) could explain the geodetic data just as well. This is because a steady-state
rheology, linear or power-law, is simply a limiting case of an appropriate Burgers rheology where the transient
viscosity is much larger than the steady-state value. On the other hand, linear Maxwell rheologies are simply
insufficient to explain the observational features.
In the following sections, we first discuss the equivalence between linear Burgers and power-law descriptions of
lithospheric rheology for the earthquake cycle, and then detail geophysical observations that may be required to
convincingly discriminate between these two rheologies. We then expound on the relationship between inferences
of average rheological parameters from crustal scales and those measured in laboratory experiments, and how a
power-law rheology is consistent with both geodetic observations and laboratory-derived flow laws. Finally, we
conclude with the implications for stress transfer and the associated assessment of regional hazard when frictional
and viscous creep are mechanically coupled.
4.1. The Effective Rheology of the Lithosphere
Geodetic investigations of lithospheric rheology, specifically the lower crust and uppermost mantle, that consider
only a relatively short time window (Δt<5years) as is typical of geodetic postseismic studies, may not be able
to distinguish between any of the rheological models discussed in this paper (linear Maxwell, linear Burgers,
and power-law). This is because postseismic geodetic observations can be reduced to two features—a spatial
pattern of cumulative postseismic deformation and the effective relaxation timescale (Figure7), and there exists
a nonunique mapping between rheological parameters from each of the discussed rheological models to these
spatial and temporal patterns of the deformation data (Figures7,8a and8b).
However, the three rheological models display diverging behavior as the observational window gets larger; this
is what we exploit during the late interseismic period. Interseismic strain localization and the stationarity of
the locking depth in time is observed in models with either a power-law rheology or a linear Burgers rheology
that approximates the effective viscosity evolution of a power-law body (Figure8d). In contrast, linear Maxwell
rheologies promote diffuse strain distributions (Figure3) which manifests as an increase in effective locking
depths late in the earthquake cycle (Figure5d), a feature that is not seen even in the best monitored strike-slip
fault systems in the world (e.g., Hussain etal.,2018). This leads us to suggest that Earth's lithosphere cannot be
well-described by a homogenous linear Maxwell body, at least over the timescale of the earthquake cycle.
These findings do not invalidate previous work on estimating the effective viscosity from postseismic, postglacial
and lake rebound deformation observations assuming a linear Maxwell rheology (e.g., Devries & Meade,2013;
England etal.,2013; Johnson & Segall, 2004; Kaufmann & Amelung, 2000; Kenner & Segall,2003; Larsen
etal.,2005; Tamisiea et al., 2007). However, the important implication is that these estimates of the average
viscosity, or viscosity structure, are tied to the observational window. This detail becomes apparent when
comparing the lithospheric viscosities estimated from processes that occur over different timescales; longer
observations windows typically show significantly higher viscosities, for example, the viscosity of the upper
mantle estimated following deglaciation (since the Last Glacial Maximum), which represents a ∼10
4year obser-
vational time window, is between 10
20 and 10
21 Pa-s (e.g., Milne etal.,2001; Tamisiea etal.,2007) while typical
viscosities estimated in the decade(s) following Mw>7 earthquakes range from 10
18 to 10
19 Pa-s (e.g., Kenner &
Segall,2003; Pollitz,2005; Ryder etal.,2007). Viscosities estimated nearly 50years after the largest earthquakes
in the 20th century appear to favor viscosities in the range 10
19−10
20 Pa-s (Freymueller etal.,2000; Khazaradze
etal., 2002; Melnick etal.,2018; Suito & Freymueller,2009). Both power-law and linear Burgers rheologies

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can help reconcile these apparently disparate viscosity estimates, since both processes produce time-dependent
viscosities which increase with time since the applied stress perturbation (Figure8d).
4.1.1. Similarities Between Power-Law and Linear Burgers Rheologies
A question that arises at this point is—how can a linear and power-law rheology satisfactorily explain deforma-
tional data throughout the earthquake cycle? The near equivalence between linear Burgers and power-law bodies
in our simulations exists because of a nonunique mapping between rheological parameters for each model and the
observational features that we use to describe the deformation time series (Figures8a–8c).
Consider the viscosity evolution of a power-law body. The power-law rheology results in a lower effective viscos-
ity during the relatively high stress and strain rate postseismic period, and the viscosity gradually increases as
stress relaxes and decays to a near time-invariant interseismic state (Figure 8d). The linear Burgers rheology
captures this same kinematic behavior through completely different dynamics. The Burgers description can be
thought of as a technique to describe nonsteady state viscous rheology, that is, there exists a finite timescale or
strain over which the system has to evolve to reach the unique mapping between stress and strain rate (Hetland
& Hager,2005; Müller,1986). In the case of a linear Burgers rheology, the initial low effective viscosity during
Figure 8. Approximating power-law rheology
(
= 1019, =3
)
with a linear Burgers body (ηM=10
20 Pa-s, ηk=3×10
18 Pa-s) for Teq=200years. (a) Cumulative
displacement for power-law and linear Burgers rheologies after 2years. The inset shows snapshots of cumulative deformation over increasing time windows of 0.5, 2,
10, 30, 50years (blue—short timescale, yellow—long timescale). (b) Relaxation time function extracted from the time series. (c) Late interseismic velocity field. (d)
Average viscosity evolution in time for both rheological models.

(Δ)=
∬

(
2,3,Δ
)
|
(
2,3,Δ
)
|d2d
3
∬|
(

2
,
3
,Δ
)
|d
2
d
3
where
| |
=
√
 2
12 + 2
13
.

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the postseismic period is a disequilibrium feature that smoothly evolves to its significantly larger steady state
viscosity (Figure8d).
While the overall kinematics predicted by the two different rheological models appear similar, the predictions
from the two models are not identical (Figure8a, inset). Although they would likely be difficult to distinguish
after considering the errors and uncertainties in typical geodetic data sets and the various models employed to fit
the data (Duputel etal.,2014; Minson etal.,2013).
4.1.2. Magnitude-Dependent Postseismic Motions
Our simulation results suggest that linear Burgers and power-law rheologies may in principle be distinguished
by the sensitivity and rate of the postseismic moment release to the magnitude of the coseismic event. For a
typical time window (Δt=2years), linear viscoelastic rheologies result in postseismic surface deformation that
is a linear function of the coseismic slip
(
∞=∞eq
)
, and thus can be normalized to produce a constant shape
(Figure7a). Similarly, the temporal evolution of this moment release is invariant of the size of the earthquake
(Figure7b). In contrast, power-law rheologies show a clear magnitude dependence, where the normalized post-
seismic deformation at the surface is smaller for small events and grows larger with increasing coseismic slip
(Figure7a). The temporal evolution of moment release is also a function of event size with smaller events having
much slower relaxation than larger events (Figure7b).
While this magnitude-dependent behavior has not been studied thoroughly, there is some evidence to suggest
the existence a magnitude-dependent pattern in postseismic observations, supporting the interpretation that lith-
ospheric deformation may follow a power-law rheology. For example, multiyear postseismic viscoelastic defor-
mation has been clearly observed and documented following MW>7 continental earthquakes (e.g., Freed &
Bürgmann,2004; Freed etal.,2010; Moore etal.,2017; Pollitz,2019; Savage & Svarc,2009; Tang etal.,2019;
Wang & Fialko, 2018; Wen et al., 2012; Zhao et al., 2021), however, observations of notable viscoelastic
deformation following slightly smaller (6.5 < MW <7.0) continental earthquakes are equivocal (e.g. Bruhat
et al., 2011; Savage et al., 1998; Wimpenny etal., 2017). Such distinction in observed postseismic behavior
for different sized earthquake ruptures may indicate a critical coseismic stress perturbation required to activate
geodetically detectable viscous flow, as would be expected from power-law rheologies (Figure7). Identifying
a clear magnitude-dependence of postseismic viscous response may be challenging given the limited historical
data available for individual fault segments, however, a careful global compilation of postseismic deformation
over a fixed time window following strike-slip fault earthquakes ranging from Mw 6 to 8 may provide further
insight to any systematic magnitude-dependent response, and help discriminate between rheological models of
the lithosphere.
4.2. What do Estimates of A and n Mean at the Lithospheric Scale?
As previously discussed, geodetic data over a single earthquake cycle is consistent with two classes of rheo-
logical models: (a) steady-state flow laws with power law exponents n≥3 and a range of A values, and (b) an
unsteady flow law with n=1, ηk/ηM<0.1, and ηM≥10
20 Pa-s. We note that for unsteady flow laws, we have only
explicitly considered the linear Burgers rheology (n=1); a power-law rheology with an additional unsteady or
transient element can exactly reproduce the observations as well. The principle of parsimony would suggest that
a steady-state power-law rheology presents a better representation of the lithosphere, but we turn to the literature
from the mineral physics community to expound on the appropriate rheological choice as well as how to interpret
what are effectively kilometer-scale averaged estimates of rheological parameters


,
 𝐴
(and
 𝐴 
) from geodetic
data.
There are two main aspects to this discussion—(a) the contribution of multiple different mechanisms to the
inferred parameters
(
𝐴 𝐴 
)
, and (b) the spatially heterogeneous variations of the parameters of various mecha-
nisms to our spatially uniform estimates of the inferred rheological parameters.
4.2.1. Averaging Over Multiple Mechanisms and Assemblages
The simplified rheology we employ in this article (Equation2) is a composite flow law, that under the assump-
tion of linear mixing would attempt to approximate a linear combination of multiple microscale processes in the
following way,

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
=
  ≈
∑
[
exp
(
−+
∗

)
−
fluid
]


(12)
This summation indicates simultaneously active processes with different values of the power-law exponent (ni),
each having material specific corresponding activation energy and volumes
(
,∗
)
, grain size dependence (mi)
and fluid phase dependence (ri). ci is a material and process-specific constant, Cf luid may refer to either the water
fugacity or melt fraction, R is the universal gas constant, T is the absolute temperature of the system, and d is a
central tendency of the grain size distribution in the sample.
Power-law rheologies for rocks with stress exponents of n∼ 3–4 are considered representative of dislocation
creep, where deformation is accommodated by the migration of dislocations and dislocation planes within the
crystal lattice (e.g., Chopra & Paterson,1981; Hirth & Kohlstedt,2003); linear rheologies indicate the diffu-
sion of vacancies and defects through the mineral grains and grain boundaries (e.g., Karato etal.,1986; Rutter
& Brodie,2004); intermediate values of n have been suggested to be related to grain boundary sliding (e.g.,
Goldsby & Kohlstedt,2001; Hansen etal.,2011), although it is important to note that this mechanism is intrin-
sically coupled to either diffusion or dislocation creep (Hansen etal., 2011; Raj & Ashby,1971). In addition
to mechanical processes, thermal effects can also be relevant to lithospheric deformation. Thermal effects are
typically thought of in terms of the steady-state geothermal gradient, but this thermal profile can be perturbed
by viscous heating during rapid shear and an associated thermal diffusion (Moore & Parsons,2015; Takeuchi &
Fialko,2013). As a consequence, the effective power law
 𝐴
inferred at the kilometer scale need not be bounded
between 1 and 4, but instead may be even higher (e.g., Kelemen & Hirth,2007).
If any of the individual parameters in Equation12 evolve with incremental strain or time, for example, temperature
or grain size (Allison & Dunham,2021; Montési & Hirth,2003), then there would not be a unique relationship
between
 𝐴
and σ until a steady state is reached. The viscous creep that would result from this equilibration process
is often called “transient creep,” and is an important motivation for invoking Burgers rheology (Chopra,1997;
Freed etal., 2012; Post, 1977). Despite the likely presence of viscous transients, we maintain that the princi-
ple of parsimony dictates that we choose steady-state power-law rheologies over Burgers rheologies for mode-
ling geodetic data. To further illustrate this preference, we draw parallels between the aforementioned transient
viscous creep and deviations from steady-state frictional strength in rock friction experiments. Unsteady evolu-
tion of the friction coefficient is captured by a state variable, θ, which is thought to represent the quality and/
or average timescale of asperity contact during frictional sliding (Marone, 1998; Scholz, 1998). Despite the
well-known importance of θ to many aspects of frictional mechanics (Scholz,2002), geodetic investigations of
frictional afterslip are rarely able to resolve the evolution of the frictional state from the data. Even when the state
evolution is identified, it is shown to quickly evolve toward steady state within a few hours and may be invisible
to typical (sampled daily) postseismic time series (Fukuda etal.,2009; Perfettini & Ampuero,2008). This argu-
ment does not obviate the existence or importance of unsteady strength evolution, but instead emphasizes that it
is not necessary to invoke an unsteady Burgers rheology when steady-state power-law rheologies can explain the
available geodetic observations. As a result, we are tempted to interpret the value of
 𝐴 ≥3
in terms of a rheology
dominated by dislocation creep, with possible contributions from thermomechanically coupled processes such as
shear heating and grain boundary sliding.
4.2.2. Averaging Over Spatially Variable Parameters
The inferred

 𝐴 
values do not only represent averages over multiple physical and chemical processes, but also
over a spatially varying set of parameters. The dominant contribution of this in Equation12 likely comes from
the depth-dependence of temperature, that is, T(x3) ∝ x3. However, our ability to geodetically infer spatially
varying rheological parameters is limited by the spatial smearing effect of elasticity as well as the apparent
homogenization of rheological properties during shear (e.g., Almeida etal.,2018; Hetland & Hager,2006; Ray &
Viesca,2019). This implies that we may at best infer a best-fitting

 𝐴 
from a single earthquake cycle, with larger
events eliciting a response from greater depths and hence a larger



∝ exp
(
−

)
. The way forward then is to use
sequences of earthquakes (events of different magnitudes and/or depth on the same fault), where each individual
earthquake may be mapped to a set of uniform

 𝐴 
but these parameters show a consistent pattern, such as a fixed
 𝐴
but


increases with increasing size or depth of the earthquake. The implication then is that spatial heterogene-
ity is necessary to explain the observations and therefore we can infer more about how the lithosphere behaves.

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4.3. Coupling Between Frictional Slip and Viscous Creep
An important implication of mechanically coupled models of fault slip and distributed deformation, such
as our simulations, is that stress-driven interactions between frictional afterslip on the fault and distributed
viscous flow in the lower crust and mantle are not independent processes, as is typically considered in many
inverse postseismic modeling studies. This simplification explicitly decouples the mechanical interactions
between frictional afterslip and viscous creep, and has been shown to systematically bias the location and
amplitude of inferred slip and strain (e.g., Muto etal.,2019; Peña etal.,2020). Our simulation results suggest
that a permissible simplification may be to treat earthquake-driven viscoelastic relaxation as an independent
process, while afterslip is driven by the coseismic stress change as well as the subsequent viscous flow of the
bulk medium (FigureA2). We highlight this by noting the amplitude and temporal evolution of afterslip is
markedly different between simulations that consider a purely elastic medium versus a viscoelastic medium
(Figure9).
Figure 9. Stress change and decomposition into contributions from fault slip and viscous shear for (a) linear Burgers rheology (ηk=3×10
18, ηM=10
20 Pa-s) with
effective viscosity ∼5×10
18 Pa-s in the plotted time window, and (b) power law rheology with n=3, A
−1=10
19. Stress is plotted at 0, 10, 20, and 30km away from
the fault at 10km depth. Total stress evolution from a nearly elastic model (linear Maxwell simulation with ηM=10
20 Pa-s) is also shown (gray). The stress evolution
over the first 5years is dominated by the viscoelastic response for linear and power-law rheologies. Additionally, due to the mechanical coupling between fault slip and
viscous shear, stress transfer from fault slip evolution in the viscoelastic simulations is significantly different from the elastic simulations.

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4.3.1. Time-Dependent Loading Rate
The effect of viscoelastic relaxation on afterslip can be thought of as a modification of the stress loading rate
along the fault. For an isolated system, the governing equation for frictional slip in response to a coseismic stress
step is

(∞−())∝
d(())
d
where f is the velocity dependent friction coefficient, k is the elastic stiffness, and v
∞
is the long-term slip rate of the fault (Marone etal.,1991). When viscoelastic relaxation of the medium is factored
in, the loading term now contains two contributions—a time-invariant contribution from v
∞, and time-dependent
stress transfer due to viscous creep in the surrounding medium (Pollitz,2017, 2012). Viscous creep is itself
a decaying function in time, with the exact decay rate being a function of the rheology (Figure3). Thus, the
effective loading rate for afterslip is no longer time-invariant (FigureA2) and the resulting time series for slip
and stress transferred to the surrounding medium can notably differ from simulations that decouple afterslip and
viscous creep (Figure9).
4.3.2. Regional Stress Interactions
The difference in time-dependent loading between purely elastic fault models and those considering viscoelastic
deformation suggests that viscoelastic interactions are an important ingredient for efforts aimed at modeling
regional tectonics and multifault interactions, particularly given that the spatial footprint of this distributed defor-
mation can be much larger than that of slip on individual faults (Figures3 and4). Viscoelastic stress interactions
have been noted to be relevant to along-strike stress transfer and timing of a recent sequence of great earthquakes
on the North Anatolian Fault (Devries & Meade,2016; Devries etal.,2017), and Southern California (e.g., Freed
& Lin, 2001). More generally, time-dependent loading alters the stress state on the fault preceding dynamic
rupture. This pre-rupture stress state has been noted to control many aspects of the rupture process from earth-
quake nucleation to rupture arrest, including the likelihood of ruptures propagating over multiple fault segments
(e.g., Lambert etal.,2021; Lambert & Lapusta,2021; Noda etal.,2009; Ulrich etal.,2019; Zheng & Rice,1998).
Time-dependent loading due to viscous creep may be particularly important when considering interactions
between major plate boundary faults and neighboring lower slip rate faults (Freed,2005; Kenner & Simons,2005).
For low slip rate faults, the loading due to the long-term tectonic loading rate, which is relatively small for low
v
∞, may be overwhelmed by the static stress transfer from a nearby earthquake and the corresponding viscous
response of the ductile lower crust and mantle (Figure9). As a result, seismicity on such low slip rate faults may
cluster in time with large earthquakes on the major plate boundary fault and may be indicative of coordinated
time-dependent loading, as opposed to an individual long-term loading rate of each fault within this system.
Future work is needed to develop more realistic treatments of fault loading in larger-scale simulations of fault
networks and models of seismic hazard (e.g., Shaw etal.,2018; Tullis etal.,2012), potentially including phys-
ically motivated approximations of viscoelastic contributions to the effective loading rate of fault populations.
5. Conclusions
Geodetic recordings of earthquake cycle deformation related to large earthquakes provide geoscientists with
one of the best opportunities to estimate the effective rheology of the lithosphere-asthenosphere system. In this
article, we showed that combining geodetic observations with numerical simulations of the earthquake cycle
translate into better estimates of rheological models and relevant parameters. Below we list a number of important
contributions and insight from this study.
1. We developed a numerical framework to model the earthquake cycle, including interactions between frictional
sliding on faults and viscous deformation of the upper mantle, that is computationally inexpensive (can be run
on personal computers). This facilitates efficient exploration of various rheological models and parameters.
2. By incorporating geodetic observations throughout the entire earthquake cycle, the ambiguity associated
with commonly used rheological models is reduced. Specifically, a homogeneous linear Maxwell viscoelastic
medium is simply inconsistent with typical geodetic observations.
3. The average viscoelastic description of the lithosphere may be that of a power-law spring-dashpot system,
although Burgers rheologies may also satisfactorily explain the data but invoke more tunable parameters.
4. Our preferred parameterization of the viscous element in this spring-dashpot system follows a steady-state
flow low of the form
 𝐴 =
, and the parameter ranges for the pre-factor 1/A range from 10
18 to 10
20 and the
power exponent n≥3. The power exponent n≥ 3 may strongly hint at dislocation creep being a dominant
process throughout the earthquake cycle. However, we caution direct interpretation of these parameters from
single earthquake relaxation studies.

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5. Differentiating between these rheologies requires geodetic observations from earthquake sequences
(events of different magnitudes). The postseismic response of power-law rheologies will show strong
magnitude-dependence while the linear Burgers body will not.
6. An important societal consequence of lithospheric viscoelasticity is to modify the spatial and temporal pattern
of stress interactions between faults and the surrounding bulk, compared to purely elastic models. This leads
to significantly stronger temporal linkage and long-distance interactions, and hence seismicity, between faults
than expected by frictional-elastic models of faults.
Appendix
A1. Equivalence Between Afterslip and Viscous Creep
To demonstrate the equivalence between rate-strengthening frictional sliding and a power-law viscous creep, we
consider a common strain rate variable v. The governing ODE for a steady-state elasto-frictional system is given
by,

(−∞)=(−)


(A1)
while for a power-law viscoelastic system, the governing ODE is

(−∞)=1

(

)
1

.
(A2)
It is clear that the equivalence between the two systems occurs as
(


)1

(this is actually the stress level in the
system) approaches a constant value over the domain of v, at least in comparison to

(−∞)
. When n=6, the
Figure A1. Approximation of frictional slip with a power-law in a simplified spring-dashpot analysis. As power-law
exponent n increases, the error in the approximation reduces. The top panel shows velocity evolution for three different values
of n while the bottom panel shows slip evolution.

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viscous strain rate changes by ∼3 orders of magnitude in some of our simulations. The resulting
(


)1

varies by
∼3 over the duration of the entire earthquake cycle. The variation by a factor of 3 also occurs slowly in time which
means that for typical geodetic observational windows, n=6 is sufficiently close to an equivalent logarithmic
system and vice-versa (FigureA1).
A2. Relative Contributions of Afterslip and Viscous Creep in a Coupled System
In an elasto-visco-frictional system that explicitly demarcates the nonelastic domain into frictional and viscous
regimes as shown in FigureA2, we show that the mechanical coupling between afterslip and viscous creep may
be reduced to a one-way coupling problem such that afterslip is driven by a time-invariant background loading as
well as time-dependent stress transfer from the viscous domain; the viscous strain rates on the other hand appear
to evolve nearly independently of afterslip (FigureA2).
The time evolution of the slip rate on a velocity-strengthening frictional fault is given as a function of the initial
condition v(0) and a relaxation timescale tR,
Figure A2. (a) Numerical modeling setup and parameters used for a mechanically coupled visco-elasto-frictional system. (b) Viscous strain rate and on-fault velocity
evolution shows the influence of viscous deformation on frictional velocity evolution. Dashed lines show theoretical frictional creep in an isolated elasto-frictional
system. (c) Surface displacement time series due to viscous (magenta) and frictional creep (green) arising from lower crustal and upper mantle deformation (shallow
creep is not considered here). Left panel shows displacement evolution at r=30, 100km (continuous line—30km, dashed line—100km). Right panel shows the
percentage contribution of viscous creep to the surface displacement time series as a function of time and space.

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
()=
(0)
(0)
∞+
−

(
1− (0)
∞
)
(A3)
The relaxation timescale here corresponds to a different model than the one discussed in Figure6. However,
considering the parameters we used in our suite of simulations (Table1), the equivalent relaxation timescale for
the frictional fault is significantly smaller than the relaxation times for at least the linear Maxwell systems.
This decaying signal appears as a straight line on a logarithmic scale, and then will smoothly transition to the
background loading velocity (FigureA2—dashed lines). However, in the presence of viscous deformation in the
mantle, frictional slip mimics the time evolution of viscous creep beyond an initial logarithmic decay (FigureA2).
The resulting displacement time series at the free surface reflects this behavior, and shows that beyond an initial
logarithmic growth of displacements due to frictional slip, this physical process is entirely masked by the surface
contributions of viscous creep (FigureA2).
Data Availability Statement
No data was used in this study. All the MATLAB code required to recreate the results of this study is available at
the Caltech data repository https://data.caltech.edu/records/20257 (DOI: 10.22002/D1.20257).
References
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Allison, K. L., & Dunham, E. M. (2021). Influence of shear heating and thermomechanical coupling on earthquake sequences and the brittle-
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Acknowledgments
This research was supported by a
TexacoPostdoctoral Fellowship awarded
to Rishav Mallick. Valere Lambert is
supported by a National Science Founda-
tion EAR Postdoctoral Fellowship.The
authors thank JGR editor Paul Tregoning,
associate editor Mike Poland, Hugo
Perfettini and an anonymous reviewer
for their review and feedback on this
manuscript. The authors are grateful to
Roland Bürgmann and Judith Hubbard
for discussions and suggestions for this
project.
Figure A3. Variation of the geodetically estimated late interseismic locking depth for a linear Maxwell rheology as a
function of the rheological parameter αM.

=


2

is a dimensionless parameter that combines rheological properties and
the recurrence time of the earthquake cycle. Small αM corresponds to a nearly elastic material, and hence minimal earthquake
cycle effects while large αM corresponds to a low viscosity material and pronounced earthquake cycle effects.

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