postseismic and interseismic displacements near a strike-slip...

Postseismic and interseismic displacements near a strike-slip fault:

A two-dimensional theory for general linear viscoelastic rheologies

E. A. Hetland and B. H. HagerDepartment of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge,Massachusetts, USA

Received 14 February 2005; revised 30 June 2005; accepted 7 July 2005; published 7 October 2005.

[1] We present an analytic solution for the deformation near an infinite strike-slip fault inan elastic layer overlying a linear viscoelastic half-space. The theory is valid for anylinear viscoelastic rheology and any earthquake sequence. This is a generalization of thework of J. C. Savage and colleagues, which only holds for models with a uniform shearmodulus, Maxwell viscoelastic lower half-space, and periodic rupture recurrence. Wedemonstrate the theory for models of an elastic layer over a Maxwell, standard linear solid,Burgers, and triviscous half-space. For each of these models, we calculate examplepostseismic displacements, interseismic displacements in a periodic earthquake sequence,and interseismic displacements for a nonperiodic earthquake sequence. Our solution is fora simple geometry; however, the model presents an elegant tool to explore the evolution ofdisplacements for relatively complex rheologies and rupture recurrence histories.

Citation: Hetland, E. A., and B. H. Hager (2005), Postseismic and interseismic displacements near a strike-slip fault: A

two-dimensional theory for general linear viscoelastic rheologies, J. Geophys. Res., 110, B10401, doi:10.1029/2005JB003689.

1. Introduction

[2] The two dimensional, time-dependent solution of theinterseismic displacements near a strike-slip fault, pro-posed by Savage and Prescott [1978], has been widelyused to guide intuition about interseismic displacementsand the earthquake cycle [e.g., Savage, 1990; Lisowski etal., 1991; Meade and Hager, 2004; Wernicke et al., 2004]as well as to model geodetic data [e.g., Dixon et al., 2002;Segall, 2002; Dixon et al., 2003]. While this model is anelegant tool for building intuition, it makes severalassumptions that probably are not applicable to the litho-sphere. In this paper, we present a more general form ofthe model.[3] Nur and Mavko [1974] demonstrated that the time-

dependent displacements following a dislocation in anelastic layer overlying a viscoelastic media could beobtained from an elastic solution by using the correspon-dence principle of viscoelasticity. Using the image solutionof Rybicki [1971] and following Nur and Mavko [1974],Savage and Prescott [1978] used the correspondence prin-ciple to solve for the surface displacements throughout aseismic cycle from a fault breaking an upper elastic layeroverlying a Maxwell viscoelastic half-space. Savage andPrescott [1978] assumed that the shear modulus in the upperlayer and half-space was identical and that successiveruptures occurred with constant repeat time and constantcoseismic displacement. Savage and Lisowski [1998] refor-mulated the solution of Savage and Prescott [1978] andSavage [2000] extended the model to the displacements atdepth, as well as at the surface; most modern calculations of

the seismic cycle model are done using one of these laterformulations. The transformation Savage and Prescott[1978] used to account for the time dependence due toMaxwell viscoelastic relaxation has been used in geometri-cally more complex models [e.g., Savage, 2000; Pollitz,2001].[4] The main prediction of the model of Savage and

Prescott [1978] is that when the seismic repeat time issmaller than twice the relaxation timescale of the Maxwellhalf-space, the velocities throughout a seismic cycle willbe approximately constant and roughly equal to thedeformation predicted by an elastic half-space model.Only when the seismic repeat time is long compared tothe Maxwell relaxation time will the surface velocitiesvary appreciably. A consequence of this behavior is that ifthe velocities before an earthquake can be described byan elastic half-space model with parameters appropriatefor the fault, then there will be little to no postseismictransient deformation. On the other hand, if there issignificant postseismic deformation, then the velocitiesacross the fault should be close to simple shear andslower than those predicted by an elastic half-spacemodel. Geodetic observations before and after largestrike-slip ruptures are often contrary to these predictions.For instance, large postseismic transients were observedfollowing the 1999 Izmit and Duzce earthquakes on theNorth Anatolian fault [Ergintav et al., 2002]; however,before the earthquakes the velocities across the fault weredescribed using an elastic half-space model appropriatefor the North Anatolian fault [Meade et al., 2002]. Weshow below that a model with a biviscous rheology, incontrast to the univiscous Maxwell rheology, may be ableto explain both of these preseismic and postseismicobservations.

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, B10401, doi:10.1029/2005JB003689, 2005

Copyright 2005 by the American Geophysical Union.0148-0227/05/2005JB003689$09.00

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[5] In this paper, we generalize the solution of Savageand Prescott [1978] to hold for any linear viscoelasticrheology and a nonperiodic fault rupture history, orearthquake sequence. In the lithosphere, the variation inshear modulus with depth has been shown to be signif-icant on surface displacements [e.g., Hearn et al., 2002],and recent studies of postseismic deformation have indi-cated that the inelastic region of the Earth has a muchricher time dependence than the single exponential decaypredicted by models with Maxwell viscoelastic rheologies[e.g., Ivins, 1996; Hearn et al., 2002; Pollitz, 2003; Freedand Burgmann, 2004]. Additionally, recent paleoseismicstudies have indicated that rarely do earthquakes occurperiodically [e.g., Grant and Sieh, 1994; Bennett et al.,2004; Weldon et al., 2004]. Since the surface displace-ments are of principle interest to modeling of geodeticdata, we present the analytic derivation of only the surfacedisplacements. However, the solution for the displacementsat depth can be achieved directly using the derivationpresented here, as the geometry factors are not dependenton the rheologies.[6] For illustrative purposes, we apply the general linear

viscoelastic theory to four viscoelastic rheologies of thelower half-space: (1) Maxwell, (2) standard linear solid(SLS), (3) Burgers, and (4) a triviscous rheology. Forbrevity, we refer to the model of an elastic crust over aMaxwell half-space as the Maxwell model, and similarly fora SLS, Burgers and triviscous half-space. The Maxwelllinear viscoelastic material is the most widely used rheologyin models of crustal deformation [e.g., Freed and Lin, 2001;Pollitz et al., 2001; Hearn et al., 2002; Kenner and Segall,2003; Johnson and Segall, 2004]. The Maxwell rheology isanalytically simple, parameterized by one number, and isimplemented in a variety of finite element modeling pro-grams used in studies of crustal deformation. The SLS isprobably the second most popular viscoelastic medium inpostseismic deformation studies [e.g., Nur and Mavko,1974; Cohen, 1982; Pollitz et al., 2000]. SLS rheologies,which are often referred to as three parameter solids, areanalytically as simple as Maxwell; however, compared tothe nonrecoverable Maxwell rheologies, SLS rheologies arefully recoverable. Pollitz et al. [2000] noted that theMaxwell rheology is contained in the SLS rheology andused the SLS as a semigeneral rheology. Researchers havealso proposed that the inelastic lithosphere should bemodeled with a biviscous rheology, using either a hetero-geneous model [Ivins, 1996] or a Burgers rheology [Pollitz,2003]. The Burgers viscoelastic rheology is often referred toas a ‘‘Burgers body.’’ In this paper we follow Findley et al.[1976] and refer to it as a ‘‘Burgers rheology,’’ or moresimply a ‘‘Burgers model.’’ Burgers rheologies have beenproposed to explain postglacial rebound [e.g., Peltier, 1985;Yuen et al., 1986; Sabadini et al., 1987], as wellas deformation experiments of mantle [e.g., Mackwellet al., 1985; Chopra, 1997] and crustal [Carter andAve’Lallemant, 1970; Smith and Carpenter, 1987] material.A Burgers rheology is capable of two phases of relaxation(one recoverable and one nonrecoverable); hence the Bur-gers rheology can be said to be biviscous. There have notbeen any studies that have proposed that a triviscousrheology is appropriate for the lithosphere; however, weinclude it for demonstrative purposes. The triviscous rheol-

ogy is composed of two recoverable phases of relaxationand one nonrecoverable phase.[7] In this paper, we derive the time-dependent solution

for the postseismic and interseismic surface displace-ments. Typically, when referring to displacements withina seismic cycle, researchers distinguish between the post-seismic displacements (early transient displacements) andinterseismic displacements (displacements later in thecycle, considered secular); in this paper we refer to alldisplacements during a seismic cycle as interseismic, andwe refer to the transient displacements after an earthquakeignoring the reloading of the fault as postseismic. Forcompleteness, we review the correspondence principleand the image solution of Rybicki [1971] under corre-spondence. We present the derivations of the analyticsolutions for general linear viscoelastic rheologies andnonperiodic earthquake sequences. We demonstrate thesolution applied to the four example viscoelastic rheolo-gies and apply the models to postseismic and interseismicdisplacements for both a periodic and a nonperiodicearthquake sequence. We construct our nonperiodic se-quence by periodically repeating a nonperiodic, finitesequence of earthquakes. As the model is antisymmetric,we only present one-half of the model throughout thispaper.

2. Background and Theory

2.1. Correspondence Principle of Viscoelasticity

[8] In two-dimensional models of strike-slip faults, onlythe shear stresses and strains are nonzero. Only consideringshear, the equation of motion for a linear viscoelastic mediais

%s ¼ 8e ð1Þ

where s and e are shear stress and engineering shear strain,respectively. % and 8 are the differential operators

% ¼Xqfk¼0

fk

dk

dtk8 ¼

Xqyk¼0

yk

dk

dtkð2Þ

where fk and yk are constants determined from therheological properties of the media [e.g., Flugge, 1967;Findley et al., 1976]. Throughout this paper we refer to fk

and yk as stress and strain coefficients, respectively, andcollectively as material coefficients. The material coeffi-cients are not defined uniquely with respect to therheological properties and different authors use differentdefinitions, achieved by redistributing rheological propertiesin equation (1). Taking the Laplace transform of equation (1)(%S ¼ 8e, where we denote L{f(t)} = f (s) to be theLaplace transform of function f(t) and L�1{f (s)} = f(t) to bethe inverse Laplace transform of f (s), and we specifyfunctions of t and s as f and f , respectively), the differentialoperators become polynomials in s

% ¼ % sð Þ ¼XqFk¼0

fksk ; 8 ¼ 8 sð Þ ¼

Xqyk¼0

yksk ; ð3Þ

B10401 HETLAND AND HAGER: GENERAL INTERSEISMIC DISPLACEMENTS

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and the transform of the equation of motion can be rewrittenas

s ¼ m sð Þe ð4Þ

where m(s) = 8=%, a ratio of two polynomials in s. Thisillustrates the well known correspondence principle ofviscoelasticity [e.g., Flugge, 1967], which states that theLaplace transform of the equations of motion of all linearviscoelastic materials are identical to the equation of motionof an elastic media (s = mee). Therefore an elastic solutioncan be used to construct a time-dependent viscoelasticsolution by replacing me with m(s) in the Laplace domain;hence we refer to m(s) as the equivalent shear modulus of aviscoelastic material.

2.2. Image Solution Under Correspondence

[9] Rybicki [1971] gave the solution for the displacementsat the surface due to a fault rupturing with displacement bfrom depth 0 � D in an elastic layer of thickness H � D andshear modulus mc overlying an elastic half-space of shearmodulus m. In the notation of Savage and Prescott [1978],the displacements are given by

u xð Þ ¼ b

p�p2� tan�1

x

DþX1n¼1

Gn WnÞ !

ð5Þ

where G is a coupling or reflection coefficient given by

G ¼ mc � mmc þ m

ð6Þ

Wn ¼ Wn x;D;Hð Þ ¼ tan�12Dx

4n2H2 � D2 þ x2

� �ð7Þ

where the plus and minus are for x > 0 and x < 0,respectively. For the remainder of this paper we onlyconsider x > 0. Wn and G contribute to the perturbations tothe half-space solution due to the image dislocations[Rybicki, 1971]. We choose to nondimensionalizeequation (7) by D, so that the nondimensional distance isx* = x/D, which when substituted into equation (7) gives

W ?n ¼ W ?

n x?; dð Þ ¼ tan�12x?

2ndð Þ2� 1þ x?2

ð8Þ

where d = H/D � 1. We show W*n for three values of d and nin Figure 1.[10] Following Nur and Mavko [1974], Savage and

Prescott [1978] demonstrated that the time-dependent dis-placements due to viscoelastic relaxation in the lower half-space could be found via the correspondence principle,where the fault rupture history is imposed by replacingthe scalar b by the function b(t). The Laplace transform ofthe image solution is

u x; sð Þ ¼ b sð Þp

p2� tan�1

x

D

� �þ 1

p

X1n¼1

b sð ÞGn sð ÞWn ð9Þ

G sð Þ ¼ mc sð Þ � m sð Þmc sð Þ þ m sð Þ ð10Þ

where mc(s) and m(s) are the equivalent shear moduli of theupper layer and lower half-space, respectively. G(s)completely describes the rheologies of the upper layer andthe half-space; hence its inverse transform describes thetime dependence of the displacements for particularrheologies due to the coupling between the upper layerand the lower half-space. The inverse transform of G(s) iscomplicated by the multiplication by b(s), which in the timedomain is the convolution

L�1 b sð ÞGn sð Þn o

¼ b tð Þ*Gn tð Þ ¼Z t

0

b zð ÞGn t � zð Þdz ð11Þ

Taking the inverse Laplace transform of equation (9), thetime-dependent displacements are given by

u x; tð Þ ¼ b tð Þp

1

2� tan�1

x

D

� �þ 1

p

X1n¼1

b tð Þ*Gn tð ÞWn; ð12Þ

while the velocities are given by the time derivative,v(x, t) = _u(x, t). Since the summation in n is infinite, we needto truncate the summation at some nfinal, the particular valueof nfinal depends on the time and distance ranges desired, aswill be described below.

2.3. &n(t) for General Viscoelastic Rheologies

[11] Savage and Prescott [1978] recognized that forthe case of an elastic layer (of shear modulus m)overlying a Maxwell linear viscoelastic half-space (ofshear modulus and viscosity m and h, respectively),equation (10) reduces to Gn(s) = [(m/2h)/(s + m/2h)]n

whose inverse Laplace transform is known. For thegeneral problem of a linear viscoelastic layer (mc(s) =

8c(s)/%c(s)) overlying a linear viscoelastic half-space

(m(s) = %(s)/%(s)), equation (10) is

Gn sð Þ ¼ 8c sð Þ% sð Þ � 8 sð Þ%c sð Þ8c sð Þ% sð Þ þ 8 sð Þ%c sð Þ

!n

ð13Þ

which is a ratio of two polynomials, the order of which isthe product of the largest orders of the differential operatorsof the upper layer and the half-space. We restrict thisstudy to the specific problem of an elastic layer (8c(s) =

Figure 1. Wn?(x?, d = H/D) for n = 1, 4, and 10.


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mc, %c(s) = 1) overlying a linear viscoelastic half-space, soGn(s) reduces to

Gn sð Þ ¼Pq

k¼0 mcfk � ykð Þsk� nPq

k¼0 mcfk þ ykð Þsk� n ð14Þ

where q = max(qy, qf), and yi = 0 and fj = 0 for i > qyand j > qf, respectively, and qy and qf are defined inequation (2). When the denominator and numerator ofequation (14) are factorable, with roots ak and bk,respectively, Gn(s) simplifies to

Gn sð Þ ¼ gnQqb

k¼1 s� bkð ÞnQqak¼1 s� akð Þn ð15Þ

where qa and qb are the number of roots and

g mcfqb

� yqb

mcfqaþ yqa

is the ratio of the leading coefficients of the polynomials ofequation (14). The units of s, ak, and bk are inverse time,and the units of g are time(qb�qa).[12] The inverse Laplace transform of equation (15) can

be identified directly by using the Laplace transformrelation

L�1 c sð Þy sð Þ

� �¼Xqk¼1

Xmk

l¼1

�kl akð Þmk � lð Þ! l � 1ð Þ! t

mk�leak t ð16Þ

where

y sð Þ ¼ s� a1ð Þm1 s� a2ð Þm2 � � � s� aq

� �mq ð17Þ

ai 6¼ aj for i 6¼ j are real, c is a polynomial of degree m1 +� � � + mq and

�kl sð Þ ¼dl�1

dsl�1c sð Þy sð Þ s� akð Þmk

� �ð18Þ

which is found by the method of partial fractions [e.g.,Churchill, 1944; Roberts and Kaufman, 1966]. The order ofthe polynomial in the numerator of equation (15) is alwaysless than or equal to the order of the polynomial in thedenominator (since the denominator always involvesaddition of positive numbers), thus qa � qb and we assume(for now) that all ak are distinct and that ak and bk are real.Using the transformation relation above, substituting mk = n,8 k 2 [1, q = qa], we find

Gn tð Þ ¼ gnXqak¼1

Xnl¼1

Wkl ai; bi; nð Þn� lð Þ! l � 1ð Þ! t

n�leak t ð19Þ

with

Wkl ai; bi; nð Þ ¼ dl�1

dsl�1�Qqb

i¼1 s� bið ÞnQqai¼1 s� aið Þn s� akð Þn

� ��s¼ak

ð20Þ

were ai and bi are the set of roots in equation (15) (e.g., ai ={aiji 2 [1, qa]}) and {� � �js signifies evaluation at s.

[13] Noting that the inverse of the roots ak are natural timescales in Gn(t), we nondimensionalize time by the absolutevalue of one of the roots, say janj, so that tjanj is thenondimensional time. Since the choice of timescale isnot unique for qa > 1, we represent the nondimensional timeas tn= tjanj.We order the set of roots with ja1j�1 < jaqa

j�1, sothat n = 1 and n = qa refer to the timescales of the fastest andslowest relaxation phases, respectively. To determine Gn(tn)from equation (10), we first nondimensionalize the dual oftime (sn = s/janj) in equation (20), yielding the relation

Wkl ai; bi; nð Þ ¼ janjn qb�qaþ1ð Þ� l�1ð Þ � dl�1

dsl�1n

Qqbi¼1 sn � bi

janj

� �nQqa

i¼1 sn � ai

janj

� �n8<:

� sn �ak

janj

� �n��sn

ð21Þ

evaluated at sn = ak/janj, so that

Wkl ai; bi; nð Þ ¼ janjn qb�qaþ1ð Þ� l�1ð Þ � Wkl

ai

janj;bijanj

; n

� �ð22Þ

The nondimensional form of Gn is then

Gn tnð Þ ¼ janjGnn tnð Þ

¼ janjgnnXqak¼1

Xnl¼1

Wklaijanj ;

bijan j ; n

� �n� lð Þ! l � 1ð Þ! � t

n�ln eaktn=jan j ð23Þ

where gn = gjanjqb�qa. Given two half-space rheologies withsets of roots {ai, bi} and {a

0i, b0i}, we nondimensionalize time

by an and a0n for each of the models. If ai/janj = a0i/ja0nj,bi/jbnj = b0i/jb0nj and gn = g

0n, then Gn

n(janjt) = Gnn(ja0njt).

[14] We define PS|to denote the product over the set of

indices S|,

PS| s;|i; pð Þ Yi2S|

s� |ið Þp ð24Þ

where |i = {|iji 2 S|}, and we adopt the shorthand notationPS|

PS|(s, |i; p), where for | = a (b), Sa = {iji 2 [1,

qa]} (Sb = {iji 2 [1, qb]}) and p = �n (+n). We define

PS|=E s;|i; pð Þ Y

i2 S|�Eð Þs� |ið Þp ð25Þ

to denote the exclusion of the indices in set E from theproduct, where (S| � E) is a set subtraction. By Leibniz’rule of differentiation of products [e.g., Boas, 1983] we find

Wkl ai; bi; nð Þ ¼Xl�1j¼0

l � 1

j

� �P jð ÞSa= kf g ak ;ai;�nð Þ

� P l�1�jð ÞSb

ak ; bi;þnð Þ ð26Þ

where

P jð Þ dj

dsjP

p

k

� � p!

p� kð Þ!k!


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are the binomial coefficients. Noting that for Gi = (s � |i),

d

dsG

pi ¼

dGi

ds

d

dGi

Gpi ¼

d

dGi

Gpi

the derivatives of P are found by application of the chainrule, which leads to the recursion relation

P mþ1ð ÞS|

s;|i; pð Þ ¼Xj2S|

Xmk¼0

m

k

� �R p; kð Þ � s� |j

� �p�k�1� P m�kð Þ

S|= jf g s; |i; pð Þ ð27Þ

for m � 0, where

R p; kð Þ ¼ p p� 1ð Þ p� 2ð Þ � � � p� kð Þ ð28Þ

Note that R(p, k) = 0 for k � p, which reflects that once p =n all derivatives of (s � |i)

n�n = 1 are zero. The exclusionnotation introduced above holds for derivatives of P overthe summation and product, i.e.,

P mþ1ð ÞS|=E ¼

Xj2 S|�Eð Þ

Xmk¼0

m

k

� �R p; kð Þ � s� |j

� �p�k�1P m�kð ÞS|= jf g[Eð Þ

ð29Þ

We implicitly assume that �i2;(� � �) = 1 andP

i2; (� � �) = 0,where ; denotes the empty set.[15] In all of the applications presented in this paper, all

roots of the denominator of equation (14) (ak) are distinct;however, when the roots are not all distinct, the abovesolution holds for mk = kkn in equation (16), where kk is thenumber of times the root ak is repeated. For a particularchoice of a viscoelastic half-space we need to identify theoperators % and8 for the media, and after verifying that theroots of equation (14) are real we directly use equation (10)to determine Gn(t).

2.4. The b(t) * &n(t) for an Earthquake Sequence

[16] Savage and Prescott [1978] noted that the faulthistory function for a periodic earthquake sequence can berepresented as the summation of steady sliding of twoquarter spaces, back slip on the fault, and episodic faultdisplacement during the earthquake. In steady sliding, eachside of the fault has block-like motion at the far-fieldvelocity, where the displacements on either side of the faultare constant with distance away from the fault (Figure 2).Steady sliding ensures that the far-field moves at the long-term fault slip rate, and in this model we do not impose far-field velocity conditions. The back slip model is such thatthe fault slips constantly with a velocity equal to thenegative of the far-field velocity, while in the episodic faultrupture model the fault instantaneously slips forward at thedesired fault rupture times. In the back slip and episodicrupture models, the interseismic displacements withrespect to the distance away from the fault are given byequation (12) (Figure 2).[17] The superposition of steady sliding, back slip and

episodic fault ruptures results in a model in which, from 0–D, the fault slips forward instantaneously during the earth-

quake and is locked at other times. At depths greater than Dthe fault slips steadily at the far-field velocity, while thecoseismic stresses diffuse in the viscoelastic half-space. Inthis model, steady sliding of the downward continuation ofthe fault loads the locked portion of the fault and drives thefar field. At depths greater than the maximum depth ofviscoelastic stress diffusion, shear is entirely localized onthe downward continuation of the fault. Whereas in a modelthat was driven by far-field velocity conditions, at sufficientdepths the deformation is simple shear, assuming there areno lower boundary conditions. For a periodic earthquakesequence, after a sufficient number of ruptures the displace-ments do not depend on the number of prior ruptures, andthe surface displacements do not depend on the steadydeformation at depth [e.g., Li and Rice, 1987; Savage,1990; Hetland and Hager, 2004]. During the first severalcycles, the displacements predicted by the deep-slip modelwill be larger than those after many ruptures, since theinitial displacements are the postseismic relaxation from thefirst earthquake plus the steady slip at depth. In a modeldriven by far-field velocity conditions, the displacementsfollowing the first few earthquakes will be smaller thanthose at steady state, since the displacements will be thepostseismic plus those of simple shear due to the far-fieldvelocities. We discuss the dependence on the steady dis-placements at depth, for both periodic and nonperiodicsequences, below.[18] Savage and Prescott [1978] only considered the back

slip of an elementary earthquake cycle, and they constructeda complete periodic earthquake sequence by superposing aseries of elementary cycles shifted in time by the period ofthe cycle. We modify the episodic back-slip model ofSavage and Prescott [1978] so that it is valid for generalnonperiodic earthquake sequences, as well as periodic ones.Savage and Prescott [1978] expressed b(t) * Gn(t) in termsof incomplete gamma functions. In this paper, we choose tofollow a different approach, avoiding the incompletegamma functions and using the Laplace transform relationgiven in equation (16).[19] For a sequence of earthquakes, where the pth rupture

occurs at time Tp with magnitude Dp, the rupture historyfunction is

bseq tð Þ ¼X1p¼0

DpH t � Tp� �

ð30Þ

where H(t � Tp) is the Heaviside step function, centered atTp: H(t � Tp) = 1 for t � Tp, and zero when t < Tp, andL{H(t � Tp)} = e�sTp/s [e.g., Boas, 1983]. For a periodic

Figure 2. Cartoon representation of the steady sliding andback-slip models, as well as the total deformation during aninterseismic period.


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earthquake sequence, Di = Di+1 and Ti+1 � Ti is constant.The episodic rupture model plus the back-slip model is

b tð Þ ¼X1p¼0

DpH t � Tp� �

� tv ð31Þ

where v is the average slip rate of the fault (v = �D/ �T , where�D and �T are the average rupture offset and recurrence time,respectively), which is also the far-field velocity. TheLaplace transform of equation (31) is

b sð Þ ¼X1p¼0

Dp

se�sTp � v

s2ð32Þ

and L{b(t) * Gn(t)} is

b sð ÞGn sð Þ ¼ gnX1p¼0

Dp

se�sTp � v

s2

" #� PSa s;ai;�nð Þ PSb s; bi;þnð Þ

ð33Þ

Since L�1{g(s) e�as} = g(t � a)H(t � a) [e.g., Boas, 1983],the inverse transform of equation (33) is

b tð Þ*Gn tð Þ ¼ gnX1p¼0

Dp G1 t � Tp;ai; bi; n� �

� H t � Tp� �

� gnvG2 t;ai; bi; nð Þ ð34Þ

where

Gm t;ai; bi; nð Þ ¼Xml¼1

W0l ai; bi; nð Þm� lð Þ! l � 1ð Þ! t

m�l

þXqak¼1

Xnl¼1

Wmkl ai; bi; nð Þ

n� lð Þ! l � 1ð Þ! tn�leak t ð35Þ

with W0l given by equation (26) with a0 = 0,

Wmkl ai; bi; nð Þ ¼

Xl�1h¼0

l � 1

h

� ��1ð Þh hþ m� 1ð Þ!

amþhk

� Wk l�hð Þ ai; bi; nð Þ ð36Þ

and Wk(l�h)(ai, bi; n) is defined in equation (26).[20] To nondimensionalize b(t) by the timescale janj�1,

we replace Tp with Tnp = janjTp in equation (31) and dividethrough by some characteristic rupture offset, D (D0p =Dp/D), so that v0n = �D0p/(Tnpþ1 � Tnp ), yielding

b tnð ÞD

¼X1p¼0

D0pH tn � Tnp� �

� v0ntn ð37Þ

For a periodic earthquake sequence, defining the period ofthe seismic cycle to be T = Tp+1 � Tp, we find thenondimensional period T janj. For a model with a Maxwellhalf-space, with mc = mm, the nondimensional period isT /2tM (see section 2.5.1), which is the parameter to asdefined by Savage and Prescott [1978], sometimes referred

to as the ‘‘Savage parameter’’. We extend the definition ofthe Savage parameter to to = T ja1j, which is the ratio of theseismic recurrence time to the relaxation timescale asso-ciated with the timescale of the fastest phase of relaxation.The Savage parameter is important in controlling theamount velocities vary with time in both periodic [Savageand Prescott, 1978] and nonperiodic [Meade and Hager,2004] earthquake cycles. Since to is related to the Wallacenumber, to gives the stability of nonperiodic rupturesequences, [e.g., Kenner and Simons, 2005]. Followingthe nondimensionalization approach in section 2.3, we findthat for an earthquake sequence, b(tn) * Gn(tn)/D is givenby equations (34)–(36), making the substitutions g ! gn,Tp! Tnp, D! D

0p, v! v0n, ai! ai/janj, bi! bi/janj, and

t ! tn. Given two models with sets of roots {ai, bi} and{a0i, b

0i}, coefficients g and g

0, and rupture displacements Dand D

0, b(tn)*Gn(tn)/D = b(t0n)*G

n(t0n)/D0.

[21] To compute only the postseismic displacements froma single earthquake of rupture magnitude D, ignoring thefar-field velocity, we use the fault rupture history inequation (30) with T0 = 0, D0 = D, and Di = 0 8 i > 0.Thus, for only the postseismic response from a singleearthquake, the convolution in equation (12) is

b tð Þ*Gn tð Þ ¼ gn DG1 t;ai; bi; nð Þ ð38Þ

and b(tn)*Gnn(tn)/D is found by substituting g! gn, D! 1,

ai ! ai/janj, bi ! bi/janj, and t! tn in equation (38) andits dependencies.

2.5. Applications to Specific Media

[22] The theory we presented in section 2.4 is valid formodels with a half-space composed of any linear viscoelas-tic rheology, given that the roots of equation (14) exist andare real. For illustration, we apply the theory applied to fourmodels: an elastic layer over Maxwell, standard linear solid(SLS), Burgers (a biviscous material), and triviscous visco-elastic half-spaces. For brevity, we refer to these models asthe Maxwell, SLS, Burgers, and triviscous model. For anymodel, we only need to determine the material coefficients(fi and yi) of the half-space material, after which allcalculations can be performed numerically. However, inthis section we determine the roots analytically in order tohighlight the method, as well as establish the timescales foreach of the models. We simplify and plot Gn(t) for a fewsimple cases.[23] In models of viscoelasticity, it is convenient to

consider mechanical analogue models composed of a con-figuration of springs (accommodating elastic deformation)and dashpots (accommodating deformation due to creep andrelaxation) [e.g., Flugge, 1967]. A spring in series with adashpot is a Maxwell viscoelastic element, representinginitial elasticity followed by a creep or relaxation phase ofdeformation. Because of the creep/relaxation of the dashpot,the deformation of the Maxwell element is nonrecoverablewith time. A spring in parallel with a dashpot is a Kelvinelement; the Kelvin element is sometimes referred to as aKelvin-Voight or Kelvin-Voigt element; in this paper wefollow Findley et al. [1976] and refer to it as a Kelvinelement. The Kelvin element is incapable of instantaneouselasticity; however, with an applied stress the dashpot will


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creep leading to a delayed elasticity. The deformation of theKelvin element is fully recoverable.2.5.1. Maxwell Half-Space[24] A Maxwell element is the conceptual model of a

Maxwell linear viscoelastic material, where the dashpot hasa viscosity of hm and the spring has a shear modulus mm(Figure 3a). The equation of motion for a linear Maxwellviscoelastic media is first-order (qf = qy = 1) and thematerial coefficients are

f0 ¼ 1 y0 ¼ 0

f1 ¼hmmm¼ tM y1 ¼ hm

ð39Þ

where tM is the Maxwell relaxation time.[25] To find the roots in equation (15), we simplify

equation (14) for a linear Maxwell viscoelastic half-space.When mc 6¼ mm, equation (14) reduces to

Gn sð Þ ¼ mc � mmmc þ mm

� �n sþ mcmmhm mc�mmð Þ

sþ mcmmhm mcþmmð Þ

" #nð40Þ

where qa = qb = 1 and

a1; b1 ¼�mcmm

hm mc � mmð Þ ð41Þ

where a and b are the addition and subtraction, respectively,and ja1j�1 is the timescale associated with the Maxwellmodel, which is always real. When mc = mm = m,

Gn sð Þ ¼ 1

2tM

� �n1

sþ 12tM

" #nð42Þ

so that qa = 1, a1 = �m/2h, and qb = 0. We then obtain Gn(t)directly from equation (10). When mc = mm = m, the timescalegiven in equation (41) reduces to 1/2tM and

Gn tð Þ ¼ 1

2tM

� �n1

n� 1ð Þ! tn�1 exp �t 1

2tM

� �ð43Þ

since

W1l ¼Xl�1j¼0

l � 1

j

� �P jð ÞSb P

l�1�jð ÞSa= 1f g ¼

1; l ¼ 1

0; l > 1

�ð44Þ

because Sb = Sa � {1} = ;, W11 = 1 and W1l = 0 for l > 1(i.e., all derivatives are zero). We can nondimensionalizeGn(t) by ja1j = m/2h

2tmð ÞGn1 t1ð Þ ¼ 1

n� 1ð Þ! tn�11 e�t1 ð45Þ

where t1 = t/2tM is dimensionless time when mc = mm. Whenmc = Xmm, timescales as (1/tM)(X/1 + X), and b1/ja1j = (1 �X)/(X + 1) and gn = (X � 1)/X for mc 6¼ mm, and when mc =mm, b1/ja1j = 1 and gn = 1; hence G1

n(t1) is constant forconstant X. Equation (45) is identical to the result obtainedby Savage and Prescott [1978] using the specific Laplacetransform pair of Erdelyi et al. [1954].[26] Gn(t) entirely describes the time response of the

deformation in the upper layer due to relaxation in thehalf-space following faulting in the upper layer. The diffu-sional nature of the Maxwell material is apparent in Gn(t)where increasing values of n describe relaxation at longertimes. Gn(t) is modulated by Wn in equation (12), and forincreasing n, Wn ! 0, so the contributions of higher modesof Gn(t) are small. We show Gn(t) for mc = mm and n = 1–40in Figure 4.2.5.2. Standard Linear Solid Half-Space[27] The standard linear solid (SLS) is conceptually

composed of a Kelvin element in series with a spring(Figure 3b). Hence the SLS is capable of instantaneouselastic deformation followed by a delayed elasticity. Theequation of motion for a SLS material is first-order (qf =qy = 1) and the material coefficients are

f0 ¼mv þ me

mvy0 ¼ me

f1 ¼hvmv¼ tK y1 ¼ me

hvmv¼ metK

ð46Þ

where me is the elastic shear modulus, mv and hv are the shearmodulus and viscosity of the Kelvin element (Figure 3), andtK is the Kelvin relaxation time, which is the timescaleleading to the delayed elasticity.[28] For an elastic layer of shear modulus mc overlying a

SLS half-space with material properties given above, whenmc 6¼ me we find Gn(s) from equation (14) to be

Gn sð Þ ¼ mc � memc þ me

� �n sþ mc mvþmeð Þ�memvhv mc�með Þ

sþ mc mvþmeð Þþmemvhv mcþmeð Þ

24

35n

ð47Þ

Figure 3. Mechanical analogue models of (a) Maxwell, (b) standard linear solid, (c) Burgers, and(d) triviscous viscoelastic materials.


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so that qa = qb = 1 and the inverse of the timescaleassociated with the SLS model is

ja1j ¼mc mv þ með Þ þ memv

hv mc þ með Þ ð48Þ

which is always real. When mc = me = m, qb = 0, qa = 1,

Gn sð Þ ¼ m2mvtK

� �n1

sþ 2mvþm2mvtK

� �n ð49Þ

and a1 = �[(2mv + m)/2mvtK]. In the case when all of theshear moduli are equal (mc = me = mv = m) a1 = �3/2tK. Asdemonstrated in section 2.5.1, when qa = 1 and qb = 0, theonly nonzero Wkl is W11 = 1, so that

Gn tð Þ ¼ 1

2tK

� �n1

n� 1ð Þ! tn�1e

�t 32tK ð50Þ

When all of the shear moduli are equal, the dimensionlesstime for the SLS is t1 = 3t/2tK, and

2tK3

� �Gn1 t1ð Þ ¼ 3n

1

n� 1ð Þ! tn�11 e�t1 ð51Þ

For the SLS half-space, Gn(t) decays much faster withrespect to n compared to the Maxwell half-space, for thesame viscosities. We show Gn(t) for mc = me and n = 1–20 inFigure 5.2.5.3. Burgers Half-Space[29] A Burgers rheology is capable of both a recoverable

and nonrecoverable relaxation phase and is considered abiviscous material. The mechanical analogue model of aBurgers rheology is a Maxwell element (with shear andviscosity of mm and hm) in series with a Kelvin element (mv,hv; Figure 3c). The equation of motion is second-order (qf =qy = 2) and the material coefficients are [e.g., Findley et al.,1976]

f0 ¼ 1 y0 ¼ 0

f1 ¼hmmmþ hm

mvþ hv

mvy1 ¼ hm

f2 ¼hmhvmmmv

y2 ¼hmhvmv

ð52Þ

The relaxation timescales of the Maxwell element (tM =hm/mm) and the Kelvin element (tK = hv/mv) appear in thematerial parameters, along with the timescale associated

Figure 5. (left) G1n(t1) and (middle) G1

n(t1)W*n for n = 1–20 using a standard linear solid viscoelastichalf-space with material mc = me, t1 = 3t/2tK, and H/D = 1. We show the square root of G1

n(t1)W*n toamplify the low magnitudes. (right) Power.


n(t1)W*n(1, 1) for n = 1–40 using a Maxwell model with mc = mm,t1 = t/2tM, and H/D = 1. We show the square root of G1

n(t1)W*n to amplify the low magnitudes. (right)RjG1

n(t1)W*n(1, 1)jdt1 (power) for each n.


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with the transfer of stress between the Kelvin and Maxwellelements (hm/mv).[30] For the case of an elastic layer overlaying a Burgers

viscoelastic half-space with mc 6¼ mm,

Gn sð Þ ¼ l2s2 þ l12þ l0

x2s2 þ x1sþ x0

� �n

¼ l2

x2

� �ns� b1ð Þn s� b2ð Þn

s� a1ð Þn s� a2ð Þn

ð53Þ

where

xi ¼ mcfi þ yi li ¼ mcfi � yi ð54Þ

The roots of the polynomials in equation (53) are

a1;2 ¼ a�;þ ¼�x1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix21 � 4x2x0

q2x2

ð55Þ

b1;2 ¼ b�;þ ¼�l1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil21 � 4l2l0

q2l2

ð56Þ

which can be verified to be real. When mc = mm, l2 = 0 andqb reduces to 1, b1 = �l0/l1, a1,2 are as in equation (55),and

Gn tð Þ ¼ gnXnl¼1

W1l ai; bi; nð Þn� lð Þ! l � 1ð Þ! t

n�lea1t

!

þ W2l ai; bi; nð Þn� lð Þ! l � 1ð Þ! t

n�lea2t

"ð57Þ

where g = li/x2, i = 1 and i = 2 for mc = mm and mc 6¼ mm,respectively. For the case when mc = mm = mv = m and hm =hv = h, defining tBB h/m, a1,2 = �2 �

ffiffiffi2

p/2tBB and

b1 = �1/2tBB. We can nondimensionalize time by eitherja1j = (2 +

ffiffiffi2

p)/2tBB or ja2j = (2�

ffiffiffi2

p)/2tBB, corresponding

to the fast and slow relaxation phases following a coseismicrupture. In Figure 6, we show Gn(t) with shear moduli equal,hm = hv, and n = 1–25. For n large, the series in equation (26)blows up due to computer precision. However, Gn(t) is quitesmall before the numerical instability is encountered and the

contribution from Gn(t) at such values of n is negligible to thedisplacements (see below).2.5.4. Triviscous Material[31] Finally, we consider a triviscous material, which is

capable of two phases of recoverable relaxation in additionto an unrecoverable phase. The material is conceptuallycomposed of a Maxwell element in series with two Kelvinelements, and we refer to the shear moduli and viscosities ofthe Maxwell and two Kelvin elements as mm, m1, and m2 andhm, h1 and h2, respectively (Figure 3d). The equation ofmotion for this triviscous material is third-order (qf = qy = 3)and the material coefficients are

f0 ¼ 1

f1 ¼hmmmþ h1

m1þ h2

m2þ hm

m1þ hm

m2

f2 ¼hmmm

h1m1þ hm

mm

h2m2þ h1

m1

h2m2þ hm

m1

h2m2þ hm

m2

h1m1

f3 ¼hmmm

h1m1

h2m2 ð58Þ

y0 ¼ 0

y1 ¼ hm

y2 ¼ hmh1m1þ h2

m2

� �

y3 ¼ hmh1m1

h2m2

Several timescales appear in the material coefficients, thetimescales of the Maxwell (tM = hm/mm) and two Kelvinelements (ti = hi/mi, i = 1, 2), as well as the crosstimescales (hm/mi, i = 1, 2).[32] For the case of an elastic layer (shear modulus

mc 6¼ mm) overlaying a triviscous viscoelastic half-space,equation (15) becomes

Gn sð Þ ¼ l3s3 þ l2s

2 þ l1sþ l0

x3s3 þ x2s2 þ x1sþ x0

� �n

¼ l3

x3

� �n Q3i¼0 s� bið ÞnQ3i¼0 s� aið Þn

ð59Þ

where xi and li are given in equations (54). The roots of thepolynomials in equation (59) can be determined analytically


n(t1)W*n for n = 1–25 using a Burgers linear viscoelastic half-space with mc = mm = mv, hm = hv, t1 = t(2 +

ffiffiffi2

p)/2tBB, and H/D = 1. We show the square root of G1

n(t1)W*nto amplify the low magnitudes. (right) Power.


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[e.g., Wang and Guo, 1989, Appendix I] or can be foundnumerically; moreover, the roots can be shown to always bereal from the realness of the relaxation modulus for atriviscous material. When mc = mm, l3 = 0 so qb reducesto 2 and bi are given by equation (56). For the case whenmc = mm = m1 = m2 = m, hm = h1 = h and h2 = h/2,

ai ¼m3h

ffiffiffiffiffi34

pcos uþ i

2p3

� �� 5

! "ð60Þ

for i = 1–3 where

u ¼ 1

3cos�1

�5934

ffiffiffiffiffi2

17

r !

b1;2 ¼�7�

ffiffiffiffiffi17

p

8

mh

Gn(t) is found directly from equation (10), which we donot reproduce here.

3. Displacements Due to Specific Fault Histories

[33] We demonstrate the theory using three fault histories:(1) postseismic displacements from a single earthquakeignoring the far-field velocity, (2) interseismic displace-ments for a periodic earthquake sequence, and (3) inter-seismic displacements for a nonperiodic earthquakesequence.

3.1. Displacements From a Single Earthquake

[34] In this section, we show the displacements andvelocities due to postseismic relaxation following a singleearthquake, for an elastic layer overlying a Maxwell, astandard linear solid, a Burgers and a triviscous half-space.For each case, the displacements are given by equation (12),and b(t) * Gn(t) is given by equation (38), where the roots(ai and bi) and the coefficients (g) are outlined insection 2.5. We compare the displacements predicted by

our analytic solution to those calculated using the finiteelement package Adina (Adina R&D, Inc). For the finiteelement calculations, we use a relatively low precisionmodel and a relatively high precision model. In both finiteelement models, the fault ruptures with a uniform slip fromthe surface to depth D and linearly tapers to zero from 1.0–1.2D, where the thickness of the elastic layer is 1.2D. Themodels are 200D by 200D, containing 11,235 finite ele-ments. We use four-node rectangular elements and a timestep of one half the shortest relaxation time in the low-precision models, while in the high-precision models weused nine node elements and a time step of one tenth of theshortest relaxation time. In the finite element calculations,the taper of slip from 1.0 to 1.2D is exact, whereas in theanalytic solutions we account for the taper by superposing40 models with uniform slip extending from 0 to 1 + j0.2/40D for j = 0–39. Since H/D is not constant for theindividual slip models, the displacements do not scale withD (see equation (8)), and we nondimensionalize distance byD purely for convenience. The rheological parameters inboth the finite element and analytic models are identical andare given below.3.1.1. Postseismic Relaxation for a Maxwell Half-Space[35] For the case of an elastic layer over a Maxwell

viscoelastic half-space, the series over n in equation (12)converges faster in the near field and at early time than inthe far field and at late time. When mc = mm, the displace-ments converge at n about 10t1 (t1 is the mechanicaltimescale of the relaxation, see section 2.5.1), in the nearfield up to 15t1 (30 Maxwell times), while at distance 15Hfrom the fault the displacements do not converge until n 40 (Figure 7). In the near-field the transient response decaysquickly, while there is a slower decay in the far field.[36] The postseismic displacements predicted by

equation (12) match the displacements from the finiteelement calculations quite well (Figure 7), with the excep-tion of immediately after the earthquake, where the finiteelement model is not able to resolve the stresses at the faulttip well. Moreover, the difference between the analytic

Figure 7. (left) Displacements following a single earthquake in a Maxwell model with H/D = 1 andvarious nfinal. (right) Displacements following a single earthquake, ua, (solid lines) compared to thedisplacements determined by finite element calculations (dashed lines; un is the high-precision model);model geometry is given in the main text. For all models mc = mm and t1 = t/2tM.


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solution and the high-precision FE calculation is much lessthan the difference with the low-precision calculation,especially during early times. The discrepancy betweenthe analytic solution and a finite element calculation withgreater element density would be less.[37] For each choice of mm and mc, the displacements due to

postseismic relaxation are distinct, since the relaxation time-scales and the gv coefficients are different. In Figure 8 weshow the postseismic displacements and velocities for mc =3mm/2, mm, and mm/2, so that timescales as 3/5tM, 1/2tM and1/3tM, respectively. The shear modulus of the continentalcrust is approximately 30 GPa, while that of the mantle isabout 70 GPa, so mc mm/2 is realistic for the continents.Assuming mc = mm results in a small, but nonnegligible,difference in the predicted displacements (Figure 8).3.1.2. Postseismic Relaxation for a Standard LinearSolid Half-Space[38] For the case of an elastic layer over a SLS visco-

elastic half-space, the series over n in equation (12) con-verges at low nfinal for all times and distances (Figure 9).The postseismic displacements predicted by the analyticsolution match the displacements from the finite element

calculation quite well (Figure 9), again with the exception ofimmediately after the earthquake.[39] For given moduli (mc, me, mv), the displacements relax

to distinct values. For mc = me, and mv = me, 2me/3, and me/3,timescales as 3/2tK, 7/4tK and 5/2tK, respectively. Thedisplacements relax to profiles resembling the coseismicprofile, exhibiting the delayed elasticity of the SLS material,where a weaker mv leads to a larger difference in coseismicand relaxed displacements (Figure 10).3.1.3. Postseismic Relaxation for a Burgers Half-Space[40] For the Burgers half-space, at large n, the series in

equation (26) blows up; however, the magnitude of Gn(t) isquite small before the numerical instability is encountered,and even out to 15D away from the fault, the displacementsconverge at n smaller than that where instability arises. Theanalytic solution matches the finite element calculationsquite well, except during early times (Figure 11).[41] When the relaxation timescale of the recoverable

Kelvin element is less than that of the nonrecoverableMaxwell element, the displacements relax rapidly immedi-ately after the earthquake, followed by a slower relaxation.The initial relaxation is similar to the response of the SLS

Figure 8. (left) Displacements and (right) velocities following a single earthquake in a Maxwell modelwith mc = (m/2)mm, t1 = (t/tM)[m/(m + 1)], and H/D = 1.

Figure 9. (left) Displacement following a single earthquake in a SLS model with H/D = 1 and variousnfinal, and (right) postseismic displacements compared to the displacements determined by finite elementcalculations with mc = mv and t1 = 3/2tK.


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half-space when hv ! hm, while it is closer to the Maxwellmodel when hv " hm (Figure 12). When time is rescaled,the initial Kelvin relaxation phase of the Burgers models issimilar to appropriately rescaled Maxwell models, and onlyin later times is the relaxation of the Maxwell element in theBurgers models apparent (Figure 12). During later times, thesecond phase of relaxation is similar to the slow relaxationof displacements predicted by a Maxwell half-space withthe same hm.3.1.4. Postseismic Relaxation for a TriviscousHalf-Space[42] For a triviscous half-space the solution again blows

up at large n, and we choose a sufficiently low nfinal to avoidthe instabilities. The postseismic displacements converge atshort distances from the fault well before nfinal; however, atlong distances from the fault and at late times the series doesnot quite converge by nfinal (Figure 13). The displacementstruncated at nfinal compare well to the displacements pre-dicted by the finite element calculation, except during earlytimes near the fault and at later times far from the fault

(Figure 13). The former is due to mesh inadequacies in thefinite element model as discussed above, whereas the latteris due to the truncation of the summation in n in equation(12). The postseismic displacements and velocities pre-dicted by the triviscous half-space are similar to those forthe Burgers half-space, except the triviscous model predictsa third relaxation phase. When the first Kelvin element inthe triviscous model is identical to the Kelvin element in theBurgers model, while the second Kelvin element inthe triviscous model is weaker than the first, early in timethe displacements relax faster compared to the Burgersmodel, while later in time the rate of relaxation is similar(Figure 14). During the intermediate times, the tertiaryrelaxation phase dominates.

3.2. Displacements Due to Periodic Earthquakes

[43] The displacements through a seismic cycle in aperiodic earthquake sequence are given by equation (12),using the convolution in equation (34), superimposed withthe steady sliding model. We demonstrate the displacements

Figure 10. (left) Displacements with respect to (right) time and (left) distance following a singleearthquake in a SLS model with me = mc, mv = (m/3)me, t1 = (t/tK)[(2m + 3)/2m], and H/D = 1. Gray line inFigure 10 (right) is the coseismic displacements, and black lines are displacements at t1 = 10, where theline style is as in the legend in Figure 10 (left).

Figure 11. (left) Displacements following a single earthquake in a Burgers model with H/D = 1 andvarious nfinal and (right) postseismic displacements compared to the displacements determined by finiteelement calculations. mc = mm = mv, hm = hv and t1 = (2 +

ffiffiffi2

p)/2tBB.


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during a periodic earthquake sequence (Figure 15) formodels of Maxwell, standard linear solid (SLS), Burgers,and triviscous half-spaces using to = 5 = 10ja1j, whereja1j�1 is the timescale associated with the fastest phase ofrelaxation (Figure 16). We use H/D = 1 in all models, mc =mm in the Maxwell model, mc = me = mv in the SLSmodel, mc = mm = mv and hm = hv in the Burgers model,and mc = mm = m1 = m2 and hm = h1 = 2h2 in the triviscousmodel. After a sufficient number of ruptures, the interseismicdisplacements are the same in all earthquake cycles [e.g.,Savage and Prescott, 1978; Li and Rice, 1987]. When thedisplacements do not depend on the particular cycle, we saythat the displacements (or velocities, etc.) are cycle invariantor that the system is at cycle invariance. The displacementsfollowing the initial fault rupture are larger than the invariantdisplacements (Figure 16) due to the addition of constantslip on the continuation of the fault at depth. In the casewhen the far field is driven by velocity boundary conditions

and steady deformation at depth is simple shear, the initialdisplacements will be smaller than the invariant displace-ments, since the initial displacements are the postseismicdisplacements plus a simple shear profile. However, duringinvariance, the surface displacements do not depend on thesteady deformation at depth [e.g., Li and Rice, 1987;Savage, 1990; Hetland and Hager, 2004].[44] For the model of the Maxwell half-space, the cycle

invariant velocities predicted by equation (12) match thosepredicted by the solution of Savage [2000] (Figure 17a).The velocities throughout an invariant seismic cycle can becharacterized as perturbations to the average velocity pro-file, which is identical to the elastic model of interseismicstrain accumulation proposed by Savage and Burford[1973] (v = (v/p)tan�1(x/D), referred to as the elastichalf-space model; Figures 17 and 18). In models with weakrheologies, where the relaxation timescale is short comparedto the interseismic period (to = Tja1j greater than about 1),

Figure 13. (left) Displacements following a single earthquake in a triviscous model for various nfinaland (right) postseismic displacements compared to the displacements determined by a finite elementcalculation with mc = mm = m1 = m2, hm = h1 = 2h2, and t1 = ja1jt, where ja1j is given in equation (60).

Figure 12. (left) Displacements through nondimensional time and (right) an instance of rescaled timefollowing a single earthquake in models with Maxwell, SLS, and Burgers viscoelastic half-spaces andD/H = 1. t1 = ja1jt, where a1 is the timescale associated with the fastest relaxation of each material. Timewas rescaled using mc = 30 GPa, the shear moduli relations in Figure 12 (left) and the indicated viscositiesfor the Maxwell, SLS, and Burgers half-spaces, where for the Burgers model hv is as in the legend inFigure 12 (left).


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the perturbations to the elastic model in the image solutionbecome significant (Figure 17). With sufficiently stiffrheologies, where the relaxation timescale is long comparedto the interseismic interval (to small), the velocitiesthroughout the cycle are identical to the elastic half-spacemodel (Figure 18). The invariant velocities in a periodicsequence are faster than the elastic half-space model early inthe cycle and slower than the elastic half-space modellater in the cycle. When to is large, the velocities forthe SLS half-space are large immediately after therupture and quickly decay to a near-constant velocity profileslightly slower than the elastic half-space velocity profile(Figure 17b). In the Burgers and triviscous models,the velocities also decay relatively rapidly early in thecycle, while they decay at a slower rate late in the cycle(Figures 17c and 17d).[45] It is possible to choose the material properties of a

Burgers rheology such that early in the seismic cycle thevelocities are similar to a Maxwell model, while later in thecycle the velocities decay slowly and are similar to anelastic half-space model with a reasonable locking depth.We illustrate such a phenomenon in Figure 19, where thevelocities in the Burgers model throughout the second halfof the cycle are close to an elastic strain accumulation modelwith an apparent locking depth of 1.2–1.9 times the actualdepth and an apparent slip rate of 90–95% of the actual sliprate (Figure 19). For the Maxwell model, the velocities latein the cycle resemble an elastic half-space model withlocking depth and slip rate 10 times and 90% of the actualvalues, respectively. Such a Burgers model may explainthe geodetic velocities near the North Anatolian faultwhere the postseismic deformation showed large transients[Ergintav et al., 2002] while the preearthquake velocitieswere described by an elastic half-space model appropriatefor the North Anatolian fault [Meade et al., 2002].

3.3. Displacements Due to a Nonperiodic Earthquake

[46] Our solution is valid for any earthquake sequence;however, we only demonstrate the solution for one partic-ular nonperiodic earthquake sequence in this paper. Weconstruct this demonstration rupture history function byperiodically repeating an elementary rupture sequence,

which we refer to as a cluster. The elementary cluster iscomposed of a group of four ruptures of magnitude D

repeated every T /2 followed by a group of four rupturesof magnitude D/2 repeated every T (Figure 15). We use T =10ja1j, where ja1j�1 is the timescale of the fastest phase ofrelaxation for each of the materials. As before, we usemc = mm in the Maxwell model, mc = me = mv in the SLSmodel, mc = mm = mv and hm = hv in the Burgers model, andmc = mm = m1 = m2 hm = h1 = 2h2 in the triviscous model, andto = 5.0 for each model.[47] Within the elementary cluster, the displacements vary

from cycle to cycle; however, after repeating the cluster asufficient number of times, the variations of displacementsthroughout the cluster are the same in all of the clusters;hence we refer to these displacements as cluster invariant(Figure 20). At cluster invariance, the variation of displace-ments within the elementary cluster is due to the modeltending toward a new, periodic, cycle invariant state (E. A.Hetland and B. H. Hager, Interseismic deformation: Cycle

Figure 14. (left) Displacements through nondimensional time and (right) an instance of rescaled timefollowing a single earthquake in models with Burgers (biviscous) and triviscous viscoelastic half-spaceswith H/D = 1, t1 = ja1jt, where a1 is the timescale associated with the fastest relaxation of each material.Time was rescaled in Figure 14 (left) using mc = 30 GPa, hm = 1019 Pa s, and relations given in Figure 14(right).

Figure 15. Six cycles of a periodic earthquake sequenceand an elementary sequence in a nonperiodic earthquakesequence. The parameter b(t) is the fault history function, Dis a reference rupture displacement, and the nondimensionaltime is t = t/T , where T is the recurrence time in theperiodic sequence.


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invariance, slip-rate, and rheology, submitted to Geochem-istry, Geophysics, Geosystems, 2005, hereinafter referred toas Hetland and Hager, submitted manuscript, 2005). Forexample, when the recurrence time first changes to T /2, thesystem adjusts toward a new cycle invariant state, which isthe same as that for a periodic system of period T /2. At thestart of the next cluster, when the recurrence time changes toT , the system then evolves toward the new cycle invariantstate characterized by T .[48] As in the periodic sequence, before the system

achieves cluster invariance, the displacements in this modelwill be different than in a model that is driven by far-fieldvelocity boundary conditions. However, during clusterinvariance, the displacements do not depend on the partic-ular distribution of steady deformation at depth. We illus-trate this by showing that the displacements predicted bythis model match those predicted by a finite element (FE)solution driven by far-field velocities (Figure 21). We usethe finite element package GeoFEST 4.3 [Lyzenga et al.,2000]. GeoFEST does not include higher-order linear vis-coelastic rheologies, so we only compare the FE calculationto the analytic Maxwell model. In the FE model, the faultuniformly breaks the entire elastic layer of thickness H andis tapered to zero displacement within the viscoelasticregion from 1.0–1.1H. Because of antisymmetry, we only

compute the displacements for x � 0, and we specify thatthe FE model is 120D # 120D. We apply free slip boundaryconditions to the top and bottom of the model, a no-displacement condition on the edge of the model containingthe fault, and a constant velocity condition on the other edgeof the model. In the analytic solution, the fault also breaksthe entire elastic layer. However, we do not account for thetaper in the analytic solution, so there is a geometricalmismatch between the analytic solution and finite elementcalculations. Furthermore, the finite element model is com-posed of 5041 four-node elements, so the numerical preci-sion is lower than the calculations presented in section 3.1.Our purpose for this comparison is only to show that theevolution of displacements throughout the elementary clus-ter during cluster invariance does not depend on theparticular model of deep deformation (Figure 21).[49] The variations of velocities during a cycle in a

nonperiodic sequence do not follow the same variation withrespect to the elastic half-space model as during cycles of aperiodic earthquake sequence [Meade and Hager, 2004;Hetland and Hager, submitted manuscript, 2005]. It is notour intention to fully characterize the complex behavior ofthe changes in displacements and velocities within thisnonperiodic earthquake sequence. However, to illustratethe large variation possible, we show the cluster invariant

Figure 16. Displacements throughout cycles within a periodic earthquake sequence with to = 5.0 andnondimensional period 10ja1j, where a1 for each material is given in section 2.5. Dashed lines are thedisplacements during the cycle following the first rupture and solid lines are displacements during aninvariant cycle. Displacements are for models with (a) Maxwell, (b) SLS, (c) Burgers, and (d) triviscousviscoelastic half-spaces, with H/D = 1 and rheological properties given in the text.


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velocities 40% through each of the cycles of the elementarycluster in Figure 22.

4. Discussion

[50] Our solution for interseismic displacements assumesthat the far field is driven by steady slip on the extension ofthe fault at depth. After Savage and Prescott [1978], we usethis model of steady slip on the fault below the locking

depth to ensure that the far field moves at the long-termfault slip rate. Below the depths of viscoelastic diffusion ofcoseismic stresses, shear is entirely concentrated on thedownward continuation of the fault. The deep deformationinitiates at time zero and is steady through time. Fullylocalized deformation at depth can be considered an end-member model of deep deformation, where simple sheardeformation would be the other end-member. A method ofachieving simple shear deformation at depth would be to

Figure 17. Cycle invariant velocities throughout a cycle within a periodic earthquake sequence withto = 5.0 = 10ja1j. Velocities are shown at t/T = 0.1, 0.4, 0.7, and 0.9 and are for models with (a) Maxwell,(b) SLS, (c) Burgers, and (d) triviscous viscoelastic half-spaces, with properties as in Figure 16. Blackdashed lines are the average velocity profile, and black dash-dotted lines in Figure 17a are velocitiescalculated using Savage [2000], and vsl is the fault slip rate.

Figure 18. Cycle invariant velocities throughout a cycle within a periodic earthquake sequence, withto = 0.5 = 10ja1j, for (a) Maxwell and (b) Burgers models. See Figure 17 caption.


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superimpose a far-field constant velocity with a sequence ofrepeated earthquakes [e.g., Bonafede et al., 1986; Pollitz,2001]; this is how finite element models are often con-structed. In this model, after the first earthquake thedisplacements would be the sum of the postseismic dis-placements and simple shear displacements from the far-field boundary conditions and will be lower than those atinvariance. Arguably, it is more geologically reasonable thatthe steady displacement profile across a newly activatedfault would be close to simple shear, as opposed to an arctangent profile predicted when the fault is loaded frombelow. However, once the model spins up to an invariantstate the interseismic displacements will not depend on thesteady deformation at depth. Li and Rice [1987], examininga periodic earthquake sequence, concluded that during cycleinvariance, the particular model of steady, deep deformationdoes not effect the interseismic surface displacements.Several other researchers have demonstrated the sameconclusion [e.g., Savage, 1990; Hetland and Hager,2004]. When extending the concept of cycle invariance tononperiodic earthquake sequences, we assume that thesequence is composed of a nonperiodic sequence containinga finite number of ruptures, repeated periodically. In es-sence, it is still a periodic sequence; however, the cycle isdefined as the elementary cluster instead of the interseismicperiod. Hence cycle invariance is the same concept ascluster invariance. With this constraint on nonperiodicearthquake sequences, the particular model of deep defor-mation is irrelevant.[51] We motivated this study, in part, from observations

of the time evolution of displacements following an earth-quake. Other mechanisms have been proposed to explainsuch observations, including poroelastic effects [e.g.,

Peltzer et al., 1998; Jonsson et al., 2003; Fialko, 2004],after slip [e.g., Marone et al., 1991; Burgmann et al., 2002;Hearn et al., 2002; Fialko, 2004; Johnson and Segall, 2004;Montesi, 2004] and nonlinear rheologies [e.g., Freedand Burgmann, 2004]. Existing models of postseismicporoelastic effects do not account for the time behavior ofdeformation, the models are simply a subtraction of aninitial state from a final state [e.g., Peltzer et al., 1998;Fialko, 2004]. Therefore any multiphase character ofrelaxation is not addressed by poroelasticity. Postseismicporoelasticity involves changes in volume [e.g., Booker,1974; Rice, 1980] and therefore cannot be considered in thismodel, which considers shear only.[52] Two classes of after-slip models have been proposed

to explain transient displacements following an earthquake.The first class of studies is when a given constitutive law isprescribed to the fault below the locking depth, such as ratestate friction [e.g., Marone et al., 1991; Hearn et al., 2002]or a viscous relaxation of the fault [e.g., Hearn et al., 2002;Johnson and Segall, 2004; Montesi, 2004]. Following therupture, the lower fault slips to relax the coseismic stresses,leading to deformation at the surface. Most of these after-slip models predict surface deformation that is not charac-terized by a single exponential relaxation phase [e.g., Hearnet al., 2002; Montesi, 2004]. The second class of after-slipmodels invert each epoch of deformation for a distributionof slip on elastic dislocation patches at depth, subject toconstraints [e.g., Shen et al., 1994; Burgmann et al., 2002].These models of after slip are able to describe the timebehavior of postseismic deformation relatively well, ac-counting for multiple relaxation phases, if present. Forinstance, to describe two phases of relaxation, essentially,one would simply require two distributions of after slip. The

Figure 19. Interseismic velocities (left) at t/T = 0.02, 0.1, 0.4 and (right) at t/T = 0.4, 0.7, and 0.98during an invariant seismic cycle (T = 20tM) for a Maxwell (mc = mm, hm = mmtM; thin solid lines) andBurgers (mc = mm = mv, hM = 10mmtM, hv = mvtM; thin dashed lines) model, with H/D = 1. The averagevelocity profile for both models (thick black line) is equivalent to an elastic half-space model (EHSM)with locking depth D and slip rate vsl. Also shown at right is an EHSM with locking depth 10D and sliprate 0.90vsl (thick black dashed line) and a range of EHSM with locking depths 1.2–1.9D and slip rates0.90–0.95vsl (shaded region).


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correspondence principle shows that in a linear system,postseismic surface deformation due to linear viscoelasticstress relaxation is equivalent to some distribution of afterslip on the downward continuation of the fault [e.g., Savage,1990]. The complete relationship between rheology anddistributions of slip is not well established, especially fora finite rupture, but it is possible that some of the deforma-tion modeled as after slip may be due to viscoelasticrelaxation, and vice versa.[53] Postseismic studies using nonlinear rheologies de-

scribe the effective viscosity as stress dependent [e.g.,Hearn et al., 2002; Freed and Burgmann, 2004]. Freedand Burgmann [2004] successfully described the postseis-mic deformation of the 1992 Landers and 1999 Hector Mineruptures using a model of nonlinear rheologies in themantle, although they assumed a very low backgroundstress level. Similarly, Hearn et al. [2002] found that inorder to describe the postseismic displacements followingthe I

:zmit rupture, the background stress level needed to be

quite low. Ivins [1996] and Pollitz [2003] both concludedthat some type of biviscous rheology was sufficient todescribe the postseismic observations considered by Freedand Burgmann [2004]. The nonlinear Andrade rheology hasbeen proposed to explain creep tests of olivine, describing acontinuous spectrum of relaxation phases [e.g., Gribb and

Figure 20. Cluster invariant displacements, 2D (H/D = 1) from the fault, throughout the cycles of theelementary cluster of a nonperiodic seismic cycle with to = 5.0. Dashed lines are the displacementsduring the short cycles and solid lines are displacements during the long cycles (Figure 15).Displacements are for models with (a) Maxwell, (b) SLS, (c) Burgers, and (d) triviscous viscoelastic half-spaces; a1 for each model is given in section 2.5. Cycle number refers to the cycle within the short- orlong-period cycles in the elementary cluster (Figure 15), ucs is the coseismic displacement, and time isrelative to the start of each cycle.

Figure 21. Cluster invariant displacements in each cyclefor a model with a Maxwell half-space (Figure 20a). Solidand dashed lines are the displacements from the analyticsolution and finite element calculation, respectively, andtime is relative to the start of each cycle.


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Cooper, 1998]. While the Andrade rheology is nonlinear intime, it is linear in stress, and the theory we present canincorporate an Andrade rheology. However, we are unableto incorporate stress-dependent rheologies into this theory.Moreover, as our solution of interseismic displacementsrelies on the superposition of models, the linearity ofrheologies with respect to stress is essential.[54] We restricted the analysis in this paper to a simple

two-dimensional geometry of a fault in an elastic layeroverlying a viscoelastic half-space, with prescribed disloca-tions on the fault. Inelastic processes of the upper seismo-genic layer have been shown to be important [e.g., Bonafedeet al., 1986; Peltzer et al., 1998; Fialko, 2004; Johnson andSegall, 2004]. Our analysis can be extended to models of aviscoelastic upper layer over a viscoelastic lower half-space,since the Laplace transform relation we present can beapplied directly to equation (13) when all rheologies arelinear viscoelastic. It is also possible to include stress-dependent slip on the fault following the analysis ofJohnson and Segall [2004]. Additionally, surface displace-ments from models with depth-dependent rheologies arequite different than when the lower region is homogeneous[e.g., Rundle, 1982; Piersanti et al., 1995; Savage, 2000],and in a subsequent paper we extend this theory to layeredviscoelastic models. Fault ruptures at given shear stresslevels can be implemented in our models following existingmethods [e.g., Bonafede et al., 1986]; however, in our

models driven by steady sliding below the locking depth,the stresses on the fault are quite different than in modelsdriven by far-field shear. Finally, three dimensional featuresof faults, such as finite length ruptures and fault geometry,strongly affect interseismic deformation, especially far fromthe fault and beyond the ends of the rupture [e.g., Rundleand Jackson, 1977; Smith and Sandwell, 2004; Meade andHager, 2005]. The model geometry we consider is notappropriate to model data over large spatial scales.

5. Conclusions

[55] We present a solution of the deformation near aninfinite strike-slip fault in an elastic layer overlying a linearviscoelastic half-space. The solution is valid for any linearviscoelastic rheology and any earthquake sequence. Weconstruct our earthquake sequences such that constant slipon the continuation of the fault below the locking depthdrives the far field and loads the fault. When the earthquakesequence is periodic or composed of the periodic repetitionof a nonperiodic, finite length earthquake sequence, theinterseismic displacements do not depend on the mechanismassumed to load the fault [e.g., Li and Rice, 1987]. Oursolution is a generalization of the work of Savageand colleagues [Savage and Prescott, 1978; Savage andLisowski, 1998; Savage, 2000]. Our solution compares wellto those solutions, as well as to finite element calculations.

Figure 22. Cluster invariant velocities 40% of the way through the cycles of an elementary cluster in anonperiodic earthquake sequence (Figure 15). Velocities are for models with (a) Maxwell, (b) SLS,(c) Burgers, and (d) triviscous viscoelastic half-spaces. Line style and color are as in Figure 20, and blackdash-dotted lines are cycle invariant velocities 40% of the way through a cycle in a periodic sequencewith period T .


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Our analytic model is only valid for a relatively simplegeometry; however, the model presents an elegant tool toexplore the evolution of displacements for relatively com-plex rheologies and rupture recurrence histories. Finally, themethod we derived to describe the time dependence due toviscoelastic relaxation may facilitate the inclusion of gen-eral viscoelastic rheologies into previously proposed 2Dmodels [e.g., Cohen, 1980; Thatcher and Rundle, 1984;Pollitz, 2001] and 3D models [e.g., Rundle and Jackson,1977; Rundle, 1982; Piersanti et al., 1995; Pollitz, 1997;Smith and Sandwell, 2004].

Notation

s, e shear stress and engineering shear strain,respectively.

%, 8 stress and strain differential operators,respectively.

fk, yk stress and strain coefficients, respectively,together material coefficients.

m shear modulus.mc shear modulus of upper elastic layer.h viscosity.

m(s) equivalent shear modulus of a viscoelasticmaterial.

u(x, t), v(x, t) displacements and velocities at the surface.D, H locking depth and thickness of the elastic

layer, respectively.d = H/D � 1, ratio of thickness of the elastic

layer to the locking depth.b(t) fault rupture history function.Gn(t) time-dependent mechanical coupling coef-

ficient.g ratio of leading factors of the denominator

and numerator of Gn(s).qa, qb number of roots of the denominator and

numerator of Gn(s), respectively.ai, bi roots of the denominator and numerator of

Gn(s), respectively.S| set of indices of roots |i.

PS(s, |i; p) product of (s � |i)p over the indices in set

S.(m, k) binomial coefficients.R(p, k) = p(p � 1)� � �(p � k).

Dp magnitude of the pth earthquake.Tp time of the pth earthquake.

H(t � a) Heaviside function H(t � a) = 0 for t < aand H(t � a) = 1 for t � a.

v, vsl average, long-term slip rate of the fault.tM, tK timescales of the Maxwell and Kelvin

elements, respectively, also referred to asthe Maxwell and Kelvin relaxation times.

tn = tjanj nondimensional time, where janj�1 aremechanical timescale of the fastest (n = 1)and the slowest (n = qa) relaxation phases;similarly, Tnp = Tpjanj is the nondimen-sional time of the pth rupture.

to = Tja1j Savage parameter, extended to generalviscoelastic rheologies.

[56] Acknowledgments. We gratefully acknowledge insightfulreviews by Maurizio Bonafede and one anonymous reviewer, as well asthe Editor John C. Mutter. We thank Rick O’Connell and Brendan Meade

for early discussion that motivated this work. We thank K. J. Bathe andAdina R&D, Inc. for the use of Adina and G. Lyzenga and J. Parker for theuse of GeoFEST. We used Matlab (The Mathworks, Inc.) to numericallyevaluate the solution presented in this paper. This research was supportedby NSF grant EAR-0346021.

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��E. A. Hetland and B. H. Hager, Department of Earth, Atmospheric and

Planetary Sciences, Massachusetts Institute of Technology, 54-610, 77Massachusetts Avenue, Cambridge, MA 02139, USA. ([email protected])


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B10401

postseismic and interseismic displacements near a strike-slip...

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