Postseismic and interseismic displacements near a strike-slip fault: A two-dimensional theory for general linear viscoelastic rheologies

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110,, doi: /2005jb003689, 2005 Postseismic and interseismic displacements near a strike-slip fault: A two-dimensional theory for general linear viscoelastic rheologies E. A. Hetland and B. H. Hager Department 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 [1] We present an analytic solution for the deformation near an infinite strike-slip fault in an elastic layer overlying a linear viscoelastic half-space. The theory is valid for any linear viscoelastic rheology and any earthquake sequence. This is a generalization of the work of J. C. Savage and colleagues, which only holds for models with a uniform shear modulus, Maxwell viscoelastic lower half-space, and periodic rupture recurrence. We demonstrate 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 example postseismic displacements, interseismic displacements in a periodic earthquake sequence, and interseismic displacements for a nonperiodic earthquake sequence. Our solution is for a simple geometry; however, the model presents an elegant tool to explore the evolution of displacements 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,, doi: /2005jb Introduction [2] The two dimensional, time-dependent solution of the interseismic displacements near a strike-slip fault, proposed by Savage and Prescott [1978], has been widely used to guide intuition about interseismic displacements and the earthquake cycle [e.g., Savage, 1990; Lisowski et al., 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 an elegant tool for building intuition, it makes several assumptions that probably are not applicable to the lithosphere. In this paper, we present a more general form of the model. [3] Nur and Mavko [1974] demonstrated that the timedependent displacements following a dislocation in an elastic layer overlying a viscoelastic media could be obtained from an elastic solution by using the correspondence principle of viscoelasticity. Using the image solution of Rybicki [1971] and following Nur and Mavko [1974], Savage and Prescott [1978] used the correspondence principle to solve for the surface displacements throughout a seismic cycle from a fault breaking an upper elastic layer overlying a Maxwell viscoelastic half-space. Savage and Prescott [1978] assumed that the shear modulus in the upper layer and half-space was identical and that successive ruptures occurred with constant repeat time and constant coseismic displacement. Savage and Lisowski [1998] reformulated the solution of Savage and Prescott [1978] and Savage [2000] extended the model to the displacements at depth, as well as at the surface; most modern calculations of Copyright 2005 by the American Geophysical Union /05/2005JB003689$09.00 the seismic cycle model are done using one of these later formulations. The transformation Savage and Prescott [1978] used to account for the time dependence due to Maxwell viscoelastic relaxation has been used in geometrically 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 is smaller than twice the relaxation timescale of the Maxwell half-space, the velocities throughout a seismic cycle will be approximately constant and roughly equal to the deformation predicted by an elastic half-space model. Only when the seismic repeat time is long compared to the Maxwell relaxation time will the surface velocities vary appreciably. A consequence of this behavior is that if the velocities before an earthquake can be described by an elastic half-space model with parameters appropriate for the fault, then there will be little to no postseismic transient deformation. On the other hand, if there is significant postseismic deformation, then the velocities across the fault should be close to simple shear and slower than those predicted by an elastic half-space model. Geodetic observations before and after large strike-slip ruptures are often contrary to these predictions. For instance, large postseismic transients were observed following the 1999 Izmit and Düzce earthquakes on the North Anatolian fault [Ergintav et al., 2002]; however, before the earthquakes the velocities across the fault were described using an elastic half-space model appropriate for the North Anatolian fault [Meade et al., 2002]. We show below that a model with a biviscous rheology, in contrast to the univiscous Maxwell rheology, may be able to explain both of these preseismic and postseismic observations. 1of21

2 [5] In this paper, we generalize the solution of Savage and Prescott [1978] to hold for any linear viscoelastic rheology and a nonperiodic fault rupture history, or earthquake sequence. In the lithosphere, the variation in shear modulus with depth has been shown to be significant on surface displacements [e.g., Hearn et al., 2002], and recent studies of postseismic deformation have indicated that the inelastic region of the Earth has a much richer time dependence than the single exponential decay predicted by models with Maxwell viscoelastic rheologies [e.g., Ivins, 1996; Hearn et al., 2002; Pollitz, 2003; Freed and Bürgmann, 2004]. Additionally, recent paleoseismic studies have indicated that rarely do earthquakes occur periodically [e.g., Grant and Sieh, 1994; Bennett et al., 2004; Weldon et al., 2004]. Since the surface displacements are of principle interest to modeling of geodetic data, we present the analytic derivation of only the surface displacements. However, the solution for the displacements at depth can be achieved directly using the derivation presented here, as the geometry factors are not dependent on the rheologies. [6] For illustrative purposes, we apply the general linear viscoelastic theory to four viscoelastic rheologies of the lower half-space: (1) Maxwell, (2) standard linear solid (SLS), (3) Burgers, and (4) a triviscous rheology. For brevity, we refer to the model of an elastic crust over a Maxwell half-space as the Maxwell model, and similarly for a SLS, Burgers and triviscous half-space. The Maxwell linear viscoelastic material is the most widely used rheology in 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 is analytically simple, parameterized by one number, and is implemented in a variety of finite element modeling programs used in studies of crustal deformation. The SLS is probably the second most popular viscoelastic medium in postseismic 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, are analytically as simple as Maxwell; however, compared to the nonrecoverable Maxwell rheologies, SLS rheologies are fully recoverable. Pollitz et al. [2000] noted that the Maxwell rheology is contained in the SLS rheology and used the SLS as a semigeneral rheology. Researchers have also proposed that the inelastic lithosphere should be modeled with a biviscous rheology, using either a heterogeneous model [Ivins, 1996] or a Burgers rheology [Pollitz, 2003]. The Burgers viscoelastic rheology is often referred to as a Burgers body. In this paper we follow Findley et al. [1976] and refer to it as a Burgers rheology, or more simply a Burgers model. Burgers rheologies have been proposed to explain postglacial rebound [e.g., Peltier, 1985; Yuen et al., 1986; Sabadini et al., 1987], as well as deformation experiments of mantle [e.g., Mackwell et al., 1985; Chopra, 1997] and crustal [Carter and Ave Lallemant, 1970; Smith and Carpenter, 1987] material. A Burgers rheology is capable of two phases of relaxation (one recoverable and one nonrecoverable); hence the Burgers rheology can be said to be biviscous. There have not been any studies that have proposed that a triviscous rheology is appropriate for the lithosphere; however, we include it for demonstrative purposes. The triviscous rheology is composed of two recoverable phases of relaxation and one nonrecoverable phase. [7] In this paper, we derive the time-dependent solution for the postseismic and interseismic surface displacements. Typically, when referring to displacements within a seismic cycle, researchers distinguish between the postseismic displacements (early transient displacements) and interseismic displacements (displacements later in the cycle, considered secular); in this paper we refer to all displacements during a seismic cycle as interseismic, and we refer to the transient displacements after an earthquake ignoring the reloading of the fault as postseismic. For completeness, we review the correspondence principle and the image solution of Rybicki [1971] under correspondence. We present the derivations of the analytic solutions for general linear viscoelastic rheologies and nonperiodic earthquake sequences. We demonstrate the solution applied to the four example viscoelastic rheologies and apply the models to postseismic and interseismic displacements for both a periodic and a nonperiodic earthquake sequence. We construct our nonperiodic sequence by periodically repeating a nonperiodic, finite sequence of earthquakes. As the model is antisymmetric, we only present one-half of the model throughout this paper. 2. Background and Theory 2.1. Correspondence Principle of Viscoelasticity [8] In two-dimensional models of strike-slip faults, only the shear stresses and strains are nonzero. Only considering shear, the equation of motion for a linear viscoelastic media is %s ¼ 8e ð1þ where s and e are shear stress and engineering shear strain, respectively. % and 8 are the differential operators % ¼ Xqf k¼0 f k d k dt k 8 ¼ Xqy k¼0 y k d k dt k where f k and y k are constants determined from the rheological properties of the media [e.g., Flügge, 1967; Findley et al., 1976]. Throughout this paper we refer to f k and y k as stress and strain coefficients, respectively, and collectively as material coefficients. The material coefficients are not defined uniquely with respect to the rheological properties and different authors use different definitions, achieved by redistributing rheological properties in equation (1). Taking the Laplace transform of equation (1) ( ^%^S ¼ ^8^e, where we denote L{f(t)} = ^f (s) to be the Laplace transform of function f(t) and L 1 {^f (s)} = f(t) tobe the inverse Laplace transform of ^f (s), and we specify functions of t and s as f and ^f, respectively), the differential operators become polynomials in s ^% ¼ ^% ðþ¼ s XqF f k s k ; k¼0 ^8 ¼ ^8 ðþ¼ s Xqy y k s k ; k¼0 ð2þ ð3þ 2of21

3 where d = H/D 1. We show W* n for three values of d and n in Figure 1. [10] Following Nur and Mavko [1974], Savage and Prescott [1978] demonstrated that the time-dependent displacements due to viscoelastic relaxation in the lower halfspace could be found via the correspondence principle, where the fault rupture history is imposed by replacing the scalar b by the function b(t). The Laplace transform of the image solution is Figure 1. W n? (x?, d = H/D) forn = 1, 4, and 10. and the transform of the equation of motion can be rewritten as ^uðx; sþ ¼ ^bðþ s p p 2 x tan 1 þ 1 D p X 1 ^G ðþ¼^m cðþ ^m s ðþ s s ^m c ðþþ^m s ðþ s n¼1 ^bðþ^g s n ðþw s n ð9þ ð10þ ^s ¼ ^m ðþ^e s where ^m(s) = ^8= ^%, a ratio of two polynomials in s. This illustrates the well known correspondence principle of viscoelasticity [e.g., Flügge, 1967], which states that the Laplace transform of the equations of motion of all linear viscoelastic materials are identical to the equation of motion of an elastic media (s = m e e). Therefore an elastic solution can be used to construct a time-dependent viscoelastic solution by replacing m e with ^m(s) in the Laplace domain; hence we refer to ^m(s) as the equivalent shear modulus of a viscoelastic material Image Solution Under Correspondence [9] Rybicki [1971] gave the solution for the displacements at the surface due to a fault rupturing with displacement b from depth 0 D in an elastic layer of thickness H D and shear modulus m c overlying an elastic half-space of shear modulus m. In the notation of Savage and Prescott [1978], the displacements are given by ux ðþ¼ b p p 2 tan 1 x D þ X1 n¼1 G n W n Þ where G is a coupling or reflection coefficient given by G ¼ m c m m c þ m W n ¼ W n ðx; D; HÞ ¼ tan 1 2Dx 4n 2 H 2 D 2 þ x 2 where the plus and minus are for x > 0 and x < 0, respectively. For the remainder of this paper we only consider x >0.W n and G contribute to the perturbations to the half-space solution due to the image dislocations [Rybicki, 1971]. We choose to nondimensionalize equation (7) by D, so that the nondimensional distance is x* = x/d, which when substituted into equation (7) gives! ð4þ ð5þ ð6þ ð7þ W n? ¼ W n? ð x? ; dþ ¼ tan 1 2x? ð2ndþ 2 ð8þ 1 þ x?2 where ^m c (s) and ^m(s) are the equivalent shear moduli of the upper layer and lower half-space, respectively. ^G(s) completely describes the rheologies of the upper layer and the half-space; hence its inverse transform describes the time dependence of the displacements for particular rheologies due to the coupling between the upper layer and the lower half-space. The inverse transform of ^G(s) is complicated by the multiplication by ^b(s), which in the time domain is the convolution L 1 n o ^bðþ^g s n ðþ s ¼ bt ðþ*g n ðþ¼ t Z t 0 bðþg z n ðt zþdz ð11þ Taking the inverse Laplace transform of equation (9), the time-dependent displacements are given by ux; ð t Þ ¼ bt ðþ p 1 2 tan 1 x D þ 1 p X 1 n¼1 bt ðþ*g n ðþw t n ; ð12þ while the velocities are given by the time derivative, v(x, t)=_u(x, t). Since the summation in n is infinite, we need to truncate the summation at some n final, the particular value of n final depends on the time and distance ranges desired, as will be described below & n (t) for General Viscoelastic Rheologies [11] Savage and Prescott [1978] recognized that for the case of an elastic layer (of shear modulus m) overlying a Maxwell linear viscoelastic half-space (of shear modulus and viscosity m and h, respectively), equation (10) reduces to ^G n (s) = [(m/2h)/(s + m/2h)] n whose inverse Laplace transform is known. For the general problem of a linear viscoelastic layer (^m c (s) = ^8 c (s)/ ^% c (s)) overlying a linear viscoelastic half-space (^m(s) = ^%(s)/ ^%(s)), equation (10) is ^G n ðþ¼ s ^8! n c ðþ^% s ðþ ^8 s ðþ^% s c ðþ s ð13þ ^8 c ðþ^% s ðþþ^8 s ðþ^% s c ðþ s which is a ratio of two polynomials, the order of which is the product of the largest orders of the differential operators of the upper layer and the half-space. We restrict this study to the specific problem of an elastic layer ( ^8 c (s) = 3of21

4 m c, ^% c (s) = 1) overlying a linear viscoelastic half-space, so ^G n (s) reduces to P q ^G n k¼0 ðþ¼ s m ð cf k y k Þs k n m ð cf k þ y k Þs k n ð14þ P q k¼0 where q = max(q y, q f ), and y i = 0 and f j = 0 for i > q y and j > q f, respectively, and q y and q f are defined in equation (2). When the denominator and numerator of equation (14) are factorable, with roots a k and b k, respectively, ^G n (s) simplifies to Q qb ^G n ðþ¼ s g n Qk¼1 qa k¼1 where q a and q b are the number of roots and g m cf qb y qb m c f qa þ y qa ð s b k ð s a kþ n ð15þ is the ratio of the leading coefficients of the polynomials of equation (14). The units of s, a k, and b k are inverse time, and the units of g are time (q b q a ). [12] The inverse Laplace transform of equation (15) can be identified directly by using the Laplace transform relation L 1 where cðþ s ¼ Xq yðþ s k¼1 X mk l¼1 Þ n kl ða k Þ ðm k lþ! ðl 1Þ! tmk l e akt ð16þ yðþ¼ s ðs a 1 Þ m1 ðs a 2 Þ m2 mq s a q ð17þ a i 6¼ a j for i 6¼ j are real, c is a polynomial of degree m m q and kl ðþ¼ s dl 1 ds l 1 cðþ s ð yðþ s s a k Þ mk ð18þ which is found by the method of partial fractions [e.g., Churchill, 1944; Roberts and Kaufman, 1966]. The order of the polynomial in the numerator of equation (15) is always less than or equal to the order of the polynomial in the denominator (since the denominator always involves addition of positive numbers), thus q a q b and we assume (for now) that all a k are distinct and that a k and b k are real. Using the transformation relation above, substituting m k = n, 8 k 2 [1, q = q a ], we find with X G n ðþ¼ t g n Xqa n k¼1 l¼1 W kl ða i ; b i ; nþ ¼ dl 1 ds l 1 W kl ða i ; b i ; nþ ðn lþ! ðl 1Þ! tn l e akt ð19þ Q qb Qi¼1 qa i¼1 ð s b i ð s a iþ n ðs a k Þ n Þ n s¼a k ð20þ were a i and b i are the set of roots in equation (15) (e.g., a i = {a i ji 2 [1, q a ]}) and {j s signifies evaluation at s. [13] Noting that the inverse of the roots a k are natural time scales in G n (t), we nondimensionalize time by the absolute value of one of the roots, say ja n j,sothattja n j is the nondimensional time. Since the choice of timescale is not unique for q a > 1, we represent the nondimensional time as t n = tja n j. We order the set of roots with ja 1 j 1 < ja qa j 1,so that n = 1 and n = q a refer to the timescales of the fastest and slowest relaxation phases, respectively. To determine G n (t n ) from equation (10), we first nondimensionalize the dual of time (s n = s/ja n j) in equation (20), yielding the relation W kl ða i ; b i ; nþ ¼ja n j nðqb qaþ1 Þ ð l 1 Þ dl 1 ds l 1 n s n a n k ja n j sn 8 < : Q qb i¼1 Q qa i¼1 s n b i s n ai ja nj evaluated at s n = a k /ja n j, so that W kl ða i ; b i ; nþ ¼ ja n j nðqb qaþ1 Þ ð l 1 Þ a i W kl ja n j ; b i ja n j ; n The nondimensional form of G n is then G n ðt n Þ ¼ja n jg n nð t nþ ¼ja n jg n n X qa k¼1 X n l¼1 n ja nj n ð21þ ð22þ a W i kl ja ; b i nj ja ; n nj ðn lþ! ðl 1Þ! tn l n e aktn=janj ð23þ where g n = gja n j q b q a. Given two half-space rheologies with sets of roots {a i, b i } and {a 0 i, b 0 i}, we nondimensionalize time by a n and a 0 n for each of the models. If a i /ja n j = a 0 i/ja 0 nj, b i /jb n j = b 0 i/jb 0 nj and g n = g 0 n, then G n n (ja n jt) =G n n (ja 0 njt). [14] We define P S to denote the product over the set of indices S, P S ðs; i ; pþ Y ðs i Þ p ð24þ i2s where i ={ i ji 2S }, and we adopt the shorthand notation P S P S (s, i ; p), where for = a (b), S a ={iji 2 [1, q a ]} (S b ={iji 2 [1, q b ]}) and p = n (+n). We define P S =Eðs; i ; pþ Y ðs i Þ p ð25þ i2ðs EÞ to denote the exclusion of the indices in set E from the product, where (S E) is a set subtraction. By Leibniz rule of differentiation of products [e.g., Boas, 1983] we find W kl ða i ; b i ; nþ ¼ Xl 1 j¼0 where l 1 j P ðl 1 jþ S b P ðþ j S a= k fgð a k; a i ; nþ ða k ; b i ; þnþ ð26þ P ðþ j dj ds j P p p! k ðp kþ!k! 4of21

5 are the binomial coefficients. Noting that for G i =(s i ), d ds Gp i ¼ dg i d G p i ¼ d G p i ds dg i dg i the derivatives of P are found by application of the chain rule, which leads to the recursion relation P ðmþ1þ S ðs; i ; pþ ¼ X X m j2s for m 0, where k¼0 P ðm kþ S = j m k R p; k ð Þ s j p k 1 fg ðs; i ; pþ ð27þ Rðp; kþ ¼ pp ð 1Þðp 2Þðp kþ 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 exclusion notation introduced above holds for derivatives of P over the summation and product, i.e., P ðmþ1þ S =E ¼ X j2ðs EÞ X m k¼0 m Rðp; kþ s k j p k 1P ðm kþ S = ðfg[e j Þ ð29þ Figure 2. Cartoon representation of the steady sliding and back-slip models, as well as the total deformation during an interseismic period. We implicitly assume that i2; () = 1 and P 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) (a k ) are distinct; however, when the roots are not all distinct, the above solution holds for m k = k k n in equation (16), where k k is the number of times the root a k is repeated. For a particular choice of a viscoelastic half-space we need to identify the operators % and 8 for the media, and after verifying that the roots of equation (14) are real we directly use equation (10) to determine G n (t) The b(t) * & n (t) for an Earthquake Sequence [16] Savage and Prescott [1978] noted that the fault history function for a periodic earthquake sequence can be represented as the summation of steady sliding of two quarter spaces, back slip on the fault, and episodic fault displacement during the earthquake. In steady sliding, each side of the fault has block-like motion at the far-field velocity, where the displacements on either side of the fault are constant with distance away from the fault (Figure 2). Steady sliding ensures that the far-field moves at the longterm fault slip rate, and in this model we do not impose farfield velocity conditions. The back slip model is such that the fault slips constantly with a velocity equal to the negative of the far-field velocity, while in the episodic fault rupture model the fault instantaneously slips forward at the desired fault rupture times. In the back slip and episodic rupture models, the interseismic displacements with respect to the distance away from the fault are given by equation (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 earthquake and is locked at other times. At depths greater than D the fault slips steadily at the far-field velocity, while the coseismic stresses diffuse in the viscoelastic half-space. In this model, steady sliding of the downward continuation of the fault loads the locked portion of the fault and drives the far field. At depths greater than the maximum depth of viscoelastic stress diffusion, shear is entirely localized on the downward continuation of the fault. Whereas in a model that was driven by far-field velocity conditions, at sufficient depths the deformation is simple shear, assuming there are no lower boundary conditions. For a periodic earthquake sequence, after a sufficient number of ruptures the displacements do not depend on the number of prior ruptures, and the surface displacements do not depend on the steady deformation at depth [e.g., Li and Rice, 1987; Savage, 1990; Hetland and Hager, 2004]. During the first several cycles, the displacements predicted by the deep-slip model will be larger than those after many ruptures, since the initial displacements are the postseismic relaxation from the first earthquake plus the steady slip at depth. In a model driven by far-field velocity conditions, the displacements following the first few earthquakes will be smaller than those at steady state, since the displacements will be the postseismic plus those of simple shear due to the far-field velocities. We discuss the dependence on the steady displacements at depth, for both periodic and nonperiodic sequences, below. [18] Savage and Prescott [1978] only considered the back slip of an elementary earthquake cycle, and they constructed a complete periodic earthquake sequence by superposing a series of elementary cycles shifted in time by the period of the cycle. We modify the episodic back-slip model of Savage and Prescott [1978] so that it is valid for general nonperiodic earthquake sequences, as well as periodic ones. Savage and Prescott [1978] expressed b(t) *G n (t) in terms of incomplete gamma functions. In this paper, we choose to follow a different approach, avoiding the incomplete gamma functions and using the Laplace transform relation given in equation (16). [19] For a sequence of earthquakes, where the pth rupture occurs at time T p with magnitude D p, the rupture history function is b seq ðþ¼ t X1 p¼0 D p H t T p ð30þ where H(t T p ) is the Heaviside step function, centered at T p : H(t T p )=1fort T p, and zero when t < T p, and L{H(t T p )} = e stp /s [e.g., Boas, 1983]. For a periodic 5of21

6 earthquake sequence, D i = D i+1 and T i+1 T i is constant. The episodic rupture model plus the back-slip model is bt ðþ¼ X1 D p H t T p tv ð31þ p¼0 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. The Laplace transform of equation (31) is p¼0 ^bðþ¼ s X1 D p v s e stp s p¼0 2 and L{b(t) *G n (t)} is " # ^bðþ^g s n ðþ¼g s n X1 D p v s e stp s 2 P Sa ð32þ ðs; a i ; nþp Sb ðs; b i ; þnþ 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 bt ðþ*g n ðþ¼g t n where X1 p¼0 G m ðt; a i ; b i ; nþ ¼ Xm l¼1 D p G 1 t T p ; a i ; b i ; n H t Tp g n v G 2 ðt; a i ; b i ; nþ ð34þ þ Xqa k¼1 W 0l ða i ; b i ; nþ ðm lþ! ðl 1 X n l¼1 Þ! tm l W m klða i ; b i ; nþ ðn lþ! ðl 1Þ! tn l e ak t ð35þ with W 0l given by equation (26) with a 0 =0, W m klða i ; b i ; nþ ¼ Xl 1 h¼0 l 1 ð 1Þ h ðh þ m 1Þ! h a mþh k W kl h ð Þ ða i ; b i ; nþ ð36þ and W k(l h) (a i, b i ; n) is defined in equation (26). [20] To nondimensionalize b(t) by the timescale ja n j 1, we replace T p with T np = ja n jt p in equation (31) and divide through by some characteristic rupture offset, D (D 0 p = D p /D), so that v 0 n = D 0 p/(t npþ1 T np ), yielding bðt n Þ D ¼ X1 p¼0 D 0 p H t n T np v 0 n t n ð37þ For a periodic earthquake sequence, defining the period of the seismic cycle to be T = T p+1 T p, we find the nondimensional period Tja n j. For a model with a Maxwell half-space, with m c = m m, the nondimensional period is T /2t M (see section 2.5.1), which is the parameter t o as defined by Savage and Prescott [1978], sometimes referred to as the Savage parameter. We extend the definition of the Savage parameter to t o = Tja 1 j, which is the ratio of the seismic recurrence time to the relaxation timescale associated with the timescale of the fastest phase of relaxation. The Savage parameter is important in controlling the amount velocities vary with time in both periodic [Savage and Prescott, 1978] and nonperiodic [Meade and Hager, 2004] earthquake cycles. Since t o is related to the Wallace number, t o gives the stability of nonperiodic rupture sequences, [e.g., Kenner and Simons, 2005]. Following the nondimensionalization approach in section 2.3, we find that for an earthquake sequence, b(t n )*G n (t n )/D is given by equations (34) (36), making the substitutions g! g n, T p! T np, D! D 0 p, v! vn, 0 a i! a i /ja n j, b i! b i /ja n j, and t! t n. Given two models with sets of roots {a i, b i } and {a 0 i, b 0 i}, coefficients g and g 0, and rupture displacements D and D 0, b(t n )*G n (t n )/D = b(tn)*g 0 n (tn)/d 0 0. [21] To compute only the postseismic displacements from a single earthquake of rupture magnitude D, ignoring the far-field velocity, we use the fault rupture history in equation (30) with T 0 =0,D 0 = D, and D i =08i >0. Thus, for only the postseismic response from a single earthquake, the convolution in equation (12) is bt ðþ*g n ðþ¼ t g n D G 1 ðt; a i ; b i ; nþ ð38þ and b(t n )*G n n (t n )/D is found by substituting g! g n, D! 1, a i! a i /ja n j, b i! b i /ja n j, and t! t n in equation (38) and its dependencies Applications to Specific Media [22] The theory we presented in section 2.4 is valid for models with a half-space composed of any linear viscoelastic rheology, given that the roots of equation (14) exist and are real. For illustration, we apply the theory applied to four models: an elastic layer over Maxwell, standard linear solid (SLS), Burgers (a biviscous material), and triviscous viscoelastic half-spaces. For brevity, we refer to these models as the Maxwell, SLS, Burgers, and triviscous model. For any model, we only need to determine the material coefficients (f i and y i ) of the half-space material, after which all calculations can be performed numerically. However, in this section we determine the roots analytically in order to highlight the method, as well as establish the timescales for each of the models. We simplify and plot G n (t) for a few simple cases. [23] In models of viscoelasticity, it is convenient to consider mechanical analogue models composed of a configuration of springs (accommodating elastic deformation) and dashpots (accommodating deformation due to creep and relaxation) [e.g., Flügge, 1967]. A spring in series with a dashpot is a Maxwell viscoelastic element, representing initial elasticity followed by a creep or relaxation phase of deformation. Because of the creep/relaxation of the dashpot, the deformation of the Maxwell element is nonrecoverable with time. A spring in parallel with a dashpot is a Kelvin element; the Kelvin element is sometimes referred to as a Kelvin-Voight or Kelvin-Voigt element; in this paper we follow Findley et al. [1976] and refer to it as a Kelvin element. The Kelvin element is incapable of instantaneous elasticity; however, with an applied stress the dashpot will 6of21

7 Figure 3. Mechanical analogue models of (a) Maxwell, (b) standard linear solid, (c) Burgers, and (d) triviscous viscoelastic materials. creep leading to a delayed elasticity. The deformation of the Kelvin element is fully recoverable Maxwell Half-Space [24] A Maxwell element is the conceptual model of a Maxwell linear viscoelastic material, where the dashpot has a viscosity of h m and the spring has a shear modulus m m (Figure 3a). The equation of motion for a linear Maxwell viscoelastic media is first-order (q f = q y = 1) and the material coefficients are f 0 ¼ 1 y 0 ¼ 0 f 1 ¼ h ð39þ m ¼ t M y m 1 ¼ h m m where t M 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 m c 6¼ m m, equation (14) reduces to ^G n ðþ¼ s where q a = q b = 1 and m c m n m s þ m cm m ð m c þ m m s þ m cm m a 1 ; b 1 ¼ m cm m h m ðm c m m Þ " # n h m m c m m Þ ð40þ h m ðm c þm m Þ ð41þ where a and b are the addition and subtraction, respectively, and ja 1 j 1 is the timescale associated with the Maxwell model, which is always real. When m c = m m = m, ^G n ðþ¼ s " # 1 n n 1 2t M s þ 1 ð42þ 2t M so that q a =1,a 1 = m/2h, and q b = 0. We then obtain G n (t) directly from equation (10). When m c = m m = m, the timescale given in equation (41) reduces to 1/2t M and since G n ðþ¼ t W 1l ¼ Xl 1 j¼0 1 n 1 1 2t M ðn 1Þ! tn 1 exp t 2t M l 1 j P ðþ j S b P ðl 1 jþ S a= 1 fg ¼ 1; l ¼ 1 0; l > 1 ð43þ ð44þ because S b = S a {1} = ;, W 11 = 1 and W 1l = 0 for l >1 (i.e., all derivatives are zero). We can nondimensionalize G n (t) byja 1 j = m/2h ð2t m ÞG n 1 t 1 ð 1Þ ¼ ðn 1 Þ! tn 1 1 e t1 ð45þ where t 1 = t/2t M is dimensionless time when m c = m m. When m c = Xm m, timescales as (1/t M )(X/1 + X), and b 1 /ja 1 j =(1 X)/(X + 1) and g n =(X 1)/X for m c 6¼ m m, and when m c = m m, b 1 /ja 1 j = 1 and g n = 1; hence G 1 n (t 1 ) is constant for constant X. Equation (45) is identical to the result obtained by Savage and Prescott [1978] using the specific Laplace transform pair of Erdélyi et al. [1954]. [26] G n (t) entirely describes the time response of the deformation in the upper layer due to relaxation in the half-space following faulting in the upper layer. The diffusional nature of the Maxwell material is apparent in G n (t) where increasing values of n describe relaxation at longer times. G n (t) is modulated by W n in equation (12), and for increasing n, W n! 0, so the contributions of higher modes of G n (t) are small. We show G n (t) form c = m m and n = 1 40 in Figure 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 instantaneous elastic deformation followed by a delayed elasticity. The equation of motion for a SLS material is first-order (q f = q y = 1) and the material coefficients are f 0 ¼ m v þ m e m v f 1 ¼ h v m v ¼ t K y 0 ¼ m e y 1 ¼ m e h v m v ¼ m e t K ð46þ where m e is the elastic shear modulus, m v and h v are the shear modulus and viscosity of the Kelvin element (Figure 3), and t K is the Kelvin relaxation time, which is the timescale leading to the delayed elasticity. [28] For an elastic layer of shear modulus m c overlying a SLS half-space with material properties given above, when m c 6¼ m e we find ^G n (s) from equation (14) to be ^G n ðþ¼ s 2 m c m n e s þ m c ðm v þm e 4 ð m c þ m e s þ m cðm v þm e 3 5 Þ m e m n v h v m c m e Þ Þþm e m v h v ðm c þm e Þ ð47þ 7of21

8 Figure 4. (left) G 1 n (t 1 ) and (middle) G 1 n (t 1 ) W* n (1, 1) for n = 1 40 using a Maxwell model with m c = m m, t 1 = t/2t M, and H/D = 1. We show the square root of G 1 n (t 1 )W* n to amplify the low magnitudes. (right) R jg1 n (t 1 )W* n (1, 1)jdt 1 (power) for each n. so that q a = q b = 1 and the inverse of the timescale associated with the SLS model is ja 1 j¼ m cðm v þ m e Þþm e m v h v ðm c þ m e Þ ð48þ which is always real. When m c = m e = m, q b =0,q a =1, ^G n m n 1 ðþ¼ s 2m v t n ð49þ K s þ 2m vþm 2m v t K and a 1 = [(2m v + m)/2m v t K ]. In the case when all of the shear moduli are equal (m c = m e = m v = m) a 1 = 3/2t K.As demonstrated in section 2.5.1, when q a = 1 and q b =0,the only nonzero W kl is W 11 = 1, so that G n ðþ¼ t 1 n 1 2t K ðn 1Þ! tn 1 e t 3 2t K ð50þ When all of the shear moduli are equal, the dimensionless time for the SLS is t 1 =3t/2t K, and 2t K G n 1ð 3 t 1Þ ¼ 3 n 1 ðn 1 Þ! tn 1 1 e t1 ð51þ For the SLS half-space, G n (t) decays much faster with respect to n compared to the Maxwell half-space, for the same viscosities. We show G n (t) form c = m e and n = 1 20 in Figure Burgers Half-Space [29] A Burgers rheology is capable of both a recoverable and nonrecoverable relaxation phase and is considered a biviscous material. The mechanical analogue model of a Burgers rheology is a Maxwell element (with shear and viscosity of m m and h m ) in series with a Kelvin element (m v, h v ; Figure 3c). The equation of motion is second-order (q f = q y = 2) and the material coefficients are [e.g., Findley et al., 1976] f 0 ¼ 1 y 0 ¼ 0 f 1 ¼ h m m m þ h m m v þ h v m v f 2 ¼ h mh v m m m v y 1 ¼ h m y 2 ¼ h mh v m v ð52þ The relaxation timescales of the Maxwell element (t M = h m /m m ) and the Kelvin element (t K = h v /m v ) appear in the material parameters, along with the timescale associated Figure 5. (left) G n 1 (t 1 ) and (middle) G n 1 (t 1 )W* n for n = 1 20 using a standard linear solid viscoelastic half-space with material m c = m e, t 1 =3t/2t K, and H/D = 1. We show the square root of G n 1 (t 1 )W* n to amplify the low magnitudes. (right) Power. 8of21

9 Figure 6. (left) G n 1 (t 1 ) and (middle) G n 1 (tp 1 )W* n for n = 1 25 using a Burgers linear viscoelastic halfspace with m c = m m = m v, h m = h v, t 1 = t(2 + ffiffi 2 )/2tBB, and H/D = 1. We show the square root of G n 1 (t 1 )W* n to amplify the low magnitudes. (right) Power. with the transfer of stress between the Kelvin and Maxwell elements (h m /m v ). [30] For the case of an elastic layer overlaying a Burgers viscoelastic half-space with m c 6¼ m m, ^G n ðþ¼ s l 2s 2 n þ l 1 2 þ l 0 x 2 s 2 ¼ l n 2 ðs b 1 Þ n ðs b 2 Þ n þ x 1 s þ x 0 x 2 ðs a 1 Þ n ðs a 2 Þ n ð53þ where x i ¼ m c f i þ y i l i ¼ m c f i y i ð54þ The roots of the polynomials in equation (53) are a 1;2 ¼ a ;þ ¼ x 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 1 4x 2x 0 2x 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 1;2 ¼ b ;þ ¼ l 1 l 2 1 4l 2l 0 2l 2 ð55þ ð56þ which can be verified to be real. When m c = m m, l 2 = 0 and q b reduces to 1, b 1 = l 0 /l 1, a 1,2 are as in equation (55), and G n ðþ¼ t g n Xn l¼1 W 1l ða i ; b i ; nþ ðn lþ! ðl 1Þ! tn l e a1t þ W 2lða i ; b i ; nþ ðn lþ! ðl 1Þ! tn l e a2t ð57þ where g = l i /x 2, i = 1 and i = 2 for m c = m m and m c 6¼ m m, respectively. For the case when m c = m m = m v = p m and h m = h v = h, defining t BB h/m, a 1,2 = 2 ffiffiffi 2 /2tBB and b 1 = 1/2t p BB. We can nondimensionalize time by either ja 1 j =(2+ ffiffi p 2 )/2tBB or ja 2 j =(2 ffiffiffi 2 )/2tBB, corresponding to the fast and slow relaxation phases following a coseismic rupture. In Figure 6, we show G n (t) with shear moduli equal, h m = h v, and n = For n large, the series in equation (26) blows up due to computer precision. However, G n (t) is quite small before the numerical instability is encountered and the contribution from G n (t) at such values of n is negligible to the displacements (see below) Triviscous Material [31] Finally, we consider a triviscous material, which is capable of two phases of recoverable relaxation in addition to an unrecoverable phase. The material is conceptually composed of a Maxwell element in series with two Kelvin elements, and we refer to the shear moduli and viscosities of the Maxwell and two Kelvin elements as m m, m 1, and m 2 and h m, h 1 and h 2, respectively (Figure 3d). The equation of motion for this triviscous material is third-order (q f = q y =3) and the material coefficients are f 0 ¼ 1 f 1 ¼ h m þ h 1 þ h 2 þ h m þ h m m m m 1 m 2 m 1 m 2 f 2 ¼ h m h 1 þ h m h 2 þ h 1 h 2 þ h m h 2 þ h m h 1 m m m 1 m m m 2 m 1 m 2 m 1 m 2 m 2 m 1 f 3 ¼ h m h 1 h 2 m m m 1 m 2 y 0 ¼ 0 y 1 ¼ h m h y 2 ¼ h 1 m þ h 2 m 1 m 2 y 3 ¼ h m h 1 m 1 h 2 m 2 ð58þ Several timescales appear in the material coefficients, the timescales of the Maxwell (t M = h m /m m ) and two Kelvin elements (t i = h i /m i, i = 1, 2), as well as the cross timescales (h m /m i, i =1,2). [32] For the case of an elastic layer (shear modulus m c 6¼ m m ) overlaying a triviscous viscoelastic half-space, equation (15) becomes ^G n ðþ¼ s l 3s 3 þ l 2 s 2 n þ l 1 s þ l 0 x 3 s 3 þ x 2 s 2 ¼ l n Q 3 3 i¼0 ð s b iþ n Q þ x 1 s þ x 0 x 3 3 i¼0 ð s a iþ n ð59þ where x i and l i are given in equations (54). The roots of the polynomials in equation (59) can be determined analytically 9of21

10 Figure 7. (left) Displacements following a single earthquake in a Maxwell model with H/D = 1 and various n final. (right) Displacements following a single earthquake, u a, (solid lines) compared to the displacements determined by finite element calculations (dashed lines; u n is the high-precision model); model geometry is given in the main text. For all models m c = m m and t 1 = t/2t M. [e.g., Wang and Guo, 1989, Appendix I] or can be found numerically; moreover, the roots can be shown to always be real from the realness of the relaxation modulus for a triviscous material. When m c = m m, l 3 =0soq b reduces to 2 and b i are given by equation (56). For the case when m c = m m = m 1 = m 2 = m, h m = h 1 = h and h 2 = h/2, a i ¼ m 3h for i = 1 3 where pffiffiffiffiffi 34 2p cos u þ i 5 3 u ¼ 1 r ffiffiffiffiffi! cos b 1;2 ¼ 7 p ffiffiffiffiffi 17 m 8 h ð60þ G n (t) is found directly from equation (10), which we do not reproduce here. 3. Displacements Due to Specific Fault Histories [33] We demonstrate the theory using three fault histories: (1) postseismic displacements from a single earthquake ignoring the far-field velocity, (2) interseismic displacements for a periodic earthquake sequence, and (3) interseismic displacements for a nonperiodic earthquake sequence Displacements From a Single Earthquake [34] In this section, we show the displacements and velocities due to postseismic relaxation following a single earthquake, for an elastic layer overlying a Maxwell, a standard linear solid, a Burgers and a triviscous half-space. For each case, the displacements are given by equation (12), and b(t) *G n (t) is given by equation (38), where the roots (a i and b i ) and the coefficients (g) are outlined in section 2.5. We compare the displacements predicted by our analytic solution to those calculated using the finite element package Adina (Adina R&D, Inc). For the finite element calculations, we use a relatively low precision model and a relatively high precision model. In both finite element models, the fault ruptures with a uniform slip from the surface to depth D and linearly tapers to zero from D, where the thickness of the elastic layer is 1.2D. The models are 200D by 200D, containing 11,235 finite elements. We use four-node rectangular elements and a time step of one half the shortest relaxation time in the lowprecision models, while in the high-precision models we used nine node elements and a time step of one tenth of the shortest relaxation time. In the finite element calculations, the taper of slip from 1.0 to 1.2D is exact, whereas in the analytic solutions we account for the taper by superposing 40 models with uniform slip extending from 0 to 1 + j0.2/ 40D for j = Since H/D is not constant for the individual slip models, the displacements do not scale with D (see equation (8)), and we nondimensionalize distance by D purely for convenience. The rheological parameters in both the finite element and analytic models are identical and are given below 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 in the far field and at late time. When m c = m m, the displacements converge at n about 10t 1 (t 1 is the mechanical timescale of the relaxation, see section 2.5.1), in the near field up to 15t 1 (30 Maxwell times), while at distance 15H from the fault the displacements do not converge until n 40 (Figure 7). In the near-field the transient response decays quickly, while there is a slower decay in the far field. [36] The postseismic displacements predicted by equation (12) match the displacements from the finite element calculations quite well (Figure 7), with the exception of immediately after the earthquake, where the finite element model is not able to resolve the stresses at the fault tip well. Moreover, the difference between the analytic 10 of 21

11 Figure 8. (left) Displacements and (right) velocities following a single earthquake in a Maxwell model with m c =(m/2)m m, t 1 =(t/t M )[m/(m + 1)], and H/D =1. solution and the high-precision FE calculation is much less than the difference with the low-precision calculation, especially during early times. The discrepancy between the analytic solution and a finite element calculation with greater element density would be less. [37] For each choice of m m and m c, the displacements due to postseismic relaxation are distinct, since the relaxation timescales and the g v coefficients are different. In Figure 8 we show the postseismic displacements and velocities for m c = 3m m /2, m m, and m m /2, so that timescales as 3/5t M,1/2t M and 1/3t M, respectively. The shear modulus of the continental crust is approximately 30 GPa, while that of the mantle is about 70 GPa, so m c m m /2 is realistic for the continents. Assuming m c = m m results in a small, but nonnegligible, difference in the predicted displacements (Figure 8) Postseismic Relaxation for a Standard Linear Solid Half-Space [38] For the case of an elastic layer over a SLS viscoelastic half-space, the series over n in equation (12) converges at low n final for all times and distances (Figure 9). The postseismic displacements predicted by the analytic solution match the displacements from the finite element calculation quite well (Figure 9), again with the exception of immediately after the earthquake. [39] For given moduli (m c, m e, m v ), the displacements relax to distinct values. For m c = m e, and m v = m e,2m e /3, and m e /3, timescales as 3/2t K, 7/4t K and 5/2t K, respectively. The displacements relax to profiles resembling the coseismic profile, exhibiting the delayed elasticity of the SLS material, where a weaker m v leads to a larger difference in coseismic and relaxed displacements (Figure 10) 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 G n (t) is quite small before the numerical instability is encountered, and even out to 15D away from the fault, the displacements converge at n smaller than that where instability arises. The analytic solution matches the finite element calculations quite well, except during early times (Figure 11). [41] When the relaxation timescale of the recoverable Kelvin element is less than that of the nonrecoverable Maxwell element, the displacements relax rapidly immediately after the earthquake, followed by a slower relaxation. The initial relaxation is similar to the response of the SLS Figure 9. (left) Displacement following a single earthquake in a SLS model with H/D = 1 and various n final, and (right) postseismic displacements compared to the displacements determined by finite element calculations with m c = m v and t 1 =3/2t K. 11 of 21

12 Figure 10. (left) Displacements with respect to (right) time and (left) distance following a single earthquake in a SLS model with m e = m c, m v =(m/3)m e, t 1 =(t/t K )[(2m + 3)/2m], and H/D = 1. Gray line in Figure 10 (right) is the coseismic displacements, and black lines are displacements at t 1 = 10, where the line style is as in the legend in Figure 10 (left). half-space when h v h m, while it is closer to the Maxwell model when h v h m (Figure 12). When time is rescaled, the initial Kelvin relaxation phase of the Burgers models is similar to appropriately rescaled Maxwell models, and only in later times is the relaxation of the Maxwell element in the Burgers models apparent (Figure 12). During later times, the second phase of relaxation is similar to the slow relaxation of displacements predicted by a Maxwell half-space with the same h m Postseismic Relaxation for a Triviscous Half-Space [42] For a triviscous half-space the solution again blows up at large n, and we choose a sufficiently low n final to avoid the instabilities. The postseismic displacements converge at short distances from the fault well before n final ; however, at long distances from the fault and at late times the series does not quite converge by n final (Figure 13). The displacements truncated at n final compare well to the displacements predicted by the finite element calculation, except during early times near the fault and at later times far from the fault (Figure 13). The former is due to mesh inadequacies in the finite element model as discussed above, whereas the latter is due to the truncation of the summation in n in equation (12). The postseismic displacements and velocities predicted by the triviscous half-space are similar to those for the Burgers half-space, except the triviscous model predicts a third relaxation phase. When the first Kelvin element in the triviscous model is identical to the Kelvin element in the Burgers model, while the second Kelvin element in the triviscous model is weaker than the first, early in time the displacements relax faster compared to the Burgers model, while later in time the rate of relaxation is similar (Figure 14). During the intermediate times, the tertiary relaxation phase dominates Displacements Due to Periodic Earthquakes [43] The displacements through a seismic cycle in a periodic earthquake sequence are given by equation (12), using the convolution in equation (34), superimposed with the steady sliding model. We demonstrate the displacements Figure 11. (left) Displacements following a single earthquake in a Burgers model with H/D = 1 and various n final and (right) postseismic displacements compared p ffiffiffito the displacements determined by finite element calculations. m c = m m = m v, h m = h v and t 1 =(2+ 2 )/2tBB. 12 of 21

13 Figure 12. (left) Displacements through nondimensional time and (right) an instance of rescaled time following a single earthquake in models with Maxwell, SLS, and Burgers viscoelastic half-spaces and D/H =1.t 1 = ja 1 jt, where a 1 is the timescale associated with the fastest relaxation of each material. Time was rescaled using m c = 30 GPa, the shear moduli relations in Figure 12 (left) and the indicated viscosities for the Maxwell, SLS, and Burgers half-spaces, where for the Burgers model h v is as in the legend in Figure 12 (left). during a periodic earthquake sequence (Figure 15) for models of Maxwell, standard linear solid (SLS), Burgers, and triviscous half-spaces using t o =5=10ja 1 j, where ja 1 j 1 is the timescale associated with the fastest phase of relaxation (Figure 16). We use H/D = 1 in all models, m c = m m in the Maxwell model, m c = m e = m v in the SLS model, m c = m m = m v and h m = h v in the Burgers model, and m c = m m = m 1 = m 2 and h m = h 1 =2h 2 in the triviscous model. After a sufficient number of ruptures, the interseismic displacements are the same in all earthquake cycles [e.g., Savage and Prescott, 1978; Li and Rice, 1987]. When the displacements do not depend on the particular cycle, we say that the displacements (or velocities, etc.) are cycle invariant or that the system is at cycle invariance. The displacements following the initial fault rupture are larger than the invariant displacements (Figure 16) due to the addition of constant slip on the continuation of the fault at depth. In the case when the far field is driven by velocity boundary conditions and steady deformation at depth is simple shear, the initial displacements will be smaller than the invariant displacements, since the initial displacements are the postseismic displacements plus a simple shear profile. However, during invariance, the surface displacements do not depend on the steady 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 those predicted by the solution of Savage [2000] (Figure 17a). The velocities throughout an invariant seismic cycle can be characterized as perturbations to the average velocity profile, which is identical to the elastic model of interseismic strain accumulation proposed by Savage and Burford [1973] (v = (v/p)tan 1 (x/d), referred to as the elastic half-space model; Figures 17 and 18). In models with weak rheologies, where the relaxation timescale is short compared to the interseismic period (t o = Tja 1 j greater than about 1), Figure 13. (left) Displacements following a single earthquake in a triviscous model for various n final and (right) postseismic displacements compared to the displacements determined by a finite element calculation with m c = m m = m 1 = m 2, h m = h 1 =2h 2, and t 1 = ja 1 jt, where ja 1 j is given in equation (60). 13 of 21