Moment rate of the 2018 Gulf of Alaska earthquake

Journal Pre-proof Moment rate of the 2018 Gulf of Alaska earthquake

Stefano Santini, Michele Dragoni PII:

S0031-9201(19)30217-1

DOI:

https://doi.org/10.1016/j.pepi.2019.106336

Reference:

PEPI 106336

To appear in: Received date:

31 July 2019

Revised date:

4 October 2019

Accepted date:

25 October 2019

Please cite this article as: S. Santini and M. Dragoni, Moment rate of the 2018 Gulf of Alaska earthquake, (2019), https://doi.org/10.1016/j.pepi.2019.106336

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© 2019 Published by Elsevier.

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Moment rate of the 2018 Gulf of Alaska earthquake Stefano Santinia,∗, Michele Dragonib a

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Dipartimento di Scienze Pure e Applicate, Universit` a di Urbino, Via Santa Chiara 27, 61029 Urbino, Italy b Dipartimento di Fisica e Astronomia, Alma Mater Studiorum Universit` a di Bologna, Viale Carlo Berti Pichat 8, 40127 Bologna, Italy

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Abstract

The 2018 Gulf of Alaska earthquake (Mw 7.9) occurred in a region of the

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Pacific plate southwest of the Alaskan subduction zone. The earthquake was a strike-slip event, with the hypocenter located at a depth of about 25 km

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and a seismic moment equal to 0.96 × 1021 Nm. Two observed moment rates

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have been obtained by the Geoscope Observatory, France, and by the United States Geological Survey (USGS). Both of them can be interpreted as due

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to the failure of two asperities on the fault surface. We consider a discrete fault model, with two asperities of different areas and strengths, and show

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that the observed moment rates can be reproduced by appropriate values of the model parameters, as inferred from the available data. A good fit to the

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observed moment rates is obtained by a sequence of three dynamic modes of the system, including a phase of simultaneous slip of the asperities. The two moment rates are however characterized by different initial conditions, in terms of different initial shear stress distributions on the fault. Shear stresses on the asperities are calculated as functions of time during the event ∗

corresponding author

Submitted to Physics of the Earth and Planetary Interiors

October 30, 2019

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and show a similar evolution in the two cases, but with different final values. The model results show that the presence of simultaneous asperity motion can significantly increase the seismic moment of a large earthquake.

Keywords: Theoretical seismology; Nonlinear dynamical systems; Fault

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mechanics; Asperity models; Seismic moment rates.

1. Introduction

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The 2018 Gulf of Alaska earthquake was a large event, with a magnitude

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Mw = 7.9, originated by a predominantly right-lateral strike slip faulting, as determined by the global CMT solution (http://www.globalcmt.org/CMTsearch.html),

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seaward of Aleutian Trench near Kodiak Island, and occurred on 23 January 2018 (Krabbenhoeft et al., 2018; Ruppert et al., 2018; Zhao et al., 2018).

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The seismic moment reported by Lay et al. (2018) was m0 = 0.96 × 1021 Nm.

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The slip distribution can be approximated as the result of the failure of two major asperities: an eastern one (asperity 1) with an average slip

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ranging from u1 = 6.7 m to 8.5 m, and a western one (asperity 2), with an average slip from u2 = 3 m to 3.8 m, as inferred from the data of the United States Geological Survey (2018) and Geoscope Observatory, France (2018) respectively. The hypocenter was located in asperity 1. The areas of the two asperities can be estimated as A1 = 2.4 × 103 km2 and A2 = 0.6 × 103 km2 , and the distance between their centroids as ` = 80 km. From the value of m0 , the 2

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total fault area is approximately A0 = 104 km2 (Fig. 1). The observed moment rates show two humps due to the asperity 1 and asperity 2 motions, with the total event duration duration from Δ t = 55 s to 60 s (Geoscope Observatory, France, 2018; United States Geological Survey, 2018). As to the tectonic environment, the relative plate velocity is estimated 1.2 × 10−15 s−1 (DeMets and Dixon, 1999).

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to be v = 6 cm a−1 for the relative plate velocity with a strain rate e˙ =

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In the present paper, the discrete fault model by Lorenzano and Dragoni

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(2018) is considered in order to reproduce the two different moment rates observed for the 2018 Gulf of Alaska earthquake.

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Lorenzano and Dragoni (2018) considered a fault with two asperities with

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different areas and strengths. The fault is treated as a dynamical system with two state variables and four dynamic modes, for which complete analytical

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solutions are provided.

The four modes correspond to stationary asperities, motion of first asper-

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ity, simultaneous motion of the two asperities and motion of second asperity. Discrete fault models with two interacting asperities have been previously

(2016).

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developed in Dragoni and Santini (2015, 2017) and Dragoni and Tallarico

When an asperity slips, the associated dislocation produces a shear stress on the fault, that is transferred to the other asperity. This stress adds to the pre-existing stress on the asperity, anticipating its failure. In turn, when this asperity slips, it transfers stress back to the first, altering its stress state, and so on. The interaction between the two asperities may result in a phase of

3

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simultaneous motion during the seismic event. The aim of the present paper is to model the two observed moment rates and seismic moments of the 2018 Gulf of Alaska earthquake. The observed moment rates of the 2018 Gulf of Alaska earthquake show that, taking into account two different asperity areas A1 and A2 , fault dynamics changes in a measure that depends on the ratio of asperity areas.

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Observations by Geoscope Observatory, France (2018) and United States Geological Survey (2018) indicate that the earthquake was due to the failure

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of two main asperities and that it can be described as a 3-mode event. The

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mode durations, the slip distribution and the moment rate function calculated on the basis of the model in order to fit the observed values.

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According to Lay et al. (2018) and Ruppert et al. (2018), the earth-

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quake rupture was more complex, involving a fault system with slip on quasi orthogonal planes. We consider this possibility by modeling the earthquake

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source as a 2-mode event on the main fault of the system, without simultaneous motion of the asperities. In doing that, we neglect the change in

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et al. (2018).

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direction of rupture during propagation that was suggested by Krabbenhoeft

2. The model

In this section, we describe the model and report a few formulae obtained in Lorenzano and Dragoni (2018), that are necessary for the following developments. We consider a discrete fault model with two asperities having areas A1 and A2 respectively, and introduce the ratio ξ= 4

A2 A1

(1)

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We assume that the fault is embedded in a shear zone that is a homogeneous and isotropic Hooke solid of width d and rigidity μ. The shear zone is placed between two tectonic plates moving at relative velocity v and is subject to a uniform strain rate e, ˙ that is determined by the values of d and v. At any given time t, the state of the fault is described by the slip deficits x(t) and y(t) of the two asperities. Fault motion is characterized by four

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dynamic modes: a sticking mode 00, corresponding to stationary asperities, and three slipping modes, corresponding to the motion of one asperity (modes

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10 and 01) or to the simultaneous motion of both asperities (mode 11).

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Three kinds of forces acting on the asperities are considered: forces due to plate motion, forces due to a difference in the slip deficits of the two asperities

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and a rate-dependent force associated with radiation damping. Accordingly,

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the asperities are subject to tangential forces

where

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˙ f1 = −K1 x − Kc (x − y) − ι1 x, 2μeA ˙ 1 , v

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K1 =

K2 =

f2 = −K2 y − Kc (y − x) − ι2 y˙ 2μeA ˙ 2 , v

Kc = μA1 A2 s

(2)

(3)

where s is the shear traction (per unit seismic moment) that the slip of one

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asperity imposes to the other, calculated at the asperity centroid. The quantities ι1 and ι2 in (2) are the impedances associated with asperity motion (Rice, 1993). The impedance, i.e. the resistance opposed by the elastic medium to asperity motion, is proportional to the asperity areas. Therefore the ratio of impedances is equal to ι2 =ξ ι1 5

(4)

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Asperity motion is controlled by friction. We assume that asperities 1 and 2 are characterized respectively by constant static frictions fs1 and fs2 and by average dynamic frictions fd1 and fd2 . Accordingly, the conditions for the failure of asperity 1 and 2 are, respectively f1 = −fs1 ,

f2 = −fs2

(5)

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We assume that the two asperities have the same constitutive equation, but different values of the intervening parameters, differing by the common factor

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β. Hence the ratio  between the dynamic and the static friction is the same

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for both asperities.

parameters Kc A2 υs , = K1 2e˙

fs2 fd2 ι1 = , γ=√ fs1 fd1 K1 μ 1 √ K1 μ1 fd2 = , V = v fs2 fs1 βξ =

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α=

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In addition to ξ, the system is described by the following nondimensional

fd1 fs1

(7)

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=

(6)

with α ≥ 0, 0 < β ≤ 1, γ ≥ 0, 0 <  < 1, V > 0, and μ1 and μ2 are the

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masses associated with the asperities, which are assumed to be proportional

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to the respective areas A1 and A2 (Fig. 1), that is A2 μ2 = = ξ. μ1 A1

(8)

Moreover we introduce the parameter α ξ

(9)

1− 1+α

(10)

α0 = We define U =2

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and U0 = 2

1− 1 + α0

(11)

which are the maximum slip amplitudes of asperities 1 and 2 in modes 10

(12)

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and 01 respectively. The nondimensional variables and time are s K1 x K1 y K1 X= , Y = , T = t μ1 fs1 fs1

The sticking region Q of the system, i.e. the set of points corresponding

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to stationary asperities, is a quadrilateral in the plane XY (Dragoni and

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Tallarico, 2016).

The kind of seismic event generated by the fault depends on initial state

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of the system. This is defined by a variable p, that is a measure of the initial

Y = X + p. The line

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stress distribution on the asperities. The orbit of the system in Q is the line

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Y = X + p0

with

(β − 1)ξ α + αξ + ξ

(14)

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p0 =

(13)

divides Q in two subsets producing events with initial mode 10 (p < p0 ) and

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01 (p > p0 ), respectively.

The subset S of Q, from which the fault produces slipping of one asperity followed by the simultaneous motion of both of them, is defined by the values of p belonging to the interval [p1 , p2 ], with p1 =

(β − 1)ξ − ακ1 U , α + αξ + ξ

p2 =

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(β − 1)ξ + αβξκ2 U 0 α + αξ + ξ

(15)

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3. The data The data concerning the 2018 Gulf of Alaska event are listed in Table 1 and the model parameters are calculated from them. The lithosphere is considered a Poisson solid with rigidity μ = 30 GPa (Holdahl and Sauber, 1994). Since asperity slip is uniform in the model, the traction s can be calculated from the formulae for a rectangular dislocation in an elastic half-

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space (Okada, 1992).

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However, the traction of a finite dislocation source is virtually indistinguishable from that of a point-like double-couple source in an unbounded

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medium, if the distance ` between the asperity centroids is greater than 3/2

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of the linear asperity size (Dragoni and Lorenzano, 2016) the linear asperity size. Since this is the case, a good approximation for s is the traction per

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unit moment due to point-like dislocations with strike-slip mechanism 5 12π`3

(16)

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s=

whence α = 0.12 and α0 = 0.48, with ξ = 0.25 (Fig. 1).

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The value of β can be estimated from the ratio u2 /u1 between the slips of the asperities (Dragoni and Santini, 2012): hence β = 0.45.

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For the ratio between dynamic and static friction, we assume a typical value  = 0.7 (e.g. Jaeger and Cook, 1976). This is consistent with the average value inferred by Fukuyama and Mizoguchi (2010) for the dynamic friction of the 2002 Denali (Alaska) earthquake, that had the same mechanism and magnitude as the 2018 event. The choice of a value for γ would require the knowledge of the absolute value of stress, that is not an observable (Kanamori, 2001). The value of γ 8

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is constrained by the choice of values for the other parameters and by the requirement that a good fit is obtained with the observed moment rate. It results γ = 1.3, corresponding to a seismic efficiency η ' 0.16 (Dragoni and Santini, 2015). In the case of 2018 Gulf of Alaska earthquake, asperity 1 fails first at time t1 , then it triggers the failure of asperity 2 at time t2 and there is a phase of

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simultaneous motion, followed by slip of asperity 2 alone at time t3 ; this is a 3-mode event, with sequence of modes 10-11-01 and duration time t4 . This

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behaviour is the consequence of the continuous stress transfer between the

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asperities occurring during rupture: it is considered in detail in section 6. In order that the first mode is 10 and the second is 11, the state of the

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system before the earthquake must be such that p1 < p < p0 . With the

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assumed values for α, β and ξ, from (14) we obtain p0 = −0.34 and from (15) p1 = −0.435 (Fig. 2). In order to reproduce the two observed moment requirements.

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rates, we consider values of p = −0.42 and −0.43, satisfying the previous

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4. Slip rates

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The orbits of the system for the two cases are shown in Fig. 3.

4.1. Failure of asperity 1 (mode 10) The slip rates are ˙ ) = U 1 + α sin ω1 T e− γ2 T ΔX(T 2ω1

(17)

ΔY˙ (T ) = 0

(18)

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where



ˉ ˉ − αY +  Uˉ1 = 2 X 1+α

with ω1 =

r

1+α−



γ2 4

(19)

(20)

The mode duration is i 1h 2ω1 Vˉ π + arctan ω1 (1 + α)Uˉ1 + γ Vˉ

of

T10 =

(21)

ΔT Δt

(22)

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k=

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In order to obtain the dimensional slip rate, we consider the ratio

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with ΔT and Δt nondimensional and dimensional event duration, we obtain ΔT t = kt Δt

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T = and

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X(kt) =

U x(t) u

(23)

(24)

where U and u are the maximum nondimensional and dimensional slip of the

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asperity 1 in mode 10.

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From (23) and from (24) we finally obtain dx(t) u dX(kt) = k dt U d(kt)

(25)

The dimensional slip rates, in the case (00 → 10), are Δx(t) ˙ = uk

γ 1+α sin ω1 kt e− 2 kt 2ω1

Δy(t) ˙ =0

(26) (27)

In the dimensional equations dots indicate differentiation with respect to t. 10

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4.2. Simultaneous asperity failure (mode 11) The slip rates are γ

˙ ) = − (c1 cos ωa T − c2 sin ωa T + c3 cos ωb T − c4 sin ωb T ) e− 2 T (28) ΔX(T   γ 1 1 ΔY˙ (T ) = − c1 cos ωa T − c2 sin ωa T − c3 cos ωb T + c4 sin ωb T e− 2 T ξ ξ (29)

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where

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ωa =

r

γ2 1− 4

ωb =

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with

γ c 2 = ωa B + A 2 γ c 4 = ωb D + C 2

and

(30)

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γ c1 = ωa A − B, 2 γ c3 = ωb C − D, 2

r

1 + α + α0 −

(31) γ2 4

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γ 1 (1 − )(1 + βξ) B, B = 2ωa 1+ξ γ ξ D, D = C= (1 − )(XP − YP ) 2ωb 1+ξ A=

(32)

(33) (34)

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Mode 11 terminates at time T = T11 , hence mode duration is T11 = min(TX , TY )

(35)

where TX and TY are respectively the smallest positive solutions of equations ˙ ) = 0, X(T

Y˙ (T ) = 0.

(36)

In order to obtain the dimensional slip rate, we consider again the ratio with ΔT and Δt nondimensional and dimensional event duration, we obtain u ˙ k Δx(t) ˙ = ΔX(kt) U 11

(37)

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Δy(t) ˙ = ΔY˙ (kt)

u k U0

(38)

substituting (28) and (29) in (37) and (38), we finally obtain γ u k (−c1 cos ωa kt + c2 sin ωa kt − c3 cos ωb kt + c4 sin ωb kt) e− 2 kt U   (39) γ u 1 1 Δy(t) ˙ = 0 k −c1 cos ωa kt + c2 sin ωa kt + c3 cos ωb kt − c4 sin ωb kt e− 2 kt U ξ ξ (40)

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Δx(t) ˙ =

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4.3. Failure of asperity 2 (mode 01)

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The slip rates are

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with

 ˉ + β  α0 X ˉ ˉ U2 = 2 Y − 1 + α0

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where

˙ )=0 ΔX(T   0 ˉ ˉ γ (1 + α ) U + γ V 2 sin ω2 T − Vˉ cos ω2 T e− 2 T ΔY˙ (T ) = 2ω2

ω2 =

r

1 + α0 −

γ2 4

(41) (42)

(43)

(44)

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The mode duration is

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T01 =

i 1h 2ω2 Vˉ π + arctan ω2 (1 + α0 )Uˉ2 + γ Vˉ

(45)

In order to obtain the dimensional slip rate, we again consider the ratio with ΔT and Δt nondimensional and dimensional event duration, we obtain Y (kt) =

U0 y(t) u

(46)

where U 0 and u are the maximum nondimensional and dimensional slip of the asperity 2 in mode 01. 12

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From (45) and from (46) we finally obtain dy(t) u dY (kt) = 0k dt U d(kt)

(47)

The dimensional slip rates, in the case (11 → 01), are (48)

  γ u (1 + α0 )Uˉ2 + γ Vˉ sin ω2 kt − Vˉ cos ω2 kt e− 2 kt Δy(t) ˙ = 0k U 2ω2

(49)

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Δx(t) ˙ =0

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The slip rates for the 2018 Gulf of Alaska event are shown in Fig. 4. The calculated values are consistent with the observed moment rates. Slip rates

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values in the order of several tens of centimeters per second are in the range

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of typical earthquake slip rates (e.g. Kasahara, 1981).

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5. Moment rates

The source function of an earthquake is usually expressed by its moment

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rate (Aki, 1966; Kanamori and Anderson, 1975; Lay and Kanamori, 1980;

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event is

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Ruff and Kanamori, 1983). In our model, the moment rate of an n-mode

˙ ) + ΔY˙ (T ) ΔX(T M˙ (T ) = M1 U

(50)

where M1 is the seismic moment that is released in a 1-mode event 10 when γ = 0. In dimensional form, the moment rate predicted by the model in case of a sequence of three slipping modes 10-11-01 (n = 3) is

m(t) ˙ = m1



 1+α Δx(t ˙ − ti ) + Δy(t ˙ − ti ) , 1 + α0 13

i = 1, 2, 3

(51)

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In order to take into account the difference in the observed mode durations Δt1 = t2 − t1 , Δt2 = t3 − t2 and Δt3 = t4 − t3 , we introduce the ratios k1 =

T10 , Δt1

k2 =

T11 , Δt2

k3 =

T01 Δt3

(52)

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From (26) and (27), from (48) and (49), from (39) and (40) the slip rates are  γ  k1 1+α sin ω1 k1 (t − t1 ) e− 2 k1 (t−t1 ) , t 1 ≤ t ≤ t2  2ω 1            k [−c1 cos ωa k2 (t − t2 ) + c2 sin ωa k2 (t − t2 )   2 (53) Δx(t−t ˙ −c3 cos ωb k2 (t − t2 ) + c4 sin ωb k2 (t − t2 )] i) =   γ    t 2 ≤ t ≤ t3 e− 2 k2 (t−t2 ) ,           0, t 3 ≤ t ≤ t4

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γ  e− 2 k2 (t−t2 ) , t 2 ≤ t ≤ t3          h i  ˉ2 +γ Vˉ  (1+α0 )U  ˉ cos ω2 k3 (t − t3 ) k sin ω k (t − t ) − V  3 2 3 3  2ω2   − γ2 k3 (t−t3 ) , t 3 ≤ t ≤ t4 e

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Δy(t ˙ − ti ) =

na

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 0, t 1 ≤ t ≤ t2             k2 [−c1 cos ωa k2 (t − t2 ) + c2 sin ωa k2 (t − t2 )      + 1ξ c3 cos ωb k2 (t − t2 ) − 1ξ c4 sin ωb k2 (t − t2 )]

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(54)

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of

The dimensional moment rate is calculated according to (51) as  1+α γ t 1 ≤ t ≤ t2 k1 2ω1 sin ω1 k1 (t − t1 ) e− 2 k1 (t−t1 ) ,          n o 
 1+α  k cos ω k (t − t ) + c sin ω k (t − t )] + 1 + [−c  2 1 a 2 2 2 a 2 2  1+α0  n  o    1+α 1 − 1ξ 1+α [−c3 cos ωb k2 (t − t2 ) + c4 sin ωb k2 (t − t2 )] 0 m(t) ˙ = m1 γ   t 2 ≤ t ≤ t3 e− 2 k2 (t−t2 ) ,         h i   ˉ2 +γ Vˉ  1+α (1+α0 )U ˉ  k sin ω k (t − t ) − V cos ω k (t − t ) 3 1+α0 2 3 3 2 3 3  2ω2    γ e− 2 k3 (t−t3 ) , t 3 ≤ t ≤ t4 (55)

re

where we set t1 = 0 and m1 is the seismic moment released by asperity 1 in

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the case γ = 0. Setting u1 = u and u2 = βu, we have

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with

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m1 = μA1 uˉ

κ1 =

(56)

u1 κ1

(57)

 γ 0 1 1 + e− 2 T 2

(58)

π ω1

(59)

uˉ =

T0 =

The function m(t), ˙ considering an average slip of asperity 1 u1 = 8.5 m and of asperity 2 u2 = 3.8 m, is shown in Fig. 5a together with the observed moment rate as reported by Geoscope Observatory, France (2018).

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Substituting the considered data in (56), we obtain m1 ' 0.94 × 1021 Nm The seismic moment is m0 =

Z

(60)

Δt

m(t) ˙ dt

(61)

0

and is equal to 1.39 × 1021 Nm, in good agreement with the observed value

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equal to 1.13 × 1021 Nm.

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The function m(t) ˙ considering a lower average slip of asperity 1 u1 = 6.7 m and of asperity 2 u2 = 3 m, is shown in Fig. 5b together with the observed

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the last data in (56), we obtain

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moment rate as reported by United States Geological Survey. Substituting

(62)

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m1 ' 0.74 × 1021 Nm

The seismic moment, m0 = 1.09 × 1021 Nm, is in good agreement with the

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value 0.96 × 1021 Nm by Lay et. al. (2018).

As a confirmation of the role of the simultaneous motion of asperities, the

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moment rate of 1964 Alaska earthquake modelled by Dragoni and Santini (2015) may be compared to the studied 2018 Gulf of Alaska earthquake.

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In the 1964 Alaska event (Mw = 9.2) two locked zones were localized at a distance of about 300 km (Ichinose et al., 2007) and the large distance probably determined the absence of a simultaneous motion. If one compares the moment rates for the two events, it is possible to note a remarkable difference: in the central time interval, the moment rate of the 1964 event has much smaller values, that were ascribed to slip of the weaker fault region (Fig. 5c). 16

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Finally, we model the case without the presence of a simultaneous motion of asperities in the main fault of a complex set of faults, as proposed by Lay et al. (2018) who calculated a seismic moment moment m0 = 0.60 × 1021 Nm. In this case, we consider a 2-mode 10-01 event, with initial condition p = −0.435, that is chosen in order to exclude simultaneous motion and to provide a reasonable fit with the observed moment rate (Fig. 6a). For the

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asperity slips, the values u1 = 4.5 m and u2 = 2 m are used, as suggested by Lay et al. (2018). The number of dynamic modes is suggested by inspection

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of the moment rate function. A 2-mode event does not involve simultaneous

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motion of the asperities, so that the moment rate as a function of time is made of two disjoint humps. It results m0 > m1 + m2 , where m1 and m2

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are the seismic moments of asperities 1 and 2 respectively, as calculated in

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our model. We obtain m1 + m2 = 0.52 × 1021 Nm, in good agreement with the value proposed by the authors. It is also possible to observe that the

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difference between m0 and m1 + m2 is about equal to the moment rates of the other three faults (Fig. 6b).

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6. Stress distribution

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We define nondimensional forces F1 =

f1 , fs1

F2 =

f2 fs1

(63)

Accordingly, in the slipping modes, the tangential forces applied to the asperities in the slip direction can be written as ˙ ) F1 (T ) = −(1 + α)X(T ) + αY (T ) − γ X(T

(64)

F2 (T ) = −(ξ + α)Y (T ) + αX(T ) − γξ Y˙ (T )

(65)

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where α is the coupling constant of asperities, γ is a parameter related to the seismic efficiency of the fault and ξ is the ratio of different asperity areas. The conditions for the failure of asperities 1 and 2 are respectively F1 = −1,

F2 = −βξ

(66)

Finally the dimensional shear stresses are

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also graphically shown in Fig. 7.

K1 u 2μe˙ u = F1 (kt) υ U A1 U

(67)

σ2 (t) = F2 (kt)

K2 u 2μe˙ u = F2 (kt) υ U A2 U

(68)

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σ1 (t) = F1 (kt)

re

where K1 and K2 are coupling constants, defined in the previous paragraph. The stresses give the same order of magnitude as the stress drop values

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obtained by Seno (2014) in Aleutian Trench (0.2 MPa < Δσ < 2 MPa). The forces and the stresses calculated in the case of 3-mode event approx-

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imating earthquake data by Geoscope Observatory, France (2018) are shown in Fig. 7a, in the case approximating earthquake data by United States

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Geological Survey (2018) are shown in Fig. 7b. 7. Discussion

Fault surfaces are characterized by inhomogeneous initial stress and friction distributions, entailing a nonuniform distribution of coseismic slip. The moment rate functions inferred from seismological data show that fault slip takes place in an irregular fashion, with a continuous stress transfer between the asperities.

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In order to reproduce the observed moment rates of the 2018 Gulf of Alaska earthquake, we considered a simplified fault model with two interacting asperities and showed that the different moment rates may result from different initial conditions on the fault. The fault model is characterized by the average values of stress, friction and slip on each asperity. Details of the stress, friction and slip distributions are neglected, so that the cal-

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culated moment rates only reproduce the envelope of the observed moment rate functions.

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The values of model parameters were inferred from the available data

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and may be affected by some uncertainty. In particular, the assumed areas of the two asperities are based on the slip distribution on the fault plane

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retrieved from seismological data. Therefore, the values of A1 and A2 are

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only approximated. The effect of a change in A1 and A2 (or in the associated masses μ1 and μ2 ) is straightforward if they are changed by the same factor

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a1 . In this case, the parameter ξ is unchanged and the only quantity that is affected in the expression (55) of moment rate is the seismic moment m1 .

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According to (56), m1 is proportional to A1 , so that the moment rate changes by the same factor a1 .

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The effect is more complicated if A1 and A2 change by different factors, say a1 and a2 respectively. In this case ξ changes by a factor a2 /a1 , entailing changes in α0 , k2 and k3 . There are also effects on the quantities p0 , p1 and p2 defining the sticking region of the system. However, if the changes in A1 and A2 are relatively small, for instance 10%, the dominant effect is still the change in m1 , with a minor distortion of the moment rate curve. A change in the values of slip amplitudes u1 and u2 has similar effects.

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If they are changed by the same factor, the parameter β is unchanged and the only effect on moment rate is through the factor m1 in (55). Otherwise, there is a change in β entailing changes in k2 , k3 , p0 , p1 and p2 , with some distortion of the moment rate curve. 8. Conclusions

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Discrete dynamical models, focusing on the large-scale properties of fault

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systems (Dragoni and Santini, 2010; 2012), have the potentiality to unveil the mechanisms controlling the evolution of such systems. The application to

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the 2018 Gulf of Alaska earthquake demonstrates that a possible mechanism

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for this earthquake was the consecutive failure of two asperities with a phase of simultaneous motion.

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The simultaneous motion of asperities may have important consequences for the energy released by large earthquakes because it changes the moment

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rate function and significantly increases the seismic moment of large events. The analytical results are in good agreement with the observed data, and the

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time differences are always smaller than 5% of the event durations (Geoscope Observatory, France, 2018; United States Geological Survey, 2018).

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An alternative model was proposed by Lay et al. (2018), who suggested a more complex fault system. In this case, the observed moment rate could be approximated by the failure of two asperities on the main fault with no simultaneous motion.

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Acknowledgments The authors are grateful to the editor Vernon Cormier and to two anonymous reviewers for helpful and constructive comments on the first version of the

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paper.

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References

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earthquake of June 16, 1964. Part 2. Estimation of earthquake movement, released energy, and stress – strain drop from the G-wave spec-

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trum, Bull. Earthquake Res. Inst., Tokyo Univ., 44, 73-88. DeMets, C., Dixon, T. H., 1999. New kinematic models for Pacific-North

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America motion from 3 Ma to present. Part 1. Evidence for steady motion and biases in the NUVEL1-A model, Geophys. Res. Lett., 26,

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1921-1924.

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Dragoni, M., and Lorenzano, E., 2016. Conditions for the occurrence of seismic sequences in a fault system, Nonlin. Proc. Geophys., 23, 419– 433.

Dragoni, M., and Santini, S., 2010. Simulation of the long-term behaviour of a fault with two asperities, Nonlin. Proc. Geophys., 17, 777–784. Dragoni, M., and Santini, S., 2012. Long-term dynamics of a fault with two asperities of different strengths, Geophys. J. Int., 191, 1457–1467. 21

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Dragoni, M., and Santini, S., 2015. A two-asperity fault model with wave radiation, Phys. Earth Planet. Int., 248, 83–93. Dragoni, M., and Santini, S., 2017. Effects of fault heterogeneity on seismic energy and spectrum, Phys. Earth Planet. Int., 273, 11–22. Dragoni, M., and Tallarico, A., 2016. Complex events in a fault model with interacting asperities, Phys. Earth Planet. Int., 257, 115-127.

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Fukuyama, E., and Mizoguchi, K., 2010. Constitutive parameters for earth-

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index.php/en/catalog/earthquake-description?seis=us2000cmy3 Holdahl, S. R., and Sauber, J., 1994. Coseismic slip in the 1964 Prince

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William Sound earthquake: A new geodetic inversion, Pure Appl. Geophys., 142, 55–82.

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Ichinose, G., Somerville, P., Thio, H. K., Graves, R. and O’Connell, D., 2007.

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Rupture process of the 1964 Prince William Sound, Alaska, earthquake from the combined inversion of seismic, tsunami, and geodetic data, J. Geophys. Res., 112, B07306. Jaeger, J. C., and Cook, N. G. W., 1976. Fundamentals of Rock Mechanics, 2nd edition, Chapman and Hall, London, pp. 585. Kanamori, H., 2001. Energy budget of earthquakes and seismic efficiency, in Earthquake Thermodynamics and Phase Transformations in the

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Earth’s Interior, Academic Press, 293–305. Kanamori, H., and Anderson, D. L., 1975. Theoretical basis of some empirical relations in seismology, Bull. Seismol. Soc. Am., 65, 1073–1095. Kasahara, K., 1981. Earthquake Mechanics, Cambridge University Press, Cambridge, USA, pp. 248. Krabbenhoeft, A., von Huene, R., Miller, J. J., Lange, D., and Vera, F.,

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2018. Strike-slip 23 January 2018 Mw 7.9 Gulf of Alaska rare intraplate

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earthquake: Complex rupture of a fracture zone system, Sci. Rep., 8, 13706, https://doi.org/10.1038/s41598-018-32071-4.

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Lay, T., and Kanamori, H., 1980. Earthquake doublets in the Solomon Is-

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lands, Phys. Earth Planet. Int., 21, 283–304. Lay, T., Ye, L., Bai, Y., Cheung, K. F., and Kanamori, H., 2018. The 2018

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MW 7.9 Gulf of Alaska earthquake: Multiple fault rupture in the Pacific plate, Geophys. Res. Lett., 45, 9542-9551. https://doi.org/10.1029/

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Lorenzano, E., and Dragoni, M., 2018. A Fault Model with Two Asperities

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of Different Areas and Strengths, Math. Geosci., 50, 697–724.

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Okada, Y., 1992. Internal deformation due to shear and tensile faults in a half-space, Bull. Seismol. Soc. Am., 82, 1018–1040. Rice, J. R., 1993. Spatio-temporal complexity of slip on a fault, J. Geophys. Res., 98, 9885–9907. Ruff, L., Kanamori, H., 1983. The rupture process and asperity distribution of three great earthquakes from long-period diffracted P-waves, Phys. Earth Pianet. Inter., 31, 202–230. 23

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Ruppert, N. A., Rollins, C., Zhang, A., Meng, L., Holtkamp, S. G., West, M. E., and Freymueller, J. T., 2018. Complex faulting and triggered rupture during the 2018 Mw 7.9 offshore Kodiak, Alaska, earthquake. Geophys. Res. Lett., 45, 7533-7541. https://doi.org/10.1029/2018GL078931 Seno, T., 2014. Stress drop as a criterion to differentiate subduction zones where Mw 9 earthquakes can occur, Tectonophysics, 621, 198–210.

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United States Geological Survey, 2018. M7.9 Alaska earthquake of January 23, 2018, Poster updated by U.S. Geological Survey National Earth-

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quake Information Center on 23 January 2018, https://earthquake.usgs.gov/

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archive/product/poster/20180123/us/1517960191614/poster.pdf Zhao, B., Qi, Y., Wang, D., Yu, J., Li, Q., and Zhang, C., 2018. Coseis-

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mic slip model of the 2018 Mw 7.9 Gulf of Alaska earthquake and its seismic hazard implications, Seismol.

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https://doi.org/10.1785/0220180141

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Lett., 90, 642-648.

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Figure 1: (a) Map of the Gulf of Alaska with the fault trace; the star indicates the epicenter. (b) The fault model with two asperities of different areas A1 and A2 , respectively; the total fault area is A0 . The slip distribution (in grey scale) is from United States Geological Survey (2018).

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Figure 2: The sticking region Q and its subset S, from which events involving the simultaneous slip of the asperities take place, considered in the case of the 2018 Gulf of Alaska earthquake (α = 0.12, β = 0.45, γ = 1.3,  = 0.7).

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Figure 3: (a) Orbit in the phase space showing a 3-mode event approximating the 2018 Gulf of Alaska earthquake data from Geoscope Observatory, France (2018), with the initial condition p = −0.42, and (b) orbit approximating the same earthquake data from United States Geological Survey (2018), with the initial condition p = −0.43 (α = 0.12, β = 0.45, γ = 1.3,  = 0.7).

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Figure 4: Slip rates Δx˙ (solid curve) and Δy˙ (dashed curve) of the asperities as functions of time, for the 3-mode events shown in Fig. 3a (a) and in Fig. 3b (b).

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Figure 5: (a) Moment rate of the event (thick curve) superimposed to that observed by Geoscope Observatory, France (2018) (dark grey filled curve); (b) moment rate of the event

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(thick curve) superimposed to that observed by United States Geological Survey (2018) (light grey filled curve). The asperity slips, considered in this model, are (a) u1 = 8.5 m and u2 = 3.8 m, and (b), u1 = 6.7 m and u2 = 3 m. (c) Moment rate of the 1964 Alaska earthquake (thick curve), calculated by Dragoni and Santini (2015), superimposed to the moment rate curve inferred by Ichinose et al. (2007) (grey filled curve).

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Figure 6: (a) Orbit in the phase space showing a 2-mode event, which modes are 10-01, with the initial condition p = −0.435; (b) moment rate of the event (thick curve), in absence of the simultaneous motion of asperities, superimposed to that modelled by Lay et al. (2018) (grey filled curve), with the main fault moment rate (red curve) of the fourfault moment rates proposed by the authors (color curves). The asperity slips, considered in this model, are u1 = 4.5 m and u2 = 2 m.

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Figure 7: Shear stresses, σ1 (solid curve) and σ2 (dashed curve), on the asperities as functions of time, for the 3-mode events shown in Fig. 3a (a) and in Fig. 3b (b).

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Table 1: Data of the 2018 Gulf of Alaska earthquake; multiple values of asperity slip, and event and model durations are due to the two different fault models.

Rigidity of the lithosphere

υ = 6 cm a−1 ,

Relative plate velocity

e˙ = 1.2 × 10−15 s−1 ,

Tectonic strain rate

A1 = 2.4 × 103 km2 ,

Fault area Area of asperity 1

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A0 = 1.0 × 104 km2 ,

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μ = 30 GPa,

Area of asperity 2

u1 = 6.7 or 8.5 m,

Average slip of asperity 1

u2 = 3.0 or 3.8 m,

Average slip of asperity 2

` = 80 km,

Distance between asperity centers

Δt = 55 or 60 s,

Duration of the event

Δt1 = 24 or 29.5 s,

Duration of mode 10

Δt2 = 12 or 6.5 s,

Duration of mode 11

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Duration of mode 01

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Δt3 = 19 or 24 s,

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A2 = 0.6 × 103 km2 ,

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Highlights • The observed moment rates of the 2018 Gulf of Alaska earthquake are reproduced by a two-asperity fault model. • Fault asperities with different areas and strengths, subject to a nonuniform initial stress, are considered.

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• The 2018 event can be represented as a sequence of three slipping modes

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including a phase of simultaneous slip of the asperities.

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• Fault slip rates and moment rates as functions of time are calculated.

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• The difference in the observed moment rates can be ascribed to different

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initial stress conditions on the fault.

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Figure 1

Figure 2

Figure 3

Figure 4

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Figure 6

Figure 7