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Three-dimensional mechanical models for the June 2000
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earthquake sequence in the South Icelandic Seismic Zone
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Loïc Dubois a, Kurt L. Feigl a, *, Dimitri Komatitsch b,
Thóra Árnadóttir c, and Freysteinn Sigmundsson c
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a
Dynamique Terrestre et Planétaire CNRS UMR 5562, Université Paul Sabatier,
Observatoire Midi-Pyrénées, 14 av. Édouard Belin, 31400 Toulouse, France
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Laboratoire de Modélisation et d'Imagerie en Géosciences CNRS UMR 5212,
Université de Pau et des Pays de l'Adour, BP 1155, 64013 Pau Cedex, France
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Nordic Volcanological Center, Institute of Earth Sciences, University of Iceland,
Reykjavik, Iceland
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Friday, February 12, 2016
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Submitted to Earth and Planetary Science Letters
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Application name Microsoft Word 2004 for Mac
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*Corresponding author, now at
Department of Geology and Geophysics
University of Wisconsin-Madison
1215 West Dayton Street
Madison, WI
53706 USA
feigl@wisc.edu
Tel. +1 608 262-8960
Fax. +1 608 262-0693
Dubois et al.
submitted 2016-02-12
TABLE OF CONTENTS
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Table of contents
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Abstract
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1. Introduction
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2. Data and Methods
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2.1 Data
2.2 Numerical Solution using the Finite Element Method
2.3 Modeling poroelastic deformation with an elastic approximation
2.4 Viscoelastic modeling
2.5 Stress transfer
3. Results
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3.1 Estimating the distribution of co-seismic fault slip
3.2 Modeling postseismic deformation with the elastic approximation for poroelastic effects
3.3 Viscoelastic model for post-seismic deformation
3.4 Stress transfer
4. Discussion and Conclusions
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4.1 Co-seismic effects
4.2 Viscoelastic effects
4.3 Poroelastic effects
4.4 Migrating seismicity
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Acknowledgments
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Supplementary Material
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SM.1 Testing against other solutions
SM.2 Estimating co-seismic slip by joint inversion of GPS and INSAR measurements
SM.3 Approximating complete poroelastic relaxation by an elastic solution
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References
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Tables
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Last page
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ABSTRACT
We analyze a sequence of earthquakes that began in June 2000 in the South Iceland Seismic Zone (SISZ)
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in light of observations of crustal deformation, seismicity, and water level, focusing on the tendency for
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seismicity to migrate from east to west. At least five processes may be involved in transferring stress from one
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fault to another: (1) propagation of seismic waves, (2) changes of static stress caused by co-seismic slip, (3)
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inter-seismic strain accumulation, (4) fluctuations in hydrological conditions, and (5) ductile flow of subcrustal
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rocks. All five of these phenomena occurred in the SISZ during the post-seismic time interval following the
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June 2000 earthquakes. By using three-dimensional finite element models with realistic geometric and rheologic
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configurations to match the observations, we test hypotheses for the underlying geophysical processes. The
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coseismic slip distribution for the Mw 6.6 June 21 mainshock is deeper when estimated in a realistic layered
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configuration with varying crustal thickness than in a homogenous half-space. Processes (1) and (2) explain
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only the aftershocks within a minute of the mainshocks. A quasi-static approximation for modeling poroelastic
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effects in process (4) by changing Poisson’s ratio from undrained to drained conditions does not provide an
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adequate fit to post-seismic deformation recorded by interferometric analysis of synthetic aperture radar images.
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The same approximation fails to explain the location of subsequent seismicity, including the June 21 mainshock.
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Although a viscoelastic model of subcrustal ductile flow (5) can increase Coulomb failure stress by more than
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10 kPa, it does so over time scales longer than a year. Accumulating interseismic strain (3) in an elastic upper
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crust that thickens eastward breaks the symmetry in the lobes where co-seismic stress exceeds the Coulomb
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failure criterion. This process may explain the tendency of subsequent earthquakes to migrate from east to west
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across the South Iceland Seismic Zone within a single sequence. It does not, however, explain their timing.
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1. INTRODUCTION
The South Iceland Seismic Zone (SISZ) is aptly named. Located on the boundary between the North
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America and Eurasia plates, it accommodates their relative motion as earthquakes. The plate boundary follows
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the spreading mid-Atlantic Ridge, comes ashore on the Reykjanes Peninsula, where it becomes a shear zone to
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reach the Western Volcanic Zone at a triple point near the Hengill volcanic center (Fig 1). There, most of the
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present-day deformation continues across the SISZ to the Eastern Volcanic Zone. Overall, the SISZ
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accommodates sinistral shear along its east-to-west trend, but most of its large (M 6) earthquakes rupture as
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right-lateral strike slip on vertical faults striking north. These parallel faults are 10 – 20 km long and ~5 km
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apart, analogous to books on a shelf [1]. Indeed, the SISZ deforms as a shear zone, accommodating
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18.9 ± 0.5 mm/yr of relative plate motion [2, 3] in a band less than 20 km wide [4]. Consequently, the
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interseismic strain rate is ~10–6 per year. Its kinematics can be described as a locked fault zone slipping 16 to
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20 mm/yr below 8 to 12 km depth with a dislocation in an elastic half-space [5]. If accumulated elastically over
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a century, this strain could be released as ~2 m of co-seismic slip. Fortunately, earthquakes in the SISZ do not
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rupture its entire 100-km length from east to west; the 20-km length of the north-south faults seems to limit their
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magnitude to 6.5 or 7.
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The seismic sequence of events began on 17 June 2000, with a main shock of magnitude Mw = 6.6 [6]. It
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produced co-seismic deformation that was measured geodetically by both GPS [7] and INSAR [8]. Three and a
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half days (81 hours) later, a second earthquake ruptured a distinct fault some 18 km to the west of the first event.
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Both earthquakes have strike-slip focal mechanisms on faults dipping nearly vertically [9]. Both earthquakes
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ruptured the ground surface in a right-lateral, en echelon pattern with an overall strike within a few degrees of
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North [10]. The slip distributions estimated from geodetic data for both earthquakes have maxima over 2 m,
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centroid depths around 6 km, down-dip widths of 10 – 12 km, and along-strike lengths of 16 – 18 km [7, 11].
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In this study, we build on previous estimates of the distribution of slip that occurred on the rupture
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surfaces of the two mainshocks in June 2000 from GPS measurements of displacements [7], INSAR recordings
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of range change [8], and a combination of both in a joint inversion [11]. All three of these studies assume an
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elastic half space with uniform values of shear modulus (or rigidity)  and Poisson’s ratio . Yet other work
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outside of Iceland suggest that this simple assumption can bias estimates of slip distribution [12, 13]. For
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example, the aftershock hypocenters located by seismological procedures are deeper than the bottom of the fault
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rupture inferred from geodetic measurements in the case of the 1994 Northridge earthquake in California [14].
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In other words, the seismological and geodetic procedures for locating co-seismic slip will yield different
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estimates if they assume different elastic models. A similar discrepancy between the deepest aftershocks and the
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bottom of the slip distribution also appears to apply to the June 2000 sequence in the SISZ. Although the slip
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diminishes to less than 1 m below 8 km depth [15], the relocated aftershock hypocenters reach down to 10 km,
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with a few as deep as 12 km [16]. The hypocentral depths estimated from seismology using a layered velocity
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model are 6.3 and 5.0 km for the June 17 and 21 mainshocks, respectively [6]. By using layered models that
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account for increasing rigidity with depth, we expect deeper slip distributions [17, 18]. An accurate estimate of
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the slip distribution is required to test the hypothesis that the M 6 earthquakes in the SISZ rupture into the lower
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crust [19]. Accurate estimates of slip distribution are also important for calculating maps of earthquakes
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triggered by stress transfer. The latter is sensitive to the former because the threshold value of 10 kPa (0.1 bar)
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usually used to evaluate the Coulomb failure criterion is several orders of magnitude smaller than the stress drop
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for a typical earthquake [20, 21].
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The earthquakes in the 2000 SISZ sequence seem to be causally related. Just after the June 17 mainshock,
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additional earthquakes struck the Reykanes Peninsula, as far as 80 km to the west of the June 17 epicenter.
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Three of these generated coseismic displacements large enough to measure by INSAR and GPS and to model
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with dislocation sources with moments equivalent to Mw = 5.3, 5.5 and 5.9 [22, 23]. These secondary
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earthquakes ruptured the surface, broke rocks, and coincided with a drop of several meters in the water level of
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Lake Klevervatn [24]. Three of these events, at 8 s, 26 s, and 30 s after the June 17 mainshock at 15:40:41 UTC
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were dynamically triggered by the seismic wave train as it propagated westward [6]. A fourth event occurred 5
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minutes after the June 17 mainshock at a distance of almost 80 km [6]. The June 17 earthquake appears to have
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triggered the June 21 earthquake, based on the increase in static Coulomb stress calculated at the June 21
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hypocenter on a vertical, north-striking fault [25]. The migration of M ≥ 6 earthquakes from east to west across
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the SISZ has been observed in previous earthquake sequences in 1732-1734, 1784, and 1896 [1]. Another M ≥ 6
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event occurred in 1912 at the east end of the SISZ [26, 27].
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Understanding the processes driving the seismicity from east to west is the second objective of this study.
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Although static Coulomb stress changes can explain the location of the triggered earthquakes, this simple theory
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cannot explain the time delay between the “source” (triggering) and “receiver” (triggered) events, as pointed out
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by Scholz [21, 28] and recently reviewed by Brodsky and Prejean [29]. At least five processes may be involved
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in transferring stress from one fault to another: (1) propagation of seismic waves, (2) changes of static stress
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caused by co-seismic slip, (3) inter-seismic strain accumulation, (4) fluctuations in hydrological conditions, and
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(5) ductile flow of subcrustal rocks. All five of these phenomena occurred in the SISZ during the post-seismic
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time interval following the June 2000 earthquakes.
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(1) The dynamic stresses caused by propagation of seismic waves apparently triggered the earthquakes
on the Reykanes Peninsula in the first minute following the June 17 mainshock, as described above [6].
(2) In the “cascading seismicity” hypothesis, one earthquake triggers the next in a kind of seismic
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“domino effect”. Changes in static stress could alter rate- and state-dependent friction, thus producing a finite
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time delay between successive earthquakes[30].
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(3) In terms of moment, the main events in the June 2000 sequence released only about a quarter of the
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strain accumulated since 1912, the date of the last major earthquake in the SISZ, assuming a constant strain rate
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over the intervening 88 years [4, 15]. Such interseismic strain accumulation could conceivably contribute to
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stress transfer. If strain accumulates at the rate of 10 -6/yr, it could increase stress by 10 kPa in less than a year,
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assuming a Young’s modulus of 50 GPa.
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(4) Fluctuations in hydrological conditions occurred in the weeks to months after the June 17 mainshock
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[31, 32]. Poroelastic effects can explain about half the post-seismic signal in an interferogram spanning June 19
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through July 24 in the SISZ [32]. The same effects might also explain stress transfer in the 2000 SISZ sequence.
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Under this hypothesis, the co-seismic perturbation to the hydrologic pressure field also alters the stress
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conditions on faults near the mainshock, triggering aftershocks in areas where increasing pressure unclamps
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faults near failure [33, 34].
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(5) Ductile flow of subcrustal rocks occurs over longer time scales of several years. For example post-
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seismic deformation has been recorded by GPS around the June 2000 faults as late as May 2004 [35]. These
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centimeter-sized displacements have been modeled using a viscoelastic Burger’s rheology in a semi-analytic
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spherical harmonic formulation [35]. Here, we use a linear Maxwell viscoelastic rheology in a Cartesian finite
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element formulation to investigate the effect of geometry, viscosity and other parameters on surface
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displacements. Although ductile flow is much too slow to explain the 86-hour time delay between the June 17
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and June 21 events, it may contribute to the time delays (~100 years) between earthquake sequences in the
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SISZ.
All these considerations motivate us to move beyond the approximation of a half-space with uniform
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elastic properties. In this study, we explore models with more realistic geometric and rheologic configurations.
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To do so, we calculate stress and displacement fields using a finite element method [e.g., 36]. In particular, we
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use the TECTON code, renovated by Charles Williams [37-39], as described below. This approach allows us to
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account for the variations in crustal thickness (Fig 1 and Fig 2) inferred from earthquake hypocenter locations,
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seismic tomography, and gravity modeling [40-44].
2. DATA AND METHODS
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2.1 Data
In this study, we consider four types of data, all of which have been analyzed and described previously.
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GPS
The complete GPS data set, including the post-seismic time series, has been described elsewhere [5, 35].
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The co-seismic displacement vectors are the same as used previously to estimate the distribution of slip on the
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faults that ruptured on June 17 and 21 [7, 11].
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INSAR
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Interferometric analysis of synthetic aperture radar images (INSAR) has recorded the crustal deformation
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that occurred between June and September, 2000, as described previously [8, 15, 22, 32]. The data set used here
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is the same as analyzed by Pedersen et al. [11]The characteristic relaxation time of the post-seismic deformation
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near the June 17 fault is of the order of a month because another interferogram spanning the second post-seismic
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month (12 August to 16 September) shows less than 15 mm of differential range change across the fault (blue
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curve in Figure 2e of Jónsson et al. [32]). About half of the post-seismic deformation observed within 5 km of
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the June 17 fault recorded by INSAR in the first month can be explained by poroelastic effects involving fluid
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flowing in the epicentral area [32].
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Water level in wells and boreholes
Fluctuations in hydrological conditions have been observed in periodic measurements of pressure and
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water level in wells and boreholes around the SISZ [31, 32]. The polarity of these changes can be well described
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by the two mainshock focal mechanisms [31, 32]. In the post-seismic interval, however, the wells recover over
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time scales varying from hours to weeks, indicating large variations in the rock-mechanical properties over
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distances as short as a few kilometers.
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Earthquake locations and focal mechanisms
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The earthquakes recorded by the SIL seismological network [19, 45] operated by the Icelandic
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Meteorological Office have been relocated using multiplet algorithms [46-48] to establish a catalog of focal
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mechanisms and locations with a precision better than 1 km in all three dimensions [16, 49, 50] as part of the
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PREPARED project [51].
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2.2 Numerical Solution using the Finite Element Method
TECTON is a software package that implements a finite element formulation based on three-dimensional
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hexahedral elements [37-39]. We have tested TECTON for accuracy by comparing its results to semi-analytic
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solutions, as described in the doctoral thesis of the first author [52] and in the Supplemental Material. We use
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the word “configuration” to describe the combination of the geometry of the mesh (topology) and distribution of
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mechanical properties (rheology) within it. These configurations appear in Fig 1, Fig 2, Table 1, and Table 2.
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Mesh
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We have developed two meshes specifically for this problem to reproduce the main interfaces in our
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problem: upper/lower crust, mantle/crust, as well as the fault rupture surfaces for the June 17 and June 21
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mainshocks. The boundary conditions on the bottom and four sides are fixed while the top surface is free. The
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region under study is 180 km by 100 km by 20 km (blue box in Fig 1). To avoid edge effects, we extend the
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meshes to 300 km by 300 km by 100 km. The elastic solution achieves numerical convergence with 70 000
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nodes, but we use 140 000 nodes to handle the time-dependence in the viscoelastic case [52]. The fault slip is
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described as relative, horizontal displacements at 690 “split nodes” [38].
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The simplest configuration is a homogeneous half-space labeled HOM. The mesh for HOM includes
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horizontal layers with constant thickness. The same mesh defines the geometry of the heterogeneous HET
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configuration. A different mesh, used in the TOM and ISL configurations, allows lateral variations in crustal
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thickness, and thus gradients in rheologic properties, as sketched in Fig 1B.
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Rheology
After defining the meshes, we now define the rheology for each configuration. All the rheologic
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configurations obey an elastic constitutive relation (Hooke’s law), with stress linearly proportional to strain. For
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each element in the mesh, we specify the Poisson’s ratio  and a shear modulus , as summarized in Fig 2 and
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Table 1.
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In the HOM and TOM configurations, we assume  0.28 and  30 GPa throughout the half-space
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[11]. In the layered HET and ISL configurations, Poisson’s ratio varies from  0.28 in the upper crust to 
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0.25 in the lower crust, and then to  0.30 in the mantle, in accordance with previous studies of Iceland [40,
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53, 54]. The thickness of these layers is constant in the HET configuration, with the base of the upper crust set
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to Hu = 5 km depth and the base of the lower crust set to Hc = 23 km depth. In the ISL configuration, the upper
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and lower crustal layers vary in thickness, as sketched in Fig 2B and Fig 2C. These two crustal layers both
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thicken and dip toward the east (Fig 1) as evident in previous three-dimensional studies [40, 41, 43, 44, 54].
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Given the density, we then use the P-wave and S-wave velocities [52] to calculate  and  (under undrained
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conditions) for each element (Fig 2).
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The shear modulus is set to  30 GPa in the HOM and TOM configurations. In the HET and ISL
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configurations, it varies from  30 GPa in the upper crust to  60 GPa in the mantle. The HOF and HEF
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configurations add weak fault zones to the HOM and HET configurations, respectively. In the elements located
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within a few kilometers of the faults, the HOF and HET configurations include values as low as  3 GPa, i.e.
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four times smaller than in the surrounding crustal rocks [55]. Such low values can be justified by previous co-
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seismic studies that have assumed values as low as  10 GPa [56] and as high as  50 GPa [57].
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Following Cocco and Rice [55, 58], we assume that the bulk modulus has the same value in the damage zone as
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in the surrounding rock. Hence we set Poisson’s ratio as high as  0.43 within the weak fault zones in the
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HEF and HOF configurations.
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A thorough sensitivity analysis suggests that the co-seismic displacement field at the surface is most
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sensitive to the value of the shear modulus (rigidity) [52]. For this parameter, a perturbation of 10 percent can
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increase the misfit to coseismic displacements measured by GPS and INSAR by as much as 50 percent. The
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shape of the displacement field is also fairly sensitive to the geometric location of the fault, particularly its
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depth. The displacement field, however, is relatively insensitive to the geometry of the subcrustal layers.
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Similarly, the absolute value of Poisson’s ratio makes little difference. A displacement field calculated with
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 = 0.25 is extremely similar to one with  = 0.28, in agreement with a previous study of the June 2000
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earthquakes [11].
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To describe the post-seismic time interval, we introduce viscosity in the lower crust below a depth of Hu,
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and in the upper mantle, below a depth of Hc.
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2.3 Modeling poroelastic deformation with an elastic approximation
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Fig 5
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To explain the post-seismic interferogram (Fig 5), Jónsson et al. invoke poroelastic effects in the first
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month following the June 17 mainshock [32]. To describe the observed signal, they employ a modeling
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approach involving changes in Poisson’s ratio . This quasi-static approximation, originally introduced by Rice
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and Cleary [59] and described in the Supplementary Material, uses only two elastic coefficients to solve a
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problem in a two-phase, poroelastic medium that requires at least two more coefficients to describe completely.
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This quasi-static model assumes a change in elastic properties, specifically Poisson’s ratio, from undrained to
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drained conditions, as for several other earthquakes in different tectonic settings [60, 61]. The validity of the
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assumptions underlying this simple, static and elastic approximation to what is actually a complex, time-
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dependent, poroelastic problem [62] will enter into our discussion below. Using all these assumptions, this
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single-parameter description approximates the complete post-seismic displacement fields generated by
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poroelastic rebound as the difference between two co-seismic fields calculated with different values of Poisson’s
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ratio. Consequently, only the change in Poisson’s ratio  undrained – drained controls the shape of the
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displacement field.
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The previous study of the 2000 SISZ sequence [32] makes strong assumptions about the spatial
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distribution of the Poisson’s ratio change . As a first approximation, it uses a half-space formulation to
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calculate the Green’s functions G such that  = 0.04 everywhere [63]. As a second approximation, it develops
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a different formulation with horizontal layers, such that the drainage occurs only in the uppermost 1.5 km of the
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crust, where  = 0.08.
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Making a third approximation is one of the goals of this study. Here we allow the Poisson’s ratio change
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 to vary horizontally as well as vertically. Indeed, the finite element method can calculate the elastic Green’s
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function G from an arbitrary geometric distribution of change in Poisson’s ratio 
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In the Supplementary Material, we conclude that = 0.04 is a reasonable average value for the change
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in Poisson’s ratio. We note, however, that this parameter trades off with the thickness of the drainage layer.
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Accordingly, we experiment with different values and geometric configurations, as listed in Table 2. To
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summarize, we consider three categories of poroelastic configurations, depending on spatial distribution of .
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In the first category,  is uniform (INV1, FOR1 and FOR2). In the second category  is a function of depth
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only (FOR3, TRY1, TRY2, TRY3, INV2, and INV3). In the third category,  varies with depth and horizontal
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position (INV4).
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2.4 Viscoelastic modeling
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To describe post-seismic deformation, we introduce a linear Maxwell viscoelastic rheology in the lower
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crust and upper mantle layers of the geometric configurations described above. Varying the viscosity in a series
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of forward models, we seek the value that minimizes the misfit to the horizontal components of the
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displacement field measured by GPS in the post-seismic interval from the time of the earthquakes through May
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2004 [5, 35].
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2.5 Stress transfer
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Using the slip distributions estimated as described above, we calculate the Coulomb stress on faults in the
Call
Fig 3
SISZ using the same notation and values as Árnadóttir et al. [25]
 C   t   f ( n 
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B
 kk )
3
(1)
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where B = 0.5 is Skempton’s coefficient, f = 0.75 is the effective coefficient of friction on the fault, t is the
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increase in shear stress projected onto the fault plane (taken positive in the sense of the slip vector), n is the
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increase in normal stress (taken positive in tension), and kk is the trace of the stress tensor.
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To evaluate the Coulomb failure criterion, we project the stress tensor onto the horizontal strike slip
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vector on a north-striking vertical “receiver” fault that typifies the SISZ focal mechanisms [25]. The result is a
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scalar Coulomb stress C. At a given location, if the value of C exceeds a critical (threshold) value
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C(critical), then we conclude that the stressing process favors the occurrence of earthquakes there. If, in fact, an
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aftershock occurs at the given location, then we count it as a “triggered” event. In this study, we assume a
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threshold value of C(critical) = 10 kPa, equivalent to the stress induced by a column of water 1 m high.
3. RESULTS
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3.1 Estimating the distribution of co-seismic fault slip
Using the method described in the Supplementary material, we estimate the distribution of co-seismic
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slip for the two mainshocks in a joint inversion of the GPS and INSAR measurements. The shape and depth of
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the slip distribution depend on the geometric and rheologic configurations used in the calculation.
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Homogenous configurations
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Using the homogeneous geometric configuration (HOM), we estimate the distribution of co-seismic slip
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on the June 17 and June 21 fault planes. It is essentially similar to that estimated using the dislocation
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formulation [15], further validating our finite-element approach. The largest differences are in the upper-most
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part of the fault, where the dislocation and finite-element methods parameterize the slip differently, as described
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above. The uncertainty in the slip estimates reaches 1 m near the surface, particularly in the section of the June
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17 fault south of the hypocenter. This result may reflect discrepancies between the INSAR and GPS
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measurements for points near the fault. Elsewhere on both faults, however, the standard deviation of the slip is
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less than 0.5 m. Most of the slip is shallow, above 6 km depth. The slip decays to less than 0.5 m below 9 km
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331
depth for the June 17 fault and 11 km depth for the June 21 fault. The centroid depths are 3.7 and 3.8 km,
332
respectively. Some of the aftershock hypocenters are located outside the high-slip areas: between 8 and 10 km
333
depth in the June 17 fault plane and between easting coordinates -8 and -6 km in the June 21 fault plane.
334
Layered configurations
335
The upper two panels of Fig 3 show the slip distribution estimated using the HET configuration. The
336
difference between the HET and HOM configurations appears in Fig 4. The slip extends slightly deeper in HET
337
than in HOM. The centroid is 0.3 km deeper for the June 17 fault and 0.5 km deeper for the June 21 fault. For
338
the June 21 fault, there is over 1 m more slip at 10 km depth in HET than in HOM. The increase in slip
339
concentrates around the hypocenter relocated at 5.0 km depth from seismology using a layered velocity model
340
[6]. This difference is statistically significant with over 95 percent confidence because it is more than two times
341
larger than the standard deviation of the HET estimate (Fig 4). The surface slip is also significantly larger in the
342
HET configuration than in the HOM estimate. The slip distributions using the ISL configuration are essentially
343
the same as those found with HET [52].
344
Configurations with fault damage zones
345
The HOF configuration adds a weak damage zone around the mainshock faults to the HOM
346
configuration. The slip estimated in HOF concentrates in the upper crust, significantly increasing the maximum
347
slip value around the June 21 hypocenter at 5 km depth (Fig 4). The slip is significantly shallower in HOF than
348
in HOM. Of all the configurations, HOF shows the steepest slip gradient and thus the highest residual stress.
349
The aftershock locations seem to correlate with these areas of high slip gradient, particularly for the southern
350
edge of the June 21 rupture. The aftershocks also concentrate at the edges of the slip distribution where the slip
351
estimates become smaller than their typical 50-cm uncertainties. Adding a damage zone to the HET
352
configuration to specify the HEF configuration also changes the slip distribution. The HEF result has
353
significantly more slip in the uppermost mesh elements and around the hypocenters in both fault planes than
354
does the HET result.
355
3.2 Modeling postseismic deformation with the elastic approximation for poroelastic effects
356
Fluids flowing in the fractured and fissured rock around the June 17 fault seem to be the most likely
357
explanation for the observed time delays: 86 hours between the two mainshocks and the month-long post-
358
seismic signature observed in the interferogram (Fig 5). To explore this possibility, we consider poroelastic
359
effects by extending the work of Jónsson et al. [32] with a series of model calculations summarized in Table 2.
360
To begin, we reproduce their results by using the homogeneous geometric configuration HOM and a Poisson’s
361
ratio contrast  = 0.04. This model calculation (FOR1) has a variance reduction of 2 = 39 percent for the
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Dubois et al.
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362
observed one-month interferogram with respect to a null model. Another such calculation, FOR2, concentrates
363
the drainage in the uppermost 2 km of crust, similar to the layered model of Jónsson et al. [32]. It increases the
364
Poisson’s contrast to  = 0.06 to achieve a variance reduction of 48 percent. Both FOR1 and FOR2 appear to
365
fit the data well in a profile across strike (A-A’ in Fig 5c), in agreement with the black and green dashed curves,
366
respectively, in Figure 2 of Jónsson et al. [32]. In map view and a profile parallel to strike (B-B’ in Fig 5b),
367
however, the deformation calculated with the finite element approach exhibits small features near the fault that
368
are not present in the dislocation calculation [32]. We interpret these features as a consequence of the increased
369
surface slip allowed in the finite element formulation. Another attempt, FOR3, using  = 0.06 in the HET
370
configuration, yields a slightly different result (2 = 40 percent), highlighting the importance of layering in
371
poroelastic models.
372
We also estimate the Poisson’s ratio change  in a trial-and-error procedure. The TRY1 configuration
373
fits the data (2 = 44 percent) as well as any other solution we could find by assigning a high value of = 0.10
374
to the shallow layer between 1 and 2 km. The TRY2 configuration achieves the same reduction with lower
375
values of  The TRY3 solution is equivalent to FOR3 (Table 2).
376
Next, we allow horizontal variations in the Poisson’s contrast  in an attempt to reproduce the
377
diminished southwest lobe of the post-seismic fringe pattern. This set of geometric configurations includes a
378
damage zone extending 2-3 km around the fault. The damage zone can be further subdivided into four quadrants
379
in two layers, adding as many as eight free parameters that we estimate using a non-negative least squares
380
method [64]. One such attempt, the INV1 solution with only one free parameter, seems reasonable because it
381
resembles FOR1 in  and 2. The INV2 solution with two free parameters and INV3 solution with five free
382
parameters, resemble TRY2 and TRY1 respectively (Table 2). An extreme attempt with 8 free parameters,
383
INV4, achieves a variance reduction of 2 = 65 percent by forcing the Poisson’s ratio contrast to the physically
384
unreasonable value of  = 0.244 in the southwest quadrant of the damage zone between 2 and 5 km depth.
385
3.3 Viscoelastic model for post-seismic deformation
386
Of the parameters in the viscoelastic model, viscosity has the most influence on the post-seismic
387
displacement field, based on a sensitivity study that varies each parameter individually [52]. By optimizing the
388
fit to the horizontal components of the GPS measurements, we estimate the viscosities in the lower crust and
389
upper mantle, albeit with large uncertainties. For the lower crust, we find the 95 percent confidence interval for
390
viscosity to fall between 2 and 20 EPa.s during the first post-seismic year, and 9 – 90 EPa.s for the interval from
391
July 2001 to May 2004. (1 exapascal-second = 1018 Pa.s). The different geometric configurations HOM, HET
392
and ISL yield similar results. For the upper mantle, the best-fitting viscosity is 1 – 30 EPa.s for the early post-
393
seismic interval and the ISL configuration. The other configurations yield lower viscosity estimates with wider
394
confidence intervals, suggesting that using a realistic geometry is particularly important when continuous GPS
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Dubois et al.
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395
observations are sparse. For the upper mantle, a viscosity of 2 – 100 EPa.s, irrespective of geometric
396
configuration, fits the 2001-2004 data. These values are compatible with a previous study that used a different
397
modeling approach to describe the same post-seismic observations [35].
398
3.4 Stress transfer
399
Next, we evaluate the consequences of various sources of stress in terms of their ability to trigger
400
earthquakes.
401
June 17 and 21 co-seismic stress
402
Does the June 17 mainshock increase the Coulomb stress on a north-striking, vertical strike-slip fault at
403
the June 21 hypocenter? Yes, the increase is 100 to 150 kPa, in the three layered configurations (HEF, HET and
404
ISL), confirming the result of a previous study [25] that assumes a homogeneous half-space equivalent to our
405
HOM configuration. In all three cases, the stress increase calculated at the June 21 hypocenter exceeds the
406
conventional threshold of 10 kPa. In map view, the area where earthquakes are favored, within the contour of
407
10 kPa, is somewhat smaller when calculated with the layered HET configuration than with the uniform HOM
408
configuration (Fig 6a).
409
Do the two mainshocks increase the Coulomb stress at the location of the aftershocks? Yes, about half
410
the aftershocks are located within the 10-kPa threshold contour. Thus we conclude that only half the
411
earthquakes after the June 17 mainshock, including the June 21 mainshock, can be explained by the co-seismic
412
stress produced by the June 17 mainshock under the Coulomb failure criterion. In other words, only half the
413
seismicity is triggered by increases in static stress. The other half must be triggered by some other, presumably
414
dynamic, process, as considered in the following paragraphs.
415
“Domino effect” of cascading aftershocks
416
We generalize the Coulomb stress calculation to include all the 14 388 earthquakes between June 17 and
417
December 31 for which reliable focal mechanisms have been estimated, except the June 21 mainshock. This
418
calculation determines the time-dependent changes in the Coulomb stress field on vertical, north-striking strike-
419
slip faults. (To save calculation time, we use the layered configuration HET and the EDGRN/EDCMP
420
implementation to calculate the Green’s functions [65]). Fig 6c shows the changes in Coulomb stress at 4 km
421
depth. This stress increase occurs within less than 10 minutes of the June 17 mainshock and remains roughly
422
constant until the end of 2000. At the June 21 hypocenter, however, the increase is only 5 kPa. The primary
423
contributor to the static stress increase is a magnitude 5 aftershock that occurred some 130 seconds after the
424
June 17 mainshock at 15:42:50 midway between the two mainshock faults [16, 49]. Accounting for this large
425
event enlarges the area of increased Coulomb stress on the June 21 fault plane.
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Dubois et al.
426
427
submitted 2016-02-12
Modeling stress transfer with the elastic approximation for poroelastic effects
The elastic approximation for modeling poroelastic effects explains the location of only the aftershocks
428
nearest to the mainshock faults. For example, even the extreme HET configuration with the INV4  values
429
does not increase the Coulomb failure stress on the June 21 fault (Fig 6b).
430
Viscoelastic processes
431
Viscoelastic processes are also capable of triggering seismicity. After four years of post-seismic
432
relaxation, the Coulomb failure stress at 4 km depth exceeds the threshold of 10 kPa over most of the SISZ (Fig
433
6d). The threshold contour for the ISL configuration is less symmetric than the one for the HET configuration,
434
highlighting the importance of a realistic geometric configuration in stress calculations.
435
Interseismic strain accumulation
436
Stress also builds up during the interseismic phase of the earthquake cycle as elastic strain accumulates
437
between large earthquakes. If the interseismic strain rate is constant in time and uniform in space, but the rigid
438
upper crust thickens from west to east, then the stress rate field will exhibit a gradient. At a given depth, say
439
5 km (on the upper/lower crust boundary near the June 21 mainshock hypocenter), the rigidity increases
440
westward because the stiffer lower crust is shallower there. Consequently, stress will be higher on the west side
441
of a fault than on the east side, thereby explaining the tendency of seismicity to migrate from east to west across
442
the SISZ. To quantify this purely geometric effect, we run a simple finite element model to mimic the constant
443
interseismic strain rate. This elastic interseismic model applies the far-field boundary conditions as 90 cm of
444
displacement at N110o on the east and west faces of the mesh to mimic 100 years of plate motion. Since the
445
strain depends on the dimension of the mesh, we cannot calibrate the absolute stress values. To isolate the stress
446
contributed by the geometric variation in crustal thickness, we run this model twice, once in the HET geometric
447
configuration with horizontal layers, and then again in the ISL configuration with the crust thickening toward
448
the east. After projecting the resultant stresses onto North-striking vertical strike-slip faults, we subtract the two
449
fields to obtain the geometric contribution to the change in Coulomb stress C. This stress field exhibits a
450
strong gradient. It is 40 kPa higher on the June 21 fault than on the June 17 fault (Fig 6e). Combined with the
451
co-seismic stress imposed by the June 17 mainshock alone, this geometric interseismic effect produces a stress
452
field that is distinctly asymmetric (Fig 6f). Its threshold contour (10 kPa) includes more of the aftershocks
453
located between 3 and 5 km depth than for the other models.
4. DISCUSSION AND CONCLUSIONS
454
455
4.1 Co-seismic effects
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Dubois et al.
456
submitted 2016-02-12
The slip distributions are significantly deeper in realistic three-dimensional geometric configurations than
457
in the conventional uniform half-space configuration. The slip diminishes to negligible values less than 50 cm
458
below 12 km depth, supporting the notion that large earthquakes in the SISZ can penetrate into the lower crust
459
[19]. The layered HET configuration is well suited for calculating coseismic deformation and stress. The
460
coseismic stress increase generated by the June 17 mainshock can explain the location of the June 21
461
mainshock, confirming a previous study [25].
462
The domino effect of cascading aftershocks following the June 17 aftershock also increases the stress on
463
the (future) location of the June 21 mainshock. Since static stress migrates at the speed of an elastic wave,
464
standard Coulomb calculations based on the June 17 mainshock or its aftershocks cannot explain the 86-hour
465
time delay between the two mainshocks.
466
4.2 Viscoelastic effects
467
The simple Maxwell viscoelastic model can explain the postseismic deformation between June 2001 and
468
May 2004 with viscosities of the order of ~10 EPa.s for both the lower crust and the upper mantle. Although the
469
uncertainties are large (an order of magnitude at 68 percent confidence), the acceptable ranges are slightly
470
higher and narrower using realistic geometric configurations. For the first year, the GPS measurements of post-
471
seismic displacements can be fit with a lower viscosity, of the order of ~1 EPa.s, albeit with large uncertainties.
472
We conclude that the available data provide only weak constraints on subcrustal viscosity, partly because the
473
poroelastic and viscoelastic effects have similar amplitude.
474
4.3 Poroelastic effects
475
None of the poroelastic models fit the geodetic data very well, despite drastic increases in the complexity
476
of the models. It seems that the elastic approximation described in the Supplementary Material is insufficient to
477
describe the post-seismic observations in this case. Even with drastic increases in the number of free parameters,
478
this approach achieves only modest reductions in variance. Nor does it account for the location of many
479
aftershocks.
480
The explanation for this shortcoming probably involves time-dependent effects. The elastic
481
approximation described above is valid only after a very long time has elapsed and the fluid flow field has
482
returned to equilibrium (drained) conditions. The elastic approximation does not reproduce the map view of the
483
post-seismic interferogram very well, particularly in the fringe lobe southwest of the June 17 rupture. There, the
484
half-space model with  = 0.04 everywhere predicts about two times more uplift than is observed. To explain
485
this discrepancy, Jónsson et al. [32] suggest that the poroelastic rebound was faster in this area than elsewhere.
486
Indeed, if the hydrological state had returned to “drained” conditions in the two days following the June 17
487
mainshock, then the poroelastic signal would not appear in the interferogram beginning June 19. The water level
Page 15
Dubois et al.
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488
in the well at Hallstun (HR-19), just south of the June 17 rupture, lends some support to this interpretation. It
489
recovers faster than other wells, suggesting locally high permeability (and thus a large change in Poisson’s ratio
490
in this area. Yet our attempts to account for such a local heterogeneity do not fit the observed fringe pattern
491
much better than the uniform half-space model. We infer that time dependence, rather than spatial
492
heterogeneity, is contributing to the misfit. We conclude that the conditions for the static elastic approximation
493
to be a valid are not fulfilled. A complete time-dependent treatment, however, would require describing the
494
material parameters with more parameters to account for permeability, porosity, etc. [62, 66, 67].
495
4.4 Migrating seismicity
496
We have found that a combination of co-seismic and accumulated interseismic stress can explain the
497
tendency of seismicity to migrate from east to west across the SISZ during a single sequence of earthquakes.
498
This new result indicates that realistic geometry, including crustal thickening, is important for modeling
499
interseismic processes.
500
This mechanism does not, however, explain the 86-hour delay observed between the June 17 and June 21
501
mainshocks. The interseismic process is quite slow, increasing Coulomb failure stress at a given point by only
502
~1 Pa per day. To explain the delay, we imagine a two-step process like the “coupled poroelastic effect” that
503
Scholz suggested to explain two successive earthquakes in the Imperial Valley, California [68]. In the first step,
504
the source shock instantaneously changes the pressure of pore fluids in the rock around the receiver fault. In the
505
second (slower) step, the hydrological flow field gradually returns to equilibrium, increasing the pore pressure,
506
unclamping the receiver fault and triggering the second earthquake. Other examples of this two-step
507
“unclogging” process have been invoked in California [29, 69, 70], Oregon [71], and elsewhere [72, 73].
508
Over longer time scales, in the ~100 year intervals between sequences of earthquakes in the SISZ (e.g.,
509
1912 to 2000), inter- and post-seismic processes seem the most plausible explanation. Indeed, we have shown
510
that a linear viscoelastic rheology can increase the Coulomb failure stress by 10 kPa at distances longer than
511
10 km in only 4 years, as to be expected from the characteristic Maxwell time of 1 to 10 years approximated by
512
the ratio of a viscosity 1 to 10 EPa.s to a rigidity of 30 GPa. Longer time scales presumably involve more
513
complicated rheologies.
514
515
ACKNOWLEDGMENTS
We thank Charles Williams for generously providing the source code for his revised version of
516
TECTON, as well as Frank Roth and his colleagues for the EDGRN/EDCMP and PSGRN/PSCMP codes. We
517
also thank Eric Hetland, Tim Masterlark, Rikke Pedersen, Kristin Vogfjord, Fred Pollitz, Sandra Richwalski,
518
Pall Einarsson and Herb Wang for helpful discussions. Grimur Bjornsson kindly explained the well data
519
graciously provided by Iceland Geosurvey and Reykjavik Energy. Kristin Vogfjord and Ragnar Stefansson
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520
provided the relocated earthquake catalog estimated by the Icelandic Meteorological Office as part of the
521
PREPARED project. This work was financed by EU projects RETINA, PREPARED and FORESIGHT as well
522
as French national programs GDR STRAINSAR and ANR HYDRO-SEISMO.
SUPPLEMENTARY MATERIAL
523
524
525
SM.1 Testing against other solutions
To evaluate the accuracy of our finite-element modeling procedure, we compare its predictions to a
526
standard analytic solution for a rectangular dislocation buried in an elastic half-space, as formulated by Okada
527
[74] and implemented in the RNGCHN code [75]. Using the coseismic slip distribution estimated in a uniform
528
half space [15], we calculate the displacement field at the surface by both methods. The average difference is
529
less than 1 mm in all three components, thus validating our finite-element approach [52].
530
To increase our confidence in the geometrically complex ISL configuration, we test its mesh in the TOM
531
configuration. Like HOM, the TOM configuration has a uniform rheology: Poisson’s ratio  0.28 and shear
532
modulus  30 GPa throughout the half-space. Like ISL, the TOM configuration uses a mesh that accounts
533
for dipping layers with variable thickness. The co-seismic surface displacement fields calculated using the HOM
534
and TOM configurations differ by less than 1 mm for all three components.
535
For the post-seismic viscoelastic models, we compare the surface displacement fields calculated using
536
two different methods. The first is our finite-element approach using the TECTON software, as described above.
537
The second is a semi-analytic Green’s function approach using the PSGRN/PSCMP software [76]. The average
538
difference in surface displacement is less than 3 mm for all three components after complete relaxation [52].
539
SM.2 Estimating co-seismic slip by joint inversion of GPS and INSAR measurements
540
To estimate the distribution of slip on the June 17 and June 21 fault planes, we use the same data set and
541
inversion algorithm as Pedersen et al. [15]. In this study, we allow more realistic geometric and rheologic
542
configurations by using the three-dimensional finite-element formulation to calculate the elastic Green’s
543
functions, i.e. the partial derivatives of the measurable quantities (displacements) with respect to the model
544
parameters (slip values in discrete fault patches). This difference in approach entails a difference in the fault
545
parameterization. The dislocation formulation specifies the slip as a constant value inside each rectangular
546
rupture “patch” on the fault plane. The parameterization in the finite-element approach assigns a slip value to
547
each “split” node. In other words, the slip distribution is discontinuous in the dislocation formulation, but
548
continuous (piece-wise planar) in the finite-element formulation. As a result, the latter leads to smoother slip
549
distributions and smaller stress concentrations than the former. The finite element formulation also allows non-
Page 17
Dubois et al.
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550
zero slip at the uppermost node located at the air-rock interface, thus avoiding a singularity in the case of an
551
earthquake with surface rupture.
552
Another technical improvement involves choosing the appropriate amount of smoothing to optimize the
553
trade-off between fitting the data and finding a reasonable slip distribution. Here we choose the value of the
554
smoothing parameter that minimizes the sum of the absolute values of the slip parameters [67].
555
To evaluate the uncertainty of the slip estimates, we use a bootstrap algorithm [77]. Accordingly, we
556
resample Nb = 1000 times the data vector randomly according to its (diagonal) covariance. For each realization
557
of the data vector, we solve the inverse problem to estimate a population of the estimates m̂ for the model
558
parameters m. Then the vector of the standard deviations of the model parameter estimates is
Nb
 (m) 
559
 (m̂
i
 E(m̂)
2
i1
Nb  1
560
where the E operator denotes the expected value.
561
SM.3 Approximating complete poroelastic relaxation by an elastic solution
562
(2)
Just after the co-seismic rupture, at time t+q, “undrained” conditions apply because the fluid in a
563
representative elementary volume has not yet reached equilibrium with an external boundary [62]. Presumably,
564
the same undrained conditions also apply during the brief time interval when a seismic wave propagates through
565
a representative elementary volume. According to previous studies [32, 60, 78], the Poisson’s ratio under
566
undrained conditions is
 (tq )  undrained
567
0.31
568
At a much later time t∞, the fluids have reached equilibrium with an external boundary, deformation is
569
controlled by the solid matrix alone, and the rocks can be considered “drained”. Under these conditions,
570
Poisson’s ratio is [32, 60, 78]
571
572
573
 (t )   drained
0.27
(3)
(4)
The differential post-seismic displacement accumulated in the interval of time between t+q and t∞ is thus
w  u(t  )  u(t q )  G   drained  G  undrained  u
(5)
574
where G is the Green’s function relating co-seismic slip u on the fault plane to displacement u at the free
575
surface. We assume the geometric configuration and the slip u to be identical in both terms. Furthermore, we
576
assume that the rigidity  is the same under drained and undrained conditions [55, 58, 62].
Page 18
Dubois et al.
577
578
579
580
581
582
583
submitted 2016-02-12
In the following, we attempt to determine reasonable values for . Under drained conditions, the value
of Poisson’s ratio is [62]
 drained 
3   B(1  undrained )
3  2 B(1  undrained )
(6)
where we set the Biot-Willis parameter  = 0.28 and the Skempton’s coefficient B = 0.5 to find drained = 0.25.
Under undrained conditions, the value of Poisson’s ratio can be calculated from the ratio of P-wave to Swave velocities Vp/Vs estimated from seismological studies
 undrained




1
1


1
2

2
 Vp 



1
 V 


s
(7)
584
In and around the SISZ, the Vp/Vs ratio is typically between 1.70 and 1.85, as measured by seismic tomography
585
[40, 79]. Bjarnason et al. [54] suggest a value of 0.29 for the undrained value for Poisson’s ratio in the SISZ.
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REFERENCES
586
587
588
[1]
589
590
591
592
[2]
[3]
593
594
[4]
595
596
[5]
597
598
[6]
599
600
[7]
601
602
603
[8]
604
605
[9]
606
607
[10]
608
609
[11]
610
611
612
[12]
613
614
[13]
615
616
[14]
617
618
619
[15]
620
621
622
[16]
623
624
625
[17]
626
627
[18]
628
629
[19]
630
631
632
[20]
633
634
[21]
635
636
[22]
637
638
[23]
639
640
641
[24]
P. Einarsson, S. Bjornsson, G. Foulger, R. Stefansson, T. Skaftadóttir, Seismicity pattern in the south Iceland
seismic zone, in: D. Simpson, P. Richards, (Eds), Earthquake Prediction, an International Review. M. Ewing Ser.
4, American Geophysical Union, Washington, D.C., 1981, pp. 141-152.
C. DeMets, R.G. Gordon, D.F. Argus, S. Stein, Current plate motions, Geophys. J. Int. 101 (1990) 425-478.
C. DeMets, R.G. Gordon, D.F. Argus, S. Stein, Effect of the recent revisions to the geomagnetic reversal time
scale on estimates of current plate motions, Geophys. Res. Lett. 21 (1994) 2191-2194.
F. Sigmundsson, P. Einarsson, R. Bilham, E. Sturkell, Rift-transform kinematics in south Iceland: deformation
from Global Positioning System measurements, 1986 to 1992, J. Geophys. Res. 100 (1995) 6235-6248.
T. Árnadóttir, W. Jiang, K.L. Feigl, H. Geirsson, E. Sturkell, Kinematic models of plate boundary deformation in
southwest Iceland derived from GPS observations, J. Geophys. Res. 111 (2006) B07402.
A. Antonioli, M.E. Belardinelli, A. Bizzarri, K.S. Vogfjord, Evidence of instantaneous dynamic triggering during
the seismic sequence of year 2000 in south Iceland, J. Geophys. Res. (Solid Earth) 111 (2006) 03302.
T. Árnadóttir, S. Hreinsdottir, G. Gudmunsson, P. Einarsson, M. Heinert, C. Völksen, Crustal deformation
measured by GPS in the South Iceland Seismic Zone due to two large earthquakes in June 2000, Geophys. Res.
Lett. 28 (2001) 4031-4033.
R. Pedersen, F. Sigmundsson, K.L. Feigl, T. Árnadóttir, Coseismic interferograms of two MS=6.6 earthquakes in
the South Iceland Seismic Zone, June 2000, Geophys. Res. Lett. 28 (2001) 3341-3344.
G. Ekström, A. Dziewonski, N. Maternovskaya, M. Nettles, Centroid-Moment Tensor
Project,http://www.seismology.harvard.edu/projects/CMT, 2005.
A. Clifton, P. Einarsson, Styles of surface rupture accompanying the June 17 and 21, 2000 earthquakes in the
South Iceland seismic zone, Tectonophysics 396 (2005) 141-159.
R. Pedersen, S. Jónsson, T. Árnadóttir, F. Sigmundsson, K.L. Feigl, Fault slip distribution of two June 2000 Mw
6.5 earthquakes in South Iceland estimated from joint inversion of InSAR and GPS measurements, Earth and
Planetary Science Letters 213 (2003) 487-502.
T. Masterlark, C. DeMets, H.F. Wang, O. Sánchez, J. Stock, Homogeneous vs heterogeneous subduction zone
models: Coseismic and postseismic deformation, Geophys. Res. Lett. 28 (2001) 4047-4050.
S. Cianetti, C. Giunchi, M. Cocco, Three-dimensional finite element modeling of stress interaction: An
application to Landers and Hector Mine fault systems, J. Geophys. Res. (Solid Earth) 110 (2005).
K.W. Hudnut, Z. Shen, M. Murray, S. McClusky, R. King, T. Herring, B. Hager, Y. Feng, P. Fang, A. Donnellan,
Y. Bock, Co-seismic displacements of the 1994 Northridge, California, earthquake, Bulletin of the Seismological
Society of America 86 (1996) 19-36.
R. Pedersen, S. Jónsson, T. Árnadóttir, F. Sigmundsson, K.L. Feigl, Fault slip distribution derived from joint
inversion of InSAR and GPS measurements of two June 2000 Mw 6.5 earthquakes in South Iceland, Earth Plan.
Sci. Lett. 213 (2003) 487-502.
S. Hjaltadottir, K.S. Vogfjord, R. Slunga, Mapping subsurface faults in southwest Iceland using relatively located
microearthquakes, EGU General assembly, Geophysical Research Abstracts 7, European Geoscience Union,
Vienna, 2005, p. 06664.
R. Cattin, P. Briole, H. Lyon-Caen, P. Bernard, P. Pinettes, Effects of superficial layers on coseismic
displacements for a dip-slip fault and geophysical implications, Geophys. J. Int. 137 (1999) 149-158.
E.H. Hearn, R. Burgmann, The effect of elastic layering on inversions of GPS data for coseismic slip and resulting
stress changes: Strike-slip earthquakes Bull. Seismol. Soc. Am. 95 (2005) 1637-1653.
R. Stefansson, R. Bodvarsson, R. Slunga, P. Einarsson, S. Jakobsdottir, H. Bungum, S. Gregersen, J. Havskov, J.
Hjelme, H. Korhonen, Earthquake prediction research in the South Iceland seismic zone and the SIL Project,
Bulletin of the Seismological Society of America 83 (1993) 696-716.
S. Steacy, D. Marsan, S.S. Nalbant, J. McCloskey, Sensitivity of static stress calculations to the earthquake slip
distribution, J. Geophys. Res. 109 (2004) doi:10.1029/2002JB002365.
C.H. Scholz, The mechanics of earthquakes and faulting, Cambridge University Press, Cambridge, U.K. ; New
York, 2002, 471 pp.
C. Pagli, R. Pedersen, F. Sigmundsson, K.L. Feigl, Triggered fault slip on June 17, 2000 on the Reykjanes
Peninsula, SW-Iceland captured by radar interferometry, Geophys. Res. Lett. 30 (2003) 6-1.
T. Árnadóttir, H. Geirsson, P. Einarsson, Coseismic stress changes and crustal deformation on the Reykjanes
Peninsula due to triggered earthquakes on 17 June 2000, J. Geophys. Res. (Solid Earth) 109 (2004) 09307.
A.E. Clifton, C. Pagli, J.F. Jonsdottir, K. Eythorsdottir, K. Vogfjord, Surface effects of triggered fault slip on
Reykjanes Peninsula, SW Iceland, Tectonophysics 369 (2003) 145-154.
Page 20
Dubois et al.
642
[25]
643
644
[26]
645
646
[27]
647
648
649
[28]
650
651
[29]
652
653
[30]
654
655
[31]
656
657
658
[32]
659
660
661
[33]
[34]
662
663
[35]
664
665
666
[36]
667
668
[37]
669
670
671
[38]
672
673
[39]
674
675
[40]
676
677
[41]
678
679
680
[42]
681
682
[43]
683
684
[44]
685
686
[45]
687
688
[46]
689
690
[47]
691
692
[48]
693
694
[49]
695
696
697
698
699
[50]
submitted 2016-02-12
T. Árnadóttir, R. Pedersen, S. Jónsson, G. Gudmundsson, Coulomb stress changes in the South Iceland Seismic
Zone due to two large earthquakes in June 2000, Geophys. Res. Lett. 30 (2003) doi:10.1029/2002GL016495.
M. Bellou, F. Bergerat, J. Angelier, C. Homberg, Geometry and segmentation mechanisms of the surface traces
associated with the 1912 Selsund Earthquake, Southern Iceland, Tectonophysics 404 (2005) 133--149.
I.T. Bjarnason, P. Cowie, M.H. Anders, L. Seeber, C.H. Scholz, The 1912 Iceland earthquake rupture: Growth
and development of a nascent transform system, Bulletin of the Seismological Society of America 83 (1993) 416435.
C.H. Scholz, The mechanics of earthquakes and faulting, Cambridge University Press, Cambridge [England] ;
New York, 1990, 439 pp.
E.E. Brodsky, S.G. Prejean, New constraints on mechanisms of remotely triggered seismicity at Long Valley
Caldera, J. Geophys. Res. (Solid Earth) 110 (2005) 04302.
S. Toda, R.S. Stein, K. Richards-Dinger, S.B. Bozkurt, Forecasting the evolution of seismicity in southern
California: Animations built on earthquake stress transfer, J. Geophys. Res. (Solid Earth) 110 (2005).
G. Bjornsson, O.G. Flovenz, K. Semundsson, E.M. Einarsson, Pressure changes in Icelandic geothermal
reservoirs associated with two large earthquakes in June 2000, Twenty-sixth workshop on geothermal reservoir
engineering SGP-TR-168, Stanford University, 2001.
S. Jónsson, P. Segall, R. Pedersen, G. Bjornsson, Post-earthquake ground movements correlated to pore-pressure
transients, Nature 424 (2003) 179 - 183.
A. Nur, J.R. Booker, Aftershocks caused by pore fluid flow?, Science 175 (1972) 885-887.
J. Noir, E. Jacques, S. Bekri, P.M. Adler, P. Tapponnier, G.C.P. King, Fluid flow triggered migration of events in
the 1989 Dobi earthquake sequence of central Afar Geophys. Res. Lett. 24 (1997) 2335-2338.
T. Árnadóttir, S. Jónsson, F. Pollitz, W. Jiang, K.L. Feigl, E. Sturkell, H. Geirsson, Post-seismic deformation
following the June 2000 earthquake sequence in the south Icelandic seismic zone, J. Geophys. Res 110 (2005)
B12308.
G. Dhatt, G. Touzot, The finite element method displayed, Wiley, Chichester [West Sussex] ; New York, 1984,
509 pp.
N.F. Stevens, G. Wadge, C.A. Williams, J.G. Morley, J.P. Muller, J.B. Murray, M. Upton, Surface movements of
emplaced lava flows measured by synthetic aperture radar interferometry, J. Geophys. Res., B, Solid Earth and
Planets 106 (2001) 11,293-211,313.
H.J. Melosh, A. Raefsky, A simple and efficient method for introducing faults into finite element computations,
Bulletin of the Seismological Society of America 71 (1981) 1391-1400.
C.A. Williams, R.M. Richardson, A rheologically layered three-dimensional model of the San Andreas fault in
central and southern California, J. Geophys. Res. 96 (1991) 16,597-516,516,623.
A. Tryggvason, S.T. Rognvaldsson, O. Flovenz, Three-dimensional imaging of the P- and S- wave velocity
structure and earthquake locations beneath Southwest Iceland, Geophys. J. Int. 151 (2002) 848-866.
R.M. Allen, G. Nolet, W.J. Morgan, K. Vogfjörd, B.H. Bergsson, P. Erlendsson, G.R. Foulger, S. Jakobsdóttir,
B.R. Julian, M. Pritchard, S. Ragnarsson, R. Stefánsson, Imaging the mantle beneath Iceland using integrated
seismological techniques, J. Geophys. Res. (Solid Earth) 107l (2002).
R.M. Allen, J. Tromp, Resolution of regional seismic models: Squeezing the Iceland anomaly, Geophysical
Journal International 161 (2005) 373-386.
F.A. Darbyshire, R.S. White, K.F. Priestley, Structure of the crust and uppermost mantle of Iceland from
combined seismic and gravity study, Earth Planet. Sci. Lett. 181 (2000) 409-428.
M.K. Kaban, O.G. Flovenz, G. Palmason, Nature of the crust-mantle transition zone and the thermal state of the
upper mantle beneath Iceland from gravity modelling, Geophys. J. Int. 149 (2002) 281--299.
R. Bövarsson, S.T. Rögnvaldsson, R. Slunga, E. Kjartansson, The SIL data acquisition system-at present and
beyond year 2000, Physics of the Earth and Planetary Interiors 113 (1999) 89-101.
R. Slunga, S.T. Rognvaldsson, R. Bodvarsson, Absolute and relative locations of similar events with application
to microearthquakes in southern Iceland, Geophysical Journal International 123 (1995) 409-419.
S.T. Rognvaldsson, R. Slunga, Single and joint fault plane solutions for microearthquakes in South Iceland,
Tectonophysics 237 (1994) 73-86.
R. Slunga, Microearthquake Analysis at Local Seismic Networks in Iceland and Sweden and Earthquake
Precursors, Lecture Notes in Earth Sciences (2003) 149-170.
K. Vogfjord, Triggered Seismicity after the June 17, Mw=6.5 Earthquake in the South Iceland Seismic Zone: The
first five minutes, EGS - AGU - EUG Joint Assembly, Abstracts from the meeting held in Nice, France, 6 - 11
April 2003, abstract #11251 (2003) 11251.
K. Vogfjord, R. Slunga, Rupture in the South Iceland Seismic Zone forced by magmatic intrusion in the Hengill
area, EGS - AGU - EUG Joint Assembly, Abstracts from the meeting held in Nice, France, 6 - 11 April 2003,
abstract #9685 (2003) 9685.
Page 21
Dubois et al.
700
[51]
701
702
[52]
703
704
[53]
705
706
707
[54]
708
709
[55]
710
711
[56]
712
713
[57]
714
715
[58]
716
717
[59]
718
719
[60]
720
721
[61]
722
723
[62]
724
725
[63]
726
727
728
[64]
[65]
729
730
[66]
731
732
[67]
733
734
735
[68]
736
737
[69]
738
739
[70]
740
741
[71]
742
743
[72]
744
745
746
747
748
[73]
749
750
751
[74]
752
753
[75]
754
755
756
757
[76]
submitted 2016-02-12
R. Stefansson, PREPARED final report, Icelandic Meterological Office, Reykjavik
http://www.vedur.is/utgafa/greinargerdir/2006/06009.pdf, 2006.
L. Dubois, Mechanical study of the June 2000 earthquake sequence with 3D finite element models: rheologic and
geometric effects (in French), Ph.D. thesis, Université Paul Sabatier, Toulouse III, France, 2006.
R.M. Allen, G. Nolet, W.J. Morgan, K. Vogfjörd, M. Nettles, G. Ekström, B.H. Bergsson, P. Erlendsson, G.R.
Foulger, S. Jakobsdóttir, B.R. Julian, M. Pritchard, S. Ragnarsson, R. Stefánsson, Plume-driven plumbing and
crustal formation in Iceland, J. Geophys. Res. (Solid Earth) 107h (2002).
I.T. Bjarnason, W. Menke, O. Flóvenz, D. Caress, Tomographic image of the mid-Atlantic plate boundary in
southwestern Iceland, J. Geophys. Res. 98 (1993) 6607-6622.
M. Cocco, J.R. Rice, Pore pressure and poroelasticity effects in Coulomb stress analysis of earthquake
interactions, J. Geophys. Res. 107 (2002) ESE 2-1 to 2-17.
G. Dal Moro, M. Zadro, Remarkable tilt-strain anomalies preceding two seismic events in Friuli (NE Italy): their
interpretation as precursors, Earth Planet. Sci. Lett. 170 (1999) 119-129.
S.E. Barrientos, S.N. Ward, The 1960 Chile earthquake: inversion for slip distribution from surface deformation,
Geophys. J. Int. 103 (1990) 589-598.
M. Cocco, J.R. Rice, Corrections to "Pore pressure and poroelasticity effects in Coulomb stress analysis of
earthquake interactions" http://esag.harvard.edu/rice/RicePubs.html 2003.
J.R. Rice, M.P. Cleary, Some basic stress-diffusion solutions for fluid-saturated elastic porous media with
compressible constituents, Reviews of Geophysics and Space Physics 14 (1976) 227-241.
G. Peltzer, P. Rosen, F. Rogez, K. Hudnut, Poroelastic rebound along the Landers 1992 earthquake surface
rupture, J. Geophys. Res. 103 (1998) 30131-30146.
Y. Fialko, Evidence of fluid-filled upper crust from observations of postseismic deformation due to the 1992 Mw
7.3 Landers earthquake, J. Geophys. Res. (Solid Earth) 109 (2004) 08401.
H.F. Wang, Theory of poroelasticity with applications to geomechanics and hydrology, Princeton University
Press, 2000.
P. Segall, Induced stresses due to fluid extraction from axisymmetric reservoirs, Pure and Applied Geophysics
139 (1992) 535-560.
C.L. Lawson, R.J. Hanson, Solving least squares problems, Prentice-Hall, 1974.
R. Wang, F. Martin, F. Roth, Computation of deformation induced by earthquakes in a multi-layered crust FORTRAN programs EDGRN/EDCMP, Comp. Geosci. 29 (2003) 195-207.
E.F. Kaaschieter, A.J.H. Frijns, Sqeezing a sponge: a three-dimensional solution in poroelasticity, Computational
Geosciences 7 (2003) 49-59.
T. Masterlark, Finite element model predictions of static deformation from dislocation sources in a subduction
zone: Sensitivities to homogeneous, isotropic, Poisson-solid, and half-space assumptions, J. Geophys. Res. 108
(2003) 2540 2510.1029/2002JB002296.
K.W. Hudnut, L. Seeber, J. Pacheco, Cross-Fault Triggering in the November 1987 Superstition Hills Earthquake
Sequence, Southern-California, Geophys. Res. Let. 16 (1989) 199-202.
T. Masterlark, H.F. Wang, Poroelastic coupling between the 1992 Landers and Big Bear Earthquakes, Geophys.
Res. Lett. 27 (2000) 3647-3650.
T. Masterlark, H.F. Wang, Transient stress-coupling between the 1992 Landers and 1999 Hector Mine, California,
earthquakes, Bull. Seism. Soc. Amer. 92 (2002) 1470-1486.
E.E. Brodsky, E. Roeloffs, D. Woodcock, I. Gall, M. Manga, A mechanism for sustained groundwater pressure
changes induced by distant earthquakes, J. Geophys. Res. (Solid Earth) 108h (2003).
D.P. Hill, P.A. Reasenberg, A. Michael, W.J. Arabaz, G. Beroza, D. Brumbaugh, J.N. Brune, R. Castro, S. Davis,
D. dePolo, W.L. Ellsworth, J. Gomberg, S. Harmsen, L. House, S.M. Jackson, M.J.S. Johnston, L. Jones, R.
Keller, S. Malone, L. Munguia, S. Nava, J.C. Pechmann, A. Sanford, R.W. Simpson, R.B. Smith, M. Stark, M.
Stickney, A. Vidal, S. Walter, V. Wong, J. Zollweg, Seismicity remotely triggered by the magnitude 7.3 Landers,
California earthquake, Science 260 (1993) 1617-1623.
S.G. Prejean, D.P. Hill, E.E. Brodsky, S.E. Hough, M.J.S. Johnston, S.D. Malone, D.H. Oppenheimer, A.M. Pitt,
K.B. Richards-Dinger, Remotely triggered seismicity on the United States west coast following the M (sub w) 7.9
Denali Fault earthquake, Bulletin of the Seismological Society of America 94 (2004) 348-359.
Y. Okada, Surface deformation to shear and tensile faults in a half-space, Bull. Seism. Soc. Am. 75 (1985) 11351154.
K.L. Feigl, E. Dupré, RNGCHN: a program to calculate displacement components from dislocations in an elastic
half-space with applications for modeling geodetic measurements of crustal deformation, Computers and
Geosciences 25 (1999) 695-704.
R. Wang, Mart, F.L. 'in, F. Roth, PSGRN/PSCMP - a new code for calculating co-postseismic deformations and
geopotential changes based on the viscoelastic-gravitational dislocation theory, Elsevier Science (2005).
Page 22
Dubois et al.
758
[77]
759
760
[78]
761
762
[79]
763
764
765
[80]
submitted 2016-02-12
B. Efron, R. Tibshirani, Bootstrap methods for standard errors, confidence intervals, and other measures of
statistical accuracy, Statistical Science 1 (1986) 54-77.
G. Peltzer, P. Rosen, F. Rogez, K. Hudnut, Postseismic rebound in fault step-overs caused by pore fluid flow,
Science 273 (1996) 1202-1204.
A.D. Miller, B.R. Julian, G.R. Foulger, Three-dimensional seismic structure and moment tensors of non-double
couple earthquakes at Hengill-Grensdalur volcanic complex, Iceland, Geophys. J. Int. 133 (1998) 309-325.
P. Einarsson, and K. Sæmundsson, Earthquake epicenters 1982–1985 and volcanic systems in Iceland, in: T.I.
Sigfusson, (Ed), Festschrift for Th. Sigurgeirsson, Menningarsjodur, Reykjavık, 1987.
766
767
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Dubois et al.
768
submitted 2016-02-12
Captions
769
Fig 1. Map showing (above) the main tectonic features of the study area from Árnadóttir et al. [23]. The
770
Reykjanes Peninsula (RP), the Hengill triple junction (He), the Western Volcanic Zone (WVZ), the South
771
Iceland Seismic Zone (SISZ), and the Eastern Volcanic Zone (EVZ) are indicated. The light shaded areas are
772
individual fissure swarms with associated central volcanoes. Mapped surface faults of Holocene age are shown
773
with black lines [80]. The epicenter locations of earthquakes on June 17, 2000 are shown with black stars. The 17
774
June main shock is labeled J17. The Reykjanes Peninsula events are 26 s afterwards on the Hvalhnúkur fault (H),
775
30 s afterwards near Lake Kleifarvatn (K), and 5 minutes afterwards at Núpshlídarháls (N) as located by geodesy
776
[23] and timed by seismology [6]. Location of the 21 June 2000 main shock is shown with a white star labeled
777
J21. The location of Reykjavik, the capital of Iceland, is noted. Inset shows a simplified map of the plate
778
boundary across Iceland, with the location of the main part of the figure indicated by the black rectangle. The
779
black arrows show the relative motion between the North America and Eurasian plates from NUVEL-1A [2, 3].
780
Block diagram (below) showing thickening of the crust from east to west across the SISZ as inferred from the
781
literature [43, 44]. The dimensions of the mesh are 300 x 300 x 100 km (whole box). Color code shows Iceland
782
(gray), the SISZ (blue rectangle), water (blue), upper crust (green), lower crust (orange), the upper mantle (red),
783
fault traces (red) for the June 17 (east) and June 21 (west) faults, and the reference point (gray dot) at the center
784
of the modeled June 21 fault trace (63.99oN, 20.71oW) for the local easting and northing coordinate system.
785
Fig 2. Geometric configuration of rheologic parameters. A) Cross section showing, as a function of depth, the
786
following rheologic parameters: density  (thick dashed line), rigidity  (thick solid line), and Poisson’s ratio 
787
(thin solid line). The thicknesses of the rheologic layers depend on the depths Hu and Hc to the bases of the upper
788
and lower crust, respectively. The configurations are composed of: a uniform half space (HOM and TOM),
789
horizontal layers with constant thickness (HET), dipping layers with variable thickness (ISL), a half-space with a
790
damage zone (HOF), and horizontal layers with a damage zone (HEF). (B) Thickness of the upper crust Hu as
791
parameterized in the TOM and ISL configurations. Contour interval is 1 km. (C) Depth to the interface between
792
lower crust and upper mantle Hc in the TOM and ISL configuration. Contour interval is 5 km. Solid lines
793
represent traces of modeled faults for June 17 (east) and June 21 (west). Dot indicates the reference point of the
794
local cartographic coordinate system, centered on the June 21 epicenter.
795
Fig 3. View of fault planes showing the slip distribution estimated for the June 17 main shock (left column) and
796
June 21 main shock (right column) in the horizontally-layered HET configuration (upper row). White stars
797
indicate the hypocenters of the mainshocks. Black dots indicate aftershocks within 1 km of each fault and 24
798
hours of the June 17 and June 21 mainshocks, respectively, as relocated by the Icelandic Meteorological Office,
Page 24
Dubois et al.
submitted 2016-02-12
799
as described in the section on data and methods. The lower panels show the 1- uncertainty of the slip estimates
800
for the June 17 (left) and June 21 (right) events.
801
Fig 4. View of fault planes of the June 17 main shock (left column) and June 20 main shock (right column)
802
showing the differences in estimated slip distribution for the HET configuration with respect to HOM (top row),
803
HOF with respect to HOM (middle row), and HEF with respect to HET (bottom row). The black lines denote the
804
significance ratio, defined as the slip estimate divided by its 1- uncertainty. The contour where this ratio equals
805
2 thus includes the area where the differences are statistically significant at the level of 95% confidence. For each
806
panel, the white circle indicates the centroid of the first slip distribution in the pair, while the black circle
807
indicates the centroid of the second, e.g., white circle is the HET centroid in the first row.
808
Fig 5. Poroelastic models for a postseismic interferogram, showing range change (the component of
809
displacement along the line of sight from the ground to satellite) in map view as observed (a) and calculated from
810
the FOR1 configuration (d). Solid black lines outline the co-seismic portion of the fault plane that ruptured
811
during the June 17 mainshock. Dashed lines indicate locations of profiles. Panel (b) shows negative range change
812
as a function of distance along east-west profile A-A’ as observed (points), and calculated from various
813
configurations: FOR3 (black line), TRY1 (green line), INV2 (blue line), and INV4 (red lines). Panel (c) shows
814
negative range change as a function of distance along north-south profile B-B’ with the same plotting
815
conventions. All range changes pertain to the first month of the post-seismic interval, from June 19 to July 24.
816
Negative range change corresponds to uplift; positive range change corresponds to subsidence.
817
Fig 6. Change in Coulomb failure stress C as a result of four candidate processes for stress transfer. The
818
contours outline the areas where the change in Coulomb failure stress C resolved on north-striking vertical
819
faults at 4 km depth exceeds the conventional threshold value of C(critical) =10 kPa. The sources of stress
820
include: (A) the co-seismic stresses generated by the June 17 mainshock calculated using configurations HOM
821
(black line) and HET (blue line); (B) poroelastic effects calculated using the elastic approximation and two
822
different configurations for the change in Poisson’s ratio : INV2 (green line) and INV4 (blue line); (C) the
823
domino effect of cascading aftershocks, (D) viscoelastic processes accumulated in mid 2004 calculated from the
824
HET (blue line) and ISL (green line) configurations (E) geometric contribution to 100 years of interseismic strain
825
accumulation obtained by subtracting the interseismic stress field calculated using the HOM geometric
826
configuration from that obtained with the ISL configuration; (F) combination of the co-seismic and the geometric
827
contribution to interseismic stress integrated over 100 years, calculated as the sum of the HET field (A) and panel
828
(E). Red lines indicate faults for the June 17 (east) and June 21 (west) mainshocks. Dots in panels (A, B, C, E, F)
829
indicate the epicenters of earthquakes between June 17 and 21, 2000 [16, 49].
Page 25
Dubois et al.
submitted 2016-02-12
830
TABLES
831
Table 1. Geometric and rheologic parameters for configurations used to estimate distributions of co-seismic slip.
832
Table 2. Parameter values for poroelastic modeling of post-seismic deformation and the variance reduction. In
833
the FOR and TRY series, the parameter values are input to forward calculations. In the INV series, the
834
parameters are estimated from the data shown in Fig 5.
835
836
LAST PAGE
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