Auxiliary Material
Lithospheric Viscoelastic Relaxation on the Contemporary Deformation Following the
1959 Mw 7.3 Hebgen Lake, Montana, Earthquake and Other Areas of the Intermountain
Seismic Belt
Wu-Lung Chang1, Robert B. Smith2, and Christine M. Puskas3
1
Department of Earth Sciences, National Central University, Chungli, Taiwan.
2
Department of Geology and Geophysics, University of Utah, Salt Lake City, Utah, USA
3
UNAVCO, Boulder, Colorado, USA
Introduction:
The auxiliary material contains three sections designed to clarify discussions in the main text
and to illustrate data fidelity.
Section 1: Baseline Measurements from EDM (trilateration) and GPS, including text and Table
S1.
Section 2: Estimation of the Background Extensional Rate of the Hebgen Lake Area, including
text, Fig. S1, and Fig. S2.
Section 3: Postseismic Relaxation of Classic Maxwell Viscoelastic Body, including text, Fig. S3,
and Fig. S4.
1
1. Baseline measurements from EDM (trilateration) and GPS
Trilateration (1973-1987) and later GPS techniques (after 1987) have been implemented to
determine the three-dimensional deformation field of the Hebgen Lake area. For trilateration
surveys, the one standard error in distance measurements is (a2+b2LEDM2)1/2, where LEDM is the
baseline length, a=3 mm, and b=0.210-6 (dimensionless) [Savage and Prescott, 1973].
Following this relation, the standard error is about 5.7 mm for a baseline length of 24-km, an
average value for the trilateration baselines used in this study.
For campaign GPS surveys, in contrast, station coordinated can be determined with an
accuracy of about 2-4 mm for the horizontal component and up to 10 mm for the vertical
component [e.g., Puskas et al., 2007]. Compared with the EDM methodology, GPS can measure
baseline length more accurately with an accuracy of ~5 mm for 100-km baseline [Dixon, 1991].
2
Table S1: Baseline measurements from EDM (trilateration) and GPS used in this study
L-L0 (m)
Error** (mm)
24147.6046=L0
24147.6100
24147.6273
24147.6337
24147.6618
0.0054
0.0227
0.0291
0.0572
5.7
5.7
5.7
5.7
5.7
1984
1987
1987
1989
1991
1993
1995
2000
24147.6764
24147.6860
24147.6927*
24147.6922*
24147.7016*
24147.7111*
24147.7140*
24147.7229*
0.0718
0.0814
0.0881
0.0876
0.0970
0.1065
0.1094
0.1183
5.7
5.7
2.5
5.3
4.0
0.7
2.2
2.3
AIRP-BIGN
1973
22921.8130=L0
(L=LEDM)
1974
1976
1978
1981
1984
1987
1987
1991
22921.8154
22921.8271
22921.8332
22921.8565
22921.8705
22921.8880
22921.8895*
22921.8941*
Baseline
Year
Measured Baseline Length (m)
EDM1 (LEDM)
GPS2 (LGPS)
AIRP-HOLM
(L=LEDM)
1973
1974
1976
1978
1981
AIRP-R161
(L=LGPS)
24147.6859
24147.6854
24147.6948
24147.7043
24147.7072
24147.7161
5.5
22921.8830
22921.8876
1987
1989
29449.4967=L0
29449.4911
0.0024
0.0141
0.0202
0.0435
0.0575
0.0750
0.0765
0.0811
5.5
5.5
5.5
5.5
5.5
5.5
3.4
1.5
-0.0056
1.8
6.6
1991
29449.5213
0.0246
1993
29449.5257
0.0290
1995
29449.5292
0.0325
2000
29449.5610
0.0643
1
Savage et al. [1993]; 2 Puskas et al. [2007];
*
LEDM is estimated from LGPS by the correction formula: LEDM=LGPS/(1-0.28310-6).
**
1- standard error for EDM and 1- weighted root mean square error for GPS.
3
2.0
1.9
2.4
3.3
Table S1: Continued
Baseline
Year
Measured Baseline Length (m)
EDM (LEDM)
GPS (LGPS)
BIGN-LION
(L=LEDM)
1973
1974
1976
1978
1981
1984
1987
LION-ROOF
(L=LEDM)
L-L0 (m)
Error (mm)
22943.2238=L0
22943.2309
22943.2408
22943.2563
22943.2730
22943.3007
0.0071
0.0170
0.0325
0.0492
0.0769
5.5
5.5
5.5
5.5
5.5
5.5
22943.3106
0.0868
5.5
0.1004
0.1105
3.7
1.7
*
1987
1991
22943.3242
22943.3343*
22943.3177
22943.3278
1973
1974
1976
1978
1981
1984
25869.5993=L0
25869.6004
25869.6086
25869.6156
25869.6458
25869.6602
0.0011
0.0093
0.0163
0.0465
0.0609
6.0
6.0
6.0
6.0
6.0
6.0
1987
25869.6614
0.0621
6.0
-0.0049
0.0460
0.0405
0.0524
0.0786
2.5
9.2
2.8
2.5
3.4
3.6
R161-D092
(L=LGPS)
1987
1989
1991
1993
1995
2000
40857.1393=L0
40857.1344
40857.1853
40857.1798
40857.1917
40857.2179
R161-O033
(L=LGPS)
1987
1989
34415.9856=L0
34415.9870
0.0014
2.7
12.7
1991
2000
34416.0137
34416.0599
0.0281
0.0743
2.5
4.2
1987
1989
1991
2000
34475.8952=L0
34475.8894
34475.9408
34475.9672
-0.0058
0.0456
0.0720
2.6
11.7
2.4
3.7
R161-V297
(L=LGPS)
4
2. Estimation of the background extensional rate of the Hebgen Lake area
To evaluate the steady-state tectonic effect across the Hebgen Lake fault zone, we first
estimated the extensional rate of a baseline between two GPS stations far from the Hebgen Lake
postseismic deformation zone, namely P461 in the relative stable Rocky Mountains and P685 in
the eastern Snake River Plain (Fig. S1). Fig. S2 reveals that the 2005-2011 average rate of
extension between these two stations is 2.7±0.8 mm/yr to the southwest (Fig. S2), similar to the
rate of 2.1±0.2 mm/yr across the eastern Snake River Plain proposed by Puskas et al. [2007].
Fig. S2 also reveals that the extension is not significant northeast of the GPS station P456,
indicating that the Hebgen Lake fault may be the boundary between the stable Rocky Mountains
and the extended eastern Snake River Plain. Therefore, by ignoring the deformation between
P461 and P456, both in the Rocky Mountains, the baseline length of ~100 km between P456 and
P685 would bring an estimate of extensional strain rate of ~0.027 strain/yr across the eastern
Snake River Plain.
5
3. Postseismic Relaxation of Classic Maxwell Viscoelastic Body
A frequently employed rheologic model for ductile rocks in the lower crust and upper mantle
is the classic Maxwell viscoelastic body [e.g., Körnig and Müller, 1989]. However, it is worth
noting that many geodetic data suggest a rapid exponential decay in postseismic deformation
immediately after the rupture, followed by a more slowly decaying (or constant) velocity at a
later period of observations [e.g., Freed and Bürgmann, 2004]. To model the difference in
short- and long-time decay rates, the Maxwell element is sometimes modified to have a nonlinear rheology, which results in a lower effective viscosity immediately after the rupture
evolving to a higher effective viscosity as the co-seismic stresses relax. The short-term transient
response and long-term steady creep can be exhibited by a Burger's body, a Maxwell in series
with a Kelvin element, that is suggested for future modeling of viscoelastic ground deformation
following large earthquakes [e.g., Pollitz, 2003a; Freed et al., 2012]. Because the trilateration
and GPS data in this study were measured in a period of 14-41 years after the Hebgen Lake
earthquake, it is reasonable to assume that the short-term effects from non-linear rheology have
minor influence to the observed postseismic deformation.
Two mechanic characters of the Maxwell rheology are justified here. First, although the
Maxwell viscosity is time-invariant, geodetic data for 2.5 years following the 1999 Hector Mine,
CA, earthquake revealed time-dependent properties of the upper mantle beneath the Mojave
Desert. Pollitz [2003] indicated a transient viscosity of 1.6 1017 Pa-s during the first 0.2 years
followed by a higher viscosity of 4.6 1018 Pa-s. This result implies a Burghers body rheology,
consisting of a Kelvin element in series with a Maxwell element, to represent mechanism of
transient and steady state relaxation, respectively. Although the Maxwell rheology with timeinvariant viscosity may be inappropriate to describe the early stage, i.e. a few years, of
6
postseismic deformation, it can be a plausible model for this study because the geodetic
observations were initiated 14 years after the Hebgen Lake earthquake and spanned a relatively
short time window of 27 years.
Second, the viscous flow of the Maxwell viscoelastic rheology is assumed to follow the
Newtonian fluid model, in which the stress is linearly proportional to the strain rate. Laboratory
experiments, however, suggest that the deformation of hot lithospheric rocks can be
characterized by power-law creep in which strain rate is proportional to stress to a power, n [e.g.,
Kirby, 1987; Karato and Wu, 1993]. Freed and Bürgmann [2004] showed that a power-law
rheology of the viscous mantle flow with n=3.5 better explains the spatial and temporal
evolution of surface deformation following the 1992 Lander and 1999 Hector Mine earthquakes.
While the power-law rheology suggests that the viscosity is a function of differential stress and
thus a function of time, the simpler Newtonian model implemented in this study may only
provide rheologic properties of the Hebgen Lake fault zone within a time window of decades
after the earthquake, when the deformation rate decreases slower then early postseismic periods.
For evaluating the postseismic viscoelastic relaxation, we used the forward modeling code
VISCO1D by Pollitz [1997] that describes the time-dependent response of a spherically stratified
elastic-viscoelastic medium to the stresses generated by an earthquake occurring in the elastic
upper-crust. The code employs a Maxwell viscoelastic fluid assessed at long time following a
stress perturbation, e.g. a time greater than the Maxwell relaxation time when the effective shear
strength approaches zero. Also, the code includes the effect of gravitational viscoelastic relaxation
that will be discussed in the following examples.
3.1 Example of one-layer model
7
Fig. S3 shows the calculated postseismic velocities at various times following a down-dip
displacement of 2 m on a 50º dipping plane, considered a good working-model for large normalfaulting earthquakes in the Basin-Range. The fault extends from the surface to 15 km, near the
nucleation depth of large earthquakes of the western U.S. interior [Doser and Smith, 1989]. The
rheologic stratification is prescribed by a 16-km-deep elastic layer overlying a Maxwell
viscoelastic half-space, with elastic constants , the bulk modulus, and , the shear modulus,
similar to those used for modeling the 1992 Landers earthquake in the central Mojave Desert,
CA [Pollitz et al., 2000].
A plausible range of viscosities, 1018, 1019, and 1020 Pa-s, were tested for the Maxwell model
that have corresponding relaxation times () of 0.5, 5 and 50 years, respectively, for a shear
modulus  of 70 GPa. Results show that the lowest viscosity of 1018 Pa-s gives the largest
surface horizontal velocity about 10 years following the earthquake. After about 10 years the
effect from the layer of 1019 Pa-s begins to dominate, while contributions from the 1020 Pa-s
layer play an important role after about 100 years. Note that after 500 years, the horizontal
surface velocities drop below 0.3 mm/yr. Such low velocities are close to the horizontal
resolution of continuous GPS velocity with uncertainties of 0.1-0.2 mm/yr from the Basin and
Range [e.g., Davis et al., 2003].
3.2 Example of two-layer model
A one-dimensional rheologic model with a viscoelastic layer sandwiched between an elastic
layer and viscoelastic half-space was used to describe lithospheric structure wherein the lower
crust lies between the brittle upper crust and ductile upper mantle [e.g., Pollitz et al., 2000]. The
8
two-layer model correlates with depths ascribed to the low seismic velocity zone above the
crust-mantle boundary, i.e., above the seismically determined Moho discontinuity.
Fig. S4 shows the viscoelastic modeling results of three two-layer models. Compared with
the responses of the one-layer model with =1019 (Fig. S3), the appearance of a layer with
viscosity 1 changes the results significantly. For the example of 1/2=1018/1019, the responses
are similar to those from the one-layer model with =1018 because the less viscous layer, 1018
Pa-s, relaxes more rapidly than the half-space, 1019 Pa-s, does, thus contributions from the halfspace are minor.
For 1/2=1020/1019, the more viscous layer damps the response from the half-space. This
model has lower postseismic velocities than the half-space model of =1019. Increasing the
layer viscosity by an order of magnitude, i.e. 1/2=1021/1019, moreover, has minor changes, less
than 10%, on surface response compared to 1/2=1020/1019 (Fig. S3). This result suggests that
the layer viscosity may be difficult to resolve if it is noticeably higher, e.g. an order of
magnitude, than the half-space viscosity. Rheologic modeling of the postseismic deformation of
the 1959 Hebgen Lake earthquake, collected between 14 and 41 years after the event, reveals
this difficulty that will be discussed in the following section.
Figs. S3 and S4 also reveal that the effect of gravitational loading tends to speed up the longwavelength component of the postseismic deformation [Pollitz, 1997]. Results of one-layer
models in Fig. S3 also indicate that the effect does not become notable on horizontal motions
until several Maxwell relaxation times, e.g. > 50 [Pollitz, 1997], after the earthquake. This
study thus did not include the gravitational effect because the Hebgen Lake geodetic data were
only observed less than 41 years, ~8 for =1019 Pa-s, after the earthquake.
9
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10
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11
Figure Captions
Fig. S1. Continuous GPS stations of the Yellowstone-Hebgen Lake area. Labeled triangles are
stations whose time series are shown in Fig. S2. Thick black lines mark the surface ruptures of
the 1959 Hebgen Lake earthquake. Dashed, dot-dashed, and solid orange lines outline the
Yellowstone calderas formed 2.1, 1.3, and 0.65 Ma, respectively [Christiansen, 2001]. Red
arrows with 2- error ellipses show 2005-2011 horizontal velocities.
Fig. S2. Time series of baseline changes across the Hebgen Lake and the eastern Snake River
Plain areas. See station locations in Fig. S1.
Fig. S3. Time-dependent postseismic horizontal motions produced by an M=7.0 scenario
normal-faulting earthquake, assuming an elastic layer overlying a viscoelastic half-space. The
velocities represent surface motions along a profile perpendicular to the fault, with the origin
corresponding to the surface fault trace. Different colors represent velocities at different time
periods, where dashed curves represent motions including the gravitational effect.
Fig. S4. Time-dependent postseismic horizontal motions produced by an M=7.0 scenario
normal-faulting earthquake, assuming an elastic layer overlying a viscoelastic layer and halfspace. See Fig. S3 for more explanations.
12