Supplement Text
Rupture Model of the 2011 Mineral, Virginia, Earthquake from Teleseismic and Regional
Waveforms
Stephen Hartzell1, Carlos Mendoza2, and Yuehua Zeng1
1
U.S. Geological Survey, Denver Federal Center, MS 966 Box 25046, Denver, Colorado 80225
2
Centro de Geociencias, Universidad Nacional Autonoma de Mexico, Campus Juriquilla,
Queretaro, Mexico
1) Comparison of Synthetic and Observed Waveforms
The comparison of observed data and synthetics for both the teleseismic and regional slip models
is shown in Figure S1 and S2, respectively.
2) Method
We utilize the linear least-squares method first applied by Hartzell and Heaton (1983). In this
approach, the fault plane is divided into rectangular subfaults with the subfault synthetics matrix,
A, times the solution vector of slip amplitudes, x, equated to the vector of observed ground
motion, b. Assuming a constant rupture velocity and stabilizing the inversion by appending
smoothing, S, and moment minimization, M, constraints results in the linear system to be solved:
FC AI FC bI
G
 S Jx  G
0 J
G
J
G
J
G
H M JK G
H0 JK
1
d
1
d
1
2
,
1
where λ1 and λ2 are linear weights, whose magnitudes control the trade-off between satisfying
the constraints and fitting the data. Cd is an a priori data covariance matrix that is used as a data
scaling matrix. The data covariance matrix is diagonal and normalizes each data record to have
peak amplitude of 1.0. Thus, each record has nearly equal weight in the inversion. The solution
vector x is obtained using the Householder reduction method that invokes a positivity constraint
on the solution (algorithm NNLS from Lawson and Hanson, 1974) so that the elements of the
vector x are greater than or equal to zero. The constraint weights, λ1 and λ2, are determined by the
empirical method of Mendoza and Hartzell (2013), who showed that an empirical relationship
exists between the values of the elements of the Cd-1A matrix and an optimum constraint weight
λ = λ1 = λ2 in Tikhonov regularization (Hansen, 1998).
3) Further Examination of Rupture Models
Additional information on the teleseismic and regional rupture models is obtained from an
examination of the time histories of the development of slip and the moment rate. Figure S3
compares the evolution of slip for the teleseismic and regional models in 0.5s snapshots in time.
Slip in the teleseismic model moves initially up dip and then to the northeast and further up dip
of the hypocenter. The final stage of the rupture includes deeper faulting at depths similar to
those seen in the regional model. The regional model initiates and propagates to the northeast
like the teleseismic model but maintaining approximately the same depth. In the final stages of
faulting slip then moves deeper.
The differences in the teleseismic and regional models may be due to the fact that we have only
one empirical Green’s function aftershock of suitable size and location. With one aftershock
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record we are limited in the depth range over which faulting can be accurately represent because
the timing of depth phases is fixed. However, this is not an indictment of the empirical Green’s
function method, which can accurately represent faulting over any depth range for which there is
a suitable distribution of aftershocks. In our case, the depth of our aftershock at 7km is
advantageously located to represent slip near the hypocentral depth of 8km. The difference
between the teleseismic and regional models is not great when one considers the small
dimensions of the fault plane and the fact that the second large source differs in depth by only
2km between the two models.
Comparison of the moment rate functions in Figure S4 for the teleseismic and regional models
sheds light on the different frequency sensitivity of the two data sets. Both models yield moment
estimates lower than the USGS long period estimate of 5.6x1024 dyne-cm (Mw 5.8) with the
regional estimate of 3.4x1024 dyne-cm being lower than the teleseismic estimate of 4.7x1024
dyne-cm. The regional moment rate function starts with a sharp initiation of slip over a time
interval of about 0.5s. In contrast the teleseismic model has more of a ramp like initiation. We
ascribe this difference to the enhanced high-frequency response of the regional data set. The
teleseismic waveforms cannot fully resolve the high-frequency initiation. The other major
difference between the two moment rate functions is in the moment release between 1 and 2s.
The teleseismic solution shows significantly greater levels of moment release in this interval.
Analogous to our above argument concerning the resolution in the initiation of rupture, we
theorize that the missing moment in the regional model is due to the diminished low frequency
content of our regional data set. These arguments are supported by the spectra in Figure S5. Here
we compare normalized spectral amplitudes for a pair of vertical components from typical
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regional (ACSO) and teleseismic (CRAG) stations at a similar azimuth. Each spectrum is based
on the same time interval of record used in the inversion. The vertical dashed lines indicate the
frequency band from 0.2 Hz to 2.0 Hz used in the regional inversion. The teleseismic waveforms
are not filtered, but are limited to a record duration of 10s. These spectra clearly show the greater
high-frequency content of the regional record and the corresponding greater low-frequency
content of the teleseismic record.
The use of a moment minimization constraint is a further consideration in the estimation of
moment. We add moment minimization and smoothing constraints to stabilize the inversion.
However, since we employ an additional step following the inversion of adjusting the average
amplitude of the synthetic waveforms to the average amplitude of the data records, the moment
minimization constraint has minimal effect on the estimated moment. As a test, we performed
several inversions with different moment minimization weights. In reducing the moment
minimization weight from our optimally estimated value to one hundredth of this value the
moment only increases by 1.5%. This reduced level has very little effect on the inversion.
Supplement Figure Captions
Figure S1: Comparison of data (solid curve) and synthetics (dashed curve) for the teleseismic
slip model in Figure 3a. All waveforms are unfiltered broadband instrumental velocity records.
Number following each trace is the ratio of the peak synthetic amplitude to the peak observed
amplitude.
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Figure S2: Comparison of data (solid curve) and synthetics (dashed curve) for the regional slip
model in Figure 3a. Waveforms are bandpass filtered from 0.2 to 2.0 Hz instrumental velocity
records, except for station CW026, which has a lowpass corner of 5.0 Hz. Records are in the
order east, north, vertical for each station grouping, except for station CW026 (69º, 339º,
vertical). Number following each trace is the ratio of the peak synthetic amplitude to the peak
observed amplitude. The low amplitude of the synthetic at station CW026 is attributed to the
simple velocity structure that does not account for turning of the rays or amplification caused by
lower near-surface velocities.
Figure S3: Rupture time histories for the teleseismic and regional models at 0.5s time intervals.
In each frame the minimum contour level is 5 cm with a contour interval of 10 cm. The peak slip
for each frame is given in the lower right hand corner.
Figure S4: Moment rate functions for the teleseismic (dashed line) and regional (solid line)
models.
Figure S5: Comparison of vertical component amplitude spectra for a pair of teleseismic
(CRAG) and regional (ACSO) records used in the inversion. Both stations lie at a similar
azimuth from the source. The vertical dashed lines indicate the frequency band used in the
regional inversion.
Supplement References
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Hansen, P. C. (1998). Rank-Deficient and Discrete Ill-Posed Problems, Numerical Aspects of
Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation,
Philadelphia, Pennsylvania.
Hartzell, S. H., and T. H. Heaton (1983). Inversion of strong ground motion and teleseismic
waveform data for the fault rupture history of the 1979 Imperial Valley, California, earthquake,
Bull. Seism. Soc. Am. 73, 1553-1583.
Lawson, C. L., and R. J. Hanson (1974). Solving Least Squares Problems, Prentice-Hall, Inc.,
Upper Saddle River, New Jersey.
Mendoza, C., and S. Hartzell (2013). Finite-fault source inversion using teleseismic P waves:
simple parameterization and rapid analysis, Bull. Seism. Soc. Am. 103, 834-844.
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