Data processing and time series analysis to obtain interseismic GPS velocities
The dual-frequency carrier phase and pseudorange data have been processed using a precise
point positioning strategy [Zumberge et al., 1997] with the GIPSY-OASIS II software (version
6.1.2) developed by NASA’s Jet Propulsion Laboratory (JPL). We use final, nonfiducial, precise
orbit and clock files generated by JPL from a global subset of core International GNSS Service
(IGS) stations as well as wide lane phase bias files that permit single-station ambiguity resolution
[Bertiger et al., 2010]. Station clocks are treated as a white noise process. The troposphere is
modeled using the Global Mapping Function, and the wet zenith delay and two gradient
parameters are estimated stochastically as random walk processes. Receiver antenna phase
center variations are accounted for using absolute phase center maps provided by IGS.
Ocean
loading parameters are obtained from http://holt.oso.chalmers.se/loading/ using the FES2004
model with a correction for the motion of the Earth’s center of mass.
We define a North America fixed reference frame using the ITRF2005/North America
relative pole of rotation defined by Altamimi [2007] and the ITRF2005 velocities for stations
distributed across the stable interior of the continent that are included in our processing. Then
we align the daily position solutions for all stations in our networks with this reference frame
using a seven-parameter Helmert transformation consisting of the three rotations, three
translations, and a scale factor that minimize the residual position of the sites used to define the
frame. Common mode scatter is typically seen among sites within a geographic region due to
factors that include errors in satellite orbit or earth orientation parameters, atmospheric effects,
and phase center mismodeling [Wdowinski et al., 1997; Dong et al., 2006]. We define a spatial
filter using the velocities of a subset of long-running and stable CGPS sites (CMBB, DIAB,
FARB, LNC1, MUSB, P314, SAOB, SODB, SUTB, and TIBB) by calculating the translation
1
required to align those sites’ daily positions with those predicted from their long-term trends.
This filter is then applied to all stations in the network on each day.
Because regional filtering requires CGPS sites that span the geographic region of interest, we
cannot define the filter for data earlier than 1999 due to insufficient site coverage. Therefore, for
most SGPS sites we use only the portion of the time series from 1999 onward. However, due to
the proximity of the SAF to the coast, relatively few GPS velocities are available to constrain its
slip rate. Therefore, we wish to leverage additional older SGPS data. We calculate the average
velocities of these sites from time series of data in the North America fixed reference frame
without regional filtering. We do the same for continuous sites in the region and then calculate
the translation required to align the CGPS velocities with and without regional filtering and
apply that translation to align the campaign site velocities to the regionally-filtered reference
frame.
We wish to estimate the average velocity over the measurement period for each GPS site in
our dataset. Noise in GPS time series exhibits temporal correlations that can be characterized by
a combination of white and colored processes as well as seasonal effects. Estimating velocities
using least squares fitting and linear propagation of errors will underestimate uncertainties,
especially for CGPS time series which can consist of thousands of observations. Therefore we
use the approach of Langbein [2004] to account for the temporally correlated noise in order to
obtain velocity estimates with more realistic uncertainties.
For continuous sites we fit each time series with a linear trend and sinusoidal terms of three
periods (annual, semi-annual, and 13.6 days, the latter following Penna et al., [2007]). Offsets
are also estimated when required due to site-specific changes to instrumentation. We determine
2
the optimal noise model which can consist of combinations of white noise, flicker noise, random
walk, power law, first-order Gauss Markov, and band-pass filtered noise as described in
Langbein [2004]. Reliably optimizing the noise model requires sufficiently long and temporally
dense time series. Therefore, we omit continuous stations whose time series have fewer than
four years’ worth of data. We also remove obvious outliers from the time series before analysis.
While the parameters defining the optimal noise model are determined simultaneously with
fitting the linear trend and seasonal terms, for operational applications we hold the noise model
parameters fixed and recalculate the velocity and seasonal amplitudes daily.
parameters are updated periodically to draw upon new observations.
The noise
The noise model
parameters used for all CGPS sites can be found in Table S1 of the Supplemental Material.
We use SGPS observations from sites with at least three surveys, where a survey typically
consists of at least two occupations lasting at least 6 hours each. Because SGPS time series
contain insufficient information for optimizing the noise model, we assume a combination of
white, flicker, and random walk noise with fixed amplitudes based on the results of Langbein
[2004] and estimate only a linear trend. For horizontal campaign observations the white, flicker,
and random walk amplitudes are 2 mm, 1 mm/(yr^0.25), and 2 mm/(yr^0.5), respectively; these
are increased by a factor of 3 for the vertical.
We wish to investigate the slip rates and creep rates of the three major strike slip faults of the
San Andreas system. In order to mitigate any deformation signal due to locking of the Cascadia
subduction zone that might be present in our data, we remove the velocities predicted from the
model of McCaffrey et al. [2007]. The subduction zone extends north from the Mendocino triple
junction (located at approximately 40.4oN). Over most of our network the magnitude of this
correction is less than 1 mm/yr, exceeding this rate at only four sites which are located at the
3
northern edge of the network where it reaches a maximum of 1.5 mm/yr. As discussed in
McCaffrey et al. [2007], velocities in our region of interest can be explained well by deformation
related to the strike slip San Andreas system without additional relative rotation of the Coast
Ranges and Sierra Nevada Great Valley blocks.
Bayesian model class selection for fault geometry parameters
Bayesian model class selection is a technique based on information theory to determine
which of many possible model designs (or "model classes") fits a set of observations best
without over-fitting the data. Here we apply this approach to identify the SAF locking depth and
the dips of the MF and BSF above their respective locking depths that best fit the GPS data. We
determine the optimal model class by maximizing the marginal likelihood, also called the
evidence in favor of a model class. (See, e.g., O'Hagan, 1994; Muto and Beck, 2008, for
background.) Minson et al. [2014] provide an analytical solution to evaluating the marginal
likelihood for each candidate fault geometry, or model class, given all potential values for the
slip parameters on the fault. In this analysis we use grids of subfaults extending from the Earth’s
surface to 9 km and 13 km depth for the MF and BSF, respectively, and model the deep slip rates
using the large deep dislocations described in Section 4.1.1. In the Discussion (Section 6.1.2) we
consider models in which the MF and BSF locking depths are also estimated via Bayesian model
class selection. In our case the model class is defined by the nonlinear parameters we wish to
estimate: SAF locking depth and the dips of the MF and BSF. The linear parameters are the
deep slip rates, creep rates, and a three-parameter translation to align data and Green’s function
reference frames. We are interested in the geometry with the maximum marginal likelihood, and
the posterior probability density function (PDF) for deep slip rates and creep rates associated
with this geometry is obtained by Bayesian linear regression.
4
Following Minson et al. [2014] we use the normal-inverse-gamma conjugate prior PDF for
the linear regression because it enables closed-form expressions for the marginal likelihood and
the posterior PDF. (The prior PDF describes the relative plausibility we assign to different
values of the model parameters in the absence of data, while the posterior PDF describes the
relative plausibility of the values of the model parameters inferred from the data given our choice
of prior PDF. Thus the posterior PDF represents the solution to our inverse problem while the
prior PDF gives our a priori constraints on the model.) A drawback of this approach is that it
requires use of Gaussian priors for deep slip rates and creep rates rather than uniform priors or
one-sided priors that could be used to minimize left-lateral slip. Therefore, we carried out this
analysis as a first step to explore the likely fault geometries which could then be held fixed in a
more general formulation of the Bayesian inversion [e.g., Minson et al., 2013] to estimate deep
slip rates and spatially-variable creep rates.
We conducted the Bayesian model class selection using a grid search over values for the SAF
locking depth and dips for the MF and BSF (Table S4). The range of SAF locking depth values
tested encompasses values inferred by other studies [e.g., Freymueller et al., 1999; Prescott et
al., 2001]. Waldhauser and Schaff [2008] did not infer dips for the MF or BSF from their
earthquake relocations. However, Castillo and Ellsworth [1993] examined absolute earthquake
locations for this region and inferred the dip of the MF to be between 50 and 75 degrees NE and
the dip of the BSF to be between 65 degrees NE and vertical. Castillo and Ellsworth’s [1993]
inferences are compatible with the diffuse relocated seismicity of Waldhauser and Schaff [2008].
Based on these two studies we test a broad range of NE dips for the MF encompassing the
minimum dips from Castillo and Ellsworth [1993] as well as allowing for the possibility that
these faults are vertical (Table S4). For both the MF and the BSF, because the locking depth is
5
held fixed, varying the dip of the seismogenic zone changes the width of the seismogenic zone as
well as the horizontal location of the top of the deep dislocation relative to the fault’s surface
trace.
The creep rate priors for both the MF and BSF were 5 +/- 3 mm/yr (1 sigma) for strike slip
and 0 +/- 1 mm/yr for dip slip motion. The strike slip deep slip rate priors were 20 +/- 5.5, 12 +/3, and 8 +/- 4 mm/yr for the SAF, MF, and BSF, respectively; the priors for deep dip slip rate
were 0 +/- 1 mm/yr for all three faults. These priors were chosen to encompass the range of
previously published slip rate and creep rate estimates at 95% confidence.
The results of this analysis indicate that, given the assumed MF and BSF locking depths, the
most likely locking depth for the SAF is 16 km and the most likely dips for the seismogenic
zones of the MF and BSF are 50 degrees NE and 90 degrees (vertical), respectively. At depths
of 8 – 10 km Waldhauser and Schaff’s [2008] relocated seismicity for the MF is offset to the NE
of the fault by an amount that varies along strike, implying that the dip is not constant. Castillo
and Ellsworth [1993] also noted an apparent change in dip from 50 degrees NE at the northern
end of the MF to ~70 degrees near Clear Lake. Therefore, we assign an average dip of 60
degrees NE for the creeping portion of the MF. Holding this value fixed in the Bayesian linear
regression does not change the most likely SAF locking depth or BSF dip.
While the Bayesian model class selection and regression for linear parameters described
above provides both the most likely fault geometry parameters and the posterior PDF of deep
slip rate and creep rate estimates for that geometry, the use of conjugate priors necessary to
obtain the analytical solution limits the choice of prior PDF for the linear parameters. Assigning
Gaussian priors to these parameters may bias the resulting posterior distributions and prevents
6
incorporation of the a priori knowledge that slip on these faults is right-lateral. Therefore, we
elect to hold the geometry fixed in a Bayesian inversion for deep slip rates and spatially variable
creep rates following the methodology of Minson et al. [2013] which allows more general priors.
As discussed in Section 6.1.2, we repeated the Bayesian model class selection with the
inclusion of two additional parameters, the locking depths of the MF and BSF. The resulting
model (defined by source geometry, deep slip rates, creep rates, and a reference frame
adjustment) is the one that best fits the GPS data in the absence of constraints from seismicity.
We used the same grid of subfaults for the MF as we had used in the Bayesian model class
selection in which the MF and BSF locking depths were held fixed (Table S4). For the BSF we
use two rows of creeping subfaults rather than three in order to test locking depths between 2 and
18 km. The parameters tested and best fitting values are summarized in Table S5.
Least squares inversion regularized with spatial smoothing
For comparison to the results obtained using the Bayesian inversion, we also carried out a
regularized, bounded least squares inversion using the same model geometry as used for the
Bayesian inversion, but with a relatively fine subfault discretization for the creeping portions of
the MF and BSF. We applied Laplacian smoothing following the approach of Murray et al.
[2001], and applied a lower bound of 0 mm/yr for creep rate and slip rate estimates to enforce
right lateral slip. The smoothing parameter was chosen using Cross Validation [Wahba, 1990].
The inversion was computed with Matlab’s lsqlin function which uses a trust-region reflective
algorithm. The creep rate is not smoothed to zero at the top or bottom edges of the creeping
portion of the MF and BSF. Figure S1 shows the resulting creep rate and slip rate estimates.
7
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