We have developed a uniform first-order method for describing
broadscale deformation patterns that is consistent with
both plate tectonics and elastic strain accumulation on
plate boundary faults [*Murray and Segall,* 2000].
We assume interseismic deformation is a superposition of
long-term average rigid-body motions on either side of
faults, defined by angular velocities of spherical plates,
and back-slip on shallow locked portions of faults.
Unlike deep-slip fault models, which poorly describe
deformation in the far-field, motions far from plate
boundary faults are consistent with those predicted by
Euler poles, while elastic strains due to shallow back-slip
remain localized to the crust adjacent to the faults and to
the fault ends at triple junctions.
This method can accommodate complex, strike-slip,
thrust, and normal fault systems, and incorporate
additional constraints from plate tectonic models and
geologic observations.

We have applied this method to deformation observed using continuous Global
Positioning System (GPS) measurements in northern California and Nevada
collected during 1993-1999. Analysis of the raw data to estimate daily site
positions and to derive site velocities relative to the stable North American
plate (NA) are described in more detail in Chapter 6 on the BARD
network. The results presented here are derived from data collected over a
slightly shorter time interval, and assume a slightly different colored noise
model. Average velocities for a subset of the longest running permanent
stations in northern California and Nevada during 1993-1999 are shown in Fig.
6.2. Stations in eastern Nevada show little motion relative to NA,
whereas the station on the Farallon Islands (FARB), 30 km offshore near San
Francisco, is moving at
mm yr^{-1} N35W. This is consistent
with the
mm yr^{-1} N33.5W motion predicted by NUVEL-1A for
the Pacific plate (PA) [*DeMets*, 1994], indicating that the network
spans nearly the entire deformation field associated with the plate boundary.

We define the velocity of a station located at the geocentric vector **r** as

The first term gives horizontal velocities due to rigid-body motion on a sphere, where the angular velocity vector of relative to an angular velocity reference frame (e.g., a fixed plate), is decomposed into a unit vector , which defines the Euler pole latitude and longitude, and a scalar magnitude , which defines the rate. The second term sums the effects of interseismic strain accumulation due to

Given that the westernmost stations in our study form a roughly linear profile
across the SAF system and their motions are predominantly parallel to predicted
PA-NA motion, we here model interseismic strain accumulation using
two-dimensional (anti-plane strain) screw dislocations, which leads to
fault-parallel velocities of the form
,
where *s* is
the uniform deep slip rate on the fault, *d* is the locking depth from the
surface, and *x* is the distance from the fault [*Savage and Burford*,
1973]. Assuming the simple geometry that plate motions and all screw
dislocations are parallel, fault-normal velocities become zero and
fault-parallel velocities, derived from Equation 25.1 in terms
of oblique colatitude
relative to a fixed
,
are

where

Deformation east of the SAF system is reasonably well characterized by rigid
block motion within the three regions: the Sierran-Great Valley (SG), and the
eastern (EB) and western (WB) regions of the Basin and Range province separated
by the Central Nevada Seismic Zone (CNSZ). We assume strain accumulation
effects are insignificant (i.e., *d*_{f} = 0 for all faults) and estimate Euler
poles and rates from the relative station motions within each (or combinations)
of the three regions. Assuming EB and WB have the same Euler pole but different
rates causes an insignificant increase in misfit, suggesting that the data do
not strongly require separate EB and WB Euler pole locations (Fig.
25.2).

Station motions within each region of our preferred kinematic solution are
consistent with rigid-body motion, with rms misfits of the horizontal
components of 1, 1, and 2 mm yr^{-1} for the EB, WB, and SG regions, respectively.
Relative motion along the boundaries between the regions varies with position.
Because EB and WB share the same Euler pole, relative motion is purely
extensional across oblique longitudinal lines, which the CNSZ closely
approximates. The predicted extension at 40N, 118W is 3 mm yr^{-1} N75W. The relative motion between WB and SG at 40N, 118W is 3
mm yr^{-1} N45W, approximately parallel to the Walker Lane-Mt. Shasta
seismicity trend, suggesting the deformation is primarily right-lateral
strike-slip. These relative motions are in general agreement with observed
earthquake mechanisms.

For deformation across the SAF system, we assume the two-dimensional model (Eq.
25.2) with motions relative to a fixed Euler pole and constant
long-term average angular velocity rates between faults (Fig. 25.3). We
use velocities of the 22 stations in the SA and SG regions parallel to the
predicted NUVEL-1A PA-NA Euler pole (48.7N, 281.8E, 0.749
Ma^{-1})
direction. We assume a geologically realistic model using 3 faults with locking
depths derived from observed seismicity: San Andreas (12.0 km), Hayward (8.5
km), and Calaveras/Concord (10.4 km). In addition to the SG and PA blocks, we
estimate angular velocity rates for blocks between the Hayward and
Calaveras/Concord faults, and between the San Andreas and Hayward faults. The
predicted deep slip rates derived from these angular velocity rates for the San
Andreas, Hayward, and Calaveras/Concord faults are 19.31.8, 11.31.9,
and 7.41.6 mm yr^{-1}, respectively, in good agreement with estimated geologic
rates (174, 92, and 53 mm yr^{-1}, respectively) [*WGCEP,* 1999].

The angular velocity-fault backslip model given by Eq. 25.1
provides a general, self-consistent description of far-field plate motions and
strain accumulation effects, which can be significant in the vicinity of
faults. It provides a framework for simultaneously estimating both angular
velocity and fault slip parameters, and for incorporating geologic and
seismological constraints. For the SAF system, we constrained
to the NUVEL-1A PA-NA Euler pole location, and used Eq.
25.2 to estimate block angular velocity rates and fault slip
parameters, while testing additional constraints on the fault locations and
locking depths, and on the long-term PA-NA rate from global plate tectonic
models. This approach can be extended to more complex, three-dimensional fault
systems (including subduction zones and extensional provinces), which can be
approximated by summing rectangular dislocations [*Okada*, 1985], and
thus provides a framework for estimating long-term plate motions over broad
regions or even globally from present-day geodetic measurements subject to
short-term earthquake-cycle effects.

DeMets, C., Effect of recent revisions to the geomagnetic reversal time scale
on estimates of current plate motions, *Geophys. Res. Lett.*, *
21*, 2191-2194, 1994.
Murray, M. H., and P. Segall, Continuous GPS measurement of Pacific-North
America plate boundary deformation in northern California and Nevada,
1993-2000, *Geophys. Res. Lett.*, *in prep.*, 2000.

Okada, Y., Surface deformation due to shear and tensile faults in a half-space,
*Bull. Seismol. Soc. Am.*, *75*, 1135-1154, 1985.

Savage, J. C., and R. O. Burford, Geodetic determination of relative plate
motion in central california, *J. Geophys. Res.*, *78*, 832-845,
1973.
Working Group on California Earthquake Probabilities, Earthquake Probabilities
in the San Francisco Bay Region: 2000 to 2030 - A Summary of Findings, *
USGS Open File Report 99-517*, 1999.

The Berkeley Seismological Laboratory, 202 McCone Hall, UC Berkeley, Berkeley CA 94720

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