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Modeling Broadscale Deformation From Plate Motions and Elastic Strain Accumulation

Mark Murray


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 $45.5 \pm 0.6$ mm yr-1 N35$^\circ $W. This is consistent with the $46.6 \pm 1.4$ mm yr-1 N33.5$^\circ $W 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.

Deformation Models

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

v({\bf r}) = \omega({\bf r}) {\bf
\hat{\Omega}(r)} \times {\bf r} + \sum^F_{f=1} ({\bf G \ast s})_f.
\end{displaymath} (25.1)

The first term gives horizontal velocities due to rigid-body motion on a sphere, where the angular velocity vector ${\bf\Omega}$ of ${\bf
r}$ relative to an angular velocity reference frame (e.g., a fixed plate), is decomposed into a unit vector ${\bf\hat{\Omega}(r)}$, which defines the Euler pole latitude and longitude, and a scalar magnitude $\omega({\bf r})$, which defines the rate. The second term sums the effects of interseismic strain accumulation due to F faults, convolving a Green's function ${\bf G}$ response of an Earth model to slip distribution ${\bf s}$ on each fault. Rigid-plate tectonic models assume $\omega({\bf r}) {\bf\hat{\Omega}(r) =
\Omega}$ is constant for all ${\bf
r}$ within each plate.

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 $v = (s/\pi) tan^{-1}(d/x)$, 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 $\phi$ relative to a fixed ${\bf\hat{\Omega}}$, are

v = a \omega(\phi) \sin\phi
- \frac{a}{\pi} \sum^F_{f=1} \Delta\omega_f
\sin\phi_f tan^{-1} \frac{d_f}{a(\phi - \phi_f)}
\end{displaymath} (25.2)

where a is the Earth's radius, and each fault, located at $\phi_f$ and locked from the surface to depth df, has a deep slip rate derived from the difference in angular velocity rates $\Delta\omega_f$ on either side of the fault. We assume that $\omega(\phi)$ is constant (i.e., rigid blocks) between faults.


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., df = 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 40$^\circ $N, 118$^\circ $W is 3 mm yr-1 N75$^\circ $W. The relative motion between WB and SG at 40$^\circ $N, 118$^\circ $W is 3 mm yr-1 N45$^\circ $W, 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 (48.7$^\circ $N, 281.8$^\circ $E, 0.749 $\deg$ 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 19.3$\pm$1.8, 11.3$\pm$1.9, and 7.4$\pm$1.6 mm yr-1, respectively, in good agreement with estimated geologic rates (17$\pm$4, 9$\pm$2, and 5$\pm$3 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 ${\bf\hat{\Omega}}$ 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.

Figure 25.1: Velocities relative to stable North America for the BARD stations and other stations operated in nearby networks. Data from October 1993 to September 1999 was processed by the BSL using GAMIT software. Ellipses show 95% confidence regions, with the predicted Pacific-North America relative plate motion in central California shown for scale. Approximate boundaries of 3 rigid blocks (SG, EB, and WB) and San Andreas fault system region (SA) are delimited by thick grey lines.
\epsfig{, width=12cm}\end{center}\end{figure*}

Figure 25.2: Euler pole locations with 50% (dashed) and 95% (solid) confidence regions. PA-NA is NUVEL-1A model. BR-NA and SG-NA are estimated from preferred model. The box shows the geographical extent of Figure 25.1.
\epsfig{, width=11cm}\end{center}\end{figure*}

Figure 25.3: Velocities relative to stable North America of stations located in a profile from the Bay Area to eastern Nevada. Velocities, with one standard deviation error bars, are parallel (top) and perpendicular (bottom) to the NUVEL-1A predicted PA-NA velocities, and distance in oblique colatitude from the NUVEL-1A PA-NA Euler pole location shown in Fig. 25.2. Solid lines indicate optimal model. The dashed line shows the model assuming the NUVEL-1A angular velocity for PA. The 3 vertical lines along the top edge show locations and relative depths of the 3 faults.
\epsfig{, width=11cm}\end{center}\end{figure*}

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