GPS exploration of the elastic properties across and within the Northern San Andreas Fault zone and heterogeneous elastic dislocation models

Romain Jolivet (Ecole Normale Superieure de Paris), Roland Bürgmann and Nicolas Houlie


The Northern San Francisco Bay Area (hereafter ``North Bay'') is sliced by three major right-lateral strike-slip faults: the northern San Andreas Fault (SAF), the Rodgers Creek Fault (RCF) and the Green Valley Fault (GVF). The RCF represents the North Bay continuation of the Hayward Fault Zone, and the GVF is the northern extension of the Concord Fault. North of the juncture with the San Gregorio Fault, geodetic and geologic data suggest a SAF slip rate of 20-25 $mm/yr$ (d'Alessio et al, 2007, Lisowski et al., 1991). Geodetically determined slip rates range from $20.2\pm1.4$ $mm/yr$ (d'Alessio et al., 2007) to $23\pm3$ $mm/yr$ (Freymueller et al., 1999). The remainder of the 40 $mm/yr$ of Pacific plate to Sierra Nevada Great Valley microplate motion is primarily accomodated by the RCF and the GVF.
Earthquake cycle deformation is commonly modeled assuming lateraly homogeneous elastic properties in the Earth's crust. First-order variations in rock elastic strength both across and within fault zones can, however, strongly impact inferences of fault slip parameters and earthquake rupture characteristics. Near Point Reyes, the SAF separates two different geologic terranes. On the east side of the fault is the Franciscan Complex, made of a mixture of Mesozoic oceanic crustal rocks and sediments, which were accreted onto the North American continent during subduction of the Farallon plate. On the west side of the SAF is the Salinian terrane, which is composed of Cretaceous granitic and metamorphic rocks, overlain by Tertiary sedimentary rocks and Quaternary fluvial terrasses. Prescott and Yu (1986) and Lisowski et al (1991) describe an asymmetric pattern along a geodetically measured surface velocity profile across to SAF at Point Reyes, which can be explained by higher rigidities to the SW of the fault. Le Pichon et al., 2005, also describes an asymmetric pattern further north along the SAF, at Point Arena, but not at Point Reyes. Chen & Freymueller, 2002, rely on near-fault strain rates determined from trilateration and GPS measurements to infer a $2$-$km$-wide near-fault compliant zone (with 50% reduced rigidity) near Bodega Bay and Tomales Bay. Here we use densily spaced GPS velocities across the SAF to evaluate changes in elastic properties and within the SAF zone.

Heterogeneous Elastic Models

Figure 2.19: Model geometry of A. deep and B. shallow CFZM. The shaded area is the weak fault zone. C. Comparison between a 10 km locking depth classic screw dislocation model (continuous line), a 10 km locking depth Shallow CFZM (long-dashed grey line) and a 10 km locking depth Deep CZFM (dashed black line) with rigidity in the 2-km-wide fault zone being reduced by 80%. D. Locking depth determined by fitting velocity profiles (400 km long with a point spacing of $0.5$ km) calculated with the CFZMs with the half-space equation 2.2. The grey dots are the best-fit locking depth for the deep CFZM and the dashed line is the corresponding polynomial fit. The black dots are the fitted locking depth for the shallow CFZM and the continuous line is the corresponding linear fit.
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The classic way to interpret a GPS-derived velocity profile across a strike-slip fault is, assuming that the movement is only horizontal, to use the screw dislocation model (Savage and Burford, 1973):

$\displaystyle v(y)=\frac{v_{max}}{\pi}atan(\frac{y}{D})$     (2.2)

Where $v$ is the predicted fault-parallel velocity of a surface point at distance $y$ from the fault and $v_{max}$ is the far field velocity. $v_{max}$ is also the slip rate on the dislocation below the locking depth $D$. This model assumes an infinite dislocation burried in a semi-infinite elastic medium. Next we consider laterally heterogeneous models that account for variation of elastic properties across and within the fault zone.
We first consider the model developed by Le Pichon et al., 2005, where the fault separates two elastic media, with different Young's modulus $E_1$ and $E_2$. They consequently use a rigidity ratio, $K$, in the following equations:

$\displaystyle y<0$ $\textstyle \Rightarrow$ $\displaystyle V(y)=KV_{max}+(\frac{2KV_{max}}{\pi})atan(\frac{y}{D})$ (2.3)
$\displaystyle y>0$ $\textstyle \Rightarrow$ $\displaystyle V(y)=KV_{max}+(\frac{2(1-K)V_{max}}{\pi})atan(\frac{y}{D})$  

Where $V(y)$ is again the velocity at a distance y from the fault, $V_{max}$ is the far field velocity, $D$ the locking depth, and $K = \frac{E_2}{E_1+E_2}$ is the asymmetry ratio.
We also evaluate the deep Compliant Fault Zone Model (CFZM) developed in Chen & Freymueller, 2002, following Rybicki and Kasahara, 1977. A low rigidity fault zone is introduced between two elastic blocks (Figure

This model (A. in Figure 2.19) is based on an infinitely deep weak fault zone. If we consider that the fault zone is weak because of damage caused by repeated earthquakes, this zone should not extend deeper than the locking depth. Therefore, we developed, using Finite Element Modeling (Chéry et al., 2001), a shallow CFZM (B. in Figure 2.19). Both models tend to localize the deformation close to the fault trace, but the shear is more localized in the shallow CFZM.
We tried to fit the computed velocity profiles obtained with both CFZMs with the classic screw dislocation model, to evaluate the trade-off between the rigidity ratio and the obtained best-fit locking depth. For both models, there is an inverse relationship between the rigidity ratio and the fitted locking depth (linear for the deep CZFM and curved for the shallow CZFM). As the difference between the CFZMs and the fitted classic models is smaller than the typical error obtained with geodetic data (typically $1$ $mm/yr$), we cannot distinguish between a shallow locking depth and a compliant fault zone, relying only on geodetic data. Thus it is important to have independent constraints on the locking depth, for instance, from the depth extent of microseismicity.

GPS velocities along the Northern San Andreas Fault

We collected GPS data in Bodega Bay and Tomales Bay, using 1996-2000 GPS measurements from Chen & Freymueller (2002) to calculate the velocities. We also used data from the Point Reyes profile, provided by the Bay Area Velocity Unification (BAVU), a compilation of the San Francisco bay area GPS velocities (d'Alessio et al., 2005). The data are processed using the GAMIT/GLOBK GPS analysis software. The site velocities are shown with respect to BARD continuous GPS station LUTZ in Figure 2.20.
A first analysis with a simple screw dislocation model, based on three parallel faults (SAF, RCF and GVF) provides a $23\pm1$ $mm/yr$ slip rate on the SAF, with a $14\pm2$ $km$ locking depth, while the whole system is accomodating $40$ $mm/yr$ of fault parallel displacement (we find a $8\pm1$ $mm/yr$ slip rate on the RCF and $9\pm1$ $mm/yr$ on the GVF)(Figure 2.20). d'Alessio et al. (2007) show that the velocity of the Farallon islands with respect to the Pacific plate is about $2.9$ $mm/yr$, consistent with our modeled velocity field. But the half-space model velocity for the Farallon Island station is $4$ to $5$ $mm/yr$ faster than the actual measured velocity. We next consider asymmetric models with a rigidity contrast across the SAF, fitting the data with equation 2.3. We find that the modeled velocity profile better matches the Farallon Islands velocity with a $0.41$ K ratio. Thus, we infer that the Salinian terrane has a rigidity $1.4$ times higher than the Franciscan complex to the east of the SAF. Our results suggest an $18$ $mm/yr$ slip rate on the SAF, with a $10$ $km$ locking depth. There is a significant trade-off between the inferred slip rate on the SAF and the rigidity contrast across the fault, with smaller rigidity contrasts leading to higher inferred slip rates.
The two networks across the SAF located further north, one in Tomales Bay and one in Bodega Bay, allow us to consider if the SAF represents a low-rigidity fault zone. Our preferred model for the Tomales Bay profile is a classic dislocation, with a $21$ $mm/yr$ slip rate on the SAF, with a $12$ $km$ locking depth. We did not explore the corresponding trade-off, but, as our data set doesn't extend far away on both side of the fault, even using the PS-SAR data from Funning et al., 2007, the determined parameters are not well constrained. In Bodega Bay, our preferred model is based on a deep CFZM, with a $28$ $mm/yr$ slip rate on the SAF, with a $15$ $km$ locking depth. The compliant zone is 40% weaker than the surrounding medium. But a classic homogeneous model with a $24$ $mm/yr$ slip rate and a $7$ $km$ locking depth on the SAF satisfies the near-field data as well, as shown by the first-order trade-off between locking depth and the compliant fault zone rigidity contrast we found in the previous section. We prefer a $15$ $km$ locking depth and, consequently, introducing this deep CFZM because of the microseismicity near the Point Reyes profile, assuming that there is no significant change in the locking depth.

Figure 2.20: A. Best fit dislocation models for the Point Reyes profile. The black dots are the fault-parallel projected GPS velocities with their associated error bars. The grey dots are the PS-SAR data from (Funning et al., 2007). The dashed line is the best classic (elastic half-space) dislocation model. The continuous line is our preferred asymmetric model with a K ratio of $0.41$ that better matches the observed velocity of the westernmost GPS site on the Farallon Islands. B. Trade off between the Locking Depth and the Slip Rate on the SAF. Contoured values are the sum of the weighted squared residuals divided by the number of data points C. Trade off between the Asymmetry Ratio and the Locking Depth on the SAF. D. Trade off between the Asymmetry Ratio and the Slip Rate on the SAF.
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This project was funded by the USGS National Earthquake Hazard Reduction Programm (NEHRP). We would like to thank J. Freymueller for his help during the GPS survey. Thanks to R. Bürgmann for having me in his team during 6 months and to all the Berkeley Active Tectonics Group.


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Chen, Q. and Freymueller, J. T., Geodetic evidence for a near-fault compliant zone along the San Andreas Fault in the San Francisco Bay Area, J. Geophys. Res., 92, 2002.

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Funning, G. J., Bürgmann, R., Ferretti, A. and Fumagalli, A. Creep on the Rodgers Creek Fault, northern San Francisco Bay Area, from 10 year ps-insar dataset. submitted, 2007.

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Savage, J. and Burford, R., Geodetic determination of relative plate motion in Central California, J. Geophys. Res., 1973

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