Heterogeneous Elastic Models

Figure 2.66: 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 34.1 . 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.

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):

$$v(y)=\frac{v_{max}}{\pi}atan(\frac{y}{D})$$ (34.1)

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:

$$y<0 \Rightarrow V(y)=KV_{max}+(\frac{2KV_{max}}{\pi})atan(\frac{y}{D})$$ (34.2)

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

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 2.66 ).

This model (A. in Figure 2.66 ) 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 2.66 ). 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 $1$ $mm.yr^{-1}$), 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.