Effect of seismic coupling on the scaling of seismicity

Abstract

S U M M A R YBased on the dynamical mass-spring model proposed by Burridge & Knopoff (1967), the effects on seismicity and the scaling exponent (i.e. the b value) of the frequency-magnitude relation due to the variation in the stiffness ratio s , defined as the ratio of the spring constant between two masses to that between a mass and the moving plate, are studied. A linearly velocity-dependent friction law is taken to control the relative motion between a mass and the plate. The distribution of the breaking strengths of the system is considered to be a fractal function. Computational results show the strong dependence of the seismicity pattern and the scaling exponent upon the stiffness ratio s. The number of large events increases with s. For s < 20, the data points of both cumulative frequency and discrete frequency versus magnitude cannot be completely interpreted by a single line. For s > 120, only the delocalized events, for which all masses of the model slide almost simultaneously in a time span, are generated. For 20 s s s 120, the data points of frequency versus magnitude in a large magnitude range follow a power-law function very well. The lower bound of the magnitude range is almost constant, but the upper bound increases with s. The b value of the cumulative frequency-magnitude relation is smaller than that of the discrete frequency-magnitude one especially for a large value of s. The b value of the power-law function for 20 S s S 120 decreases with increasing s in the following forms: b -s-2'3 for the cumulative frequencymagnitude relation and b -s ~ 1' 2 for the discrete frequency-magnitude one. In addition, the variations in several other parameters of the friction law also lead to different b values, but the distributions of b versus s still follow the previously mentioned relations.

Slip of a Two-degree-of-freedom Spring-slider Model in the Presence of Slip-dependent Friction and Viscosity

“…He stressed that instability occurs when the ratio reaches unity, i.e., K f ≈K s . His ratio is similar to s=K/L defined by Wang [1995].…”

“…Figures 4 and 5 suggest that strong coupling (with large s) between two fault segments is more capable of generating a larger-sized event than week coupling (with small s). Hence, for an earthquake fault consisting of a few segments it is easier to generate a larger-sized event from larger s than from smaller s. Of course, the value of s of generating a larger-sized event increases with the number of sliders [Wang, 1995]. From numerical simulations, Wang [1995] obtained a power-law correlation between the b-value of the Gutenberg-Richter frequency-magnitude law and s: b~s -2/3 for the cumulative frequency and b~s -1/2 for the discrete frequency.…”

“…It is difficult to directly evaluate the value of s, because it is not easy to measure L. Wang [1995] showed that for a higher-order system with the number of sliders being larger than 100, the value of s which will be appropriate for seismicity simulations ranges from 20 to 120. As mentioned above, large s more easily produces self-organization in a system and thus there is strong coupling between two sliders than small s. Strong coupling makes the two sliders move almost simultaneously.…”

“…Hence, the leaf spring L can change the total energy in the system. Therefore, the stiffness ratio s=K/L is a significant parameter representing the level of conservation of energy in the system [Wang, 1995]. In this study, s is taken to represent seismic coupling.…”

“…Several simple friction laws have been taken into account by some researchers, for examples, a velocity-dependent, weakening-hardening friction law [Burridge and Knopoff, 1967]; a purely nonlinearly velocity-weakening friction law [Carlson and Langer, 1989]; a piecewise, linearly velocity-dependent weakening-hardening friction law [Wang, 1995[Wang, , 1996[Wang, , 2012; a nonlinearly velocity-weakening friction law [Noda et al, 2009]; a displacement softening-hardening friction law [Cao and Aki, 1984/85]; and a piecewise, linearly slip-dependent friction law [Ionescu and Campillo, 1999;and Urata et al, 2008]. Cochard and Madariaga [1994] and Madariaga and Cochard [1994] assumed that purely velocity-dependent friction models can lead to unphysical phenomena or mathematically ill-posed problems.…”