A study of rupture length vs. moment for synthetic earthquakes

From the events synthesized from the one-dimensional dynamical mass-spring model proposed byBurridge andKnopoff (1967), the relation between rupture length and earthquake momentM is studied for various model parameters. The earthquake moment is defined to be the total displacement of a connected set of mass elements which slide during an event. A parameter stiffness ratios is defined as the ratio of the spring constant between the two mass elements to that between one mass element and the moving plate. The velocity-dependent friction law (including weakening and hardening processes) is taken to control the sliding of a mass element. The distribution of the breaking strengths over the system is considered to be a fractal function. The cases for severals values and different velocity-dependent friction laws with different decreasing ratesr_w of the frictional force with sliding velocity are studied numerically. The weakening process of the frictional force from the static one to the dynamic one obviously affects theM– relation. Meanwhile, a rapid weakening process rather than a slow weakening process can result in aM– relation, which is comparable to the observed one. Although an increase in thes value can yield an increase in the upper bound of the value and the number of events with largeM and values, the scaling of theM– relation is not affected by the change of thes value. For the cases in this study, the theoretical –M relations for small events withM<1 are almost in the form: M^1/2, while those for large events withM>1 have a scaling exponent less than but close to 1. In addition, the fractal dimension, the friction drop ratio and the roughness of the distribution of the breaking strengths over the fault surface are the minor parameters influencing the –M relation. A comparison between the theoreticalM– relation and the observed one for strike-slip earthquakes shows that for large events the theoreticalM– relation is quite consistent with the observed one, while for small events there is a one-order difference in the two relations. For the one-dimensional model, the decreasing rate of the dynamic frictional force with velocity is the main factor in affecting the characteristic value of the earthquake moment, at which the scaling of theM– relation changes.