Métodos de recuento en placa

Seismic inversions have indicated that most earthquake ruptures propagate in a pulselike mode [Heaton, 1990] in which a point on the fault slips for only a short amount of time compared to the total event duration. This is to be contrasted with the other mode of rupture, the crack-like mode, in which the rupture once started would continue to expand

coherently and would not stop until the rupture reaches the outermost boundaries of the fault where arrest waves are reflected causing the cessation of rupture at the interior points. Accordingly, in the crack-like mode, a point on the fault will continue to slip for a duration that is comparable to the overall event time [Fig.(4.1)].

A possible mechanism for the fast healing of the slip behind the rupture front in the pulse-like mode of rupture was suggested by Heaton (1990) to be strong velocity- weakening friction. This is a frictional constitutive formulation in which the frictional strength of the interface varies rapidly with variations in the slip rate [Fig.(4.2)]. This means that as the slip rate increases, the frictional strength drops significantly and vice versa. Accordingly, after an initial nucleation phase in which the rupture develops as a small coherent crack, if the slip rate behind the rupture front starts decreasing, say due to a low prestress value [See for example Zheng and Rice, 1998], the friction will start increasing and ultimately the friction will be high enough to stop the rupture. High velocity frictional experiments in the last few years [e.g. Tullis et al., 2003 and Beeler et al., 2008] have confirmed such friction-slip rate dependence.

Fig. (4.1): Schematic diagram showing the difference between crack-like (Top) and pulse-like (Bottom) ruptures

Fig.(4.2): An example of a strong velocity-weakening friction formulation (blue) based on the flash-heating formulation of Rice (1999) along with the classical Dieterich-Ruina rate and state frictional formulation (red). Note the drastic difference between the friction coefficients in the two formulations at seismic slip rates (around 1m/s). This has important implications on the dynamic rupture [Heaton 1990, Zheng and Rice, 1998].

With the advancement of computational power and the development of more efficient numerical schemes (e.g., Lapusta et. al 2000), modelers could go beyond simulation of ruptures with the classical slip weakening friction law or the Dieterich-Ruina frictional formulation and start to consider the more realistic and stronger weakening that occurs coseismically, as well as the faster rate-dependent healing that is provided by the strong- velocity friction laws. Zheng and Rice (1998) showed that low prestress is required to

favor slip pulses over cracks under rate and state friction laws with strong velocity- weakening character. Aagaard and Heaton (2009) showed that the strong velocity- weakening friction and heterogeneous prestress lead to the propagation of localized and unsteady ruptures in the form of slip pulses that are capable of sustaining prestress heterogeneity after the earthquake is over. In Chapter 5, we show that under certain conditions in nucleation and prestress, a slip pulse could propagate steadily on a frictional interface with strong velocity-weakening behavior, and that pulses are very sensitive to variations in the existing prestress, which has important implications on slip complexity and heterogeneity sustainability in the real crust.

Nonetheless, the available computational resources are still insufficient for simulating rupture scenarios in 3D settings or with frictional parameters like those inferred from laboratory experiments. While we can currently use mesh cells as small as 100 m in 3D simulations or 1m in 2D anti-plane ruptures, a model with realistic frictional parameters will require grid spacing on the sub-centimeter level. The only numerical simulation with lab-derived frictional parameters that has been done so far was by Noda, Dunham and Rice (2009). They could not go beyond a 32 m 2D anti-plane fault and the parallelized computation lasts for nearly a month (Hiroyuki Noda, personal communication). If we would like to carry over this experiment to larger faults, it is obvious that the

computational capabilities need to be increased by several orders of magnitude in order for the calculations to end in a reasonable amount of time. This is infeasible, at least in the short run.

With the current limitation in the computational resources, there is an increased

interest in developing physically based approximate models that can simulate reliably the main rupture variables, such as the final slip or the slip rise time, with a reasonable computational cost. In the language of dynamical systems, this usually lies within the class of problems of dimension reduction; how to take a high dimensional multi-degrees of freedom system and describe its dynamics satisfactorily with only a few degrees of freedom. In the context of pulse-like ruptures we ask the following question: is it possible to exploit the localized nature of the slip pulse in designing computational methods that can reproduce some of the macroscopic rupture variables, such as the final slip, without conducting the full dynamic simulations? An affirmative answer to this question is useful in many ways. For instance, since many of the rupture variables are thought to be

correlated to each other, e.g., the final slip, the rise time, the pulse maximum slip rate,etc., knowledge of the final slip would facilitate estimating the other variables and consequently help in the fast generation of ground motion scenarios. Another useful application would be in exploring the long-time evolution of the prestress in different fault models, a problem that is computationally intractable with the current computational resources. If, however, a reliable method could simulate single-event ruptures quickly then it would be a premise for fast simulation of earthquake cycles.

It is the purpose of this chapter to explore the possibility of an affirmative answer to the question of the existence of reduced-order models for pulselike ruptures. To investigate the answer we consider a simple 1D discrete rupture model and study its evolution and the characteristics of the slip pulse generated by it. The discrete model represents the

extreme limit where all elastic interactions are short ranged and there is no radiated wave field. While this is a clear limitation for the model, it has the advantage of making us focus on pulse-related variables and gives us an intuitive picture of what is going to happen as the pulse becomes more and more sensitive to the local conditions rather than to the perturbations carried by the wave field. We will show that in the context of our discrete model, it is possible to track the evolution of energy in the system using an ordinary differential equation, which, upon its solution, yields the final slip in an event given the distribution of the existing prestress and some information about the friction. The extension to the continuum problem would require the inclusion of the far-field radiated energy. We discuss some applications of the equation and the possible extension to the continuum towards the end of the chapter.