VOLUME 88, NUMBER 23 PHYSICAL REVIEW LETTERS 10 JUNE 2002 Foreshocks and Aftershocks in the Olami-Feder-Christensen Model Stefan Hergarten* and Horst J. Neugebauer † Geodynamics-Physics of the Lithosphere, University of Bonn, Germany (Received 25 January 2002; published 24 May 2002) With the help of numerical simulations we show that the established Olami-Feder-Christensen earth- quake model exhibits sequences of foreshocks and aftershocks; this behavior has not been recognized in previous studies. Our results are consistent with Omori’s empirical law, but the exponents predicted by the model are lower than observed in nature. The occurrence of foreshocks and aftershocks can be attributed to the nonconservative character of the Olami-Feder-Christensen model. DOI: 10.1103/PhysRevLett.88.238501 PACS numbers: 91.30.– f, 05.45.Ra, 05.65.+b Foreshocks and aftershocks.—The occurrence of after- shocks belongs to the most distinctive phenomena in seis- mology. Large earthquakes are often accompanied by a series of aftershocks; both their frequency and strength decrease through time. The decrease in frequency was first quantified more than 100 years ago by Omori [1,2]. Omori’s law states that the rate of aftershocks decays with some power of the time after the main shock: R a t 2 t M  2p , (1) where t M is the time when the main shock occurred, R a is the number of earthquakes per unit time (regardless of their magnitude), and p  1. Integrating Omori’s law over time shows that it predicts an infinite number of aftershocks. Therefore, Eq. (1) must be either truncated for t ! t M (if p $ 1) or cut off for t ! ` (if p # 1). A similar relationship was found for the number of fore- shocks [3–5]: R f t M 2 t  2q , (2) where q  1, too. However, the total number of fore- shocks is considerably lower than the number of after- shocks, so that sequences of foreshocks are often less clear than sequences of aftershocks. The Olami-Feder-Christensen model.— Since the 1960s, spring-block models have become powerful tools for un- derstanding earthquake dynamics. The first model of this type was proposed by Burridge and Knopoff [6]; it de- scribes the stick-slip motion of a one-dimensional chain of blocks which are connected with each other and with a rigid driver plate by springs. Several extensions of this seminal model were proposed [7,8]. The rapid increase of computer capacity and the transition from sets of ordinary differential equations to cellular automata led to further progress with respect to reliable statistics. A first step in this direction was performed by Rundle and Jackson [9], while several others followed [10–13]. The Olami-Feder- Christensen (OFC) model [13] is perhaps the most promi- nent representative of this family of models. It refers to a two-dimensional, rectangular array of blocks which are connected with each other and with a driving plate moving at a predefined velocity. If the force acting on any of the blocks exceeds the maximum static friction, this block be- comes unstable and moves instantaneously to a new equi- librium position where the sum of all forces acting on it vanishes. As a result, a part of the force acting on the block is transferred to its neighbors, so that some of the neigh- bors may become unstable. This may lead to avalanches. In a nondimensional formalism, the OFC model hinges on the following rules. (i) Long-term driving as long as all blocks remain stable: ≠F i ≠t  1 as long as F i , 1. (3) (ii) Instantaneous relaxation of unstable blocks: F j ! F j 1aF i for j [ N i  , F i ! 0 if F i $ 1. (4) For simplicity, the blocks are numbered with a single in- dex i , and N i  denotes the nearest neighborhood of the block i . F i is the force acting on block i , normalized with respect to the maximum static friction force. The transmis- sion parameter a depends on the strengths of the springs; it is not larger than 0.25. OFC recognized that the model exhibits self-organized criticality (SOC) [14 – 16] not only in the conservative case a  0.25, but also in the nonconservative case a, 0.25. They obtained a size distribution of the earthquakes con- sistent with the empirical Gutenberg-Richter law [17] for a  0.2. Our simulations of the OFC model have shown that the time needed until the quasisteady state with critical prop- erties is approached depends on the boundary conditions. We found that this time is shorter for free boundary condi- tions than for the widely used boundary conditions where the system is confined by a rigid frame. However, we have checked that the results presented in the following are not strongly affected by the boundary conditions. Un- der free boundary conditions, the transmission parameter a in Eq. (4) at the boundaries differs from that in the bulk according to a i  1 ni 1k . Here n i is the actual number of neighbors of the block i , i.e., 2, 3, or 4. The parameter k denotes the elastic constant of the upper leaf springs, mea- sured relatively to that of the other springs. Nontrivial temporal behavior was soon recognized in the OFC model. OFC found long-term correlations between 238501-1 0031-9007 02 88(23) 238501(4)$20.00 © 2002 The American Physical Society 238501-1