## OFC Model - Mark Naylor

This study investigates the evolution of far from equilibrium systems, with application to earthquake predictability models.

### Background

The Earth's crust apparantly drives itself to a state of near criticality. This prompt the question of to what degree should earthquakes be predicatable, given that a truely critical system is unpredictable. The OFC slider block model has previously been used to investigate the statistical properties of such far from equilibrium systems. The first part of this study is investigating the fundamental behaviour of the nearest neighbour OFC model and it application to the earthquake problem.### The OFC slider block model

The OFC model is a cellular automaton, designed to simulate stress loading and rupture cycles (Olami*et al*; 1992). A lattice is uniformly loaded until a cell is above its threshold strength. At this point it ruptures, propagating a proportion of its stress onto its nearest neighbours. If these cells now loaded above their threshold strength, they will also rupture. This process continues until no cells are above their threshold strength. All ruptured cells have uniform stress at the start of the next loading cycle. For the normalised, homogeneous model the ruptured cells are reset to 1 and the threshold strength is uniformly 1.

### Java model

The source code for the OFC model developed for this study is freely available below. The code is written in Java and should run on any platform. There are currently GUI and batch interfaces, an applet interface is being developed.#### Design issues

##### Generation of neighbour lists

This has been documented under toolkit.cellNeighbourLists.##### Time parallel vs. time continuous

There are two different update schemes that can be employed. The time parallel scheme updates using a constant increment size and multiple ruptures are allowed during each update. The time continuous scheme only loads to the next event, which should ensure only one rupture source per update. The time continuous scheme is much slower to implement, but probably represents reality better. However, the time parallel scheme can be used provided it can be demonstrated that the interesting results do not differ significantly from the time continuous scheme.

##### Rupture area count

The simplest way to measure rupture areas is to count the total number of adjacent cells which have ruptured. However, using the time parallel scheme it is possible for several ruptures to occur simultaneously. This introduces a problem in counting the rupture areas in that two or more ruptures may be in contact with each other. When this happens, rather than counting two small ruptures, one large rupture will be counted. This becomes more of a problem for high beta with large rupture areas, which are more likely to intercept. This has the effect of biasing the area frequency statistics of many previous studies, increasing the number of large ruptures. It is frequently these large ruptures that the area-frequency analyses hinge upon.

A more intellegent area counting algorithm can distinguish between different ruptures by identifying each ruptured element back to its source rupture and counting the number of element which originated from each source. This dramatically reduces the number of large events and reduces the evidence of a power law dependence of the area-frequency statistics in the slider-bloack OFC model.

Comparing the area-frequency statistics derived from both the simple and intellegent counting algorithms provides a technique to determine a small enough constant update increment to minimise the effect of rupture area interception for a given beta.

#### Sources

## Class File | ## Role |

Cell11.java | The cell data object |

CellularAutonome11.java | Describes lattice interactions of an array of cells |

LomnitzAutonome.java | Describes the time update mechanism of the model |

finalGUI_0_1.java | GUI interface (execuatble) |

BatchFromStrippedGUI2.java | Batch interface (execuatble) |

ReadBatchInput2.java | Class which reads in data for the batch interface class |

Link to the OFC model javadoc.

To compile the batch or gui sources:

javac -O -classpath . BatchFromStrippedGUI2.java

javac -O -classpath . finalGUI_0_1.java

#### GUI Mode

The gui interface is executed using:

java -classpath . GUIbatch1

#### Batch Mode

The table below summarises the list of parameters that can be varied when the model is run in batch mode.

## Parameter | ## Type | ## Default | ## Effect |

fileNameRoot | String | data | Output filename |

Nx | int | 200 | Cell dimension |

neighbourNumber | int | 4 | Lattice dimension |

beta | double | 0.4 | Conservation factor |

maxStrength | double | 1.0 | Maximum cell strength |

minStrength | double | 1.0 | Minimum cell strength |

relaxationRange | double | 0.0 | The absolute range of random values, above 0, that each cell can be reset to |

percentageNoise | double | 0.0 | Adds random noise about the loading stress as a proportion of the stress increment |

loadingMechanism | String | constantLoading | constantLoading, loadToRupture |

usePeriodicBC | boolean | false | If true, periodic BC are used, otherwise openen BC |

runtimeDuration | int | 20000 | Number of loading cycles in a run. To be superceded by total loading stress. |

asymmetricFactor | double | 1 | Introduces asymmetric redistribution to the square lattice, the default is for a symmetric redistribution. (Not supported be hexagonal lattice) |

The batch interface is executed using:

java -classpath . BatchFromStrippedGUI2 batchParameterFile.txt

### Relevant literature

*Bak, P. and Tang, C. (1989) Earthquakes as a self-organized critical phenomena*

The original BT slider block model.

*
Boulter, C.J. and Miller, G. (2003)Nonuniversality and scaling breakdown in a nonconservative earthquake model*

*Christensen, K. and Olami, Z. (1992)Scaling, phase transitions and nonuniversality in a self-organized critical cellular-automaton model*

*Grassberger, P. (1994)Efficient large-scale simulations of a uniformly driven system*

Identifies a transition around approximately beta/4=0.18 (beta=0.72). This approximate threshold is defined to be a broad transition between approximately ordered and strongly disordered asymptopic states.

*Main, I.G., O'Brien, G. and Henderson, R. (2000)Statistical physics of earthquakes: Comparison of distribution exponents for source area and potential energy and the dynamic emergence of log-periodic energy quanta*

Identifies quanta in the connecting-spring distribution. Interprets a transition at beta=0.8 as the percolation threshold.

*Olami, Z., Feder, H.J.S. and Christensen, K. (1992)Self-organized criticality in a continuous, nonconservative cellular automaton modelling earthquakes*

The original OFC paper that extends the BT model.