Of overlapping Cantor sets and earthquakes: Analysis of the discrete Chakrabarti-Stinchcombe model

TL;DR

This paper provides an exact analysis of a discrete Cantor set model for earthquakes, revealing probabilistic patterns in overlap magnitudes and their sequences, which could inform earthquake magnitude predictions.

Contribution

It offers an exact analytical solution for the discrete Chakrabarti-Stinchcombe model, including overlap distributions and conditional probabilities, advancing understanding of fractal fault models.

Findings

Overlap magnitude distribution follows a binomial distribution with p=1/3.

Conditional probability of subsequent overlaps follows a binomial distribution with p=1/2.

Model does not reproduce Gutenberg-Richter law but suggests probabilistic fault behavior.

Abstract

We report an exact analysis of a discrete form of the Chakrabarti-Stinchcombe model for earthquakes [Physica A \textbf{270}, 27 (1999)] which considers a pairof dynamically overlapping finite generations of the Cantor set as a prototype of geological faults. In this model the $n$-th generation of the Cantor set shifts on its replica in discrete steps of the length of a line segment in that generation and periodic boundary conditions are assumed. We determine the general form of time sequences for the constant magnitude overlaps and hence obtain the complete time-series of overlaps by the superposition of these sequences for all overlap magnitudes. From the time-series we derive the exact frequency distribution of the overlap magnitudes. The corresponding probability distribution of the logarithm of overlap magnitudes for the $n$-th generation is found to assume the form of the binomial distribution for $n$ Bernoulli trials with probability 1/3 for the success of each trial. For an arbitrary pair of consecutive overlaps in the time-series where the magnitude of the earlier overlap is known, we find that the magnitude of the later overlap can be determined with a definite probability; the conditional probability for each possible magnitude of the later overlap follows the binomial distribution for $k$ Bernoulli trials with probability 1/2 for the success of each trial and the number $k$ is determined by the magnitude of the earlier overlap. Though this model does not produce the Gutenberg-Richter law for earthquakes, our results indicate that the fractal structure of faults admits a probabilistic prediction of earthquake magnitudes.