Generalizations of power-law distributions applicable to sampled fault-trace lengths: model choice, parameter estimation and caveats

Abstract

Summary It has often been observed that fault-trace lengths tend to follow a power-law or Pareto distribution, at least for sufficiently large lengths. A very common method of fitting this type of model to data consists of plotting on log–log axes the number of faults with trace length greater than x against x, and reading off the slope of the resulting approximate straight line. We demonstrate that maximum likelihood is a more efficient and less biased method of estimating the power-law exponent. A further complication is that this log–log plot is often curved, suggesting that the power-law distribution is not a complete description of the data. In this paper we review the literature on probability distributions with Pareto behaviour for long trace lengths, but not necessarily for short trace lengths. The Feller–Pareto distribution is an attractive family within this class, with many well-known statistical distributions as special cases. We use maximum likelihood to fit the Feller–Pareto distribution to a sample of 1034 fault-trace lengths from the South Yorkshire coalfields. We conclude that the Burr III model superficially provides a satisfactory fit to these data. We also discuss an interpretation of the Feller–Pareto model in terms of a particular type of observational bias on data generated from the power-law distribution. However, there are a number of complications to be considered. In particular, geometrical sampling biases, stereological effects and spatial structure in the data mean that a rigorous analysis is not straightforward. We suggest ways in which future data collection and analysis may address some of these problems. If our sampling protocols and estimation procedures are adopted, geoscientists should be able to estimate the power-law exponent more accurately and more objectively than with current ad hoc procedures, and with more direct relevance to strain calculations and other geophysical applications. Furthermore, our recommended method of estimation, maximum likelihood, provides point estimates and associated standard errors of the unknown parameters, and is efficient, consistent and relatively straightforward to apply.