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

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 o n log-log axes the number of faults with trace length greater than x agains t 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, suggestin g 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 trac e lengths. The Feller-Pareto distribution is an attractive family within th is 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 conclu de that the Burr III model superficially provides a satisfactory fit to the se data. We also discuss an interpretation of the Feller-Pareto model in te rms of a particular type of observational bias on data generated from the p ower-law distribution. However, there are a number of complications to be considered. In particula r, geometrical sampling biases, stereological effects and spatial structure in the data mean that a rigorous analysis is not straightforward. We sugge st ways in which future data collection and analysis may address some of th ese problems. If our sampling protocols and estimation procedures are adopted, geoscienti sts should be able to estimate the power-law exponent more accurately and m ore objectively than with current ad hoc procedures, and with more direct r elevance to strain calculations and other geophysical applications. Further more, our recommended method of estimation, maximum likelihood, provides po int estimates and associated standard errors of the unknown parameters, and is efficient, consistent and relatively straightforward to apply.