Fractal dimension confidence interval estimation of epicentral distributions

Estimates of the fractal dimension of hypocentral distributions require evaluating the range of independent variables in which fractal parameters exhibit a power law. Systematic and accidental errors are produced mainly by the subjective selection of this range, the insufficiency of data sets as well as by hypocenter mislocations. Therefore it is very important to determine the confidence intervals which are associated with fractal dimension estimates. The effects of various sources of errors are studied using different geometric clusters of epicenters, which have been synthetically generated using a multicluster algorithm with different hierarchical levels, so as to reproduce some characteristics of the patterns typical of real epicenter distributions. Subsequently, groups of differently sized subsets of synthetic epicenters were obtained by randomly sampling each distribution. Confidence intervals of fractal dimensions were thus calculated using all the estimates obtained for the various subsets. This procedure was also tested on real seismic data, consisting of epicentral distributions in three Sicilian areas and five clusters of mining-induced seismic events (Wujek coal mine, Poland). In that analysis both correlation dimensions and their confidence intervals were taken into account.