Earthquake spatial distribution: the correlation dimension

We review methods for determining the fractal dimensions of earthquake epicentres and hypocentres, paying special attention to the problem of errors, biases and systematic effects. Among effects considered are earthquake location errors, boundary effects, inhomogeneity of depth distribution and temporal dependence. In particular, the correlation dimension of earthquake spatial distribution is discussed, techniques for its evaluation presented, and results for several earthquake catalogues are analysed. We show that practically any value for the correlation dimension can be obtained if many errors and inhomogeneities in observational data as well as deficiencies in data processing are not properly considered. It is likely that such technical difficulties are intensified when one attempts to evaluate multifractal measures of dimension. Taking into account possible errors and biases, we conclude that the fractal dimension for shallow seismicity asymptotically approaches 2.20 +/- 0.05 for a catalogue time span of decades and perhaps centuries. The value of the correlation dimension declines to 1.8-1.9 for intermediate events (depth interval 71-280 km) and to 1.5-1.6 for deeper ones.