Dimension estimates of earthquake epicentres and hypocentres

We use the Hill estimator to estimate the correlation dimension of epicentres and hypocentres from four earthquake catalogues: Wellington micro earthquakes, New Zealand catalogue, Kanto micro earthquakes, and the Japanese (JMA) catalogue. We wanted to determine if the fracturing process is fractal, and whether it is different in the four selected regions. We found that the spatial pattern of tabulated shallow events in Japan is more tightly clustered than in New Zealand, while for deeper events the spatial pattern is similar In both regions, tabulated shallow events are more clustered than tabulated deeper events. It may appear that if one had a sufficiently large amount of data (enabling one to look at sufficiently small interpoint distances), the dimension of the epicentres and hypocentres would appear to be two or three, respectively; however, this may not be a true reflection of the fracturing process. Estimates that indicate that the fracturing tends ultimately to fill the entire space are probably caused by measurement errors in hypocentre locations. Similarly, for large values of interpoint distances, the power law exponent tends to be underestimated because of the boundary effect. When these two effects are sufficiently severe, they tend to merge, and it is difficult to determine power law exponents. These biases are not peculiar to the estimation procedure we have used.