Monofractal or multifractal: a case study of spatial distribution of mining-induced seismic activity
Abstract. Using finite data sets and limited size of study volumes may result in significant spurious effects when estimating the scaling properties of various physical processes. These effects are examined with an example featuring the spatial distribution of induced seismic activity in Creighton Mine (northern Ontario, Canada). The events studied in the present work occurred during a three-month period, March-May 1992, within a volume of approximate size 400 x 400 x 180 m3. Two sets of microearthquake locations are studied: Data Set 1 (14,338 events) and Data Set 2 (1654 events). Data Set 1 includes the more accurately located events and amounts to about 30 per cent of all recorded data. Data Set 2 represents a portion of the first data set that is formed by the most accurately located and the strongest microearthquakes.
The spatial distribution of events in the two data sets is examined for scaling behaviour using the method of generalized correlation integrals featuring various moments q. From these, generalized correlation dimensions are estimated using the slope method. Similar estimates are made for randomly generated point sets using the same numbers of events and the same study volumes as for the real data. Uniform and monofractal random distributions are used for these simulations. In addition, samples from the real data are randomly extracted and the dimension spectra for these are examined as well.
The spectra for the uniform and monofractal random generations show spurious multifractality due only to the use of finite numbers of data points and limited size of study volume. Comparing these with the spectra of dimensions for Data Set 1 and Data Set 2 allows us to estimate the bias likely to be present in the estimates for the real data. The strong multifractality suggested by the spectrum for Data Set 2 appears to be largely spurious; the spatial distribution, while different from uniform, could originate from a monofractal process. The spatial distribution of microearthquakes in Data Set 1 is either monofractal as well, or only weakly multifractal. In all similar studies, comparisons of result from real data and simulated point sets may help distinguish between genuine and artificial multifractality, without necessarily resorting to large number of data.
