EFFECT OF LIMITED DATA SETS IN EVALUATING THE SCALING PROPERTIES OF SPATIALLY DISTRIBUTED DATA - AN EXAMPLE FROM MINING-INDUCED SEISMIC ACTIVITY

The estimates of the scaling properties of various geophysical systems may be significantly affected by the use of limited data sets featuri ng finite numbers of data, finite sizes of study volumes, and measurem ent errors. These effects are illustrated through the spatial distribu tion of induced seismic activity in Creighton Mine (northern Ontario, Canada). The events studied occurred during a three-month period, Marc h-May 1992, within a volume of approximate size 400 x 400 x 180 m(3). Two data sets are considered, the first one consisting of the most acc urately located microearthquakes (14 338 events), and the second one i ncluding the portion of the first set that features the strongest micr oearthquakes (1654 events). The scaling properties of the spatial dist ribution of these events are studied using generalized correlation int egrals. From these, generalized correlation dimensions are estimated u sing the slope method. The dimension spectra are examined for the real data sets, randomly generated point sets featuring uniform and monofr actal distributions and mimicking the limitations of the real data, an d samples randomly extracted from the real data through a bootstrap pr ocedure. The random simulations indicate that the uniform and monofrac tal random distributions can show spurious multifractality due only to the use of limited data sets. The re-sampling procedure demonstrates that is is possible to work effectively with small data sets. A compar ison of the results from the real data, random point sets, and the re- sampled real data makes it possible to conclude that: (1) the bias in the estimates of the correlation dimensions from limited data sets can be readily evaluated, making it unnecessary to work with ever-increas ing data sets; (2) correlation dimensions estimated from data sets fea turing different limitations cannot be directly compared, neither is i t recommended to assign specific physical meanings to their numerical values; (3) the strong multifractality suggested by the real dimension spectra in this study appears to be mainly spurious in character; (4) the spatial distribution of the larger microearthquakes, while differ ent from a uniform distribution, could originate from a monofractal pr ocess; (5) the spatial distribution of the smaller microearthquakes is either monofractal or only weakly multifractal; and (6) small data se ts can be effectively used to observe temporal variations in the scali ng properties that may be associated with the occurrence of larger eve nts.