Effect of limited data sets in evaluating the scaling properties of spatially distributed data: An example from mining-induced seismic activity

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Abstract

The stimates of the scaling properties of various geophysical systems may be significantly affected by the use of limited data sets featuring finite numbers of data, finite sizes of study volumes, and measurement errors. These effects are illustrated through the spatial distribution of induced seismic activity in Creighton Mine (northern Ontario, Canada). The events studied occurred during a three-month period, March-May 1992, within a volume of approximate size 400 × 400 × 180 m3. Two data sets are considered, the first one consisting of the most accurately located microearthquakes (14 338 events), and the second one including the portion of the first set that features the strongest microearthquakes (1654 events). The scaling properties of the spatial distribution of these events are studied using generalized correlation integrals. From these, generalized correlation dimensions are estimated using the slope method. The dimension spectra are examined for the real data sets, randomly generated point sets featuring uniform and monofractal distributions and mimicking the limitations of the real data, and samples randomly extracted from the real data through a bootstrap procedure. The random simulations indicate that the uniform and and monofractal random distributions can show spurious multifractality due only to the use limited data sets. The re-sampling procedure demonstrates that is is possible to work effectively with small data sets. A comparison 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-increasing data sets; (2) correlation dimensions estimated from data sets featuring different limitations cannot be directly compared, neither is it 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 different from a uniform distribution, could originate from a monofractal process; (5) the spatial distribution of the smaller microearthquakes is either monofractal or only weakly multifractal; and (6) small data sets can be effectively used to observe temporal variations in the scaling properties that may be associated with the occurrence of larger events.

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APA

Eneva, M. (1996). Effect of limited data sets in evaluating the scaling properties of spatially distributed data: An example from mining-induced seismic activity. Geophysical Journal International, 124(3), 773–786. https://doi.org/10.1111/j.1365-246X.1996.tb05637.x

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