Sampling power-law distributions

Abstract

Power-law distributions describe many phenomena related to rock fracture. Data collected to measure the parameters of such distributions only represent samples from some underlying population. Without proper consideration of the scale and size limitations of such data, estimates of the population parameters, particularly the exponent D, are likely to be biased. A Monte Carlo simulation of the sampling and analysis process has been made, to test the accuracy of the most common methods of analysis and to quantify the confidence interval for D. The cumulative graph is almost always biased by the scale limitations of the data and can appear non-linear, even when the sample is ideally power law. An iterative correction procedure is outlined which is generally successful in giving unbiased estimates of D. A standard discrete frequency graph has been found to be highly inaccurate, and its use is not recommended. The methods normally used for earthquake magnitudes, such as a discrete frequency graph of logs of values and various maximum likelihood formulations can be used for other types of data, and with care accurate results are possible. Empirical equations are given for the confidence limits on estimates of D, as a function of sample size, the scale range of the data and the method of analysis used. The predictions of the simulations are found to match the results from real sample D-value distributions. The application of the analysis techniques is illustrated with data examples from earthquake and fault population studies.

Influence of rift obliquity on fault-population systematics: results of experimental clay models

“…We eliminated measurements at the small end of the dataset (left-hand truncation of Pickering et al, 1995), recognizing that too many small faults fall below the resolution of the imaging technique, and plotted the data in log-linear space. For models of ar308 we are con®dent that long faults were thoroughly sampled and that, therefore, these data should not be considered censored.…”

Influence of Landscape Coverage on Measuring Spatial and Length Properties of Rock Fracture Networks: Insights from Numerical Simulation

“…Actually we can also see from Fig. 5b that the landscape coverage can lead to a curvature at the small scale in the density distribution of fracture lengths, which may be related to the socalled ''truncation effect'' (Pickering et al 1995;Bonnet et al 2001) that is always present in the length distribution plot of outcrop data (Davy 1993;Odling 1997;Odling et al 1999;Bour et al 2002;Davy et al 2010;Le Garzic et al 2011;Bertrand et al 2015;Lei et al 2015;Lei and Wang 2016).…”

“…The density term a is related to the total number of fractures in the system and varies as a function of fracture orientations (Davy et al 2010). The length exponent a defines the relative proportion of large and small fractures (Davy 1993;Pickering et al 1995). The extent of the power law relation is bounded by an upper limit l max that is probably related to the thickness of the crust, and a lower limit l min that is constrained by a physical length scale (e.g.…”

“…It can be seen that the fracture network still follows a fractal spatial distribution after landscape coverage. The power law length exponent a can be derived from the cumulative distribution or density distribution of fracture lengths (Davy 1993;Pickering et al 1995). Figure 5b shows an example of the density distribution of lengths of a fracture network pattern before and after landscape cover of different ratios.…”

Practicalities of extrapolating one‐dimensional fault and fracture size‐frequency distributions to higher‐dimensional samples

“…However, a similar type of concave curve can be the result of plotting poorly sampled data from a power law distribution, where truncation, censoring, and finite range effects have led to a flattening of the curve for small x and a steep fall off for large x [Pickering et al, 1995]. Thus data from a truly exponential distribution can mistakenly be assumed to follow a power law, using the center part of the log-log plot to fit a power law exponent to the data.…”