SAMPLING POWER-LAW DISTRIBUTIONS

Power-law distributions describe many phenomena related to rock fractu re. Data collected to measure the parameters of such distributions onl y represent samples from some underlying population. Without proper co nsideration 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 pr ocess has been made, to test the accuracy of the most common methods o f analysis and to quantify the confidence interval for D. The cumulati ve graph is almost always biased by the scale limitations of the data and can appear nan-linear, even when the sample is ideally power law. An iterative correction procedure is outlined which is generally succe ssful in giving unbiased estimates of D. A standard discrete frequency graph has been found to be highly inaccurate, and its use is not reco mmended. The methods normally used for earthquake magnitudes, such as a discrete frequency graph of logs of values and various maximum likel ihood formulations can be used for other types of data, and with care accurate results are possible. Empirical equations are given for the c onfidence limits on estimates of D, as a function of sample size, the scale range of the data and the method of analysis used. The predictio ns of the simulations are found to match the results from real sample D-value distributions. The application of the analysis techniques is i llustrated with data examples from earthquake and fault population stu dies.