Power law models for rockfall frequency-magnitude distributions: review and identification of factors that influence the scaling exponent

Power laws fit to rockfall frequency-magnitude distributions are commonly used to summarize rockfall inventories, but uncertainty remains regarding which variables control the shape of the distribution, whether by exerting influence on rockfall activity itself or on our ability to measure rockfall activity. In addition, the current literature lacks concise summaries of background on the frequency-magnitude distribution for rockfalls and power law fitting. To help address these knowledge gaps, we present a new review of the rockfall frequency literature designed to collect the basic concepts, methods, and applications of the rockfall frequency-magnitude curve in one place, followed by a meta-analysis of 46 rockfall inventories. We re-fit power laws to each inventory based on the maximum likelihood estimate of the scaling exponent and the cutoff volume and used Analysis of Variance and regression to test for relationships between 11 independent physical and systematic variables and the scaling exponent, applying both single-predictor and two-predictor models. Notable relationships with the scaling exponent were observed for rockmass condition, geology, and maximum inventory volume. Higher scaling exponent values were associated with higher quality rockmasses, sedimentary rocks, and with smaller maximum rockfall volumes. Climate, data collection frequency, and data collection method also appear to have some influence on the scaling exponent, since higher scaling exponents were associated with temperate climates and inventories with shorter temporal extent and methods that involved few or no revisits to the slope. Relationships between the scaling exponent and slope angle, slope aspect, number of rockfalls in the inventory, record length, and minimum inventory volume are more ambiguous due to the noise inherent in comparing many studies together. In line with previous work, this study reinforces that sampling large volumes is important to obtaining an accurate distribution, and that the spatial scale of the inventory affects the likelihood of obtaining these measurements. We conclude with discussion of the results and recommendations for future work.