Size Distribution of Seismic Events in Mines

Earth and Space Science, 2019

The size or energy of diverse structures or phenomena in geoscience appears to follow power law distributions. A rigorous statistical analysis of such observations is tricky, though. Observables can span several orders of magnitude, but the range for which the power law may be valid is typically truncated, usually because the smallest events are too tiny to be detected and the largest ones are limited by the system size. We revisit several examples of proposed power law distributions dealing with potentially damaging natural phenomena. Adequate fits of the distributions of sizes are especially important in these cases, given that they may be used to assess long-term hazard. After reviewing the theoretical background for power law distributions, we improve an objective statistical fitting method and apply it to diverse data sets. The method is described in full detail, and it is easy to implement. Our analysis elucidates the range of validity of the power law fit and the corresponding exponent and whether a power law tail is improved by a truncated lognormal. We confirm that impact fireballs and Californian earthquakes show untruncated power law behavior, whereas global earthquakes follow a double power law. Rain precipitation over space and time and tropical cyclones show a truncated power law regime. Karst sinkholes and wildfires, in contrast, are better described by truncated lognormals, although wildfires also may show power law regimes. Our conclusions only apply to the analyzed data sets but show the potential of applying this robust statistical technique in the future.

The European Physical Journal B, 1999

We discuss the pertinency of the log-Weibull model in the statistical understanding of energy release for earthquake magnitude data. This model has many interesting features, the most remarkable of which being: depending on the value α > 0 of the deformation index of the source, it may present tails ranging from moderately heavy (α < 1) to very heavy (with tail index zero as α > 1), through hyperbolic (power law) for the critical value α = 1. Under this model (for which a precise tail study is supplied), the occurrence of power laws appears as a critical phenomenon: this reinforces the current trend predicting that some departure from the ideal (strictly scaling fractal) model may be ubiquitous. Having applied an affine transformation in the logarithmic scale, quantile estimation and the Kolmogorov-Smirnov statistics are used to fit the log-Weibull distribution to a realization of an iid sample. This enables to decide whether the upper tail of the phenomenon under study is light/heavy/very heavy. A comparative study of recorded French and Japanese earthquake magnitudes suggests that they exhibit comparable tail behaviour, albeit with different centrality and dispersion parameters.

Despite its limitations the power law size distribution hazard estimates offers obvious benefits. It provides a useful association between its parameters and seismic hazard related factors. The �-value correlates positively with the stiffness of the system, with rock mass and mine layout heterogeneities, with the predictability of larger events and with the influence of small events on stress transfer in the rock mass. It also gives an insight into the fundamental scaling relation between seismic potency and radiated seismic energy, logE = (�P =�E)log P + (1=�E)log (�E=�P ), where �P , �P , �E, �E are parameters of the potency, P, and the energy, E, power laws respectively. Importantly, it is simple and reasonably well understood by mine seismologists and geotechnical practitioners. Because seismic events in mines follow production it is useful to distinguish between the observed hazard, estimated in the time domain, and seismic hazard potential which, due to the intermittent nature...

Tectonophysics, 2008

It is well-documented that a variety of factors controlling the rockmass fracturing process in mines often results in a complexity of mining event size distribution. In such cases, the estimation of the probability functions of source size parameterizations, with the use of ...

Journal of Geophysical Research: Solid Earth

Nature, 2005

The earthquake size distribution follows, in most instances, a power law 1,2 , with the slope of this power law, the 'b value', commonly used to describe the relative occurrence of large and small events (a high b value indicates a larger proportion of small earthquakes, and vice versa). Statistically significant variations of b values have been measured in laboratory experiments, mines and various tectonic regimes such as subducting slabs, near magma chambers, along fault zones and in aftershock zones 3 . However, it has remained uncertain whether these differences are due to differing stress regimes, as it was questionable that samples in small volumes (such as in laboratory specimens, mines and the shallow Earth's crust) are representative of earthquakes in general. Given the lack of physical understanding of these differences, the observation that b values approach the constant 1 if large volumes are sampled 4 was interpreted to indicate that b 5 1 is a universal constant for earthquakes in general 5 . Here we show that the b value varies systematically for different styles of faulting. We find that normal faulting events have the highest b values, thrust events the lowest and strike-slip events intermediate values. Given that thrust faults tend to be under higher stress than normal faults we infer that the b value acts as a stress meter that depends inversely on differential stress.

2017

We develop new empirical scaling laws for rupture width W, rupture length L, rupture area A, and average slip D, based on a large database of rupture models. The database incorporates recent earthquake source models in a wide magnitude range (Mw 5.4–9.2) and events of various faulting styles. We apply general orthogonal regression, instead of ordinary least-squares regression, to account for measurement errors of all variables and to obtain mutually self-consistent relationships. We observe that L grows more rapidly with Mw compared to W. The fault-aspect ratio (L=W) tends to increase with fault dip, which generally increases from reversefaulting, to normal-faulting, to strike-slip events. At the same time, subduction-interface earthquakes have significantly higherW (hence a larger rupture area A) compared to other faulting regimes. For strike-slip events, the growth of W with Mw is strongly inhibited, whereas the scaling of L agrees with the L-model behavior (D correlated with L). ...

Bulletin of the Seismological Society of America, 2017

We develop new empirical scaling laws for rupture width W, rupture length L, rupture area A and average slip D, based on a large database of rupture models. The database incorporates recent earthquake source models in a wide magnitude-range (MW 5.4 -9.2), and events of various faulting styles. We apply general orthogonal regression, instead of ordinary least squares regression, to account for measurement errors for all variables and to obtain mutually self-consistent relationships. We observe that L grows more rapidly with MW, compared to W. The fault-aspect ratio (L/W) tends to increase with fault dip, which generally increases from reversefaulting, normal-faulting, to strike-slip events. At the same time, subduction-interface earthquakes have significantly higher W (hence larger rupture area A) compared to other faulting regimes. For strike-slip events, the growth of W with MW is strongly inhibited, while the scaling of L agrees with the L-model behavior (D correlated with L). However, at a regional scale where seismogenic depth is essentially fixed, the scaling behavior corresponds to the W-model (D not correlated with L). A consistent scaling behavior of MW-log10 A with slope~1.0 is found, except for normal-faulting events. Interestingly, the ratio D/W (a proxy for average stress-drop) tends to increase with MW, except for shallow crustal reverse-faulting events, suggesting the possibility of scale-dependent stress-drop. The observed variations in source-scaling properties for different faulting regimes can be interpreted in terms of geological and seismological factors. We find substantial differences between our new scaling relationships and those of previous studies. Therefore, our study provides critical updates on source-scaling relations needed in seismic-tsunami hazard analysis and engineering applications. Online Material: Figures depicting regression analysis, normality probability plots and comparisons between different source-scaling relationships, and tables listing rupture models and different earthquake source-scaling relationships.

Physical review letters, 2003

1983