Rare events in a log-Weibull scenario - Application to earthquake magnitude data (original) (raw)
Related papers
Pure and Applied Geophysics, 2008
We develop a new method for the statistical estimation of the tail of the distribution of earthquake sizes recorded in the Harvard catalog of seismic moments converted to m Wmagnitudes (1977-2004 and 1977-2006). For this, we suggest a new parametric model for the distribution of main shock magnitudes, which is composed of two branches, the pure Gutenberg-Richter distribution up to an upper magnitude threshold m 1 , followed by another branch with a maximum upper magnitude bound M max , which we refer to as the two-branch model. We find that the number of main events in the catalog (N = 3975 for 1977-2004 and N=4193 for 1977-2006) is insufficient for a direct estimation of the parameters of this model, due to the inherent instability of the estimation problem. This problem is likely to be the same for any other twobranch model. This inherent limitation can be explained by the fact that only a small fraction of the empirical data populates the second branch. We then show that using the set of maximum magnitudes (the set of T-maxima) in windows of duration T days provides a significant improvement, in particular (i) by minimizing the negative impact of time-clustering of foreshock / main shock / aftershock sequences in the estimation of the tail of magnitude distribution, and (ii) by providing via a simulation method reliable estimates of the biases in the Moment estimation procedure (which turns out to be more efficient than the Maximum Likelihood estimation). We propose a method for the determination of the optimal choice of the T-value minimizing the Mean Square Error of the estimation of the form parameter of the GEV distribution approximating the sample distribution of T-maxima, which yields T optimal =500 days. We have estimated the following quantiles of the distribution of T-maxima for the whole period 1977-2006: Q 16% (M max)= 9.3, Q 50% (M max)= 9.7 and Q 84% (M max) = 10.3. Finally, we suggest two more stable statistical characteristics of the tail of the distribution of earthquake magnitudes: the quantile Q T (q) of a high probability level q for the T-maxima, and the probability of exceedence of a high threshold magnitude ρ T (m*) = P{ m k ≥ m*}. We obtained the following sample estimates for the global Harvard catalog T Q (q=0.98) = 8.6 ± 0.2 and T ! (8) = 0.13-0.20. The comparison between our estimates for the two periods 1977-2004 and 1977-2006, where the later period included the great Sumatra earthquake 24.12.2004, m W =9.0 confirms the instability of the estimation of the parameter M max and the stability of Q T (q) and ρ T (m*) = P{ m k ≥ m*}.
The Weibull - Log Weibull Distribution for Interoccurrence Times of Earthquakes
2008
By analyzing the Japan Meteorological Agency (JMA) seismic catalog for different tectonic settings, we have found that the probability distributions of time intervals between successive earthquakes --interoccurrence times-- can be described by the superposition of the Weibull distribution and the log-Weibull distribution. In particular, the distribution of large earthquakes obeys the Weibull distribution with the exponent α_1 <1, indicating the fact that the sequence of large earthquakes is not a Poisson process. It is found that the ratio of the Weibull distribution to the probability distribution of the interoccurrence time gradually increases with increase in the threshold of magnitude. Our results infer that Weibull statistics and log-Weibull statistics coexist in the interoccurrence time statistics, and that the change of the distribution is considered as the change of the dominant distribution. In this case, the dominant distribution changes from the log-Weibull distributio...
Journal of Geophysical Research, 2000
We investigate the relationship between the size distribution of earthquake rupture area and the underlying elastic potential energy distribution in a cellular automaton model for earthquake dynamics. The frequency-rupture area distribution has the form n(S)-S'exp(-S?So) and the s. ystem potential energy distribution from the elastic Hamiltonian has the form n(E)-EVexp(-EA)), both gamma distributions. Here n(S) reduces to the Gutenberg-Richter frequency-magnitude law, with slope b-'r, in the limit that the correlation length •, related to the characteristic source size So, tends to infinity. The form of the energy distribution is consistent with a statistical mechanical mo_del with I degrees of freedom, where v=(/-2)/2 and 0 is proportional to the mean energy per site E. We examine the effect of the local energy conservation factor • and the degree of material heterogeneity (quenched disorder) on the distribution parameters, which vary systematically with the controlling variables. The inferred correlation length increases systematically with increasing material homogeneity and with increasing •. The thermal parame_.ter 0 varies systematically between the leaf springs and the connecting springs, and is proportional to E as predicted. For heterogeneous faults, z-1 stays relatively constant, consistent with field observation, and So increases with increasing • or decreasing heterogeneity. In contrast, smooth faults produce a systematic decrease in z with respect to • and So remains relatively constant. For high • approximately log-periodic quanta emerge spontaneously from the dynamics in the form of modulations on the energy distribution. The output energy for both types of fault shows a transition from strongly quasi-periodic temporal fluctuations for strong dissipation, to more chaotic fluctuations for more conservative models. Only strongly heterogeneous faults show the small fluctuations in energy strictly required by models of self-organized criticality. One of the main lines of attack on this problem is the development of numerical models for earthquakes which reproduce the observed scaling properties ]. The scaling properties of such complex, nonlinear systems are "emergent" in the sense that they cannot be predicted linearly from the local physical interactions. Instead, they result from fundamental probabilistic and statistical mechanical constraints based on the cooperative response of the system for a large number of connected elements. One of the properties of this class of discrete numerical models, consistent with analytical theories discussed in section 2, is the emergence and maintenance of broad-bandwidth scale invariance in space and time .
Size Distribution of Seismic Events in Mines
2012
As for earthquakes, the sizes of seismic events induced by mining are, within a certain range, power law distributed: N (≥ R) = αR −β , where N (≥ R) is the number of events not smaller than R, as measured by seismic potency P , moment M or radiated energy E, α measures the activity rate and β is the exponent, or the β-value. We analysed different data sets of seismicity related to underground hard rock mining with differing geological structures, mining layout, extraction ratio, depth and rate of mining. The exponent β correlates positively with the stiffness of the system (the ability to resist seismic deformation with increasing stresses), i.e. the stiffer the system the higher the exponent. As mining progresses and the overall stiffness of the rock mass degrades the parameter α tends to increase and β tends to decrease. At high mining rates we observed a negative correlation between β and the fractal dimension of the hypocentres. The uncertainty or unpredictability of R, as measured by Shannon entropy, increased with decreasing β. For the three data sets analysed in this paper none of the traditional size distribution parameters, namely: α, β or P max1 = α 1/β , managed to rate seismic hazard consistently and reliably. However, all parameters incorporating volume mined, V m , rated hazard appropriately. Since the rate of rock extraction that drives the seismic rock mass response to mining varies, the most conclusive parameters to quantify seismic hazard are those incorporating volume mined. In almost all cases the data deviates from the classical power law. At the lower end of the size spectrum the observed deviations are mainly due to contamination of data with blasts or due to bad seismological processing, otherwise there is a remarkable fit down to the lowest observable event. At the high end of the scale deviations are rather the rule than the exception, and they are most frequently convex, but in some cases concave. This has serious implications for seismic hazard assessment. Therefore, we show a relation, based on the upper-truncated power law distribution, to estimate the size of the next record breaking event. This relation is a function of β, which in turn is a function of the volume of rock extracted or to be extracted.
Variations in earthquake-size distribution across different stress regimes
Nature, 2005
The earthquake size distribution follows, in most instances, a power law, 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
Is it necessary to construct empirical distributions of maximum earthquake magnitudes?
This study was "provoked" by the recent publication of Howell (1981) who proposed two empirical relations: the unlimited logarithmic and limited tanh distributions for largest earthquake magnitudes. He found that the logarithmic relation fits the world data better than the Gumbel III distribution, while the tanh equation gives the best fit to the data at large magnitudes for regions such as the contiguous United States and Alaska.
Modeling of Extremal Earthquakes
CIM Series in Mathematical Sciences, 2015
Natural hazards, such as big earthquakes, affect the lives of thousands of people at all levels. Extreme-value analysis is an area of statistical analysis particularly concerned with the systematic study of extremes, providing an useful insight to fields where extreme values are probable to occur. The characterization of the extreme seismic activity is a fundamental basis for risk investigation and safety evaluation. Here we study large earthquakes in the scope of the Extreme Value Theory. We focus on the tails of the seismic moment distributions and we propose to estimate relevant parameters, like the tail index and high order quantiles using the geometric-type estimators. In this work we combine two approaches, namely an exploratory oriented analysis and an inferential study. The validity of the assumptions required are verified, and both geometric-type and Hill estimators are applied for the tail index and quantile estimation. A comparison between the estimators is performed, and their application to the considered problem is illustrated and discussed in the corresponding context.