Bootstrap determination of the reliability of b-values: an assessment of statistical estimators with synthetic magnitude series

Abstract

International audienceWe consider some practical issues of the determination of the b-value of sequences of magnitudes with the bootstrap method for short series of length L and various quantization levels Dm of the magnitude. Preliminary Monte Carlo tests performed with Dm ¼ 0 demonstrate the superiority of the maximum likelihood estimator bMLE, and the inconsistency of the, yet often used, bLR estimator defined as the least-squares slope of the experimental Gutenberg-Richter curve. The Monte Carlo tests are also applied to an estimator, bKS, which minimizes the Kolmogorov-Smirnov distance between the cumulative distribution of magnitudes and a power-law model. Monte Carlo tests of discrete versions of the bMLE and bKS estimators are done for Dm ¼ f0:1; 0:2; 0:3g and used as reference to evaluate the performance of the bootstrap determination of b. We show that all estimators provide b estimates within 10 % error for L C 100 and if a large number, n = 2 9 105, of bootstrapped sample series is used. A resolution test done with Dm ¼ 0:1 reveals that a clear distinction between b = 0.8, 1.0, and 1.2 is obtained if L= 200

Relationship between the Frequency Magnitude Distribution and the Visibility Graph in the Synthetic Seismicity Generated by a Simple Stick-Slip System with Asperities

“…It can be observed that the b -value obtained for the first run R1 is lower than that obtained for the fifth run R5 for each granularity degree. The b -value controls the proportion of earthquakes with different magnitudes [17] , therefore a higher b -values indicates a relatively higher number of small events than large events and vice versa. In our experiments, the magnitude of the events is the length of the displacement during a slip.…”

Estimation of the return period of rockfalls according to the block size

“…Referring to the Generalized Pareto Distribution, the degree of precision of the estimates of the parameters is determined through a bootstrap analysis (Efron and Tibshirani, 1994;Bengoubou-Valerius and Gibert, 2013). This analysis allows to determine the variance and the confidence bounds of the parameters of the GPD that fit each reduced list F * of the previously considered rockfall sites.…”

Short-term volcano-tectonic earthquake forecasts based on a moving mean recurrence time algorithm: the El Hierro seismo-volcanic crisis experience

“…Several other cases in which the completeness is maintained at the lowest measurable value during the duration of the VT swarm suggest that the latter condition may be real. For example, Benoit and McNutt (1996) report an increase of the minimum magnitude when the VT activity approaches a mean magnitude of 3. Likewise, De la Cruz-Reyna et al 2008and Hurst et al (2014) report a similar behaviour for the VT seismicity of Popocatépetl and Tongariro volcanoes respectively.…”

“…4). Currently, VT earthquake swarms are classified according to three family types proposed by Benoit and McNutt (1996), and later updated by Farrell et al (2009) and Brumbaugh et al (2014): type Amainshock and aftershocks; type B -foreshocks, mainshock, and aftershocks; and type C -swarms without a mainshock. Further examination of the lower part of Fig.…”

“…The answer to the second question on timing is related to the onset and evolution of a volcanic crisis. Such crises are generally characterized by VT seismic swarms, lasting an average of 5.5 days in the case of El Hierro, concentrated in regions within or around the volcanic system, and generally defined as "a sequence of events closely clustered in time and space" (Benoit and McNutt, 1996). Since such swarms may include events of damaging magnitudes, it is important to detect the non-random components of the swarm evolution, as it contains factors linking different stages of the process that may allow major VT events to be forecast.…”