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Mathematical Methods Of Statistics

On generalise l'estimateur bien connu de Hill de l'indice d'une fonction de reparatition avec queue de variation reguliere a une estimation de l'indice d'une loi de valeurs extremes. On demontre la convergence et la normalite asymptotique. On utilise l'estimateur pour certaines estimations comme celle d'une quantile elevee et d'un point d'extremite

A moment estimator for the index of an extreme-value distribution

In seismology, a wealth of data-processing programs are available, and a seismic observatory typically makes use of several. A common problem is the lack of a proper database structure which prevents effective use of the data, particularly with data from different sources. A primary goal of SeisAn is to organize data from all kinds of seismic stations into a simple database and to provide most of the tools needed for routine processing. The SeisAn database is a way of organizing data by using the directory and file structure, and is not a relational data system with general access techniques such as SQL, which usually is meant by the term “database.” In addition, the intention is to facilitate research tasks by integrating additional programs to work directly on the database.

A second goal is that SeisAn must work in an identical manner under Sun (Solaris-1 and -2), Linux, and MS-Windows, and...

SeisAn Earthquake Analysis Software

Power-law frequency distributions characterize a wide array of natural phenomena. In ecology, biology, and many physical and social sciences, the exponents of these power laws are estimated to draw inference about the processes underlying the phenomenon, to test theoretical models, and to scale up from local observations to global patterns. Therefore, it is essential that these exponents be estimated accurately. Unfortunately, the binning-based methods traditionally used in ecology and other disciplines perform quite poorly. Here we discuss more sophisticated methods for fitting these exponents based on cumulative distribution functions and maximum likelihood estimation. We illustrate their superior performance at estimating known exponents and provide details on how and when ecologists should use them. Our results confirm that maximum likelihood estimation outperforms other methods in both accuracy and precision. Because of the use of biased statistical methods for estimating the exponent, the conclusions of several recently published papers should be revisited.

/pdf/on-estimating-the-exponent-of-power-law-frequency-35cbay6z8v.pdf

On estimating the exponent of power-law frequency distributions

Following the 2009 L'Aquila earthquake, the Dipartimento della Protezione Civile Italiana (DPC), appointed an International Commission on Earthquake Forecasting for Civil Protection (ICEF) to report on the current state of knowledge of short-term prediction and forecasting of tectonic earthquakes and indicate guidelines for utilization of possible forerunners of large earthquakes to drive civil protection actions, including the use of probabilistic seismic hazard analysis in the wake of a large earthquake. The ICEF reviewed research on earthquake prediction and forecasting, drawing from developments in seismically active regions worldwide. A prediction is defined as a deterministic statement that a future earthquake will or will not occur in a particular geographic region, time window, and magnitude range, whereas a forecast gives a probability (greater than zero but less than one) that such an event will occur. Earthquake predictability, the degree to which the future occurrence of earthquakes can be determined from the observable behavior of earthquake systems, is poorly understood. This lack of understanding is reflected in the inability to reliably predict large earthquakes in seismically active regions on short time scales. Most proposed prediction methods rely on the concept of a diagnostic precursor; i.e., some kind of signal observable before earthquakes that indicates with high probability the location, time, and magnitude of an impending event. Precursor methods reviewed here include changes in strain rates, seismic wave speeds, and electrical conductivity; variations of radon concentrations in groundwater, soil, and air; fluctuations in groundwater levels; electromagnetic variations near and above Earth's surface; thermal anomalies; anomalous animal behavior; and seismicity patterns. The search for diagnostic precursors has not yet produced a successful short-term prediction scheme. Therefore, this report focuses on operational earthquake forecasting as the principle means for gathering and disseminating authoritative information about time-dependent seismic hazards to help communities prepare for potentially destructive earthquakes. On short time scales of days and weeks, earthquake sequences show clustering in space and time, as indicated by the aftershocks triggered by large events. Statistical descriptions of clustering explain many features observed in seismicity catalogs, and they can be used to construct forecasts that indicate how earthquake probabilities change over the short term. Properly applied, short-term forecasts have operational utility; for example, in anticipating aftershocks that follow large earthquakes. Although the value of long-term forecasts for ensuring seismic safety is clear, the interpretation of short-term forecasts is problematic, because earthquake probabilities may vary over orders of magnitude but typically remain low in an absolute sense (< 1% per day). Translating such low-probability forecasts into effective decision-making is a difficult challenge. Reports on the current utilization operational forecasting in earthquake risk management were compiled for six countries with high seismic risk: China, Greece, Italy, Japan, Russia, United States. Long-term models are currently the most important forecasting tools for civil protection against earthquake damage, because they guide earthquake safety provisions of building codes, performance-based seismic design, and other risk-reducing engineering practices, such as retrofitting to correct design flaws in older buildings. Short-term forecasting of aftershocks is practiced by several countries among those surveyed, but operational earthquake forecasting has not been fully implemented (i.e., regularly updated and on a national scale) in any of them. Based on the experience accumulated in seismically active regions, the ICEF has provided to DPC a set of recommendations on the utilization of operational forecasting in Italy, which may also be useful in other countries. The public should be provided with open sources of information about the short-term probabilities of future earthquakes that are authoritative, scientific, consistent, and timely. Advisories should be based on operationally qualified, regularly updated seismicity forecasting systems that have been rigorously reviewed and updated by experts in the creation, delivery, and utility of earthquake information. The quality of all operational models should be evaluated for reliability and skill by retrospective testing, and they should be under continuous prospective testing against established long-term forecasts and alternative time-dependent models. Alert procedures should be standardized to facilitate decisions at different levels of government and among the public. Earthquake probability thresholds should be established to guide alert levels based on objective analysis of costs and benefits, as well as the less tangible aspects of value-of-information, such as gains in psychological preparedness and resilience. The principles of effective public communication established by social science research should be applied to the delivery of seismic hazard information.

/pdf/operational-earthquake-forecasting-state-of-knowledge-and-4t0v3rfrup.pdf

OPERATIONAL EARTHQUAKE FORECASTING. State of Knowledge and Guidelines for Utilization

This paper provides a generic equation for the evaluation of the maximum earthquake magnitude m
 max for a given seismogenic zone or entire region. The equation is capable of generating solutions in different forms, depending on the assumptions of the statistical distribution model and/or the available information regarding past seismicity. It includes the cases (i) when earthquake magnitudes are distributed according to the doubly-truncated Gutenberg-Richter relation, (ii) when the empirical magnitude distribution deviates moderately from the Gutenberg-Richter relation, and (iii) when no specific type of magnitude distribution is assumed. Both synthetic, Monte-Carlo simulated seismic event catalogues, and actual data from Southern California, are used to demonstrate the procedures given for the evaluation of m
 max. The three estimates of m
 max for Southern California, obtained by the three procedures mentioned above, are respectively: 8.32 ± 0.43, 8.31 ± 0.42 and 8.34 ± 0.45. All three estimates are nearly identical, although higher than the value 7.99 obtained by Field et al. (1999). In general, since the third procedure is non-parametric and does not require specification of the functional form of the magnitude distribution, its estimate of the maximum earthquake magnitude m
 max is considered more reliable than the other two which are based on the Gutenberg-Richter relation.

Estimation of the Maximum Earthquake Magnitude, m max

The maximum likelihood estimation of earthquake hazard parameters (maximum regional magnitude, m max , earthquake activity rate λ, and b parameter in the Gutenberg-Richter equation) is extended to the case of mixed data containing large historical events and recent complete observations. The method accepts variable quality of complete data in different parts of a catalog with different threshold magnitude values. As an illustration, the procedure is applied for the estimation of seismicity parameters in the area of Calabria and eastern Sicily.

Estimation of earthquake hazard parameters from incomplete data files. Part I. Utilization of extreme and complete catalogs with different threshold magnitudes

The maximum likelihood estimation of earthquake hazard parameters (maximum regional magnitude m max , activity rate λ, and the Gutenberg - Richter parameter b ) from incomplete data files is extended to the case of uncertain magnitude values. Two models of uncertainty are considered. In the first one, earthquake magnitude is specified by two values, the lower and the upper magnitude limit. It is assumed that such an interval contains the real, unknown magnitude. In the second model, uncertainty of earthquake magnitude is defined in the same way as it was proposed by Tinti and Mulargia (1985): the departure of the observed (apparent) magnitude from the true, unknown value is distributed normally. The proposed approach allows the combination of catalog parts of different quality, e.g., those where the assessment of magnitude is questionable and those with magnitudes determined very precisely. As an illustration, the proposed procedures are applied for the estimation of seismicity parameters in western Norway with adjacent sea areas.

Estimation of earthquake hazard parameters from incomplete data files. Part II. Incorporation of magnitude heterogeneity

—A new methodology for probabilistic seismic hazard analysis (PSHA) is described. The approach combines the best features of the "deductive" (Cornell, 1968) and "historical" (Veneziano et al., 1984) procedures. It can be called a "parametric-historic" procedure.¶The maximum regional magnitude m

 max is of paramount importance in this approach and Part I of our work presents some of the statistical techniques which can be used for its evaluation. The work is an analysis of parametric procedures for the evaluation of m

 max, when the form of the magnitude distribution is specified. For each of the formulae given there are notes on its origin, assumptions made of its derivation, and some comparisons. The statistical concepts of bias and variance are considered for each formula, and appropriate expressions for these are also given. Also, following Knopoff and Kagan (1977), we shall demonstrate why there must be a finite upper bound to the largest seismic event if the Gutenberg-Richter frequency-magnitude relation is accepted.

Parametric-historic Procedure for Probabilistic Seismic Hazard Analysis Part I: Estimation of Maximum Regional Magnitude m max

Higher mode surface wave theory is used to model the vertical component Lg wave observed in eastern North America at regional distances. Tests of the model are made to determine whether it is capable of describing empirical spectral scaling laws, spatial attenuation, and peak time domain Lg amplitudes. It is found that a simple crustal model and a rough estimate of crustal QB are all that are required to accomplish this. Good results are obtained if the average crustal Qp is equated to the coda Q of the same frequency. INTRODUCTION The Lg wave is a wave observed on short-period seismograms of local and regional seismic events which travels with a group velocity of approximately 3.5 km/sec in the Central United States. It is usually the largest arrival on the seismogram, which makes it suitable for defining a magnitude scale (Nuttli, 1973; Herrmann and Nuttli, 1982). As part of developing the magnitude scale, there have been many studies on the spatial attenuation of the Lg wave (Nuttli, 1973; Street, 1976; Bollinger, 1979; Singh and Herrmann, 1982). The Lg wave is usually thought of as a superposition of higher mode surface waves propagating within the crustal wave guide. Herrmann and Nuttli (1975a, b) attempted to deterministically model observed Lg amplitudes in terms of focal mechanism, focal depth, and seismic moment using surface wave techniques. They were limited by their inability to compute surface wave eigenfunctions at frequencies greater than 0.5 Hz. However, their results were encouraging. Harkrider (1979), drawing upon the work of Apsel (1979), discovered a way to compute surface wave energy integrals of the higher modes very precisely. Bache

Andrzej Kijko

Papers

Estimation of the Maximum Earthquake Magnitude, m max

Estimation of earthquake hazard parameters from incomplete data files. Part I. Utilization of extreme and complete catalogs with different threshold magnitudes

Estimation of earthquake hazard parameters from incomplete data files. Part II. Incorporation of magnitude heterogeneity

Parametric-historic Procedure for Probabilistic Seismic Hazard Analysis Part I: Estimation of Maximum Regional Magnitude m max

Modeling some empirical vertical component Lg relations