Earthquake Size Distribution: Power-Law with Exponent Beta = 1/2?

We propose that the widely observed and universal Gutenberg-Richter relation is a mathematical consequence of the critical branching nature of earthquake process in a brittle fracture environment. These arguments, though preliminary, are confirmed by recent investigations of the seismic moment distribution in global earthquake catalogs and by the results on the distribution in crystals of dislocation avalanche sizes. We consider possible systematic and random errors in determining earthquake size, especially its seismic moment. These effects increase the estimate of the parameter beta of the power-law distribution of earthquake sizes. In particular, we find that estimated beta-values may be inflated by 1-3% because relative moment uncertainties decrease with increasing earthquake size. Moreover, earthquake clustering greatly influences the beta-parameter. If clusters (aftershock sequences) are taken as the entity to be studied, then the exponent value for their size distribution would decrease by 5-10%. The complexity of any earthquake source also inflates the estimated beta-value by at least 3-7%. The centroid depth distribution also should influence the beta-value, an approximate calculation suggests that the exponent value may be increased by 2-6%. Taking all these effects into account, we propose that the recently obtained beta-value of 0.63 could be reduced to about 0.52--0.56: near the universal constant value (1/2) predicted by theoretical arguments. We also consider possible consequences of the universal beta-value and its relevance for theoretical and practical understanding of earthquake occurrence in various tectonic and Earth structure environments. Using comparative crystal deformation results may help us understand the generation of seismic tremors and slow earthquakes and illuminate the transition from brittle fracture to plastic flow.Comment: 46 pages, 2 tables, 11 figures 53 pages, 2 tables, 12 figure

Testing long-term earthquake forecasts: likelihood methods and error diagrams

We propose a new method to test the effectiveness of a spatial point process forecast based on a log-likelihood score for predicted point density and the information gain for events that actually occurred in the test period. The method largely avoids simulation use and allows us to calculate the information score for each event or set of events as well as the standard error of each forecast. As the number of predicted events increases, the score distribution approaches the Gaussian law. The degree of its similarity to the Gaussian distribution can be measured by the computed coefficients of skewness and kurtosis. To display the forecasted point density and the point events, we use an event concentration diagram or a variant of the Error Diagram (ED). We demonstrate the application of the method by using our long-term forecast of seismicity in two western Pacific regions. We compare the ED for these regions with simplified diagrams based on two-segment approximations. Since the earthquakes in these regions are concentrated in narrow subduction belts, using the forecast density as a template or baseline for the ED is a more convenient display technique. We also show, using simulated event occurrence, that some proposed criteria for measuring forecast effectiveness at EDs would be strongly biased for a small event number.Comment: 31 pages text, 3 tables, 10 figure

Double-Couple Earthquake Source: Symmetry and Rotation

We consider statistical analysis of double couple (DC) earthquake focal mechanism orientation. The symmetry of DC changes with its geometrical properties, and the number of 3-D rotations one DC source can be transformed into another depends on its symmetry. Four rotations exist in a general case of DC with the nodal-plane ambiguity, two transformations if the fault plane is known, and one rotation if the sides of the fault plane are known. The symmetry of rotated objects is extensively analyzed in statistical material texture studies, and we apply their results to analyzing DC orientation. We consider theoretical probability distributions which can be used to approximate observational patterns of focal mechanisms. Uniform random rotation distributions for various DC sources are discussed, as well as two non-uniform distributions: the rotational Cauchy and von Mises-Fisher. We discuss how parameters of these rotations can be estimated by a statistical analysis of earthquake source properties in global seismicity. We also show how earthquake focal mechanism orientations can be displayed on the Rodrigues vector space.Comment: 40 pages, 14 figures, 1 tabl

Statistical Distributions of Earthquake Numbers: Consequence of Branching Process

We discuss various statistical distributions of earthquake numbers. Previously we derived several discrete distributions to describe earthquake numbers for the branching model of earthquake occurrence: these distributions are the Poisson, geometric, logarithmic, and the negative binomial (NBD). The theoretical model is the `birth and immigration' population process. The first three distributions above can be considered special cases of the NBD. In particular, a point branching process along the magnitude (or log seismic moment) axis with independent events (immigrants) explains the magnitude/moment-frequency relation and the NBD of earthquake counts in large time/space windows, as well as the dependence of the NBD parameters on the magnitude threshold (magnitude of an earthquake catalog completeness). We discuss applying these distributions, especially the NBD, to approximate event numbers in earthquake catalogs. There are many different representations of the NBD. Most can be traced either to the Pascal distribution or to the mixture of the Poisson distribution with the gamma law. We discuss advantages and drawbacks of both representations for statistical analysis of earthquake catalogs. We also consider applying the NBD to earthquake forecasts and describe the limits of the application for the given equations. In contrast to the one-parameter Poisson distribution so widely used to describe earthquake occurrence, the NBD has two parameters. The second parameter can be used to characterize clustering or over-dispersion of a process. We determine the parameter values and their uncertainties for several local and global catalogs, and their subdivisions in various time intervals, magnitude thresholds, spatial windows, and tectonic categories.Comment: 50 pages,15 Figs, 3 Table

On geometric complexity of earthquake focal zone and fault system: A statistical study

We discuss various methods used to investigate the geometric complexity of earthquakes and earthquake faults, based both on a point-source representation and the study of interrelations between earthquake focal mechanisms. We briefly review the seismic moment tensor formalism and discuss in some detail the representation of double-couple (DC) earthquake sources by normalized quaternions. Non-DC earthquake sources like the CLVD focal mechanism are also considered. We obtain the characterization of the earthquake complex source caused by summation of disoriented DC sources. We show that commonly defined geometrical fault barriers correspond to the sources without any CLVD component. We analyze the CMT global earthquake catalog to examine whether the focal mechanism distribution suggests that the CLVD component is likely to be zero in tectonic earthquakes. Although some indications support this conjecture, we need more extensive and significantly more accurate data to answer this question fully.Comment: 53 pages text, 12 figure

Random stress and Omori's law

We consider two statistical regularities that were used to explain Omori's law of the aftershock rate decay: the Levy and Inverse Gaussian (IGD) distributions. These distributions are thought to describe stress behavior influenced by various random factors: post-earthquake stress time history is described by a Brownian motion. Both distributions decay to zero for time intervals close to zero. But this feature contradicts the high immediate aftershock level according to Omori's law. We propose that these statistical distributions are influenced by the power-law stress distribution near the earthquake focal zone and we derive new distributions as a mixture of power-law stress with the exponent psi and Levy as well as IGD distributions. Such new distributions describe the resulting inter-earthquake time intervals and closely resemble Omori's law. The new Levy distribution has a pure power-law form with the exponent -(1+psi/2) and the mixed IGD has two exponents: the same as Levy for small time intervals and -(1+psi) for longer times. For even longer time intervals this power-law behavior should be replaced by a uniform seismicity rate corresponding to the long-term tectonic deformation. We compute these background rates using our former analysis of earthquake size distribution and its connection to plate tectonics. We analyze several earthquake catalogs to confirm and illustrate our theoretical results. Finally, we discuss how the parameters of random stress dynamics can be determined through a more detailed statistical analysis of earthquake occurrence or by new laboratory experiments