This paper addresses two problems: the random rotation of double-couple (DC) earthquake sources and the display of earthquake focal mechanisms. We consider several equivalent representations for DC sources and their properties and provide mathematical expressions for their mutual transformation. Obviously, a 3-D rotation of any object is more intricate than a 2-D rotation. Any rotation of a DC source is further complicated by its symmetry properties. Applying statistical tests to a DC distribution often requires one to compare it to a completely (or uniform) random DC pattern. We review several methods for obtaining random distribution of DC orientation; some of these seemingly natural techniques yield an incorrect result. The DC random rotation problem is closely connected to displays of focal mechanisms. In such displays, a strike or an azimuth of a focal mechanism can be neglected; hence, we are confronted with mapping a 2-D distribution onto a flat surface. We review different methods for such displays and discuss more specifically how to project a random focal mechanism distribution on a flat 2-D display with uniform probability density. Such displays can be used to analyse earthquake patterns statistically in various tectonic regions.

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.

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 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.

Several reasons for the failure can be proposed: 1. The multidimensional character of seismicity: time, space, and earthquake focal mechanism need to be modeled. The latter is a symmetric second-rank tensor of a special kind. 2. The intrinsic randomness of earthquake occurrence, necessitating the use of stochastic point processes and appropriate complex statistical techniques. 3. The scale-invariant or fractal properties of earthquake processes; the theory of random stable or heavy-tailed variables is significantly more difficult than that of Gaussian variables and is only now being developed. Earthquake process theory should be capable of being renormalized. 4. Statistical distributions of earthquake sizes, earthquake temporal interactions, spatial patterns and focal mechanisms are largely universal. The values of major parameters are similar for earthquakes in various tectonic zones. The universality of these distributions will enable a better foundation for earthquake process theory. 5. The quality of current earthquake data statistical analysis is low. Since little or no study of random and systematic errors is performed, most published statistical results are artifacts. 6. During earthquake rupture, propagation focal mechanisms sometimes undergo large 3-D rotations. These rotations require non-commutative algebra (e.g., quaternions and gauge theory) for accurate models of earthquake occurrence. 7. These phenomenological and theoretical difficulties are not limited to earthquakes: any fracture of brittle materials, tensile or shear, would encounter similar problems.

[1] The purpose of this paper is to interpret slip statistics in a framework of extended earthquake sources. We first discuss deformation pattern of the Earth's surface from earthquakes and suggest that continuum versus block motion controversy can be reconciled by a model of the fractal distribution of seismic sources. We consider earthquake slip statistical distributions as they can be inferred from seismic moment-frequency relations and geometrical scaling for earthquakes. Using various assumptions on temporal earthquake occurrence, these distributions are synthesized to evaluate the accuracy of geologic fault slip determinations and to estimate uncertainties in long-term earthquake patterns based on paleoseismic data. Because the seismic moment distribution is a power law (Pareto), a major part of the total seismic moment is released by major earthquakes, M >= 10(19.5) N m ( moment magnitude m >= 7); for these large earthquakes the rupture is confined to the upper brittle crust layer. We review the various moment-frequency and earthquake scaling relationships and apply them to infer the slip distribution at area- and site-specific regions. Simulating the seismic moment and strain accumulation process demonstrates that some synthetics can be interpreted as examples of a quasiperiodic sequence. We demonstrate the application of the derived slip statistical relations by analyzing the slip distribution and history of the San Andreas fault at Wrightwood, California.

We review methods for determining the fractal dimensions of earthquake epicentres and hypocentres, paying special attention to the problem of errors, biases and systematic effects. Among effects considered are earthquake location errors, boundary effects, inhomogeneity of depth distribution and temporal dependence. In particular, the correlation dimension of earthquake spatial distribution is discussed, techniques for its evaluation presented, and results for several earthquake catalogues are analysed. We show that practically any value for the correlation dimension can be obtained if many errors and inhomogeneities in observational data as well as deficiencies in data processing are not properly considered. It is likely that such technical difficulties are intensified when one attempts to evaluate multifractal measures of dimension. Taking into account possible errors and biases, we conclude that the fractal dimension for shallow seismicity asymptotically approaches 2.20 +/- 0.05 for a catalogue time span of decades and perhaps centuries. The value of the correlation dimension declines to 1.8-1.9 for intermediate events (depth interval 71-280 km) and to 1.5-1.6 for deeper ones.

We derive new, simplified formulae for evaluating the 3-D angle of earthquake double-couple (DC) rotation. The complexity of the derived equations depends on both accuracy requirements for angle evaluation and the completeness of desired solutions. The solutions are simpler than my previously proposed algorithm based on the quaternion representation designed in 1991. We discuss advantages and disadvantages of both approaches. These new expressions can be written in a few lines of computer code and used to compare both DC solutions obtained by different methods and variations of earthquake focal mechanisms in space and time.

We compare several modem (1977-present) worldwide earthquake catalogs to infer their completeness, earthquake origin time and hypocenter location accuracy, magnitude/scalar seismic moment errors, and difference between individual focal mechanism/moment tensor solutions. The Harvard centroid moment tensor (CMT), US Geological Survey (USGS) MT, USGS first-motion (FM) focal mechanism, PDE and ISC catalogs have been analyzed and compared. The catalogs' completeness and accuracy vary in time and depend on earthquake depth and tectonic environment. We propose a new statistical method for evaluating catalog completeness and show the results for the CMT dataset. A difference in frequency range of seismic waves used in earthquake processing leads to varying degrees of catalog completeness for foreshocks and aftershocks close in time. Earthquake origin time versus centroid time as well as hypocenter location versus centroid location can be explained well by earthquake scaling relations. Comparing moment magnitudes and regular earthquake magnitudes yields estimated magnitude uncertainties and shows that latter magnitudes poorly estimate earthquake size for large events. Moment errors reported in the CMT solutions are well correlated with the CMT/GS-MT magnitude difference, and hence indicate magnitude uncertainty well. A normalized seismic moment tensor has 4 d.f. and its accuracy can be represented as the non-double-couple (non-DC) component-value, the 3-D angle (Phi) of DC source rotation, and a position of the rotation pole. Our results suggest that a routinely determined non-DC component is in most cases only an artifact. The distribution of the Phi-value varies over catalog time, earthquake depth, focal mechanism, and magnitude. The seismic moment errors and the value of the non-DC component are indicative of the Phi-value; for the best solutions, the 3-D angle in the CMT catalog is on the order of 5-7degrees. The CMT catalog is obviously the best dataset in completeness and accuracy of its detailed solutions. Our results specifying uncertainties and completeness of global earthquake catalogs, can be used in studies of geodynamic processes, tectonic deformation associated with earthquakes, earthquake stress analysis and in many other applications of earthquake catalog data. Seismogram interpretation techniques can be reviewed and possibly revised in light of these results. (C) 2002 Elsevier Science B.V. All rights reserved.

We propose a new method to test the performance 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 find forward relation between the error diagram curve and the information score as well as inverse relation for one simple model of point spatial fields. We again show that the error diagram is more informative than the likelihood ratio.

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.

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 catalogue completeness). We discuss applying these distributions, especially the NBD, to approximate event numbers in earthquake catalogues. 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 catalogues. 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 catalogues, and their subdivisions in various time intervals, magnitude thresholds, spatial windows, and tectonic categories. The theoretical model of how the clustering parameter depends on the corner (maximum) magnitude can be used to predict future earthquake number distribution in regions where very large earthquakes have not yet occurred.