Search

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 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.

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 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.

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 discuss two methods for measuring the effectiveness of earthquake prediction algorithms: The information score based on the likelihood ratio and error diagrams. For both of these methods, closed form expressions are obtained for the renewal process based on the gamma and lognormal distributions. The error diagram is more informative than the likelihood ratio and uniquely specifies the information score. We derive an expression connecting the information score and error diagrams. We then obtain the estimate of the region bounds in the error diagram for any value of the information score. We discuss how these preliminary results can be extended for more realistic models of earthquake occurrence.

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.

Least-squares linear regression is so popular that it is sometimes applied without checking whether its basic requirements are satisfied. In particular, in studying earthquake phenomena, the conditions (a) that the uncertainty on the independent variable is at least one order of magnitude smaller than the one on the dependent variable, (b) that both data and uncertainties are normally distributed and (c) that residuals are constant are at times disregarded. This may easily lead to wrong results. As an alternative to least squares, when the ratio between errors on the independent and the dependent variable can be estimated, orthogonal regression can be applied. We test the performance of orthogonal regression in its general form against Gaussian and non-Gaussian data and error distributions and compare it with standard least-square regression. General orthogonal regression is found to be superior or equal to the standard least squares in all the cases investigated and its use is recommended. We also compare the performance of orthogonal regression versus standard regression when, as often happens in the literature, the ratio between errors on the independent and the dependent variables cannot be estimated and is arbitrarily set to 1. We apply these results to magnitude scale conversion, which is a common problem in seismology, with important implications in seismic hazard evaluation, and analyse it through specific tests. Our analysis concludes that the commonly used standard regression may induce systematic errors in magnitude conversion as high as 0.3-0.4, and, even more importantly, this can introduce apparent catalogue incompleteness, as well as a heavy bias in estimates of the slope of the frequency-magnitude distributions. All this can be avoided by using the general orthogonal regression in magnitude conversions.

In his paper Kossobokov investigates the efficiency of our short-term forecast for two western Pacific regions. Although we agree with the basic results of his evaluation that the forecast statistics is much better than a random guess, we have reservations about his definition of earthquake prediction, some of his tests, and his interpretation of the test results. We distinguish between deterministic earthquake predictions and statistical forecasts. We argue that some techniques used by Kossobokov may not be appropriate for testing our forecasts and discuss other testing methods, based on the likelihood function. We demonstrate that Kossobokov's null hypothesis may be biased, and this bias can influence some of his conclusions. We show that contrary to Kossobokov's statement, our algorithm predicts mainshocks when they are preceded by foreshocks.

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.

We compare the source time functions (i.e., moment release rates) of three large California mainshocks with the seismic moment release rates during their aftershock sequences. Aftershock moment release rates, computed by summing aftershock moments in time intervals, follow a power-law time dependence similar to Omori's law from minutes to months after the mainshock; furthermore, in contrast to the previously observed saturation in numbers of aftershocks shortly after the mainshock rupture, no such saturation is seen in the aftershock moment release rates, which are dominated by the largest aftershocks. We argue that the observed saturation in aftershock numbers described by the 'time offset' parameter c in Omori's law is likely an artefact due to the underreporting of small aftershocks, which is related to the difficulty of detecting large numbers of small aftershocks in the mainshock coda. We further propose that it is more natural for c to be negative (i.e. singularity follows the onset of mainshock rupture) than positive (singularity precedes onset of rupture). To make a more general comparison of mainshock rupture process and aftershock moment rates, we then scale mainshock time functions to equalize the effects of the varied seismic moments. For the three California mainshocks, we compare the scaled time functions with similarly scaled aftershock moment rates. Finally, we compare global averages of scaled time functions of many shallow events to the average scaled aftershock moment release rate for six California mainshocks. In each of these comparisons, the extrapolation, using Omori's law, of the aftershock moment rates back in time to the onset of the mainshock rupture indicates that the temporal intensity of the aftershock moment release is about 1.5 orders of magnitude less than the maximum reached by the mainshock rupture. This may be due to the differing amplitudes and relative importance of static and dynamic stresses in aftershock initiation compared to mainshock rupture propagation.

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.

[1] We compute the stress tensor in the upper crust of southern California as a function of time and compare observed seismicity with the estimated stress at the time of each earthquake. Several recent developments make it possible to do this much more realistically than before: ( 1) a wealth of new geodetic and geologic data for southern California and ( 2) a catalog of moment tensors for all earthquakes with magnitudes larger than 6 since 1850 and larger than 5 since 1910. We model crustal deformation using both updated geodetic data and geologically determined fault slip rates. We subdivide the crust into elastic blocks, delineated by faults which move freely at a constant rate below a locking depth with a rate determined by the relative block motion. We compute normal and shear stresses on nodal planes for each earthquake in the catalog. We consider stress increments from previous earthquakes ("seismic stress'') and aseismic tectonic stress, both separately and in combination. The locations and mechanisms of earthquakes are best correlated with the aseismic shear stress. Including the cumulative coseismic effects from past earthquakes does not significantly improve the correlation. Correlations between normal stress and earthquakes are always very sensitive to the start date of the catalog, whether we exclude earthquakes very close to others and whether we evaluate stress at the hypocenter or throughout the rupture surface of an earthquake. Although the correlation of tectonic stress with earthquake triggering is robust, other results are unstable apparently because the catalog has so few earthquakes.

[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.

[1] We estimate the relative importance of small and large earthquakes for static stress changes and for earthquake triggering, assuming that earthquakes are triggered by static stress changes and that earthquakes are located on a fractal network of dimension D. This model predicts that both the number of events triggered by an earthquake of magnitude m and the stress change induced by this earthquake at the location of other earthquakes increase with m as similar to 10(Dm/2). The stronger the spatial clustering, the larger the influence of small earthquakes on stress changes at the location of a future event as well as earthquake triggering. If earthquake magnitudes follow the Gutenberg-Richter law with b > D/2, small earthquakes collectively dominate stress transfer and earthquake triggering because their greater frequency overcomes their smaller individual triggering potential. Using a southern California catalog, we observe that the rate of seismicity triggered by an earthquake of magnitude m increases with m as 10(alpha m), where alpha = 1.05 +/- 0.05. We also find that the magnitude distribution of triggered earthquakes is independent of the triggering earthquake's magnitude m. When alpha approximate to b, small earthquakes are roughly as important to earthquake triggering as larger ones. We evaluate the fractal correlation dimension D of hypocenters using two relocated catalogs for southern California. The value of D measured for distances 0.1 < r < 5 km is D = 1.54 for the Shearer et al. catalog and D = 1.73 for the Hauksson et al. catalog. The value of D reflects both the structure of the fault network and the nature of earthquake interactions. By considering only those earthquake pairs with interevent times larger than 1000 days, we can largely remove the effects of short-term clustering. Then D approximate to 2, close to the value D = 2 alpha = 2.1 predicted by assuming that earthquake triggering is due to static stress. The value D approximate to 2b implies that small earthquakes are as important as larger ones for stress transfers between earthquakes and that considering stress changes induced by small earthquakes should improve models of earthquake interactions.

Heavy tailed random variables (rvs) have proven to be an essential element in modeling a wide variety of natural and human induced processes, and the sums of heavy tailed rvs represent a particularly important construct in such models. Oriented toward both geophysical and statistical audiences, this paper discusses the appearance of the Pareto law in seismology and addresses the problem of the statistical approximation for the sums of independent rvs with common Pareto distribution F(x)=1 - xα for 1/2 < α < 2. Such variables have infinite second moment which prevents one from using the Central Limit Theorem to solve the problem. This paper presents five approximation techniques for the Pareto sums and discusses their respective accuracy. The main focus is on the median and the upper and lower quantiles of the sum?s distribution. Two of the proposed approximations are based on the Generalized Central Limit Theorem, which establishes the general limit for the sums of independent identically distributed rvs in terms of stable distributions; these approximations work well for large numbers of summands. Another approximation, which replaces the sum with its maximal summand, has less than 10% relative error for the upper quantiles when α < 1. A more elaborate approach considers the two largest observations separately from the rest of the observations, and yields a relative error under 1% for the upper quantiles and less than 5% for the median. The last approximation is specially tailored for the lower quantiles, and involves reducing the non-Gaussian problem to its Gaussian equivalent; it too yields errors less than 1%. Approximation of the observed cumulative seismic moment in California illustrates developed methods.

[1] McCann et al. [1979] published a widely cited "seismic gap'' model ascribing earthquake potential categories to 125 zones surrounding the Pacific Rim. Nishenko [ 1991] published an updated and revised version including probability estimates of characteristic earthquakes with specified magnitudes within each zone. These forecasts are now more than 20 and 10 years old, respectively, and sufficient data now exist to test them rather conclusively. For the McCann et al. forecast, we count the number of qualifying earthquakes in the several categories of zones. We assume a hypothetical probability consistent with the gap model ( e. g., red zones have twice the probability of green zones) and test against the null hypothesis that all zones have equal probability. The gap hypothesis can be rejected at a high confidence level. Contrary to the forecast of McCann et al., the data suggest that the real seismic potential is lower in the gaps than in other segments, and plate boundary zones are not made safer by recent earthquakes. For the 1991 Nishenko hypothesis, we test the number of filled zones, the likelihood scores of the observed and simulated catalogs, and the likelihood ratio of the gap hypothesis to a Poissonian null hypothesis. For earthquakes equal to or larger than the characteristic magnitude, the new seismic gap hypothesis failed at the 95% confidence level in both the number and ratio tests. If we lower the magnitude threshold by 0.5 for qualifying earthquakes, the new gap hypothesis passes the number test but fails in both the likelihood and likelihood ratio tests at the 95% confidence level.

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.

The tapered (or generalized) Pareto distribution, also called the modified Gutenberg--Richter law, has been used to model the sizes of earthquakes. Unfortunately, maximum likelihood estimates of the cutoff parameter are substantially biased. Alternative estimates for the cutoff parameter are presented, and their properties discussed.