Yan. Y. Kagan

We study the distributions of earthquake numbers in two global earthquake catalogues: Global Centroid-Moment Tensor and Preliminary Determinations of Epicenters. The properties of these distributions are especially required to develop the number test for our forecasts of future seismic activity rate, tested by the Collaboratory for Study of Earthquake Predictability (CSEP). A common assumption, as used in the CSEP tests, is that the numbers are described by the Poisson distribution. It is clear, however, that the Poisson assumption for the earthquake number distribution is incorrect, especially for the catalogues with a lower magnitude threshold. In contrast to the one-parameter Poisson distribution so widely used to describe earthquake occurrences, the negative-binomial distribution (NBD) has two parameters. The second parameter can be used to characterize the clustering or overdispersion of a process. We also introduce and study a more complex three-parameter beta negative-binomial distribution. We investigate the dependence of parameters for both Poisson and NBD distributions on the catalogue magnitude threshold and on temporal subdivision of catalogue duration. First, we study whether the Poisson law can be statistically rejected for various catalogue subdivisions. We find that for most cases of interest, the Poisson distribution can be shown to be rejected statistically at a high significance level in favour of the NBD. Thereafter, we investigate whether these distributions fit the observed distributions of seismicity. For this purpose, we study upper statistical moments of earthquake numbers (skewness and kurtosis) and compare them to the theoretical values for both distributions. Empirical values for the skewness and the kurtosis increase for the smaller magnitude threshold and increase with even greater intensity for small temporal subdivision of catalogues. The Poisson distribution for large rate values approaches the Gaussian law, therefore its skewness and kurtosis both tend to zero for large earthquake rates: for the Gaussian law, these values are identically zero. A calculation of the NBD skewness and kurtosis levels based on the values of the first two statistical moments of the distribution, shows rapid increase of these upper moments levels. However, the observed catalogue values of skewness and kurtosis are rising even faster. This means that for small time intervals, the earthquake number distribution is even more heavy-tailed than the NBD predicts. Therefore for small time intervals, we propose using empirical number distributions appropriately smoothed for testing forecasted earthquake numbers.

We have obtained new results in the statistical analysis of global earthquake catalogues with special attention to the largest earthquakes, and we examined the statistical behaviour of earthquake rate variations. These results can serve as an input for updating our recent earthquake forecast, known as the ‘Global Earthquake Activity Rate 1’ model (GEAR1), which is based on past earthquakes and geodetic strain rates. The GEAR1 forecast is expressed as the rate density of all earthquakes above magnitude 5.8 within 70 km of sea level everywhere on earth at 0.1 × 0.1 degree resolution, and it is currently being tested by the Collaboratory for Study of Earthquake Predictability. The seismic component of the present model is based on a smoothed version of the Global Centroid Moment Tensor (GCMT) catalogue from 1977 through 2013. The tectonic component is based on the Global Strain Rate Map, a ‘General Earthquake Model’ (GEM) product. The forecast was optimized to fit the GCMT data from 2005 through 2012, but it also fit well the earthquake locations from 1918 to 1976 reported in the International Seismological Centre-Global Earthquake Model (ISC-GEM) global catalogue of instrumental and pre-instrumental magnitude determinations. We have improved the recent forecast by optimizing the treatment of larger magnitudes and including a longer duration (1918–2011) ISC-GEM catalogue of large earthquakes to estimate smoothed seismicity. We revised our estimates of upper magnitude limits, described as corner magnitudes, based on the massive earthquakes since 2004 and the seismic moment conservation principle. The new corner magnitude estimates are somewhat larger than but consistent with our previous estimates. For major subduction zones we find the best estimates of corner magnitude to be in the range 8.9 to 9.6 and consistent with a uniform average of 9.35. Statistical estimates tend to grow with time as larger earthquakes occur. However, by using the moment conservation principle that equates the seismic moment rate with the tectonic moment rate inferred from geodesy and geology, we obtain a consistent estimate of the corner moment largely independent of seismic history. These evaluations confirm the above-mentioned corner magnitude value. The new estimates of corner magnitudes are important both for the forecast part based on seismicity as well as the part based on geodetic strain rates. We examine rate variations as expressed by annual earthquake numbers. Earthquakes larger than magnitude 6.5 obey the Poisson distribution. For smaller events the negative-binomial distribution fits much better because it allows for earthquake clustering.

In our paper published earlier we discussed forecasts of earthquake focal mechanism and ways to test the forecast efficiency. Several verification methods were proposed, but they were based on ad hoc, empirical assumptions, thus their performance is questionable. We apply a conventional likelihood method to measure the skill of earthquake focal mechanism orientation forecasts. The advantage of such an approach is that earthquake rate prediction can be adequately combined with focal mechanism forecast, if both are based on the likelihood scores, resulting in a general forecast optimization. We measure the difference between two double-couple sources as the minimum rotation angle that transforms one into the other. We measure the uncertainty of a focal mechanism forecast (the variability), and the difference between observed and forecasted orientations (the prediction error), in terms of these minimum rotation angles. To calculate the likelihood score we need to compare actual forecasts or occurrences of predicted events with the null hypothesis that the mechanism's 3-D orientation is random (or equally probable). For 3-D rotation the random rotation angle distribution is not uniform. To better understand the resulting complexities, we calculate the information (likelihood) score for two theoretical rotational distributions (Cauchy and von Mises-Fisher), which are used to approximate earthquake source orientation pattern. We then calculate the likelihood score for earthquake source forecasts and for their validation by future seismicity data. Several issues need to be explored when analyzing observational results: their dependence on forecast and data resolution, internal dependence of scores on forecasted angle and random variability of likelihood scores. Here, we propose a simple tentative solution but extensive theoretical and statistical analysis is needed.

In our paper published earlier we discussed forecasts of earthquake focal mechanism and ways to test the forecast efficiency. Several verification methods were proposed, but they were based on ad-hoc, empirical assumptions, thus their performance is questionable. In this work we apply a conventional likelihood method to measure a skill of forecast. The advantage of such an approach is that earthquake rate prediction can in principle be adequately combined with focal mechanism forecast, if both are based on the likelihood scores, resulting in a general forecast optimization. To calculate the likelihood score we need to compare actual forecasts or occurrences of predicted events with the null hypothesis that the mechanism's 3-D orientation is random. For double-couple source orientation the random probability distribution function is not uniform, which complicates the calculation of the likelihood value. To better understand the resulting complexities we calculate the information (likelihood) score for two rotational distributions (Cauchy and von Mises-Fisher), which are used to approximate earthquake source orientation pattern. We then calculate the likelihood score for earthquake source forecasts and for their validation by future seismicity data. Several issues need to be explored when analyzing observational results: their dependence on forecast and data resolution, internal dependence of scores on forecasted angle, and random variability of likelihood scores. In this work we propose a preliminary solution to these complex problems, as these issues need to be explored by a more extensive statistical analysis.

Forecasts of the focal mechanisms of future shallow (depth 0–70 km) earthquakes are important for seismic hazard estimates and Coulomb stress, and other models of earthquake occurrence. Here we report on a high-resolution global forecast of earthquake rate density as a function of location, magnitude and focal mechanism. In previous publications we reported forecasts of 0.5° spatial resolution, covering the latitude range from −75° to +75°, based on the Global Central Moment Tensor earthquake catalogue. In the new forecasts we have improved the spatial resolution to 0.1° and the latitude range from pole to pole. Our focal mechanism estimates require distance-weighted combinations of observed focal mechanisms within 1000 km of each gridpoint. Simultaneously, we calculate an average rotation angle between the forecasted mechanism and all the surrounding mechanisms, using the method of Kagan & Jackson proposed in 1994. This average angle reveals the level of tectonic complexity of a region and indicates the accuracy of the prediction. The procedure becomes problematical where longitude lines are not approximately parallel, and where shallow earthquakes are so sparse that an adequate sample spans very large distances. North or south of 75°, the azimuths of points 1000 km away may vary by about 35°. We solved this problem by calculating focal mechanisms on a plane tangent to the Earth's surface at each forecast point, correcting for the rotation of the longitude lines at the locations of earthquakes included in the averaging. The corrections are negligible between −30° and +30° latitude, but outside that band uncorrected rotations can be significantly off. Improved forecasts at 0.5° and 0.1° resolution are posted at http://eq.ess.ucla.edu/kagan/glob_gcmt_index.html.

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 analysed in statistical crystallographic texture studies, and we apply their results to analysing 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.

We consider three questions related to the 2011 Tohoku mega‐earthquake: (1) Why was the event size so grossly underestimated by Japan’s national hazard map? (2) How should we evaluate the chances of giant earthquakes in subduction zones? (3) What is the repeat time for magnitude 9 earthquakes off the Tohoku coast? The maximum earthquake size is often guessed from the available history of earthquakes, a method known for its significant downward bias. We show that historical magnitudes systematically underestimate this maximum size of future events, but the discrepancy shrinks with time. There are two quantitative methods for estimating the corner magnitude in any region: a statistical analysis of the available earthquake record and the moment conservation principle. However, for individual zones the statistical method is usually ineffective in estimating the maximum magnitude; only the lower limit can be evaluated. The moment conservation technique, which we prefer, matches the tectonic deformation rate to that predicted by earthquakes with a truncated or tapered magnitude–frequency distribution. For subduction zones, the seismic or historical record is insufficient to constrain either the maximum or corner magnitude. However, the moment conservation principle yields consistent estimates: for all the subduction zones the corner magnitude is of the order 9.0–9.7. Moreover, moment conservation indicates that variations in estimated corner magnitude among subduction zones are not statistically significant. Another moment conservation method, applied at a point on a major fault or plate boundary, also suggests that magnitude 9 events are required to explain observed displacement rates at least for the Tohoku area. The global rate of magnitude 9 earthquakes in subduction zones, predicted from statistical analysis of seismicity as well as from moment conservation, is about five per century; five actually happened.

Unfortunately, working scientists sometimes reflexively continue to use "buzz phrases" grounded in once prevalent paradigms that have been subsequently refuted. This can impede both earthquake research and hazard mitigation. Well-worn seismological buzz phrases include "earthquake cycle," "seismic cycle," "seismic gap," and "characteristic earthquake." They all assume that there are sequences of earthquakes that are nearly identical except for the times of their occurrence. If so, the complex process of earthquake occurrence could be reduced to a description of one "characteristic" earthquake plus the times of the others in the sequence. A common additional assumption is that characteristic earthquakes dominate the displacement on fault or plate boundary "segments." The "seismic gap" (or the effectively equivalent "seismic cycle") model depends entirely on the "characteristic" assumption, with the added assumption that characteristic earthquakes are quasi-periodic. However, since the 1990s numerous statistical tests have failed to support characteristic earthquake and seismic gap models, and the 2004 Sumatra earthquake and 2011 Tohoku earthquake both ripped through several supposed segment boundaries. Earthquake scientists should scrap ideas that have been rejected by objective testing or are too vague to be testable.