Andrew F. Bell

Bufe

Daub

Ekstrom

Gutenberg

Ian G. Main

Kagan

Leonard

Main

Malamud

Mark Naylor

Michael

Naylor

Shearer

Simons

Stein

Turcotte

The Gutenberg-Richter (GR) frequency-magnitude relation is a fundamental empirical law of seismology, but its form remains uncertain for rare extreme events. Here, we show that the temporal evolution of model likelihoods and parameters for the frequency-magnitude distribution of the global Harvard Centroid Moment Tensor catalog is inconsistent with an unbounded GR relation, despite if being the preferred model at the current time. During the recent spate of 12 great earthquakes in the last 8years, record-breaking events result in profound steps in favor of the unbounded GR relation. However, between such events the preferred model gradually converges to the tapered GR relation, and the form of the convergence cannot be explained by random sampling of an unbounded GR distribution. The convergence properties are consistent with a global catalog composed of superposed randomly-sampled regional catalogs, each with different upper bounds, many of which have not yet sampled their largest event.</p

Bell, Andrew F.

Naylor, Mark

Main, Ian G.

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Convergence of the frequency-size distribution of globalearthquakesCitation for published version:Bell, AF, Naylor, M & Main, IG 2013, 'Convergence of the frequency-size distribution of global earthquakes'Geophysical Research Letters, vol 40, no. 11, pp. 2585-2589. DOI: 10.1002/grl.50416Digital Object Identifier (DOI):10.1002/grl.50416Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Publisher's PDF, also known as Version of recordPublished In:Geophysical Research LettersPublisher Rights Statement:©2013. American Geophysical Union. All Rights Reserved.General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 28. Apr. 2017Convergence of the frequency-size distribution of global earthquakesAndrew F. Bell,1 Mark Naylor,1 and Ian G. Main1Received 14 February 2013; revised 26 March 2013; accepted 26 March 2013; published 7 June 2013.[1] The Gutenberg-Richter (GR) frequency-magnituderelation is a fundamental empirical law of seismology,but its form remains uncertain for rare extreme events.Here, we show that the temporal evolution of modellikelihoods and parameters for the frequency-magnitudedistribution of the global Harvard Centroid MomentTensor catalog is inconsistent with an unbounded GRrelation, despite if being the preferred model at thecurrent time. During the recent spate of 12 greatearthquakes in the last 8 years, record-breaking eventsresult in profound steps in favor of the unbounded GRrelation. However, between such events the preferredmodel gradually converges to the tapered GR relation,and the form of the convergence cannot be explained byrandom sampling of an unbounded GR distribution. Theconvergence properties are consistent with a globalcatalog composed of superposed randomly-sampledregional catalogs, each with different upper bounds,many of which have not yet sampled their largest event.Citation: Bell, A. F., M. Naylor, and I. G. Main (2013),Convergence of the frequency-size distribution of global earthquakes,Geophys. Res. Lett., 40, 2585–2589, doi:10.1002/grl.50416.1. Introduction[2] In common with many complex, near-critical sys-tems, the earthquake frequency-size distribution takesthe form of a power-law in energy or seismic moment,at least at small to intermediate scales [Turcotte, 1997].This distribution is known as the Gutenberg-Richter(GR) relation [Gutenberg and Richter, 1949], usuallyexpressed as log(F) = a bm, where F is the total numberof earthquakes of magnitude m or greater, and a and bare model parameters. Magnitude is related to seismicmoment M by log(M) =A +Bm, whence F(M) ~Mb, withb= b/B [Turcotte, 1997]. The precise form of this distri-bution at large scales is far more uncertain. Physically,we expect an upper bound or taper [Kagan and Jackson,2000], with a maximum earthquake size limited by fac-tors such as the long-term tectonic deformation rate[Main, 1996], stress regime, seismogenic thickness, andfault zone geometry. This bound is usually modeled bymodifying the GR relation (MGR), speciﬁcally addingan exponential tail or taper to the cumulative form F(M) ~Mbexp(M/θ), where θ is a characteristic or “corner”moment [Turcotte, 1997]. Until the largest earthquakesare sampled, the MGR distribution will continue to looklike the unbounded GR. Since smaller magnitudes aresampled more frequently, the rate will tend to convergefrom below (i.e., rates measured over short time windowsare, on average, likely to be lower than the true rate, evenin a stationary system) [Main et al., 2008; Naylor et al.,2009]. As a consequence, apparent “surprises” [Lay,2012] (local record-breaking events in size or rate) canoccur as the sample size increases, with signiﬁcant impli-cations for assessing uncertainties in probabilistic earth-quake hazard assessment [Stein et al., 2012]. This paperis concerned with the statistical evidence that can be usedto identify when earthquakes magnitude samples startdeviating from GR to constrain maximum magnitudes.[3] The large number of great earthquakes in recentyears has given rise to speculation of non-stationary clus-tering compared to the global average in the last century[Bufe and Perkins, 2005]. Although this clustering is not(yet) thought to be statistically signiﬁcant [Michael,2011; Daub et al., 2012; Shearer and Stark, 2012], it isalso not clear that the global earthquake catalog for thedigital era is sufﬁciently well sampled for a referencebaseline of average properties to be determined [Mainet al., 2008] or to address epistemic uncertainties suchas the optimal form of the frequency-size relation. Thecentral limit theorem ensures that the standard error onestimates of a mean of a population decreases with sam-ple size, provided the moments of the distribution areﬁnite. However, the unbounded GR fails this criterionas the distribution does not have a ﬁnite mean [Malamudand Turcotte, 1999; Naylor et al., 2009]; importantly,the MGR is indistinguishable from GR prior to the taperbeing resolved. For such distributions, statistical prefe-rence for one model at any given snapshot may not bea sufﬁcient criterion to believe that model.[4] Here, we examine the temporal evolution of likeli-hoods and parameters for the unbounded GR andbounded MGR models for the digital era in order toresolve the form of the earthquake frequency-size distri-bution at large scales. We analyze the Harvard CentroidMoment Tensor (CMT) catalog [Ekstrom et al., 2012]between 1 January 1977 and 31 July 2012. This cataloghas a shorter duration and higher completeness magnitudethan others, but has more consistent moment estimationand reporting. The moment of the 2004 Sumatra-Andaman Islands Earthquake (SAIE) is underestimatedin the standard CMT catalog because the point sourceapproximation is no longer valid [Ekstrom et al., 2012].Higher estimates would not inﬂuence our results and weprefer to use this value for consistency. We use a conser-vative minimum moment magnitude threshold of 5.75(supporting information, Figure S1) for consistency withthe previous studies [Main et al., 2008] and to ensurecompleteness and homogeneity of the catalog, especiallyAdditional supporting information may be found in the online version of thisarticle.1School of GeoSciences, University of Edinburgh, Edinburgh, UK.Corresponding author: A. F. Bell, School of GeoSciences, University ofEdinburgh, Edinburgh, UK. (a.bell@ed.ac.uk)©2013. American Geophysical Union. All Rights Reserved.0094-8276/13/10.1002/grl.504162585GEOPHYSICAL RESEARCH LETTERS, VOL. 40, 2585–2589, doi:10.1002/grl.50416, 2013in its early part [Kagan, 1997]. Parameters of the GR andMGR models are estimated using the maximum-likelihood method [Kagan and Jackson, 2000]. We preferapplying the cumulative form (the MGR as deﬁned in themain text) rather than the gamma distribution of the prob-ability density function [Kagan and Jackson, 2000].Parameter uncertainties are estimated directly from thedistribution of likelihoods (supporting information, FigureS2), rather than the Hessian matrix of the maximum like-lihood estimate (MLE) optimization, and are stronglyasymmetric.[5] Prior to the record-breaking December 2004 Sumatra-Andaman Islands Earthquake [Lay et al., 2005] (SAIE), themean monthly earthquake rate and moment rate hadapparently converged to stable values [Main et al., 2008](conﬁrmed in Figure 1). Since then, more events haveoccurred, including the record-breaking Tohuku earthquakein March 2011 [Simons et al., 2011] and its aftershocks. Inthis period, the mean event and moment rate have continuedto rise, with standard deviations roughly proportional to theirrespective means. This lack of convergence to a central limitis consistent with under-sampling of the MGR distributionand demonstrates that we must wait longer to reliablyconstrain these model parameters.[6] In contrast, the maximum likelihood estimate (MLE)of the exponent b in the GR and MGR models becamestable (within error) by 1990 (Figure 2a). The 95% conﬁ-dence intervals for b converge with time and are even in-sensitive to model choice since mid-2004. However, thecorner moment θ of the best-ﬁt MGR model (expressedas an equivalent moment magnitude) shows signiﬁcantsteps up from 8.0 to 8.6, and then to 9.0, associated withthe SAIE and the Tohoku earthquakes, respectively.Before 1985 and after the SAIE, the upper 95% conﬁ-dence interval for θ cannot be constrained by the data.[7] The analysis of Main et al. [2008] is updated inFigure 2b using the Bayesian Information Criterion(BIC) to distinguish between the competing GR andMGR models, introducing an appropriate penalty for theadditional parameter θ in the MGR model. The best ﬁtcurves and data are shown in color codes at three times:(1) 30 June 1999, (2) 31 December 2006, and (3) 31 July2012. With these snapshots, the GR is the current best-ﬁtmodel, although in 1999 the MGR model was preferred.[8] We examine this change in preference in moredetail by plotting the values of the model selection crite-rion ΔBIC(MGR-GR) continuously as a function of time(Figure 3b). We use a modiﬁed Bayesian InformationCriterion [Leonard and Hsu, 1999], BIC =2ln(L) + k lnFigure 1. (a) Mean monthly event rate (black triangles)above a threshold magnitude and its standard deviation (bluecircles). (b) Mean monthly moment rate (black triangles) andits standard deviation (blue circles) in Nm. The verticalcolored lines correspond to times 30 June 1999 (green), 31December 2006 (black)—to allow comparison with theresults of Main et al. [2008]—and 31 July 2012 (red).Figure 2. (a) Maximum likelihood estimates of b of theGR (black triangles) and MGR models (red squares), θ ofthe MGR model (blue circles), and their 95% conﬁdencelimits (dashed lines). The color code for the three snapshottimes (vertical lines) are as in Figure 1. (b) Best ﬁt incremen-tal frequency-moment distribution for the three times (solidlines) and their data (dots). To separate the distributions forvisual inspection, data are not normalized to the catalogduration up to that point.BELL ET AL.: EARTHQUAKE FREQUENCY-SIZE DISTRIBUTION2586(n), where L is the likelihood function, n is the totalnumber of events, and k is the number of model parame-ters. The difference is then ΔBIC(MGR-GR) =2(Δln(L)) + ln(n). In this notation, the preferred model has thelower BIC. The GR model is initially preferred (ΔBIC(MGR-GR)> 0). Values of ΔBIC(MGR-GR) then trend sys-tematically towards a preference for the MGR model(ΔBIC(MGR-GR)< 0) in the late 1980s, ending in ΔBIC(MGR-GR)8 (or an equivalent likelihood ratio of2981), i.e., clearly in favor of the MGR model just beforethe record-breaking SAIE (Figure 3a). The preferencesuddenly reverts back to the GR model after the SAIEand is increased by the Tohoku earthquake (the nextrecord-breaking event), such that ΔBIC(MGR-GR) +4 oran equivalent likelihood ration of 54 in favor of the GRmodel.[9] The convergence properties in Figure 3b are clearlyvolatile, so it may be premature to conclude that the cur-rent best ﬁt model will ultimately be the best, given thecurrent state of incomplete sampling revealed by Figures 1and 2. To test this, we compare the observed sequentialvariation in ΔBIC(MGR-GR) with that expected froma randomly-sampled GR distribution (Figure 4a). Wegenerate 100 random GR catalogs of with the samenumber of events as the real ﬁltered CMT catalog. Eachsimulation uses a value of b randomly selected accordingto the probability density determined from the likelihoodfunction for the CMT catalog up to 31 July 2012. Thesimulated values show moderate variability and areanalytically bounded in the positive direction by ΔBIC(MGR-GR) = ln(n), where n is the total number of eventsin the catalog up to that point. As the two-parameterGR model cannot ﬁt the data any better than the three-parameter MGR model, so L ≤ 1, the most favorablescenario for the GR model is that Δln(L) = 0. Notably,ΔBIC(MGR-GR) rarely prefers the MGR and only reachesmodestly negative values when it does not. We ﬁnd sim-ilar results when using higher magnitude thresholds(supporting information, Figure S3).[10] Using 10,000 simulations at the three comparisontimes, we only once ﬁnd values of ΔBIC(MGR-GR) as lowas that observed prior to the SAIE (Fig 4b), and even afterthe SAIE and Tohoku earthquakes, the values of ΔBIC(MGR-GR) remain well below average. We are only ableto observe the large negative values of ΔBIC(MGR-GR) inthe real data in a period where there are no or only onesmall record-breaking changes in the maximum recordedearthquake (Figure 3a), an associated broad minimum inthe average seismic moment release rate (Figure 1), andduring a period of relative quiescence in great earthquakes[Shearer and Stark, 2012]. Clearly, the order of earth-quake occurrence plays a key role in our ability to distin-guish between these two models (supporting information,Figure S4). Too few m ≥ 8 earthquakes were observedprior to the SAIE for the data to be consistent with theGR model favored for the catalog today (supporting infor-mation, Figure S5). Thus, it is extremely unlikely that theearthquake frequency-size distribution before the SAIEarises because of an under-sampled GR model.[11] In summary, apart from the seismic b-value, nosimple statistical metrics or model parameters haveconverged to their long-term limits for the global CMTcatalog. The unbounded GR relation is currently identi-ﬁed by the BIC as the best model for the frequency-Figure 3. (a) Plot of seismic moment M0 in Nm as a function of time for all earthquakes in the global CMT catalog. Thelower threshold (light blue dashed line) represents the completeness threshold (Mw 5.75) and the upper threshold (blackdashed line) is determined by the size of the largest earthquake up to that point. The Mw 8.0, 22 June 1977 Tonga Islands;Mw 8.3, 19 August 1977 Sumbawa Island; Mw 8.4, 23 June 2001 Peru; Mw 9.0, 26 December 2004 Sumatra; and Mw 9.1,11 March 2011 Japan earthquakes are all record-breaking events. (b) Temporal evolution of ΔBIC(MGR-GR). The same snap-shot times corresponding to Figures 1 and 2 are shown as vertical lines.BELL ET AL.: EARTHQUAKE FREQUENCY-SIZE DISTRIBUTION2587magnitude distribution, but the temporal evolution ofΔBIC(MGR-GR) is inconsistent with random samplingfrom an unbounded GR distribution, indicating a prefer-ence for the MGR model that is very unlikely to occurby chance. The earthquakes occurring up to the SAIErepresent a well-sampled global MGR distribution,where the corner magnitude, θ, is Mw 8.0. However,the SAIE changed the fundamental statistics of theglobal earthquake catalog by sampling a record-breaking event from a distinct population of earthquakeswhere the truncation magnitude is signiﬁcantly higherthan that observed previously. As the mean seismicmoment release rate and event rate stabilize with time,the best model will return to an MGR relation with ahigher truncation magnitude. The frequency-size distri-bution of the global earthquake catalog is therefore bestexplained as a mixture distribution, sampled from a largenumber of regional MGR distributions with spatiallyvarying model parameters, in particular the event rateand corner moment. These ﬁndings are fundamental toconstraining insurance risk derived from the largestevents; if the catalog has converged, the risk isconstrained; if GR remains the preferred model, theexposure to single events will continue to grow.[12] Acknowledgments. A.B. was funded by the NERC standard awardEFFORT (Ne/H02297x/1). M.N. was funded by a Scottish Government andRoyal Society of Edinburgh fellowship. We thank Sandy Steacy and an ano-nymous reviewer for their constructive comments.[13] The Editor thanks Sandy Steacy and an anonymous reviewer fortheir assistance in evaluating this paper.ReferencesBufe, C. G., and D. M. Perkins (2005), Evidence for a global seismic-moment release sequence, Bull. Seismol. Soc. Am., 95(3), 833–843,doi:10.1785/0120040110.Daub, E. G., E. Ben-Naim, R. A. Guyer, and P. A. Johnson (2012),Are megaquakes clustered?, Geophys. Res. Lett., 39, doi:10.1029/2012gl051465.Ekstrom, G., M. Nettles, and A. M. Dziewonski (2012), The global CMTproject 2004–2010: Centroid-moment tensors for 13,017 earthquakes,Phys. Earth Planet. Inter., 200, 1–9, doi:10.1016/j.pepi.2012.04.002.Gutenberg, B., and C. F. Richter (1949), Seismicity of the Earthand Associated Phenomena, Princeton University Press, Princeton,New Jersey.Kagan, Y. Y. (1997), Seismic moment-frequency relation for shallow earth-quakes: Regional comparison, J Geophys Res B Solid Earth Planets,102(B2), 2835–2852, doi:10.1029/96jb03386.Kagan, Y. Y., and D. D. Jackson (2000), Probabilistic forecastingof earthquakes, Geophys. J. Int., 143(2), 438–453, doi:10.1046/j.1365-246X.2000.01267.X.Lay, T., et al. (2005), The great Sumatra-Andaman earthquake of 26December 2004, Science, 308(5725), 1127–1133, doi:10.1126/science.1112250.Figure 4. (a) The difference in BIC (MGR-GR) as a function of event number for the global CMT catalog (heavy blackline) and 100 simulated catalogs with GR distribution, beta-value for each sequence is chosen on the basis of its likelihoodin the real data set. The color code for the three snapshot times (vertical lines) are as in Figure 1; (b) Frequency distributionof ΔBIC values from 10000 GR synthetic catalogs at (from left to right): 30 June 1999; 31 December 2006; and 31 July2012. 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Touati (2009), Quantifying uncertainty inmean earthquake interevent times for a ﬁnite sample, J Geophys Res BSolid Earth Planets, 114, doi:10.1029/2008jb005870.Shearer, P. M., and P. B. Stark (2012), Global risk of big earthquakes hasnot recently increased, Proc. Natl. Acad. Sci. U.S.A., 109(3), 717–721,doi:10.1073/pnas.1118525109.Simons, M., et al (2011), The 2011 magnitude 9.0 Tohoku-Oki Earthquake:Mosaicking the megathrust from seconds to centuries, Science,332(6036), 1421–1425, doi:10.1126/science.1206731.Stein, S., R. J. Geller, and M. A. Liu (2012), Why earthquake hazard mapsoften fail and what to do about it, Tectonophysics, 562, 1–25.doi:10.1016/j.tecto.2012.06.047.Turcotte, D. (1997), Fractals and Chaos in Geology and Geophysics,Cambridge University Press, Cambridge, UK.BELL ET AL.: EARTHQUAKE FREQUENCY-SIZE DISTRIBUTION2589

Convergence of the frequency-size distribution of global earthquakes

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