# Tohoku Earthquake: A Surprise?

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<ul><li><p>Tohoku Earthquake: A Surprise?</p><p>by Yan Y. Kagan and David D. Jackson</p><p>Abstract We consider three questions related to the 2011 Tohoku mega-earthquake:(1) Why was the event size so grossly underestimated by Japans 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? Themaximum earthquake size is often guessed from the available history of earthquakes, amethod known for its significant downward bias. We show that historical magnitudessystematically underestimate this maximum size of future events, but the discrepancyshrinks with time. There are two quantitative methods for estimating the corner mag-nitude in any region: a statistical analysis of the available earthquake record and themoment conservation principle. However, for individual zones the statistical method isusually ineffective in estimating the maximum magnitude; only the lower limit can beevaluated. The moment conservation technique, which we prefer, matches the tectonicdeformation rate to that predicted by earthquakes with a truncated or tapered magni-tudefrequency distribution. For subduction zones, the seismic or historical record isinsufficient to constrain either the maximum or corner magnitude. However, the mo-ment conservation principle yields consistent estimates: for all the subduction zonesthe corner magnitude is of the order 9.09.7. Moreover, moment conservation indi-cates that variations in estimated corner magnitude among subduction zones are notstatistically significant. Another moment conservation method, applied at a point ona major fault or plate boundary, also suggests that magnitude 9 events are requiredto explain observed displacement rates at least for the Tohoku area. The global rate ofmagnitude 9 earthquakes in subduction zones, predicted from statistical analysis ofseismicity as well as from moment conservation, is about five per century; fiveactually happened.</p><p>Introduction</p><p>The magnitude 9.1 Tohoku, Japan, earthquake of11 March 2011 and the ensuing tsunami near the east coastof Honshu caused nearly 20,000 deaths and more than300 billion dollars in damage, ranking as one of the worstnatural disasters ever recorded (Geller, 2011; Hayes et al.,2011; Simons et al., 2011; Stein et al., 2011). The great dif-ference between the expected and observed earthquake mag-nitudes contributed to this enormous damage. The estimatedmaximum magnitude for the Tohoku area (around 7.7) wasproposed in the official hazard map (Headquarters for Earth-quake Research Promotion, 2005; Simons et al., 2011).</p><p>Several other estimates of the maximum size earthquakein the Tohoku area had been published before 2011. Ruff andKanamori (1980, fig. A1) suggested, basing their analysison historical data, age of subducting plate, and plate-ratesystematics, that the maximum magnitude is 8.28.35. Wes-nousky et al. (1984; their table 1, fig. 8) applied the charac-teristic earthquake hypothesis to estimate the maximumearthquake size and expected occurrence time for theseevents in three zones approximately covering the Tohoku</p><p>earthquake rupture area. Nishenko (1991, fig. 23 andpp. 234235) defines the magnitude of characteristic earth-quakes in northeast Japan, zones J4J6 as 7.17.7. However,he does mention anm 8:1 earthquake in 1611. Minoura et al.(2001) suggest that the 869 Jogan earthquake occurred on afault 240 85 km and was m 8:3. A similar estimate of theJogan earthquake magnitude is proposed by Sugawara et al.(2012). On the basis of the historical and instrumental cata-log analysis, Grunewald and Stein (2006) suggest the maxi-mum magnitude m 8:4 for the Tokyo area. Koravos et al.(2006) provide an estimate 7 < mmax < 8, based on histori-cal earthquakes since 599 A.D., again mentioning a fewearthquakes with magnitude slightly larger. Annaka et al.(2007) suggest mmax 8:5 based on historical earthquakessince 1611. Rikitake and Aida (1988; their table 1, fig. 5)did not expect the Tohoku area tsunami exceeding 7 metersand defined the maximum magnitude for zones IIIV to bewithin 7.47.9. The Evaluation of Major Subduction-zoneEarthquake(s) 2008 PDF file at the website (see Data andResources) defines probabilities of major earthquakes in</p><p>1181</p><p>Bulletin of the Seismological Society of America, Vol. 103, No. 2B, pp. 11811194, May 2013, doi: 10.1785/0120120110</p></li><li><p>subduction zones. For the northeast part of Honshu Island,the maximum magnitudes are in the range of 6.88.2. Nanjoet al. (2011, p. 159) suggest that one should expect magni-tudes up to about 8 or larger for [Japan] offshore events.These opinions of the Japanese and international researchersconfirm that the maximum earthquake size in the Tohokuarea was dramatically underestimated before 11 March 2011.</p><p>Several quantitative estimates of the maximum possibleearthquakes in the subduction zones had been published be-fore the Tohoku event (Kagan, 1997; Kagan and Jackson,2000; Bird and Kagan, 2004; McCaffrey, 2007, 2008; Kaganet al., 2010). In these publications, the upper magnitudeparameter was determined to be within a wide rangefrom 8.5 to 9.6.</p><p>Two quantitative methods have been deployed to esti-mate the upper magnitude limit: a statistical determinationof the magnitudemoment-frequency parameters from earth-quake data alone, and a moment conservation principle inwhich the total moment rate is estimated from geologic orgeodetic deformation rates (Kagan, 1997).</p><p>The distinction between maximum and corner magnitudesrefers to different approaches in modeling the magnitudefrequency relationship for large earthquakes. In one approach,the classical GutenbergRichter magnitude distribution ismodified by truncating it at an upper limit called the maxi-mummagnitude. In another approach, an exponential taper isapplied to the moment distribution derived from the classicalmagnitude distribution. For the tapered GutenbergRichter(TGR) distribution, the corner moment is the value at whichthe modeled cumulative rate is reduced to 1=e of the classicalrate, and the corner magnitude is that which correspondsto the corner moment. In either approach, the rate of earth-quakes of any magnitude and the total seismic moment ratecan be computed from three observable parameters: the rateof earthquakes at the lower magnitude threshold, the b-valueor asymptotic slope of the magnitude distribution, and themaximum or corner magnitude. Conversely, the maximumor corner magnitude may be determined if the threshold rate,b-value, and total moment rate are known.</p><p>We generally shun the use of maximum magnitude be-cause there is no scientific evidence that it really represents amaximum possible magnitude, although it is frequently in-terpreted that way. Where the two approaches lead to similarconclusions, we will generalize the discussion by referring tothe upper magnitude parameter.</p><p>In the following text we use the term mmax to signifythe upper limit of the magnitude variable in a likelihoodmap or the limit of integration in an equation to compute atheoretical tectonic moment arising from earthquakes. As wewill see next, in different approximations of earthquake sizedistribution, mmax may have various forms, though theirestimated numerical values are usually close. We do not treatmmax as a firm limit, but we use it as a convenient generalreference because many other researchers do so.</p><p>The statistical estimate of the maximum magnitude forglobal earthquakes, including subduction zones and other</p><p>tectonic regions, yielded the values mmax 8:3 (Kagan andJackson, 2000). The moment conservation provided an esti-mate for subduction zones mmax 8:5 8:7 0:3 (Kagan,1997, 2002b). Moreover, the maximum earthquake size wasshown to be indistinguishable, at least statistically, for all thesubduction zones studied. Applying a combined statisticalestimate and moment conservation principle, Bird and Kagan(2004) estimated the corner magnitude to be about 9.6. Aswe explain next, the difference between the previously pre-sented estimates (8.6 versus 9.6) is caused mainly by variousassumptions about the tectonic motion parameters. Thesemmax determinations, combined with the observation of verylarge (m 9:0) events in the other subduction zones (McCaf-frey, 2007, 2008; Stein and Okal, 2007, 2011), should havewarned of such a possible earthquake in any major subduc-tion zone, including the Sumatra and Tohoku areas.</p><p>In the Evaluation of Earthquake Size Distribution sectionwe consider two statistical distributions for the earthquakescalar seismic moment and statistical methods for evaluatingtheir parameters. The Seismic Moment Conservation Princi-ple section discusses the seismic moment conservation prin-ciple and its implementation for determining the uppermagnitude parameter. We then demonstrate how these tech-niques for size evaluation work in subduction zones, showingthat m 9:09:7 earthquakes can be expected in all majorzones, including the Tohoku area. For the Tohoku area, theapproximate recurrence interval for m 9:0 earthquakes ison the order of 350 years (Kagan and Jackson, 2012; seethe Discussion section in this paper). By the term recurrenceinterval we do not imply that large earthquakes occur cycli-cally or quasi-periodically; contrary to that, we presentedevidence that all earthquakes, including the large ones, areclustered in time and space (Kagan and Jackson, 1999; seethe Discussion section in this paper).</p><p>Evaluation of Earthquake Size Distributionfor Subduction Zones</p><p>Earthquake Catalogs</p><p>We studied earthquake distributions and clustering forthe global CMT catalog of moment tensor inversions com-piled by the GCMT group (Ekstrm et al., 2005; Ekstrm,2007; Nettles et al., 2011). The present catalog containsmore than 33,000 earthquake entries for the period 1 January1977 to 31 December 2010. The event size is characterizedby a scalar seismic moment M.</p><p>We also analyzed the 19001999 Centennial catalog byEngdahl and Villaseor (2002). The catalog is completedown to magnitude 6.5 (MS=mB or their equivalent) duringthe period 19001963 and to 5.5 from 19641999. Up toeight different magnitudes for each earthquake are listed inthe catalog. We use the maximum of the available magni-tudes in the Centennial catalog as a substitute for the momentmagnitude and construct a moment-frequency histogram.</p><p>1182 Y. Y. Kagan and D. D. Jackson</p></li><li><p>There are 1623 shallow earthquakes in the catalog withm 7:0; of these 30 events have m 8:5.</p><p>Seismic Moment/Magnitude Statistical Distributions</p><p>In analyzing earthquakes here, we use the scalar seismicmoment M directly, but for easy comparison and display, weconvert it into an approximate moment magnitude using therelationship (Hanks, 1992)</p><p>mw 2</p><p>3log10 M C; (1)</p><p>where C 9:0, if moment M is measured in newton meters(Nm), and C 16:0 for momentM expressed in dyncm asin the GCMT catalog. Equation (1) provides a unique map-ping from magnitude to moment, so where appropriate wewill use both of the same subscripts. Thus, mmax implies acorresponding Mmax, etc.</p><p>Because we are using the moment magnitude almostexclusively, later we omit the subscript in mw. Unless spe-cifically indicated, we use for consistency the moment mag-nitude calculated as in equation (1) with the scalar seismicmoment from the GCMT catalog.</p><p>In this work we consider two statistical distributions ofthe scalar seismic moment: (1) the truncated GutenbergRichter (GR) or equivalently truncated Pareto distribution, inwhich the upper magnitude parameter is the maximum mag-nitude, and (2) the gamma distribution (Kagan, 2002a,b),in which the upper magnitude parameter is the cornermagnitude mcg.</p><p>For the truncated Pareto distribution, the probabilitydensity function (PDF) is</p><p>M MxpM</p><p>t</p><p>Mxp MtM1 for Mt M Mxp: (2)</p><p>Here Mxp is the upper truncation parameter, Mt is the lowermoment threshold (the smallest moment above which thecatalog can be considered to be complete), and is the indexparameter of the distribution. Note that 2</p><p>3b, where b is</p><p>the familiar b-value of the GutenbergRichter distribution(Gutenberg and Richter, 1954, pp. 1625).</p><p>The gamma distribution has the PDF</p><p>M C1 M</p><p>Mt=M expMt M=Mcg;for Mt M < ; (3)</p><p>where Mcg is the corner moment parameter controlling thedistribution in the upper ranges of M (the corner moment),and C is a normalizing coefficient. Specifically,</p><p>C 1 Mt=Mcg expMt=Mcg1 ;Mt=Mcg; (4)</p><p>where is the incomplete gamma function (Bateman andErdelyi, 1953). ForMcg Mt, the coefficient C 1. Below</p><p>we simplify the notationMxp andMcg asMx andMc, respec-tively, keeping the notation Mmax to represent either Mxpor Mcg as appropriate. Thus, each distribution is controlledby two parameters: its slope for small and moderate earth-quakes, , and its maximum or corner moment Mx or Mcdescribing the behavior of the largest earthquakes. Equa-tions (3) and (4) are normalized distributions. Both need amultiplicative constant, the threshold earthquake rate, to cal-culate rates at any magnitude above the threshold.</p><p>In Figure 1 we show the magnitudefrequency curvesfor shallow earthquakes (depth less or equal to 70 km) inthe Flinn-Engdahl (Flinn et al., 1974; Young et al., 1996)JapanKurileKamchatka zone 19 for the period 19772010 (i.e., before the Tohoku event). Similar curves for thetime period before and after 1 January 2011 for the rupturezone of the Tohoku earthquake are shown in our companionpaper (Kagan and Jackson, 2012, figs. 1, 2). Knowing the dis-tribution of the seismic moment, one can calculate occurrencerates for earthquakes of any size, so we need a reliable stat-istical technique to determine the parameters of a distribution.</p><p>Likelihood Evaluation of Distribution Parameters</p><p>We applied the likelihood method to obtain estimatesof and mc (Kagan, 1997, 2002a). Figure 2 displays the</p><p>Figure 1. Solid line, the number of earthquakes in the Flinn-Engdahl (FE) zone 19 (JapanKurileKamchatka) with the momentmagnitude (m) larger than or equal to m as a function of m for theshallow earthquakes in the 19772010 GCMT catalog. Magnitudethreshold mt 5:8; the total number of events is 425. The unre-stricted GutenbergRichter law is shown by a dotted line (Kagan,2002a). Dashed lines show two tapered GR (TGR) distributions:the GR law restricted at large magnitudes by an exponential taperwith a corner magnitude. Left line is for the corner magnitudemsc 8:7 evaluated by the maximum likelihood method using the earth-quake statistical record (with no upper limit, see Fig. 2). Right line isfor the corner moment estimate mmc 9:4 based on the momentconservation (see Table 1). The slope of the linear part of the curvescorresponds to 0:610. The color version of this figure is avail-able only in the electronic edition.</p><p>Tohoku Earthquake: A Surprise? 1183</p></li><li><p>map of the log-likelihood function for two parameters of thegamma distribution. The -value (around 0:61 0:038)can be determined from the plot with sufficient accuracy(Fig. 2), but the corner magnitude evaluation encounters seri-ous difficulties. The upper cont...</p></li></ul>

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