PostScript file created: December 21, 2011; time 1069 minutes TOHOKU EARTHQUAKE: A SURPRISE? Yan Y. Kagan and David D. Jackson

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1 PostScript file created: December 21, 2011; time 1069 minutes TOHOKU EARTHQUAKE: A SURPRISE? Yan Y. Kagan and David D. Jackson Department of Earth and Space Sciences, University of California, Los Angeles, California , USA; s: ykagan@ucla.edu, david.d.jackson@ucla.edu Abstract. We consider three issues related to the 2011 Tohoku mega-earthquake: (1) how to evaluate the earthquake maximum size in subduction zones and why the event size was so grossly under-estimated for the Tohoku-Oki area, (2) what is the repeat time for the largest earthquakes in this area, and (3) what are the possibilities of numerical short-term forecasts during the 2011 earthquake sequence in the Tohoku area. The maximum earthquake size is usually guessed basing on the available history of earthquakes, the method known for its significant downward bias. We make an estimate of this bias: historical magnitudes underestimate the maximum/corner magnitude but discrepancy shrinks with time. There are two quantitative methods which can be applied to estimate the maximum earthquake size in any region: a statistical analysis of the available earthquake record and the moment conservation principle. The latter technique studies how much of the tectonic deformation rate is released by earthquakes. Both of these methods have been developed by the authors since For the subduction zones, the seismic or historical record is not sufficient to provide a reliable statistical measure of the maximum earthquake. However, the moment conservation principle yields consistent estimates: for all the subduction zones the maximum moment magnitude is of the order , this is the value suggested by various measurements. Moreover, the latter method indicates that for all major subduction zones the maximum earthquake size 1

2 is statistically indistinguishable. Another moment conservation method comparing the site-specific deformation rate and its release by earthquakes rupturing the site, also suggests that the maximum earthquake size should be of the order m9. Since 1977 we have developed statistical short- and long-term earthquake forecasts to predict the earthquake rate per area, time, and magnitude unit. For worldwide seismicity as well as for several seismically active regions these forecasts are posted on our web sites. We have carried out long- and short-term forecasts for Japan and the surrounding areas using the GCMT catalog starting in For forecasts based on the GCMT catalog, the expected earthquake focal mechanisms are also evaluated. Long-term forecasts indicate that the repeat time for the m9 earthquake in the Tohoku area is of the order of 350 years: this estimate is confirmed by the seismicity levels recorded by the GCMT catalog. We have archived several forecasts made before and after the Tohoku earthquake, they are displayed in diagrams and tables in this paper. The long-term rate estimates indicate that, as expected, the forecasted rate changed only by a few percent after the Tohoku earthquake, whereas due to the foreshocks, the short-term rate increased by a factor of more than 100 before the mainshock event as compared to the long-term rate. After the Tohoku mega-earthquake the rate increased by a factor of more than These results suggest that an operational earthquake forecasting strategy needs to be developed to take the increase of the short-term rates into account. Short running title: Tohoku earthquake prediction Key words: Probabilistic forecasting; Probability distributions; Earthquake interaction, forecasting, and prediction; Seismicity and tectonics; Statistical seismology; Dynamics: seismotectonics; Subduction zones; Maximum/corner magnitude. 2

3 1 Introduction The 11 March 2011 Tohoku, Japan, magnitude 9.1 earthquake and the ensuing tsunami near the east coast of the island of Honshu caused nearly 20,000 deaths and more than 300 billion dollars in damage, resulting in the worst natural disaster ever recorded (Hayes et al., 2011; Simons et al., 2011; Geller, 2011; Stein et al., 2011). The major issue in the enormous damage was a great difference between the expected and the observed earthquake magnitudes. The maximum magnitude size for Tohoku area (around 7.7) was proposed in the official hazard map (Geller, 2011; Stein et al., 2011; Simons et al., 2011). Nanjo et al. (2011) suggest that one should expect magnitudes up to about 8 or larger for [Japan] offshore events. Several quantitative estimates of maximum possible earthquakes in subduction zones had been published before the Tohoku event (Kagan, 1997a; Kagan and Jackson, 1994; Bird and Kagan, 2004; Kagan et al., 2010). In these publications the maximum size of these earthquakes was determined to be within a range m8.5 m9.6. Two quantitative methods have been deployed to estimate the maximum size of an earthquake: a statistical determination of the magnitude moment/frequency parameters and a moment conservation principle. The former technique employs standard statistical parameter estimation to evaluate two parameters of the earthquake size distribution: the b-value and the maximum magnitude (Kagan, 2002a; 2002b). The second method works by comparing the estimates of tectonic deformation at plate boundaries with a similar estimate of the seismic moment release (Kagan, 1997a). Below we use the term m max to signify the upper limit of the magnitude variable in a likelihood map or the limit of integration in an equation to compute a theoretical tectonic moment arising from earthquakes. As we will see below, in different approximations of earthquake size distribution m max may have various forms, though usually their estimated 3

4 numerical values are close. We do not treat m max as a firm limit, but we use it for a convenient general reference because many other researchers use it this way. The statistical estimate of the maximum magnitude for global earthquakes, including subduction zones and other tectonic regions, yielded the values m max 8.3 (Kagan and Jackson, 1994; 2000). The moment conservation provided an estimate for subduction zones m max = ± 0.3 (Kagan, 1997a; 1999; 2002b). Moreover, the maximum earthquake size was shown to be the same, at least statistically, for all the studied subduction zones. Applying a combination of the statistical estimate and the moment conservation principle, Bird and Kagan (2004) obtained the estimate m max = As we explain below, the difference between the above estimates (8.6 vs 9.6) is caused mainly by various assumptions about the tectonic motion parameters. These m max determinations, combined with the observation of very large (m 9.0) events in the other subduction zones (McCaffrey, 2008), could have served as a warning of a possibility of such an earthquake in any major subduction zone including the Sumatra and the Tohoku area. In Section 2 below we consider two statistical distributions for the earthquake scalar seismic moment and statistical methods for evaluating their parameters. Section 3 discusses the seismic moment conservation principle and its implementation for determining the maximum earthquake size. We then demonstrate an application of these techniques for size evaluation in subduction zones, showing that m9.0 m9.6 earthquakes can be expected in all major zones, including the Tohoku area. For the Tohoku area the approximate recurrence interval for m 9.0 earthquake is of the order of 350 years (Section 4.1). Sections 4.1 and 4.2 show the long- and short-term earthquake forecasts during the Tohoku sequence. Section 4.2 illustrates the possibility of an operational type calculation of short-term earthquake rates for short intervals (a few days) after a major earthquake exceeding the long-term rates by times. 4

5 2 Evaluation of earthquake size distribution for subduction zones 2.1 Earthquake catalogs We studied earthquake distributions and clustering for the global CMT catalog of moment tensor inversions compiled by the GCMT group (Ekström et al., 2005; Ekström, 2007; Nettles et al., 2011). The present catalog contains more than 33,000 earthquake entries for the period 1977/1/1 to 2010/12/31. The earthquake size is characterized by a scalar seismic moment M. We also analyse the Centennial ( ) catalog by Engdahl & Villaseñor (2002). The catalog is complete to magnitude 6.5 (M S /m B or their equivalent) during the period and 5.5 (M S /m B or their equivalent) during the period. Up to 8 different magnitudes for each earthquake are listed in the catalog. We use the maximum of the available magnitudes in the Centennial catalog as a substitute for the moment magnitude and construct a moment-frequency histogram. There are 1623 shallow earthquakes in the catalog with m 7.0; of these 30 events have m Seismic moment/magnitude statistical distributions In analyzing earthquakes here we use the scalar seismic moment M directly, but for easy comparison and display we convert it into an approximate moment magnitude using the relationship (Hanks, 1992) m W = 2 3 ( log 10 M C ), (1) where C = 9.0, if moment M is measured in Newton m (Nm), and C = 16.0 for moment M expressed in dyne-cm as in the GCMT catalog. Since we are using the moment magnitude 5

6 almost exclusively, later we omit the subscript in m W. Unless specifically indicated, we use the moment magnitude calculated as in (1) with the scalar seismic moment from the GCMT catalog. In this work we consider two statistical distributions of the scalar seismic moment: (a) the truncated Gutenberg-Richter (G-R) or equivalently truncated Pareto distribution, and (b) the gamma distribution (Kagan, 2002a; 2002b). For the truncated Pareto distribution the probability density (pdf) is φ(m) = M β xpm β t M β xp M β t βm 1 β for M t M M xp. (2) Here M xp is the upper truncation parameter, M t is the moment threshold: the smallest moment above which the catalogue can be considered to be complete, β is the index parameter of the distribution; β = 2 b, b is a familiar b-value of the Gutenberg-Richter distribution 3 (G-R, Gutenberg and Richter, 1954, pp ). The gamma distribution has the pdf φ(m) = C 1 β M (M t/m) β exp[(m t M)/M cg ], for M t M <, (3) where M cg is a corner moment parameter controlling the distribution in the upper ranges of M ( the corner moment ) and C is a normalizing coefficient. C = 1 (M t /M cg ) β exp(m t /M cg ) Γ(1 β, M t /M cg ), (4) where Γ is a gamma function (Bateman & Erdelyi 1953). For M cg >> M t the coefficient C 1. Later we would usually simplify the notation M px and M cg as M x and M c, respectively, we keep notation M max for a general purpose. Thus, both distributions are controlled by two parameters: its slope for small and moderate earthquakes, β, and the maximum/corner moment M x or M c, that describes the distributions behavior at their tails. 6

7 In Fig. 1 we show the moment-frequency curves for shallow earthquakes (the depth is less or equal to 70 km) in the Japan-Kurile-Kamchatka (#19) Flinn-Engdahl (Flinn et al., 1974; Young et al., 1996) zone. Knowing the distribution of the seismic moment, one can calculate occurrence rates for earthquakes of any size, so we need a reliable statistical technique to determine the parameters of a distribution. 2.3 Likelihood evaluation of distribution parameters We applied the likelihood method to obtain estimates of β and m c (Kagan, 1997a). Fig. 2 displays the map of the log-likelihood function for two parameters of the gamma distribution. The β-value (around 0.61 ± 0.038) can be determined from the plot with sufficient accuracy (Fig. 2), but the corner magnitude evaluation encounters serious difficulties. The upper contour of the 95% confidence area in the likelihood map is not well-constrained and extends to infinity. This means that only the lower limit for m c, around 8.2 can be reliably evaluated with the available data, hence the maximum likelihood estimate (m c = 8.7) is not wellconstrained by the map. However, even the lower limit for m c is significantly higher than the maximum magnitude size (around 7.7) proposed by the official hazard map for the Tohoku area (Geller, 2011; Stein et al., 2011), thus a simple statistical test could have indicated a possible size of a major earthquake. Fig. 2 is similar to the likelihood map constructed by Bird and Kagan (2004, Fig. 7B, see also Table 5) for all subduction zones taken together. The calculation in this plot has been made with the tapered G-R distribution (TGR), for β = 0.64 the m c -values are 0.39 lower (Kagan, 2002b; see also Kagan and Jackson, 2000, Fig. 2 and its discussion) than those for the gamma distribution (2). From Bird and Kagan s map the lower limit for m max or m cm is about 9.0, but even the complete 20-th century earthquake record is still insufficient to obtain the upper statistical limit. Since, as Kagan (1997a) has shown that all 7

8 subduction zones should have the same maximum magnitude distribution, this result implies that m cg = 9.4 is the minimum estimate for the maximum magnitude in the major zones. Geller (2011) and Stein et al. (2011) suggest that a major reason for a gross underestimating of the maximum magnitude for Japanese earthquakes is that leading seismologists accept the flawed characteristic magnitude/seismic gap model (see, for example, Annaka et al., 2007). This model suggests that a fault length can be subdivided into segments and a maximum allowable event on such a segment is limited either by its length or by the available historic/instrumental record. Jackson and Kagan (2011) summarize theoretical and observational arguments against the characteristic magnitude/seismic gap model. Simons et al. (2011) also present some evidence contradicting the hypothesis. 3 Seismic moment conservation principle We try to estimate the upper bound of the seismic moment-frequency relation, using the moment conservation principle as another, more effective method for determining the maximum/corner magnitude. Quantitative plate tectonics and space geodetic methods currently provide a numerical estimate of the tectonic deformation rate for all major tectonic plate boundaries as well as the continental regions of significant distributed deformation (Bird and Kagan, 2004; Kagan et al., 2010). We compare these estimates with a similar estimate of the seismic moment release. The seismic moment rate depends on three variables (see Eq. 6 below) 1. The number of earthquakes in a region (N), 2. The β-value (or b-value) of the G-R relation, 3. The value of the maximum (corner) magnitude m c. The tectonic moment rate Ṁ T depends on the following three variables which are not 8

9 well-known 1. The width of seismogenic zone (W km), 2. The seismic efficiency (coupling) coefficient (χ %), 3. The value of the shear modulus (µ 30-49GPa). Ṁ T = χ µ W L u, (5) where u is the average slip, L is the length of a fault (compare Eq. 13 by Kagan, 2002b). 3.1 Area-specific conservation principle The discussion in this Subsection is based on our previous papers (Kagan, 2002a; 2002b). In those papers we consider two theoretical moment-frequency models: (a) a truncated Pareto distribution; and (b) the gamma distribution. Ṁ s = α 0 M β 0 β 1 β M 1 β x ξ p = α 0 M β 0 β 1 β M 1 β c Γ(2 β) ξ g, (6) where Ṁs is the seismic moment rate and α 0 is the rate of occurrence for earthquakes with moment larger or equal M 0 ; α 0 = N/ T with T as the catalog duration. Coefficient ξ is a correction factor needed if the distribution is left-truncated close to the maximum or the corner moment; under usual circumstances it equals 1.0. We assume that the two theoretical laws (6) describe a distribution with the same moment rate Ṁ s release and the seismic rate of occurrence α 0. Using (6), relations between the maximum or corner moments can be specified M xp β 1/(1 β) = M cg [β Γ(2 β)] 1/(1 β). (7) The gamma function Γ(2 β) changes slowly in the range of β-values encountered in the moment-frequency relations: for values of β in the interval 1/2 to 2/3, the gamma function is 9

10 Γ(2 1/2) = or Γ(2 2/3) = Therefore, the difference between the values of the maximum/corner magnitudes is relatively small: for southern California our calculations (Kagan, 2002b, Fig. 2) yield the magnitude values m x = 8.35, and m c = 8.45 for two distributions shown in (Eqs. 2 and 3). Fig. 3 shows the β-values determined for 18 Flinn-Engdahl zones (Gutenberg and Richter, 1954, Fig. 1; Flinn et al., 1974; Young et al., 1996) corresponding to major subduction zones. These zones have been selected because the regions had been defined before the GCMT catalog started, thus eliminating selection bias. It is also easier to replicate our results (the programs and tables for the Flinn-Engdahl zones are publicly available, see Section Data and Resources ). In this plot we use the GCMT catalog at the same temporal interval as in our previous paper (Kagan, 1997a). In Fig. 4 the catalog duration is extended to the end of Both plots demonstrate that (a) the β-values do not depend significantly on the catalog duration, though their standard errors decrease with the duration and earthquake numbers increase; (b) the β-values are approximately the same for all the zones, and the hypothesis of the values equality cannot be statistically rejected (Kagan, 1997a). Figs. 5, 6, and 7 show the distribution of the corner magnitude obtained using Eq. 6 for the earthquake catalogs of different time duration. The estimates of m c for all the diagrams in all subduction zones are statistically indistinguishable. This means that all subduction zones can be expected to have the same maximum/corner magnitude. The estimate of m c depends on the parameter value used to calculate the tectonic moment rate (Eq. 6): for β = 2/3, the change of any parameters by a factor of two implies an increase or decrease of the m c by about 0.6 (Kagan, 2002b, Eq. 17). We see this influence by comparing Table 1 with the results for the subduction zones in a similar Table 1 in Kagan (1997a), where the parameters used for calculation of tectonic rate were: W = 30 km, µ = 30 GPa, χ = 1.0. The difference in the m c estimates in the two tables is chiefly 10

11 caused by the change in the above parameters. We also compare the m c -values for the same zones in Figs. 5, 6, and 7. The values in Figs. 6 and 7 differ significantly, by 0.5 and more in five zones: Kermadec, Fiji, Japan- Ryukyu, Sunda, and Andaman (FE12, 13, 20, 24, 46). This is not surprizing as the magnitudes in the Centennial catalog were determined with large random and systematic errors. In addition, because of the high magnitude threshold, the earthquake numbers have significant random fluctuations. When matching up Figs. 5 and 6, only one zone, Andaman-Sumatra, shows the m c difference of about 0.9. This is due to the 2004 Sumatra earthquake and its aftershocks which significantly increased the total number of events to 143 in 34 years (Table 1) vs 22 earthquakes in 19.5 years in Table 1 by Kagan (1997a). The annual rate increase is by a factor 3.15, that for β = 2/3 corresponds to m c decrease of 1.0; in our m c calculations (6) we use the β-values determined for each zone. As we mentioned, the m c -values should be approximately the same for all the subduction zones considered. Thus, the hypothesis that m c in all subduction zones is verified by comparing the theoretical estimates with the actually measured magnitude values in several subduction zones. For example, a m9.0 earthquake occurred in zone #19 (Kamchatka, Russia) on November 4, 1952, confirming that this subduction zone could experience large earthquakes in excess of m9. We also calculate parameter values for the Tohoku area (latitudes N, longitudes E, the length of subduction trench zone is around 620 km in this spherical rectangle). The value of the tectonic moment accumulation rate attributed to the subduction of the Pacific plate is compatible with the estimate proposed by Ozawa et al. (2011) in the Japan trench area from latitude 36 N to 39.5 N ( Nm/y). Ozawa et al. (2011) also suggest using µ = 40 GPa for Japan trench. Whereas the values of the parameters β and m c are approximately the same for the 11

12 Tohoku area and Flinn-Engdahl zone #19, the other entries for these rows differ significantly. This means that the maximum observed earthquakes or the ratio of the seismic rate to tectonic rates are subject to great variation and cannot reliably be used to characterize the area seismicity. During about 110 years of the instrumental seismic record, five zones have experienced earthquakes with the magnitude equal or in excess of m9. Figs. 5 7 also show that with the catalog duration increase the average maximum measured magnitude approaches the average estimate of m c, suggesting that if the available earthquake record duration were comparable to the recurrence time of the largest earthquake (a few hundred years), the difference between the observed maximum magnitude (m o ) and m c would largely disappear. Using the parameter values for the moment-frequency distribution determined by Bird and Kagan (2004, Table 5) for all the subduction zones (b = 0.96, M t = Nm, α t = eq/y, m c = 9.58), we calculate the number of m 9 events expected to occur worldwide during a century N(m > 9) = α t ( ) = 5.2. (8) This number agrees well with the five large (m 9.0) earthquakes registered during the last 100 years (see Figs. 5 7). The evaluations of the distribution parameters (Bird and Kagan, 2004) have been made before the occurrence of two recent giant earthquakes, thus the almost perfect correspondence can be considered as coincidence. Fig. 8 demonstrates again the influence of the catalog duration on the ratio of the seismic rate to the tectonic rate for different subduction zones. The ratio is below one for a shorter catalog, whereas it increases to a value close to 1.0 for a longer list. This increase is caused mainly by a few large earthquakes that occurred in South America and Sumatra regions. As Zaliapin et al. (2005) show, a sum of the scalar earthquake moments behaves highly 12

13 irregularly due to its power-law distribution. 3.2 Geometric self-similarity of earthquake rupture Fig. 9 displays an update of Fig. 6a by Kagan (2002c, see also similar diagrams in Kagan, 2011). Two recent mega-earthquakes are included in the plot, the 2004 Sumatra and the Tohoku events the right-hand symbols in the diagram. Despite the differences in the aftershock zone length, that has been emphasized to in many publications, the symbols for these earthquakes are not far away vertically; the difference is smaller than a scatter for moderate earthquakes. The seeming contradiction of their size evaluation is caused by various techniques employed in measuring the rupture size. In our work we use the same measurement method for all earthquakes, namely, a fit of the aftershock spatial scatter by a two-dimensional Gaussian distribution (Kagan, 2002c). For both events we obtain the aftershock zone size (2 σ confidence area length) 905 and 533 km, respectively. Comparison of the regression results from Fig. 9 with those of Fig. 6a by Kagan (2002c) demonstrates stability of the scaling parameters and their evaluation robustness. The earthquake numbers increase almost by a factor of two since year Moreover, in Fig. 9 there are three major (m 8.8) events, whereas the largest earthquake in Fig. 6a is m8.4. However, the values of regression coefficients in both datasets are essentially the same. In Fig. 9 two regression curves approximate the aftershock zone length versus the magnitude: the linear and the quadratic. There is practically no difference between these fits. No observable scaling break or saturation occurs for the largest earthquakes, thus the earthquake geometrical focal zones are self-similar. Assuming self-similarity, we adopted the following scaling for the average length (L), the average downdip width (W ), and the average slip (U) as a function of the moment (M): the length (L) is proportional to the cube root of the moment: L 3 M, implying a self-similarity of the earthquake rupture pattern, i.e., W and 13

14 U are also proportional to 3 M. The above results imply that the earthquake slip penetrates below the seismogenic layer into an underlying zone and that a deep slip occurs during large earthquakes. Shaw and Wesnousky (2008) note that a significant coseismic slip occurs below seismogenic layer. If the downdip seismogenic width W changes for the largest earthquakes, it may influence the calculation of tectonic rate ṀT (see Eq. 5). 3.3 Site-specific conservation principle Many attempts have been made to compare the slip budget at the subduction zones with its release by earthquakes. McCaffrey (2008) compared the slip values at the global subduction zones with their release by earthquakes spanning the whole length of a zone. He found that practically for all zones m9 and greater earthquakes are possible with the recurrence times of the order of a few centuries. In particular, for Japan McCaffrey (2008) calculates the maximum moment M max as M max = µ u L Z max / sin δ, (9) where u is the average slip, Z max is the maximum depth of the slip (40 km used), and δ is the average fault dip angle (taken to be 22 for Japan), implying W = km. The recurrence time for the maximum earthquake is T = u / f χ ν, (10) where ν is the plate motion rate, f is the fraction of the total seismic moment in m9 earthquakes, and u = L. The parameter f is taken to be equal to 1 β (apparently using results for the characteristic earthquake distribution by Kagan, 2002a; 2002b). By taking L = 654 km, β = 0.57, ν = mm/y, χ = 1, McCaffrey obtains M max = Nm, i.e., m max = 9.0, and T 532 y. 14

15 Simons et al. (2011) considered how the slip in the Tohoku area of the order ν = cm/y can be accommodated by subduction earthquakes, and proposed that only the occurrence of very large events similar to the Tohoku m9.1 earthquake can explain such a displacement rate. However, it is only the earthquakes of a specific large magnitude (assumed to be a maximum possible) are included in the slip budget calculations. However, the events smaller than the maximum earthquake also contribute to the slip budget; all earthquakes need to be taken into account in site-specific calculations. A discussion in this Subsection is based broadly on one of our previous papers (Kagan, 2005). Several issues need to be taken into account in site-specific slip calculations: 1. The form of the general (area-specific) distribution of the earthquake size, for the simplicity of calculations we take it as the truncated Pareto distribution (see Eq. 2). 2. The site-specific moment distribution large earthquakes have a bigger chance to intersect a site, hence the moment distribution is different from area-specific. 3. The geometric scaling of earthquake rupture. As described earlier, length-width-slip are scale-invariant, i.e., for earthquake of magnitude m: L m, W m, u m 3 M. 4. The earthquake rupture geometric self-similarity implies that the earthquake depth distribution would be different for small versus large shocks: at least for strike-slip earthquakes large events would penetrate below the seismogenic layer. For thrust and normal events the consequences of geometric self-similarity are not clear, their depth distribution has not been sufficiently studied. 5. Most of the small earthquakes do not reach the Earth surface and therefore do not contribute to the surface fault slip. The contribution of small earthquakes needs to be properly computed. Because of our insufficient knowledge of the slip distribution with depth we calculate the 15

16 maximum earthquake size for several special simple cases and their combinations. An earthquake of moment M (magnitude m) is specified as M = µ L m W m u m. (11) Using the results shown in Fig. 9, we presume for an earthquake of magnitude m = 7.0 or moment M = Nm, that L 7 = 60 km, W 7 = 10 km, and u 7 = 1.76 m or u 7 = m, depending on the value of the shear modulus: µ = 30 GPa or µ = 49 GPa, respectively. The slip distribution over the fault plane is highly non-uniform in horizontal direction (Manighetti et al., 2005; 2009). Kagan (2005, [51]) argued that the slip of large earthquakes should catch up on the slip deficit at the Earth s surface left by smaller events, thus the slip of large events must be larger at the surface than in the middle of a seismogenic zone. This may imply that the seismic efficiency coefficient (χ) may also change with depth. However, since we lack reliable data, for our approximate calculations we take slip to be uniform over a rectangle L m W m. We specify earthquake magnitude m rupture dimensions as L m = L 7 ( m+9 / ) 1/3. (12) Analogous expressions are used for W m and u m. In the first simple case of the maximum earthquake size calculation, we suppose that earthquakes of all sizes are distributed uniformly over the subducting slab of width W. Then, as we mentioned above, for the earthquake with magnitude m the surface slip contribution u m would be u m = u m W m /W for W W m, (13) accounting for the fact that only few of the smaller earthquakes would reach the surface. 16

17 In deriving formulas for site-specific surface displacement U s, we simplify Kagan s (2005) results, taking into account the self-similarity of earthquake rupture (12) U s = λ m u m L m 1.5 b, (14) where λ m = α m /L is the rate of earthquakes with magnitude greater or equal m per one km of a subduction zone. If W m = W is due to assumed scaling, the resulting U s would be always the same, since increase in u m and L m would be compensated by a decrease in λ m. Making the order-of-magnitude calculations, we take α 7 = (425/34) = (see Fig. 1) and L = 3000 km; if the magnitude is not close to the maximum, α m scales with magnitude m as 10 b (7 m) (see Eq. 2). Then for b = 1 and m = 7 we obtain U s = 7 mm/y, for m = 8 slip rate is U s = 22 mm/y, and U s = 70 mm/y for m = 9. If we replace µ = 30 GPa by µ = 49 GPa, then for m = 9 the slip is U s = 43 mm/y. All these values are smaller than U s = 82.5 mm/y suggested by Simons et al. (2011) for the Tohoku area. As another simple case of the maximum earthquake size calculation, we presume that all surface slip is due to earthquakes (m f ) exceeding a certain size (like earthquakes larger than m8) which always rupture through the Earth surface. In effect, we suppose that small earthquakes do not contribute to the surface slip. Such a model may be appropriate for strike-skip faults as in California, where very few small earthquakes occur near the surface (Kagan, 2005). It is possible that even for subduction zones this model would produce more correct results. Then for b 1.0 we obtain U s = λ f u f L f 1.0 b [ 10 (b 1.0)(mx m f ) 1 ], (15) and for b = 1 U s = λ f u f L f log 10 (m x m f ). (16) 17

18 Several calculations can be made with these formulas to get a rough estimate of the maximum magnitude m x needed to obtain the slip rate U s = 82.5 mm/y. For b = 1 and µ = 30 GPa, m x = m f Therefore, if we assume m f = 7.0, m x = 8.02 and for m f = 8.0, as more appropriate for subduction zones, m x = These values depend, of course, on the presumed parameters of the earthquake rupture. If, for example, instead of the rupture parameters shown near Eq. 11, we take the values suggested by Kagan (2005): L 7 = 37.5 km, W 7 = 15 km, and u 7 = 1.87 m, the estimate of the maximum magnitude changes to m x = m f A similar increase of m x occurs if we modify µ: for µ = 49 GPa, m x = m f If we change the b-value, for instance, b = 0.9, but keep µ = 30 GPa, then m x = m f As the third case, we consider a combination of two models (Eqs. 14 and 16): we suppose that in the upper part of the subduction boundary with the width W f, small earthquakes are distributed uniformly, whereas large earthquakes m m f penetrate deeper. Thus, the total slip would be a sum of two terms, one reflecting the contribution of small and moderate events and the other for the large earthquakes. Then taking b = 1, m f = 8, W f = 31.6 km, and µ = 49 GPa, we obtain m x = The calculations in this section are not as reliable as in the previous Subsection 3.1; unfortunately the distribution of the slip and the earthquake depth has not yet been studied as thoroughly as the area-specific magnitude/moment frequency relation. However, these approximate computations imply that the maximum magnitude in the Flinn-Engdahl zone #19 and in the Tohoku area is around 9.0, i.e., much greater than was assumed in the Japanese hazard maps. 18

19 4 Long- and short-term earthquake forecasts during the Tohoku sequence 4.1 Recurrence intervals Since 1977 we developed statistical models of seismicity which fit a catalog of earthquake times, locations, and seismic moments, and subsequently we base forecasts on these models. The forecasts are produced in two formats: long- and short-term (Kagan and Knopoff, 1977; Kagan and Jackson, 1994; Jackson and Kagan, 1999; Kagan and Jackson, 2000; 2011) and presently they predict the earthquake rate per area, time, magnitude unit, and focal mechanism. Several earthquake catalogs are used in our forecasts, the GCMT catalog was used most frequently as it employs relatively consistent methods and reports tensor focal mechanisms. The forecasts including those for the north-west Pacific area covering Japan, are posted on our Web site: kagan/predictions index.html. Figs. 10 and 11 show long-term forecasts for the north-west Pacific area; one forecast is calculated before the 2011 Tohoku sequence started, the other one week after the megaearthquake. These forecasts are calculated around midnight Los Angeles time. Their description can be found in our publications (Jackson and Kagan, 1999; Kagan and Jackson, 1994; 2000; 2011). There is little difference between these forecasts long-term forecasts do not depend strongly on current events. Plate 1 in Kagan and Jackson (1994) and Fig. 8a in Kagan and Jackson (2000) display previous forecasts for the Appearance of both plots is similar to Figs. 10 and 11. In Table 2, we display earthquake forecast rates around the epicenter of the m7.4 Tohoku foreshock. The ratio of the short- to long-term rates (the last column) rises sharply both after the foreshock and after the mainshock. Conclusions similar to those in the previous 19

20 paragraph can be drawn from Table 2: the maximum long-term rates change only by a few tens of a percent. The predicted focal mechanisms are also essentially the same for the center of the focal area. To calculate the earthquake long-term rates for the extended area we can integrate the tables over the desired surface, or as a better option, calculate an ensemble of the seismograms for each point of interest on the Earth s surface. This can be accomplished by using a forecasted tensor focal mechanism, such as shown in Table 2. These seismograms can be used to calculate probable damage to any structure due to earthquake waves. Such calculations are superior in usefulness to earthquake hazard maps (such as the Japanese map shown, for example, by Geller, 2011 and by Stein et al., 2011). Hazard maps display an intensity of shaking estimated for one particular wave period, whereas synthetic seismograms allow calculating a probability of any structural damage or collapse, depending on the structure s mechanical properties. Another advantage of earthquake rate forecasts is that they are easily tested for effectiveness (Kagan & Jackson, 2000; 2011) by comparing their predictions with future earthquakes. This testing information is readily available from earthquake catalogs. Hazard maps are more difficult to verify; data on the ground motion intensity are more scarce, especially in the less populated territories. Moreover, these maps may fail for two reasons: an incorrect seismicity model or an incorrect attenuation relation, thus it is difficult to find out the cause of poor performance. As an illustration, using simple methods, we make an approximate estimate of the longterm recurrence rate for large earthquakes in the Tohoku area. In the GCMT catalog, the number of earthquakes with M 5.8 in a spherical rectangle N, E, covering the rupture area of the Tohoku event, is 109 for years (see Table 1). If we assume that the corner magnitude is well above m9.0 (similar to m9.6 for subduction zones, see Bird 20

21 and Kagan, 2004), the repeat time for the m9 and larger events in this rectangle depends on the assumed b-value and is between 300 and 370 years. Uchida and Matsuzawa (2011) suggest a recurrence interval for m9 events years. Simons et al. (2011) propose a year interval. This interval estimate is partly based on the observation of the Jogan earthquake of 13 July 869 and its tsunami. Even if we assume that this Jogan event was similar in magnitude to the 2011 earthquake and no other such earthquakes have occurred meanwhile (see more discussion in Koketsu and Yokota, 2011), for the Poisson occurrence the probability of such an interval is of the order 3-5%. Moreover, the observation of only one inter-event interval does not constrain the recurrence time of these mega-earthquakes in a really meaningful way; the interval can be as small as a few hundred years or as large as tens of thousands of years. Only the moment conservation principle and the tapered G-R distribution (TGR) distribution provide a reasonable estimate for this interval. 4.2 Short-term forecasts Figs display short-term forecasts for the north-west Pacific area produced during the initial period of the Tohoku sequence. Fig. 12, calculated before m7.4 foreshock, shows a few weakly red spots in those places where earthquakes occurred during the previous weeks and months. The short-term rate in these spots is usually of the order of a few percent or a few tens of a percent compared to the long-term rate (see also Table 2). The predicted earthquake rates in the neighbourhood of the future Tohoku event increased strongly with the occurrence of a m7.4 foreshock (Fig. 13). As Table 2 demonstrate, just before the Tohoku earthquake, the forecasted rate was about 100 times higher than the long-term rate. The area of significantly increased probability covers the northern part of the Honshu 21

22 Island following the Tohoku mega-earthquake occurrence (Fig. 14). The size of the area hardly decreased one week later (Fig. 15) mostly due to the Tohoku aftershocks. Although only around 5-10% few shallow earthquakes are preceded by foreshocks, the results shown in Figs suggest that an operational earthquake forecasting strategy needs to be developed (Jordan and Jones, 2010; van Stiphout et al., 2010; Jordan et al., 2011) to take the increase of short-term rates into account. 5 Discussion It is commonly believed that after a large earthquake the focal area of an earthquake has been destressed (see, for example, Matthews et al., 2002) thus lowering the probability of a new large event in this place, though it can increase in nearby zones. This reasoning goes back to the flawed seismic gap/characteristic earthquake model (Jackson and Kagan, 2011). Kagan and Jackson (1999) showed that earthquakes as large as 7.5 and larger often occur in practically the same area soon after the occurrence of a previous earthquake. Table 3 displays pairs of shallow earthquakes m 7.5 epicentroid of which are closer than their focal zone size (an update of Table 1 by Kagan and Jackson, 1999, or Table 1 by Kagan, 2011). The Table includes three earthquake pairs (see # 3, 24, 25) of the Tohoku sequence and demonstrates that strong shocks tend to repeat in the focal zones of previous events. Michael (2011) shows that earthquakes as large as m8.5 are clustered in time and space, thus an occurrence of such a big event does not protect its focal area from the giant next shock. Stein et al. (2011) suggest the following forecast requirements: ideally a forecast should anticipate total economic and casualty losses due to earthquakes. Over a relatively short time period earthquake damage would seemingly reach the maximum not for the rare very large events, but for the m7 m8 shocks (England & Jackson, 2011). But over long-term, 22

23 expected or average losses would peak for the largest m8 m9 events; though their rates are low, the total average damage for one event increases faster than the probability of these earthquakes decreases. Since the expected economic and other losses peak for the strongest earthquakes (Kagan, 1997b), it is more important to predict disastrous earthquakes than small ones. However, the loss calculations (Molchan & Kagan, 1992; Kagan, 1997b) are very uncertain, because major losses are often caused by unexpected secondary earthquake effects. Therefore, a prediction of the largest earthquakes is important, hence prediction schemes that do not specify the earthquake size are of restricted practical use. However, if the maximum earthquake is over-predicted, it diverts resources unnecessarily (Stein et al., 2011). Any forecast scheme that extrapolates the past instrumental seismicity record would predict future moderate earthquakes reasonably well. However as the history of the Tohoku area shows, we need a different tool to forecast the largest events. In our forecasts we consider the earthquake rate to be independent of the earthquake size distribution, so the latter needs to be specified separately. Why is it so difficult to determine the maximum earthquake size for the subduction zones and their recurrence period? This question is especially important after two unpredicted giant earthquakes: the 2004 Sumatra and the 2011 Tohoku. The available earthquake record is short which makes it difficult to obtain this information by simple observation. As indicated earlier, the seismic moment conservation principle can provide an answer to the above questions. The general idea of the moment conservation was suggested some time ago (Brune, 1968; Wyss, 1973). However, without the knowledge of the earthquake size distribution, the calculation of the maximum earthquake moment size (M max ) is still difficult and leads to uncertain or contradictory results. The classical G-R relation is not helpful in this respect because it lacks the specification of M max. Only a modification of the 23

24 G-R law, that introduces the limiting upper moment could provide a tool to quantitatively derive M max or its variants. Kagan and Jackson (2000) and Kagan (2002a, 2002b) propose such distributions defined by two parameters, β and variants of m max. The application of these distributions allows us also to solve the problem of evaluating the recurrence period for these large earthquakes. Determining the maximum earthquake size either by historical/instrumental observations or by the qualitative analogies does not provide such an estimate: a similar earthquake may occur hundreds or tens of thousand years later. Fig. 1 shows how using statistical distributions may facilitate such calculations. As we discussed in Subsections 3.1 and 3.3, the moment conservation principle allows to quantitatively determine the maximum earthquake size. In this respect area-specific calculations provide a more precise size evaluation for many tectonic zones and, most importantly to show that the subduction zones could have the same maximum earthquake size (Kagan, 1997a). Although the determination of M max by comparing tectonic and seismic rates is not yet sufficiently accurate for our purposes, giving m max in the range of 8.5 to 9.7, comparing these estimates to the number of largest earthquakes in the subduction zones during the last 110 years definitely argues for the larger of the above values. The site-specific calculations are not yet as accurate and reliable as the area-specific, and the computation for several of the subduction zones has not been performed. However, even the approximate estimates in Subsection 3.3 suggest that m9 is an appropriate earthquake size for the Tohoku area. In conclusion, we would like to determine the upper magnitude limit for the subduction zones as well as recurrence intervals for such earthquakes. Two upper global estimates can be calculated: for the gamma distribution, we take the values from Table 1 m c = 9.36 ± 0.27 to get the 95% upper limit m c = 9.9. Bird and Kagan (2004, Table 5) determined for the tapered G-R (TGR) distribution m cm = , and the 95% upper limit m cm =

25 For the sake of simplicity we take m max = Calculations similar to (8) can be made to obtain an approximate estimate of the average inter-earthquake period. From Fig. 1b by Kagan (2002a) one can estimate the return period as it differs from the regular G-R law: the gamma distribution cumulative function at m c is below the G-R line by a factor of about 10, for the TGR distribution the factor is e. Thus, for the gamma distribution the recurrence time for the global occurrence of the m 10.0 earthquake is about 1750 years; for the TGR distribution this period is about 475 years. Of course, the distributions in these calculations are extrapolated beyond the limit of their parameters evaluation range, but the above recurrence periods provide a rough idea how big such earthquakes can be and how frequently they can occur worldwide. For Flinn-Engdahl #19 zone, the m 10.0 earthquake could repeat in about 9,000 or 32,000 years for the TGR and the gamma distributions, respectively. The rupture length of the m10.0 event can be estimated from Fig. 9: at about 2100 km it is comparable to the 3000 km length of zone #19. These long recurrence periods indicate that it would be difficult to find displacement traces for these earthquakes in paleo-seismic investigations. 6 Conclusions 1. The major cause for excessive fatalities and economic losses during the worst global natural disaster in the Tohoku-Oki area was a gross under-estimation of the maximum earthquake magnitude (m max ) and its recurrence interval. 2. We discuss several methods for determining the maximum earthquake size in the subduction zones: historical/instrumental data, statistical evaluation of the seismic moment/frequency relation, and two techniques based on the moment/slip conservation princi- 25

26 ple. 3. The historical/instrumental record usually provides an estimate of m max that is significantly lower when compared to new earthquakes and to the evaluations by other methods. 4. The statistical method, when applied to individual subduction zones, yields a quantitative estimate of the earthquake size distribution parameters, but due to small number of the recorded large earthquakes, it lacks sufficient resolution in the evaluation of m max. The statistical method produces a reasonable estimate of the lower limit for m max only for global seismicity. 5. The seismic moment conservation principle provides the best estimate of the maximum earthquake size. The most important information obtained by this technique is the absence of statically significant variation in m max among the major subduction zones. This fact combined with the very large (m 9.0) known events in the other subduction zones in the 20-th century should have served as a warning of a possibility of such an earthquake in the Tohoku area. 6. Seismic slip conservation calculations also suggest that to explain the mm/y slip in the Tohoku area, earthquakes with magnitudes in excess of m8.5 should be considered. 7. Long-term forecasts based on the optimal smoothing of seismicity in and around Japan suggest that the recurrence period for the m9 earthquakes is of the order of 350 years in the Tohoku area. 8. Short-term forecasts can provide time-dependent information for aftershocks occurrence. In some cases, if foreshocks are present, as in the Tohoku sequence, mainshock rates can be predicted. Therefore, these forecasts can be used for developing an operational earthquake forecasting strategy. 26

27 7 Data and Resources The global CMT catalog of moment tensor inversions compiled by the GCMT group is available at (last accessed December 2011). The Centennial catalog by Engdahl & Villaseñor (2002) is available at (last accessed December 2011). Flinn-Engdahl Regions are explained and their coordinates as well as FORTRAN files to process them are available at ftp://hazards.cr.usgs.gov/feregion/fe_1995/ (last accessed December 2011). The forecasts including those for the north-west Pacific area covering Japan, are posted on our Web site: (last accessed December 2011). Acknowledgments We are grateful to Peter Bird and Paul Davis for useful discussion and suggestions. The authors appreciate support from the National Science Foundation through grants EAR and EAR , as well as from the Southern California Earthquake Center (SCEC). SCEC is funded by NSF Cooperative Agreement EAR and USGS Cooperative Agreement 02HQAG0008. Publication 0000, SCEC. 27

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