Parameters of Tsunami Source Versus Earthquake Magnitude

Transcription

1 Pure Appl. Geophys. 168 (2011), Ó 2011 Springer Basel AG DOI /s Pure and Applied Geophysics Parameters of Tsunami Source Versus Earthquake Magnitude ANNA V. BOLSHAKOVA 1 and MIKHAIL A. NOSOV 1 Abstract This study employs an empirical-analytical approach in combination with Monte-Carlo method to establish relationships between earthquake moment magnitude and upper limits of tsunami source parameters: double-amplitude of vertical bottom deformation, displaced water volume, and potential energy of initial elevation. The approach is based on the Okada solution for a finite rectangular fault and empirical scaling laws for earthquake sources. Results are compared to empirical and theoretical relationships published previously. Parameters of some recent tsunami sources, those for which USGS provides slip distribution data, are considered in light of the established relationships. Key words: Bottom earthquake, moment magnitude, tsunami source, co-seismic deformation, initial elevation. 1. Introduction Strong bottom earthquakes are the most widespread cause of devastating tsunami waves. In order to understand the nature of tsunamis and other oceanic phenomena of seismotectonic origin it is useful to define simple general relationships between parameters of a tsunami source and the magnitude of its associated earthquake. In practice, such relationships may support prompt assessment of characteristics of a seismotectonic tsunami for a given earthquake magnitude. Relationships between earthquake magnitude and source parameters are widely used in seismology (e.g. KANAMORI and ANDERSON, 1975; SATO, 1979; WELLS and COPPERSMITH, 1994; OKADA, 1995; KANAMORI and BRODSKY, 2004). Numerous attempts have been undertaken to establish similar relationships between magnitude and tsunami source parameters. Without 1 Faculty of Physics, M.V. Lomonosov Moscow State University, Moscow, Russia. annabolshakova@list.ru; nosov@phys.msu.ru claiming to present a full list, we shall mention several publications: (IIDA, 1963; HATORI, 1970; YAMASHITA and SATO, 1974; ALEKSEEV and GUSYAKOV, 1976; WARD, 1980; KAJIURA, 1981; DOTSENKO and SOLOVIEV, 1990; OKAL 1988, 2003). Typically, the energy of a tsunami is related to earthquake magnitude. Additional relationships that have been investigated include amplitude of vertical bottom deformation (or initial elevation) versus magnitude and the area of a tsunami source (or mean radius) versus magnitude. In this study, we shall consider an additional parameter water volume displaced by co-seismic bottom deformation. Along with the tsunami energy, the displaced water volume represents an integral of motion in the hydrodynamic tsunami problem. Although it represents an important measure of tsunami strength, this parameter is rarely mentioned in publications (e.g. GRILLI et al., 2007; NOSOV and KOLESOV, 2009). How the displaced water volume is related to earthquake magnitude has not been previously investigated. In tsunami simulation (e.g. TITOV and GONZALEZ 1997; KOWALIK et al., 2005; GISLER, 2008, LEVIN and NOSOV, 2008), it is common practice to calculate coseismic deformations of the ocean bottom in the source area, making use of the analytical solution of the stationary problem of elasticity theory, presented in (OKADA, 1985). In spite of the fact that the Okada formulae are analytical expressions, it is next to impossible to derive any general relationships from them analytically, as the formulae are quite cumbersome and contain numerous input variables. The Monte-Carlo method was employed in (OKADA, 1995) to relate earthquake magnitude and co-seismic crustal deformation. However, relationships obtained by Okada are hardly applicable for tsunami sources because his approach, which was based on the

2 2024 A. V. Bolshakova, M. A. Nosov Pure Appl. Geophys. solution for a point double-couple source, fails when the hypocentral distance is comparable to or smaller than the source region size. In the present study, in contrast to (OKADA, 1995), we shall consider a finite rectangular fault this statement of the problem is more appropriate to tsunami generation. Like OKADA (1995), we shall apply a sort of Monte-Carlo method to relate earthquake magnitude and upper limits of amplitude of vertical bottom deformation, displaced water volume, and tsunami energy. Newly obtained relationships will be compared with existing empirical and theoretical dependences. A secondary, though important, goal of this article is to examihne parameters of some recent tsunami sources in light of the general relationships. 2. Relations of Earthquake Magnitude and Tsunami Empiric relations for the mean radius of the tsunami source R TS (km) and for the maximum of the initial elevation n 0 (m) had been derived for magnitudes within the range 6.7 \ M \ 8.5 in (DOTSENKO and SOLOVIEV, 1990): log 10 R TS ¼ ð0:50 0:07Þ M ð2:1 0:6Þ ð1þ log 10 n 0 ¼ ð0:8 0:1ÞM ð5:6 1:0Þ ð2þ The estimates for confidence intervals in (1) and (2) correspond to 80% probability. Note that formula (2) seems to yield overestimated values in cases of large magnitude. For example, for the catastrophic tsunamigenic earthquake that occurred on 26 December 2004, and the magnitude of which was M w = 9.1 (USGS), formula (2) gives the value of n 0 & 48 m. By contrast, more realistic calculations performed by GRILLI et al.(2007) with use of Okada s solution, give a much smaller maximum vertical seafloor subsidence uplift of -3.8 to 8.6 m. So, due to their limited magnitude range of validity and extensive confidence intervals, the regressions by Dotsenko and Soloviev turn out to be unhelpful in many cases. Using the Monte-Carlo method, OKADA (1995), among other dependencies, obtained the following relation between the maximum vertical deformation U z (cm) and parameters of a point double-couple source, such as the hypocentral distance, R (km) and magnitude, M: log 10 U z ¼ 1:5 M 2 log 10 R 5:96: ð3þ A strong tsunamigenic earthquake cannot be considered as a point-like event, especially in the near-field region within the tsunami source area. Thus, as already mentioned above, expression (3) cannot be directly applied to a tsunami source. KAJIURA (1981) derived theoretically the following relation between the upper limit of tsunami energy and the moment magnitude: log 10 E TS ðjþ ¼2:0 M w 2:46 ð4þ Comparing formula (4) with the well-known energy-magnitude relationship defined in terms of M w by KANAMORI (1977) log 10 E EQ ðjþ ¼1:5M w þ 4:8; ð5þ Eq. (5) for the ratio of tsunami energy and earthquake energy, Kajiura obtained the following estimate: log 10 E TS =E EQ ¼ 0:5M w 7:26 ð6þ According to formula (6), the ratio E TS /E EQ increases with an increase of magnitude, however, even for the strongest earthquakes it remains less than 1% (E TS /E EQ & for M w = 9.5). Using normal mode theory, OKAL (2003) analytically derived the following relation between the energy of a far-field tsunami generated by a dislocation of moment M 0 (dyn 9 cm): E TS ðergs) ¼ 7: M 4=3 0 ð7þ Introducing SI units in formula (7) and expressing the moment via moment-magnitude as M w = 2/3 log 10 M (KANAMORI, 1977), one readily obtains the following link: log 10 E TS ðjþ ¼2:0M w 1:66 ð8þ Expressions (4) and (8), though derived by absolutely different approaches, exhibit an impressive similarity. However, Kajiura s formula (4) always gives a value of the tsunami energy approximately 6.3 times smaller than that of Okal s formula (8). As for the accuracy of the formulae for tsunami energy, as already mentioned by OKAL (2003), a universal experimental method for measuring the energy of

3 Vol. 168, (2011) Parameters of Tsunami Source Versus Earthquake Magnitude 2025 large transoceanic tsunamis does not exist. In our opinion, the only reliable way to estimate the energy of a tsunami consists in computation of the potential energy of initial elevation. This open question regarding the tsunami energy in its relation to earthquake magnitude serves as an additional motivation for this study. 3. Method of Calculation In simulation of tsunamis of seismotectonic origin, an earthquake is traditionally considered to instantly displace a water column forming a perturbation on its free surface. Then, the assumption is made that the shape of the perturbation is fully similar to the vertical component of the residual deformation of the bottom. The perturbation of the water surfaces thus obtained, so-called initial elevation, is applied as an initial condition in resolving the problem of tsunami propagation (e.g. TITOV and GONZALEZ, 1997; KOWALIK et al., 2005; GISLER, 2008; LEVIN and NOSOV, 2008). Though easy-to-use in practice, this traditional approach turns out to be inaccurate in some cases. Imperfectness may be attributed to smoothing of water surface perturbations as compared with the bottom deformations (TANIOKA and SENO, 2001; SAITO and FURUMURA, 2009; NOSOV and KOLESOV, 2010), contribution of horizontal deformation of a sloping (non-horizontal) bottom (TANIOKA and SATAKE, 1996; NOSOV and KOLESOV, 2009), effects of dynamics of bottom deformations (HAMMACK, 1973; NOSOV, 1998) and effects of water compressibility (NOSOV, 1999; NOSOV and KOLESOV, 2007). However, the traditional approach more or less adequately reproduces the main effect responsible for seismotectonic tsunami generation displacement of the water. Therefore, in what follows, we restrict our consideration to the traditional assumption that the initial elevation is equal to the vertical residual bottom deformation. In that way, the hydrodynamic part of the problem remains beyond consideration. An additional reason to ignore the water column is that the potential energy of initial elevation, due to smoothing of perturbations of water surface, tends to decrease as the ocean depth increases (NOSOV and KOLESOV, 2009). So, the upper limit of the potential energy is reached at minimal, i.e. at zero, depth. We take the Cartesian reference system as shown in Fig. 1. The elastic medium occupies the region of z B 0. The Ox axis is taken to be parallel to the strike direction of a finite rectangular fault, i.e. along length L of the fault. We assume Burgers vector D~ ¼ ðu 1 ; U 2 ; 0Þ lies in the fault plane. So U 1 and U 2 represent strike-slip and dip-slip components respectively; the tensile component, directed normal to the fault plane, assumed equal to zero, U 3 = 0. In the general case, a dislocation is determined by two angles: the dip angle d and the rake (slip) angle h. In the case of uniform movement along a rectangular fault plane, residual deformations of the ocean bottom can be calculated making use of the analytical solution of the stationary problem of elasticity theory, as presented in (OKADA, 1985). The Okada formulae allow obtaining a vector field of bottom deformation due to inclined shear fault in an isotropic homogeneous elastic half-space. In the modeling case under consideration, for calculation of the bottom deformations the following Figure 1 Geometry of the source model of an earthquake. L is the length of the fault plane, W is the width of the fault plane, D~ is the Burgers vector, d is the dip angle, h is the rake angle, h is the depth of the upper edge of the fault plane

4 2026 A. V. Bolshakova, M. A. Nosov Pure Appl. Geophys. set of eight parameters is required: length of the Burgers vector D, dip and rake angles, length L and width W of the fault plane, depth of the source h (we consider depth of the upper edge of the fault plane), and Lame constants l and k. The multiplicity of the input parameters required for calculation of the bottom displacement exhibits a serious obstacle to deriving the desired relationships. So, in order to reduce number of the input parameters, we must introduce additional relations. First, we involve the scaling laws by KANAMORI and ANDERSON (1975): L/W = 2, D/L = Then, considering the definition of the seismic moment M 0 = ldlw (N 9 m) and the well-known link between the earthquake moment and momentmagnitude M w = 2/3 log 10 M (KANAMORI, 1977), we arrive at the following formulae: log 10 L ðkm) ¼ 0:5 M w A L log 10 W ðkm) ¼ 0:5 M w A W log 10 D ðmþ ¼0:5 M w A D ð9þ ð10þ ð11þ Due to variation of the shear modulus l within the limits Pa the coefficients in formulae (9) (11) undergo insignificant changes within the ranges A L = , A W = , A D = In what follows we shall take the shear modulus as l = Pa (usual value for crustal faults). It is worth noting that formulae (9) (11) are virtually the same as the relations used in tsunami forecasts (Handbook for Tsunami Forecast in the Japan Sea, 2001). The Lame constants l and k enter into Okada s expressions in the form of a combination, which for practical calculations is conveniently expressed via the respective velocities of longitudinal and transverse seismic waves c p and c s, v l k þ l ¼ c2 s c 2 p c2 s ð12þ As can be gathered from the CRUST2.0 model (BASSIN et al., 2000), the value v lies within limits from 0.3 to 0.5. In our calculations we shall assume the ratio is v = Taking into account relations (9) (12), we arrive at the following reduced set of four input parameters: magnitude M w, depth h, dip and rake angles. In calculating of the vertical component of bottom deformations, g z (x, y) all the input parameters were chosen randomly from the following intervals: 0 \ h \ 30 km, 6.5 \ M w \ 9.5, -90 \ h \ 90, 0 \ d \ 90. The cases of h = 100 and 300 km were examined separately. Of all the possible parameters of a tsunami source, we shall consider only those parameters which can be unambiguously calculated from the vertical component of co-seismic bottom deformations. There are three of these: double-amplitude of vertical bottom deformation, displaced water volume and potential energy of the initial elevation. It must be stressed that the tsunami source area cannot be unambiguously calculated from the co-seismic deformations. In order to calculate this parameter it is necessary to introduce an additional quantity i.e. a level of deformation separating the tsunami source from the external domain. So, from the function g z (x, y) we determine the following parameters: 1. double-amplitude of vertical bottom deformation: g max ¼ Max½g z ðx; yþš Min½g z ðx; yþš ð13þ 2. displaced water volume: ZZ V ¼ g z ðx; yþ dx dy ð14þ 3. potential energy of the initial elevation: E ts ¼ q g ZZ g 2 zðx; yþ dx dy 2 ð15þ where g is acceleration due to gravity and q is density of water (we assume q = 1,000 kg/m 3 ). The integration in formulas (14) and (15) was performed over a domain where vertical deformation g z exhibited a noticeable value. Precisely, the integration domain was - L - h \ x \ 2L? h, - W - h \ y \ 2 W? h. In contrast to the simple model of uniform slip along a rectangular fault we used in the Monte-Carlo simulations, the realistic slip distribution along the fault plane turns out to be non-uniform. For some recent strong seismic events, the slip distributions (Finite Fault) are available in digital form from the USGS site

5 Vol. 168, (2011) Parameters of Tsunami Source Versus Earthquake Magnitude 2027 world/historical.php. Among all the available events we selected 15 bottom earthquakes. The selected events are listed in the Table 1. For a finite fault event the fault surface is divided into a finite number of rectangular elements, for each of which the slip vector D~ is determined. The bottom deformation caused by each of these rectangular elements was calculated by the Okada formulae. Ultimately, the contributions of all elements are summed up. The ratio v for a given finite fault event were determined via seismic wave velocities c p and c s taken from the CRUST2.0 model (BASSIN et al., 2000). In our calculations we used a value of v averaged along the fault plane. For all the finite fault events we calculated the maximum uplift and subsidence, the double-amplitude of vertical bottom deformation, the displaced water volume and the potential energy of initial elevation. All the calculations were carried out in accordance with formulae (13) (15). Co-seismic deformation data were computed on a grid with space increments of 1 angular min. The integration domain was chosen so that it fully covered all the noticeable vertical deformations (normally, degrees). Finalizing this section, we introduce the simple formulae we used for estimation of the displaced volume and the potential energy of initial elevation from the empirical relationships by DOTSENKO and SOLOVIEV (1990) V ¼ pr 2 TS n 0 ð16þ E TS ¼ q g 2 n2 0 p R2 TS ð17þ 4. Results and Discussion Results of Monte-Carlo calculations are presented in Figs. 2, 3 and 4 by small gray and black points. The light grey points stand for shallow earthquakes, where the depth of the upper edge of the fault plane falls within the range of h = 0 30 km. The dark grey points stand for events with depth of h = 100 km, and black points stand for events with depth of h = 300 km. Due to the significant influence on the bottom deformation of parameters of seismic source, one can observe an essential data scattering. However, the clusters of points obey some special regularities. First, the double-amplitude and the energy of a tsunami noticeably decrease as the depth of its seismic source h increases. This corresponds to the wellknown fact that tsunamis are most often generated by shallow earthquakes. Second, the displaced water volume exhibits no dependence on the depth h. This remarkable feature corresponds well to the imaginary empirical formula obtained by OKADA (1995) for a point double-couple source (see Eq. (3)): Table 1 List of tsunamigenic earthquakes under consideration Date Region M w Uplift (m) Subsidence (m) g max (m) V (m 3 ) E (J) 1 26 December 2004 Sumatra Andaman Islands July 2006 South of Java, Indonesia November 2006 Kuril Islands January 2007 East of the Kuril Islands April 2007 Solomon Islands August 2007 Near the Coast of the Central Peru September 2007 Kepulauan Mentawai region, Indonesia September 2007 Southern Sumatra, Indonesia November 2007 Antofagasta, Chile February 2008 Simeulue, Indonesia September 2008 Kermadec Islands, New Zealand January 2009 Near the North Coast of Papua, Indonesia July 2009 Off West Coast of the South Island, New Zealand September 2009 Samoa Islands region September 2009 Southern Sumatra, Indonesia

6 2028 A. V. Bolshakova, M. A. Nosov Pure Appl. Geophys. Figure 2 Double-amplitude of co-seismic bottom deformation in tsunami source versus moment magnitude. Line 1 stands for the empiric relation (2), dotted lines 1a and 1b show 80% confidence intervals. Line 2 shows the upper limit calculated with use of formula (18). Solid line denoted as Slip depicts the amount of slip calculated with use of formula (11) Figure 4 Potential energy of the initial surface elevation in tsunami source versus moment magnitude. Line 1 stands for the energy of the tsunami calculated with use of formula (17) from empiric relations (1) and (2), dotted lines 1a and 1b show 80% confidence intervals. Thick black line denoted as TS shows the upper limit calculated with use of formula (20). Black line denoted as EQ stands for energy of the earthquake estimated from relation (5). White black line 2 and dashed light-grey line 3 depict the relations obtained by Kajiura (1981) and by Okal (2003), respectively Figure 3 Displaced water volume in tsunami source versus moment magnitude. Line 1 stands for the volume calculated with use of formula (16) from empiric relations (1) and (2), dotted lines 1a and 1b show 80% confidence intervals. Line 2 shows the upper limit calculated with use of formula (19) U z * R -2, where U z is the amount of vertical static deformation and R is the hypocentral distance. Let us consider elementary displaced volume, dv = U z dxdy. Introducing spherical coordinates and putting its origin in the hypocenter, we readily obtain dxdy = f(u, w) R 2 dudw, where f(u, w) is a trigonometric expression. Therefore, the elementary displaced volume does not depend on the hypocentral distance, R, i.e., it does not depend on the depth of the seismic source. As a result, the whole displaced volume turns out to be independent of the depth of the seismic source. Taking into account that the Okada formulae for a finite rectangular fault are the result of integration of expressions for a point doublecouple source, one can easily explain the independence of the displaced water volume from the depth of seismic source, h. Third, all the clusters of points have rather sharp upper boundaries, whereas lower boundaries turn out to be relatively diffusive. This feature is a direct consequence of the existence of certain upper limits of all the parameters under consideration and the presentation of data in logarithmic scale. Analysis of the clusters of points gives us grounds to suggest the following estimates for upper limits of

7 Vol. 168, (2011) Parameters of Tsunami Source Versus Earthquake Magnitude 2029 tsunami source parameters such as double-amplitude of vertical bottom deformation, displaced water volume and potential energy of initial elevation: log 10 g max ðmþ ¼0:5 M w 3:4 log 10 Vðm 3 Þ¼1:5 M w 1:8 log 10 E TS ðjþ ¼2:0 M w 1:7 ð18þ ð19þ ð20þ Combining formulae (18) and (19), we arrive at a useful estimation p for mean radius of tsunami source (R TS ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ): V=g max log 10 R TS ðkmþ ¼0:5 M w 2:2 ð21þ It is worth noting that estimation (21) is rather close to the empirical relationship (1) obtained by DOTSENKO and SOLOVIEV (1990). According to formulae (18) (21), parameters of tsunami source at M w = 6.5 take the following values: g max = 0.71 m, V = m 3, E TS = J, R TS = 11.2 km, whereas at M w = 9.5: g max = 22.4 m,v = m 3, E TS = J, R TS = 355 km. Newly obtained estimates (18) (20) are shown in Figs. 2, 3 and 4 by thick solid lines. Relationships based on the empirical links (1), (2) by DOTSENKO and SOLOVIEV (1990) and on the formulae (16), (17) are depicted in Figs. 2, 3 and 4 by thin black lines. Black dashed lines (1a, 1b) stand for the confidence intervals. From the figures, it can be seen that estimates based on the empirical links by Dotsenko and Soloviev always overestimate tsunami source parameters prescribed by formulae (18) (20). However, bearing in mind the huge confidence intervals, one may declare that the Monte-Carlo calculations fit the empiric relations more or less well. Slip amount in earthquake source versus magnitude is depicted in Fig. 2 by a black solid line of medium thickness. It can be seen from Fig. 2, and it is reasonable from physical point of view, that the amount of slip, D is always larger than the double amplitude, g max (approx. 1.5 times). Estimates of tsunami energy versus earthquake magnitude obtained by Kajiura (4) and by Okal (8) are shown on Fig. 4 by solid white-black lines (2) and by dashed light grey lines (3), respectively. Our relationship (20) is virtually similar to Okal s formula. At the same time, Kajiura s formula (4) always noticeably underestimates (approx. 6.3 times) the amount of tsunami energy as compared with relationship (20). The difference can be explained as follows. First, in the study (KAJIURA 1981), all computations were carried out for the range of the relative depth of upper rim of the fault, 0.04 B h/l B In our computations, the minimal depth of the upper rim was set as zero. The maximal values of tsunami energy are obviously associated with the smallest depth. Second, 30 years ago, probably due to a lack of computational capability, integration was performed in a limited region delineated by g/g max C 0.1. This also may lead to underestimation of tsunami energy. Comparing relations (5) and (20), we obtain the following estimate for the ratio of tsunami energy and earthquake energy: log 10 E TS =E EQ ¼ 0:5M w 6:5 ð22þ According to formula (22), a tsunami takes from 0.05% (M w = 6.5) to 1.8% (M w = 9.5) of the earthquake energy. For the reasons mentioned above, KAJIURA (1981) provided a different estimation for this ratio (see Eq. (6)). It is worth noting that, in reality, the ratio E TS /E EQ may turn out to be much smaller, because formula (20) gives the upper limit of the tsunami energy. Real events are depicted in Figs. 2, 3 and 4 by black rhombuses. Digits on the rhombus represent the numbers of events in the same order as they are listed in Table 1. It can be seen that parameters of real tsunami sources, in most cases, turn out to be within the clusters of points, i.e. below the upper limits prescribed by the formulae (18) (20). Nevertheless, there are two exceptions: #4 East of the Kuril Islands (13 January 2007) and #14 Samoa Islands region (29 October 2009). Both events are characterized by essential concentration of slip within a narrow area on a fault plane in combination with significant dip angle (58 o and 57 o respectively). As a result, corresponding co-seismic deformations in the tsunami sources are narrow and large amplitude subsidence of the bottom. So, in the case of a locally concentrated slip, relationship (18) obtained under the condition of uniform slip distribution may slightly underestimate amplitude of bottom deformation. As for the displaced water volume and the tsunami

8 2030 A. V. Bolshakova, M. A. Nosov Pure Appl. Geophys. energy, relationships (19) and (20) provide correct estimations of the upper limits. 5. Conclusions We carried out Monte-Carlo simulation of vertical co-seismic deformations in a tsunami source on the basis of the Okada solution for a finite rectangular fault and empirical scaling laws for the earthquake source. Parameters of seismic source were chosen randomly from certain realistic intervals. In each particular case the following three parameters of tsunami source were computed: double-amplitude of vertical bottom deformation, displaced water volume, and potential energy of the initial elevation. The computed values were plotted against earthquake magnitude. Analysis of clusters of points allows us to suggest relationships between earthquake moment magnitude and upper limits of the parameters of the tsunami source. The obtained relationships were compared to some existing empirical and theoretical dependencies. Results turned out to be in reasonable agreement, except in some details. In particular, it was found that empirical relations by DOTSENKO and SOLOVIEV (1990) give overestimated values, especially for earthquakes of large magnitude. Our estimation of the upper limit of tsunami energy versus magnitude turned out to be in perfect agreement with the relation derived by OKAL (2003), whereas KAJIURA (1981) reported a value of tsunami energy approximately 6.3 times smaller than our estimations. The ratio of tsunami energy and earthquake energy was also estimated as a function of earthquake magnitude. This ratio increases with the increase of magnitude. In any case, the fraction of tsunami energy remains rather negligible: the tsunami takes from 0.05% (M w = 6.5) to 1.8% (M w = 9.5) of earthquake energy. The results of the Monte-Carlo simulation were superimposed with data on 15 recent tsunamigenic earthquakes with known slip distributions (USGS Finite Fault). It was shown that parameters of realistic tsunami sources, such as displaced water volume and potential energy, do not exceed the upper limits prescribed by the obtained relationships. However, double amplitude of bottom deformation turns out to exceed the theoretical upper limit in two cases. These seismic events are characterized by highly concentrated slip and by significant dip angle. Acknowledgements This work was supported by the Russian Foundation for Basic Research, projects and We are grateful to USGS for providing the Finite Fault data. REFERENCES ALEKSEEV, A.S. and GUSYAKOV, V.K. (1976), Numerical modeling of tsunami and seismic surface wave generation by a submarine earthquake, Proc. Tsunami Res. Symp. (eds. HEATH R.A. and CRESWELL M.M.) Roy. Soc. New Zealand, Wellington, BASSIN, C., LASKE, G., and MASTERS, G., (2000), The Current Limits of Resolution for Surface Wave Tomography in North America, EOS Trans. AGU, 81, F897. DOTSENKO, S.F., SOLOVIEV, S.L. (1990), Mathematical modeling of tsunami excitation process by displacement of the ocean bottom, Tsunami researches (in Russian), 4, 8 20, Moscow. GISLER, G.R. (2008), Tsunami simulations. Annu. Rev. Fluid Mech., 40, GRILLI, S.T., IOUALALEN, J.M., KIRBY, J.T., WATTS, P., ASAVANT, J., and SHI, F. (2007), Source Constraints and Model Simulation of the December 26, 2004, Indian Ocean Tsunami, Journal of Ocean Engineering, 133(6), HAMMACK, J.L. (1973), A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech., 60, Handbook for Tsunami Forecast in the Japan Sea, (Earthquake and Tsunami Observation Division, Seismological and Volcanological Department, Japan Meteorological Agency. 22 P. 2001). HATORI, T. (1970), Vertical crustal deformation and tsunami energy, Bulletin of the Earthquake Research Institute, 48, IIDA, K. (1963), Magnitude, energy and generation mechanism of tsunamis and a catalogue of earthquakes associated with tsunamis, Proc. Tsunami Meet. Assoc 10th Pacific Sci Congr., 1961, I.U.G.G. Monogr. No. 24, KAJIURA, K. (1981), Tsunami energy in relation to parameters of the earthquake fault model, Bulletin of the Earthquake Research Institute, 56, KANAMORI, H (1977), The energy release in great earthquakes, J. Geophys. Res., 82, KANAMORI, H., ANDERSON, D.L. (1975), Theoretical basis of some empirical relations in seismology, Bulletin of the Seismological Society of America, 65, KANAMORI, H., BRODSKY, E.E. (2004), The physics of earthquakes, Rep. Prog. Phys., 67, KOWALIK, Z., KNIGHT, W., LOGAN, T., and WHITMORE, P. (2005), Numerical modeling of the global tsunami: Indonesian tsunami of December , Sci. Tsunami Hazard, 23(1), 40 56

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