Synthetic tsunami mareograms for realistic oceanic models

Transcription

1 Geophys. J. Int. (2) 4, Syntheti tsunami mareograms for realisti oeani models Giuliano F. Panza,,2 Fabio Romanelli,3 and Tatiana. Yanovskaya2,4 Dipartimento di Sienze della T erra, V ia E. Weiss 4, 3427, T rieste, Italy. panza@univ.trieste.it 2 International Centre for T heoretial Physis, SAND Group, T rieste, Italy 3 CNR-Gruppo Nazionale per la Difesa dei T erremoti, V ia Nizza, 28-I-98, Rome, Italy 4 Institute of Physis, St Petersburg University, Petrodvoretz, 9894, St Petersburg, Russia Aepted 999 Deember 2. Reeived 999 Deember ; in original form 999 Marh 23 SUMMARY We study how the tsunami mode is generated by a saled double-ouple seismi soure, and how it propagates in realisti oeani models. The method developed and used is the diret extension to tsunami waves propagating in multilayered oeani media of the well-known Haskell method. The most intensive tsunamis may be expeted from soures loated within the rust in the deep-water parts of the oean. The extension to laterally heterogeneous strutures shows that, if the thikness of the oean liquid layer diminishes, the maximum amplitude of the tsunami wave train inreases. Key words: lateral heterogeneity, syntheti mareograms, tsunamis. that of Ward (98). Comer s (984) method is based on a formalism similar to those used for the study of Rayleigh waves, but suffers from some limitations, for example the omputed examples are for a half-spae underlying a single inompressible liquid layer and the exitation problem is studied using a variational approah for a uniform oean depth. Okal (988) studied the influene of the seismi soure para- meters, for example depth, foal mehanism, moment, diretivity, on the generation of tsunami, using the modal approah for laterally homogeneous, strutural models. The method developed and used in this work is a diret extension to tsunami waves propagating in multilayered oeani media of the well-known Haskell method (Haskell 953), whih has been suessfully employed for the study of Rayleigh and Love waves in multilayered strutures and for the on- strution of broad-band syntheti seismograms (e.g. Panza 985; Panza & Suhadol 987; Florsh et al. 99). We solve the equations of elasti motion for the ase when a onstant gravitational field ats in the downward diretion in ultilayered fluid, whih is in sliding ontat with ultilayered solid half-spae where we assume that only elasti fores at. We an thus use the effiient algorithms valid for flat, multilayered oeani models for the prodution of syntheti mareo- grams due to the exitation, by seismi soures, of the tsunami mode propagating in oeani strutures. This method is then extended to laterally heterogeneous strutures in order to study the effet of the variation of the oean bottom. Suh realisti syntheti tsunami mareograms not only are important for a subsequent inversion proess for some important physial parameters (see, e.g. Ritsema et al. 995), but play a relevant role for a quik system of tsunami warning, sine a database of syntheti parameters (e.g. maximum height at various loations) an be onstruted before the event ours. INTRODUCTION If a seismi wave is propagating in a solid medium, the main restoring fore involved is the interation between adjaent partiles; that is, the elasti fore. When fluid media are onsidered this is not always true: the restoring fore at some frequenies an be primarily due to gravity. Among the gravity waves that an propagate near the oeani surfae the most important ones, due to their amplitudes, are tsunamis. Suh waves an be distinguished from other gravity waves by the way in whih they are generated. In fat, most tsunamis are exited by submarine earthquakes, and an be onsidered as seismi sea waves (Ward 989). Owing to their generation mehanism, periods and wavelengths assoiated with tsunamis are longer than those assoiated with ordinary sea waves, and for large submarine earthquakes their amplitudes an be very impressive, espeially when tsunamis approah the shorelines. In this work we study how the tsunami mode is generated by a saled double-ouple seismi soure, and how it propagates in realisti models of oeani media. Several authors have takled the problem of how the tsunami mode is exited by a seismi soure for a fully oupled oean and solid earth model. Pod yapol sky (968, 97) was the first to onsider the tsunami mode as one of the normal modes of the Earth. Ward (98, 98, 982a,b) was the first to apply the free-osillation formalism for a finite body to a spherially symmetri oeani model, and he showed how the energy of a tsunami mode is partitioned among gravitational and elasti omponents for fluid and solid media. Comer (984) onsidered the same problem but using a flat, laterally homogeneous model of the Earth. He studied the exitation of the tsunami mode as a propagating mode, and the theoretial development of the solution is different from RAS Downloaded from on 3 September 28

2 Syntheti tsunami mareograms PHASE VELOCITIES AND EIGENFUNCTIONS OF TSUNAMI WAVES IN MULTILAYERED OCEANIC STRUCTURES The mathematial formulation we adopt for the problem is similar to the one used by Comer (984), exept that we take into aount the elasti fores in the liquid medium in order to obtain the same form of solution for the displaement in the solid and liquid layers. In order to take into aount the effets due to the gravity and elasti fores we follow Pod yapol sky (968; 97): the gravitational fore is assumed to be uniform and ating in the vertial diretion, i.e. gr e, leading to a stati displaement z field whih an be regarded as an equilibrium state. The variation of this fore due to the dynami displaement field, u(x, t), is aused only by the variation of the density, i.e. by Dr/r= VΩu. Then the equations of motion an be written as a2v (VΩu) g e VΩu= 2u z t2 in the liquid layers, and a2v (VΩu) ge z VΩu b2v (V u)= 2u t2 () (2) Figure. Referene system for the laterally homogeneous (-D) model. in the solid layers, where a is the veloity of P waves, b is the veloity of S waves, and u=(u,,w) is the displaement vetor. Eqs () and (2) an be derived from eq. (8.59) of Aki & Rihards (98), if gravitational aeleration g is onstant, the gravity potential, U, is equal to g r z, and onsequently the perturbation of U is g r w. At the free surfae of the fluid, the pressure p, whih in the liquid is due both to the elasti fore and to the hydrostati pressure, has to vanish; that is, In ondition (5b), the terms orresponding to the hydrostati pressure are negleted sine the vertial displaement in the solid is everywhere muh smaller than its average value in the liquid, and the gravity term in eq. (2) will be negleted later; thus, s is the normal stress due only to the elasti fores. At the interfae between the mth and the (m+)th solid layers, all of the omponents of displaement and of stress are ontinuous; that is, p[z +w(z )]=p(z )+ dp w w (z )=w (z ), l l l dzk l m m m+ m z l u (z )=u (z ), = r a2 VΩu +r gw =, (3) m m m+ m (6) l l l l l z l s (z )=s (z ), m m m+ m where the seond-order terms are negleted. t (z )=t (z ), At the interfaes between the liquid layers, the pressure and m m m+ m the vertial omponent of motion have to be ontinuous. If the jth and ( j+)th liquid layers are onsidered, the ontinuity for m N. onditions should be satisfied at the perturbed interfae, i.e. at The solution of the equation of motion in the jth liquid z=z +w(z ), but for the vertial omponent of displaea harmoni wave propagating along the x-axis with angular layer, due to both elasti and gravitational fores, in terms of ment it is equivalent to onsider the unperturbed interfae, i.e. z=z. The onditions of ontinuity between the jth and frequeny v and phase veloity, is ( j+)th liquid layers give w (z )=w (z ), u (x, z, t)= ia2 v C A exp A g 2() z r a2 VΩu +r gw z = r a2 VΩu +r gw z. At the liquid solid boundary (zeroth interfae aording to Fig. ), we have w (z )=w (z ), r a2 VΩu (z )=s (z ), (4) (5a) (5b) w (x, z, t)= exp A g () z D exp[i(vt kx)], v2 C g () A exp A g 2() z g 2() exp A g () z D exp[i(vt kx)], =t (z ). (5) (7) 2 RAS, GJI 4, Downloaded from on 3 September 28

3 5 G. F. Panza, F. Romanelli and T.. Yanovskaya where k is the horizontal wavenumber, i is the imaginary unit, and g = vy g, () 2 where r am= G i[ ( /)2]/2 if >, [( /)2 ]/2 if <. Introduing the boundary onditions (see Appendix A), the g =vy g, (8) 2() 2a dispersion equation an be onstruted and the eigenvalues y2 = a2 2 + g2 determined following standard methods (see Appendix ). 4a2 v2. One the phase veloity is determined, the oeffiients in the expressions (7) and (9) for the eigenfuntions an be determined by a reursive proedure. The oeffiients for the fluid and The solution of the equation of motion in the mth solid the solid layers are alulated separately, eah with its own layer is arbitrary normalization. Then from any boundary ondition u (x, z, t) at the liquid solid boundary (for example, the one orresponding to the ontinuity of the vertial displaement, 5a), the m = Gia2 m oeffiients are redued to the same normalization. v C C m exp A g 2m z +D m exp A g m z D For the solid layers, the equation (see Appendix ) ir bm v C E m exp A vr bm z +F m exp A vr bm z exp[i(vt kx)], w (x, z, t) m = G v2 C A g m + gb2 m a (a2 b2 m m m ) C m exp A g 2m z Ag 2m + gb2 m a (a2 b2 m m m ) D m exp A g m z D where b2 m v C E m exp A vr bm z exp[i(vt kx)], +F m exp A vr bm z DH DH (9a) where the matries L and K are defined in Appendix A. When the oeffiients for the last ( j=) liquid layer are deter- mined, the oeffiients in the solid and in the fluid layers are mathed by equating the vertial displaements at the liquid solid boundary, and from this ondition the normalization oeffiient, for all the liquid layers, an be determined. r bm= G i[ ( /)2]/2 if >, (a) [(b /)2 ]/2 if <b. m m Gravity has no effet on the shear waves (9a), and it is easy to estimate that the effet of gravity on the total displaement is minor. Therefore in the following we neglet the gravity fore in the equations of motion for the solid medium. The formulas (9a) then redue to u m (x, z, t) = Gia2 m v C C m exp A vr am z ir bm v C E m exp A vr bm z exp[i(vt kx)], w (x, z, t) m C C m exp A vr am z v C E m exp A vr bm z = Gr am v b2 m exp[i(vt kx)], +D m exp A vr am z +F m exp A vr bm z +D m exp A vr am z +F m exp A vr bm z D DH D DH (9b) (b) CL D 2 D N na L C N = () E N is used for the alulation of the oeffiients of the eigenfuntions in the half-spae C, E. One of them may be assumed n n to be equal to, and the oeffiients for the upper layers are determined using Haskell (953) layer matries up to the first solid layer. For the fluid layers we start from the oeffiients of the uppermost layer, whih are determined from eq. (3) orresponding to the ondition of free surfae: S la A l =, (2) l where the S matrix is defined in Appendix A. The oeffiients for the lower liquid layers are determined reursively from the equation A A =L L K A A, (3) 3 TSUNAMI WAVES EXCITED Y SEISMIC SOURCES For a laterally homogeneous oeani model, one the eigenvalues and the eigenfuntions are determined it is possible to alulate the syntheti mareogram due to the exitation of the tsunami mode by a double-ouple seismi soure by using the asymptoti expression for a harmoni wave (Levshin et al. 989): U= exp( ip/4) 8p exp[iv(t X/)] Rx X v v I g u v g I, (4) where v g is the group veloity, X is the epientral distane, u=u(z, v)=u(z, v)e x +w(z, v)e z, R= R(v) exp[i arg(r(v))] is the Fourier transform of the soure time funtion, and x 2 RAS, GJI 4, Downloaded from on 3 September 28

4 Syntheti tsunami mareograms 5 represents the part of the exitation (radiation pattern) that is a funtion of the foal mehanism and of the soure depth, h s. The expression for x for a double-ouple point soure and the expression for I, the energy integral, are given in Appendix C For weakly laterally heterogeneous media the asymptoti expression for the harmoni wave beomes (Levshin et al. 989) U= exp( ip/4) 8p exp[iv(t t)] J Rx C v v I g D s C u v g I D r, (5) where the subsripts s and r indiate that the quantities are alulated at the soure and at the reeiver, respetively. Eq. (5) is obtained by traing the ray onneting the projetions of the soure and the reeiver site onto the (x, y)-plane. The expressions for the geometrial spreading, J(x, y), and the traveltime at the reeiver, t(x, y), whih replae X/ and X in (4), respetively, are given in Appendix D. The other spetral quantities appearing in (5) an be alulated by determining loally, for the points in the (x, y)-plane, the eigenvalues and the relevant eigenfuntions. We assume that the struture is varying only along the diretion of propagation; that is, along the x-axis. This restrition is not important when the phase veloity is hanging approximately linearly along the path. In fat, as is shown in Appendix D, the differene between the geometrial spreading (and the traveltime) alulated for our model and for odel in whih the angle between the diretion of propagation and the diretion of variation of the struture is not zero is negligible for a wide range of angles. Therefore, in a 2-D medium, the phase veloity, whih enters into the alulations for t and J, an be onsidered a funtion of x only. Sine (x) annot be omputed exatly, we assume that varies linearly between the values orresponding to the strutural models limited by the vertial planes, parallel to the ( y, z)-plane, passing through x=x (=), x,,x (=X) (see Fig. 2a). At n eah point x (i=,, n), the phase veloity of the tsunami i mode,,, an be evaluated using the formalism of n Setion 2. The formulas for t and J are given in Appendix D. From a omparison of (4) and (5), it is straightforward to obtain the expression for the so-alled shoaling fator ; that is, the orretion term that an be applied to the tsunami amplitudes, given by (4), to aount for the lateral variability of the oean depth. Considering the ratio between the amplitudes of the vertial omponents of motion at the free surfae (i.e. at z=) for two different sites, and 2 at distanes X and X, respetively, in a laterally varying oeani model, from 2 (5) one has K W(X 2,,v) W(X,,v)K = C w(, v) 2 v g I w(, v) v g I 2 D J J 2. (6) If the oean bottom is assumed to be rigid and the fluid is inompressible, the eigenfuntions beome (see eq. 7) w(z, v)=w(, v) sinh[k(h z)], sinh(kh) u(z, v)= iw(, v) osh[k(h z)], sinh(kh) (7) Figure 2. Referene system for the laterally heterogeneous model. (a) Lateral view: the star and the triangle represent, respetively, the soure and the reeiver site position; (b) top view; () top view orresponding to the model shown in (a). where H is the oean depth. Under this assumption, using (7), the energy integral (see Appendix C) beomes rw2(, v) I = sinh2(kh) sinh(2kh ). (8) 2k Furthermore, at long periods (e.g. see Fig. 4) one has #v g # gh, rgw2(, v) I #, v2 (9) and the first term on the right-hand side of eq. (6) beomes 4S H. (2) H 2 2 RAS, GJI 4, Downloaded from on 3 September 28

5 52 G. F. Panza, F. Romanelli and T.. Yanovskaya Thus, if the variations in the geometrial spreading and in 5 km from a double-ouple soure with a seismi moment the arrival times are negleted, that is J =J, the effet of the of 3 N m and a foal depth of 4 km from the free surfae 2 lateral variation of the oean depth expressed in eq. (6) (i.e. km deep in the solid media). The strike reeiver angle beomes equivalent (Ward, personal ommuniation) to the orresponds to aximum of the radiation pattern, and this simplified form of the shoaling fator, known as Green s Law hoie is used for all the omputations. The strutural model (e.g. Dingemans 997, p. 76), given by (2). that is adopted in the omputations orresponds to the one used by Comer (984): a 4-km liquid layer (a=.5 km s, r=. g m 3) over a solid half-spae with a=7.5 km s, 4 EXAMPLE CALCULATIONS IN THE b=4. km s, r=3. g m 3. A general harateristi is that FREQUENCY DOMAIN the dip-slip mehanism (suffix ds in Fig. 3a) is more effetive The expressions (4) and (5), together with the formulae in the generation of tsunami motion than strike-slip, while the shown in Appendix C, reveal immediately that the exitation radial omponent of motion is greater than the vertial one proess of the tsunami mode by a seismi soure is dependent by a fator of about 3. on several parameters. In this work we assume that the slip These results are similar to those obtained by Okal (988), ours on a vertial fault, and therefore we onsider dip-slip who also found that ehanism orresponding to an up-dip and strike-slip as fundamental mehanisms. For an extended slip on a fault dipping at 45 (T45) is more effetive than dipslip when the soure is in the upper solid region; however, for study of the influene of the foal mehanism, foal depth, and other parameters on the generation of tsunamis, the reader is the foal depths that are onsidered in this work, the results referred to Okal (988). for the T45 and dip-slip soure are pratially idential (see Fig. 3(a) shows the amplitude spetra, for the radial and the Figs 7 and 9 in Okal 988). vertial omponents of motion, alulated at a distane of Fig. 3( b) shows the amplitude spetra for various thiknesses of the liquid layer: 4, 6 and 8 km. The parameters of the solid half-spae are the same as in Fig. 3(a), the foal mehanism onsidered is dip-slip, and the soure is loated km deep in the solid layers. The most important effet is due to the dependene of the phase and the group veloities on the thikness of the water layer, whih is shown in Fig. 4 (.f. Fig. of Ward 98). To study the effet of the layering of the solid struture, the soure radiation spetra are alulated for three solid models: model orresponds to the homogeneous half-spae used by Comer (984); model 2 is the same as model with a sedimentary layer (a=3.5 km s, b=. km s, r=.5 g m 3) km thik on top of the half-spae; and model 3 has a rustal layer (a=5.2 km s, b=3. km s, r=2.6 g m 3) 2 km thik over the half-spae (a=8. km s, b=4.7 km s, r=3.2 g m 3). The thikness of the oean is fixed at 4 km, while the seismi soure orresponds to a dip-slip foal mehanism. Fig. 5 shows the amplitude spetra for the vertial omponent of motion, alulated using model and model 3. The soure reeiver distane is 5 km, and two foal depths, Figure 3. Amplitude spetra alulated for a double-ouple soure (3 N m seismi moment) 5 km from the reeiver. (a) Radial (x) and vertial (z) omponents of motion for pure strike-slip (ss) and pure dip-slip (ds) mehanisms; ( b) radial and vertial omponents for a liquid layer 4, 6 and 8 km thik. Figure 4. Phase- (p) and group- (g) veloity dispersion urve of the tsunami mode for three values of the thikness of the liquid layer: 4 (solid lines), 6 (dashed lines) and 8 (dotted lines) km. 2 RAS, GJI 4, Downloaded from on 3 September 28

6 Syntheti tsunami mareograms 53 Figure 5. Amplitude spetra alulated for a dip-slip double-ouple soure (3 N m seismi moment) using model and model 3 (see text for the elasti parameter speifiation). The soure reeiver distane is 5 km and two foal depths, 9 and 4 km (seond part of the aronym), are shown. 9 and 4 km, are shown. While the influene of the soure reeiver distane on the amplitude spetra is a sale fator, the ombination of the soure-depth effet with the solid layering is more interesting. Fig. 5 shows that earthquakes within the rust an exite the tsunami mode more intensively than an deeper earthquakes, and that the lower the S-wave veloity at the depth of the soure, the larger the exitation of the tsunami mode. Calulations arried out for various ombinations of a and b show that the parameter ontrolling the amplitude of the tsunami waves is the S-wave veloity rather than the P-wave veloity. For soures loated below the sediment layer, the differenes in the amplitude spetra for models and 2 are negligible, and in Fig. 5 the urves alulated for model 2 would exatly overlap those for model. This an be explained by omparing (see Figs 6a and b) the eigenfuntions for the three models as funtions of depth. The frequeny hosen (.7 Hz, i.e. T =43 s) orresponds approximately to the maxima of the amplitude spetra shown in Fig. 5. The urves shown in Fig. 6 are in good agreement with those shown by Ward (98, Figs 3 and 4) for a period of 5 s, and the same onlusions were obtained by Okal (988), who onsidered a rupture propagating in a sedimentary layer. 5 SYNTHETIC MAREOGRAMS Fig. 7 shows the syntheti signals, obtained from (4) and (5), for a dip-slip mehanism with M =8.2; the foal depths are w h =4 km (Fig. 6a) and h =34 km (Fig. 6b). We aount s s for soure finiteness by properly weighting the point-soure spetrum using the saling laws of Gusev (983), as reported in Aki (987). The strutural model is model 3 of Setion 4. For eah of the two soure reeiver distanes onsidered, the upper trae refers to the -D model and the lower trae to a laterally varying model. In the laterally varying model the liquid layer beomes thinner with inreasing distane from the soure, with a gradient of.75, and the uppermost solid layer ompensates for this thinning. At 5 km from the soure, the effet of the lateral variation of the sea bottom is negligible: Figure 6. (a) Eigenfuntions of the radial (solid line) and vertial (dotted line, normalized to at the free surfae) omponents of motion at a frequeny of.7 Hz, in the fluid. The urves for models, 2 and 3 overlap exatly; (b) eigenfuntions in the solid layers. this is not surprising sine Dh, the differene in the thiknesses of the liquid layer between the -D and the 2-D models, is only km. At distanes greater than 2 km from the soure, Dh is equal to 3.5 km, and the effets due to the variable bathymetry are learly visible: () the 2-D wave train arrives later than the -D wave train; and (2) the amplitudes of the 2-D signals are generally larger. These effets an be explained by the dispersion of the tsunami mode: for a large range of periods ( 2 s approximately) the phase and group veloities are diretly proportional to the square root of the thikness of the liquid layer (see Fig. 4), and when the tsunami mode propagates in strutures with a thinning liquid layer the differenes between the veloities at the various periods beome smaller, making the wave train more onentrated in time. The other fator that auses the slower deay of the amplitudes of the tsunami wave train is the onservation of the energy flux, sine as the wave train propagates it is spread over a thinner liquid layer. 2 RAS, GJI 4, Downloaded from on 3 September 28

7 54 G. F. Panza, F. Romanelli and T.. Yanovskaya Figure 8. Curves of the maximum height of the alulated tsunami signal (vertial omponent) versus the epientral distane. The urve labels denote the -D model ( or 3) and the magnitude (M w ) adopted in the alulations. The symbols denote the data of Abe (995) for two magnitudes (squares for M s =8., irles for M s =7.7). epientral distane of 2 km, using both model and model 3 of Setion 4, and a dip-slip soure with the foal depth equal to 4 km and M w equal to 8. and 7.7. As a further example, we show some syntheti mareograms alulated for the tsunami mode propagating in a realisti model. The sheme of the laterally heterogeneous model onsidered is shown in Fig. 9(a), and represents a possible senario for a tsunami exited by an earthquake generated at a subdution zone. Fig. 9(b) shows the syntheti mareograms (vertial omponent) alulated at various distanes along the setion shown in Fig. 9(a) for a lateral extension of zone C equal to 5 km. The elasti parameters of the rustal layer and of the half-spae are those of model 3. The soure has a dip-slip foal mehanism, foal depth of 6 km and M w =8.. In Fig. we show the maximum amplitude of the tsunami wave train (vertial omponent) versus epientral distane, X, for various ombinations of magnitude, mehanism and foal depth, whih are alulated using the strutural model of Fig. 9(a). A omparison of the urves of Fig. and those shown in Fig. 8 indiates that the lateral variations of the sea bottom an strongly modify the height of the tsunami wave trains, in terms of amplifiation or de-amplifiation aording to the thinning or the thikening of the liquid layers. Figure 7. Syntheti signals for the tsunami mode (vertial omponent) exited by a dip-slip mehanism with M =2.2 2 N m. (a) h =4 km; s (b) h =34 km. For eah soure reeiver distane, X, the upper trae s refers to the -D model and the lower trae to the 2-D model. In order to ompare the theoretial parameters that an be extrated from syntheti signals with experimental ones, we show in Fig. 8 the data set of Abe (995), with the maximumamplitude urves (vertial omponent) alulated up to an 6 CONCLUSIONS The method developed allows us to alulate syntheti signals due to the tsunami mode, exited by a saled double-ouple soure and propagating into an oean, of variable thikness, fully oupled with the solid earth. The inrease of the tsunami wave-train maximum amplitude in shallow water is aused not only by the onservation of the energy flux for a given frequeny and then a distribution of the flux over a thinner water layer, but also by the derease of the veloity dispersion at the frequenies orresponding to the maximum radiation. The two main pratial appliations of our results are as follows. () Inversion of experimental tsunami reordings for some soure parameters. The effetiveness of suh an approah depends 2 RAS, GJI 4, Downloaded from on 3 September 28

8 Syntheti tsunami mareograms 55 Figure. Curves of the maximum height of the alulated tsunami signal (vertial omponent) versus the epientral distane. Eah aronym shows the foal mehanism (ds indiates dip-slip; ss, strike-slip), the foal depth (km) and the magnitude (M w ) adopted in the alulations. The model adopted in the alulations is shown in Fig. 9(a). ACKNOWLEDGMENTS We are grateful to Dr Stephen Ward, whose useful omments allowed us to larify the basi assumptions of our modelling and who suggested the onnetion with Green s Law, and to Prof. Emile A. Okal for a ritial review. We aknowledge finanial support from GNDT (ontrat PF54), MURST (4 per ent and 6 per ent funds), and EEC Contrat EV5V-CT REFERENCES Abe, K., 995. Estimate of tsunami run-up heights from earthquake magnitudes, in T sunami: Progress in Predition, Disaster Prevention and Warning, pp. 2 35, eds Tsuhiya, Y. & Shuto, N., Kluwer, Dordreht. Aki, K., 987. Strong motion seismology, in Strong Ground Motion Seismology, NATO ASI Series, Series C: Mathematial and Physial Sienes, Vol. 24, pp. 3 39, eds Erdik, M. & Toksöz, M., Reidel, Dordreht. Aki, K. & Rihards, P.G., 98. Quantitative Seismology, Freeman, San Franiso. en-menahem, A. & Harkrider, D.., 964. Radiation patterns of seismi surfae waves from buried dipolar point soures in a flat stratified Earth, J. geophys. Res., 69, Figure 9. (a) Sketh of the laterally heterogeneous model for a realisti Comer, R.P., 984. The tsunami mode of a flat earth and its exitation senario. The numbers refer to the thikness (km) of the water and of by earthquake soures, Geophys. J. R. astr. So., 77, 27. the rustal layer, and to the lateral extension of eah zone. ( b) Syntheti Dingemans, M.W., 997. Water Wave Propagation Over Uneven mareograms (vertial omponent) alulated at various distanes along ottoms, Part I L inear Wave Propagation, World Sientifi, Singapore. the setion shown in (a). The extent of zone C is 5 km. Florsh, N., Fäh, D., Suhadol, P. & Panza, G.F., 99. Complete syntheti seismograms for high-frequeny multimode SH-waves, on the availability of tsunami measurements that are not Paageoph, 36, heavily affeted by unknown tide-gauge instrumental response Gusev, A.A., 983. Desriptive statistial model of earthquake soure or by harbour resonanes: when tsunami reords from pressure radiation and its appliation to an estimation of short period strong gauges in open oean are available, some very long-period motion, Geophys. J. R. astr. So., 74, soure harateristis an be revealed (Ritsema et al. 995). Haskell, N.A., 953. The dispersion of surfae waves in multilayered (2) Constrution of a database for a quik system of tsunami media, ull. seism. So. Am., 43, Knopoff, L., 964. A matrix method for elasti wave problems, ull. warning. The alulated tsunami signals allow us to determine seism. So. Am., 54, easily, for any soure of interest, the maximum height of Levshin, A.L., Yanovskaya, T.., Lander, A.V., ukhin,.g., tsunami motion expeted at various loations, taking into armin, M.P., Ratnikova, L.I. & Its, E.N., 989. Surfae waves in aount the loal variations of the sea bottom. These om- media with weak lateral inhomogeneity, in Seismi Surfae Waves in putations an of ourse be made before the event ours, thus a L aterally Inhomogeneous Earth, pp , ed. Keilis-orok, V.I., optimizing the effiieny of any warning system. Kluwer, Dordreht. 2 RAS, GJI 4, Downloaded from on 3 September 28

9 56 G. F. Panza, F. Romanelli and T.. Yanovskaya Okal, E.A., 988. Seismi parameters ontrolling far-field tsunami pressure is gw (r r ). Therefore we may assume that amplitudes: a review, Natural Hazards,, at the boundary z=z there is ontinuity of the quantity Panza, G.F., 985. Syntheti seismograms: the Rayleigh waves modal r a2 (VΩu ) gr w. summation, J. Geophys., 58, Using (7) and omitting the fator exp[i(vt kr)], one Panza, G.F. & Suhadol, P., 987. Complete strong motion synthetis, obtains that in Seismi Strong Motion Synthetis, pp , ed. olt,.a., Aademi Press, Orlando. Panza, G.F., Shwab, F.A. & Knopoff, L., 973. Multimode surfae (A3) waves for seleted foal mehanisms. I. Dip-slip soures on a vertial fault plane, Geophys. J. R. astr. So., 34, Pod yapol sky, G.S., 968. Generation of long periodi gravitational wave in the oean by seismi soure in the rust, Izvestia AN SSSR, Fizika Zemli,, 7 24 (in Russian). Pod yapol sky, G.S., 97. Generation of the tsunami wave by the earthquake, in T sunamis in the Paifi Oean, pp. 9 32, ed. A kw p =L K A A, where =A L a y g 2v y + g 2v r a2 A + gg () v2 r A a2 + gg 2() a v2, H g 2() K =Aexp A (A4) exp A H g (), (A5) Adams, W.M., East west Center Press, Honolulu. Ritsema, J., Ward, S. & Gonzalez, F.I., 995. Inversion of deep-oean tsunami reords for gulf of Alaska earthquake parameters, ull. seism. So. Am., 85, Shwab, F.A., 97. Surfae-wave dispersion omputations: Knopoff s method, ull. seism. So. Am., 6, Shwab, F.A. & Knopoff, L., 972. Fast surfae wave and free mode omputations, in Methods in Computational Physis, pp. 86 8, ed. olt,.a., Aademi Press, New York, Shwab, F., Nakanishi, K., Cusito, M., Panza, G.F., Liang, G. & Frez, J., 984. Surfae-wave omputations and the synthesis of theoretial seismograms at high frequenies, ull. seism. So. Am., 74, Ward, S.N., 98. Relationships of tsunami generation and an earth- quake soure, J. Phys. Earth, 28, Ward, S.N., 98. On tsunami nuleation: I. A point soure, J. geophys. Res., 86, Ward, S.N., 982a. On tsunami nuleation: II. An instantaneous modulated line soure, Phys. Earth planet. Inter., 27, Ward, S.N., 982b. Earthquake mehanisms and tsunami generation: the Kurile Islands event of Otober 3, 963, ull. seism. So. Am., 72, Ward, S.N., 989. Tsunamis, in T he Enylopaedia of Solid Earth Geophysis, pp , ed. James, D.E., Van Nostrand Reinhold Company, New York. and H is the thikness of the jth liquid layer. The layer matrix, whih relates the vertial omponent of displaement and the normal stress in the jth liquid layer to the same quantities in the ( j+)th liquid layer, is D =L K L. (A6) The relationship between the oeffiients in (7) for the uppermost liquid layer and those for the first liquid layer an be found using (A3) and (A6): A A l l =L l D l D l+ D 2 L A A, APPENDIX A: OUNDARY CONDITIONS and the boundary ondition (A) an be written The boundary ondition at the free surfae, eq. (3), an be written as a2 VΩu gw =, (A) l l l or, in matrix form, S la A l l = Ar l a2 la gg ( l) a l v2 r l a2 la +gg 2( l) a l v2a A l l =. (A2) The boundary onditions at the interfaes between liquid layers orrespond to the ontinuity of the vertial omponent of displaement, w, and of the pressure, p, at the interfae z +w (sine in an ideal fluid the ondition of ontinuity of the tangential omponent of displaement is not required). We assume that the variation in depth of the interfae has no effet on the boundary ondition of the displaement, and we aept ontinuity of w at z=z instead of z=(z +w ), while for p the variation is taken into aount at z=z. The pressure inrement due to the variation of the hydrostati S l L l D l D l+ D 2 L A A =M A A =. (A7) (A8) The expression for the pressure in the last liquid layer (-st in Fig. ) is p =r a2 VΩu =R AA, (A9) and the expression for the vertial omponent of displaement an be written in the same form: kw = a v (g ( ) =P AA. g 2( ) )K A A (A) The boundary onditions (eqs 5a to 5b) at the liquid solid interfae an be written, using (9) (A9) and (A), in the 2 RAS, GJI 4, Downloaded from on 3 September 28

10 following matrix notation: A r a a r a a b2 2 r a2 A 2b2 2 r A a2 2b2 2 2m b r b b2 2 2m b r b AC F = A p p 2 r r 2 A A. (A) From (A8), ontaining the matrix M=(m, m ), one obtains 2 A = m 2 K, =m K, (A2) where K is an unknown multiplier, and the right-hand side of (A) an be written in the following form: FK= A f f 2 K= A p m 2 +p 2 m r m 2 +r 2 m K. D E (A3) The two equations (A) an be redued to one for the oeffiients of the first solid layer, and onsidering the ondition (5) one has D CAC E F =, (A4) where C is a (2 4) matrix, and the problem is now formally equivalent to that for a system of solid layers, where (A4) replaes the ondition of free surfae, and the dispersion funtion an be onstruted using either the Haskell (953) or Knopoff (964) method. APPENDIX : METHODS HASKELL AND KNOPOFF After some manipulations, eq. (A4) an be transformed to AC D C E F =2f r2 A 4 ( ) q/(r 2) D AC E F =, qr 2 Syntheti tsunami mareograms 57 onditions (6) to expressions (7) we obtain L m K mac m D m E m F m=l m+ac m+ D m+ E F m+, =A where the matries L m and K m are a2 a2 m m 2 2 a r m am r am L m 2m m r 2m a r am m m am r a2 ( ) r a2 ( ) m m m m m m r bm b2 m 2 r bm b2 m 2 (3), m ( ) m ( ) m m m m 2m b r 2m b r m m bm m m bm vr H am m K =Aexp A exp Avr am H m m exp A vr bm H m exp Avr bm H m. (4) Using (3) reursively, the relationship between the oeffiients of (9) in the first solid layer and the oeffiients in the half-spae (nth layer) an be written as AC D E F =L D 2 D n L n A C n E n, (5) where the matrix L (4 2) ontains the first and the third n olumns of the matrix L, and the layer matrix is D =L K m m m L (m=2,, n ). Using (A3), one obtains m (3), where all of the matries appearing are real. ( ) APPENDIX C: THE RADIATION PATTERN AND THE ENERGY INTEGRAL () where r is the density of the uppermost solid layer, =2b2 /2 and q= f 2 / f. Following Shwab (97), the matrix T an be extrated from (): T =( ( ) ( )2 2 q/r 2 ( )), (2) and Knopoff s algorithm (Shwab & Knopoff 972) an be immediately applied. In order to derive the dispersion equation by the Haskell (953) method, we start from (A3). At the interfae between the mth and (m+)th solid layers, applying the boundary 2 RAS, GJI 4, In eq. (4), x represents the azimuthal dependene of the exitation fator, whih for a double-ouple point soure (e.g. en-menhaem & Harkrider 964) is x(h, Q)=d +i(d sin Q+d os Q)+d sin 2Q+d os 2Q, s and d = 2 (h ) sin l sin 2d, s d = C(h s ) sin l os 2d, d = C(h ) os l os d, 2 s d =A(h ) os l sin d, 3 s d = 4 2 A(h ) sin l sin 2d, s (C) (C2) Downloaded from on 3 September 28

11 58 G. F. Panza, F. Romanelli and T.. Yanovskaya where Q is the angle between the strike of the fault and the diretion epientre station measured ounter-lokwise, h is s the foal depth, d is the dip angle, and l is the rake angle. The funtions of h that appear in (C2) depend on the values s assumed by the eigenfuntions at the hypoentre: A(h s )= iku(h s ), (h s )= iku(h s ) A3 4 b2(h s ) a2(h s ) 2 zk C w iku(h ) b2(h s A 2 s ) a2(h hs s )D, C(h )= u ikw(h ). s zk s hs (C3) Any possible radiation pattern an be obtained by the ombination of three elementary onfigurations (e.g. Aki & Rihards (98), pp. 7 8): a strike-slip mehanism on a vertial fault, a dip-slip mehanism on a vertial fault, and a purely thrusting mehanism on a fault dipping at d=p/4. The syntheti signal an be obtained with three signifiant digits using eq. (4) as long as the ondition kx> is satisfied (Panza et al. 973). The quantity I in (4) is the energy integral defined as I = 2 P 2 r[(u*)2+w2]dz, (C4) (see eq in Aki & Rihards 98), and, for ultilayered half-spae formed by homogeneous layers (see Fig. ), expression (C4) an be written as I = n+l 2 r i= iap H i (u*)2dz+ P H i w2dz. (C5) i i Using (7) and (9) for liquid and solid layers, respetively, one an easily solve the integrals in (C4) analytially, sine only trigonometri or hyperboli funtions are involved (Shwab et al. 984). and, as shown in Fig. 2(b) (X, Y ) is the position of the reeiver site, p is the ray parameter, i.e. sin h/, whih is onstant along the ray-mode trajetory, and h is the angle between the diretion of propagation and the x-axis (h at x =). If the phase veloity, (x), varies linearly along the x-axis, i.e. = (+ex), then at x=x Y = P X sin h (+ex) sin2 h (+ex)2 dx= (os h os h), (D4) e sin h t= P X dx (+ex) sin2 h (+ex)2 = [osh (/sin h) osh (/sin h )] e, (D5) J= (os h os h) = Y. e sin2 h sin h (D6) If the wave is propagating along the x-axis, so that Y = at x=x, one has to onsider the onditions h, h, and eqs (D5) to (D6) beome t= log(+ex) e, (D7) J= ex A+ X. (D8) 2 If (x) is a pieewise linear funtion (see Fig. 2), the traveltime is given by a sum of (D5) or (D7), and the geometrial spreading, (D6) and (D8), beomes h os h n sin h J=Gos n i= n i= y i os h i os h i h, A + e i x i 2 x i h =, where the y (i=,, n) are alulated using formula (D4) at i x=x. i To justify the assumption that h an be onsidered equal APPENDIX D: FORMULAE FOR THE to zero, we alulate the traveltimes and the geometrial GEOMETRICAL SPREADING AND THE spreading using (D5) to (D7) and (D6) to (D8) under the TRAVELTIME assumption that the veloity varies in the diretion OX. Assuming that the elasti parameters are varying horizontally Assuming a veloity variation similar to that used in this study only along the x-diretion, i.e. that the phase veloity is =(x), (from.2 km s at x= to.4 km s at X=2 km), the traveltime at the reeiver, t(x, Y ), and the geometrial the differene in the geometrial spreading evaluated by (D9) spreading, J(X, Y ), are given by is about 5 per ent for h =46, and per ent for h =64. Suh a differene is obtained at low frequenies, for whih the t= P X dx differene in veloities due to the derease of the thikness (x) p2(x)2(x), (D) of the water layer is maximum. For higher frequenies, the differene in veloities dereases, and the effet is even smaller. (D2) Estimation of the differene in the arrival times for the same J= Y h os h, models shows that the signal arrives s earlier if h =25, where and s earlier if h =72. This does not lead to a signifiant differene with respet to the signals alulated for normal Y = P X p(x)(x) inidene, even if the take-off angle with respet to the x-axis p2(x)2(x) dx, (D3) is suffiiently large (~6 7 ). (D9) 2 RAS, GJI 4, Downloaded from on 3 September 28