Acoustic resonant oscillations between the atmosphere and the solid earth during the 1991 Mt. Pinatubo eruption

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 5,, di:0.029/200jb007747, 200 Acustic esnant scillatins between the atmsphee and the slid eath duing the 99 Mt. Pinatub euptin Shing Watada and Hi Kanami 2 Received June 200; evised 23 Septembe 200; accepted 5 Octbe 200; published 2 Decembe 200. [] Lng peid hamnic Rayleigh waves wee bseved n seismmetes duing the 99 Mt. Pinatub euptin in the Philippines. The amplitude spectum f the Rayleigh waves shws tw distinct peaks at peids f abut 230 and 270 s. In the Eath s atmsphee, lng wavelength standing acustic waves ae bunded in a lw sund velcity channel between the themsphee and the gund. The Rayleigh waves and the fundamental and fist vetne f atmspheic acustic waves tapped in the lw sund velcity channels have appximately the same hizntal wavelength and fequency at peids f 230 and 270 s, espectively, i.e., the atmsphee and the slid eath satisfy the cnditin f acustic esnant scillatins. The standing atmspheic lng wavelength acustic waves set ff by the euptin selectively excited seismic spheidal mdes nea the esnant peid thugh acustic esnant cupling and esulted in hamnic Rayleigh waves. In cntast, gavity waves and Lamb waves (atmspheic bunday waves) d nt cuple t the gund efficiently and ae nt easily bseved as gund distubance n seismgams duing vlcanic euptins. Citatin: Watada, S., and H. Kanami (200), Acustic esnant scillatins between the atmsphee and the slid eath duing the 99 Mt. Pinatub euptin, J. Gephys. Res., 5,, di:0.029/200jb Intductin [2] A maj euptin f Mt. Pinatub (5.4 N E) in the Philippines ccued n 5 June 99. Kanami and Mi [992] and Widme and Zün [992] ecgnized that lng peid gund mtins duing the euptin ae dminated by scillaty mvements at distinct peids and ae ecded at many wldwide seismgaphic netwks. They cnfimed fm thei gup and phase velcities and the etgade gund paticle mtin that the waves ae fundamental mde Rayleigh waves adiated fm Mt. Pinatub. The bseved wave tains have tw distinct peaks at 230 and 270 s in the amplitude specta. Widme and Zün [992] als epted that simila bichmatic Rayleigh waves with peids f 95 and 266 s wee adiated fm the 4 Apil 982 El Chichón euptin. [3] Vaius mechanisms f the geneatin f the hamnic waves have been ppsed. Widme and Zün [992] suggested that a feedback system between lcal atmspheic scillatins and the euptin pcess ae espnsible f the excitatin f Rayleigh waves. Kanami and Mi [992] intepeted them as the seismic Rayleigh waves excited by atmspheic scillatins set ff by the euptin. [4] If the suce is the acustic esnance f a magma chambe as suggested by Widme and Zün [992], the Eathquake Reseach Institute, Univesity f Tky, Tky, Japan. 2 Seismlgical Labaty, Califnia Institute f Technlgy, Pasadena, Califnia, USA. Cpyight 200 by the Ameican Gephysical Unin /0/200JB size f the chambe equied t explain the bseved lngpeid scillatins is at least seveal hunded kilmetes and is unealistic. [5] Kanami and Mi [992] epted that the suce phase f the bseved Rayleigh waves is azimuthally independent and the scillaty pessue change bseved nea Mt. Pinatub (D = 2 km) was abut 3.5 mba ( ba = 0 5 Pa). They btained the time histy f the atmspheic lading nea the vlcan by decnvlving the bseved Rayleigh waves with the synthetic gund mtin cmputed f a delta functin vetical single fce. The amplitude f the atmspheic lading fce is abut N(N= 0 5 dyne). Assuming that the pessue change ccued simultaneusly ve a cicula aea, they ughly estimated the adius f the aea t be abut 40 km. [6] Kanami et al. [994] shwed that a nea suce seismgam, which pesumably wked as a bagam, ecded duing the 980 Mt. St. Helens euptin displayed a simila scillatin with a peid f abut 300 s. They tied t explain these spectal peaks bseved at Mt. Pinatub and Mt. St. Helens in tems f chaacteistic mdes f atmspheic scillatins. They shwed that in an isthemal atmsphee tw distinct mdes ae excited by a pint suce in the atmsphee. They als shwed that the tw peaks, ne at the acustic cutff fequency and ne at a fequency less than the buyancy fequency, appea in the amplitude specta f nea suce synthetic bagams. Based n these studies, they intepeted that the peaks f the Rayleigh wave specta bseved f the euptin f Mt. Pinatub and f the seismgam ecded nea the Mt. St. Helens euptin cespnd t these tw chaacteistic fequencies in the atmsphee. f20

2 [7] Lng peid ai waves in the atmsphee, with a peid f abut 3 30 min, fm vaius suces such as vlcanic euptins [Pekeis, 948; Pess and Hakide, 966; Hakide and Pess, 967; Mikum and Blt, 985], a meteite impact [Pekeis, 939], gund defmatin assciated with a lage eathquake [Mikum, 968; Mikum et al., 2008] and nuclea explsins in the ai have been studied by many investigats [e.g., Geges, 968; Piece and Psey, 970]. These bseved waves ae ppagating acustic and gavity waves in the fa field fm the suce and nne f these studies addessed the enegy cupling f atmspheic waves t seismic Rayleigh waves nea the suce. Lgnnné et al. [998] cmputed a lng peid gund mtin f a suce in the ai using the nmal mde methd. A eview f lngpeid ai waves fm seismic suces is fund by Mikum and Watada [200], and a eview f atmspheic/inspheic signals fm eathquakes is given by Lgnnné [200]. [8] In this pape, we investigate () the acustic cupling between seismic Rayleigh waves and atmspheic waves and (2) the excitatin f Rayleigh waves by a pint suce in the atmsphee using the Eath s nmal mdes cmputed f a ealistic spheically symmetic eath mdel with the slid (elastic) eath, the cean and the atmsphee (Figue ). The bjective is t undestand the excitatin mechanism f the hamnic gund mtins geneated duing the vlcanic euptins f Mt. Pinatub and Mt. St. Helens and the enegy cupling between the waves in the atmsphee and the slid eath. We will examine the simple qualitative mdel f Kanami and Mi [992] and an excitatin mdel in an isthemal atmsphee studied by Kanami et al. [994] 2. Methd [9] We emply the nmal mde methd which has been widely used in seismlgy. The theies have been pesented by Lve [9], Alteman et al. [959], Takeuchi and Sait [972], Gilbet [980], Sait [988], and Wdhuse [988]. Eath mdels used in these studies ae bunded at the suface by the gund the cean suface, and the atmsphee has been neglected. [0] We extend the eath mdel t include the atmsphee. A tp bunday is placed abitaily in the uppe atmsphee. We include a fluid laye in the mdel in the same way as we d the ute ce and the cean layes. A simila appach is taken by Lgnnné et al. [998], Atu et al. [2004] and Kbayashi [2007] f the study f cupled scillatins f the atmsphee and the slid eath with a igus teatment f a adiatin bunday cnditin. In the pesent study thee types f uppe bunday cnditins, stess fee, n displacement (i.e., igid) and adiatin (i.e., n eflectin), ae cnsideed. A vlcanic euptin is simply mdeled with an istpic pint suce in the atmsphee. Symbls used in this pape ae listed in the Ntatin sectin at the end f this pape. 2.. Equatin f Mtin [] We igne the effects f tatin f the Eath such as the centifugal fce and the Cilis fce because the peid f waves we ae inteested in is much shte than day. We neglect the advectin f the backgund medium such as wind in the atmsphee and themal diffusin and adiatin pcesses. We assume that fluid is inviscid. We cnside that the Eath is spheically symmetic. [2] Small scillatin pblems in fluid, such as tsunami and sla scillatins ae histically fmulated with the Euleian appach. Seismlgy teats shea and stain in elastic slid and fluid media and thus the Lagangian appach is suitable [Aki and Richads, 980, Bx 2.3]. The nmal mde pblems in seismlgy have been teated with the Lagangian appach. The diffeential equatins f the eigenfunctins Z i, i =, 2, 5, and 6, in a fluid cmmnly used in seismlgy [Sait, 988; Wdhuse, 988] ae 0 whee A ij s ae given by Z 0 Z d Z 2 B C Z 5 A ¼ A Z 2 B Z 5 A ; Z 6 Z 6 ðþ 0 þ llþ ð Þg llþ! 2 2 ð Þ llþ ð Þ! 2 2! 2 2 0! 2 þ llþ ð Þ! 2 2 g 2 4 g llþ ð Þg g llþ ð Þ! 2 2! 2 2 l þ 4G : ð2þ 4G 0 l 4G C ðl þ Þ þ llþ ð Þ g! 2 2 llþ ð Þ A llþ ð Þ l! 2 2! 2 2 using a igus nmal mde they f a ealistic elastic eath mdel suunded by the cean and the atmsphee. () g () and () ae efeence spheical mdel paametes and ae functins f adius,. Quantities with a nught indicate that the quantity is petubed duing the scillaty mtin. We edeive the simultaneus diffeential equatins () and (2) f the nmal mde f the fluid egin f the Eath in Appendix A fm the Lagangian appach. Definitins f the symbls in equatin (2) ae fund in the Ntatin sectin. These equatins ae a special case f the me geneal six simultaneus fist de diffeential equatins f a slid mdel. In a fluid Z 4, the eigenfunctin f the shea tactin acting n a suface nmal t the adial axis, vanishes eveywhee because the shea mdulus is ze. Z 3, the eigenfunctin f the hizntal displacement, is expessed explicitly as a linea cmbinatin f Z, Z 2 and Z 5, whee Z is the eigenfunctin f the vetical displacement, Z 2 is the eigenfunctin f the nmal tactin acting n a suface nmal t the adial axis, Z 5 is the eigenfunctin f the gavitatinal ptential petubatin. By eliminating Z 3 the diffeential equatins ae educed t equatins () and (2) [Takeuchi and Sait, 972; Sait, 988]. Z 6 is the eigenfunctin f the cmbinatin 2f20

3 buyancy. When a pacel in a gavitatinally statified fluid is displaced vetically, the pacel expands cntacts adiabatically accding t the ambient vetical pessue pfile in the fluid and changes its density. The density cntast between the displaced pacel and the suunding fluid mateial is the igin f buyancy and the expessin f this tem suppsedly includes the vetical density gadient. Hweve, the equatins abve ae a set f cect equatins as discussed by Piece [966] and indeed the dispesin elatin f an isthemal hizntally statified medium [e.g., Gill, 982, sectin 6.4] can be deived, as we will see in sectin 2.2, fm these equatins by igning the gavity ptential petubatin and using a flat Eath appximatin f an isthemal atmsphee. Figue. Atmspheic mdels used in this study. Black line is the U.S. Standad Atmsphee [976] mdel. Gay line is a lcal atmspheic mdel nea Pinatub duing euptin, cnstucted fm data by Champin et al. [985] (5 N July mdel belw 90 km, 0 N June mdel between 90 and 0 km, 600 K exsphee mdel abve 0 km). Atmspheic density deceases quickly upwad with a scale height f abut 7 km. At 00 km, the density is abut ne millinth f the gund atmspheic density. The exspheic tempeatue f U.S. Standad Atmsphee [976] appaches 000 K. Slid eath mdel is PREM, nt istpic PREM, including a hmgeneus cean laye with a sund velcity f.45 km/s, and the uppemst custal laye with V p f 5.8 km/s and V s f 3.2 km/s. f the ptential petubatin and its adial gadient and the vetical displacement, and has a cnvenient fm t define the bunday cnditin f the gavity ptential petubatin. See Appendix A f the exact definitins. [3] In equatin (2), nte that even if mdel paametes,, g, culd have adial gadients the A ij s d nt cntain the adial gadient f the eath mdel paametes and we can avid numeical diffeentiatins f the mdel paametes. This is the advantage f using the Lagangian petubatin f the density and the tactin. In cntast, if we tk the Euleian petubatin f the density and the tactin, the 2.2. Lcal Behavi f Eigenfunctins [4] In this sectin we investigate the lcal behavi f the eigenfunctins. F example, in 2D Catesian cdinates (x, z), the lcal behavi f sund waves expessed by e i(wt kxx kzz), is examined by the signs f k x 2 and k z 2 gvened by the dispesin f sund waves k z 2 + k x 2 = w 2 /c 2. F a given fequency w, sund velcity c, and wavenumbe in the x diectin k x, k z 2 can be negative psitive. If k z 2 is psitive, the sund wave ppagates in the z diectin. If negative, the sund wave becmes evanescent, i.e., the wave is phibited fm ppagatin and the wave amplitude decays gws in the z diectin. In the high fequency and sht wavelength limit, the effect f gavity n the equatin f mtin can be igned and an eigenfunctin f atmspheic waves behaves like a sund wave. At lwfequency gavity plays an imptant le. F atmspheic waves, the gavity petubatin caused by the density vaiatin is vey small cmpaed t the efeence gavity, elasticity and buyancy fces. [5] An eigenfunctin in a spheical gemety with an angula de l at adius has a eal hizntal wavenumbe k h expessed by k h =(l +0.5)/ [e.g., Aki and Richads, 980, p. 356]. The adial behavi f the eigenfunctin is gvened by the lcal wavenumbe in the diectin which is examined fm the lcal dispesin elatin. Nte that the investigatin f lcal behavi f the eigenfunctins is based n vaius appximatins and is nt used f the cmputatin f eigenfunctins. Igning the gavity ptential petubatin but including the efeence gavity g [Cwling, 94], the diffeential equatins () and (2) becme d Z d Z 2 0 ¼ þ llþ ð Þg llþ! 2 2 ð Þ! 2 2 B! 2 þ llþ ð Þ! 2 2 g 2 4 g llþ ð Þg A Z Z 2! 2 2 ¼ B Z : ð3þ Z 2 definitins f Z 2 wuld change and the cespnding new A ij wuld have the adial gadients f the mdel paametes. The vanishing f the adial gadients f mdel paametes is due t the chice f the Lagangian petubatins f density and tactin and the Euleian petubatin f gavity ptential. At the fist sight this seems t cntadict the intuitin that in a fluid the adial gadient f density shuld play an imptant le in the equatin f mtin as a suce f The vetical vaiatin f P and define a lcal scale height H s given by H s ¼ d ln P d d ln : ð4þ d Thugh the middle and lwe atmsphee, H s is bunded between 6.4 km and 8.4 km at heights, z, fm 0 t 86 km 3f20

4 and H s is 50 km at z 300 km [Gill, 982, sectin 3.5]. In de t examine the lcal behavi f eigenfunctins fm the cefficient matix, the apid change f the cefficient matix as a functin f adius is nt desiable. Because and ae the mst apidly changing vaiables in equatin (3), and adial gadient f g is much smalle than thse f and, the vetical scale length f the B 2 and B 2 tems in equatin (3) is abut that f, fm equatin (4), H s. On the the hand the vetical scale length f the G is the ati f the specific heats, and defines the fllwing elatinships ¼ GP ¼ c 2 : ð5þ In the abve, G, M, and R ae cnstants, and M, U m, c 2 and c ae functins f. We ewite equatin (3) using the dimensinless quantities d d Z Z 2 ¼ 0 GU m M 2 4R 6 2 þ llþ ð Þ c 2 þ llþ ð Þ c 2 c 3 4 c 2 4R 6 GU m M 2 2 c 2 V g llþ ð Þc 2 llþ ð Þ C A Z Z 2 2 c : ð6þ eigenfunctins can be as lage as 00 km, the thickness f the atmspheic laye. [6] Hee we intduce nndimensinal quantities which ae finite as appaches = R whee the bunday cnditin is impsed. Fllwing Unn et al. [989], the nndimensinal quantities, U m, V g, c, s ae whee U m ¼ d ln M d ln ¼ 4 3 ¼ 3 M ; V g ¼ g c 2 ¼ GM c 2 ¼ ; GH s c ¼ 3 M R R 3 ¼ ðrþ M ðþ ; M ¼ 2 ¼ Z We have used G defined by 0 R3 GM R! 2 ; ðþ4 2 d; ð5þ ð6þ ð7þ ð8þ ð9þ ðþ¼ M ; ð0þ g ðþ¼ GM 2 ; ðþ g dg d ¼ 43 M 2 ¼ U m 2; ð2þ ¼ U m M 4 3: G ð3þ : ð4þ s Changing the dependent vaiables fm (Z, Z 2 )t(x, X 2 ) which ae defined by we btain X ¼ Z ¼ U; X 2 ¼ Z 2 g ¼ 4c R 6 GU m M 2Z 2; ð7þ ð8þ 0 d þ llþ ð Þ X c ¼ 2 c V g llþ ð Þ 2 B d X 2 2 þ llþ ð Þ 2 c þ K U m llþ ð Þ A X ; X 2 c 2 c ð9þ whee ¼ C X X 2 ; ð20þ K ¼ d ln ¼ : d H s ð2þ Nw in the cefficients C ij f X and X 2, apidly changing mdel paametes and in the diectin disappea and nly c, V g, K and U m which change slwly with exist. Heeafte we assume that the change f C ij is s small cmpaed t the change f eigenfunctins X and X 2 that C ij s ae lcally cnstant. In the atmsphee c is clse t, U m is clse t ze and V g has a finite value f the de f 00 t 000. Since f() = a is the slutin f a df()/d = af() type diffeential equatin, we assume that X and X 2 lcally depend n in plynmial fm as l. Replacing c by and U m by 0, the chaacteistic equatin f equatin (20) is simplified t 2 ðc þ C 22 Þ þ C C 22 C 2 C 2 ¼ 0 ð22þ 2 ð2 þ KÞ llþ ð Þ V 2 g 2 K þ V g Vg 2 V gðk þ 4Þþ3 þ K ¼ 0 ð23þ 4f20

5 and its slutins ae whee ¼ 2 2 þ K p ffiffiffi ; ð24þ 2þ4 llþ ð Þ ¼ K 2V g þ 4 V g 2 K þ V g : ð25þ Using the nmalized buyancy fequency A* 2 A* ¼ N 2 ¼ d g d þ g c 2 ; ð26þ Fm equatin (3) t equatin (32) we have appximated an expnential functin f, exp( a), by a pwe functin f, (e/ ) a. Tw functins have the same value and gadient at =, whee a is a cnstant and is the adius whee the eigenfunctin is lcally examined. If g > 0, tw evanescent slutins f the gwing and deceasing mdal enegy with altitude exist. At the tp bunday the slutin f the gwing mdal enegy assciated with l + is ejected and the mdal enegy deceases expnentially with altitude nea the tp because we assume the suce f the mdal enegy is belw the tp bunday. If g < 0, the tw ppagating wave type slutins assciated with l + and l epesent waves whse vetical phase velcities ae dwnwad and upwad, espectively, because we assumed that eigenfunctins have time dependence e iwt. F a given atmspheic mdel, g is a functin f lcal mdel paametes and the peid and the hizntal wavelength f the mde, and can be witten as ð;!;k h Þ ¼ 4V g 2 2 llþ ð Þ 2 K þ V g V g 2 2 K 2V g þ 4 ; ð33þ 4V g ¼ 4V g : ð34þ s (, k h ) and s 2 (, k h ) ae the ts f the ight hand side f equatin (33). Because llþ ð Þ ðk 2V g þ4þ 2 Vg >0, 4Vg > 0 and f a stably statified fluid ¼ K V g ; ð27þ K V g ¼ A* ¼ N 2 g > 0 ð35þ whee N is the buyancy fequency ( Bunt Väisälä fequency), equatin (25) is expessed as 2þ4 llþ ð Þ ¼ A* V g þ 4 V g 2 A* : ð28þ Unn et al. [989] wh used the Euleian pessue petubatin as a dependent vaiable instead f the Lagangian pessue petubatin eached the same chaacteistic equatin. In this chaacteistic equatin the adial gadient f mdel paametes is indeed included thugh K = /H s. The enegy density f a mde, E = ( du dt ) (du dt ) = w 2 u 2, whee dentes the cmplex cnjugate, is given by 2 E ¼! 2 juj 2 þllþ ð ÞjVj 2 Yl m ð; Þ y Yl m ð; Þ; ð29þ / j! j 2 jx j 2 2 þ llþ ð Þ 2 4 ðjx X 2 jþ!; 2 ð30þ / K 2 2 ¼ ffiffi p ; / e ffiffi p : ð3þ ð32þ hlds, the tw psitive eal fequencies s and s 2 (s < s 2 ) always exist f any subadiabatic atmspheic mdel. As we see belw, s (, k h ) cespnds t the buyancy fequency as k h, and s 2 (, k h ) cespnds t the acustic cutff fequency as k h 0. Waves ae evanescent f s 2 > s > s and ae ppagating vetically f s 2 < s s > s, theefe s and s 2 ae called the cutff fequencies. We have the fllwing asympttic values f the egime bundaies (g = 0) between the ppagating egime and the evanescent egime in the fequency hizntal wavenumbe dmain: when l! 0; 2! 0 and 2 2! A* þ A* V 2 g þ 4 ; ð36þ 4V g equivalently when l!; 2! A* and 2 2! llþ ð Þ ; ð37þ V g when l! 0;! 2! 0 and! 2 2! N 2 þ c 2 g 2H s c 2 þ 2 2 ; R ð38þ when l!;! 2! N 2 and! 2 2! llþ ð Þc2 R 2 : ð39þ These lcal cutff fequencies ae indeed the same as thse f an isthemal atmsphee [e.g., Gill, 982, Figue 6.8] if R H s when k h!0;! 2! 0 and!2 2!N 2 þc 2 g 2 2H s c 2 ¼ when k h!;! 2! N 2 and! 2 2! c2 k 2 h ; c 2 ¼ N 2 A ; 2H s ð40þ whee N A is the acustic cutff fequency and k h is the hizntal wavenumbe. [7] The ate f vetical flw f enegy pe unit aea, F, is given by, e.g., Gill [982, equatin (4.6.4) neglecting viscsity] the adial cmpnent f the eal pat f F ¼ p 0 þ 2 jvj2 v y ; ð4þ 5f20

6 whee v = du dt, Lagangian velcity f the mateial element. F a small scillatin, F is appximated [Watada, 2009] by F = p ( du dt ) ReðF Þ ¼ Re i!p 0 u y ; ð42þ h / Re i! g 2 X X y X 2X y i : ð43þ Eliminating X 2 by equatin (20) and using plynmial expessin f X we have V g llþ ð Þ h ReðF Þ / Re i! g i V g j X j 2 ; 2 2 ð44þ / Imð Þ; ð45þ p / Im ffiffiffi ; ð46þ whee the uppe and lwe signs cespnd t the dwnwad (l + ) slutin and upwad (l ) slutin, espectively. Intepeting equatin (46), we have the fllwing esults. Vetical enegy flw des nt exist when g 0, i.e., whee waves ae lcally evanescent. The p ffiffiffi sign f vetical enegy flw depends n the sign f Im( ) and the sign f (V g l(l + )/s 2 ). When V g > l(l + )/s 2 equivalently w 2 > L 2 l, vetical wave enegy flw, upwad dwnwad, equies vetical phase ppagatin, upwad (l ) dwnwad (l + ), espectively. Hee we have intduced the Lamb fequency L l defined by L 2 l ¼ llþ ð Þ 2 c 2 : ð47þ F lage l, w 2 > L l 2 cespnds t ppagating acustic waves. Since we eject the slutin with dwnwad enegy flw, an apppiate slutin always has an upwad vetical phase velcity (l slutin). When V g < l(l + )/s 2 equivalently w 2 < L l 2, wave enegy flw upwad dwnwad equies phase ppagatin dwnwad upwad, espectively. Similaly ejecting the dwnwad enegy flw, an apppiate slutin always has a dwnwad vetical phase velcity (l + slutin). The ppsing diectin between the vetical phase velcity and the vetical enegy flw is chaacteistic f ppagating gavity waves in bth cmpessible and incmpessible fluid [e.g., Gill, 982, sectin 6.4; Watada, 2009]. [8] Thee is a mdal banch cespnding t suface gavity waves indicated by tsunami in Figue 6 f an eath mdel with the cean laye at the tp. This tsunami banch csses the ppagating gavity wave egime and the evanescent egime f the atmspheic waves in the fequency hizntal wavenumbe dmain (Figue 2c). The cupling between tsunami in the cean and the ppagating gavity waves in the atmsphee is discussed in sectin 7. [9] In summay, acustic mdes (w > w 2 ), which appach sund waves in the sht hizntal wavelength limit, have an upwad vetical phase velcity and upwad enegy tanspt. Evanescent mdes (w 2 > w > w ) have ze upwad vetical phase velcity and n vetical enegy tanspt. Gavity mdes (w > w), whse eigenpeid appaches the buyancy fequency in the sht wavelength limit, have a dwnwad vetical phase velcity and upwad enegy tanspt. The lcally defined quantity g in equatins (25), (28) and (33) diagnses the lcal behavi f waves in the vetical diectin f a given peid and hizntal wavelength. In the egin whee g is negative, waves ae tapped ppagating depending n the bunday cnditin. In the egin whee g is psitive, waves becme evanescent and the mdal enegy density decays inceases expnentially in the vetical diectin. Analyses in this sectin ae valid when the vetical scale length f waves is shte than that f g. As the vetical scale length f g appaches the vetical wavelength f atmspheic waves, the analyses becme less accuate Cmputatin f Nmal Mde Cnventinal Uppe Bunday Cnditins [20] Unde the Cwling appximatin [Cwling, 94], the fee uppe bunday cnditin is satisfied by setting Z 2 t ze, and the nnvetical displacement bunday igid bunday cnditin is satisfied by setting Z t ze. When the gavitatinal ptential petubatin is included, an additinal bunday cnditin Z 6 = 0 must be impsed. The deteminant f the tw sets f dependent slutins at the bunday is cmputed. The bunday cnditin is satisfied when the deteminant vanishes [Takeuchi and Sait, 972] Radiatin Uppe Bunday Cnditin [2] The adiatin bunday cnditin impses a elatin between the eigenfunctins. Once l is evaluated at the adiatin bunday by equatin (24), fm equatins (7), (8) and (20) f an evanescent mde with w 2 > w > w, Z 2 ðc ÞZ þ C 2 ¼ 0 g C 2 Z þ ðc 22 Þ Z 2 ¼ 0 g ð48þ ð49þ is the bunday cnditin. The eigenfequencies and eigenfunctins ae eal. F an acustic mde with w > w 2 a gavity mde with w < w, Z 2 ðc ÞZ þ C 2 ¼ 0 g C 2 Z þ ðc 22 Þ Z 2 ¼ 0 g ð50þ ð5þ is the bunday cnditin. The eigenfequencies and the eigenfunctins becme cmplex. We chse l = l + l t satisfy the adiatin bunday cnditin that inhibits dwnwad pffiffiffi enegy p flw ffiffiffi thugh the uppe bunday. If we define with Im ( ) 0, fm the discussin in sectin 2.2, l is the apppiate bunday cnditin f acustic mdes, and l + is the apppiate bunday cnditin f gavity mdes. The bunday cnditin f the ppagating mdes is teated a little diffeently by Unn et al. [989]. They set the bunday cnditin whee mdal enegy density E in 6f20

7 Figue 2. Nmal mdes f the U.S. Standad Atmsphee [976] mdel in Figue. The tp bunday at 200 km is a stess fee suface bunday and the bttm is a igid bunday. Thee maj gups f eigenmdes exist. Acustic mdes ae the sund waves in the atmsphee. Gavity mdes exist f a fluid bdy unde gavity with subadiabatic density statificatin, such as the Eath s atmsphee. Lamb waves ae bunday waves which tavel alng the gund with the speed f sund. (a) Hizntal wavelength vesus hizntal gup velcity. Acustic and gavity wave egimes velap. (b) Peid vesus hizntal phase velcity f cmpaisn with ealy studies. The hizntal phase velcity is cmputed by w e /(l + 0.5), whee e = 637 km. (c) Angula de vesus fequency. The bxed egin is magnified in Figue 7. equatin (32) des nt gw decay lcally by adjusting the value f l ±. Thei set f cmplex eigenfequency s, cmplex g and cmplex l d nt satisfy the chaacteistic equatin (equatins (22), (23), (24) and (25)) Lwe Bunday Cnditins [22] T cmpute atmspheic mdes neglecting the cean and the slid eath, the bttm bunday f the atmsphee is assumed igid by setting Z t 0. F mdes whse eigenfunctins stat beneath the cean gund suface, such as seismic Rayleigh waves and the mdes f an eath mdel including the atmsphee, the bunday values f eigenfunctins at a stating level ae btained fm the analytic slutin f an istpic hmgeneus fluid slid bdy whse mdel paametes ae the same as thse at the bunday [Pekeis and Jasch, 958; Takeuchi and Sait, 972]. The apppiate stating level f the integatin is whee the amplitudes f the eigenfunctins becme s small that the ati f amplitude at the stating level elative t the maximum amplitude eaches a peset cmputatinal accuacy. Such a staing level is ughly estimated by integating the adial wavenumbe f a mde fm the stating level t the tuning level f the cespnding seismic ay. F example, an acustic/seismic spheical wave with angula de l has a vetical wavenumbe k at which is given by k 2 = w 2 /c() 2 l(l + )/ 2. Abve, at, belw the tuning pint f the acustic/seismic ay cespnding t this spheical wave, k 2 is eal, ze, imaginay, espectively. Belw the tuning pint, the wave amplitude decay is ughly estimated by exp( R pffiffiffiffiffiffiffiffiffiffi t 2 s k d), whee s and t ae the integatin stating and ay tuning adii, espectively Integatin [23] We use a shting methd t cmpute the nmal mdes. The diffeential equatins f a fluid bdy (equatins () and (2)) ae integated fm a stating level t a bunday level. The cmpund matix methd [Takeuchi and Sait, 972], equivalently the mins methd [Wdhuse, 988], is nt used. In a slid, the equatins f mtin ae expessed t six ( fu if the Cwling appximatin is used) simultaneus fistde diffeential equatins. The cnnectin f eigenfunctins at the slid fluid bunday is descibed by Takeuchi and Sait [972]. The cnnectin f eigenfunctins at the fluid fluid bunday at the suface f the cean is similaly deived. F the adiatin uppe bunday cnditin cmplex eigenfequencies and eigenfunctins ae btained by a cmplex t seach [Fiedman, 966]. We emply an adaptive step size cntl Runge Kutta integat [Pess et al., 992]. Mdel paametes ae given as a table at discete pints. Between the gid pints the mdel paametes ae linealy inteplated. Since the pgam cntls the integatin step size, a unifm accuacy f the eigenfunctins can be maintained easily. F an eigenfunctin with the fee suface bunday igid bunday cnditin, the numeical integatin accuacy is cudely tested by the enegy integals [Takeuchi and Sait, 972]. [24] When the integatin is pefmed in a elatively thick egin whee g has lage psitive values and the mdal enegy is deceasing in the diectin f integatin, the integatin becmes numeically unstable because the unwanted slutin gws expnentially [Pess et al., 992]. Hweve, the eigenvalue is accuately cmputed f this distted eigenfunctin because this gwing cmpnent is the the slutin with the same eigenvalue [Jensen et al., 994]. F 7f20

8 gavity mdes with lage angula de numbes, the mdal enegy is tapped smetimes lcally, e.g., nea the mespause, and the ne way integatin becmes unstable. F these cases we integate fm the bttm bunday upwad and fm the tp bunday dwnwad and match the tw sets f eigenfunctins at a level whee the mdal enegy is tapped. F lw de acustic mdes we can stably integate bth upwad and dwnwad by the adaptive step size Runge Kutta integatin because g is elatively small. The hizntal gup velcity is btained by numeical diffeentiatin alng the mdal banch, if the bunday cnditin is fee igid, by integal elatins [Takeuchi and Sait, 972]. 3. Nmal Mdes in the Atmsphee [25] Figue shws the acustic velcity pfile f the atmsphee mdels used in this study. The sund velcity is cmputed fm the tempeatue and the pessue by c 2 = G P /. We assume that G is cnstant,.4, the value f ideal diatmic mlecule gas [e.g., Gill, 982]. We csschecked the eigenfequencies f atmspheic mdes cmputed by the Runge Kutta methd in sectin with thse cmputed using the Haskell matix methd [Pess and Hakide, 962], and gup velcities cmputed by the methd in sectin with thse cmputed using the patial deivatives f the Haskell matix [Hakide, 964]. The Haskell matix methd is applied f an atmsphee mdel cmpsed f stacked isthemal layes. Figue 2 shws the nmal mdes f the U.S. Standad Atmsphee [976] mdel. The cutff fequencies cmputed at z = 00 km and 200 km ae pltted in Figue 3. In the nntating spheically symmetic atmsphee, thee types f waves exist, acustic waves, Lamb waves and intenal gavity waves. Many efeences n atmspheic acustic waves, the Lamb waves and intenal gavity waves can be fund in, f example, Lamb [90], Bee [974] and Geges [968]. The hizntal gup velcity f a mde can be ughly btained fm the slpe f the banch in the (fequency angula de numbe) plt (Figue 2c). All the cmputed nmal mdes f the atmsphee fall int these thee categies except suface gavity waves alng a deep wate wave like banch in the atmsphee which is an atifact f the tp fee suface bunday. These atificial mdes ae excluded fm Figue 2. [26] In the fequency band in which the hamnic seismic suface waves wee bseved duing the euptin f Mt. Pinatub, abut s, all thee types f atmspheic waves can exist. They ae lng wavelength acustic waves nea the acustic cutff fequency, Lamb waves and sht wavelength gavity waves nea the buyancy fequency, the Bunt Väisälä fequency. Kanami et al. [994] suggested that the acustic mdes nea the acustic cutff fequency and the gavity mdes nea the buyancy fequency can be excited efficiently by a vlcanic euptin. The efficiency f the cupling between the atmspheic waves excited by a vlcanic euptin and the seismic waves (and tsunami if including the cean) in the slid eath ae cntlled by the wave peid and the wavelength alng the inteface. Efficient cupling is expected between tw waves abve and belw the inteface when they have a cmmn phase velcity alng the inteface. Gup velcity alne cannt be used as an indicat f the cupling efficiency between tw waves. F the lng wavelength acustic mdes nea the acustic cutff, the lnge the wavelength, the lage the hizntal phase velcity and the smalle the hizntal gup velcity. F example in Figue 2, at angula de 30 hizntal wavelength 300 km, the gavest thee acustic mdes have a hizntal phase velcity f abut 4 5 km/s and a hizntal gup velcity f abut 0 m/s. Sht wavelength gavity mdes, f example at l = 2000 (wavelength = 20 km), nea the cutff have small hizntal phase velcity 00 m/s and hizntal gup velcity 0 m/s. [27] The wavefms in the fa field bagaphic ecds [e.g., Pess and Hakide, 962; Hakide, 964; Mikum, 968; Mikum and Blt, 985; Mikum et al., 2008] wee dminated by the Lamb waves, which ppagate at the speed f sund. A few banches f acustic and gavity mdes, whse gup and phase velcities ae clse t the sund velcity in the atmsphee, ae equied in the cmputatin t simulate the bagaphic bsevatins [Hakide, 964]. In this pape, we call all atmspheic mdes whse mdal enegy is cncentated twad the suface f the gund, the cean, Lamb wave mdes. The Lamb mde banch csses the acustic and gavity banches (Figue 2a). [28] Kanami and Mi [992] and Kanami et al. [994] epted hamnic bagaphic pessue changes ecded duing the euptins f Kakata 883, Mt. St. Helens 980 and Mt. Pinatub 99. Table summaizes hamnic scillatins fund in bagams and seismgams duing these maj vlcanic euptins. [29] The mdal enegy density f lng wavelength (l = 30, wavelength = 300 km) acustic mdes nea the cutff fequency is shwn in Figue 4. The gavest mde with a peid f 329 s has its enegy nly in the themsphee. F this peid and hizntal wavelength, a standing ppagating wave may exist in the themsphee and the tpsphee whee g is negative. T have a standing wave in a negative g well, we equie fm equatin (32) pffiffiffiffiffiffi h ¼ n; ð52þ whee is the aveage f g in the negative g egin, h is the vetical extent f the negative egin and n is a psitive intege. The negative g egin in the tpsphee des nt hld a standing wave because the vetical scale f the tpsphee is t small t have a standing wave in this egin and nly a ppagating wave exists in the themsphee f this mdel. When we take the tp bunday at z = 00 km, this mde disappeas. [30] The secnd gavest mde with a peid f 274 s, which is clse t ne f the bseved peids (Table ), is a tapped mde. The psitive g egin in the mespause, the lw velcity channel which fms the bunday laye between the themsphee and messphee, wks as a waveeflecting wall f this mde and the mdal enegy is tapped belw the mespause and abve the gund (Figue ). [3] The thid gavest mde with a peid f 236 s, which is als clse t ne f the bsevatins, has a lage enegy cncentatin belw the themsphee whee g has lage negative values than in the uppe atmsphee. Because g is negative eveywhee, the mdal enegy can leak int the uppe atmsphee. F this type f mde a adiatin bunday cnditin is necessay. 8f20

9 Figue 3. Cutff fequencies w and w 2 (w < w 2 )ftheu.s. Standad Atmsphee [976] mdel cmputed at z = 00 km and 200 km. In the egin whee w 2 > w > w, waves becme evanescent. In the egin whee w > w 2 w < w,ppagating acustic gavity waves exist, espectively. The cutff fequencies change mainly because atmspheic tempeatue changes with altitude. As altitude changes, the wave chaacteistics f (l, w) culd change fm a ppagating wave t an evanescent wave and vice vesa. [32] The enegy distibutin f highe fequency acustic mdes changes with altitude like a cs 2 standing wave type scillatin. This standing wave type scillatin is an atifact caused by the eflectin at the tp fee suface bunday; if we emve the eflecting bunday at the tp, then the mdal enegy will smthly ppagate upwad dwnwad thugh the bunday. The scillatin f tapped mdes epesents the evebeatin f sund waves in the negative g well and may be a suce f the tempal hamnic lading n the gund. At highe fequencies, because upwad shtpeid acustic waves d nt cme dwn by eflectin, acustic evebeatin in the atmsphee may nt exist and cannt be a suce f hamnic lading. The hamnic gund mtins with peids f 270 s and 230 s ecded n seismgams thughut the wld ae mst likely the tapped acustic mdes that cupled t the gund. [33] The sht wavelength gavitatinal mdes (l = 2000, wavelength = 20 km) (Figue 4) have cncentated mdal enegy at highe altitude nea z = 00 km except ne mde. The mde with a peid f 300 s has enegy cncentatin in the statsphee. A lw altitude suce such as a vlcanic euptin may pefeentially excite this mde and the peidic pessue change with a peid f 300 s may be ecded at the gund level. Because f its small hizntal gup velcity, the pessue change assciated with this gavity mde will be bseved nea the vlcan. The atmspheic scillatin with a peid nea 300 s bseved by a seismmete which wked as a bagaph nea Mt. St. Helens just afte the explsive euptin in 980 may be this type f gavity mde tapped in the statsphee. A satellite infaed image f Mt. Pinatub duing the 99 euptin shws [Kanami et al., 994, Figue ] a cncentic patten f tempeatue distubance with wavelength f seveal tens f kilmetes aund the vlcan extending ve a distance f abut 400 km. If we assume that the waves in the satellite image have the peid f abut 300 s, the cncentic patten pbably epesents the gavity waves excited by the vlcanic euptin, because acustic waves and Lamb waves shuld have much shte peids at the wavelength f seveal tens kilmetes (Figue 2). [34] The chice f the tp bunday type and its lcatin ae nt imptant if a mde is natually tapped, ducted, belw the tp bunday by wall layes which have psitive g. Ou numeical cmputatin cnfims that the 270 s acustic mde always exists f any type f tp bunday if placed at highe than 00 km (Figue 5). The mde always hits a natual bunday nea z = 00 km and its mdal enegy is cnfined between z = 0 and abut 00 km. [35] The lcal atmspheic stuctue can deviate fm the standad atmsphee mdel diunally, seasnally, gegaphically and latitudinally [Champin et al., 985]. In the themsphee the tempeatue is mainly cntlled by the sla activity [COSPAR Wking Gup IV, 972]. These vaiatins may esult in the change f nt nly eigenpeid but als the enegy density distibutin f mdes; as a esult a tapped mde may becme a ppagating mde and vice vesa. In Table 2 we list the esult f cmputatin f an atmspheic stuctue clse t the lcal mdel duing the euptin f Mt. Pinatub in 99 pltted in Figue. [36] The lcal atmspheic mdel nea Mt. Pinatub in June has tw distinct featues. Lw tempeatue, abut 0 K lwe than the aveage at the bttm f the statsphee, and high themspheic tempeatue, abut 500 K highe. We chse the high themspheic tempeatue because the sla activity was high in 99. The eigenpeids f the fu tapped acustic mdes change slightly. Tw gavity mdes with enegy tapped in the statsphee exist with distinct eigenfequencies. A highe tempeatue in the themsphee esults in the incease f g, but still negative, and the mdal enegy density deceases me nea z = 200 km than thse f the standad atmsphee, and the secnd vetne f the acustic mde becmes nealy tapped. We speculate that the bseved peid 95 s duing the 982 El Chichón euptin may be a tapped acustic mde when the sla activity was extemely high. In fact, in 982 its cycle was nea the peak. Hweve, the easn why the amplitude spectum peak nea 230 s is missing in the seismgaphic bsevatin duing the 982 El Chichón euptin emains unexplained. 4. Nmal Mdes in the Eath [37] We use the Peliminay Refeence Eath Mdel (PREM) f Dziewnski and Andesn [98] f the slid eath. The nmal mdes f PREM ae shwn in Figue 6. Table. Obsevatins f Hamnic Atmspheic Oscillatins and Hamnic Gund Mtin Duing Maj Vlcanic Euptins a Euptin Peid Obsevatin Methd Distance 99 Pinatub 270 s, 230 s seismgaph teleseism 982 El Chichón 95 s, 266 s seismgaph teleseism 980 Mt. St. Helens 300 s seismgaph wked nea suce as bagaph 883 Kakata 300 s bagaph nea suce a Cmpiled by Kanami et al. [994]. 9f20

10 Figue 4. Mdal enegy density ( U 2 + l(l +) V 2 ) 2 f acustic mdes l = 30 (equivalent hizntal wavelength 300 km) and gavity mdes l = 2000 (equivalent hizntal wavelength 20 km) with a peid f abut s with a fee suface bunday cnditin. The lcal chaacteistic functin g in equatin (33) is cmputed as a functin f adius f the eigenpeid and angula de f each mde. Hee g > 0 is seen as a ptential wall whee mdal enegy evades and the eigenfunctin becmes evanescent. Hee g < 0 is seen as a ptential well whee mdal enegy is tapped. The mdal enegy f each mde is nmalized t the maximum value. Hee g is als nmalized t its maximum. The scale f g is at the tp, but the unit is abitay. A ntable featue in Figue 6 is the tsunami banch with a nealy cnstant hizntal phase velcity f abut 70 m/s. This mde is a suface gavity wave in the tp cean laye f the PREM eath mdel with a thickness f h = 3 km. The tsunami banch and seismic banches ae sepaated well in the (fequency de numbe) dmain (Figue 6). F the tsunami mde a simple lng wave appximatin pffiffiffiffiffiffiffi gives a nndispesive hizntal phase velcity g h which is clse t the hizntal phase velcity f the tsunami mdes cmputed f PREM. At vey lng wavelength, the hizntal phase velcity slightly deceases. F example, at l = 0 the hizntal phase velcity deceases by abut 3% fm 70 m/s. At lng peid the Cilis effect shuld be taken int accunt. At sht wavelength, the dispesin elatin f wate waves deviates fm the shallw wate appximatin and appaches the deep wate appximatin, and its hizntal phase velcity deceases accding t pffiffiffiffiffiffiffiffiffiffiffi g =k h. The easn why the tw waves, Rayleigh waves and tsunamis with simila eigenfequencies, d nt cuple efficiently is that the wavelength f the excess pessue field at the cean bttm caused by tsunamis is diffeent fm the wavelength f Rayleigh waves by me than an de f magnitude. Tw waves having an inteface between them ae able t cuple efficiently when bth the peid and the hizntal wavelength f the tw waves ae the same. This cmputatin cnfims the study by Cme [984] wh shwed that f a flat eath mdel, Rayleigh waves and tsunamis ae pactically uncupled ve the entie peids f seismic waves, and the elasticity f the Eath can be safely igned f cmputatin f tsunami. This weak cupling between seismic suface waves and tsunamis explains why the tsunamis in the cean ae epted as a lading suce f fced defmatin f the Eath s suface [Yuan et al., 2005], but neve epted as a distinct excitatin suce f 0 f 20

11 Table 2. Cmpaisn f the Eigenmdes f the Tw Atmspheic Mdels a Peid (s) Mde Type Standad Atmsphee Lcal at Mt. Pinatub acustic acustic acustic acustic gavity , 290 a U.S. Standad Atmsphee [976] and lcal mdel nea Mt. Pinatub atmsphee in Figue. The same bunday cnditins used in Figue 2 ae applied. Acustic mde with l = 30 and gavity mde with l = 2000 ae cmputed. appximatin. If we take the adiatin bunday cnditin, we used the Cwling appximatin f all types f mdes. Figue 5. The vaiatin f the eigenpeid f atmspheic acustic mdes (l = 30) f the altitude change f the tp bunday. The numbe f each line is the vetne index numbe. (tp) F igid suface bunday. (bttm) F fee suface bunday. Lng wavelength atmspheic acustic mdes tend t have a natual eigenpeid f 270 s. The natual eigenpeid is ielevant t the tp bunday types and the altitudes f the impsed tp bunday. 5. Resnance Mechanism Between the Atmsphee and the Slid Eath [39] The acustic cupling between the atmsphee and the slid eath has been cnsideed t be vey small. The atmsphee is teated as a vacuum by seismlgists and the suface f the Eath is a igid bunday f atmspheic scientists because f the lage acustic impedance cntast; (c) atmsphee /(c) cust is abut the de f 0 4. The hizntal phase velcity f acustic waves in the atmsphee diffes fm that f seismic waves by de f magnitude. Hweve, in a few cases gund mtins induced by the atmspheic acustic cupling at a peid f seismic nmal mdes have been bseved n seismgams. Seismic waves with a peid f abut 240 s excited by the pessue pulse nea a nuclea explsin in the ai wee bseved by a gund tiltmete [Ben Menahem and Singh, 98] at a teleseismic distance. Hakide et al. [974] cmputed theetical seismgams fm atmspheic pint suces f the funda- suface waves f the slid eath. Hweve, it shuld be nted that tsunamis ae indeed excited by the lcal defmatin at the cean bttm assciated with eathquakes. In this case the initial distubance f tsunami is caused by a sudden uplift subsidence f the cean suface displaced fm a gavitatinal equiptential suface and is nt elated t the wave cupling. [38] The adptin f the Cwling appximatin changes the eigenpeid f the nnadial spheidal nmal mdes. The eigenpeids f the fundamental spheidal mdes with and withut the Cwling appximatin f a slid eath mdel ae shwn in Table 3. The fequency f fundamental spheidal mdes which we ae inteested in, 3 5 mhz s, changes by less than 0.3%. The meit f the use f the Cwling appximatin is that we can educe the cmputatinal time and impve the cmputatinal accuacy because we have fewe dependent vaiables. The Cwling appximatin is used f the lage angula de gavity wave mdes which becme unstable easily. Acustic and Lamb wave mdes ae cmputed withut the Cwling Figue 6. Spheidal nmal mdes in the PREM [Dziewnski and Andesn, 98] eath mdel which has a 3 km cean laye at the tp that allws wate waves. Seismic waves and tsunami waves ae almst decupled. High angula de nmal mdes l > 50 and tsunami mdes ae cmputed using the Cwling appximatin. The bxed egin is magnified in Figue 7. f 20

12 Table 3. Eigenpeid f Fundamental Spheidal Mdes f the PREM Eath Mdel in s a Angula Ode Cwling Appximate b Exact c a The physical dispesin is included in bth cases. b Ignes the gavity ptential petubatin but includes the efeence gavity. c Includes the effect f self gavity. mental Rayleigh waves with a peid f less than 60 s at teleseismic distances. Lng peid hamnic Rayleigh wave mtin assciated with the 99 euptin f Mt. Pinatub (5.4 N E) in the Philippines, wee epted by Kanami and Mi [992] and Widme and Zün [992], and with the 982 El Chichón euptin by Widme and Zün [992] (Table ). [40] We can expect efficient cupling between the slid eath and the atmsphee if seismic mdes and the atmspheic mdes fall int the same egin f the fequencywavenumbe (w k) dmain. In fact, a cmmn egin f atmspheic mdes and seismic mdes exists nea the peid f abut s (Figue 7), the peid f the bseved Rayleigh waves. As discussed in sectin 3, an atmspheic acustic mde nea the peid s with a small angula de has a small hizntal gup velcity and the mdal enegy density is tapped in the atmsphee belw the themsphee. Once a lng wavelength atmspheic acustic mde is excited by a pint suce in the ai, the mdal enegy is cnfined in the atmsphee hizntally and vetically. The tapped acustic mde can efficiently excite a Rayleigh wave that has the same eigenfequency and hizntal wavelength t thse f the acustic mde. 6. Cmpaisn f Synthetic Gund Mtin With Obsevatins [4] Since the bseved suface waves ae fundamental mde Rayleigh waves [Kanami and Mi, 992; Widme and Zün, 992], we cmpute the nmal mdes nea the fundamental spheidal mde banch (Figue 8) f a spheically symmetic eath mdel including the cean and atmsphee. These waves epesent seismic Rayleigh waves and acustic waves in the atmsphee. The nmal mdes f a cupled system using a cmbined eath mdel autmatically takes int accunt the mechanical cupling between the atmsphee, cean and slid eath. The cmputed eigenfequencies f the cmbined mdel deviate little fm the uncupled eigenfequencies except when tw mdes with the same angula de have vey clse eigenfequencies. [42] The mdal enegy density f the fundamental spheidal mdes f the cmbined eath mdel is shwn in Figue 9. Nte that the pltted enegy density in the atmsphee is magnified by 00. The atmspheic pat f the enegy density f Rayleigh waves is usually less than % f the maximum enegy density in the slid eath. Cnvesely the acustic mdes, which ae nt shwn in Figue 9, als have little enegy penetatin in the slid eath. F a few exceptinal seismic spheidal mdes with a peid clse t Figue 7. Ovelay f atmspheic mdes (bx in Figue 2) and seismic mdes (bx in Figue 6). Slid tiangles and pen cicles ae fundamental and highe spheidal mdes and atmspheic acustic mdes, espectively. Stng cupling between the Rayleigh waves and the atmspheic acustic waves is expected at the tw bld cicles aund tiangles because f the pximity f the hizntal phase velcities. 2 f 20

13 q ð s Þ S q ¼ b 0 M þ 2U qð s Þ llþ ð ÞV qð s Þ ; s s s ffiffiffiffiffiffiffiffiffiffiffiffi 2l þ b 0 ¼ ; ð57þ 4 b ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l þ Þðl þ Þl ; ð58þ 2 4 Figue 8. Fequency angula de plt f nmal mdes f the cmbined eath mdel. Nmal mdes f the cmbined PREM eath mdel and U.S. Standad Atmsphee [976]. Only mdes inside the plygn ae used in the cmputatin f synthetic spectum in Figue. the tapped atmspheic acustic mdes, the peak amplitude f the enegy density in the atmsphee is as lage as % f the peak density amplitude in the slid eath. The small but significant mdal enegy distibutin f 272 s and 237 s spheidal mdes in the atmsphee suggests that the 272 s and 237 s mdes ae pefeentially excited by a suce in the lwe atmsphee cmpaed t the spheidal mdes because the excitatin amplitude f an eigenmde by a pint suce is ughly pptinal t the enegy density amplitude at the lcatin f the suce. [43] The amplitude in the ai f a seismic spheidal mde can be lage because f the vey small density in the uppe atmsphee. Fm the ati f the amplitude peaks f the mdal enegy density in the ai and the slid eath, %, and the ati f mateial density at the peaks, 0 0, we get u at z = 00 km /u at z = 50 km 0 4. When the mde has an amplitude f mm at the gund, the amplitude f this mde at z = 00 km is abut cm and much smalle than the hizntal wavelength f this mde f abut 000 km me. At z = 00 km the linea amplitude they f small scillatins is still valid. [44] We cmpute the synthetic seismgams using the eigenfunctins f the cmbined eath mdel f the subset f mdes shwn in Figue 8. The excitatin suce is mdeled by an istpic pint suce with a mment M (thee thgnal diples each having a diple mment f M ) which is lcated at ( s, 0, 0) and vaies as a step functin in time. [45] The displacement at (,, ) in the epicental cdinates is given by [Wdhuse and Ginius, 982] u ð;; ;tþ ¼ X q u ð;; ;tþ ¼ X q b 0 U q ðþy l 00 ð; ÞS q e i!kt ; ð53þ b V q ðþ Y l 0 ð; ÞþYl 0 ð; Þ Sq e i!kt ; and Y l n m (, ) ae the genealized spheical hamnics defined by Phinney and Buidge [973] and q dentes a multiplet mde. The eigenfunctins used hee ae cmputed with the fee ze adial displacement suface bunday cnditin and nmalized by Z R 0! 2 U 2 2 q þ llþ ð ÞV q 2 ¼ : ð59þ The iginal badband seismgam ecded n the day f the Pinatub euptin is shwn in Figue 0. The amplitude spectum f the synthetic gund mtin is pltted in Figue. As we expected, the spheidal mdes nea 270 s and 230 s ae pefeentially excited and the bseved amplitude ati f the tw peaks nea 230 s and 270 s is epduced in the synthetic. We did nt include the mdes with peids shte than 200 s because in the eal Eath, the atmspheic eigenfunctin becmes a ppagating acustic wave (sectin 3) and the excitatin f the mde by a suce in the ppagating egin will be small cmpaed t the excitatin by a suce in the standing wave egin. We have nt used a nmal mde appach t cmpute synthetics f the adiatin bunday cnditin as ppsed by Yamamua and Kawakatsu [998]. [46] A highe altitude suce yields lage gund mtin because the enegy density f the stngly cupled mdes (230s and 270s) inceases with altitude in the tpsphee and the statsphee (up t 50 km) and these mdes ae excited me efficiently by a suce at highe altitude. In u synthetic expeiments, the amplitudes f the gund mtin excited by a suce lcated at, 2, 4 and 8 km in the ai elative t thse excited by a suce at 0.5 km, ae.,.2,.5 and 2.2, espectively. The wavefms f hamnic scillatins have little dependence n the suce altitude. [47] Seismic mment has been used t expess the magnitude f eathquakes in seismlgy; hweve, the suce desciptin in tems f enegy is me apppiate f vlcanic euptins. The vlcanic euptin suce culd be mdeled by a nn istpic diple mment suce, f example by a cmbinatin f diples with diffeent mments, but f simplicity we use an istpic diple suce in this pape. We cnside a small spheical suce vlume with a adius aund the pint suce in a fluid. If we use an istpic diple mment suce the displacement at the suface f the suce vlume is given by ð54þ u ð;; ;tþ ¼ 0; ð55þ u ¼ M 4 2 ; ð60þ 3 f 20