Analysis of Surface Wave Propagation Based on the Thin Layered Element Method

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1 ABSTACT : Analyi of Surface Wave Propagation Baed on the Thin ayered Element Method H. AKAGAWA 1 and S. AKAI 1 Engineer, Technical Centre, Oyo Corporation, Ibaraki, Japan Profeor, Graduate School of Engineering, Chiba Univerity, Chiba, Japan nakagawa-hiroto@oyonet.oyo.co.jp, nakai@faculty.chiba-u.jp In thi paper, a tudy i preented to obtain eigenvalue of urface wave, uch a ayleigh and ove wave, baed on the two-dimenional thin layered element method. The formulation include the computation of phae velocitie, mode hape and amplitude repone function (medium repone of ayleigh and ove wave propagating in a layered half-pace. By comparing the computed value with the one, it i concluded that the thin layered element method can be ued a a very good alternative to the olution method. In addition to the eigenvalue, H/V amplitude ratio of the urface wave have been examined by thi method, in which a ynthei of higher mode contribution of both ayleigh and ove wave i formulated. The propoed method wa applied to a ite where a PS-logging i available. Finally, wave propagation in an irregular ground due to an incident ayleigh wave i formulated baed on the finite element method. It wa found that wave propagation i ignificantly influenced by the irregularity. KEYWODS: Surface Wave, Thin ayered Element Method, H/V Spectra, Microtremor, Finite Element Method, Irregular Ground 1. I TODUCTIO There are a number of method available that are ued for etimating dynamic propertie of the ground. Among other, microtremor meaurement i one of the implet and the mot inexpenive way and ha been conducted extenively. It i accepted that the peak of it horizontal-to-vertical pectral ratio (H/V pectrum obtained from microtremor meaurement correpond to the natural frequency of a urface ground. Tokimatu and Arai (1998 proved that the H/V pectra of microtremor can be explained by the theory of urface wave propagation, including higher mode in addition to the mode. Although the cloed form olution to the eigenvalue problem of urface wave propagation i available, difficultie are encountered when the oil profile i complex. In uch a cae, a numerical approach can be ued. Waa (197 and other reearcher have propoed a finite element approach to the wave propagation problem, in which the oil layer i divided into a number of ublayer and the force-diplacement relationhip i obtained at the vertical boundary baed on the mode uperpoition method. Thi technique, known a the thin layered element method, ha been ued in the two-dimenional finite element analye a the energy tranmitting boundary. However, thi method can alo be applied to eigenvalue problem of urface wave propagation.. THI AYEED EEME T METHOD In thi tudy, a finite element technique ha been applied to the wave propagation problem due to an excitation force and a urface wave incidence. A lumped ma finite element formulation for a multi-layered ytem with a rigid bae wa developed by ymer(197 and wa extended by Waa(197 to include a conitent ma formulation and love wave. Thi method i briefly reviewed below..1. Eigenvalue Problem The layered ytem i treated a a continuum in the horizontal direction but i dicretized in the vertical direction by auming that the diplacement i continuou at each interface and varie linearly or curvilinearly

2 within each layer. In thi tudy, quadratic element are ued. Baed on the formulation given by Waa(197, the equation of motion for the layered ytem may be written a: ( k [ A] + [ G] [ M] { v} = { } ( k [ A] + k[ B] i+ [ G] [ M] { v} = { } ω (.1 ω (. v i a vector containing layer interface and mid-point node amplitude for quadratic element a M are matrice aembled by the addition of layer ubmatrice, k i the wave number, ubcript preent ayleigh wave, and ubcript preent ove wave. Eq. (.1 and (. are the quadratic eigenvalue problem. A numerical technique can be applied to find eigenvalue and correponding eigenvector. Emphai can be placed on the fact that among thee eigenvalue there exit phyical mode which correpond to actual ayleigh and ove wave. In which, { } hown in Figure 1, [ A ],[ B ],[ G ],and [ ].. ine oad Excitation and Correponding Amplitude Function The load-diplacement relationhip, i.e. the dynamic tiffne matrix, at a vertical boundary of a emi-infinite layered region i called a tranmitting boundary and i given for the region extending toward right a: T [ ] i[ A][ V][ K][ V] [ A] 1 [ ] = i[ A][ V][ K][ V] [ D] = (.3 + where [ V ] i a matrix which contain mode hape in it column, [ ] number on the diagonal and [ D ] i related to [ B ] through [ B] [ D] T [ D] region i analogou to that of a right layered region. The dynamic tiffne matrix, [ ] region may be computed from [ ] (.4 K i a diagonal matrix which contain wave =. The analyi of a left layered, of a left layered by changing only the ign of all the coefficient that relate horizontal force to vertical diplacement or vertical force to horizontal diplacement. Thu, [ ] [ ] the vertical boundary, { } + i the tiffne matrix at the vertical boundary x=. Thu, we can calculate the diplacement at u, when an external force act at x=. The mode participation factor,{ α }, can be obtained a follow: 1 { } = [ V] { u} 1 { } = [ ] { } α (.5 α V u (.6 It i worthy of note that the participation factor of the phyical mode repreent their medium repone (Harkrider(1964, or amplitude function. y ine Force x 1 V S V P 3 Node igid Bae z Element 4 1 Velocity[m/] Figure 1 Thin layered element method Figure Wave Velocity Profile at ite N

3 .3. H/V Spectra due to ayleigh and ove Wave By conulting the work done by Tokimatu and Arai(1998, the horizontal-to-vertical diplacement ratio at the ground urface, i.e. the H/V pectra, due to ayleigh and ove wave can be computed from the mode hape and the participation factor by the following expreion: { } ( α { 1 γ ( vx vz } x z ( ( v v 1+ γ ( v v + x z ( H / V = α (.7 ( α ( vx vz ( ( α ( ( vx vz γ = α (.8 ( ( H / V = ( H / V 1+ 1 (.9 Where, γ i an H/V ratio of excitation force and i a ayleigh to ove wave amplitude ratio in horizontal motion. To examine the applicability and the limitation of Eq.(.7-(.9 to the H/V pectrum of microtremor, three component microtremor data oberved at a ite in Chiba city, ay ite N, are ued in thi tudy. It wave velocity profile i hown in Figure. Mode hape for the frequency of 8 Hz are hown in Figure 3. Phae velocitie, medium repone, and H/V pectra are hown in Figure 4. Here, i et to.7, referring to Tokimatu and Arai(1998. By comparing the computed value with the one, it i concluded that the thin layered element method can be ued a a very good alternative to the olution method. 1 Fundamental ove mode at frequency of 8 Hz TM Nomalized amplitude 1 1t higher ove mode at frequency of 8 Hz eal part of TM Imag. part of TM Nomalized amplitude 1 Fundamental ayleigh mode at frequency of 8Hz (u x (u z e(u x of TM Im(u x of TM e(u z of TM Im(u z of TM Nomalized amplitude 1 1t higher ayleigh mode at frequency of 8 Hz (u x (u z e(u x of TM Im(u x of TM e(u z of TM Im(u z of TM Nomalized amplitude Figure 3 Comparion of mode hape(x= for of 8Hz at ite N Phae velocity [m/] 6 4 Fundamental 1t higher ayleigh ove ayleigh nd ove ayleigh ove TM H/V ratio 6 4 TM oberved Medium epone ove TM 1t nd Medium epone ayleigh TM 1t 5 1 nd 15 Figure 4 Comparion of phae velocitie, medium repone, and H/V pectra at ite N 3. I CIDE T AYEIGH WAVE I A IEGUA GOU D A number of tudie have been made on the urface wave propagation baed on the hypothei that the oil medium i horizontally layered. The applicability of H/V pectra of microtremor to etimation of dynamic propertie of the ground i uually dicued in thi context. However, we often encounter a ituation in which it i difficult to aume horizontal layering. Complex landform i a typical example of thi. In thi tudy, a two-dimenional finite element approach in conjunction with the thin layered element method decribed earlier

4 ha been ued to invetigate urface (ayleigh wave propagation in an irregular ground. dicued by Drake(197 and Uebayahi(6. A imilar topic wa 3.1. Method of Analyi The analyi method i baed on the o-called ubtructure approach in elatodynamic. In thi approach, an infinite medium i divided into a near field and a far field. In the analyi of the near field, the impedance matrix of the far field i attached to the near field at it boundary and the driving force due to an incident wave from the far field i applied to the boundary. Suppoe a eparation of a oil model into three part along the vertical boundarie a hown in Figure 5, the driving force due to an incident ayleigh wave propagating from left can be expreed a: { f } [ ] { v} { p} = α (3.1 c + In which, [ ] mode hape of the incident ayleigh wave and { p} i an impedance matrix of a left layered oil, α i a mode participation factor, { } v i a i a traction due to the incident wave at the boundary of the left layered oil. 3.. Analyi Model In thi tudy, a valley haped irregular ground i conidered, a hown in Figure 5. The oil conit of two layer and their boundary i horizontal. The underlying layer i aumed a a half-pace by changing it thickne a H = 4V f where V i it hear wave velocity and f i a frequency of analyi. The P-wave velocity and the denity are the ame for both layer; Vp =15m/, ρ =17kg/m 3. The urface layer wa partitioned into 5 element for the vertical direction and the underlying layer wa divided into. In order to uppre body wave reflection from the bottom, relatively large damping of 3% to 5% i aumed in the lower part of the model. Theoretical value of H/V ratio and the phae velocity of ayleigh wave at x= and x=6 for the frequency of 1 Hz are hown in Table 1. When a two layered oil i aumed, the ayleigh wave ha three mode for the terrace part and two mode for the valley part, for the frequency of 1 Hz. The phae velocity of a uperpoed mode i obtained by referring to Tokimatu et al(199. TM x= (incident ayleigh wave FEM x=15 x=1 TM [] 15m 9m 15m [] x=16 1m x=138 8m V=m/ Node 4 x V / f (variable V=4m/ Figure 5 Finite element model igid Bae Table 1 Theoretical value of H/V ratio and phae velocity of ayleigh wave at x= and x=6 for 1Hz terrace (x=;surface layer thickne = m valley (x=6;surface layer thickne = 8m mode H/V ratio phae velocity[m/] H/V ratio phae velocity[m/] t higher nd higher uperpoed

5 3.3. Propagation of an Incident ayleigh wave Figure 6 how the wave propagation in a valley haped irregular ground due to an incident ayleigh wave from the left layered oil. The figure repreent the reult of two cae: the mode incidence and the firt higher mode incidence, repectively. Each cae ha four different kind of reult: cro ectional ditribution of the amplitude and the phae of horizontal and vertical diplacement. eft to thee ditribution diagram are the diplacement ditribution of the incident ayleigh wave along the vertical axi. There are a number of point that can be made from thi figure. Firt, it i clearly een from the phae diagram that body wave are generated a reflected wave from the lope at the left-hand ide of the valley. Thi occur for each mode incidence. Striped pattern are alo clearly een in the amplitude diagram, which i conidered a a reult of the interference of body wave and the incident ayleigh wave. Next, if we look at the central part of the valley, it i noticed that the phae pattern become vertical a the ditance from the lefthand lope increae. Thi explain that the body wave attenuate rapidly compared to the urface wave becaue of large damping. In thi cae, the urface wave dominate in the right ide of the valley. Finally, if we look at the right-mot part of the valley, it i een that body wave are again generated from the lope. However, the phae pattern are fairly complex compared to the left-hand ide of the valley. Thi implie that the propagating wave on the right-hand ide of the valley i not a imple a a ingle mode wave propagation Phae Velocity The effect of irregularity of the ground on the urface wave propagation can be examined in detail from the viewpoint of it phae velocitie, by looking into the reult decribed above. Figure 7 how the phae velocity, a a function of the ditance from the left-mot boundary of the model. The bottom of the valley pan 876 meter from the ditance of 16[m] through 138[m]. Here, the phae velocity wa computed from the phae difference between adjacent two node and wa given by averaging two component obtained from horizontal and vertical diplacement. By reffering to Table 1, it i found that on the terrace in the left-hand ide, the phae velocity roughly correpond to that of a two layered ground with the ame oil profile a the terrace, for all cae of wave mode incidence. Variation of the value on the terrace can be attributed to the exitence of body wave reflected from the lope. More variation for higher mode may be reulted from dicretization error. 3.5.Contribution of ayleigh Wave for 1 Hz et u look at the lowland. The phae velocity varie a lot near the left-hand lope due to body wave generated from the lope. The variation decreae a the ditance from the lope increae. In order to examine thi tendency, a contribution ratio of ayleigh wave to nodal diplacement along the ground urface of the lowland (valley wa computed baed on Eq.(.6. The computation proce i a follow: firt compute mode participation factor { α} by ubtituting nodal diplacement { u } into Eq.(.6, then eparate the modal contribution by inverting Eq.(.6 a: { } [ ]{ } u = V α (3. Notice that there exit only two mode of ayleigh wave for the two layered oil with the ame oil profile a the valley part of the model and for the frequency of 1 Hz. Thu, the nodal diplacement can be expreed a a um of the contribution of the and firt higher mode of ayleigh wave and the body wave. The reult are hown in Figure 8. A can be een from the figure that the urface wave dominate a the ditance from the lope increae. The fluctuation of the finite element reult i conidered to be due to the body wave generated from the left-hand lope. It i alo found from the figure that the higher mode prevail over the mode in the valley, which can be confirmed by the fact that the phae velocity in the right ide of the valley i cloe to that of the firt higher mode of a two layered oil a hown in Figure 7.

6 Depth[m] Depth[m] 4 6 X= V= m/ V= 4m/ Diplacement[m] X= Diplacement[m] e(u x Im(u x e(u z Im(u z V= m/ V= 4m/ e(u x Im(u x e(u z Im(u z 1 Horizontal amplitude ( mode incidene 18 1 Horizontal phae (1t higher mode incidene 18 1 Vetical amplitude ( mode incidene 18 1 Vertical phae ( mode incidene Ditance [m] 1 Horizontal amplitude (1t higher mode incidene 18 1 Horizontal phae (1t higher mode incidene 18 1 Vetical amplitude (1t higher mode incidene 18 1 Vertical phae (1t higher mode incidene Ditance [m] Dip. [m] 1e-1 7.5e-11 5e-11.5e-11. Phae [rad] π π/ -π/ -π Dip. [m] 1e-1 7.5e-11 5e-11.5e-11. Phae [rad] π Figure 6 ayleigh wave propagation in a valley haped irregular ground for the frequency of 1Hz π/ -π/ -π Phae velocity [m/] 6 4 mode incidence 1t higher mode incidence nd higher mode incidence Ditance[m] Figure 7 Phae velocity along the ground urface for the frequency of 1Hz Diplacement [m] Diplacement [m] FEM-eult FEM-eult 1t higher 1t higher Horizontal freq. =1 [Hz] Vertical freq. =1 [Hz] Ditance [m] Figure 8 Contribution of the ayleigh wave for the frequency of 1Hz (Fundamental mode incidence

7 3.6.Contribution of ayleigh Wave : Frequency Characteritic Next, let u examine how the contribution of ayleigh wave change when the frequency change. Figure 9 how the frequency dependency of the contribution of each mode to the total diplacement for variou location on the ground urface of the lowland (valley. The location x=16 correpond to the left-mot point of the lowland and at the foot of the left-hand lope. The location x=138 correpond to the right-mot point of the lowland and at the foot of the right-hand lope. Other location are ituated in-between. It i clearly een from Figure 9 that the mode determine the total diplacement (FEM-eult in the figure in the lower frequency range up to 1 Hz, where the firt higher mode appear. It i alo clear that, once the firt higher mode appear, it dominate over the mode. Thi tendency become clear a the ditance from the lope increae. The contribution of ayleigh wave mode i relatively mall near the lope, epecially near the natural frequency of the urface layer (6.5 Hz, which mean that the effect of generated body wave i large in thi area. Thi reult i compatible with the one out by Tokimatu and Tamura( Horizontal Diplacement [m] FEM-eult 1t higher x= x=45 FEM-eult 1t higher x=6 FEM-eult 1t higher x=75 FEM-eult 1t higher FEM-eult x=138 1t higher Vertical Diplacement [m] FEM-eult 1t higher x= x=45 FEM-eult 1t higher x=6 FEM-eult 1t higher x=75 FEM-eult 1t higher x=138 FEM-eult 1t higher Figure 9 Contribution of the ayleigh wave ( mode incidence uperpoed 1t higher 15 H/V ratio > Ditance [m] Figure 1 H/V pectrum along the ground urface ( mode incidence

8 Figure 1 how the H/V pectrum ditribution along the ground urface of the model. The horizontal axi repreent ditance, the vertical axi repreent frequency, and the color expree H/V ratio of the diplacement. Three color bar on the right-hand ide repreent the H/V pectra of the, firt higher and uperpoed mode of a two-layered oil correponding to the lowland. It i found from thi figure that the peak frequency and the ratio itelf of H/V pectrum vary a lot in term of the location and the frequency. The peak frequency in the low frequency range roughly coincide with that of a mode of a two-layered oil. However, thi tendency become unclear a the location move to the right. In the right-hand ide of the lowland, the H/V pectrum in the high frequency range almot coincide with that of the firt higher mode of the two-layered oil. Thi confirm again that the firt higher mode dominate in thi part. It mut be noted that thi phenomenon differ from what the theory baed on the parallel layer aumption predict, in which the mode dominate even in the high frequency range. The effect of irregularity of the ground need to be further examined from thi viewpoint. 4. CO CUSIO S From thi tudy, the following concluion can be made: (1 H/V pectra of a horizontally layered medium a well a the urface wave mode can be obtained baed on the thin layered element method. ( When an incident ayleigh wave propagate toward an irregular ground, different mode of ayleigh wave a well a body wave are generated. (3 A the ditance from the irregularity increae the urface wave dominate over the body wave. (4 In the high frequency range, higher mode dominate over the mode, which differ from the prediction baed on a parallel layer aumption. ACK OWEDGME TS The author wih to expre their gratitude to Dr. Maanori Saito for the ue of hi program for the analyi of urface wave. EFEE CES Drake,. A.(197. ove and ayleigh Wave in Nonhorizontally ayered Media, Bull. Seim. Soc. Am., Vol6, No.5, pp Harkrider, D. G.(1964. Surface Wave in Multilayered Elatic Media I. ayleigh and ove Wave from Buried Source in a Multilayered Elatic Half-Space, Bull. Seim. Soc. Am., Vol.54, No.., pp ymer, J.(197. umped Ma Method for ayleigh Wave, Bull. Seim. Soc. Am., Vol.6, No.1,pp Tokimatu, K., Tamura, S., Kojima, H.(199. Effect of Multiple Mode on ayleigh Wave Diperion Characteritic, J. Geotech. Eng., ASCE, Vol.118, No.1, pp Tokimatu, K. and Tamura, S. (1995. Contribution of ayleigh and Body Wave to Diplacement Induced by Vertical Point Force on a ayered Elatic Half-pace, J. Struct. Contr. Eng., AIJ, 476, pp (in Japanee. Tokimatu, K. and Arai, H.(1998. Effect of ayleigh to ove Wave Amplitude atio on Microtremor Horizontal -to-vertical Spectral, J. Struct. Contr. Eng., AIJ, 511, pp (in Japanee. Uebayahi, H.(6. Inverion of Microeim Horizontal-to-Vertical Spectral atio for Determining the Boundary Shape of a Bain Structure -Verification by numerical experiment-, J. Struct. Contr. Eng., AIJ, 63, pp (in Japanee. Waa.G.(197. Analyi Method for Footing Vibration through ayered Media,Tech.eport S-71-14,eport 3,Univ. of Calif. Berkeley.