P-SV Wave-Field Connection Technique for Regional Wave Propagation Simulation

Transcription

1 Bulletin of the Seimological Society of America, Vol. 95, No. 4, pp , Augut 2005, doi: / P-SV Wave-Field Connection Technique for Regional Wave Propagation Simulation by Zengxi Ge, Li-Yun Fu,* and Ru-Shan Wu Abtract A boundary element (BE) method i developed to calculate the twodimenional P-SV elatic repone for crutal wave guide with irregular topographic feature. To imulate long-range propagation of regional wave, a connection technique i propoed to avoid large matrix inverion that become formidable for longrange, high-frequency problem. By uing thi technique, a long crutal wave guide can be divided into relatively horter ection, and the BE method can be ued ection by ection to model the effect of rough topography on wave propagation at extended regional ditance. The validity of the technique i teted by comparion with a direct calculation. Numerical imulation with thi cheme how that rough topography can catter the P and Rayleigh wave and attenuate the energy propagating in the wave guide. Thi method can be ued in computing the ite effect on ite uch a canyon, mountain, and valley. The connection technique expand thi method to deal with large earth model with irregular topography. Introduction Regional phae have long been recognized a an important iue in the tudy of large-cale crutal tructure, mall-cale crutal heterogeneitie, eimic ource, and analyi of underground exploion and earthquake. To improve path correction and verify empirical obervation, numerical imulation are needed to etimate the path effect on regional wave propagation. A number of regional numerical modeling method have been developed to model Lg propagation behavior and path effect, depending on the complexity of the crut heterogeneity in the region conidered. Finite-difference (FD) and finite-element (FE) method are univeral numerical technique for imulation in general heterogeneou media. Becaue of the computational intenity, thee full-waveform method are often limited to low frequencie for Lg imulation at regional ditance. To reduce computation time, ome alternative and flexible approache, for example, ray diagram (Botock and Kennett, 1990; Keer et al., 1996) and dynamic raytracing method (Gibon and Campillo, 1994), have been ued to obtain intuitive undertanding of path effect on Lg propagation. For long-range Lg-wave propagation, Wu et al. (1996, 2000) introduced a half-pace GSP (generalized creen propagator) for modeling the main characteritic of Lg (2D SH cae) in heterogeneou crutal wave guide. The method wa ex- *Preent addre: Intitute of Geology and Geophyic, Chinee Academy of Science, P.O. Box 9825, Beijing , People Republic of China. lfu@mail.igca.ac.cn tended to include the cae of irregular urface topographie (Wu and Wu, 2001) Baed on obervation and numerical experiment, large-cale crutal tructure with variation in free urface and Moho topography control the principal characteritic of Lg propagation. In contrat to the FD and FE method, the boundary integral method are more uitable for modeling complex reflection and tranmiion effect acro largecale crutal tructure with rugged topography. For intance, the boundary element (BE) method provide a geometrically accurate decription of irregular interface. Becaue the BE method i formulated in term of integral along boundarie, the traction-free condition for a rugged free urface i eaily and naturally treated. However, for regional wave guide up to thouand of kilometer, the BE method in it original form lead to very large matrice to be inverted and it implementation i prohibitively expenive. In a previou paper (Fu and Wu, 2001), a SH-wavefield connection technique wa developed by which the BE method can be ued ection by ection for an event at farregional ditance up to everal thouand kilometer; the method take the output of the previou ection a the input of the next ection to complete the entire crutal wave-guide computation. Full-wave BE modeling i implemented within each ection. The wave-field connection technique i ued to couple the field calculated in two adjoining ection. The diviion of ection i baed on the complexity of crutal tructure with a criterion of minimizing the poible multiple backcattering between ection. Thi approach for 1375

2 1376 Z. Ge, L.-Y. Fu, and R.-S. Wu long regional waveguide lead to ignificant computational aving in time and memory compared with the whole waveguide BE method. The ection-by-ection approach alo lead to a hybrid modeling cheme of BE and GSP. The BE method i implemented in the frequency domain and ha a kernel function compatible with the GSP method. In the hybrid cheme, the time-conuming BE method can be ued to handle the ection with complicated boundary tructure and evere urface topographie. Subequently, the output will be ued a the input to the GSP method for modeling ection with a large volume of moderately heterogeneou media and mild topographie. The hybrid method ha been applied to two crutal wave-guide model from the Tibet region, one with Lg blockage and another without blockage (Fu and Wu, 2001). The object of thi article i to develop a P-SV wavefield connection technique for imulating elatic wave propagation in regional crutal waveguide. We firt introduce an integral equation repreentation for the crutal waveguide problem. We then briefly decribe the principle of the elatic BE method and validate the computation program uing previouly publihed reult. We develop the P-SV wave-field connection technique and tet it by uing numerical experiment. Finally, we apply the ection-by-ection approach to Lg-propagation imulation in long regional wave guide. Boundary Integral Equation for Crutal Wave Guide Conider 2D teady-tate elatic wave propagation in a implified crutal wave guide X 1 bounded by a rough free urface C 1 and an irregular Moho interface C 2. Figure 1 depict the geometry of the problem. The wave-guide medium i iotropic and homogeneou, decribed by the Lame contant (k 1 and l 1 ) and denity (q 1 ). The diplacement vector u(r) at a location r(x, z) atifie the following elatic wave equation: 2 2 l u(r) (k l) u(r) qx u(r) f(r, x) (1) where f(r, x) i the body force occupying a region X. u(r) alo atifie the traction-free boundary condition on C 1 and the continuity condition of diplacement and traction acro the Moho. The medium in the mantle X 2 i decribed by k 2, l 2, and q 2. We add two artificial boundarie C at the two truncated edge of the wave guide to form a cloed olution domain C C 1 C 2 C. Baed on the repreentation theorem (Aki and Richard, 1980), equation (1) can be tranformed into the following integral formulation: C(r)u(r) [u(r)r(r, r) t(r)g(r, r)]dc(r) C f(r, x)g(r, r)dx(r) (2) X Figure 1. Geometry of a implified crutal wave guide with irregular topography and Moho interface. where t(r) i the traction vector, the coefficient C(r) generally depend on the local geometry at r, and G(r, r) and R(r, r) are the fundamental olution (Green tenor) for diplacement and traction, repectively. C(r), R(r, r) and G(r, r) are 2 2 matrice for 2D problem. In thi equation, r i the poition of obervation point and r i the poition of cattering point. For implicity, the ource ditribution conit of a point ource at r 0 located inide X 1. The ource integral over X in the right ide of equation (2) can be reduced to G(r,r)f(r,x)dr G(r,r )f(x) (3) 0 X where f (x) i the ource pectrum vector. The diplacement Green tenor G(r, r) atifie: 2 [(k l) l ]G(r,r) 2 qx G(r,r) d(r r) (4) with the olution given by More and Fehbach (1953). The traction Green tenor R(r, r) can be derived from G(r, r) by uing Hooke law, R(r,r) ki( G(r,r)) l(g(r,r) G(r,r)), (5) where I i the unit dyadic and G(r, r)d i the tranpoe of DG(r, r) with repect to the correponding coordinate, ee Wu (1989, Appendix). For analytic expreion of G and R in an iotropic homogeneou elatic medium, thee Green tenor for 2D problem can be expreed a (Fu, 1996): i Gij (wdij ci c) j 4l i w ij (dijck nk cj n i) 4 r r 2 (c n 2c ccn ) 2 cccn r r k w c i n l r r r j i j i j k k i y k k

3 P-SV Wave-Field Connection Technique for Regional Wave Propagation Simulation 1377 where i, j 1, 2, n i the outward normal of the boundary C, and (1) 1 (1) b (1) w H 0 (kr) H 1 (kr) H 1 (kpr) kr 2 b (1) (1) 2 2 p 2 H (k r) H (k r) with k x/b, k p x/, r r r and c j (r j r j )/r. The boundary integral repreentation for wave propagation naturally atifie the Sommerfeld radiation boundary condition that are impoed on the far-field behavior at infinity. No wave come back to X 1 through C, that i, the following integral on C for the interior problem vanihe: C [R(r,r)u(r) G(r,r)t(r,r)]dr 0, r X (6) 1 For numerical calculation, truncating the wave guide i neceary and an infinite boundary element aborbing boundary technique (Fu and Wu, 2000) ha to be applied to the element at the truncating point on C 1 and C 2. Conidering equation (3) and (6), and applying the traction-free condition to the free urface C 1, equation (2) for the interior problem in the crut (region X 1 ) i implified to C(r)u(r) C1 R(r, r)u(r)dr [R(r, r)u(r) G(r, r)t(r)]dr (7) C2 G(r, r )f(x) 0 Equation (7) i a tarting point for numerical implementation for wave-propagation imulation. To olve u(r) and t(r) on C 2 we mut build the correponding boundary integral equation in X 2. The following integral formulation can be etablihed for r X 2 bounded by a cloed urface C C 2 C : C(r)u(r) [R(r,r)u(r) G(r,r)t(r)]dr 0 (8) C Boundary Element Method for Elatic Wave Simulation The boundary integral repreentation decribed previouly can be ued to calculate the field at any point inide X 1 once u(r) and t(r) are known on the boundarie. We ue the BE method to olve the value of u(r) and t(r) onthe boundarie. We dicretize C 1 into L 1 element and N 1 node, and C 2 into L 2 element and N 2 node. The total node number i N. By uing the linear interpolation hape function (n) in an element between the node I 1 and I 2, the variable u(r) and t(r) are approximated by the linear combination of their node value over the element, for example (ee Fig 1): I 2 l l li 1 u(n) u(r ) (n), (10) where n and l denote the local coordinate and local node index of an element. Uing the following Kronecker delta function notation relating the local node code l of an element to the global node code j we have 0, l j d, (11) ij 1, l j N I 2 l l ij j1 li 1 u(n) u(r ) (n)d, (12) Letting C i C(r i ) and u i u(r i ), equation (7) for i 1 to N i dicretized into Cu N L I 1 2 i i (r i,r(n)) l(n)dr(n) dijuj j1 e1 li1 C e L2 I2 (r i,r(n)) l(n)dr(n) dljuj e1 li1 C e L2 I2 G(r i,r(n)) l(n)dr(n) dljtj e1 li1 C e Similarly, the integration over the tranparent artificial boundary C vanihe for the interior problem, implifying equation (8) a G(r,r )f(x) i 0 which can be further rewritten a (13) C(r)u(r) [R(r,r)u(r) G(r,r)t(r)]dr 0. (9) C2 Equation (8) and (9) provide a decription of the field through the crutal wave guide, making poible the imultaneou evaluation of the unknown [u(r) onc 1 and u(r) and t(r) onc 2 ] by uing the continuity of diplacement and traction acro C 2. N (1) (1) (2) (2) (2) (2) ij j ij j ij j i 0 j1 with h u h u g t G(r,r )f(x), (14) L1 I2 (1) (1) ij j l ij j ij e1 li1 Ce h [R(r,r(n)) (n)dr(n)]d C d, (15)

4 1378 Z. Ge, L.-Y. Fu, and R.-S. Wu L2 I2 (2) (2) ij i l ij j ij e1 li1 Ce h [R(r,r(n)) (n)dr(n)]d C d, (16) L2 I2 (2) gij [G(r i,r(n)) l(n)dr(n)]d ij, (17) e1 li1 Ce (1) In thi expreion u j i the diplacement vector on the free urface C 1 and u (2) j and tj (2) are the diplacement and traction vector on the Moho C 2. The coefficient matrice h (1) ij, hij (2) (2) and g ij, obtained by numerically integrating the product of the Green tenor with interpolation hape function over element, denote a concentrated force generated at the jth cattering point on C and applied at the ith obervation point. For i 1toN, equation (14) can be further compacted a a matrix equation: (1) (1) (2) (2) (2) (2) H u H u G t F, (18) where F qg (r i, r 0 )f(x). Similarly, equation (9) for i 1 to N can be dicretized and compacted a (2) (2) (2) (2) H u G t 0, (19) where the coefficient matrice, H (2) and G (2), are calculated by uing the medium propertie of the mantle X 2. A imultaneou ytem of matrix equation (18) and (19) can be aembled by uing the continuity of diplacement and traction acro C 2, The computation program of the elatic wave BE method decribed previouly i teted by dimenionle frequency repone of a emicircular canyon of radiu a. Previouly publihed reult (Sánchez-Sema et al., 1985; Sánchez-Sema and Campillo, 1991) for thi typical topographic tructure are ued for comparion. The two harp edge at x a provide a crucial target to validate variou numerical method. Figure 2 how the comparion with good agreement in both horizontal and vertical amplitude between our reult (olid line) and the Sánchez-Sema and Campillo olution for vertically incident plane P wave and for variou normalized frequencie g defined a g 2a / l, where l i the wavelength. Poion ratio i aumed to be 1/3. We can ee ome minor departure becaue of different element approximation ued by thee two numerical method. Fewer element per wavelength will reduce the ize of the reultant coefficient matrice. To determine an applicable element number per wavelength, the comparion for 0 Incidence and g 1.0 are given in Figure 3 for different dicretization rate of point per wavelength to dicretize the arc of the emicircular canyon. We ee that a ampling at three point per wavelength could be ued for general application. The preceding comparion confirm the validity of our formulation and computation code. Later we will ue thi method to validate the connection technique for a larger model. We now demontrate the applicability of our program by yntheizing wave propagation through a ingle- u (2) (2) ũ (2) (2) t t (20) by which the unknown [u(r)onc 1 and u(r) and t(r)onc 2 ] can be found. Thee matrice in equation (18) and (19) are full with complex coefficient that are function of frequency, material property, and geometry. The BE method decribed previouly can be directly extended to complex geological tructure with multiple region (Fu, 1996) for explorationoriented eimic modeling. Alvarez-Rubio et al. (2004) applied the BE method to ite effect aement of laterally varying layered media. Becaue a large number of matrix operation are involved and the matrix for each frequency component mut be inverted, the BE method i computationally intenive at high frequencie for far-regional wave guide. To improve computation peed, a frequency-dependant element dimenion technique can be adopted in the program implementation (Fu and Mu, 1994). Bouchon et al. (1995) uggeted a threhold criterion approach to make the coefficient matrice parer by removing very mall entrie in the BE coefficient matrice. An efficient modification to the BE method can be made uing the ection-by-ection approach with the wave-field connection technique developed in thi tudy. Figure 2. The horizontal and vertical amplitude repone by our method (dotted line) and Sanchez- Sema and Campillo (1991) (olid line) of a emicircular canyon topography to vertically incident P wave for variou dimenionle frequencie g.

5 P-SV Wave-Field Connection Technique for Regional Wave Propagation Simulation 1379 Figure 3. The horizontal and vertical amplitude repone of a emicircular canyon topography to vertically incident P wave for g 1.0 with the olid line denoting the olution with 21 point per wave length and the dot denoting the pecified ampling rate. Figure 4. A ingle-layer flat crutal wave guide with an exploive ource at 2 km depth. The wave generated in the wave guide are grouped into three ytem: the firt directly from the ource to the receiver (olid line), the econd off the free urface (dahed line), and the third off the Moho (dotted line). layer crutal wave guide with flat free urface. The homogeneou wave guide hown in Figure 4 i 100 km long and 32 km thick, overlaying a flat mantle half-pace. The point ource (P-wave exploive ource) i located on the left boundary at 2.0 km depth. Figure 5 how the ynthetic eimogram calculated in the frequency range of Hz, with receiver at 1-km pacing along a vertical profile at the ditance of 80 km from the ource. We ee that for elatic wave propagation thi imple wave guide with both the flat topography and flat Moho lead to complex uperpoition of variou wave, demontrating the development of converted wave and the formation of guided wave a repetitive reflection at both the free urface and Moho. In term of propagation path (ee Fig. 4), we can identify three major ytem of wave generated in the wave guide. A hown in Figure 5, the direct P wave from the ource to receiver carrie a major part of the ource energy in thi flat boundary wave guide. The P-wave incident on the free urface and the Moho at oblique angle generate two ytem of wave, one et off the free urface and another et off the Moho. Thee two ytem of wave and their converted wave bounce back and forth between the free urface and the Moho. Becaue no cattering mechanim i preent in the wave guide, the contructive interference of the repeatedly reflected wave preent a checkerboard-like pattern (Jih, 1996) that adequately explain the formation of crutal guided wave either a multiple reflection or a higher mode.

6 1380 Z. Ge, L.-Y. Fu, and R.-S. Wu Figure 5. Synthetic eimogram for receiver at 1-km pacing along a vertical profile at 80 km for a point ource. The horizontal (left panel) and vertical (right panel) component are computed in the frequency range 0 3.5Hz. P-SV Wave-Field Connection Technique We aim to develop a ection-by-ection approach for imulating wave propagation in regional wave guide to reduce the computation cot of extremely large matrix operation. The wave-field connection technique couple the field between two adjacent ection. The connection configuration i illutrated in Figure 6a. An artificial boundary C AB i introduced a a wave-field connection boundary to a crutal wave guide coniting of an irregular free urface C 1 and an interface C 2. The wave guide i divided into four ubdomain, X 1 and X 2 in the crut, and Xl and X2 in the mantle. The left boundarie of the firt ection X 1 and X l, and the right boundarie of the econd ection X 2 and X 2 are aumed to extend to infinity at left and right, repectively. The field in X 1 and X l are calculated from the ource uing boundary element method and the output wave field u 0 (r) on the connection boundary C AB will be ued to atify the boundary condition acro C AB when the BE method i ued to calculate wave propagation in X 2 and X 2. The output field are received along the next connection interface C CD and will be ued a the input to the next propagation. With the initial field u 0 (r) known on the connection boundary C AB, we analyze wave propagation in X 2 and X 2 by building boundary integral equation for wave propagation in thee ection. Apparently, all the pace point in X 2 and X 2 except thoe on C AB only receive the cattered field. Therefore, the total field u(r) on the boundarie of X 2 and X 2 i compoed of u 0(r) u (r) r CAB u(r). (21) u (r) r C Applying the traction-free condition to the free urface C 1, the boundary integral equation can be built for r X 2 and r C AB AB C(r)u (r) C 1 C 2 CAB R(r,r)u (r)dr (r,r)[u (r) u (r)]dr (22) 0 CCD (r,r)u (r)dr The artificial boundarie C AB and C CD are aumed to be tranparent, implying the outward radiation of energy acro C AB and C CD alway hould be in the outward direction with no reflection returning to X 2. Therefore, there i no energy contribution cattering from C AB and C CD, that i, [R(r,r)u (r) G(r,r)t (r)]dr 0 r X (23) 2 CAB and [R(r,r)u (r) G(r,r)t (r)]dr 0 r X (24) 2 CCD Scattering from the artificial truncated point, A, B, C, and D, can be handled by uing an infinite-element aborbing boundary technique (Fu and Wu, 2000). Therefore, equation (22) i reduced to C(r)u (r) C1 R(r,r)u (r)dr [(r,r)[u (r) G(r,r)t (r)]dr (25) C2 CAB [(G(r,r)t (r) (r,r)u (r)]dr 0 0

7 P-SV Wave-Field Connection Technique for Regional Wave Propagation Simulation 1381 Figure 6. The boundary connection technique. (a) The diagram how the connection formulation. (b) The vertical incident wave field along the connecting boundary ABB. (c) Comparion of the horizontal component of the wave field along CDD calculated directly from the ource (thick line) and thoe calculated by uing the connection technique (thin line). (d) Comparion of the vertical component. In equation (25), both the initial diplacement and traction on the connection boundary are aumed to be known. u 0 (r) can be calculated from the previou ection, and incident traction t 0 (r) can alo be calculated from the previou ection. Alternatively, we can ue the elatic Rayleigh integral (Wu, 1989), which contain only either the diplacement or the traction field. We can reduce the urface integral in the right ide of equation (25) to the elatic wave Rayleigh integral that only contain the initial diplacement u 0 (r): C(r)u (r) C1 R(r,r)u (r)dr C2 [(r,r)u (r) G(r,r)t (r)]dr (26) 2 CAB [(r,r)u (r)]dr. 0 We ee that the unknown in equation (26) are u (r) onc 1 and u (r) and t (r) onc 2. To olve for u (r) and t (r), we mut build the correponding boundary integral equation in ubdomain X 2 : C(r)u (r) R(r,r)u (r)dr (r,r)[(u (r) 0 C2 CBB u (r)]dr (r,r)u 0(r)dr CDD (27) G(r,r)t (r)dr G(r,r)[(t (r) 0 C2 CBB G(r,r)t 0(r)dr, CDD t (r)]dr where a ufficiently long boundary C BB i ued with it end et a an infinite element. Similarly, becaue there i no di-

8 1382 Z. Ge, L.-Y. Fu, and R.-S. Wu continuity acro C BB and C DD, we can aume that C BB and C DD are tranparent and reduce equation (27) to C(r)u (r) [R(r,r)u (r) G(r,r)t (r)]dr C2 (28) [G(r,r)t (r) (r,r)u (r)]dr. 0 0 CBB The integration term containing the initial traction t 0 (r) in the right ide of equation (28) can be removed by uing the elatic wave Rayleigh integral repreentation, yielding C(r)u (r) [R(r,r)u (r) G(r,r)t (r)]dr C2 (29) 2 CBB (r,r)u (r)]dr. 0 The continuity of the diplacement and traction acro interface C BD i ued when equation (26) and (29) are combined to olve the problem. By olving the joint boundary integral equation of X 2 and X 2, we can obtain the wave field u (r) onc 1, u (r) and t (r) onc 2. The oberved field along C CD i calculated explicitly from the field on the boundarie. Similar to equation (18) and (19), equation (26) and (29) can be expreed in the matrix form and (1) (1) (2) (2) (2) (2) H u H u G t F, (30) (2) (2) (2) (2) H ũ G t F, (31) with the continuou condition acro C 2 u (2) (2) ũ (2) (2) t t (32) By uing thi technique, wave propagation in a long regional wave guide can be partitioned into everal ection contribution for great aving in both CPU time and memory. For example, if we divide a large model into N ection, we can reduce the ize of matrix by N time. Thu the total compute time will be reduced by N * (1/N) 3 1/N 2 time. Validation for P-SV Connection Technique To validate the connection technique, we preent a comparion between the wave field obtained by uing the BE method to directly calculate wave propagation from the ource to the obervation urface C CD and the wave field calculated by the connection cheme (Fig. 6). In both cae the ource time function are the ame, with a dominant frequency of 1 Hz. Firt the intermediate wave field u 0 (r) on C AB (hown in Fig. 6b) calculated from the ource i ued a the incident field for wave propagation in X 2 and X 2. The dominant arrival for the incident field at C AB conit of direct P wave, ps, pp, and multireflection between the two layer. Then the BE method i ued to calculate wave propagation from C AB to C CD. More multiply reflected wave between the free urface and the interface can be clearly een in the eimogram at C CD. The excellent agreement between the wave field calculated by the two method hown in Figure 6c and 6d confirm the validity of the connection technique. Although thi model i only deigned to tet the validity of the connection technique, we can ee computation aving. For a ingle-proceor Pentium IV 2.0 Hz computer, it take about 2 hr to calculate the wave field along C CD directly from the ource, wherea it take only 45 min to compute the wave field along C CD with the connection technique. The memory requirement i alo reduced by four time. Numerical Example and Application In thi ection, we ue the BE connection technique given previouly to imulate the wave field for everal model with rough topographie. The firt model hown in the top panel of Figure 7 i a two-layer crutal model with an irregular topography. There are two dicontinuou interface located at the depth of 15 km and 37 km, repectively, with the propertie lited in Table 1. The correlation length of the topographic fluctuation i 5 km and the root-mean-quare amplitude i 0.5 km. The receiver are along the urface. An exploive ource i located at the depth of 2 km, and the ource time function i a Ricker wavelet with a dominant frequency of 1 Hz. Figure 7 how the ynthetic eimogram received along the rough urface. Comparing with the wave field (Fig. 8) for the model with a flat urface, we can ee that the amplitude of the direct P and Rayleigh wave are diminihed becaue of the cattering effect of the rough topography. Both the forward-cattering and backwardcattering wave can be een in Figure 7. The top panel of Figure 9 how four different topographic curve ued with the crutal model of Figure 7. The ynthetic eimogram are calculated for the exponential and Gauian random topographie, repectively, with each including two rm amplitude of 0.5 km and 0.25 km. The energy attenuation curve correponding to thee topographie are calculated with increaing propagation ditance and are hown in the lower panel of Figure 9. Thee energy Table 1 Propertie of Crut Model Depth (km) V p (km/ec) V (km/ec) P (g/cm 3 ) Infinity

9 P-SV Wave-Field Connection Technique for Regional Wave Propagation Simulation 1383 Figure 7. Synthetic eimogram for a two-layered model with random topography. The ource i located at the depth of 2 km. The topographic fluctuation ha an exponential correlation function with a rm of 0.5 km and a correlation length of 5 km. We can ee both forward-cattering and backcattering in the eimogram. Figure 8. Synthetic eimogram for the model hown in Figure 7 but with flat urface. Multiply-reflected wave and Rayleigh wave can be clearly een in the figure.

10 1384 Z. Ge, L.-Y. Fu, and R.-S. Wu Figure 9. Energy attenuation for earth model with different topographie. On the top are the topographie ued in our calculation, the thick olid line i a Gauian random topography with a rm of 0.5 km, the thin olid line i a Gauian random topography with a rm of 0.25 km, the thick dahed line i exponential random topography with rm 0.5 km, and the thin dahed line i exponential random topography with rm 0.25 km, while the reference model (flat urface) i hown by the dotted line. On the bottom are the energy attenuation curve for the whole time window, the line have the ame meaning a thoe in the top figure. attenuation curve manifet different degree of topographycattering attenuation. We can ee that the exponential topographie lead to more effective energy attenuation than Gauian topographie, and alo the larger the topographic fluctuation i, the tronger i the topographic cattering. Next we apply the connection technique to regional wave propagation imulation in another ynthetic model. A hown in Figure 10a, the model i a laterally varying crutal wave guide, 600 km long and 32 km thick, overlaying a mantle half-pace. The model i divided into three 200- km-long egment. The firt egment contain a Gauian hill which i given by h(x) h 2 (x x 0) 0 exp 2 where x 0 62 km, h 0 4 km, r km. The econd egment preent a thinned wave guide with the 7-km-high tep. The third egment contain an exponential random topography with correlation length of 5 km and rm height of 0.6 km. The point ource i located at 8 km depth. The receiver are along both the free urface and vertical profile that lice thee wave guide. The ynthetic eimogram are calculated for the frequency range of Hz. We firt compute wave propagation from the ource to produce the 2r

11 P-SV Wave-Field Connection Technique for Regional Wave Propagation Simulation 1385 Figure 10. Application to a large model. (a) The model. (b) Horizontal wave field on C 1. (c) Horizontal wave field on C 2. (d) Horizontal wave field on C 3. (e) Horizontal wave field along the urface. (f) Energy ditribution along the urface. incident wave field on the firt connection boundary C 1 hown in Figure 10b, where the multiply reflected wave, converted wave, head wave, and Rayleigh wave can all be clearly een. Subequently the BE method i ued to calculate wave propagation in the econd ection and obtain the wave field on the econd connection boundary C 2 hown in Figure 10c. Finally, we ue the wave field on C 2 a the incident field and calculate the wave field in the third ection. Figure 10d how the horizontal wave field on C 3, where the cattering effect of the random topography i obviou in that the Rayleigh wave are motly cattered and ome coda wave appear. The wave field received, repectively, along the free urface of the three ection are put together and hown in Figure 10e. Figure 10f i the energy ditribution curve along the entire free urface, where a relatively low energy appear around the hill and the attenuation curve varie dramatically along the random topography. The topography ha a great effect on the energy attenuation in the wave guide.

12 1386 Z. Ge, L.-Y. Fu, and R.-S. Wu Concluion and Dicuion We preent the boundary element method for the 2D P- SV elatic wave problem. To model the effect of irregular topography on regional wave propagation, we further developed a connection technique to imulate long-range wave propagation ection by ection. Numerical comparion with independent method indicated that the preented method including the connection technique i accurate for regional wave imulation. Numerical reult how that irregular topography can attenuate the total wave energy propagating in the crutal waveguide. The boundary element method can be ued in computing the ite effect on ite uch a canyon, mountain, and valley. The connection technique expand thi method to deal with large earth model with irregular topography. To extend the boundary element method to 3D cae, everal change have to be made. Firt, the Green function of diplacement and traction of the 3D cae hould be ued. Second, becaue the ingularity of the 3D Green function i 1/r and 1/r 2 while in 2D cae thoe are 1/r 1/2 and 1/r, the integral hould be treated more carefully. Third, the ize of matrice to olve the problem will be quared to the 2D cae, becaue thi method include the matrix inverion, the requirement of memory and computation time will be enlarged. Extenion to 3D cae will be conducted in the future. Acknowledgment Thi work wa done while we were working in the Intitute of Geophyic and Planetary Phyic, Univerity of California, Santa Cruz. The reearch i upported by the Air Force Reearch Laboratory under Grant no. DTRA01-01-C The facility upport from the W. M. Keck Foundation i alo acknowledged. Reference Alvarez-Rubioa, S., F. J. Sanchez-Sema, J. J. Benitoc, and E. Alarcond (2004). The direct boundary element method: 2D ite effect aement on laterally varying layered media (methodology), Soil Dyn. Earthquake Eng. 24, Aki, K., and P. G. Richard (1980). Quantitative Seimology. Theory and Method. W. H. Freeman, San Francico. Botock, M. G., and B. L. N. Kennett (1990). The effect of three-dimenional tructure on Lg propagation pattern, Geophy. J. Int. 101, Bouchon, M., C. A. Schultz, and M. N. Toköz (1995). A fat implementation of boundary integral equation method to calculate the propagation of eimic wave in laterally varying layered media, Bull. Seim. Soc. Am. 85, Fu, L. Y. (1996). 3-D boundary element eimic modeling in complex geology, in 66th Ann. Internat. Mtg., Soc. Expl. Geophy., Expanded Abtract, Fu, L. Y., and Y. G. Mu (1994). Boundary element method for elatic wave forward modeling, Acta Geophy. Sinica, 37, Fu, L. Y., and R. S. Wu (2000). Infinite boundary element aborbing boundary for wave propagation imulation, Geophyic 65, Fu, L. Y., and R. S. Wu (2001). A hybrid BE-GS method for modeling regional wave propagation, Pure Appl. Geophy. 158, Gibon, R. L., Jr., and M. Campillo (1994). Numerical imulation of highand low-frequency Lg-wave propagation, Geophy. J. Int. 118, Jih, R. S. (1996). Waveguide effect of large-cale tructural variation, anelatic attenuation, and random heterogeneity on SV Lg propagation: a finite-difference modeling tudy, in Proc. 18th Annual Seimic Reearch Smpoium on Monitoring a Comprehenive Tet Ban Treaty, Keer, H., G. Nolet, and F. A. Dahlen (1996). Ray theoretical analyi of Lg, Bull. Seim. Soc. Am. 86, More, P. M., and H. Fehbach (1953). Method of Theoretical Phyic, McGraw-Hill, New York. Sánchez-Sema, F. J., M. A. Bravo, and I. Herrera (1985). Surface motion of topographical irregularitie for incident P, SV, and Rayleigh wave, Bull. Seim. Soc. Am. 75, Sánchez-Sema, F. J., and M. Campillo (1991). Diffraction of P, SV and Rayleigh wave by topographic feature: a boundary integral formulation, Bull. Seim. Soc. Am. 81, Wu, R. S. (1989). Repreentation integral for elatic wave propagation containing either the diplacement term or the tre term alone, Phy. Rev. Lett. 62, Wu, R. S., S. Jin, and X. B. Xie (1996). Synthetic eimogram in heterogeneou crutal waveguide uing creen propagator, in Proc. 18th Annual Seimic Reearch Sympoium on Monitoring a Comprehenive Tet Ban Treaty, Wu, R. S., S. Jin, and X. B. Xie (2000). Seimic wave propagation and cattering in heterogeneou crutal waveguide uing creen propagator. I: SH wave, Bull. Seim. Soc. Am. 90, Wu, X. Y., and R. S. Wu (2001). Lg wave imulation in heterogeneou crut with irregular topography uing halfpace creen propagator, Geophy. J. Int. 146, Intitute of Geophyic and Planetary Phyic Univerity of California 1156 High Street Santa Cruz, California (Z.G., L.-Y.F., R.-S.W.) Laboratory of Computational Geodynamic Department of Geophyic, School of Earth and Space Science Peking Univerity Beijing , People Republic of China zxgai@pku.edu.cn (Z.G.) Manucript received 20 Augut 2004.