Seismic ray path variations in a 3D global velocity model

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1 Physics of the Earth and Planetary Interiors 141 (2004) Seismic ray path variations in a 3D global velocity model Dapeng Zhao, Jianshe Lei Geodynamics Research Center, Ehime University, Matsuyama , Japan Received 5 December 2002; accepted 22 November 2003 Abstract A three-dimensional (3D) ray tracing technique is used to investigate ray path variations of P, PcP, pp and PP phases in a global tomographic model with P wave velocity changing in three dimensions and with lateral depth variations of the Moho, 410 and 660 km discontinuities. The results show that ray paths in the 3D velocity model deviate considerably from those in the average 1D model. For a PcP wave in Western Pacific to East Asia where the high-velocity (1 2%) Pacific slab is subducting beneath the Eurasian continent, the ray path change amounts to 27 km. For a PcP ray in South Pacific where very slow ( 2%) velocity anomalies (the Pacific superplume) exist in the whole mantle, the maximum ray path deviation amounts to 77 km. Ray paths of other phases (P, pp, PP) are also displaced by tens of kilometers. Changes in travel time are as large as 3.9 s. These results suggest that although the maximal velocity anomalies of the global tomographic model are only 1 2%, rays passing through regions with strong lateral heterogeneity (in velocity and/or discontinuity topography) can have significant deviations from those in a 1D model because rays have very long trajectories in the global case. If the blocks or grid nodes adopted for inversion are relatively large (3 5 ) and only a low-resolution 3D model is estimated, 1D ray tracing may be feasible. But if fine blocks or grid nodes are used to determine a high-resolution model, 3D ray tracing becomes necessary and important for the global tomography Elsevier B.V. All rights reserved. Keywords: Seismic tomography; Ray tracing; Seismic discontinuity; Subducting slabs; Mantle plumes 1. Introduction During the last two decades, seismic tomography has become the most powerful tool to explore the heterogeneous structure of the Earth. Many three-dimensional (3D) Earth models have been produced that have greatly advanced our knowledge about the structure and dynamics of the Earth s interior. Continuing efforts, however, are needed to improve the theoretical aspects of seismic tomography as well as its applications. Seismic ray tracing, Corresponding author. Tel.: ; fax: addresses: zhao@sci.ehime-u.ac.jp (D. Zhao), leijs@sci.ehime-u.ac.jp (J. Lei). the computation of ray paths and travel times in a seismic velocity model, is one of the key elements in traveltime tomographic imagings. The accuracy of ray tracing directly affects the final results of a tomographic inversion. Most of local and regional tomographic studies have already used 3D ray tracing schemes to compute travel times and ray paths (see a recent review by Zhao, 2001b). Because usually millions of travel time data are used, most of the global tomographic studies are still using one-dimensional (1D) ray tracing methods, that is, ray paths are only calculated for a 1D velocity model where velocity changes with depth alone, and the ray paths are fixed during the inversion process; ray path variations due to the lateral velocity heterogeneity are not considered /$ see front matter 2003 Elsevier B.V. All rights reserved. doi: /j.pepi

2 154 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) With the remarkable advances in computer technology, 3D ray tracing methods have become feasible in large tomographic computations. Recently, a few researchers have started to apply 3D ray tracing schemes in their global tomographic studies (e.g., Bijwaard and Spakman, 2000; Gorbatov et al., 2001; Zhao, 2001a). Bijwaard and Spakman (1999) examined ray path variations of first P waves in a 3D mantle velocity model. They found that for a direct P wave with an epicentral distance of 87, the ray path is displaced by nearly 100 km and the travel time change is larger than 2 s in their 3D velocity model. The changes in trajectory for other well used later phases, such as pp, PP, and PcP, in a 3D global velocity model are still unclear. Studying this subject is important for not only theoretical aspects of seismology but also practical imagings of the Earth s heterogeneous structure. The Moho depth ranges from about 10 km under oceans to km under the continents (Mooney et al., 1998). The 410 and 660 km discontinuities exhibit depth variations of up to 36 km on a global scale (Flanagan and Shearer, 1998). Such large depth changes of the discontinuities would certainly affect the travel times and ray paths of seismic waves. However, the changes in travel time and ray path due to the discontinuity topography have not been reported in literature, including Bijwaard and Spakman (1999). In this work, we have attempted to use a 3D ray tracing method to investigate ray path variations of P, pp, PP, and PcP phases due to the lateral velocity variations (Zhao, 2001a) as well as the topography of the Moho, 410 and 660 km discontinuities (Mooney et al., 1998; Flanagan and Shearer, 1998). 2. Three-dimensional ray tracing In this work, we have used a modified version (Zhao, 2001a) of the 3D ray tracing technique originally developed by Zhao et al. (1992). Its principle is shown schematically in Fig. 1. This ray tracing scheme can deal with a velocity model that contains several velocity discontinuities of complex geometry and with 3D velocity variations everywhere in the model. Consider a case as shown in Fig. 1a where a complexly shaped seismic velocity discontinuity, MM,exists and velocities on both sides of MM are continuous and change in three dimensions. A and B are two Fig. 1. (a) Schematic illustration of the scheme using Snell s law to find the location of intersection C between a seismic ray (ACB) and a discontinuity MM. (b) Schematic illustration of the 3D ray tracing method used in this study. The thick straight line denotes the initial ray path. The dashed, dotted and thin solid lines denote ray tracing processes using Snell s law, the pseudo-bending scheme, and Snell s law again, respectively. See text for details. points on different sides of MM. Velocities at A and B are V a and V b, respectively. C is the intersection between a seismic ray AB and the discontinuity MM. The velocities at C on the two sides of MM are V c1 and V c2, respectively. We take the arithmetic average V 1 of V a and V c1 and arithmetic average V 2 of V b and V c2 : V 1 = V a + V c1 2 V 2 = V b + V c2 2 (1) (2)

3 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) to represent the mean velocity around AC and BC, respectively. This approximation may be poor when A and B are far away from MM. When A and B are close to MM, however, the approximation will become sufficiently good. In order to find the location of point C, we first estimate the locations of two points, A and B, where AA and BB are perpendicular to MM. The point C should exist within the section of MM between A and B. Then we use the bisection method to gradually narrow the range in which C exists until we find the location of C which satisfies Snell s law: sin θ 1 = sin θ 2 (3) V 1 V 2 with sufficient accuracy. Fig. 1b shows the ray tracing algorithm schematically for the case with three discontinuities. The triangle denotes a station and the star a hypocenter. The straight line A1 A5 connecting the station and hypocenter is assumed to be the initial ray path. The points on the ray are divided into two types: one is the intersections between the ray and the discontinuities (A2, A3, A4), which we call discontinuous points (DPs); the other is the points in the continuous medium bounded by two adjacent discontinuities (B1, B2, B3, B4), which we call continuous points (CPs). The principle of the ray tracing algorithm is that Snell s law is used to perturb the DPs (Fig. 1a), while the pseudo-bending technique (Um and Thurber, 1987) is adopted to perturb the CPs. In Fig. 1b, we first use Snell s law to find new DPs A 2 from A1 and A3, A 3 from A2 and A4, and A 4 from A3 and A5. Then we use the pseudo-bending to find CPs B1 from A1 and A 2, B2 from A 2 and A 3, B3 from A 3 and A 4, and B4 from A 4 and A5. Then again using Snell s law we find A 2 from B1 and B2, A 3 from B2 and B3, and A 4 from B3 and B4, and so on. After a number of iterations the ray path converges to its true location. It is found that this algorithm is very efficient and works very well for 3D seismic ray tracing in local to regional scales (epicentral distance less than about 1000 km) (Zhao et al., 1992, 1994). In Zhao (2001a) and the present study, we found that the original routine of this algorithm developed by Zhao et al. (1992) gives less accurate results for mantle rays in a global scale (P, PP, PcP and pp, etc.); the epicentral distance is as large as 100 (1 = km). After careful examinations we found that the problem was caused by the enhancement factor used in the pseudo-bending technique (Um and Thurber, 1987). The enhancement factor is adopted to quicken the convergence of the ray tracing process. We found that the optimal value of the enhancement factor is dependent on the length of seismic rays. The optimal enhancement factor for the local and regional scale is inadequate for seismic rays in the global scale. After numerous tests we determined the optimal values of the enhancement factor for different lengths of the mantle seismic rays with different focal depths. Our modified version of the 3D ray tracing algorithm gives very accurate results with the absolute computational error in travel time smaller than 0.05 s, which is considered to be accurate enough because the uncertainty of arrival time pickings is generally 0.1 s or greater for the mantle rays in the global scale. Our new 3D ray tracing code can deal with not only the first P and S rays but also the later phases such as pp, PP, PcP and Pdiff, etc. 3. 3D global velocity model We examined seismic rays along many cross sections using the 3D ray tracing technique and the global P wave tomography model of Zhao (2001a). Here we show examples of the 3D ray tracing for three regions: Western Pacific to East Asia, South Pacific, and Tibet (Fig. 2). The subducting Pacific slab is clearly imaged in the upper mantle beneath the Japan Island and the Sea of Japan (Fig. 3a). Intermediate-depth and deep earthquakes form a clear Wadati Benioff zone within the high-velocity (high-v) subducting slab. The slab becomes stagnant in the mantle transition zone under Northeast China. Moderately fast anomalies in the lower mantle represent pieces of the old Pacific slab which are collapsing down to core mantle boundary (CMB) after stagnating in the transition zone for a long time (Maruyama, 1994; Zhao, 2004). Prominent low-velocity (low-v) anomalies are visible under the active Changbai volcano and the Datong Quaternary volcano, which may be caused by the deep dehydration process of the subducting Pacific slab and the convective circulation process in the mantle wedge (Tatsumi et al., 1990; Zhao et al., 1994, 1997; Zhao,

4 156 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) Fig. 2. Map showing locations of three vertical cross sections (AB, CD and EF) in which seismic ray paths are investigated in this work. Stars and open triangles denote hypothesized earthquake hypocenters and seismic stations, respectively. Solid triangles denote the surface hotspot volcanoes except for those in East China (WV, Wudalianchi volcano; DV, Datong volcano; CV, Changbai volcano). TA, Tahiti hotspot. 2004). Hence the active intraplate volcanism in Northeast China is not hotspot like Hawaii but a kind of back-arc volcanism. Slow anomalies are also visible under the Pacific slab in the transition zone and the lower mantle, which represent upwellings either related to a mantle plume or caused by a local mantle convection associated with the subduction of the Pacific slab (Zhao, 2004). Several hotspot volcanoes are located on the South Pacific Superswell (McNutt, 1998) (Fig. 2). A huge slow anomaly with a lateral extent of over 1000 km is visible in the entire mantle from the surface down to the CMB (Fig. 3b), which represents the Pacific superplume as has been detected by many previous studies with various geophysical and geochemical approaches (Su et al., 1994; Maruyama, 1994; McNutt, 1998; Condie, 2001; Zhao, 2001a; Tanaka, 2002). Detailed resolution analyses confirmed that the major structural features in Fig. 3 as mentioned above are reliable (Zhao, 2004). The geodynamic implications of the tomographic results (Fig. 3) are discussed in detail by Zhao (2004). 4. Influence of lateral velocity variations Fig. 3 shows the 1D and 3D ray paths of P, pp, PP and PcP phases along the AB and CD profiles. A focal depth of 300 km is adopted, though any other depth values are feasible. Here 3D ray path means the ray trajectory that is determined by applying the 3D ray tracing method (Fig. 1) to a velocity model with 3D P wave velocity variations determined by Zhao (2001a), while all the velocity discontinuities (the Moho at 35, 410 and 660 km) are spherical interfaces without lateral depth variations. The 1D ray path denotes that determined by using the ray tracing scheme to a 1D velocity model that is the average of the 3D model for each depth. The average 1D velocity model is nearly the same as the iasp91 Earth model (Kennett and Engdahl, 1991), the difference between them is less than 1%. To see the ray path variations more clearly, we plot the differences between the 1D and 3D rays in the horizontal and vertical directions as well as the total displacement in Figs The straight lines show the 1D rays, while the curved lines show the 3D rays. P wave velocity images along the 1D ray paths in the horizontal and vertical directions are also shown (Figs. 4 7). The changes in travel time and ray path for the rays along profile AB are listed in Table 1. The results for those in profile CD are listed in Table 2. In addition to the travel time (T11) spent by a 1D ray path in the 1D velocity model and that (T33) taken by a 3D ray path in the 3D velocity model, we also computed the travel times (T13) of 1D rays with velocities of the 3D

5 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) Fig. 3. 1D and 3D ray paths of P, pp, PP and PcP phases and P wave velocity images (Zhao, 2001a) along the profile AB (a, c) and CD (b) in Fig. 2. Blue and red colors denote fast and slow velocities, respectively. The dotted and solid lines denote 1D and 3D rays, respectively. The two dashed lines show the 410 and 660 km discontinuities. The Moho is taken as a spherical surface at 35 km depth. Solid triangles denote active volcanoes, small white dots in (a) and (c) show earthquakes that occurred within a 150 km width of profile AB. Stars and open triangles denote hypothesized earthquake hypocenters and seismic stations, respectively. JA, Japan arc; JT, Japan trench. See text for details. model and that (T31) of 3D rays with velocities of the 1D model. In most of the previous global tomographic inversions (e.g., Inoue et al., 1990), ray paths are fixed to those in a 1D velocity model, while travel times are updated by using the 3D velocity distribution obtained in each of the iterations during the inversion process. The differences between the four travel times (T11, T33, T13 and T31) for the four types of phases (P, pp, PP, and PcP) are shown in Tables 1 and 2. For the rays along profile AB (Fig. 3a), the travel time differences are relatively small (<0.2 s), and the ray path deviations are generally less than 33 km. Table 1 Travel time and ray path changes for rays along profile AB Phase T11 T13 (s) T11 T31 (s) T13 T33 (s) T11 T33 (s) DMH (km) DMV (km) DMT (km) DS3D-1D (km) P PcP pp PP T ij denotes travel time of an id ray path in a jd velocity model. DMH, DMV and DMT denote the maximum horizontal, vertical and total ray path deviations, respectively. DS3D-1D denotes the difference in ray length between 3D and 1D ray paths. See text for details.

6 158 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) Fig. 4. Difference in ray path (in km) between the 1D and 3D rays vs. epicentral distance (in degree) for direct P phase (a c) and pp phase (d f) along the profile AB as shown in Fig. 3a. The straight and curved lines denote the 1D and 3D rays, respectively. (a) Horizontal, (b) vertical and (c) total ray deviations for direct P wave. (d) Horizontal, (e) vertical and (f) total ray deviations for pp wave. P wave velocity images along the 1D ray paths are shown in gray scale. The velocity perturbation scale is shown at the bottom. Table 2 Travel time and ray path changes for rays along profile CD Phase T11 T13 (s) T11 T31 (s) T13 T33 (s) T11 T33 (s) DMH (km) DMV (km) DMT (km) DS3D-1D (km) P PcP pp PP T ij denotes travel time of an id ray path in a jd velocity model. DMH, DMV and DMT denote the maximum horizontal, vertical and total ray path deviations, respectively. DS3D-1D denotes the difference in ray length between 3D and 1D ray paths. See text for details.

7 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) Fig. 5. The same as Fig. 4 but for PcP (a c) and PP rays (d f). These relatively small differences in both travel time and ray path are related to the alternate low- and high-velocity anomalies along the rays (see Figs. 3a, 4 and 5). For the rays along profile CD, however, the travel time differences become much larger, amounting to 3.1 s, because of the huge slow anomalies associated with the Pacific superplume. The differential times (T13 T33) amount to 0.7 s, exceeding the picking accuracy of arrival time data recorded by short-period instruments ( s), as is the case for the International Seismological Center (ISC) data sets which were used by many global tomographic studies including Zhao (2001a). These results suggest that 1D ray tracing could cause large calculation errors for rays which pass through strong velocity anomalies. We can also see in Tables 1 and 2 that T33 is smaller than T13 for all the cases, indicating that 3D ray tracing is always better than 1D ray tracing. Ray trajectories are sensitive to lateral and vertical velocity gradients, and they are apt to move toward

8 160 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) Fig. 6. The same as Fig. 4 but for direct P (a c) and pp rays (d f) along the profile CD in Fig. 3b. faster areas in a 3D velocity field. This is clearly visible in Fig. 3. Because of the slow anomaly under the Changbai volcano, the PP bounce point at the surface is moved by 16 km toward the west (Fig. 3a). The easternmost segment of PP is elevated by 27 km to escape the slow anomaly under the subducting Pacific slab (Figs. 3a and 5e). Because of the high-v stagnant slab in the mantle transition zone, the western portion of the long segment of pp is elevated by 12 km (Figs. 3a and 4e). The PcP bounce point at the CMB is moved by 18 km westward from a low-v zone to a high-v patch in the lowermost mantle (Fig. 3a). The most significant displacement in ray path occurs for the northern segment of PcP under South Pacific (Figs. 3b and 7c). The horizontal, vertical and total deviations in ray path are 26, 48, and 77 km, respectively (Fig. 7a c). The ray is displaced from the large slow Pacific superplume toward fast areas in the north (Figs. 3b and 7c). The northern segment of pp is moved by 32 km toward a high-v zone in the north

9 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) Fig. 7. The same as Fig. 6 but for PcP (a c) and PP rays (d f). (Figs. 3b and 6e). In addition, the PP bounce point at the surface is displaced by 20 km toward the north to escape the slow anomaly under Tahiti. Most of the PP and pp rays propagate a long distance in and around the mantle transition zone and their paths show complex changes of significant amounts because of the existence of high-v stagnant slabs and low-v mantle plumes in the region. Thus, calculating travel times and ray paths of PP and pp phases precisely using a 3D ray tracing method becomes particularly important and necessary. Fig. 3c shows five 1D and 3D rays for direct P waves passing through the upper mantle and transition zone under the Western Pacific region; their epicentral distances are 14,17,20,25 and 29, respectively. The hypothesized hypocenter is located near the Japan trench with a focal depth of 75 km. The high-v subducting slab and the low-v anomalies in the mantle

10 162 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) Fig. 8. Ray paths of P, pp and PP phases along the profiles AB (a) and CD (b) in Fig. 2. The dotted lines denote rays in the 1D iasp91 model. The solid lines show the rays in a heterogeneous model with lateral depth variations of the Moho discontinuity. The 410 and 660 km discontinuities are assumed to be spherical interfaces without lateral depth variations. The dashed lines denote the 410 and 660 km discontinuities and the global-average Moho discontinuity at 35 km depth. The Moho depth variations (Mooney et al., 1998) along the profiles AB and CD are shown in thin solid lines. The iasp91 Earth model is used for the 1D velocities in each layer. Lateral variations in seismic velocity are not considered. wedge and below the slab cause significant changes in the travel times and ray paths. For the rays P1 to P5 (Fig. 3c), the travel time difference (T11 T33) is 0.64, 0.11, 1.58, 2.24 and 1.11 s, respectively, and the maximum displacement in ray path is 6, 19, 10, 9 and 8 km, respectively. The rays bend toward the high-v slab. 5. Influence of the discontinuity topography The changes in seismic ray paths caused by the depth variations of the Moho, 410 and 660 km discontinuities along the three profiles in Fig. 2 are shown in Figs The models by Mooney et al. (1998) and Flanagan and Shearer (1998) are used for the Fig. 9. The same as Fig. 8 but for three direct P phases along the profile EF in Fig. 2. The focal depth is 80 km.

11 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) Fig. 10. The same as Fig. 8 but for rays (solid lines) in a velocity model with a flat Moho at 35 km depth and with lateral depth variations of the 410 and 660 km discontinuities from the model of Flanagan and Shearer (1998). Fig. 11. The same as Fig. 8 but for rays (solid lines) in a velocity model with lateral depth variations of the Moho, 410 and 660 km discontinuities from the models of Mooney et al. (1998) and Flanagan and Shearer (1998).

12 164 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) topography of the three discontinuities. The iasp91 Earth model (Kennett and Engdahl, 1991) is used for the 1D P wave velocities in the crust and mantle; lateral velocity variations are not considered. In Figs. 8 11, ray path changes in the models with depth variations of the discontinuities are compared with those in the 1D iasp91 Earth model where the Moho (35 km depth), 410 and 660 km discontinuities are taken as spherical interfaces without lateral depth variations. In Figs. 8 and 9, ray path changes caused only by the Moho topography are examined. The Moho depth changes from 12 to 38 km along the profile AB, and from 11 to 14 km along the profile CD. The Moho topography caused the surface bounce point of the PP ray displaced by 34 km in Fig. 8a and 40 km in Fig. 8b. The maximum change in the pp ray path is 17 km along profile AB and 11 km along profile CD. The P rays are little affected by the Moho topography, though their travel times are changed significantly. In Fig. 8a, the travel time changes for P, pp and PP rays are 0.91, 0.91, and 1.10 s, respectively. In Fig. 8b, the corresponding travel time changes are 0.91, 3.02, and 3.87 s, respectively. Along the profile EF passing through the Himalaya mountains (Fig. 2), the Moho depth changes from 39 to 60 km (Fig. 9). For rays P1, P2 and P3 in Fig. 9, the travel time changes are 1.43, 2.13 and 1.16 s, and the maximum deviations in ray paths are 10, 17 and 9 km, respectively. Ray path changes caused by the topography of the 410 and 660 discontinuities are shown in Fig. 10. The Moho is flat with a depth of 35 km. The ray path of direct P wave is only slightly affected by the 410 and 660 km topography; the path deviations are less than 6 km in both AB and CD profiles. The maximum path deviations of pp and PP rays are 9 and 34 km in Fig. 10a, and 13 and 11 km in Fig. 10b, respectively. The travel time changes of P, pp and PP rays are 0.27, 0.42 and 1.00 s in Fig. 10a, and 0.12, 0.10 and 0.44 s in Fig. 10b, respectively. Fig. 12. The same as Fig. 3 but for rays (solid lines) in a velocity model with lateral depth variations of the Moho, 410 and 660 km discontinuities from the models of Mooney et al. (1998) and Flanagan and Shearer (1998).

13 D. Zhao, J. Lei / Physics of the Earth and Planetary Interiors 141 (2004) Fig. 11 shows the ray path changes caused by the lateral depth variations of all the three velocity discontinuities (the Moho, 410 and 660). The maximum path changes of P, pp and PP rays are 4, 19 and 28 km in profile AB, and 6, 6 and 30 km in profile CD, respectively. The corresponding changes in travel times are 0.64, 0.52 and 0.29 s in Fig. 11a, and 0.80, 2.95 and 3.42 s in Fig. 11b, respectively. 6. Composite effect of velocity changes and discontinuity topography Fig. 12 shows P, pp and PP rays in the model of Zhao (2001a) where both P wave velocity changes in three dimensions and the Moho, 410 and 660 km discontinuities have lateral depth variations as revealed by Mooney et al. (1998) and Flanagan and Shearer (1998). These 3D ray paths (solid lines in Fig. 12) are compared with those in the 1D iasp91 model (dotted lines in Fig. 12). The maximum path differences between the 1D and 3D rays for P, pp and PP phases are 12, 16 and 26 km in Fig. 12a, and 46, 38 and 32 km in Fig. 12b, respectively. The corresponding travel time changes are 0.59, 0.73 and 0.14 s in Fig. 12a, and 0.95, 1.62 and 0.45 s in Fig. 12b, respectively. 7. Discussion and conclusions Tracing seismic rays in a heterogeneous Earth model precisely and quickly is essential in tomographic studies because usually a great number of arrival times are used and, for each arrival, ray tracing is employed several times to conduct hypocenter relocation and structure determination. Although several efficient algorithms for 3D ray tracing have been developed (see Zhao, 2001b) and there have been great advances in computer technology, continuing efforts are still needed to search better ray tracing schemes in order to use more data to image the Earth structure in greater details and in higher accuracy. Our present results suggest that although the maximal velocity anomalies of the 3D global model are only 1 2%, rays passing through those very heterogeneous regions (such as subducting slabs and mantle plumes) can have deviations from the 1D rays amounting to 77 km because of the very long ray trajectories in the global case. Since damping and smoothing are usually applied in the tomographic inversions, the amplitude of velocity anomalies in the tomographic models is generally underestimated. Actual amplitudes of mantle heterogeneities would be greater. For example, high-resolution (5 30 km) tomographic imagings of local and regional scales produce velocity anomalies of up to 6% for P wave and 10% for S-wave (Zhao et al., 1992, 1997). Unlike the global case, in local or regional scale studies, arrival time data have high quality and ray path coverage is good, thus heavy damping and smoothing as in the global tomography can be avoided, and so not only the pattern but also the amplitude of velocity heterogeneity can be recovered well. With the improvement of spatial resolution of global tomography, greater amplitudes of velocity anomalies are expected. Thus ray paths in the mantle would have larger deviations. The significant lateral depth variations of the Moho, 410 and 660 km discontinuities (e.g., Mooney et al., 1998; Flanagan and Shearer, 1998) make the changes in ray path and travel time more complex. All these necessitate the use of 3D ray tracing in the future generation of global tomographic inversions. Recently, the computation of frequency-dependent travel times and Frechet kernels in a heterogeneous Earth model has been paid much attention (e.g., Dahlen et al., 2000; Hung et al., 2000; Zhao et al., 2000). Although it will be a challenge to apply the new approach to conduct tomographic inversions with a great number of travel time data, it may enable us to better understand the propagational behaviors of seismic waves in the heterogeneous interior of the Earth. Acknowledgements This work was partially supported by a grant to D. Zhao (Kiban-B No ) and a grant to E. Takahashi (Tokyo Institute of Technology) and D. Zhao (Special grant No ) from Ministry of Education and Science, Japan. K. Creager and two anonymous referees provided thoughtful comments which improved the manuscript. References Bijwaard, H., Spakman, W., Fast kinematic ray tracing of first- and later-arriving global seismic phases. Geophys. J. Int. 139,

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