Pure and Applied Geophysics

, Volume 174, Issue 7, pp 2733–2761 | Cite as

Fast Computation of Global Sensitivity Kernel Database Based on Spectral-Element Simulations

  • Elliott Sales de AndradeEmail author
  • Qinya Liu


Finite-frequency sensitivity kernels, a theoretical improvement from simple infinitely thin ray paths, have been used extensively in recent global and regional tomographic inversions. These sensitivity kernels provide more consistent and accurate interpretation of a growing number of broadband measurements, and are critical in mapping 3D heterogeneous structures of the mantle. Based on Born approximation, the calculation of sensitivity kernels requires the interaction of the forward wavefield and an adjoint wavefield generated by placing adjoint sources at stations. Both fields can be obtained accurately through numerical simulations of seismic wave propagation, particularly important for kernels of phases that cannot be sufficiently described by ray theory (such as core-diffracted waves). However, the total number of forward and adjoint numerical simulations required to build kernels for individual source–receiver pairs and to form the design matrix for classical tomography is computationally unaffordable. In this paper, we take advantage of the symmetry of 1D reference models, perform moment tensor forward and point force adjoint spectral-element simulations, and save six-component strain fields only on the equatorial plane based on the open-source spectral-element simulation package, SPECFEM3D_GLOBE. Sensitivity kernels for seismic phases at any epicentral distance can be efficiently computed by combining forward and adjoint strain wavefields from the saved strain field database, which significantly reduces both the number of simulations and the amount of storage required for global tomographic problems. Based on this technique, we compute traveltime, amplitude and/or boundary kernels of isotropic and radially anisotropic elastic parameters for various (\(P\), \(S\), \(P_{\mathrm{diff}}\), \(S_{\mathrm{diff}}\), depth, surface-reflected, surface wave, S 660 S boundary, etc.) phases for 1D ak135 model, in preparation for future global tomographic inversions.


Seismic wave simulations body waves surface waves and free oscillations seismic tomography computational seismology theoretical seismology 



The authors wish to acknowledge the developers of the SPECFEM3D_GLOBE software and ObsPy for their continuing work. Computations were performed on the Sandybridge supercomputer at the SciNet HPC Consortium. SciNet is funded by the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund—Research Excellence; and the University of Toronto. The ray paths shown in the figures in Sect. 5 are computed based on ObsPy (Krischer et al. 2015; The ObsPy Development Team 2016). The authors also recognize support from the NSERC G8 Research Councils Initiative on Multilateral Research Funding and the Discovery Grant No. 487237.


  1. Aki, K., & Richards, P. G. (2002). Quantitative Seismology (2nd ed.). Sausalito: University Science Books.Google Scholar
  2. Becker, T. W., Kellogg, J. B., Ekström, G., & O’Connell, R. J. (2003). Comparison of azimuthal seismic anisotropy from surface waves and finite strain from global mantle-circulation models. Geophysical Journal International, 155(2), 696–714. doi: 10.1046/j.1365-246X.2003.02085.x.CrossRefGoogle Scholar
  3. Born, M., & Wolf, E. (1970). Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (4th ed.). Oxford: Pergamon Press.Google Scholar
  4. Capdeville, Y., Chaljub, E., & Montagner, J. P. (2003). Coupling the spectral element method with a modal solution for elastic wave propagation in global Earth models. Geophysical Journal International, 152(1), 34–67.CrossRefGoogle Scholar
  5. Carrington, L., Komatitsch, D., Laurenzano, M., Tikir, M.M., Michéa, D., Le Goff, N., Snavely, A., & Tromp, J. (2008). High-frequency simulations of global seismic wave propagation using SPECFEM3D-GLOBE on 62K processors. In: Proceedings of the 2008 ACM/IEEE conference on supercomputing, SC ’08 (pp. 60:1–60:11). Piscataway: IEEE Press. doi: 10.1145/1413370.1413432
  6. Chaljub, E., Komatitsch, D., Vilotte, J. P., Capdeville, Y., Valette, B., & Festa, G. (2007). Spectral-element analysis in seismology. In: Advances in geophysics (Vol. 48, pp. 365–419). Amsterdam: ElsevierGoogle Scholar
  7. Chang, S. J., Ferreira, A. M. G., Ritsema, J., van Heijst, H. J., & Woodhouse, J. H. (2015). Joint inversion for global isotropic and radially anisotropic mantle structure including crustal thickness perturbations. Journal of Geophysical Research: Solid Earth, 120(6), 4278–4300. doi: 10.1002/2014JB011824.Google Scholar
  8. Chen, M., & Tromp, J. (2007). Theoretical and numerical investigations of global and regional seismic wave propagation in weakly anisotropic earth models. Geophysical Journal International, 168(3), 1130–1152. doi: 10.1111/j.1365-246X.2006.03218.x.CrossRefGoogle Scholar
  9. Chen, M., Huang, H., Yao, H., van der Hilst, R., & Niu, F. (2014). Low wave speed zones in the crust beneath SE Tibet revealed by ambient noise adjoint tomography. Geophysical Research Letters, 41(2), 334–340. doi: 10.1002/2013GL058476.CrossRefGoogle Scholar
  10. Chen, M., Niu, F., Liu, Q., Tromp, J., & Zheng, X. (2015). Multiparameter adjoint tomography of the crust and upper mantle beneath East Asia: 1. Model construction and comparisons. Journal of Geophysical Research: Solid Earth, 120(3), 1762–1786. doi: 10.1002/2014JB011638.Google Scholar
  11. Chen, P., Jordan, T. H., & Zhao, L. (2007). Full three-dimensional tomography: A comparison between the scattering-integral and adjoint-wavefield methods. Geophysical Journal International, 170(1), 175–181. doi: 10.1111/j.1365-246X.2007.03429.x.CrossRefGoogle Scholar
  12. Chen, P., Jordan, T. H., & Lee, E. J. (2010). Perturbation kernels for generalized seismological data functionals (GSDF): Perturbation kernels for GSDF. Geophysical Journal International, 183(2), 869–883. doi: 10.1111/j.1365-246X.2010.04758.x.CrossRefGoogle Scholar
  13. Colombi, A., Nissen-Meyer, T., Boschi, L., & Giardini, D. (2012). Seismic waveform sensitivity to global boundary topography. Geophysical Journal International, 191(2), 832–848. doi: 10.1111/j.1365-246X.2012.05660.x.CrossRefGoogle Scholar
  14. Cupillard, P., Delavaud, E., Burgos, G., Festa, G., Vilotte, J. P., Capdeville, Y., et al. (2012). RegSEM: A versatile code based on the spectral element method to compute seismic wave propagation at the regional scale: RegSEM: A regional spectral element code. Geophysical Journal International, 188(3), 1203–1220. doi: 10.1111/j.1365-246X.2011.05311.x.CrossRefGoogle Scholar
  15. Dahlen, F. A. (2005). Finite-frequency sensitivity kernels for boundary topography perturbations. Geophysical Journal International, 162(2), 525–540. doi: 10.1111/j.1365-246X.2005.02682.x.CrossRefGoogle Scholar
  16. Dahlen, F. A., & Tromp, J. (1998). Theoretical Global Seismology. Princeton: Princeton University Press.Google Scholar
  17. Dahlen, F. A., Hung, S. H., & Nolet, G. (2000). Fréchet kernels for finite-frequency traveltimes—I. Theory. Geophysical Journal International, 141(1), 157–174. doi: 10.1046/j.1365-246X.2000.00070.x.CrossRefGoogle Scholar
  18. de Vos, D., Paulssen, H., & Fichtner, A. (2013). Finite-frequency sensitivity kernels for two-station surface wave measurements. Geophysical Journal International, 194(2), 1042–1049. doi: 10.1093/gji/ggt144.CrossRefGoogle Scholar
  19. Dziewoński, A. M., & Anderson, D. L. (1981). Preliminary reference Earth model. Physics of the Earth and Planetary Interiors, 25(4), 297–356. doi: 10.1016/0031-9201(81)90046-7.CrossRefGoogle Scholar
  20. Ekström, G., & Dziewoński, A. M. (1998). The unique anisotropy of the Pacific upper mantle. Nature, 394(6689), 168–172. doi: 10.1038/28148.CrossRefGoogle Scholar
  21. Favier, N., & Chevrot, S. (2003). Sensitivity kernels for shear wave splitting in transverse isotropic media. Geophysical Journal International, 153(1), 213–228.CrossRefGoogle Scholar
  22. Fichtner, A. (2009). Full seismic waveform inversion for structural and source parameters. PhD thesis, Ludwig-Maximilians-Universität München, Munich, GermanyGoogle Scholar
  23. Fichtner, A., & van Leeuwen, T. (2015). Resolution analysis by random probing. Journal of Geophysical Research: Solid Earth, 120(8), 5549–5573. doi: 10.1002/2015JB012106.Google Scholar
  24. Fichtner, A., & Villaseñor, A. (2015). Crust and upper mantle of the western Mediterranean—Constraints from full-waveform inversion. Earth and Planetary Science Letters, 428, 52–62. doi: 10.1016/j.epsl.2015.07.038.CrossRefGoogle Scholar
  25. Fichtner, A., Kennett, B. L. N., Igel, H., & Bunge, H. P. (2008). Theoretical background for continental- and global-scale full-waveform inversion in the time–frequency domain. Geophysical Journal International, 175(2), 665–685. doi: 10.1111/j.1365-246X.2008.03923.x.CrossRefGoogle Scholar
  26. Fichtner, A., Igel, H., Bunge, H. P., & Kennett, B. L. (2009). Simulation and inversion of seismic wave propagation on continental scales based on a spectral-element method. Journal of Numerical Analysis, Industrial and Applied Mathematics, 4(1–2), 11–22.Google Scholar
  27. Fuji, N., Chevrot, S., Zhao, L., Geller, R. J., & Kawai, K. (2012). Finite-frequency structural sensitivities of short-period compressional body waves: 3-D Fréchet kernels for high frequencies. Geophysical Journal International, 190(1), 522–540. doi: 10.1111/j.1365-246X.2012.05495.x.CrossRefGoogle Scholar
  28. Gokhberg, A., & Fichtner, A. (2016). Full-waveform inversion on heterogeneous HPC systems. Computers & Geosciences, 89, 260–268. doi: 10.1016/j.cageo.2015.12.013.CrossRefGoogle Scholar
  29. Grand, S. P. (1994). Mantle shear structure beneath the Americas and surrounding oceans. Journal of Geophysical Research: Solid Earth, 99(B6), 11591–11621. doi: 10.1029/94JB00042.CrossRefGoogle Scholar
  30. Grand, S. P. (2002). Mantle shear-wave tomography and the fate of subducted slabs. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 360(1800), 2475–2491. doi: 10.1098/rsta.2002.1077.CrossRefGoogle Scholar
  31. Gu, Y. J., Dziewoński, A. M., Su, W., & Ekström, G. (2001). Models of the mantle shear velocity and discontinuities in the pattern of lateral heterogeneities. Journal of Geophysical Research, 106(B6):11169–11199. doi: 10.1029/2001JB000340
  32. Gu, Y. J., Dziewoński, A. M., & Ekström, G. (2003). Simultaneous inversion for mantle shear velocity and topography of transition zone discontinuities. Geophysical Journal International, 154(2), 559–583. doi: 10.1046/j.1365-246X.2003.01967.x.CrossRefGoogle Scholar
  33. Guttenberg, B. (1960). The shadow of the Earth’s core. Journal of Geophysical Research, 65(3), 1013–1020. doi: 10.1029/JZ065i003p01013.CrossRefGoogle Scholar
  34. Herraiz, M., & Espinosa, A. F. (1987). Coda waves: A review. Pure and Applied Geophysics, 125(4), 499–577. doi: 10.1007/BF00879572.CrossRefGoogle Scholar
  35. Hosseini, K., & Sigloch, K. (2015). Multifrequency measurements of core-diffracted P waves (P\(_{{\rm diff}}\)) for global waveform tomography. Geophysical Journal International, 203(1), 506–521. doi: 10.1093/gji/ggv298.CrossRefGoogle Scholar
  36. Houser, C., Masters, T. G., Shearer, P. M., & Laske, G. (2008). Shear and compressional velocity models of the mantle from cluster analysis of long-period waveforms. Geophysical Journal International, 174(1), 195–212. doi: 10.1111/j.1365-246X.2008.03763.x.CrossRefGoogle Scholar
  37. Hudson, J. A., & Heritage, J. R. (1981). The use of the Born approximation in seismic scattering problems. Geophysical Journal of the Royal Astronomical Society, 66(1), 221–240. doi: 10.1111/j.1365-246X.1981.tb05954.x.CrossRefGoogle Scholar
  38. Hung, S. H., Dahlen, F. A., & Nolet, G. (2000). Fréchet kernels for finite-frequency traveltimes—II. Examples. Geophysical Journal International, 141(1), 175–203. doi: 10.1046/j.1365-246X.2000.00072.x.CrossRefGoogle Scholar
  39. Kárason, H., & van der Hilst, R. D. (2000). Constraints on mantle convection from seismic tomography. In M. A. Richards, R. G. Gordon, & R. D. van der Hilst (Eds.), The History and Dynamics of Global Plate Motion, Geophysical Monograph Series (Vol. 121, pp. 277–288). Washington, D. C.: American Geophysical Union.CrossRefGoogle Scholar
  40. Kennett, B. L. N., Engdahl, E. R., & Buland, R. P. (1995). Constraints on seismic velocities in the Earth from traveltimes. Geophysical Journal International, 122(1), 108–124. doi: 10.1111/j.1365-246X.1995.tb03540.x.CrossRefGoogle Scholar
  41. Komatitsch, D., & Tromp, J. (1999). Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophysical Journal International, 139, 806–822. doi: 10.1046/j.1365-246x.1999.00967.x.CrossRefGoogle Scholar
  42. Komatitsch, D., & Tromp, J. (2002a). Spectral-element simulations of global seismic wave propagation—I. Validation. Geophysical Journal International, 149, 390–412. doi: 10.1046/j.1365-246X.2002.01653.x.CrossRefGoogle Scholar
  43. Komatitsch, D., & Tromp, J. (2002b). Spectral-element simulations of global seismic wave propagation—II. Three-dimensional models, oceans, rotation and self-gravitation. Geophysical Journal International, 150, 303–318. doi: 10.1046/j.1365-246X.2002.01716.x.CrossRefGoogle Scholar
  44. Komatitsch, D., Tsuboi, S., Ji, C., & Tromp, J. (2003). A 14.6 billion degrees of freedom, 5 teraflops, 2.5 terabyte earthquake simulation on the Earth simulator. In: SC ’03 Proceedings of the 2003 ACM/IEEE conference on supercomputing (p. 4). New York: ACM Press. doi: 10.1145/1048935.1050155
  45. Komatitsch, D., Liu, Q., Tromp, J., Süss, P., Stidham, C., & Shaw, J. H. (2004). Simulations of ground motion in the Los Angeles basin based upon the spectral-element method. Bulletin of the Seismological Society of America, 94(1), 187–206. doi: 10.1785/0120030077.CrossRefGoogle Scholar
  46. Komatitsch, D., Michéa, D., & Erlebacher, G. (2009). Porting a high-order finite-element earthquake modeling application to NVIDIA graphics cards using CUDA. Journal of Parallel and Distributed Computing, 69(5), 451–460. doi: 10.1016/j.jpdc.2009.01.006.CrossRefGoogle Scholar
  47. Komatitsch, D., Erlebacher, G., Göddeke, D., & Michéa, D. (2010). High-order finite-element seismic wave propagation modeling with MPI on a large GPU cluster. Journal of Computational Physics, 229(20), 7692–7714. doi: 10.1016/ Scholar
  48. Komatitsch, D., Vinnik, L. P., & Chevrot, S. (2010). \(SH_{{\rm diff}} - SV_{{\rm diff}}\) splitting in an isotropic Earth. Journal of Geophysical Research, 115(B7). doi: 10.1029/2009JB006795.
  49. Komatitsch, D., Xie, Z., Bozdağ, E., Sales de Andrade, E., Peter, D., Liu, Q., et al. (2016). Anelastic sensitivity kernels with parsimonious storage for adjoint tomography and full waveform inversion. Geophysical Journal International, 206(3), 1467–1478. doi: 10.1093/gji/ggw224.CrossRefGoogle Scholar
  50. Krischer, L., Megies, T., Barsch, R., Beyreuther, M., Lecocq, T., Caudron, C., & Wassermann, J. (2015). ObsPy: A bridge for seismology into the scientific Python ecosystem. Computational Science & Discovery, 8(1):014003. doi: 10.1088/1749-4699/8/1/014003
  51. Kustowski, B., Ekström, & G., Dziewoński, A. M. (2008a). Anisotropic shear-wave velocity structure of the Earth’s mantle: A global model. Journal of Geophysical Research, 113(B6):B06306. doi: 10.1029/2007JB005169
  52. Kustowski, B., Ekström, G., & Dziewoński, A. M. (2008b). The shear-wave velocity structure in the upper mantle beneath Eurasia. Geophysical Journal International, 174(3), 978–992. doi: 10.1111/j.1365-246X.2008.03865.x.CrossRefGoogle Scholar
  53. Lawrence, J. F., & Shearer, P. M. (2006). A global study of transition zone thickness using receiver functions. Journal of Geophysical Research: Solid Earth, 111(B6):B06307. doi: 10.1029/2005JB003973
  54. Lee, E. J., Chen, P., Jordan, T. H., Maechling, P. B., Denolle, M. A. M., & Beroza, G. C. (2014). Full-3-D tomography for crustal structure in Southern California based on the scattering-integral and the adjoint-wavefield methods. Journal of Geophysical Research: Solid Earth, 119(8), 6421–6451. doi: 10.1002/2014JB011346.Google Scholar
  55. Lekić, V., Cottaar, S., Dziewoński, A. M., & Romanowicz, B. (2012). Cluster analysis of global lower mantle tomography: A new class of structure and implications for chemical heterogeneity. Earth and Planetary Science Letters, 357–358, 68–77. doi: 10.1016/j.epsl.2012.09.014.Google Scholar
  56. Li, X. D., & Romanowicz, B. (1996). Global mantle shear velocity model developed using nonlinear asymptotic coupling theory. Journal of Geophysical Research: Solid Earth, 101(B10):22245–22272. doi: 10.1029/96JB01306
  57. Liu, Q., & Gu, Y. J. (2012). Seismic imaging: From classical to adjoint tomography. Tectonophysics, 566–567, 31–66. doi: 10.1016/j.tecto.2012.07.006.CrossRefGoogle Scholar
  58. Liu, Q., & Tromp, J. (2006). Finite-frequency kernels based on adjoint methods. Bulletin of the Seismological Society of America, 96, 2383–2397. doi: 10.1785/0120060041.CrossRefGoogle Scholar
  59. Liu, Q., & Tromp, J. (2008). Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods. Geophysical Journal International, 174, 265–286. doi: 10.1111/j.1365-246X.2008.03798.x.CrossRefGoogle Scholar
  60. Loken, C., Gruner, D., Groer, L., Peltier, R., Bunn, N., Craig, M., et al. (2010). SciNet: Lessons learned from building a power-efficient top-20 system and data centre. Journal of Physics: Conference Series, 256(012), 026. doi: 10.1088/1742-6596/256/1/012026.Google Scholar
  61. Love, A. E. H. (1927). A Treatise on the Mathematical Theory of Elasticity (4th ed.). Cambridge: Cambridge University Press.Google Scholar
  62. Luo, Y., & Schuster, G. T. (1991). Wave-equation traveltime inversion. Geophysics, 56(5), 645–653. doi: 10.1190/1.1443081.CrossRefGoogle Scholar
  63. Manners, U. J. (2008). Investigating the structure of the core-mantle boundary region using S and P diffracted waves. Ph.D. dissertation, University of California, San DiegoGoogle Scholar
  64. Marquering, H., Dahlen, F. A., & Nolet, G. (1999). Three-dimensional sensitivity kernels for finite-frequency traveltimes: The banana-doughnut paradox. Geophysical Journal International, 137(3), 805–815. doi: 10.1046/j.1365-246x.1999.00837.x.CrossRefGoogle Scholar
  65. Masters, T. G., Laske, G., Bolton, H., & Dziewoński, A. M. (2000). The relative behavior of shear velocity, bulk sound speed, and compressional velocity in the mantle: Implications for chemical and thermal structure. In: Karato, S., Forte, A., Liebermann, R., Masters, T. G., Stixrude, L. (eds.) Geophysical Monograph Series (Vol. 117, pp 63–87). Washington, D. C.: American Geophysical UnionGoogle Scholar
  66. Mercerat, E. D., & Nolet, G. (2012). Comparison of ray- and adjoint-based sensitivity kernels for body-wave seismic tomography. Geophysical Research Letters, 39(12). doi: 10.1029/2012GL052002
  67. Meschede, M., & Romanowicz, B. (2015). Lateral heterogeneity scales in regional and global upper mantle shear velocity models. Geophysical Journal International, 200(2), 1076–1093. doi: 10.1093/gji/ggu424.CrossRefGoogle Scholar
  68. Mégnin, C., & Romanowicz, B. (2000). The three-dimensional shear velocity structure of the mantle from the inversion of body, surface and higher-mode waveforms. Geophysical Journal International, 143(3), 709–728. doi: 10.1046/j.1365-246X.2000.00298.x.CrossRefGoogle Scholar
  69. Montelli, R., Nolet, G., Dahlen, F. A., Masters, T. G., Engdahl, E. R., & Hung, S. H. (2004). Finite-frequency tomography reveals a variety of plumes in the mantle. Science, 303(5656), 338–343. doi: 10.1126/science.1092485.CrossRefGoogle Scholar
  70. Moulik, P., & Ekström, G. (2014). An anisotropic shear velocity model of the Earth’s mantle using normal modes, body waves, surface waves and long-period waveforms. Geophysical Journal International, 199(3), 1713–1738. doi: 10.1093/gji/ggu356.CrossRefGoogle Scholar
  71. Nissen-Meyer, T., Dahlen, F. A., & Fournier, A. (2007a). Spherical-earth Fréchet sensitivity kernels. Geophysical Journal International, 168(3), 1051–1066. doi: 10.1111/j.1365-246X.2006.03123.x.CrossRefGoogle Scholar
  72. Nissen-Meyer, T., Fournier, A., & Dahlen, F. A. (2007b). A two-dimensional spectral-element method for computing spherical-earth seismograms— I. Moment-tensor source. Geophysical Journal International, 168(3), 1067–1092. doi: 10.1111/j.1365-246X.2006.03121.x.CrossRefGoogle Scholar
  73. Nissen-Meyer, T., Fournier, A., & Dahlen, F. A. (2008). A 2-D spectral-element method for computing spherical-earth seismograms—II. Waves in solid–fluid media. Geophysical Journal International, 174(3), 873–888. doi: 10.1111/j.1365-246X.2008.03813.x.CrossRefGoogle Scholar
  74. Nissen-Meyer, T., van Driel, M., Stãhler, S. C., Hosseini, K., Hempel, S., Auer, L., et al. (2014). AxiSEM: Broadband 3-D seismic wavefields in axisymmetric media. Solid Earth, 5(1), 425–445. doi: 10.5194/se-5-425-2014.CrossRefGoogle Scholar
  75. Nolet, G. (2008). A breviary of seismic tomography: Imaging the interior of the earth and sun. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  76. Panning, M. P., & Nolet, G. (2008). Surface wave tomography for azimuthal anisotropy in a strongly reduced parameter space. Geophysical Journal International, 174(2), 629–648. doi: 10.1111/j.1365-246X.2008.03833.x.CrossRefGoogle Scholar
  77. Peter, D., Komatitsch, D., Luo, Y., Martin, R., Le, Goff N., Casarotti, E., et al. (2011). Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes: SPECFEM3D Version 2.0 ‘Sesame’. Geophysical Journal International, 186(2), 721–739. doi: 10.1111/j.1365-246X.2011.05044.x.CrossRefGoogle Scholar
  78. Ritsema, J., Deuss, A., van Heijst, H. J., & Woodhouse, J. H. (2011). S40RTS: A degree-40 shear-velocity model for the mantle from new Rayleigh wave dispersion, teleseismic traveltime and normal-mode splitting function measurements. Geophysical Journal International, 184(3), 1223–1236. doi: 10.1111/j.1365-246X.2010.04884.x.CrossRefGoogle Scholar
  79. Sales de Andrade, E., Liu, Q., Ma, Z., Manners, U. J., Lee-Varisco, E., & Masters, G. (2014). An updated one-degree seismic tomographic model based on a sensitivity kernel database. AGU Fall Meeting abstracts-s 33.Google Scholar
  80. Schumacher, F., & Friederich, W. (2015). The modularized software package ASKI—Full waveform inversion based on waveform sensitivity kernels utilizing external seismic wave propagation codes, AGU Fall Meeting Abstracts.Google Scholar
  81. Shen, Y., Zhang, Z., & Zhao, L. (2008). Component-dependent Fréchet sensitivity kernels and utility of three-component seismic records. Bulletin of the Seismological Society of America, 98(5), 2517–2525. doi: 10.1785/0120070283.CrossRefGoogle Scholar
  82. Sieminski, A., Liu, Q., Trampert, J., & Tromp, J. (2007a). Finite-frequency sensitivity of body waves to anisotropy based upon adjoint methods. Geophysical Journal International, 171(1), 368–389. doi: 10.1111/j.1365-246X.2007.03528.x.CrossRefGoogle Scholar
  83. Sieminski, A., Liu, Q., Trampert, J., & Tromp, J. (2007b). Finite-frequency sensitivity of surface waves to anisotropy based upon adjoint methods. Geophysical Journal International, 168(3), 1153–1174. doi: 10.1111/j.1365-246X.2006.03261.x.CrossRefGoogle Scholar
  84. Sigloch, K., McQuarrie, N., & Nolet, G. (2008). Two-stage subduction history under North America inferred from multiple-frequency tomography. Nature Geoscience, 1(7), 458–462. doi: 10.1038/ngeo231.CrossRefGoogle Scholar
  85. Simmons, N. A., Forte, A. M., Boschi, L., & Grand, S. P. (2010). GyPSuM: A joint tomographic model of mantle density and seismic wave speeds. Journal of Geophysical Research, 115(B12). doi: 10.1029/2010JB007631.
  86. Tape, C., Liu, Q., Maggi, A., & Tromp, J. (2009). Adjoint tomography of the Southern California crust. Science, 325(5943), 988–992. doi: 10.1126/science.1175298.CrossRefGoogle Scholar
  87. The ObsPy Development Team (2016) ObsPy 1.0.1. doi: 10.5281/zenodo.48254
  88. Tian, Y., Sigloch, K., & Nolet, G. (2009). Multiple-frequency SH-wave tomography of the western US upper mantle. Geophysical Journal International, 178(3), 1384–1402. doi: 10.1111/j.1365-246X.2009.04225.x.CrossRefGoogle Scholar
  89. To, A., Fukao, Y., & Tsuboi, S. (2011). Evidence for a thick and localized ultra low shear velocity zone at the base of the mantle beneath the central Pacific. Physics of the Earth and Planetary Interiors, 184(3–4), 119–133. doi: 10.1016/j.pepi.2010.10.015.CrossRefGoogle Scholar
  90. Tong, P., Yang, D., & Hua, B. (2011). High accuracy wave simulation—Revised derivation, numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation. International Journal of Solids and Structures, 48(1), 56–70. doi: 10.1016/j.ijsolstr.2010.09.003.CrossRefGoogle Scholar
  91. Tromp, J., Tape, C., & Liu, Q. (2005). Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophysical Journal International, 160(1), 195–216. doi: 10.1111/j.1365-246X.2004.02453.x.CrossRefGoogle Scholar
  92. Tromp, J., Komatitsch, D., & Liu, Q. (2008). Spectral-element and adjoint methods in seismology. Communications in Computational Physics, 3(1), 1–32.Google Scholar
  93. Valenzuela, R. W., & Wysession, M. E. (1998). Illuminating the base of the mantle with diffracted waves. In M. Gurnis, M. E. Wysession, E. Knittle, & B. A. Buffett (Eds.), Geodynamics series (Vol. 28, pp. 57–71). Washington, D. C.: American Geophysical Union.Google Scholar
  94. Wen, L., Silver, P., James, D., & Kuehnel, R. (2001). Seismic evidence for a thermo-chemical boundary at the base of the Earth’s mantle. Earth and Planetary Science Letters, 189(3–4), 141–153. doi: 10.1016/S0012-821X(01)00365-X.CrossRefGoogle Scholar
  95. Yang, H. Y., Zhao, L., & Hung, S. H. (2010). Synthetic seismograms by normal-mode summation: A new derivation and numerical examples: Normal-mode synthetic seismograms. Geophysical Journal International, 183(3), 1613–1632. doi: 10.1111/j.1365-246X.2010.04820.x.CrossRefGoogle Scholar
  96. Yuan, H., & Romanowicz, B. (2010). Lithospheric layering in the North American craton. Nature, 466(7310), 1063–1068. doi: 10.1038/nature09332.CrossRefGoogle Scholar
  97. Zaroli, C., Debayle, E., & Sambridge, M. (2010). Frequency-dependent effects on global S-wave traveltimes: Wavefront-healing, scattering and attenuation: Global multiple-frequency S-wave traveltimes. Geophysical Journal International, 182(2), 1025–1042. doi: 10.1111/j.1365-246X.2010.04667.x.CrossRefGoogle Scholar
  98. Zhao, D. (2001). New advances of seismic tomography and its applications to subduction zones and earthquake fault zones: A review. The Island Arc, 10(1), 68–84. doi: 10.1046/j.1440-1738.2001.00291.x.CrossRefGoogle Scholar
  99. Zhao, L., & Chevrot, S. (2011a). An efficient and flexible approach to the calculation of three-dimensional full-wave Fréchet kernels for seismic tomography—I. Theory. Geophysical Journal International, 185, 922–938. doi: 10.1111/j.1365-246X.2011.04983.x.CrossRefGoogle Scholar
  100. Zhao, L., & Chevrot, S. (2011b). An efficient and flexible approach to the calculation of three-dimensional full-wave Fréchet kernels for seismic tomography—II. Numerical results. Geophysical Journal International, 185(2), 939–954. doi: 10.1111/j.1365-246X.2011.04984.x.CrossRefGoogle Scholar
  101. Zhao, L., Jordan, T. H., & Chapman, C. H. (2000). Three-dimensional Fréchet differential kernels for seismic delay times. Geophysical Journal International, 141(3), 558–576. doi: 10.1046/j.1365-246x.2000.00085.x.CrossRefGoogle Scholar
  102. Zhao, L., Chen, P., & Jordan, T. H. (2006). Strain Green’s tensors, reciprocity, and their applications to seismic source and structure studies. Bulletin of the Seismological Society of America, 96(5), 1753–1763. doi: 10.1785/0120050253.CrossRefGoogle Scholar
  103. Zhou, Y., Nolet, G., Dahlen, F.A., & Laske, G. (2006). Global upper-mantle structure from finite-frequency surface-wave tomography. Journal of Geophysical Research, 111(B4). doi: 10.1029/2005JB003677.
  104. Zhou, Y., Liu, Q., & Tromp, J. (2011). Surface wave sensitivity: Mode summation versus adjoint SEM. Geophysical Journal International, 187(3), 1560–1576. doi: 10.1111/j.1365-246X.2011.05212.x.CrossRefGoogle Scholar
  105. Zhu, H., Bozdağ, E., Peter, D., & Tromp, J. (2012). Structure of the European upper mantle revealed by adjoint tomography. Nature Geoscience, 5(7), 493–498. doi: 10.1038/ngeo1501.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of TorontoTorontoCanada
  2. 2.Department of Earth SciencesUniversity of TorontoONCanada

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