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Recursive travel-time inversion: A tool for real-time seismic tomography

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A new recursive inverse scheme is applied to a currently popular problem named seismic travel-time tomography, in order to enhance the efficiency and reliability in obtaining a new velocity model if a small number of new data are added to a large data set in the past. In comparison with conventional inverse schemes in seismic tomography, either least-squares or iterative types, this scheme does not require large amounts of matrix-type computations but utilizes the amount of modification in model parameters responsible for each new data set. We also introduce the computation of a collocation travel time (i.e., from a given station to every grid point) for the reference velocity model inverted by the data for all the past events, using a ray tracing scheme called the Huygens’ method (Saito, 2001), suitable to computations prior to a new event. Combining the above information already stored with the recursive inverse scheme, we can obtain a new or updated velocity model immediately after a new event takes place, because a temporal interval between two events is usually very long in a given local area. Since the model is revised at each recursive step, we perform ray tracings with the updated reference model to get more accurate ray paths and travel times than the conventional inversion schemes that use all the ray tracings for the same reference model. We first showed the validity and stability of the proposed method with synthetic data. We then applied the new approach to the P-wave travel-time data recorded in the Hidaka, south-central Hokkaido, Japan, region, and compared our result with other previous results. Our result shares the overall feature with the previous ones. In addition, a new low-velocity zone is detected in the east of the Hidaka mountains at the depth of 10 km, corresponding to the collision zone of two arcs, due to the use of the updated reference velocity model at each recursive step. We also confirmed that the order of data does not affect the final result, so that the present approach is shown as an appropriate tool for so-called real-time seismic tomography: a updated velocity model is immediately obtained at each time that a new event takes place, in order to monitor temporal variations of model parameters such as velocity structure on the real-time basis.


  1. Aki, K., Overview, in Seismic Tomography: Theory and Practice, edited by H. M. Iyer and K. Hirahara, pp. 1–8, Chapman & Hall, London, 1993.

  2. Aki, K. and W. H. K. Lee, Determination of three-dimensional velocity anomalies under a seismic array using first P arrival times from local earthquakes: 1. A homogeneous initial model, J. Geophys. Res., 81, 4381–4399, 1976.

  3. Aki, K. and P. G. Richards, Quantitative Seismology, vol. 2, pp. 641–719, W. H. Freeman and Comp., San Francisco, 1980.

  4. Aki, K., A. Christofferson, and E. S. Husebye, Determination of the three-dimensional seismic structure of the lithosphere, J. Geophys. Res., 82, 277–296, 1977.

  5. Ding, Z., Y. Zhao, H. Ren, J. S. Nelson and Z. Chen, Real-time phase-resolved optical coherence tomography and optical Doppler tomography, Opt. Exp., 10, 236–245, 2002.

  6. Foulger, G. R., B. R. Julian, A. M. Pitt, D. P. Hill, P. E. Malin, and E. Shalev, Three-dimensional crustal structure of Long Valley caldera, California, and evidence for the migration of CO2 under Mammoth Mountain, J. Geophys. Res., 108(B3), 2147, doi:10.1029/2000JB000041, 2003.

  7. GPS Research Group, in The National Center for Atmospheric Research & the UCAR Office of Programs,, 2004.

  8. Hofmann-Wellenhof, B., H. Lichtenegger, and J. Collins, Global Positioning System: Theory and Practice, 5th Rev., 383 pp., Springer Verlag, Wien, 2001.

  9. Iwasaki, T., O. Ozel, T. Moriya, S. Sakai, S. Suzuki, G. Aoki, T. Maeda, and T. Iidaka, Lateral structural variations across a collision zone in central Hokkaido, Japan as revealed from seismic refraction profilings, Geophy. J. Int., 132, 435–457, 1998.

  10. Iwasaki, T. and Research Group of HOKKAIDO TRANSECT, Crustal section across the Hidaka collision zone, Hokkaido, Japan as inferred from seismic refraction/reflection profiling, in Abst. XXIII Gen. Assem. Int. Union Geod. Geophys., SS03/07A/A02-007, 2003.

  11. Iyer, H. M. and K. Hirahara (eds), Seismic Tomography: Theory and Practice, 842 pp., Chapman & Hall, London, 1993.

  12. Katsumata, K. et al. (64 authors), Distribution of Hypocenters and Focal Mechanisms in and around the Hidaka Arc-Arc Collision Zone Revealed by a Dense Temporary Seismic Network, Bull. Earthq. Res. Inst., 77, 199–223, 2002 (In Japanese with English abstract).

  13. Katsumata, K., N. Wada, and M. Kasahara, Newly imaged shape of the deep seismic zone within the subducting Pacific plate beneath the Hokkaido corner, Japan-Kurile arc-arc junction, J. Geophys. Res., 108(B12), 2565, doi:10.1029/2002JB002175, 2003.

  14. Miyamachi, H. and T. Moriya, Velocity structure beneath the Hidaka Mountains in Hokkaido, Japan, J. Phys. Earth, 32, 13–42, 1984.

  15. Miyamachi, H. and T. Moriya, Velocity structure and aftershock distribution of the 1982 Urakawaoki earthquake, J. Phys. Earth, 35, 309–326, 1987.

  16. Moriya, T., H. Miyamachi, O. Ozel, N. Ozel, T. Iwasaki, and M. Kasahara, Collision structure and seismotectonics of the Hidaka Mountains, Hokkaido, Japan, Struc. Geol., 42, 15–30, 1997 (in Japanese with English abstract).

  17. Moriya, T., H. Okada, T. Matsushima, S. Asano, T. Yoshi, and A. Ikami, Collision structure in the upper crust beneath the southwestern foot of the Hidaka Mountains, Hokkaido, Japan, as derived from explosion seismic observations, Tectonophys., 290, 181–196, 1998.

  18. Murai, Y., S. Akiyama, K. Katsumata, T. Takanami, T. Yamashina, T. Watanabe, I. Cho, M. Tanaka, A. Kuwano, N. Wada, H. Shimamura, I. Furuya, D. Zhao, and R. Sanda, Delamination structure imaged in the source area of the 1982 Urakawa-oki earthquake, Geophys. Res. Lett., 30, 1490, doi:10.1029/2002GL016459, 2003.

  19. Nishi, K., A three-dimensional robust seismic ray tracer for volcanic regions, Earth Planets Space, 53, 101–109, 2001.

  20. Nishigami, K., Deep crustal heterogeneity along and around the San Andreas fault system in central California and its relation to the segmentation, J. Geophys. Res., 105, 7983–7998, 2000.

  21. Niu, F., P. G. Silver, R. Nadaeu, and T. McEvilly, Stress-induced migration of seismic scatters associated with the 1993 Parkfield aseismic transient, Nature, 426, 544–548, 2003.

  22. Nolet, G. (ed.), Seismic Tomography: With Applications in Global Seismology and Exploration Geophysics, Kluwer Academic Pub., 386 pp., 1987.

  23. Paige, C. and M. A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares, Assoc. Comput. Mach. Trans. Math. Software, 8, 43–71, 1982.

  24. Park, B. H., M. D. Pierce, B. Cense, and J. F. de Boer, Real-time multifunctional optical coherence tomography, Opt. Exp., 11, 782–793, 2003.

  25. Qin, F., Y. Luo, K. Olsen, W. Cai, and G. T. Schuster, Finite-difference solution of the eikonal equation along expanding wavefronts, Geophysics, 57, 478–487, 1992.

  26. Rawlinson, N. and M. Sambridge, Wavefront evolution in strongly heterogeneous layered media using the fast marching method, Geophys. J. Int., 156, 631–647, 2004.

  27. Rodgers, C. D., Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation, Rev. Geophys. Space Phys., 14, 609–624, 1976.

  28. Saito, H., Seismic Traveltime Tomography for Shallow Subsurface Explorations, Ph.D. thesis, 126 pp., Hokkaido University, Sapporo, 2001.

  29. Sen, M. and P. L. Stoffa, Global Optimization Methods in Geophysical Inversion, Elsevier, 281 pp., 1995.

  30. Shearer, P. M., Introduction to Seismology, Cambridge Univ. Press, 260 pp., 1999.

  31. Taira, T. and K. Yomogida, Imaging of three-dimensional small-scale heterogeneities in the Hidaka, Japan region: Coda spectral analysis, Geo-phys. J. Int., 158, 998–1008, 2004.

  32. Takanami, T., Three-dimensional seismic structure of the crust and upper mantle beneath the orogenic belts in southern Hokkaido, Japan, J. Phys. Earth, 30, 87–104, 1982.

  33. Tsumura, N., H. Ikawa, T. Ikawa, M. Shinohara, T. Ito, K. Arita, T. Moriya, G. Kimura, and T. Ikawa, Delamination-wedge structure beneath Hi-daka collision zone, central Hokkaido, Japan inferred from seismic reflection profiling, Geophys. Res. Lett., 26, 1057–1060, 1999.

  34. Vidale, J. E., Finite-difference calculation of traveltimes in three dimensions, Geophysics, 55, 521–526, 1990.

  35. Wunsch, C., Using data with models: Ill-posed problems, in Oceano-graphic and Geophysical Tomography, edited by Y. Desaubies, A. Tarantola, and J. Zinn-Justin, pp. 203–248, North-Holland, Amsterdam, 1990.

  36. Zeng, Y., Deterministic and stochastic modeling of the high-frequency Seismic wave generation and propagation in the lithosphere, Ph.D. thesis, Univ. of South. Calif., Los Angeles, 1991.

  37. Zhao, D., A. Hasegawa, and S. Horiuchi, Tomographic imaging of P and S wave velocity structure beneath northeastern Japan, J. Geophys. Res., 97, 19909–19928, 1992.

  38. Zvyagin, A. V., J. B. FitzGerald, K. K. M. B. D. Silva, and D. D. Sampson, Real-time detection technique for Doppler optical coherence tomography, Opt. Lett., 25, 1645–1647, 2000.

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Correspondence to Kiyoshi Yomogida.

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Ogiso, M., Yomogida, K. & Katsumata, K. Recursive travel-time inversion: A tool for real-time seismic tomography. Earth Planet Sp 57, 477–489 (2005) doi:10.1186/BF03352582

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Key words

  • Seismic tomography
  • recursive inverse scheme
  • velocity structure
  • Hidaka mountains