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A three-dimensional robust seismic ray tracer for volcanic regions


Seismic velocity structure of volcanic region is highly heterogeneous so that seismic ray tracer used in this field should be robust for the velocity heterogeneity. From this view point a three-dimensional robust seismic ray tracer, effective in any complicated velocity structure, is developed by using a hybrid scheme of the shortest path calculation and the downhill simplex optimization method. The node configuration necessary for the three-dimensional shortest path calculation is newly presented. Validity and efficiency of the calculation are examined by synthetic tests. A travel time accuracy of less than 0.1%, and a rms ray path error of 0.05 km are achieved. Calculations for the synthetic velocity models and checkerboard testing show the effectiveness of this ray tracer in practical situations. The present ray tracer is recommended to be used in travel time tomography in highly heterogeneous velocity structure such as volcanic regions.


  • Cheng, N. and L. House, Minimum traveltime calculation in 3-D graph theory, Geophysics, 61, 1895–1898, 1996.

    Article  Google Scholar 

  • Dijkstra, E. W., A note on two problems in connection with graph, Numer. Math., 1, 269–271, 1959.

    Article  Google Scholar 

  • Fisher, R. and J. M. Lees, Shortest path ray tracing with sparse graphs, Geophysics, 58, 987–996, 1993.

    Article  Google Scholar 

  • Gallo, G. and S. Pallottino, Shortest path methods: A unifying approach: Mathematical Programming Study, 26, 38–64, 1986.

    Article  Google Scholar 

  • Klimes, L. and M. Kvasnicka, 3-D network ray tracing, Geophys. J. Int., 116, 726–738, 1994.

    Article  Google Scholar 

  • Moser, T. J., Shortest path calculation of seismic rays, Geophysics, 56, 59–67, 1991.

    Article  Google Scholar 

  • Nakanishi, I. and K. Yamaguchi, A numerical experiment on nonlinear image reconstruction from the first-arrival times for two-dimensional island arc structure, J. Phys. Earth, 34, 195–201, 1986.

    Article  Google Scholar 

  • Nelder, J. A. and R. Mead, A simplex method for function minimization, Computer Journal, 7, 308–313, 1965.

    Article  Google Scholar 

  • Podvin, P. and I. Lecomte, Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools, Geophys. J. Int., 105, 271–284, 1991.

    Article  Google Scholar 

  • Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: the art of scientific computing, 2nd edition, pp. 402–406, Cambridge University Press, 1992.

  • Reshef, M. and D. Kosloff, Migration of commom-shot gathers, Geophysics, 51, 324–331, 1986.

    Article  Google Scholar 

  • Saito, H., Travel times and ray paths of first arrival seismic waves: computation method based on Huygens’ principle, in Expanded abstracts, 59th Annual Int, SEG Meeting, pp. 244–247, Soc. Explor. geophys., Tulsa, Oklahoma, 1989.

    Google Scholar 

  • Saito, H., 3-D ray tracing method based on Huygens’ principle, in Expanded abstracts, 60th Annual Int, SEG Meeting, pp. 1024–1027, Soc. Explor. geophys., Tulsa, Oklahoma, 1990.

    Google Scholar 

  • Sheriff, R. E. and L. P. Geldart, Exploration Seismology Second Edition, pp. 93–95, Cambridge University Press, 1995.

  • Sudo, Y., An attenuating structure beneath the Aso Caldera determined from the propagation of seismic waves, Bull. Volcanol., 53, 99–111, 1991.

    Article  Google Scholar 

  • Sudo, Y., Project Aso98 (Seismic explosions): Seismic explosion around active crater, Abstracts 1999 Japan Earth and Planetary Science Joint Meeting, Vb-022, 1999.

  • Um, J. and C. Thurber, A fast algorithm for two point seismic ray tracing, Bull. Seism. Soc. Am., 77, 972–986, 1987.

    Google Scholar 

  • van Trier, J. and W. W. Symes, Upwind finite-difference calculation of traveltimes, Geophysics, 56, 812–821, 1991.

    Article  Google Scholar 

  • Vidale, J., Finite-difference calculation of traveltimes, Bull. Seis. Soc. Am.78, 2062–2076, 1988.

    Google Scholar 

  • Zhang, J. and M. N. Toksöz, Nonlinear refraction travel time tomography, Geophysics, 63, 1726–1737, 1998.

    Article  Google Scholar 

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Correspondence to Kiyoshi Nishi.

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Nishi, K. A three-dimensional robust seismic ray tracer for volcanic regions. Earth Planet Sp 53, 101–109 (2001).

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  • Travel Time
  • Block Size
  • Velocity Structure
  • Simplex Method
  • Hybrid Scheme