KSCE Journal of Civil Engineering

, Volume 23, Issue 9, pp 4102–4112 | Cite as

Simulation of Earthquake Motion Phase considering Its Fractal and Auto-covariance Features

  • Adam Ahmed Abdelrahman
  • Tadanobu Sato
  • Chunfeng WanEmail author
  • Lei Zhao
Structural Engineering


The earthquake motion phase (EMP) is decomposed into linear delay and fluctuation parts. In this paper, the peculiar stochastic characteristics of the fluctuation part of the phase (FPP) are discussed. First, we show that the FPP has self-afSne similarity and should be expressed as a fractal stochastic process by using several observed earthquake motion time histories, as well as the FPP has a long term memory in the frequency domain. Moreover, the possibility of simulating FPP using the simple fractional Brownian motion (fBm) is discussed and conclude that this is not possible. To overcome this problem, we develop a new stochastic process, the modified fBm that is able to simulate a stochastically rigorous sample FPP. This newly developed algorithm represents the phase characteristics of the observed EMP well.


phase power law of variance auto-covariance modified fractional stochastic process fractional Brownian motion Hurst index 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to acknowledge Japan Meteorological Agency and National Research for Earth and Disaster Resilience, as well as the Peer NGA Strong Motion Database, to provide valuable observed earthquake records. We also acknowledge the supports from JSPS, Grant-in-Aid for Scientific Research #18K04334, the National Natural Science Foundation of China (No.51578140) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD, No. CE02).


  1. Abdelrahman, A. A., Sato, T., Wan, C., and Wu, Z. (2019). “Definition of yield seismic coefficient spectrum considering the uncertainty of the earthquake motion phase.” Applied Sciences, Vol. 9, No. 11, p. 2254, DOI: 10.3390/app9112254.CrossRefGoogle Scholar
  2. Biagini, F., Hu, Y., Oksendal, B., and Zhang, T. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications, Springer Science & Business Media, London, UK.CrossRefzbMATHGoogle Scholar
  3. Boore, D. M. (2003). “Phase derivatives and simulation of strong ground motions.” Bulletin of the Seismological Society of America, Vol. 93, No. 3, pp. 1132–1143, DOI: 10.1785/0120020196.CrossRefGoogle Scholar
  4. Cohen, L. (1995). Time - Frequency analysis, Vol. 778, Prentice Hall, Upper Saddle River, NJ, USA.Google Scholar
  5. Falconer, K. (2014). Fractal geometry: Mathematical foundations and applications. John Wiley and Sons, Hoboken, NJ, USA.zbMATHGoogle Scholar
  6. Izumi, M., Watanabe, T., and Katukura, H. (1980). “Interrelation of fault mechanisms, phase inclinations and nonstationarities of seismic waves.” Proc. 7th World Conference on Earthquake Engineering, Istanbul, Turkey, Vol. 1, pp. 89–96.Google Scholar
  7. Katukura, H. (1978). “A study on the phase properties of seismic waves.” Proc. 5th Japan Earthquake Engineering Symposium, Japan Society of Civil Engineers, Shinjuku, Tokyo, Japan, pp. 209–216.Google Scholar
  8. Katukura, K. (1983). “A fundamental study on the phase properties of seismic waves.” Journal of Structural and Construction Engineering, Transactions of AIJ, 327, pp. 20–27.Google Scholar
  9. Mandelbrot, B. B. and Van, J. W (1968). “Fractional Brownian motions, fractional noises and applications.” SIAM Review, Vol. 10, No. 4, pp. 422–437, DOI: 10.2307/2027184.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Meyer, Y. (1992). Wavelets and operators (Vol. 1), Cambridge University Press, Cambridge, UK.Google Scholar
  11. Murono, Y, Sato, T., and Murakami, M. (2002). “Modeling of phase spectra for near-fault earthquake motions.” Proc. of 12th European Conf. on Earthquake Eng., Paris, France.Google Scholar
  12. Nigam, N. C. (1982). “Phase properties of a class of random processes.” Earthquake Engineering and Structural Dynamics, Vol. 10, No. 5, pp. 711–717, DOI: 10.1002/eqe.4290100508.CrossRefGoogle Scholar
  13. Ohsaki, Y. (1979). “On the significance of phase content in earthquake ground motions.” Earthquake Engineering and Structural Dynamics, Vol. 7, No. 5, pp. 427–439, DOI: 10.1002/eqe.4290070504.CrossRefGoogle Scholar
  14. Papoulis, A. (1962). The Fourier integral and its applications. McGraw-Hill, New York, NY, USA.zbMATHGoogle Scholar
  15. Peng, Y. and Li, J. (2014). “Stochastic modeling for starting-time of phase evolution of random seismic ground motions.” Theoretical and Applied Mechanics Letters, Vol. 4, No. 1, p. 013009, DOI: 10.1063/2.1401309.CrossRefGoogle Scholar
  16. Sato, T., Murono, Y., and Nishimura, A. (2002). “Phase spectrum modeling to simulate design earthquake motion.” Journal of Natural Disaster Science, Vol. 24, No. 2, pp. 91–100.Google Scholar
  17. Satoh, T., Uetake, T., and Sugawara, Y. (1996). “A study on envelope characteristics of strong motions in a period range of 1 to 15 seconds by using group delay time.” Proc. 11th World Conference on Earthquake Engineering, WCEE, Acapulco, Mexico.Google Scholar
  18. Tanaka, K. and Sato, T. (2017). “Evaluation of inhomogeneous structures in seismic propagation path in Japan based on the fractal characteristic of observed earthquake motion phase.” Proc. 16th World Conference on Earthquake Engineering, WCEE, Acapulco, Mexico, Paper No. 1420.Google Scholar
  19. Waezi, Z. and Rofooei, F. R. (2017). “Stochastic non-stationary model for ground motion simulation based on higher-order crossing of linear time variant systems.” Journal of Earthquake Engineering, Vol. 21, No. 1, pp. 123–150, DOI: 10.1080/13632469.2016.1149894.CrossRefGoogle Scholar
  20. Yang, D. and Zhang, C. (2013). “Fractal characterization and frequency properties of near-fault ground motions.” Earthquake Engineering and Engineering Vibration, Vol. 2, No. 4, pp. 503–518, DOI: 10.1007/s11803-013-0192-y.CrossRefGoogle Scholar
  21. Yang, D., Zhang, C., and Liu, Y. (2015). “Multifractal characteristic analysis of near-fault earthquake ground motions.” Soil Dynamics and Earthquake Engineering, Vol. 72, pp. 12–23, DOI: 10.1016/ j.soildyn.2015.01.020.CrossRefGoogle Scholar
  22. Zhang, C., Sato, T., and Lu, L. Y. (2011). “A phase model of earthquake motions based on stochastic differential equation.” KSCE Journal of Civil Engineering, KSCE, Vol. 15, No. 1, pp. 161–166, DOI: 10.1007/sl2205-011-1074-3.CrossRefGoogle Scholar

Copyright information

© Korean Society of Civil Engineers 2019

Authors and Affiliations

  1. 1.Key Laboratory of Concrete and Prestressed Concrete Structure of Ministry of EducationSoutheast UniversityNanjingChina
  2. 2.Civil Engineering DepartmentNyala UniversitySouth DarfurSudan

Personalised recommendations