Simulation of Standing and Propagating Sea Waves with Three-Dimensional ARMA Model

  • Ivan GankevichEmail author
  • Alexander Degtyarev
Part of the Springer Oceanography book series (SPRINGEROCEAN)


Simulation of sea waves is a problem appearing in the framework of developing software-based ship motion modelling applications. These applications generally use linear wave theory to generate small-amplitude waves programmatically and determine impact of external excitations on a ship hull. Using linear wave theory is feasible for ocean waves, but is not accurate for shallow-water and storm waves. To cope with these shortcomings we introduce autoregressive moving-average (ARMA) model, which is widely known in oceanography, but rarely used for sea wave modelling. The new model allows to generate waves of arbitrary amplitudes, is accurate for both shallow and deep water, and its software implementation shows superior performance by relying on fast Fourier transform family of algorithms. Integral characteristics of wavy surface produced by ARMA model are verified against the ones of real sea surface. Despite all its advantages, ARMA model requires a new method to determine wave pressures, an instance of which is included in the chapter.


  1. 1.
    St. Denis, M., & Pierson, W. J., Jr. (1953). On the motions of ships in confused seas. Technical report, New York University, Bronx School of Engineering and ScienceGoogle Scholar
  2. 2.
    Rosenblatt, M. (1956). A random model of the sea surface generated by a hurricane. Technical report, DTIC DocumentGoogle Scholar
  3. 3.
    Sveshnikov, A. A. (1959). Mathematics Akademii Mechanics and Engineering, 3, 32.Google Scholar
  4. 4.
    Longuet-Higgins, M. S. (1957). Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 249(966), 321.Google Scholar
  5. 5.
    Kochin, N., Kibel, I., & Roze, N. (1966). Theoretical hydrodynamics (in Russian). FizMatLit.Google Scholar
  6. 6.
    Beck, R. F., Reed, A. M., Sclavounos, P. D., & Hutchison, B. L. (2001). Transactions-Society of Naval Architects and Marine Engineers, 109, 1.Google Scholar
  7. 7.
    Box, G. E., & Jenkins, G. M. (1976). Time series analysis: Forecasting and control (revised ed.). Holden-Day.Google Scholar
  8. 8.
    Kostecki, M. (1972). Stochastic model of sea waves. Ph.D. thesis, CTO, Gdansk.Google Scholar
  9. 9.
    Rozhkov, V. A., & Trapeznikov, Y. A. (1990). Probabilistic models of oceanographic processes. Leningrad: Gidrometeoizdat.Google Scholar
  10. 10.
    Gurgenidze, A. T., & Trapeznikov, Y. A. (1988). Probabilistic model of wind waves (pp. 8–23). Leningrad: Gidrometeoizdat.Google Scholar
  11. 11.
    Spanos, P. D. (1982). ARMA algorithms for ocean spectral analysis. University of Texas at Austin, Engineering Mechanics Research Laboratory.Google Scholar
  12. 12.
    Spanos, P. D., & Zeldin, B. (1996). Earthquake Engineering and Structural Dynamics, 25(5), 497.CrossRefGoogle Scholar
  13. 13.
    Fusco, F., & Ringwood, J. V. (2010). IEEE Transactions on Sustainable Energy, 1(2), 99.CrossRefGoogle Scholar
  14. 14.
    Degtyarev, A., & Gankevich, I. (2012). Proceedings of 11th International Conference on Stability of Ships and Ocean Vehicles, Athens (pp. 841–852)Google Scholar
  15. 15.
    Degtyarev, A. B., & Podoliakin, A. B. (1998). Proceedings of II International Conferences on Shipbuilding (ISC’98), Saint-Petersburg (Vol. V, pp. 416–423).Google Scholar
  16. 16.
    Degtyarev, A., & Boukhanovsky, A. (1997). Analysis of peculiarities of ship-environmental interaction. Technical report, 09-97-1AB-1VA, Strathclyde University, Ship Stability Research Center, Glasgow.Google Scholar
  17. 17.
    Boccotti, P. (1983). Meccanica, 18(4), 205.CrossRefGoogle Scholar
  18. 18.
    Degtyarev, A. B., & Reed, A. M. (2011). Proceedings of the 12th International Ship Stability Work-shop.Google Scholar
  19. 19.
    Degtyarev, A. B., & Reed, A. M. (2013). International Shipbuilding Progress, 60(1–4), 523.Google Scholar
  20. 20.
    Boukhanovsky, A. V. (1997). Probabilistic modeling of wind wave fields taking into account their heterogeneity and nonstationarity. Ph.D. thesis, Saint Petersburg State University.Google Scholar
  21. 21.
    Wolfram Research Inc. (2016). Mathematica. Champaign, Illinois.Google Scholar
  22. 22.
    Matsumoto, M., & Nishimura, T. (1998). ACM Transactions on Modeling and Computer Simulation (TOMACS), 8(1), 3.CrossRefGoogle Scholar
  23. 23.
    Matsumoto, M., & Nishimura, T. (1998). Monte Carlo and Quasi-Monte Carlo Methods, 2000, 56.Google Scholar
  24. 24.
    Oppenheim, A. V., Schafer, R. W., Buck, J. R., et al. (1989). Discrete-time signal processing (Vol. 2). Englewood Cliffs, NJ: Prentice Hall.Google Scholar
  25. 25.
    Svoboda, D. (2011). Image Analysis and Processing–ICIAP 2011 (pp. 453–462). Springer.Google Scholar
  26. 26.
    Pavel, K., & David, S. (2013). Algorithms for efficient computation of convolution. INTECH Open Access Publisher.Google Scholar
  27. 27.
    Veldhuizen, T. L., & Jernigan, M. E. (1997). International Conference on Computing in Object-Oriented Parallel Environments (pp. 49–56). Springer.Google Scholar
  28. 28.
    Veldhuizen, T. (2000). Computer Science Technical Report, 542, 60.Google Scholar
  29. 29.
    Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Alken, P., et al. (2009). GNU scientific library reference manual. In B. Gough (Ed.) (3rd ed.). Network Theory Ltd.Google Scholar
  30. 30.
    Goto, K., & Van De Geijn, R. (2008). ACM Transactions on Mathematical Software (TOMS), 35(1), 4.CrossRefGoogle Scholar
  31. 31.
    Goto, K., & Geijn, R. A. (2008). ACM Transactions on Mathematical Software (TOMS), 34(3), 12.CrossRefGoogle Scholar
  32. 32.
    Kilgard, M. J. (1996). The OpenGL Utility Toolkit (GLUT) Programming Interface.
  33. 33.
    Fabri, A., & Pion, S. (2009). Proceedings of the 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (pp. 538–539). ACM.Google Scholar
  34. 34.
    clFFT developers. clFFT: OpenCL Fast Fourier Transforms (FFTs).

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Saint Petersburg State UniversityPetergof, Saint PetersburgRussia

Personalised recommendations