Skip to main content

Statistical analysis of three-dimensional run-up heights using Gaussian process emulator of particle method


This study presents that an emulator of a particle method has the potential to be applied for the statistical analysis of tsunami run-up phenomena. The emulator follows a Gaussian process to model a particle method specifically for estimating wave heights of tsunami run-up in front of buildings on the ground. In general, Gaussian processes have the advantage of designing statistical emulators to reduce the computational cost required by simulators. Although statistical analysis using computational models requires a considerable number of simulations, Gaussian process emulators can address this challenge. In contrast, particle methods are advantageous for simulating free-surface flow problems including tsunami run-up. The mesh-free methods discretize the Navier–Stokes and continuity equations without mesh generations as opposed to mesh methods, and thus they can simulate tsunami behaviors near ground buildings. In this study, we simplify tsunami run-up as three-dimensional dam-break problems where the collapse of a water column owing to gravity moves in a slope and impacts on two buildings. Although these problems are not exactly tsunami run-up phenomena, the study intends to show the possibility of applying the Gaussian process emulator for such phenomena. The sensitivity analysis of the wave heights is carried out using the emulator to observe how the run-up heights are influenced by the initial size of the water column. Consequently, it predicts the tendency of the wave heights based on the initial settings and demonstrates the effectiveness of the emulator for the run-up analysis. In addition, this study illustrates the frequency and quartiles of the run-up heights near ground structures, which can be applied for tsunami evacuation planning.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Tsunami wo yosoku suru shikumi. Japan Meteorological Agency. Accessed 10 Nov 2020 (in Japanese)

  2. 2.

    Arikawa T (2015) Consideration of characteristics of pressure on seawall by solitary waves based on hydraulic experiments. J Japan Soc Civ Eng Ser B2 (Coast Eng) 71(2):I\_889-I\_894 (in Japanese)

    Google Scholar 

  3. 3.

    Synolakis C (1987) The run-up of solitary waves. J Fluid Mech 185:523–545

    Article  Google Scholar 

  4. 4.

    Synolakis C (1991) Tsunami run-up on steep slopes: how good linear theory really is. Nat Hazards 4:221–234

    Article  Google Scholar 

  5. 5.

    Asakura R, Iwase K, Ikeya T, Takao M, Kaneto T, Fujii, N, Ohmori M (2003) The tsunami wave force acting on land structures. In: Proceedings of the 28th international conference on coastal engineering, pp 1191–1202 (in Japanese)

  6. 6.

    Arimitsu T, Ooe K, Kawasaki K (2012) Hydraulic experiment on evaluation method of tsunami wave pressure using inundation depth and velocity in front of land structure. J Japan Soc Civ Eng Ser B2 (Coast Eng) 68(2):I\_776-I\_780 (in Japanese)

    Google Scholar 

  7. 7.

    Asai M, Goda T, Oguni K, Isobe D, Kashiyama K, Isshiki M (2014) Evaluation of tsunami fluid force acted on tsunami refuge building by using a stabilized ISPH. J Japan Soc Civ Eng Ser A2 (Appl Mech (AM)) 70(2):I\_649-I\_658 (in Japanese)

    Google Scholar 

  8. 8.

    Suwa T, Imamura F, Sugawara D (2014) Development of a tsunami simulator integrating the smoothed-particle hydrodynamics method and the nonlinear shallow water wave model. J Japan Soc Civ Eng Ser A2 (Appl Mech (AM)) 70(2):I\_016-I\_020 (in Japanese)

    Google Scholar 

  9. 9.

    Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013–1024

    Article  Google Scholar 

  10. 10.

    Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Notices R Astron Soc 181(3):375–389

    Article  Google Scholar 

  11. 11.

    Monaghan JJ (1992) Smoothed particle hydrodynamics. Annu Rev Astron Astrophys 30:543–574

    Article  Google Scholar 

  12. 12.

    Cummins S, Rudman M (1999) An SPH projection method. J Comput Phys 152(2):584–607

    MathSciNet  Article  Google Scholar 

  13. 13.

    Lo EYM, Shao S (2002) Simulation of near-shore solitary wave mechanics by an incompressible SPH method. Appl Ocean Res 24(5):275–286

    Article  Google Scholar 

  14. 14.

    Shao S, Lo EYM (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv Water Resour 26(7):787–800

    Article  Google Scholar 

  15. 15.

    Koshizuka S, Oka Y (1996) Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123:421–434

    Article  Google Scholar 

  16. 16.

    Koshizuka S, Nobe A, Oka Y (1998) Numerical analysis of breaking waves using the moving particle semi-implicit method. Int J Numer Method Fluids 26:751–769

    Article  Google Scholar 

  17. 17.

    Koshizuka S, Shibata K, Kondo M, Matsunaga T (2018) Moving particle semi-implicit method: a meshfree particle method for fluid dynamics. Academic Press, Chippenham

    Google Scholar 

  18. 18.

    Shakibaeinia A, Jin Y (2010) A weakly compressible MPS method for modeling of open-boundary free-surface flow. Int J Numer Methods Fluids 63(10):1208–1232

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Oochi M, Koshizuka S, Sakai M (2010) Explicit MPS algorithm for free surface flow analysis. Trans Japan Soc Comput Eng Science 2010:20100013 (in Japanese)

    Google Scholar 

  20. 20.

    Oochi M, Yamada Y, Koshizuka S, Sakai M (2011) Validation of pressure calculation in Explicit MPS method. Trans Japan Soc Comput Eng Sci 2011:20110002 (in Japanese)

    Google Scholar 

  21. 21.

    Yamada Y, Sakai M, Mizutani S, Koshizuka S, Oochi M, Murozono K (2011) Numerical simulation of three-dimensional free-surface flows with explicit moving particle simulation method. Trans At Energy Soc Japan 10(3):185–193 (in Japanese)

    Article  Google Scholar 

  22. 22.

    Ferrari A, Dumbser M, Toro EF (2009) A new 3D parallel SPH scheme for free surface flows. Comput Fluids 38:1203–1217

    MathSciNet  Article  Google Scholar 

  23. 23.

    Chow AD, Rogers BD, Lind SJ, Stansby PK (2018) Incompressible SPH (ISPH) with fast Poisson solver on a GPU. Comput Phys Commun 226:81–103

    Article  Google Scholar 

  24. 24.

    Shibata K, Koshizuka S, Masaie I (2016) Cost reduction of particle simulations by an ellipsoidal particle model. Comput Methods Appl Mech Eng 307(1):411–450

    MathSciNet  Article  Google Scholar 

  25. 25.

    Shibata K, Koshizuka S, Matsunaga T, Masaie I (2017) The overlapping particle technique for multi-resolution simulation of particle methods. Comput Methods Appl Mech Eng 325(1):434–462

    MathSciNet  Article  Google Scholar 

  26. 26.

    Murotani K, Oochi M, Fujisawa T, Koshizuka S, Yoshimura S (2012) Distributed memory parallel algorithm for Explicit MPS using ParMETIS. Trans Japan Soc Comput Eng Sci 2012:20120012 (in Japanese)

    Google Scholar 

  27. 27.

    Murotani K, Koshizuka S, Tamai T, Shibata K, Mitsume N, Yoshimura S, Tanaka S, Hasegawa K, Nagai N, Fujisawa T (2014) Development of hierarchical domain decomposition explicit MPS method and application to large-scale tsunami analysis with floating objects. J Adv Simul Sci Eng 1(1):16–35

    Google Scholar 

  28. 28.

    Mizuno Y, Mitsume N, Yamada T, Yoshimura S (2019) Time-based dynamic load balancing algorithm for domain decomposition with particle method adopting three-dimensional polygon-wall boundary model. J Adv Simul Sci Eng 6(2):282–297

    Google Scholar 

  29. 29.

    O’Hagan A (2006) Bayesian analysis of computer code outputs: a tutorial. Reliab Eng Syst Saf 91(10–11):1290–1300

    Article  Google Scholar 

  30. 30.

    Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. The MIT Press, Massachusetts

    MATH  Google Scholar 

  31. 31.

    Rougier J (2008) Efficient emulators for multivariate deterministic functions. J Comput Graph Stat 17(4):827–843

    MathSciNet  Article  Google Scholar 

  32. 32.

    Rougier J, Maute A, Guillas S, Richmond AD (2009) Expert knowledge and multivariate emulation: the thermosphere-ionosphere electrodynamics general circulation model (TIE-GCM). Spec Issue Comput Model 51(4):414–424

    MathSciNet  Google Scholar 

  33. 33.

    Igarashi Y, Hori T, Murata S, Sato K, Baba T, Okada M (2016) Maximum tsunami height prediction using pressure gauge data by a Gaussian process at Owase in the Kii Peninsula, Japan. Mar Geophys Res 37(4):361–370

    Article  Google Scholar 

  34. 34.

    Oakley J, O’Hagan A (2002) Bayesian inference for the uncertainty distribution of computer model outputs. Biometrika 89(4):769–784

    Article  Google Scholar 

  35. 35.

    Oakley J, O’Hagan A (2004) Probabilistic sensitivity analysis of complex models: a Bayesian approach. J R Stat Soc Stat Methodol Ser B 66(3):751–769

    MathSciNet  Article  Google Scholar 

  36. 36.

    Sarri A, Guillas S, Dias F (2012) Statistical emulation of a tsunami model for sensitivity analysis and uncertainty quantification. Nat Hazards Earth Syst Sci 12(6):2003–2018

    Article  Google Scholar 

  37. 37.

    Mizuno Y, Koshizuka S (2020) Gaussian process emulation of particle method for estimating free-surface heights. J Fluid Sci Technol 15(3):JFST0021

    Article  Google Scholar 

  38. 38.

    Grezio A, Babeyko A, Baptista M, Behrens J, Costa A, Davies G, Geist EL, Glimsdal S, González FI, Griffin J, Harbitz CB, LeVeque RJ, Lorito S, Løvholt F, Omira R, Mueller C, Paris R, Parsons T, Polet J, Power W, Selva J, Sørensen MB, Thio H (2017) Probabilistic tsunami hazard analysis: multiple sources and global applications. Rev Geophys 55(4):1158–1198

    Article  Google Scholar 

  39. 39.

    Kohavi R (1995) A study of cross-validation and bootstrap for accuracy estimation and model selection. In: Proceedings of the 14th international joint conference on artificial intelligence, vol 2, pp 1137–1143

Download references


This work was supported by JSPS KAKENHI Grant Numbers JP18K04576, JP21J12302.

Author information



Corresponding author

Correspondence to Yoshiki Mizuno.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mizuno, Y., Shibata, K. & Koshizuka, S. Statistical analysis of three-dimensional run-up heights using Gaussian process emulator of particle method. Comp. Part. Mech. (2021).

Download citation


  • EMPS method
  • Free-surface flow
  • Gaussian process regression
  • Computational speed
  • Sensitivity analysis
  • Statistical emulation
  • Tsunami evacuation planning