AIMS Geosciences, 2015, 1(1): 41-78. doi: 10.3934/geosci.2015.1.41

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A Hybrid Monte Carlo Sampling Filter for Non-Gaussian Data Assimilation

Computational Science Laboratory, Department of Computer Science, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24060, USA

Data assimilation combines information from models, measurements, and priors to obtain improved estimates of the state of a dynamical system such as the atmosphere. Ensemble-based data assimilation approaches such as the Ensemble Kalman filter (EnKF) have gained wide popularity due to their simple formulation, ease of implementation, and good practical results. Many of these methods are derived under the assumption that the underlying probability distributions are Gaussian. It is well accepted, however, that the Gaussianity assumption is too restrictive when applied to large nonlinear models, nonlinear observation operators, and large levels of uncertainty. When the Gaussianity assumptions are severely violated, the performance of EnKF variations degrades. This paper proposes a new ensemble-based data assimilation method, named the sampling filter, which obtains the analysis by sampling directly from the posterior distribution. The sampling strategy is based on a Hybrid Monte Carlo (HMC) approach that can handle non-Gaussian probability distributions. Numerical experiments are carried out using the Lorenz-96 model and observation operators with different levels of non-linearity and differentiability. The proposed filter is also tested with shallow water model on a sphere with linear observation operator. Numerical results show that the sampling filter performs well even in highly nonlinear situations where the traditional filters diverge.
  Article Metrics


1. Ades M, Van Leeuwen PJ, (2015) The equivalent-weights particle filter in a high-dimensional system . Quarterly J Royal Meteorological Society Volume: 141, 484–503.

2. Alexander F, Eyink G, Restrepo J, (2005) Accelerated Monte Carlo for optimal estimation of time series. J Statistical Physics Volume: 119, 1331-1345.

3. Attia A, Sandu A, (2014) A Sampling Filter for Non-Gaussian Data Assimilation. Cornell University, arXiv Preprint arXiv:1403.7137. [Available from]

4. Anderson JL, (1996) A method for producing and evaluating probabilistic forecasts from ensemble model integrations. J Climate Volume: 9, 1518-1530.

5. Anderson JL, (2001) An ensemble adjustment Kalman filter for data assimilation. Monthly Weather Rev Volume: 129, 2884-2903.

6. Bennett A, Chua B, (1994) Open-ocean modeling as an inverse problem: the primitive equations. Monthly Weather Rev Volume: 122, 1326–1336.

7. Beskos A, Pillai N, Roberts G, Sanz-Serna JM, Stuart A, (2013) Optimal tuning of the hybrid Monte Carlo algorithm. Bernoulli Volume: 19, 1501-1534.

8. Beskos A, Pinski FJ, Sanz-Serna JM, Stuart A, (2011) Hybrid Monte Carlo on Hilbert spaces. Stochastic Processes Applications Volume: 121.

9. Bishop CH, Etherton BJ, and Majumdar SJ, (2001) Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Monthly Weather Rev Volume: 129, 420-436.

10. Blanes S, Casas F, Sanz-Serna JM, (2014) Numerical integrators for the hybrid Monte Carlo method. SIAM J on Scientific Computing Volume: 36, A1556-A1580.

11. Burgers G, Van Leeuwen PJ, Evensen G, (1998) Analysis scheme in the ensemble Kalman filter. Monthly Weather Rev Volume: 126, 1719-1724.

12. Chorin A, Morzfeld M, Tu X, (2010) Implicit particle filters for data assimilation. Communications in Applied Mathematics and Computational Sci Volume: 5, 221-240.

13. Cohn SE, (1997) An introduction to estimation theory. J Meteorological Society of Japan Volume: 75, 257-288.

14. S. L. Cotter and M. Dashti and A. M. Stuart, (2012) Variational data assimilation using targeted random walks Inter J numerical methods in fluids Volume: 68, 403-421.

15. Doucet A, De Freitas N, Gordon NJ, (2001) An introduction to sequential Monte Carlo methods. Series Statistics For Engineering and Information Sci. Springer.

16. Duane S, Kennedy AD, Pendleton BJ, and Roweth D, (1987) Hybrid Monte Carlo. Physics Letters B Volume: 195, 216-222.

17. Evensen G, (1994) Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J Geophysical Res: Oceans Volume: 99, 10143-10162.

18. Evensen G, (2003) The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dynamics Volume: 53, 343-367.

19. Evensen G, (2007) Data assimilation: The ensemble Kalman filter. Springer.

20. Fisher M, Courtier P, (1995) Estimating the covariance matrices of analysis and forecast error in variational data assimilation. European Center for Medium-Range Weather Forecasts

21. Girolami M, Calderhead B, (2011) Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J Royal Statistical Society: Series B (Statistical Methodology) Volume: 73, 123-214.

22. Gordon Neil, Salmond D, Smith A, (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F (Radar and Signal Processing) Volume: 140, 107-113.

23. Gu Y, Oliver D, (2001) An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. Spe Journal, 12:438–446. Society of Petroleum Engineers.

24. Hamill TM, Whitaker JS, Snyder C, (2001) Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Monthly Weather Rev Volume: 129, 2776-2790.

25. Hoffman MD, Gelman A, (2014) The no-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. The J Machine Learning Res Volume: 15(1), 1593–1623.

26. Houtekamer PL, Mitchell HL, (1998) Data assimilation using an ensemble Kalman filter technique. Monthly Weather Rev Volume: 126, 796-811.

27. Houtekamer PL, Mitchell HL, (2001) A sequential ensemble Kalman filter for atmospheric data assimilation. Monthly Weather Rev Volume: 129, 123-137.

28. Jardak M, Navon IM, Zupanski M, (2010) Comparison of sequential data assimilation methods for the Kuramoto-Sivashinsky equation. Inter J Numerical Methods in Fluids Volume: 62, 374-402.

29. Jardak M, Navon IM, Zupanski M, (2013) Comparison of Ensemble Data Assimilation methods for the shallow water equations model in the presence of nonlinear observation operators. Submitted to Tellus

30. Kalman RE, (1960) A new approach to linear filtering and prediction problems. J Fluids Engineering Volume: 82, 35-45.

31. Kalman RE, Bucy RS, (1961) New results in linear filtering and prediction theory. J Fluids Engineering Volume: 83, 95-108.

32. Kalnay E, (2002) Atmospheric modeling, data assimilation and predictability. Cambridge University Press

33. Kalnay E, Yang S-C, (2010) Accelerating the spin-up of ensemble Kalman filtering. Quarterly J Royal Meteorological Society Volume: 136, 1644-1651.

34. Kitagawa G, (1996) Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J Computational and Graphical Statistics Volume: 5, 1-25.

35. Law K and Stuart A, (2012) Evaluating data assimilation algorithms. Monthly Weather Rev Volume: 140, 3757-3782.

36. Liu JS, (2008) Monte Carlo strategies in scientific computing. Springer.

37. Lorenc AC, (1986) Analysis methods for numerical weather prediction. Quarterly J Royal Meteorological Society Volume: 112, 1177-1194.

38. Lorenz EN, (1996) Predictability: A problem partly solved. Proc. Seminar on Predictability Volume: 1.

39. Lorenz EN, Emanuel KA, (1998) Optimal sites for supplementary weather observations: Simulation with a small model. J Atmospheric Sciences Volume: 55, 399-414.

40. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E, (1953) Equation of state calculations by fast computing machines. J Chemical Phys Volume: 21, 1087-1092.

41. Nakano S, Ueno G,Higuchi T, (2007) Merging particle filter for sequential data assimilation. Nonlinear Processes in Geophys Volume: 14, 395-408.

42. Neal RM, (1993) Probabilistic inference using Markov chain Monte Carlo methods. Department of Computer Science, University of Toronto Toronto, Ontario, Canada.

43. Neal RM, (2011) MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo.

44. Neta B, Giraldo FX, Navon IM, (1997) Analysis of the Turkel-Zwas scheme for the twodimensional shallow water equations in spherical coordinates. J Computational Phys Volume: 133, 102-112.

45. Nino Ruiz ED, Sandu A, Anderson JL, (2014) An efficient implementation of the ensemble Kalman filter based on an iterative Sherman–Morrison formula. Statistics and Computing Volume: 1-17.

46. Ott E, Hunt BR, Szunyogh I, Zimin AV, Kostelich EJ, Corazza M, Kalnay E, Patil DJ, Yorke JA, (2004) A local ensemble kalman filter for atmospheric data assimilation. Tellus A Volume: 56, 415-428.

47. Sakov P, Oliver D, Bertino L. 2012. An iterative EnKF for strongly nonlinear systems. Monthly Weather Rev Volume: 140, 1988-2004.

48. Sanz-Serna JM, (2014) Markov chain Monte Carlo and numerical differential equations. Current Challenges in Stability Issues for Numerical Differential Equations Volume: 39-88.

49. Sanz-Serna JM , Calvo MP, (1994) Numerical Hamiltonian problems Applied Mathema Mathematical Computation Volume: 7. Chapman & Hall London.

50. Simon E, Bertino L, (2009) Application of the Gaussian anamorphosis to assimilation in a 3-D coupled physical-ecosystem model of the North Atlantic with the EnKF: a twin experiment. Ocean Science Volume: 5, 495-510.

51. Snyder C, Bengtsson T, Bickel P, Anderson J, (2008) Obstacles to high-dimensional particle filtering. Monthly Weather Rev Volume: 136, 4629-4640.

52. St-Cyr A, Jablonowski C, Dennis JM, Tufo HM, Thomas SJ, (2007) A comparison of two shallow water models with nonconforming adaptive grids. Monthly Weather Rev Volume: 136, 1898-1922.

53. Talagrand O, Vautard R, Strauss B, (1997) Evaluation of probabilistic prediction systems. Proc. ECMWF Workshop on Predictability Volume: 1-25.

54. Tierney L, (1994) Markov chains for exploring posterior distributions. The Ann Statistics Volume: 1701-1728.

55. Tippett MK, Anderson JL, Bishop CH, Hamill TM, Whitaker JS, (2003) Ensemble square root filters. Monthly Weather Rev Volume: 131, 1485-1490.

56. Van Leeuwen PJ, (2009) Particle filtering in geophysical systems. Monthly Weather Rev Volume: 137, 4089-4114.

57. Van Leeuwen PJ, (2010) Nonlinear data assimilation in geosciences: an extremely efficient particle filter. Quarterly J Royal Meteorological Society Volume: 136, 1991-1999.

58. Van Leeuwen PJ, (2011) Efficient nonlinear data-assimilation in geophysical fluid dynamics. Computers & Fluids Volume: 46, 52-58.

59. Whitaker JS, Hamill TM, (2002) Ensemble data assimilation without perturbed observations. Monthly Weather Rev Volume: 130, 1913-1924.

60. Zupanski M, (2005) Maximum likelihood ensemble filter: Theoretical aspects. Monthly Weather Rev Volume: 133, 1710-1726.

61. Zupanski M, Navon IM, Zupanski D, (2008) The maximum likelihood ensemble filter as a nondifferentiable minimization algorithm. Quarterly J Royal Meteorological Society Volume: 134, 1039-1050.

Copyright Info: © 2015, Ahmed Attia, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved