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

Attia Ahmed; Sandu Adrian; Attia Ahmed; Sandu Adrian

doi:10.3934/geosci.2015.1.41

AIMS Geosciences

2015, Volume 3, Issue 1: 41-78. doi: 10.3934/geosci.2015.1.41

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Research article

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

Attia Ahmed ^,,
Sandu Adrian

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

Received: 18 September 2015 Accepted: 14 December 2015 Published: 21 December 2015

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.
- Data assimilation,
- variational methods,
- ensemble filters,
- Markov chain,
- hybrid Monte-Carlo
Citation: Attia Ahmed, Sandu Adrian. A Hybrid Monte Carlo Sampling Filter for Non-Gaussian Data Assimilation[J]. AIMS Geosciences, 2015, 3(1): 41-78. doi: 10.3934/geosci.2015.1.41

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Abstract

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.

References

[1]	Ades M, Van Leeuwen PJ (2015) The equivalent-weights particle filter in a high-dimensional system.Quarterly J Royal Meteorological Society 141: 484-503.
[2]	Alexander F, Eyink G, Restrepo J (2005) Accelerated Monte Carlo for optimal estimation of time series.J Statistical Physics 119: 1331-1345.
[3]	Attia A, Sandu A (2014) A Sampling Filter for Non-Gaussian Data Assimilation.Cornell University, arXiv Preprint arXiv .
[4]	Anderson JL (1996) A method for producing and evaluating probabilistic forecasts from ensemble model integrations.J Climate 9: 1518-1530.
[5]	Anderson JL (2001) An ensemble adjustment Kalman filter for data assimilation.Monthly Weather Rev 129: 2884-2903.
[6]	Bennett A, Chua B (1994) Open-ocean modeling as an inverse problem: the primitive equations.Monthly Weather Rev 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 19: 1501-1534.
[8]	Beskos A, Pinski FJ, Sanz-Serna JM, Stuart A (2011) Hybrid Monte Carlo on Hilbert spaces.Stochastic Processes Applications 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 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 36: A1556-A1580.
[11]	Burgers G, Van Leeuwen PJ, Evensen G (1998) Analysis scheme in the ensemble Kalman filter.Monthly Weather Rev 126: 1719-1724.
[12]	Chorin A, Morzfeld M, Tu X (2010) Implicit particle filters for data assimilation.Communications in Applied Mathematics and Computational Sci 5: 221-240.
[13]	Cohn SE (1997) An introduction to estimation theory.J Meteorological Society of Japan 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 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 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 99: 10143-10162.
[18]	Evensen G (2003) The ensemble Kalman filter: Theoretical formulation and practical implementation.Ocean Dynamics 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) 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) 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.
[24]	Hamill TM, Whitaker JS, Snyder C (2001) Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter.Monthly Weather Rev 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 15: 1593-1623.
[26]	Houtekamer PL, Mitchell HL (1998) Data assimilation using an ensemble Kalman filter technique.Monthly Weather Rev 126: 796-811.
[27]	Houtekamer PL, Mitchell HL (2001) A sequential ensemble Kalman filter for atmospheric data assimilation.Monthly Weather Rev 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 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 82: 35-45.
[31]	Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory.J Fluids Engineering 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 136: 1644-1651.
[34]	Kitagawa G (1996) Monte Carlo filter and smoother for non-Gaussian nonlinear state space models.J Computational and Graphical Statistics 5: 1-25.
[35]	Law K and Stuart A (2012) Evaluating data assimilation algorithms.Monthly Weather Rev 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 112: 1177-1194.
[38]	Lorenz EN (1996) Predictability: A problem partly solved. Proc.Seminar on Predictability 1.
[39]	Lorenz EN, Emanuel KA (1998) Optimal sites for supplementary weather observations: Simulation with a small model.J Atmospheric Sciences 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 21: 1087-1092.
[41]	Nakano S, Ueno G, Higuchi T (2007) Merging particle filter for sequential data assimilation.Nonlinear Processes in Geophys 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 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 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 56: 415-428.
[47]	Sakov P, Oliver D, Bertino L (2012) An iterative EnKF for strongly nonlinear systems.Monthly Weather Rev 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 39-88.
[49]	Sanz-Serna JM, Calvo MP (1994) Numerical Hamiltonian problems Applied Mathema Mathematical Computation.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 5: 495-510.
[51]	Snyder C, Bengtsson T, Bickel P, Anderson J (2008) Obstacles to high-dimensional particle filtering.Monthly Weather Rev 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 136: 1898-1922.
[53]	Talagrand O, Vautard R, Strauss B (1997) Evaluation of probabilistic prediction systems.Proc. ECMWF Workshop on Predictability 1-25.
[54]	Tierney L (1994) Markov chains for exploring posterior distributions.The Ann Statistics 1701-1728.
[55]	Tippett MK, Anderson JL, Bishop CH, Hamill TM, Whitaker JS (2003) Ensemble square root filters.Monthly Weather Rev 131: 1485-1490.
[56]	Van Leeuwen PJ (2009) Particle filtering in geophysical systems.Monthly Weather Rev 137: 4089-4114.
[57]	Van Leeuwen PJ (2010) Nonlinear data assimilation in geosciences: an extremely efficient particle filter.Quarterly J Royal Meteorological Society 136: 1991-1999.
[58]	Van Leeuwen PJ (2011) Efficient nonlinear data-assimilation in geophysical fluid dynamics.Computers & Fluids 46: 52-58.
[59]	Whitaker JS, Hamill TM (2002) Ensemble data assimilation without perturbed observations.Monthly Weather Rev 130: 1913-1924.
[60]	Zupanski M (2005) Maximum likelihood ensemble filter: Theoretical aspects.Monthly Weather Rev 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 134: 1039-1050.

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