Conditions for successful data assimilation

We show, using idealized models, that numerical data assimilation can be successful only if an effective dimension of the problem is not excessive. This effective dimension depends on the noise in the model and the data, and in physically reasonable problems it can be moderate even when the number of variables is huge. We then analyze several data assimilation algorithms, including particle filters and variational methods. We show that well-designed particle filters can solve most of those data assimilation problems that can be solved in principle, and compare the conditions under which variational methods can succeed to the conditions required of particle filters. We also discuss the limitations of our analysis.

[1]  Jun S. Liu,et al.  Blind Deconvolution via Sequential Imputations , 1995 .

[2]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[3]  Young Soo Moon,et al.  Bounds in algebraic Riccati and Lyapunov equations: a survey and some new results , 1996 .

[4]  James Martin,et al.  A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion , 2013, SIAM J. Sci. Comput..

[5]  W. Budgell,et al.  Ocean Data Assimilation and the Moan Filter: Spatial Regularity , 1987 .

[6]  E. Vanden-Eijnden,et al.  Data Assimilation in the Low Noise Regime with Application to the Kuroshio , 2012, 1202.4952.

[7]  N. Komaroff,et al.  Iterative matrix bounds and computational solutions to the discrete algebraic Riccati equation , 1994, IEEE Trans. Autom. Control..

[9]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[10]  P. J. van Leeuwen,et al.  A variance-minimizing filter for large-scale applications , 2003 .

[11]  M. Kalos,et al.  Monte Carlo methods , 1986 .

[12]  N. Komaroff,et al.  Upper bounds for the solution of the discrete Riccati equation , 1992 .

[13]  Peter Jan,et al.  Particle Filtering in Geophysical Systems , 2009 .

[14]  Mary F. Wheeler,et al.  Stochastic collocation and mixed finite elements for flow in porous media , 2008 .

[15]  A. Chorin,et al.  Stochastic Tools in Mathematics and Science , 2005 .

[16]  D. S. McCormick,et al.  Accuracy and stability of filters for dissipative PDEs , 2012, 1203.5845.

[17]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[18]  P. Bickel,et al.  Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems , 2008, 0805.3034.

[19]  A. Chorin,et al.  Implicit particle filtering for models with partial noise, and an application to geomagnetic data assimilation , 2011, 1109.3664.

[20]  P. Bickel,et al.  Curse-of-dimensionality revisited : Collapse of importance sampling in very large scale systems , 2005 .

[21]  G. Evensen.  Data Assimilation: The Ensemble Kalman Filter , 2006 .

[22]  Matthias Morzfeld,et al.  Implicit particle filters for data assimilation , 2010, 1005.4002.

[23]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[24]  L. Bailey.  The Kalman Filter , 2010 .

[25]  A. Chorin,et al.  Implicit Particle Methods and Their Connection with Variational Data Assimilation , 2012, 1205.1830.

[26]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. Ii: Numerical Results , 2007 .

[27]  Lance M. Leslie,et al.  Tropical Cyclone Prediction Using a Barotropic Model Initialized by a Generalized Inverse Method , 1993 .

[28]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. I: Theory , 2007 .

[29]  R. Adler.  The Geometry of Random Fields , 2009 .

[30]  Robert N. Miller,et al.  A Kalman Filter Analysis of Sea Level Height in the Tropical Pacific , 1989 .

[31]  P. Bickel,et al.  Obstacles to High-Dimensional Particle Filtering , 2008 .

[32]  P. Leeuwen,et al.  Nonlinear data assimilation in geosciences: an extremely efficient particle filter , 2010 .

[33]  P. Bickel,et al.  Sharp failure rates for the bootstrap particle filter in high dimensions , 2008, 0805.3287.

[34]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[35]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[36]  M. Bocquet,et al.  Beyond Gaussian Statistical Modeling in Geophysical Data Assimilation , 2010 .

[37]  J. Whitaker,et al.  Ensemble Square Root Filters , 2003, Statistical Methods for Climate Scientists.

[38]  Nancy Nichols,et al.  Conditioning of incremental variational data assimilation, with application to the Met Office system , 2011 .

[39]  Antonio J. Busalacchi,et al.  Sea surface topography fields of the tropical Pacific , 1995 .

[40]  Dongxiao Zhang,et al.  An efficient, high-order perturbation approach for flow in random porous media via Karhunen-Loève and polynomial expansions , 2004 .

[41]  Matthias Morzfeld,et al.  A random map implementation of implicit filters , 2011, J. Comput. Phys..

[42]  Brad Weir,et al.  Implicit Estimation of Ecological Model Parameters , 2013, Bulletin of mathematical biology.

[43]  C. Snyder.  Particle filters, the "optimal" proposal and high-dimensio nal systems , 2011 .

[44]  Nancy Nichols,et al.  Conditioning and preconditioning of the variational data assimilation problem , 2011 .

[45]  M. Adès,et al.  An exploration of the equivalent weights particle filter , 2013 .

[46]  Fredrik Gustafsson,et al.  Particle Filters , 2015, Encyclopedia of Systems and Control.

[47]  A. Chorin,et al.  Implicit sampling for particle filters , 2009, Proceedings of the National Academy of Sciences.

[48]  Peter J. Bickel,et al.  Comparison of Ensemble Kalman Filters under Non-Gaussianity , 2010 .

[49]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[50]  Jonathan Weare,et al.  Particle filtering with path sampling and an application to a bimodal ocean current model , 2009, J. Comput. Phys..