Dynamic Data Assimilation: A historical view

1. Dynamic Data Assimilation:A historical view S. Lakshmivarahan School of Computer Science University of Oklahoma Norman

2. What is data assimilation? • Fitting models to data - Statistics • Models can be empirical or based on causality • Inverse problems: y = f(x) in Geophysical domain • Computing y from x is the forward problem • Computing x from y is the inverse problem • Identification/Parameter (IC, BC, physical parameters) estimation problems • State estimation

3. Other examples • Diagnosis of diseases from symptoms • CSI Miami • NTSB trying to pin down the causes for the collapse of the I-35 bridge in Minnesota or the TWA flight that crashed shortly after take-off

4. Why combine model and data? • Let x1 and x2 be two pieces of information about an unknown x with p1 and p2 be their variances • A linear estimate for x = a1x1 + a2x2 • Optimal (min. var) values for ai are: • a1 = p2 / (p1 +p2) and a2 = p1/ (p1 + p2) • Var(x) = p1p2/(p1 +p2) < min (p1, p2)

5. Dynamic model • Dynamic model: xk+1 = M[ xk ] + wk+1 • Model space is Rn • For simplicity consider discrete time - k • M denotes the model – linear or nonlinear • Randomness in models enters from two sources - initial condition x0 and forcing wk • wk denotes the model errors • This class of models are called Markov models in Rn • Analysis of continuous time needs a good working knowledge of stochastic calculus

6. Randomness in initial condition only and no random forcing • Complete characterization of the state of a stochastic system is given the time evolution of its probability density in the state space Rn • Solution to the evolution of the probability density of xt when there is randomness only in the initial condition is given by the Liouville’s equation • This is a (deterministic) PDF relating to the evolution of pt(xt), well known in fluid dynamics

7. Random forcing + random initial condition • In this case, the evolution of the probability density of xt is given by the well known (deterministic) PDEs called Kolmogorov’s forward equations (1930’s) • This equation is a generalization of theLiouville’s equationin the sense that when there is no random forcing, it reduces to Liouville’s equation

8. When data comes into play • Zk = h(xk) + vk • Observation space is Rm • h: map of the model space into Observation space – can be linear or nonlinear • When h varies with time as it happens with some of the ecological modeling, hk(xk) is used

9. Essence of data assimilation:Fusion of data with model • Bayesian situation naturally comes into play • Model forecast is used as the prior • Data represents the new information • Combined to obtain posterior estimate • This combined estimate has lesser variance/covariance – witness the first example

10. Where it all began? • Began with Gauss in 1801 with his discovery of the method of least squares • C. Gauss (1809) Theory of motion of heavenly bodies moving about the sun in conic sections, Dover edition published in 1963

11. Kolmogorov-Wiener era – 1940’s • Norbert Wiener in USA and Kolmogorov in the former USSR began a systematic analysis of the filtering, prediction and smoothing problems • Wiener, N. (1949) Extrapolation, interpolation andsmoothing of stationary time series with engineeringapplications, Wiley [This was originally published in 1942 as a classified defense document] Also available from MIT press • Kolmogorov, A. N. ((1941) “Interpolation, extrapolation of stationary random series”, Bulletin of the Academy ofSciences, USSR, vol 5 [English translation by RAND corporation memorandum RM-3090-PR, April 1962]

12. The modern Kalman- Bucy Era – since 1960 • The previous methods for data assimilation were off-line techniques • Space travel presented much needed impetus to the development of on-line or sequential methods • Kalman, R. E. (1960) “ A new approach to linear filtering and prediction problems” ASME Transactions, Journal of BasicEngineering, Series D, vol 82, pp 35-45 • Kalman R. E. and R. S. Bucy ( 1961) “New results in linear filtering and prediction theory”, ibid, vol 83, pp 95-108 • These some of the most quoted papers in all of the literature

13. Nonlinear filtering is the general version of the data assimilation problem • The evolution of the probability density of the optimal estimate is described by a (stochastic) PDE called Kushner-Zakai equation • So, by late 1960’s complete solution to the nonlinear filtering problem which is the problem of assimilation noisy data into nonlinear stochastic dynamic models • Except in very special cases these equations are very difficult to solve even numerically • These equations are further generalization of the well Kolmogorov’s forward equation • There is a natural nesting of problems and solutions: • Liouville < Kolmogorov < Kushner-Zakai

14. Nonlinear filtering • Kushner (1962) “On the differential equations staisfied by the conditional probability densities of Markov processes with applications”, SIAM Journal on Control, vol 2, pp 106-119 • Zakai, M (1969) “On the optimal filtering of diffusion processes”, Warsch. Und Ver. Gebiete, vol 11, 230-243 • There is a growing literature on the stochastic PDE today • P.L Chow (2007) Stochastic Partial Differential Equations, Chapman Hall/CRC Press, Boca Raton, FL

15. Classical approximations • Difficulty with solving Kushner-Zakai type equations forced us to look for easily computable moments ofthe state probability density, pt(xt) • But again the moment closure problem forces us to settle for moment approximation • First-order (extended Kalman) filter approximates the mean and covariance of xt using the first derivative term in the Taylor series of M(xt) around the known analysis • Second-order filter – improved approximation using the first two derivatives in the Taylor series • Since the 1960’s, these approximations have been applied to solve problems in Aerospace and Economics with considerable success • Applications of Kalman filtering in Meteorology began only early 1980’s

16. Ensemble Kalman filtering: an alternative to classical moment approximation • In meteorology, the problems sizes are rather large: n = a few tens of millions and m = a few millions • Curse of dimensionality prevents application of the classical moment approximations • Computation of the forecast and its covariance (resulting from n3 complexity) is very time consuming • To circumvent this difficulty, the notion of the (Monte Carlo type) ensemblefiltering was introduced by G. Evensen (1994) in Meteorology • There has been an explosion of literature on exsemble Kalman filtering in Meteorology • G. Evensen (2007) Data Assimilation: The ensembleKalman filter, Springer Verlag 279 pages

17. A Historical note • Monte-Carlo type filtering has been around within the systems science community for quite some times • Hanschin, J. E. and D. Q. Mayne (1969) Monte Carlo techniques to estimate the conditional expectation in multi-stage nonlinear filtering, International Journal ofControl, vol 9, pp 547-559 • Tanizaki, H. (1996) Nonlinear filters: estimationand applications, Springer Verlag, NY (second edition)

18. Modern era: Beyond Kalman filtering • In the extended Kalman, we need to compute the Jacobian of the model map M(x) which is an n*n matrix of partial derivatives of the components of M(x) • In the second order model we would need the Hessian (second partial-derivatives) of the components of M(x) • If the model size is large, this may be very cumbersome • Since the mid 1990’s, interest in the derivative free filtering came into prominence

19. Unscented Kalman filtering • This is based on the use of deterministic ensemble – either a symmetric ensemble of size (2n+1) or asymmetric ensemble of size (n+1) • Using this ensemble, moments of the model forecast can be computed up to third and/or fourth order accuracy • These are better than first and second order filters described earlier • This filters may be very useful for low dimensional ecological models • Julier, S., J. Uhlmann and H.F. Durrant-Whyte (2002) A new emthod for the nonlinear trasformation of mean and covariance in filters and estimators, IEEE Transactionson Automatic Control, vol 45, 477-482

20. Particle filters: a return to Monte Carlo • By using very clever sampling techniques, this class draws samples directly from a representation of the posterior distribution with out actually computing the posterior distributions. • This is based on the idea of using a proposal density in place of the actual posterior density • There are different versions of this algorithms • Markov Chain Monte Carlo algorithm is well known to this community

21. Particle filters • Doucet, A, N. de Freitas and N. Gordon (Editors) (2001) Sequential Monte Carlo Methods inPractice, Springer • Ristic, B, S. Arulambalam, and N. Gordon (2004) Beyond KalmanFilter: Particle Filters and Tracking Applications, Arch Tech House, Boston

22. Reference • J. M. Lewis, S. Lakshmivarahan and S. K. Dhall (2006) Dynamic Data Assimilation, Cambridge University Press • This book grew out of teaching graduate level courses at OU and provides a comprehensive introduction to all aspects of Data assimilation • Classical retrieval algorithms • 3D Var based on Bayesian frame work • 4D Var – Adjoint methods • Linear and nonlinear filtering • Ensemble/ reduced rank filters • Predictability – deterministic and stochastic aspects

23. Research and educational mission • Offer on-site intense short courses on specific aspects • Conduct hands on workshops • Work closely with researchers in this area in pushing the frontiers of the science