Dynamic Data Assimilation: A historical view

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

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### Dynamic Data Assimilation:A historical view

S. Lakshmivarahan

School of Computer Science

University of Oklahoma

Norman

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
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
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)
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
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
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
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
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
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
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]
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
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
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
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
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
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)
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
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
• 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
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
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
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