Dynamic Data Assimilation:A historical view

S. LakshmivarahanSchool of Computer Science

University of OklahomaNorman

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 the Liouville’s equation in 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 and smoothing of stationary time series with engineering applications, 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 of Sciences, 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 Basic Engineering, 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 of the 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) ensemble filtering 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 ensemble Kalman 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 of Control, vol 9, pp 547-559

• Tanizaki, H. (1996) Nonlinear filters: estimation and 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 Transactions on Automatic Control, vol 45, 477-482

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

Particle filters

• Doucet, A, N. de Freitas and N. Gordon (Editors) (2001) Sequential Monte Carlo Methods in Practice, Springer

• Ristic, B, S. Arulambalam, and N. Gordon (2004) Beyond Kalman Filter: 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