Hw d we slve it and what des the slutin l lie? KF/PFs ffer slutins t dynamical systems, nnlinear in general, using predictin and update as data becmes available. Tracing in time r space ffers an ideal framewr fr studying KF/PF. The Mdel Cnsider the discrete, linear system, where x +1 = M x + w, = 0, 1,,..., (1 x R n is the state vectr at time t M R n n is the state transitin matrix (mapping frm time t t t +1 r mdel {w R n ; = 0, 1,,...} is a white, Gaussian sequence, with w N(0, Q, ften referred t as mdel errr Q R n n is a symmetric psitive definite cvariance matrix (nwn as the mdel errr cvariance matrix. 4 f 3 Sme f the fllwing slides are frm: Sarah Dance, University f Reading
The Observatins We als have discrete, linear bservatins that satisfy y = H x + v, = 1,, 3,..., ( where y R p is the vectr f actual measurements r bservatins at time t H R n p is the bservatin peratr. Nte that this is nt in general a square matrix. {v R p ; = 1,,...} is a white, Gaussian sequence, with v N(0, R, ften referred t as bservatin errr. R R p p is a symmetric psitive definite cvariance matrix (nwn as the bservatin errr cvariance matrix. We assume that the initial state, x 0 and the nise vectrs at each step, {w }, {v }, are assumed mutually independent. 5 f 3 Summary f the Kalman filter Predictin step Mean update: bx +1 = M bx Cvariance update: P +1 = M P M T + Q. Observatin update step Mean update: bx = bx 1 + K (y H bx 1 Kalman gain: K = P 1 H T (H P 1 H T + R 1 Cvariance update: P =(I K H P 1. 16 f 3
CAGLARS HOMEWORK Predictin step We first derive the equatin fr ne-step predictin f the mean using the state prpagatin mdel (1. mesurement x +1 = Mx + Bw x +1 = E[x +1 y 1,..., y ] = Mˆx 9 f 3 bx +1 = E [x +1 y 1,...y ], +1 = E [M x + w ], = M bx (5 CAGLARS HOMEWORK The ne step predictin f the cvariance is defined by, i P +1 = E h(x +1 bx +1 (x +1 bx +1 T y 1,...y. (6 Exercise: Using the state prpagatin mdel, (1, and ne-step predictin f the mean, (5, shw that P +1 = M P M T + Q. (7 mesurement x +1 = Mx + Bw P +1 = E[(x +1 ˆx +1 (x +1 ˆx +1 T y 1,..., y ] = MP M T + BQB T 10 f 3
Bayesian Framewr m : mdel parameter vectr (unnwn parameters t be estimated d : data vectr relating t m via an equatin h(. d = h(m + nise Classical parameter estimatin framewr: Unnwn but deterministic m Bayesian parameter estimatin framewr: Unnwn and randm variable m Bayes Frmula p(m, d = p(m dp(d = p(d mp(m POSTERIOR LIKELIHOOD PRIOR p(m d = p(d m p(m p(d = p(d m p(m p(d m p(mdm EVIDENCE Bayes p(m d p(d mp(m Sequential updates Cnsider d cnsisting f tw independent data set p(d 1, d = p(d 1 p(d p(m d = p(m d 1, d = p(d d 1, mp(m d 1 p(d d 1 = p(d d 1, m p(d p(d 1 mp(m p(d 1 p(d mp(d 1 mp(m Generalizing p(m d N p(d i mp(m i=1 Thus, in principle with n measurement equatin, yu can update sequentially r just at nce
Inversin, Filtering and Smthing p(x t y t : Inversin, Only bservatins at time t p(x t y 1:t : Filter, Observatins frm time 1: t p(x t y 1:T : Smther, Observatins frm time 1: T x y = f 1( x 1, v = h ( x, w ˆx x -1 x - previus states A Single Kalman Iteratin x x = F 1x 1 y = H x + w ~ N (ˆ x, P p( x x 1, x!, x0 y, y 1, y p( x! 1 + v ˆx previus states ˆ x -1 x - x 1 1. Predict the mean ˆ using previus histry. p( x x 1 x 1 { x x 1} x p( x x dx xˆ 1 E = 1 = P 1. Predict the cvariance using previus histry. 3. Crrect/update the mean using new data y p( x Y xˆ { x Y } = x p( x Y dx = E P 4. Crrect/update the cvariance using y PREDICT UPDATE! p( x 1 Y 1 p( x Y 1 p( x Y! PREDICTOR-CORRECTOR DENSITY PROPAGATOR
Peter s slide frm last wee Prduct(f(Gaussians=Gaussian:( One data pint prblem Fr the general linear inverse prblem we wuld have ½ p(m exp 1 ¾ Prir: (m mt Cm 1 (m m ½ p(d m exp 1 ¾ Lielihd: (d GmT Cd 1 (d Gm Psterir PDF ½ exp 1 ¾ [(d GmT Cd 1 (d Gm+(mmT Cm 1 (m m] # exp$ 1 % m ˆm S 1 = G T C d 1 G + C m 1 ˆm = G T C d 1 G + C m 1 [ ] T S 1 [ m ˆm ] & ' ( ( 1 ( G T C 1 d d + C 1 m m 0 = m 0 + G T C 1 1 ( d G + C m 1 G T C 1 d d Gm 0 60 ( DATA ASSIMILATION
Basic estimatin thery Observatin: T 0 = T + e 0! First guess: T m = T + e m! E{e 0 } = 0! E{e m } = 0! E{e 0 } = s 0! E{e m } = s m! E{e 0 e m } = 0! Assume a linear best estimate: T n = a T 0 + b T m! with T n = T + e n.! Find a and b such that! 1 E{e n } = 0 E{e n } minimal!! 1 Gives: E{e n } = E{T n -T} = E{aT 0 +bt m -T} =! = E{ae 0 +be m + (a+b-1 T} = (a+b-1 T = 0! Hence b=1-a.! Basic estimatin thery E{e n } minimal gives:!! E{e n } = E{(T n T } = E{(aT 0 + bt m T } =! = E{(ae 0 +be m } = a E{e 0 } + b E{e m } =! = a s 0 + (1-a s m! This has t be minimal, s the derivative wrt a has t be zer:! a s 0 - (1-a s m = 0, s (s 0 + s m a s m = 0, hence:!! a = s m! and! s 0 + s m! b = 1-a = s 0!! s 0 + s m! s n = E{e n } = s 4 m s 0 + s 4 0 s m! = s 0 s m!! (s 0 + s m! s 0 + s m!
Basic estimatin thery Slutin: T n = s m! T 0 + s 0! T m! s 0 + s m! s 0 + s m! and! 1! = 1! +! 1! s n! s 0! s m! Nte: s n smaller than s 0 and s m!! Best Linear Unbiased Estimate BLUE! Just least squares!!!! Tutrial Lecture: Data Assimilatin Adrian Sandu Cmputatinal Science Labratry Virginia Plytechnic Institute and State University
Data assimilatin fuses infrmatin frm (1 prir, ( mdel, (3 bservatins t btain cnsistent descriptin f a physical system Chemical inetics Transprt Meterlgy Optimal analysis state CTM Data Assimilatin Observatins Aersls Emissins Targeted Observ. Imprved: frecasts science field experiment design mdels emissin estimates Sep. 7, 011. SAMSI UQ Methdlgy Wrshp. Surce f infrmatin #1: the prir encapsulates ur current nwledge abut the state f the system Sep. 7, 011. SAMSI UQ Methdlgy Wrshp.
Crrect mdels f bacgrund (prir errrs are very imprtant fr data assimilatin Bacgrund errr representatin determines the spread f infrmatin, and impacts the assimilatin results Needs: high ran, capture dynamic dependencies, efficient cmputatins Traditinally estimated empirically (NMC, Hllingswrth-Lnnberg 1. Tensr prducts f 1d crrelatins, decreasing with distance (Singh et al, 010. Multilateral AR mdel (Cnstantinescu et al 007 3. Hybrid methds in the cntext f 4D-Var (Cheng et al, 009 [Cnstantinescu and Sandu, 007] Sep. 7, 011. SAMSI UQ Methdlgy Wrshp. Surce f infrmatin #: the mdel encapsulates ur nwledge abut the physical laws that gvern the evlutin f the system Hw large are the mdels f interest? Typically O(10 8 variables, and O(10 different physical prcesses Picture: L. Isasen (http://www.ecmwf.int Sep. 7, 011. SAMSI UQ Methdlgy Wrshp.
Surce f infrmatin #3: the bservatins are sparse and nisy snapshts f reality Hw many bservatins? ECMWF: O(10 7 Sep. 7, 011. SAMSI UQ Methdlgy Wrshp. Lars Isasen (http://www.ecmwf.int Sme cnventinal and remte data surces used at ECMWF fr numerical weather predictin Lars Isasen (http://www.ecmwf.int SYNOP/METAR/SHIP: pres., wind, RH Aircraft: wind, temperature 13 Sunders: NOAA AMSU-A/B, HIRS, AIRS, Gestatinary, 4 IR and 5 winds Sep. 7, 011. SAMSI UQ Methdlgy Wrshp.
T allw mdel-data cmparisn, bservatin peratrs map the mdel state space t bservatin space Mdel T and q H Mdel Radiance J cmpare Observatin Satellite Radiance Sep. 7, 011. SAMSI UQ Methdlgy Wrshp. Lars Isasen (http://www.ecmwf.int Result f DA: the analysis, which encapsulates ur enhanced nwledge abut the state f the system Sep. 7, 011. SAMSI UQ Methdlgy Wrshp.
Result f DA: the analysis, which encapsulates ur enhanced nwledge abut the state f the system Sep. 7, 011. SAMSI UQ Methdlgy Wrshp. Result f DA: the analysis, which encapsulates ur enhanced nwledge abut the state f the system Sep. 7, 011. SAMSI UQ Methdlgy Wrshp.
The ensemble Kalman filter (EnKF is based n EKF, and uses a MC apprach t prpagate cvariances [Picture frm J.L. Andersn] x M x η bi ( ai ( ( i 1 t t 1 1 ( ( T b 1 K b i b b i b K i 1 P x x x x Sequential apprach t DA: Incrprates data in successin Sep. 7, 011. SAMSI UQ Methdlgy Wrshp. The ensemble Kalman filter (EnKF is based n EKF, and uses a MC apprach t prpagate cvariances [Picture frm J.L. Andersn] b T 1 K b( i b b( i b T H H K i 1 b T 1 K b( i b b( i b K i 1 H H H H PH x x x x HPH x x x x Sep. 7, 011. SAMSI UQ Methdlgy Wrshp. T
The ensemble Kalman filter (EnKF is based n EKF, and uses a MC apprach t prpagate cvariances [Picture frm J.L. Andersn] x x K y H x a b b Sep. 7, 011. SAMSI UQ Methdlgy Wrshp. The ensemble Kalman filter (EnKF is based n EKF, and uses a MC apprach t prpagate cvariances [Picture frm J.L. Andersn] x x K y H x a b b Sep. 7, 011. SAMSI UQ Methdlgy Wrshp.
The ensemble Kalman filter (EnKF is based n EKF, and uses a MC apprach t prpagate cvariances [Picture frm J.L. Andersn] Sep. 7, 011. SAMSI UQ Methdlgy Wrshp.