A variance limiting Kalman filter for data assimilation: I. Sparse observational grids II. Model error

Transcription

1 A variance limiting Kalman filter for data assimilation: I. Sparse observational grids II. Model error Georg Gottwald, Lewis Mitchell, Sebastian Reich* University of Sydney, *Universität Potsdam Durham, August 6th, 2011

2 Observational Data Forecast Model Data Assimilation

3 Observational Data Forecast Model Data Assimilation Climatological Information

4 I. Sparse observational grids Typically only a very small number of observations O(10 4 )/O(10 7 ) are available compared to the number of gridpoints for the model O(10 9 ) large unobserved regions!

5 I. Sparse observational grids Typically only a very small number of observations O(10 4 )/O(10 7 ) are available compared to the number of gridpoints for the model O(10 9 ) large unobserved regions!

6 I. Sparse observational grids Typically only a very small number of observations O(10 4 )/O(10 7 ) are available compared to the number of gridpoints for the model O(10 9 ) large unobserved regions!

7 Sparse observational grids What is the effect of the sparsity of observations? The obvious: We don t have much information Overestimation of error covariances (exacerbated by finite ensemble sizes) (Whitaker et al. 2009) The subtle: We create spurious correlations and unbalanced flow (Kepert 2009)

8 Sparse observational grids What is the effect of the sparsity of observations? The obvious: We don t have much information Overestimation of error covariances (exacerbated by finite ensemble sizes) (Whitaker et al. 2009) The subtle: We create spurious correlations and unbalanced flow (Kepert 2009) Applications are sparse observational networks re-analysis of climate when direct observations are not available (mesosphere) slow-fast systems

9 Sparse observational grids Our particular perspective here: Proper (noisy) observations are available for some variables (observables) but not for other unresolved variables, for which only their statistical climatic behaviour such as their variance and their mean is available (pseudo-observables). Question: How can the statistical information available for some data which are otherwise not observable, be effectively incorporated into data assimilation to control overestimation?

10 Setting Assume an N-dimensional dynamical system whose dynamics is given by ż = f(z) with the state variable z R N (no model error for now).

11 Setting Assume an N-dimensional dynamical system whose dynamics is given by ż = f(z) with the state variable z R N (no model error for now). Assume that the state space is decomposable according to z = (x,y) with observables x R n and pseudo-observables y R m and n + m = N.

12 observables Observations x obs at observation times t n = n t obs observation operator H : R N R n x obs (t i ) = Hz(t i ) + r obs (t i ) with observational noise r obs r obs N(0,R obs ) with error covariance matrix R obs

13 observables Observations x obs at observation times t n = n t obs observation operator H : R N R n x obs (t i ) = Hz(t i ) + r obs (t i ) with observational noise r obs r obs N(0,R obs ) with error covariance matrix R obs pseudo-observables Assume climatic knowledge about the pseudo-observables y (mean a target and variance A target ) pseudo-observation operator h : R N R m R w is the unknown error covariance matrix associated with the pseudo-observables

14 observables Observations x obs at observation times t n = n t obs observation operator H : R N R n x obs (t i ) = Hz(t i ) + r obs (t i ) with observational noise r obs r obs N(0,R obs ) with error covariance matrix R obs pseudo-observables Assume climatic knowledge about the pseudo-observables y (mean a target and variance A target ) pseudo-observation operator h : R N R m R w is the unknown error covariance matrix associated with the pseudo-observables Question: How do we choose/find the error covariance matrix R w?

15 observables Observations x obs at observation times t n = n t obs observation operator H : R N R n x obs (t i ) = Hz(t i ) + r obs (t i ) with observational noise r obs r obs N(0,R obs ) with error covariance matrix R obs pseudo-observables Assume climatic knowledge about the pseudo-observables y (mean a target and variance A target ) pseudo-observation operator h : R N R m R w is the unknown error covariance matrix associated with the pseudo-observables Question: How do we choose/find the error covariance matrix R w? (The naive first guess R w = A target is wrong)

16 The Variance Limiting Kalman Filter (VLKF) An ensemble (Evensen, 1996) with k members z k Z = [z 1,z 2,...,z k ] R N k is propagated by the full nonlinear dynamics Ż = F(Z), Z(0) = Z b. The ensemble is split into its mean z and its ensemble deviation matrix Z Step 1: Forecast step Z f = F(Z b ) P f = 1 k 1 Z f (t)[z f (t)]t Remark: P f (t) is rank-deficient for k < N (N 10 9 and k 100)

17 The Variance Limiting Kalman Filter (VLKF) Step 2: Analysis step: Minimise the cost function S(z) = 1 2 (z z f) T P 1 f (z z f) (x obs Hz) T R 1 obs (x obs Hz) (a target hz) T R 1 w (a target hz)

18 The Variance Limiting Kalman Filter (VLKF) Step 2: Analysis step: Minimise the cost function S(z) = 1 2 (z z f) T P 1 f (z z f) (x obs Hz) T R 1 obs (x obs Hz) (a target hz) T R 1 w (a target hz) z a = z f K obs [H z f x obs ] K w [h z f a target ] where K obs = P a H T R 1 obs K w = P a h T R 1 w with the covariance of the analysis P a = ( ) 1 P 1 f + H T R 1 obs H + ht R 1 w h

19 The Variance Limiting Kalman Filter (VLKF) Step 2: Analysis step Constraining the variance of the pseudo-observable hz is done by requiring Introducing P a 1 = P 1 f hp a h T = A target + H T R 1 obsh, we obtain R 1 w = A 1 target ( hp a h T) 1

20 The Variance Limiting Kalman Filter (VLKF) Step 2: Analysis step Constraining the variance of the pseudo-observable hz is done by requiring Introducing P a 1 = P 1 f hp a h T = A target + H T R 1 obsh, we obtain R 1 w = A 1 target ( hp a h T) 1 The naive expectation R w = A target is true only for {R obs,p f } A target

21 The Variance Limiting Kalman Filter (VLKF) Step 2: Analysis step Constraining the variance of the pseudo-observable hz is done by requiring Introducing P a 1 = P 1 f hp a h T = A target + H T R 1 obsh, we obtain R 1 w = A 1 target ( hp a h T) 1 The naive expectation R w = A target is true only for {R obs,p f } A target For sufficiently small background error covariance P f, the error covariance R w is not positive definite ( switch ): Update only overestimated eigendirections with hp a h T > A target

22 The Variance Limiting Kalman Filter (VLKF) Step 3: Update of the ensemble The ensemble needs to be consistent with P a = 1 [ ] k 1 Z a Z T a Method of ensemble square root filters: Ensemble transform Kalman filter (EnTKF) (Tippett et al 2003): Z a = Z fs with S Rk k Ensemble adjustment Kalman filter (EnAKF) (Anderson 2001): Z a = AZ f with A RN N

23 The Variance Limiting Kalman Filter (VLKF) Step 3: Update of the ensemble The ensemble needs to be consistent with P a = 1 [ ] k 1 Z a Z T a Method of ensemble square root filters: Ensemble transform Kalman filter (EnTKF) (Tippett et al 2003): Z a = Z fs with S Rk k Ensemble adjustment Kalman filter (EnAKF) (Anderson 2001): Z a = AZ f with A RN N Step 4: Update of the forecast Set Z b = Z a to propagate the ensemble forward again with the full dynamics to the next observation time

24 Summary of VLKF Step 1: Forecast step Z f = F(Z b ) P f = 1 k 1 Z f(t)[z f(t)] T Step 2: Analysis step K obs = P ah T R 1 obs, Step 3: Update of the ensemble z a = z f + K obs (x obs H z f ) + K w(a target h z f ) K w = P ah T R 1 w, P a = The ensemble needs to be consistent with Step 4: Update of the forecast R 1 w = A 1 target (hp ah T ) 1 P a = 1 k 1 Z a ˆZ a T P 1 f + H T R 1 obsh +h T R 1 w h Set Z b = Z a to propagate the ensemble forward again with the full dynamics to the next observation time. 1

25 Analytical linear toy model Consider the system of coupled linear oscillators z = (x,y) with x R 2, y R 2 dz = Mzdt Γzdt + S dw t + Czdt with S = «ωxj 0 M = 0 ω yj «C = σxi 0 0 σ yi 0 λj 0 0 «γxi 0 Γ = 0 γ yi «J = «. Introducing the propagator L(t) = exp ((M Γ + C)t), the solution can be obtained using Itô s formula z(t) = L(t)z 0 + S t 0 L(t s)dw s

26 Analytical linear toy model We can calculate the mean m(t) = L(t)z 0, and covariance Σ(t) = S (2Γ C) 1 (I exp( (2Γ C) t)) S T, where C = 0 λj λj 0 «. The climatic mean m clim R 4 and covariance matrix Σ clim R 4 4 are then obtained in the limit of t as and m clim = lim t m(t) = 0, Σ clim = lim t Σ(t) = S (2Γ C) 1 S T. Remark: The coupling has to be sufficiently small with λ 2 < 4γ x γ y.

27 Analytical linear toy model We will investigate the variance constrained Kalman filter for this toy model: Under what conditions is R w positive definite and the variance constraint will be switched on? When does the VLKF yield skill improvement compared to the standard ETKF?

28 Analytical linear toy model We will investigate the variance constrained Kalman filter for this toy model: Under what conditions is R w positive definite and the variance constraint will be switched on? When does the VLKF yield skill improvement compared to the standard ETKF? 1. For an ensemble the covariance of the forecast is calculated by averaging over realizations of the Brownian motion and over the ensemble P f (t i+1 ) = L( t obs )P a (t i )L T ( t obs ) + Σ( t obs ) Introduce filter inflation δ 1 Restrict to small observation times t obs 1 Then P a (t i ) P f (t i+1 )

29 Analytical linear toy model t obs (δ) > δ(λ2 4γ xγ y) + 4γ xγ y 2γ x(1 + γ 2 y) t obs For δ > 1 we can have t obs < 0 (4γ xγ y λ 2 > 0) t obs (δ = 1) > t obs (δ > 1). R 1 w / t obs > 0 at t obs = 0. For λ 1 or γ x 1, our expression for t obs (δ) may not be consistent with the assumption of small observation times t obs 1 For t obs and large R obs, we have P f Σ clim and the variance constraint should not be switched on, but in numerical simulations it is?

30 Analytical linear toy model Finite size effect (no inflation): R obs = 0.25 R obs = 2 max(svd(hpah T )) k

31 Analytical linear toy model 2. When does the VLKF yield skill improvement when compared to the standard ETKF? Filter skill E = E t,dw z a(t i) z truth (t i) 2 G E t denotes temporal average over analyzes cycles, and averaging over Brownian paths. The norm ab G = a T Gb can be employed with G = I for overall skill, G = H T H for the observables only, and G = h T h for the pseudo-observables only. E ETKF = E t (I K obs H)L( t obs ) ξ ti 1 2 G + E t (I K obs H)η ti 2 G + E t K obs r obs 2 G E VLKF = E t (I K obs H)L( t obs ) ξ ti 1 2 G + E t (I K obs H)η ti 2 G + E t K obs r obs 2 G with the mutually independent, normally distributed random variables ξ ti = z a(t i) z truth (t i) N(0,P a(t i)) η ti = S Z ti r obs N(0,R obs ) t i 1 L( t obs s) dw s N(0,Σ( t obs ))

32 Analytical linear toy model Skill improvement for the pseudo-observables S = E ETKF /E VLKF Remarks: S > 1 for either γ y or γ x 0 Suggests that the skill is controlled by the ratio of the time scales of the observed and the unobserved variables S/ R obs > 0 at R obs = 0 (effective slowed down relaxation towards equilibrium of the observed variables) S/ δ > S γ y S γ y

33 Lorenz-96 model Consider z R N (typically N = 40): dz i dt = z i 1 (z i+1 z i 2 ) z i + F i = 1,,N z i±n = z i This is a paradigmatic model for the midlatitude atmosphere: has forcing F has linear damping non-linear terms conserve the energy 1 2 i z i 2

34 Lorenz-96 model Consider a latitudinal ring in the midlatitudes with a circumference of roughly 30,000 km. At those latitudes the doubling time is roughly days: For N = 40 and F = 8: 1 time unit 5 days distance between observation stations: 750 km 30, 000/40 km observation stations z i

35 Lorenz-96 model Instead of 40 observations z z 1 z 2 z3 z 39 z z 4 z 37 z5 z 36 z 34 z 35 z 33 z 6z7 z 32 z 31 z 30 z 29 z 28 z 27 z 15 z 26 z 16 z 25 z 17 z 24 z z z 18 23z22 z z20 z 8 z 9 z 10 z 11 z 12 z 13 z 14

36 Lorenz-96 model Observe only every N obs = 5 component z 40 z 5 z 35 z 10 z 30 z 15 z 25 z 20

37 Lorenz-96 model The pseudo-observables contain the prior climatic knowledge: a target = µ clim and A target = σ 2 clim I with µ clim = 2.34 and σ clim = 3.6 measured from a long time trajectory N obs = 6, t obs = 4 hours, R obs = (0.25σ clim ) 2 I z t (days) z t (days) ETKF

38 Lorenz-96 model The pseudo-observables contain the prior climatic knowledge: a target = µ clim and A target = σ 2 clim I with µ clim = 2.34 and σ clim = 3.6 measured from a long time trajectory N obs = 6, t obs = 4 hours, R obs = (0.25σ clim ) 2 I z t (days) z t (days) z t (days) ETKF z t (days) VLKF

39 Lorenz-96 model Quantify the skill improvement by the r.m.s error E = 1 T z a (l t obs ) z truth (l t obs ) TD 2 l=1 R obs = (0.25σ clim ) 2 I Best performance of VLKF over ETKF for: small t obs N obs = 4 S 2.6 N = 3 obs N = 4 obs t obs N = 5 obs N = 6 obs

40 Lorenz-96 model How is the skill distributed over S N = 3 obs N = 4 obs N = 5 obs N = 6 obs S 1.8 N obs = 3 N = 4 obs N obs = 5 N = 6 obs ε t obs observables N = 3 obs 1 N = 4 obs N = 5 obs N obs = 6 ε t obs pseudo-observables 10 N = 3 obs N = 4 obs N = 5 obs N = 6 obs t obs t obs

41 Lorenz-96 model There is an order of magnitude difference between the RMS errors for the observables and the pseudo-observables for large N obs. This suggests that the information of the observed variables does not travel too far away from the observational sites Total RMS error for each site i, i = 1,2,,40 if only site i = 21 is observed. RMS i Remark:The advective time scale of the Lorenz-96 system is much smaller than t obs which explains why the skill is not equally distributed over the sites, and why, especially for large values of N obs we observe a big difference between the site-averaged skills of the observed and unobserved variables.

42 Lorenz-96 model Dependency on observational noise level R obs = (η σ clim ) 2 I, N obs = E E E η η t obs = t obs = 0.05 t obs = 0.25 (1 hour) (2 hours) (5 hours) η Red: Observations Green: ETKF Blue: VLKF

43 Lorenz-96 model VLKF produces significant skill in sparse observational grids for small observation intervals (< 6 hours) the larger the observational noise the better As before, the increased skill is a finite size effect: 14 max(svd(hpah T )) k

44 Lorenz-96 model: Filter divergence and blow-up (Harlim & Majda (2010)) for small R obs ETKF N obs x x h 6 h 9 h 12 h 15 h 18 h 21 h 24 h 27 h 30 h τ obs VLKF N obs h 6 h 9 h 12 h 15 h 18 h 21 h 24 h 27 h 30 h τ obs

45 Lorenz-96 model: Filter divergence and blow-up (Harlim & Majda (2010)) for small R obs ETKF N obs x x h 6 h 9 h 12 h 15 h 18 h 21 h 24 h 27 h 30 h τ obs VLKF N obs h 6 h 9 h 12 h 15 h 18 h 21 h 24 h 27 h 30 h τ obs (GAG, Mitchell & Reich, MWR 2011, in press)

46 II. Model Error Numerical codes tend to overestimate noise at the grid resolution. This is typically controlled via artificial viscosity. For example, to control unwanted gravity wave activity severe divergence damping is introduced to the equations of motion to stabilize the numerical scheme causing an underestimation of the error covariances (Durran 1999). Sometimes we want conservation properties (J. Thuburn 2008). Can one use numerical forecast models which are not artificially damped and control the resulting overestimation of the forecast covariance within the data assimilation procedure? dz i dt = z i 1(z i+1 z i 2 ) γz i + F Truth: γ = 1 Forecast model: γ < 1 Now we will be interested in the case of t obs 1

47 Model Error z1 5 0 t obs = 24 hours γ = 0.5: 10 5 z1 0 t obs = 48 hours z Reproducing the truth t z Reproducing the statistics t

48 Model Error z1 5 0 t obs = 24 hours γ = 0.5: 10 5 z1 0 t obs = 48 hours z Reproducing the truth t z Reproducing the statistics t z a γ ( za za ) γ

49 Model Error P f : 15 s H s h N obs N obs P a : 0.75 s H s h N obs N obs γ = 1 γ = 0.9 γ = 0.8 ( t obs = 48hrs) continuous: ETKF dashed: VLKF

50 Model Error P f : 15 s H s h N obs N obs P a : 0.75 s H s h N obs N obs γ = 1 γ = 0.9 γ = 0.8 ( t obs = 48hrs) continuous: ETKF dashed: VLKF increased sparsity and model error lead to overestimation covariances of observables are also limited for VLKF

51 Model Error We consider performance over standard ETKF S = EE E V, and over using the poor man s analysis of observations and climatology S Ŝ E = Ê E E, Ŝ E ŜV = Ê E V Ŝ V γ γ γ t obs = 24hrs t obs = 36hrs t obs = 48hrs

52 Model Error We consider performance over standard ETKF S = EE E V, and over using the poor man s analysis of observations and climatology S Ŝ E = Ê E E, Ŝ E ŜV = Ê E V Ŝ V γ γ γ t obs = 24hrs t obs = 36hrs t obs = 48hrs Trade-off: The smaller γ, the better skill over ETKF, but the less skill compared with poor man s analysis

53 Model Error How is the RMS error distributed over the observables and the pseudo-observables ( t obs = 48hrs)? E o γ γ γ E uo γ γ γ N obs = 3 N obs = 4 N obs = 8 (ETKF/VLKF)

54 Summary and outlook We have here derived a variance limiting Kalman filter (VLKF) which adaptively damps unrealistic excitation of ensemble spread in underresolved regions applied this filter to a sparse observational grid has better skill than ETKF for small ( 6h) observation times has better skill for observables and pseudo-observables is stabilizing and avoids filter divergencies such as blow-up is robust to incomplete knowledge of the climatic mean and variance applied this filter to model error (underdamping) has better skill than ETKF for large observation intervals ( 36h) has worse or equal skill for observed variables trade-off between superior skill over ETKF and being better than observations/climatology We will apply this filter to slow-fast systems where fast degrees of freedom do not need to be tracking investigate blow-up further

55 Model Error Dependency of skill on sparsity N obs 4 4 VLKF E ETKF N obs N obs

56 Model Error Dependency of skill on noise error R obs = (ησ clim ) 2 I 1.15 S γ Ŝ E Ŝ V γ γ