Nonlinear error dynamics for cycled data assimilation methods

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Nonlinear error dynamics for cycled data assimilation methods A J F Moodey 1, A S Lawless 1,2, P J van Leeuwen 2, R W E Potthast 1,3 1 Department of Mathematics and Statistics, University of Reading, UK. 2 Department of Meteorology, University of Reading, UK. 3 Deutscher Wetterdienst (DWD), Offenbach am Main, Germany. January 15, 2013

Outline 1 Bacground 2 Data assimilation 3 Nonlinear error evolution 4 Numerical example 5 Final remars

Outline 1 Bacground 2 Data assimilation 3 Nonlinear error evolution 4 Numerical example 5 Final remars

Dynamical system Dynamical system: nonlinear dynamical flow and a linear measurement equation. Let M (t) : X X and M : X X be a true nonlinear model operator and a modelled nonlinear model operator respectively. All operators are mapping a state x (t) 1 X discretely onto its state x (t) for N 0 for the Hilbert space (X, X ). The operator M : X X is modelled, such that M ( x (t) ) = M (t) ( x (t) ) +ζ +1, (1) where ζ +1 is some additive noise which we call model error and is bounded by some constant υ > 0 for all time t for N 0.

Observations Let H be an injective linear time-invariant compact observation operator, such that H : X Y, for Hilbert spaces (X, X ) and (Y, Y ). Let y (t) Y be the true observations (measurements) located at discrete times t linearly, such that y (t) = H (t) x (t) = Hx (t) = y η, (2) where η is some additive noise that we call the observation error and is bounded by some constant δ > 0 for all time t for N 0. It is possible to carry through the same analysis using a noise term on H modelled, ( H H (t)) x (t) = ω. (3)

A couple of definitions Definition (p.10 in Kress 1999) An inner product space, which is complete with respect to the norm x := x,x 1/2, (4) for all x X, is called a Hilbert space. Definition (Definition 7.1 in Rynne and Youngson 2007) Let X and Y be normed space. An operator H L(X,Y) is compact if, for any bounded sequence (x n ) in X, the sequence (Hx n ) in Y contains a convergent subsequence.

Aim Develop theory demonstrating the asymptotic stability of a cycled data assimilation scheme with an ill-posed observation operator in a nonlinear infinite dimensional setting. Wor within the framewor of data assimilation scheme that employ static covariances, such as three dimensional variational data assimilation (3DVar). Here we extend previous linear results from R W E Potthast, A J F Moodey, A S Lawless and P J van Leeuwen 2012.

Stability We define the analysis error as the difference between the analysis and the true state of the system, such that e := x (a) x (t), (5) where x (t) represents the true state of the system at time t for N 0. We will call a data assimilation scheme stable if given some constant C > 0 e X C (6) as with some appropriate norm X. The analysis error in other fields is nown as state reconstruction error, observer error, estimation error, etc.

Outline 1 Bacground 2 Data assimilation 3 Nonlinear error evolution 4 Numerical example 5 Final remars

Operator equation Mathematically, we interpret the data assimilation tas as seeing x (a) at every assimilation step t, N 0 to solve an operator equation of the first ind Hx (a) = y. (7) This operator equation represents a Fredholm integral equation of the first ind and is ill-posed when the dimension of the state space X is infinite.

Ill-posedness Hadamard defined that a well-posed problem must satisfy: 1 There exists a solution to the problem (existence). 2 There is at most one solution to the problem (uniqueness). 3 The solution depends continuously on the data (stability). If a problem does not satisfy all three of these condition, it is ill-posed in the sense of Hadamard. Nashed defined that an operator equation is called well-posed if 1 the set of observations is a closed set, that is if H(X) is closed. If a problem does not satisfy this property, then it is ill-posed in the sense of Nashed (Nashed 1981, Nashed 1987).

Ill-posed observation operator Theorem (Theorem 15.4 in Kress 1999) Let X and Y be normed spaces and H L(X,Y) be a compact operator. If X has infinite dimension then H cannot have a bounded inverse and the operator equation of the first ind is ill-posed. Regularization methods exist to provide a stable approximate solution to the ill-posed problem. Tihonov-Phillips regularization shifts the the eigenvalues of the operator H H by a regularization parameter α, where H is the adjoint to the operator H.

Cycled Tihonov-Phillips regularization Definition Given measurements y Y for N 0 and an initial guess x (b), the objective of cycled Tihonov-Phillips regularization is to see an estimate, x (a) that minimises the functional, J (CTP) (x ) := α x x (b) 2 + y l 2 Hx 2 l 2, (8) with respect to x for the l 2 norm, where x (b) = M 1 (x (a) 1 ) and α > 0.

Cycled Tihonov-Phillips regularization Theorem If H is linear, the minimiser x (a) to (8) is given by ( )) x (a) = M 1 x (a) 1 + R α (y HM 1 x (a) 1, (9) given x (b) = M 1 (x (a) 1 ), where R α = (αi +H H) 1 H (10) is nown as the Tihonov-Phillips inverse, with an adjoint H and a regularization parameter α. Here (9) is what we characterise as cycled Tihonov-Phillips regularization.

3DVar Theorem If H is linear then the minimiser x (a) to J (3D) (x ) = B 1( ) x x (b),x x (b) l 2 + R 1 (y Hx ),y Hx, (11) l 2 is given by where x (a) = x (b) + K ( y Hx (b) ), (12) K := BH (HBH +R) 1 (13) is the Kalman gain and H is the adjoint with respect to the Euclidean inner product.

Equivalence Theorem (Theorem 2.1 in Marx and Potthast 2012) For the Euclidean inner product and the weighted norms, B 1 :=,B 1 on X and, R 1 :=,R 1 on Y (14) for self-adjoint, positive definite operators B and R, the Kalman gain K for 3DVar corresponds to the Tihonov-Phillips inverse R α where its adjoint is given by for α = 1. H = αbh R 1 (15)

Outline 1 Bacground 2 Data assimilation 3 Nonlinear error evolution 4 Numerical example 5 Final remars

Full deterministic analysis update From x (a) = M 1 ( x (a) 1 )+R α (y HM 1 ( we can derive ( ( e = N (M 1 x (a) 1 ) M 1 where e := x (a) noise terms. x (t) x (t) 1 x (a) 1 )), (16) )) +Nζ + R α η, (17), N := I R αh and ζ and η are the

Full deterministic analysis update As a first attempt, ( e N M 1 Assumption x (a) 1 ) M 1 ( x (t) 1) + N υ+ Rα δ. The nonlinear mapping M : X X is Lipschitz continuous with a global Lipschitz constant such that given any a,b X, (18) M (a) M (b) K a b (19) where K K, the global Lipschitz constant for all time t, for N 0. Therefore we obtain e ν e 1 + N υ + R α δ, (20) where ν := K N given the global Lipschitz constant K.

Stability result Theorem (Moodey et al. 2013) For the Hilbert space (X, B 1), let the model error ζ, N 0 be bounded by υ > 0. Let the observation error η, N 0 be bounded by δ > 0. If the nonlinear model operator M : X X is Lipschitz continuous and satisfies then the analysis error evolution e := e is estimated by for N 0. If ν < 1 then 1 e ν e 0 + ν l ( N υ + R α δ), (21) lim sup l=0 e N υ + R α δ. (22) 1 ν

Stability result Lemma (Potthast et al. 2012) Let X and Y be Hilbert spaces and let H L(X,Y) be an injective compact linear operator, then the operator norm of the regularized reconstruction error operator is given by N = I R α H = 1. (23) This means to obtain ν := K N < 1, the model operator M must be strictly damping, that is the global Lipschitz constant K < 1.

Stability result Lemma (Potthast et al. 2012) Let X and Y be Hilbert spaces and let H L(X,Y) be an injective compact linear operator, then the operator norm of the regularized reconstruction error operator is given by N = I R α H = 1. (23) This means to obtain ν := K N < 1, the model operator M must be strictly damping, that is the global Lipschitz constant K < 1. Question: Can we do better than this?

State space decomposition We use the singular system of the observation operator to split the state space. Let (µ i,ϕ i,g i ) be the singular system of H. Let P (1) and P (2) be orthogonal projection operators, such that P (1) : X span{ϕ i,i n} and P (2) : X span{ϕ i,i > n}, (24) for i,n N. We define the following orthogonal subspaces X (1) := span{ϕ 1,...,ϕ n } and X (2) := span{ϕ n+1,...,ϕ }. (25)

Nonlinear error evolution Then we can expand our error evolution as follows, e = N (P (1) +P (2))( ( ( )) M 1 x (a) 1 ) M 1 x (t) 1 where +Nζ + R α η (26) ( ( ) ( )) = N X (1) M (1) 1 x (a) 1 M (1) 1 x (t) 1 +Nζ ( ( ) ( )) +N X (2) M (2) 1 x (a) 1 M (2) 1 x (t) 1 + R α η, (27) M (1) ( ) := P(1) M ( ) and M (2) ( ) := P(2) M ( ). (28)

Nonlinear error evolution Taing norms and rearranging we have, ( ( ) e = N X (1) M (1) 1 x (a) 1 ( ( + N X (2) M (2) 1 x (a) 1 M (1) 1 ( ) M (2) 1 ( ) K (1) 1 N X (1) +K(2) 1 N X (2) x (t) 1 ( x (t) 1 )) +Nζ )) + R α η (29) x (a) 1 x(t) 1 + Nζ + R α η (30) where we assume Lipschitz continuity, M (j) 1 (x(a) 1 ) M(j) 1 (x(t) 1 ) (j) K 1 x (a) 1 x(t) 1 (31) for j = 1,2, with restrictions according to the singular system of H.

Norm estimates Lemma (Potthast et al. 2012) Let (X, B 1) be a Hilbert space with weighted norm and let H be an injective linear compact observation operator. Then, by choosing the regularization parameter α > 0 sufficiently small we can find a parameter 0 < ρ < 1, such that N X (1) ρ < 1. Lemma (Potthast et al. 2012) Let (X, B 1) be a Hilbert space with weighted norm and let H be an injective linear compact observation operator, then the operator norm of the regularized reconstruction error operator is given by N X (2) = 1. (32)

Dissipation We require that ν < 1. Applying our norm estimates we have that ν := K (1) N X (1) +K (2) N X (2) K (1) ρ+k (2). (33) The nonlinear system M has to be damping in X (2) for all time. Definition A nonlinear system M, N 0, is dissipative with respect to H if it is Lipschitz continuous and damping with respect to higher spectral modes of H, in the sense that M (2) satisfies M (2) (a) M(2) (b) (2) K a b (34) a,b X, where K (2) K (2) < 1 uniformly for N 0.

Final result Under this assumption that M is dissipative with respect to H, we can choose the regularization parameter α > 0 small enough, such that ρ < 1 K(2) K (1), (35) to achieve a stable cycled scheme. We are now able to summarise this result in the following theorem. Theorem (Moodey et al. 2013) Let (X, B 1) be a Hilbert space with weighted norm. Let the nonlinear system M : X X be Lipschitz continuous and dissipative with respect to higher spectral modes of H. Then, for regularization parameter α > 0 sufficiently small, we have ν := K (1) N X (1) +K (2) N X (2) < 1. Then, lim sup e N υ + R α δ. (36) 1 ν

Outline 1 Bacground 2 Data assimilation 3 Nonlinear error evolution 4 Numerical example 5 Final remars

Lorenz 63 equations The Lorenz 63 equations are as follows, dx = σ(x y), dt (37) dy = ρx y xz, dt (38) dz = xy βz, dt (39) where typically σ, ρ and β are nown as the Prandtl number, the Rayleigh number and a non-dimensional wave number respectively. We choose the classical parameters, σ = 10, ρ = 28 and β = 8/3 and discretise the system using a fourth order Runge-Kutta approximation.

Numerical experiment We set up a twin experiment and begin with an initial condition, ( ) x (t) 0,y(t) 0,z(t) 0 = ( 5.8696, 6.7824, 22.3356), (40) We produce a run of the system until time t = 100 with a step-size h = 0.01, which we call a truth run. Now we create observations at every tenth time-step from the truth run by adding random normally distributed noise with zero mean and standard deviation σ (o) = 2/40. The bacground state is calculated in the same way at initial time t 0 with zero mean and standard deviation σ (b) = 1/400 such that, ( ) x (b) 0,y (b) 0,z (b) 0 = ( 5.8674, 6.7860, 22.3338). (41) Now we calculate e = (I R α H) ( ( )) (M 1 x (a) 1 ) M 1 x (t) 1 + R α η. (42)

Numerical experiment We calculate a sampled bacground error covariance between the bacground state and true state over the whole trajectory such that B = 117.6325 117.6374 2.3513 117.6374 152.6906 2.0838 2.3513 2.0838 110.8491. (43) We simulate the consequence of an ill-posed observation operator H with a random 3 3 matrix with its last singular value µ 3 = 10 8 such that H = 0.4267 0.5220 0.5059 0.8384 0.7453 1.6690 0.4105 1.6187 0.0610. (44) Therefore, H is severely ill-conditioned with a condition number, κ = 2.1051 10 8.

Numerical experiment 10 2 10 2 Error Integral 10 1 10 0 10 1 Error Integral 10 3 10 4 10 5 10 2 10 10 10 5 10 0 α (a) 10 6 10 10 10 5 10 0 α (b) Figure: (a) l 2 norm of the analysis error e l 2 integrated for all assimilation time t for = 1,...,1000, varying the regularization parameter, α. (b) Weighted norm of the analysis error e B 1 integrated for all assimilation time t for = 1,...,1000, varying the regularization parameter, α.

Numerical experiment 10 2 10 1 50 40 Error 10 0 10 1 10 2 z 30 20 10 10 3 0 200 400 600 800 1000 (a) 0 20 10 0 10 20 x (b) Figure: (a) l 2 norm of the analysis error e l 2 as the scheme is cycled for index, with regularization parameter α = 200, which corresponds to 3DVar. (b) Trajectories in state space for t 200:220.

Numerical experiment z 50 40 30 20 10 20 10 0 10 20 x (a) z 50 40 30 20 10 0 20 10 0 10 20 x (b) Figure: (a) and (b) Trajectories in state space for t 774:805 and t 858:895 respectively.

Numerical experiment 10 2 10 1 50 40 Error 10 0 10 1 10 2 z 30 20 10 10 3 0 200 400 600 800 1000 (a) 0 20 10 0 10 20 x (b) Figure: (a) l 2 norm of the analysis error e l 2 as the scheme is cycled for the index, with regularization parameter α = 2, an inflation in the bacground variance of 100%. (b) Trajectories in state space for t 200:220.

Numerical experiment 10 2 10 1 50 40 Error 10 0 10 1 10 2 z 30 20 10 10 3 0 200 400 600 800 1000 (a) 0 30 20 10 0 10 20 x (b) Figure: (a) l 2 norm of the analysis error e l 2 as the scheme is cycled for the index, with regularization parameter α = 10 10, an inflation in the bacground variance of 2 10 12 %. (b) Trajectories in state space for t 200:220.

Numerical experiment 10 4 10 4 10 2 10 2 Error 10 0 Error 10 0 10 2 10 2 0 20 40 60 80 100 (a) 10 4 0 20 40 60 80 100 (b) Figure: (a) l 2 norm of the analysis error, e l 2 as the scheme is cycled for the index with regularization parameter, α = 10 6. (b) Weighted norm of the analysis error, e B 1 as the scheme is cycled for the index with regularization parameter, α = 10 6. Solid line: Nonlinear analysis error. Dashed line: Linear bound. Dotted line: Asymptotic limit.

Outline 1 Bacground 2 Data assimilation 3 Nonlinear error evolution 4 Numerical example 5 Final remars

Conclusion and future wor Conclusion We have developed new stability results for cycled data assimilation schemes with an ill-posed observation operator. Under weighted norms, the choice of the α was crucial to eep in the stable range. Dissipation in the nonlinear model dynamics is necessary for our stability result to hold. Future wor Investigate the analysis error in data assimilation scheme that employ an update in the bacground error covariance.

References R Kress. Linear integral equations. Applied mathematical sciences. Springer, 1999. B Marx and R W E Potthast. On instabilities in data assimilation algorithms, GEM, 3, 253-278, 2012. M Z Nashed. Operator-theoretic and computational approaches to ill-posed problems with applications to antenna theory, IEEE Transactions on Antennas and Propagation, 29, 220-231, 1981. M Z Nashed. A new approach to classification and regularization of ill-posed operator equations, Inverse and ill-posed problems, 53-75, 1987. B P Rynne and M A Youngson. Linear functional analysis. Springer, 2007.

References A J F Moodey, A S Lawless, R W E Potthast and P J van Leeuwen Nonlinear error dynamics for cycled data assimilation methods, Inverse Problems, 29, 025002, 2013. R W E Potthast, A J F Moodey, A S Lawless and P J van Leeuwen On error dynamics and instability in data assimilation, Submitted for publication, preprint available: MPS-2012-05 at http://www.reading.ac.u/maths-andstats/research/maths-preprints.aspx.

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