Model Uncertainty Quantification for Data Assimilation in partially observed Lorenz 96

Transcription

1 Model Uncertainty Quantification for Data Assimilation in partially observed Lorenz 96 Sahani Pathiraja, Peter Jan Van Leeuwen Institut für Mathematik Universität Potsdam With thanks: Sebastian Reich, Georg Gottwald

2 Motivation Uncertainty Quantification important for successful DA Main focus: Ensemble Data Assimilation Model Uncertainty due to unresolved sub-grid scale processes

3 Problem Setting System states x j evolve according to the following stochastic difference equation: x j = M(x j 1 ) + η j η j is an additive stochastic model error.

4 Problem Setting System states x j evolve according to the following stochastic difference equation: x j = M(x j 1 ) + η j η j is an additive stochastic model error. Observations of the system are available in the form of: y j = Hx j + ε j ε j N(0, R)

5 Problem Setting System states x j evolve according to the following stochastic difference equation: x j = M(x j 1 ) + η j η j is an additive stochastic model error. Observations of the system are available in the form of: y j = Hx j + ε j ε j N(0, R) Aim: estimate p(x j y j ) (i.e. filtering).

6 2 Layer Lorenz 96 Figure: Two Layer Lorenz 96 system with 8 coarse scale variables and 32 fine scale variables (taken from Arnold et al., 2013)

7 Aim Build on existing stochastic and deterministic parameterization techniques: e.g. Wilks (2005), Crommelin & Vanden-Eijnden (2008), Kwasniok (2012), Arnold et al. (2013), Lu et al. (2017)

8 Aim Build on existing stochastic and deterministic parameterization techniques: e.g. Wilks (2005), Crommelin & Vanden-Eijnden (2008), Kwasniok (2012), Arnold et al. (2013), Lu et al. (2017) Develop method to estimate distribution of η for the following conditions: No dynamical equations for the fine scale process Only partial observations of coarse scale process available

9 2 Layer Lorenz 96 dx k dt dy l,k dt = X k 1 (X k 2 X k+1 ) X k + F + h x L L Y l,k ; k {1,..., K} = 1 ξ ( Y l+1,k(y l+2,k Y l 1,k ) Y l,k + h y X k ; l {1,..., L} l=1

10 Assumptions 1. States are directly but partially observed (i.e. H is a non-square (0, 1) matrix)

11 Assumptions 1. States are directly but partially observed (i.e. H is a non-square (0, 1) matrix) 2. Model error η j depends on some informative variable (e.g. x j 1 or some reduced form of it)

12 Assumptions 1. States are directly but partially observed (i.e. H is a non-square (0, 1) matrix) 2. Model error η j depends on some informative variable (e.g. x j 1 or some reduced form of it) 3. ε j << η j

13 Assumptions 1. States are directly but partially observed (i.e. H is a non-square (0, 1) matrix) 2. Model error η j depends on some informative variable (e.g. x j 1 or some reduced form of it) 3. ε j << η j 4. Error statistics are the same at each point in time and space (translation invariance): p(η j [k] x j 1 ) = p(η b [l] x b 1 ) k, j, b, l

14 Proposed Method

15 Proposed Error Estimation Method

16 Proposed Error Estimation Method ˆη 1 o = y 1 HM(ˆx 0 ) ˆη 1 = H T ˆη 1 o + [H ] T ˆη 1 u ˆx 1 = M(ˆx 0 ) + ˆη 1 ˆη 2 o = y 2 HM(ˆx 1 ).

17 Proposed Error Estimation Method Minimize total conditional variance: Var(η j [k] x j 1 [k])dx j 1 [k]

18 Proposed Error Estimation Method Minimize total conditional variance: Var(η j [k] x j 1 [k])dx j 1 [k] N b i=1 SV (η j [k] x j 1 [k] = x i 1 ) + SV (η j [k] x j 1 [k] = x i ) x i 2

19 Benchmark Method (B1) Analysis Increment Based Method x fi j = M(x ai j 1) αη i j η i j N(b m, P m )

20 Benchmark Method (B1) Analysis Increment Based Method x fi j = M(x ai j 1) αη i j η i j N(b m, P m ) where: P = 1 N 1 α = tuning parameter δx a j b m = 1 N N j=1 δx a j N ] ] T [δxj a b m [δxj a b m j=1 = 1 n n i=1 ( xj ai xj fi )

21 Benchmark Method (B2) Long Window Weak Constraint 4dVar based Method: t+u J(η t:t+u ) = ηj T Qη j where: x j = M(x j 1 ) + η j j=t t+u + (Hx j y j ) T R 1 (Hx j y j ) j=t All other aspects same as proposed approach

22 Numerical Experiments - L96 Chaotic Two Layer Lorenz 96: dx k dt dy l,k dt = X k 1 (X k 2 X k+1 ) X k + F + h x L L Y l,k ; k {1,..., K} = 1 ξ ( Y l+1,k(y l+2,k Y l 1,k ) Y l,k + h y X k ; l {1,..., L} l=1 Parameter 1 ξ 128 Case Study 1 - large time scale sep. Case Study 2 - small time scale sep h x h y 1 1 J K 9 9 F 10 14

23 Numerical Experiments - L96 Observation Details: every 2nd X k is measured 0.02 & 0.04 MTU for Case Study 1 and 2 respectively Negligible observation error: R = 10 7 I

24 Error Estimation Results - Case Study 1

25 Error Estimation Results - Case Study 2

26 Why conditional variance minimization? Figure: Snapshot of J Q values for method B2, proposed and true data for Case Study 2

27 Impact on Assimilation/Forecasts Data Assimilation setup: Ensemble Transform Kalman Filter (ETKF) (Wang & Bishop, 2004) ensemble size (n) = 1000 observation frequency - as per estimation period assimilation length observation intervals

28 Forecast Skill

29 Non-Gaussian Uncertainties Figure: Example Evolution of Forecast density in Case Study 1

30 Summary Proposed method for quantifying model uncertainty due to unresolved sub-grid scale processes Difficult conditions: No dynamical equations of fine scale process and partial observations of coarse scale process Improved representation of model uncertainty with minimal prior knowledge

31 Further Work/Extensions Extension for including Obs Error Scalability

32 Acknowledgements This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1294 Data Assimilation We gratefully acknowledge Professor Georg Gottwald and Professor Sebastian Reich for thoughtful discussions on this work.

33 References Arnold, H. M., Moroz, I. M. and Palmer, T. N. (2013) Stochastic parametrizations and model uncertainty in the Lorenz 96 system, Philosophical Transactions of the Royal Society A, 371. doi: dx.doi.org/ /rsta Crommelin, D. and Vanden-Eijnden, E. (2008) Subgrid-Scale Parameterization with Conditional Markov Chains, Journal of the Atmospheric Sciences, 65(8), pp doi: /2008JAS Kwasniok, F. (2012) Data-based stochastic subgrid-scale parametrization: an approach using cluster-weighted modelling, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370(1962), pp doi: /rsta Lu, F., Tu, X., Chorin, A. J., Lu, F., Tu, X. and Chorin, A. J. (2017) Accounting for model error from unresolved scales in ensemble Kalman filters by stochastic parametrization, Monthly Weather Review, p. MWR-D doi: /MWR-D Mitchell, L. and Carrassi, A. (2015) Accounting for model error due to unresolved scales within ensemble Kalman filtering, Quarterly Journal of the Royal Meteorological Society, 141(689), pp doi: /qj Tremolet, Y. (2006) Accounting for an imperfect model in 4D-Var, Quarterly Journal of the Royal Meteorological Society, 132(621), pp doi: /qj Wang, X., Bishop, C. H. and Julier, S. J. (2004) Which Is Better, an Ensemble of Positive-Negative Pairs or a Centered Spherical Simplex Ensemble, Bulletin of the American Meteorological Society, pp doi: / (2004) :WIBAEO 2.0.CO;2. Wilks, D. S. (2005) Effects of stochastic parameterizations in the Lorenz 96 system, Quarterly Journal of the Royal Meteorological Society, 131. doi: /qj