Model error in geophysical data assimilation Some (older and new) ideas Alberto Carrassi Nansen Environmental and Remote Sensing Center, Norway Geophysical Institute, University of Bergen, Norway With: P. Ailliot (Un. Brest, FR), M. Bocquet (ENPC-CEREA, FR), C. Grudzien (UNV-Reno, USA), M. Lucini (Un. Reading, UK), L. Mitchell (Un. Adelaide, AUS), T. Miyoshi (RIKEN, JP), M. Pulido (Un. Reading, UK), P. Raanes (NORCE, NO), P. Tandeo (IMT, FR), S. Vannitsem (RMI, BE), Y. Zhen (IMT, FR). Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 1 / 20
DA and model error The impact of model error The impact of model error ◮ For years model error impacts on NWP predictions was considered small compared to the (growth of) i.c. error, and thus often neglected in DA. ◮ The amelioration of the i.c. & the increase of the forecast horizons (seasonal-to-interannual) led to a larger impact of the model error on prediction skill. ◮ In DA it often manifests as underestimation of the estimate state error co-variance ⇒ Inflation . ◮ Particularly on long timescales, model error becomes evident through the emergence of biases. ECMWF IFS model coupled with NEMO ocean model. Sea surface forecast bias (Years 14–23). Figure from Magnusson et al. , 2012 Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 2 / 20
DA and model error The posing of the problem Posing of the problem: Nonlinear Gaussian state-space model It is usually assumed an HMM such as: x k = M k : k − 1 ( x k − 1 , λ ) + η k , y k = H k ( x k ) + ǫ k . (1) ◮ x k ∈ R m and λ ∈ R p are the model state and parameter vectors respectively. ◮ y k ∈ R d are noisy observations related to the system’s state via the, generally nonlinear, observation operator , H : R m → R d ◮ M k : k − 1 : R m → R m is usually a nonlinear, possibly chaotic , function from time t k − 1 to t k . ◮ The model and the observational errors, η k and ǫ k , are usually assumed to be uncorrelated in time, mutually independent, and Gaussian distributed: η k ∼ N ( 0 , Q k ) and ǫ k ∼ N ( 0 , R k ) Given the multiple sources of model error a stochastic approach is generally used. An accurate estimate of the model error covariance, Q k , is necessary. Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 3 / 20
DA and model error 1D illustration The importance of a good Q - 1D illustration Perfect Q Univariate, linear case. x k =0 . 95 x k − 1 + η k (2) Under-estimated Q y k = x k + ǫ k (3) with η k ∼ N (0 , Q t ) and ǫ k ∼ N (0 , R t ) ◮ Promote the use of inflation. Over-estimated Q Tandeo et al , 2019 - Under review Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 4 / 20
DA and model error 1D illustration The importance of a good || Q / R || ratio - 1D illustration ◮ It is the ratio Q/R that matters for the accuracy of the state estimate. Tandeo et al , 2019 - Under review ◮ Good Q/R (no matter the individual estimates of Q and R ) suffices to get good RMSE ◮ However it impacts differently the uncertainty quantification ( i.e. coverage probability). Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 5 / 20
DA and model error 1D illustration The importance of simultaneously estimating Q and R - 1D illustration ◮ Estimate Q or R with the Expectation Maximization (EM) (Shumway and Stoffer, 1982) ◮ Figure from Tandeo et al , 2019 - Under Review It is not possible to fully compensate for the misrepresentation of Q / R by optimizing R / Q ⇒ The best is to estimate Q and R simultaneously. Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 6 / 20
DA and model error Estimating Q: key obstacles and objectives Estimating Q : key obstacles and objectives Large variety of possible error sources (incorrect parametrizations of physical processes, numerical discretizations, unresolved scales, etc..) The amount of available data insufficient to realistically describe the model error statistics, i.e. dim ( y ) = d ≪ dim ( x ) = m . Lack of a general framework for model error dynamics (as opposed to the dynamics of the i.c. error). What this talk is about : 1 Is the white-noise assumption always a good one? 2 Can we efficiently estimate Q k along with the system state? 3 On one mechanism behind the need for the ultimate therapy: Inflation. Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 7 / 20
DA and model error Time-correlated model error Time-correlated model error - Formulation Let assume to have the model: d x ( t ) = f ( x , λ ) d t used to describe the true process: dˆ x ( t ) dˆ y ( t ) = ˆ ′ ) = ˆ ′ ) f (ˆ x , ˆ y , λ h (ˆ x , ˆ y , λ d t d t ′ − λ parametric error. ◮ ˆ ′ ) : unresolved scale; ∆ λ = λ h (ˆ x , ˆ y , λ The evolution of the error covariance in the resolved scale: � t � t ′ < [ f ( x , λ ) − ˆ y , λ ′ )][ f ( x , λ ) − ˆ P ( t ) = < δ x 0 δ x T y , λ ′ )] > T 0 > + d τ d τ f (ˆ x , ˆ f (ˆ x , ˆ (4) t 0 t 0 ◮ The important factor controlling the evolution is the difference between the velocity fields, the tendencies f ( x , λ ) − ˆ f (ˆ x , ˆ y , λ ) Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 8 / 20
DA and model error Time-correlated model error Time-correlated model error - Formulation ◮ The evolution equation for the model error covariance cannot be implemented in high dimension. ◮ A suitable approximation can be obtained for short-time ( e.g. the assimilation window). ′ )] T ( t 1 − t 2 ) 2 + O (3) Q ( t 1 , t 2 ) � [ f ( x , λ ) − ˆ ′ )][ f ( x , λ ) − ˆ f (ˆ x , ˆ y , λ f (ˆ x , ˆ y , λ (5) ◮ The difference between the model and the nature tendencies, f ( x , λ ) − ˆ ′ ) is treated as f (ˆ x , ˆ y , λ being correlated in time. ◮ The white-noise case would correspond to the terms f ( x , λ ) − ˆ ′ ) being delta-correlated f (ˆ x , ˆ y , λ and the short-time evolution would be bound to be linear. Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 9 / 20
DA and model error Time-correlated model error How to estimate the model-to-nature tendencies difference Making use of the reanalysis f ) T > t 2 ⇒ Q t ≈ < ( f − ˆ f )( f − ˆ ◮ Needs to estimate the statistics of the velocity fields discrepancy. ◮ Use of the analysis increments from a reanalysis data-set assumed to be the “truth”: d t ≈ x f r ( t + τ r ) − x a − x a r ( t + τ r ) − x a = δ x a f = d x d t − dˆ r ( t ) r ( t ) x f − ˆ r ⇒ τ r τ r τ r T > τ 2 Q ( t ) ≈ < δ x a r δ x a r τ 2 r with τ r reanalysis assimilation interval and τ current assimilation interval. Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 10 / 20
DA and model error Time-correlated model error EnKF with short-time correlated model error ◮ L96 two scales. Neglect the fast scales in the model and observe 12/36 points on the coarse scale. ◮ ETKF (Bishop et al , 2001) with “best tuned” multiplicative inflation and localization (red line). ◮ ETKF with model error matrix Q estimated using the short-time approximation and the re-analysis (ETKF-TC, green line). ◮ ETKF with time-varying model error, randomly sampled from the reanalysis-increment statistics (ETKF-TV blue line) such that x f η k ∼ N ( ¯ i = M ( x a i ) + η i τ δ x a r , Q ) i = 1 , ..., N τ r Mitchell and Carrassi, 2015 Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 11 / 20
DA and model error Time-correlated model error 4DVar with short-time correlated model error Minimize the cost-function: � τ � τ ( δ x t 1 ) T Q − 1 2 J = t 1 t 2 ( δ x t 2 )d t 1 d t 2 + ... / tr = 10% 0.25 0 0 mean quadratic error 0.2 Model Lorenz 3-variables . Strong-constraint - Assume perfect 0.15 model. 0.1 Weak constraint 4DVar with 0.05 uncorrelated model error : Q t = α B (blue) or Q t = Q ( t ) 2 (red marks) 0 10 3 10 2 10 1 10 0 10 1 10 2 10 3 Short-time weak constraint 4DVar with correlated model error - Q ( t 1 , t 2 ) ≈ Q 0 ( t 1 )( t 2 ) Carrassi and Vannitsem, 2010 Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 12 / 20
DA and model error Estimate Q using data Time-batch estimated model error covariance ◮ The idea (Pulido et al , 2018) is to maximize the log-likelihood of the data ( model evidence ) as a function of the parameter θ � l ( θ ) = ln p ( x K :0 , y K :1 | θ )d x K :0 where θ can be λ , R or Q . ◮ Inserting an arbitrary PDF q ( x K :0 ) and using the Jensen inequality we have � p ( x K :0 , y K :1 | θ ) � � l ( θ ) ≥ q ( x K :0 ) ln d x K :0 ≡ Q ( q, θ ) q ( x K :0 ) and the equality holds when q ( x K :0 ) = p ( x K :0 | y K :1 , θ ) that is the PDF maximizing Q ( q, θ ) and a lower bound for l ( θ ) . ◮ p ( x K :0 | y K :1 , θ ) can be obtained as the outcome of a DA procedure ( e.g. EnKF, EnKS ...) Model error in DA - EnKF workshop 3 rd June 2019 Carrassi et al. 13 / 20
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