Uncertainty Reduction in Atmospheric Composition Models by Chemical - PowerPoint PPT Presentation

Uncertainty Reduction in Atmospheric Composition Models by Chemical Data Assimilation Adrian Sandu Computational Science Laboratory Virginia Polytechnic Institute and State University Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO

Information feedback loops between CTMs and observations: data assimilation and targeted meas. Transport Optimal analysis state Meteorology Chemical kinetics Observations CTM Data Assimilation Aerosols Targeted Observ. Improved: Emissions • forecasts • science • field experiment design • models • emission estimates Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO

What is data assimilation? The fusion of information from: prior knowledge, 1. imperfect model predictions, and 2. sparse and noisy data, 3. to obtain a consistent description of the state of a physical system, such as the atmosphere. Lars Isaksen (http://www.ecmwf.int) Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO

Source of information #1: The prior encapsulates our current knowledge of the state ◮ The background (prior) probability density: P b ( x ) ◮ The current best estimate: apriori (background) state x b . ◮ Typical assumption on random background errors ε b = x b − S ( x true ) ∈ N ( 0 , B ) . ◮ With many nonlinear models the normality assumption is difficult to justify, but is nevertheless widely used because of its convenience. Data assimilation. General view of data assimilation. Sources of information [7/21] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)

Source of information #2: The model encapsulates our knowledge about physical and chemical laws that govern the evolution of the system ◮ The model evolves an initial state x 0 ∈ R n to future times x i = M t 0 → t i ( x 0 ) . � 10 7 � ◮ Typical size of chemical transport models: n ∈ O variables. ◮ The model is imperfect � � � � x true x true S = M t i − 1 → t i · S − η i , i i − 1 where η i is the model error in step i . Data assimilation. General view of data assimilation. Sources of information [8/21] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)

Source of information #3: The observations are sparse and noisy snapshots of reality ◮ Measurements y i ∈ R m ( m ≪ n ) taken at times t 1 , . . . , t N y i = H t � � � � x true − ε instrument S ( x true − ε obs = H i = 1 , · · · , N . ) , i i i i ◮ Observation operators ◮ H t : physical space → observation space, while ◮ H : the model space → observation space. ◮ The observation error � � − H t � � ε obs ε instrument S ( x true x true + H = ) i i i i � �� � � �� � instrument error representativeness error ◮ Typical assumptions: ε obs ε obs , ε obs ∈ N ( 0 , R i ) ; independent for t i � = t j . i i j Data assimilation. General view of data assimilation. Sources of information [9/21] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)

Result of data assimilation: The analysis encapsulates our enhanced knowledge of the state ◮ The analysis (posterior) probability density P a ( x ) : P a ( x ) = P ( x | y ) = P ( y | x ) · P b ( x ) . Bayes: P ( y ) ◮ The best state estimate x a is called the aposteriori, or the analysis . ◮ Analysis estimation errors ε a = x a − S ( x true ) characterized by ◮ analysis mean error (bias) β a = E a [ ε a ] ◮ analysis error covariance matrix A = E a � ( ε a − β a ) ( ε a − β a ) T � ∈ R n × n ◮ In the Gaussian, linear case, Bayes posterior admits an analytical solution by Kalman filter formulas Data assimilation. General view of data assimilation. Sources of information [10/21] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)

Extended Kalman filter ◮ The observations are considered successively at times t 1 , · · · , t N . ◮ The background state at t i given by the model forecast: x b i ≡ x f i = M t i − 1 → t i · x a i − 1 . ◮ Model is imperfect, but is assumed unbiased η i ∈ N ( 0 , Q i ) ◮ Model error η i and solution error ε a i − 1 are assumed independent; solution error small, propagated by linearized model M = M ′ ( x ) O ( n 3 ) : i − 1 M T B i ≡ P f i = M t i − 1 → t i P a t i → t i − 1 + Q i . ◮ EKF analysis uses H i = H ′ ( x f i ) : � � x a i = x f y i − H ( x f O ( nm ) : i + K i i ) O ( nm 2 + n 2 m + m 3 ) : � � − 1 K i = P f i H T H i P f i H T i + R i i O ( n 2 m + n 3 ) : A i ≡ P a i = ( I − K i H i ) P f i . Data assimilation. The Bayesian framework. [11/21] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)

Practical Kalman filter methods ◮ EKF is not practical for very large systems ◮ Suboptimal KF approximate the covariance matrices e.g., � distance { gridpoint ( ℓ ) , gridpoint ( k ) } 2 / L 2 � B ( ℓ ) , ( k ) = σ ( ℓ ) σ ( k ) exp ◮ Ensemble Kalman filters (EnKF) use a Monte-Carlo approach � � x f x a i [ e ] = M t i − 1 → t i i − 1 [ e ] + η i [ e ] , e = 1 , . . . , E ���� model error � � x a x f y i + ε obs [ e ] − H i ( x f i [ e ] = i [ e ] + K i i [ e ]) , e = 1 , . . . , E . i ◮ Error covariances P f i , P a i estimated from statistical samples ◮ EnKF issues: rank-deficiency of the estimated P f i ◮ EnKF strengths: capture non-linear dynamics, doesn’t need TLM, ADJ, accounts for model errors, almost ideally parallelizable Data assimilation. The Bayesian framework. [12/21] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)

Maximum aposteriori estimator Maximum aposteriori estimator (MAP) defined by x a = arg max P a ( x ) = arg min J ( x ) = − ln P a ( x ) . J ( x ) , x x Using Bayes and assumptions for background, observation errors: − ln P a ( x ) = − ln P b ( x ) − ln P ( y | x ) + const J ( x ) = 1 x − x b � T B − 1 � + 1 2 ( H ( x ) − y ) T R − 1 ( H ( x ) − y ) � x − x b � = ˙ 2 Optimization by gradient-based numerical procedure ∇ x J ( x a ) = B − 1 � x a − x b � + H T R ( H ( x a ) − y ) ; H = H ( x b ) . Hessian of cost function approximates inverse analysis covariance x , x J = B − 1 + H T R − 1 H ≈ A − 1 . ∇ 2 Data assimilation. The Bayesian framework. [13/21] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)

Four dimensional variational data assimilation (4D-Var) I ◮ All observations at all times t 1 , · · · , t N are considered simultaneously ◮ The control variables (parameters p , initial conditions x 0 , boundary conditions, etc) uniquely determine the state of the system at all future times ◮ 4D-Var MAP estimate via model-constrained optimization problem N 1 + 1 � � � � 2 �H ( x i ) − y i � 2 � x 0 − x b J ( x 0 ) = B − 1 R − 1 0 2 2 0 i i = 1 x a = arg min J ( x 0 ) 0 subject to: x i = M t 0 → t i ( x 0 ) , i = 1 , · · · , N ◮ Formulation can be easily extended to other model parameters Data assimilation. The Bayesian framework. [14/21] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)

Four dimensional variational data assimilation (4D-Var) II ◮ The large scale optimization problem is solved in a reduced space using a gradient-based technique. ◮ The 4D-Var gradient reads � ∂ x i N � T � � � ∇J x 0 ( x 0 ) = B − 1 x 0 − x b H T i R − 1 + ( H ( x i ) − y i ) 0 0 i ∂ x 0 i = 1 ◮ Needs linearized observation operators H i = H ′ ( x i ) ◮ Needs the transposed sensitivity matrix ( ∂ x i /∂ x 0 ) T ∈ ❘ n × n ◮ Adjoint models efficiently compute the transposed sensitivity matrix times vector products ◮ The construction of an adjoint model is a nontrivial task. Data assimilation. The Bayesian framework. [15/21] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)

Correct models of background errors are of great importance for data assimilation • Background error representation determines the spread of information, and impacts the assimilation results • Needs: high rank, capture dynamic dependencies, efficient computations • Traditionally estimated empirically (NMC, Hollingsworth-Lonnberg) 1. Tensor products of 1d correlations, decreasing with distance (Singh et al, 2010) 2. Multilateral AR model of background errors based on “monotonic TLM discretizations” (Constantinescu et al 2007) 3. Hybrid methods in the context of 4D-Var (Cheng et al, 2007) [Constantinescu and Sandu, 2007] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO

What is the effect of mis-specification of inputs? (Daescu, 2008) Consider a verification functional Ψ( x a v ) defined on the optimal solution at a future time t v . Ψ is a measure of the forecast error. What is the impact of small errors in the specification of covariances, background, and observation data? �� � � − 1 R − 1 ∇ 2 ∇ y i Ψ = H i M t 0 → t i ∇ x 0 Ψ x 0 , x 0 J i � � R − 1 ( H ( x a ∇ R i (:) Ψ = i ) − y i ) ⊗ ∇ y i Ψ i �� � � − 1 B − 1 ∇ 2 ∇ x b Ψ = x 0 , x 0 J ∇ x 0 Ψ 0 � � B − 1 0 ( x a 0 − x b ∇ B 0 (:) Ψ = 0 ) ⊗ ∇ x b Ψ Data assimilation. The Bayesian framework. [21/25] Aug. 4, 2011. IFIP UQ Workshop, Boulder, CO. (http://csl.cs.vt.edu)