Some challenges of four-dimensional data assimilation problems Juan - PowerPoint PPT Presentation

Some challenges of four-dimensional data assimilation problems Juan Carlos De los Reyes Centro de Modelización Matemática (MODEMAT) Escuela Politécnica Nacional de Ecuador New trends in PDE constrained optimization Linz, 2019 1 / 30

Contenidos Motivation 1 Analysis of variational data assimilation 2 Optimal placement problem 3 Numerical results 4 Summary of topics 5

Outline Motivation 1 Analysis of variational data assimilation 2 Optimal placement problem 3 Numerical results 4 Summary of topics 5

Motivation Data assimilation in Numerical Weather Prediction Data assimilation methods aim at finding a good initial condition of the athmospheric system in order to get better weather forecasts; 3 / 30

Motivation Data assimilation in Numerical Weather Prediction Data assimilation methods aim at finding a good initial condition of the athmospheric system in order to get better weather forecasts; Information can be obtained mainly from ground stations, radionsonds or satellite images; 3 / 30

Motivation Data assimilation in Numerical Weather Prediction Data assimilation methods aim at finding a good initial condition of the athmospheric system in order to get better weather forecasts; Information can be obtained mainly from ground stations, radionsonds or satellite images; Reconstruction results depend strongly on the number of observations, which can be very limited in some cases. 3 / 30

Equations of the athmosphere Basic model ∂ z + uv tan( φ ) ∂ u ∂ t = − u ∂ u ∂ x − v ∂ u ∂ y − w ∂ u − uw a − 1 ∂ p ∂ x − 2 Ω( w cos( φ ) − v sin( φ )) + Fr x a ρ ∂ z + u 2 tan( φ ) ∂ v ∂ t = − u ∂ v ∂ x − v ∂ v ∂ y − w ∂ v − uw a − 1 ∂ p ∂ y − 2 Ω u sin( φ ) + Fr y ρ a ∂ z + u 2 + v 2 ∂ w ∂ t = − u ∂ w ∂ x − v ∂ w ∂ y − w ∂ w − 1 ∂ p ∂ z + 2 Ω u cos( φ ) − g + Fr z a ρ ∂ T ∂ t = − u ∂ T ∂ x − v ∂ T ∂ y + ( γ − γ d ) w + 1 dH c p dt � ∂ u � ∂ρ ∂ t = − u ∂ρ ∂ x − v ∂ρ ∂ y − w ∂ρ ∂ x + ∂ v ∂ y + ∂ w ∂ z − ρ ∂ z ∂ q ∂ t = − u ∂ q ∂ x − v ∂ q ∂ y − w ∂ q ∂ z + Q v + Boundary conditions + Initial conditions 4 / 30

Equations of the athmosphere Basic model ∂ z + uv tan( φ ) ∂ u ∂ t = − u ∂ u ∂ x − v ∂ u ∂ y − w ∂ u − uw a − 1 ∂ p ∂ x − 2 Ω( w cos( φ ) − v sin( φ )) + Fr x a ρ ∂ z + u 2 tan( φ ) ∂ v ∂ t = − u ∂ v ∂ x − v ∂ v ∂ y − w ∂ v − uw a − 1 ∂ p ∂ y − 2 Ω u sin( φ ) + Fr y ρ a ∂ z + u 2 + v 2 ∂ w ∂ t = − u ∂ w ∂ x − v ∂ w ∂ y − w ∂ w − 1 ∂ p ∂ z + 2 Ω u cos( φ ) − g + Fr z a ρ ∂ T ∂ t = − u ∂ T ∂ x − v ∂ T ∂ y + ( γ − γ d ) w + 1 dH c p dt � ∂ u � ∂ρ ∂ t = − u ∂ρ ∂ x − v ∂ρ ∂ y − w ∂ρ ∂ x + ∂ v ∂ y + ∂ w ∂ z − ρ ∂ z ∂ q ∂ t = − u ∂ q ∂ x − v ∂ q ∂ y − w ∂ q ∂ z + Q v + Boundary conditions + Initial conditions 4 / 30

Approaches to data assimilation Approaches based on static covariance matrices Optimal interpolation 3D-Var Recent approaches based on dynamic covariance matrices 4D-Var Extended Kalman filters Ensemble methods Eugenia Kalnay Atmospheric Modeling, Data Assimilation and Predictability Cambridge Univ. Press, 2002. 5 / 30

Optimal interpolation Kalman filter u b : background information vector u a : “analysis” (estimation of the state) z : observation vector z = Hu + v , where H is an observation operator and v is the observation error all variables are assumed to be Gaussian Kalman filter u a = u b + K ( z − Hu b ) , where K = BH T ( HBH T + R ) − 1 is the Kalman gain corresponding to the linear unbiased estimator of minimim variance. 6 / 30

Intuition of OI 7 / 30

3D-Var MAP estimation The data assimilation problem may be treated via a Bayesian approach to find the maximum-a-posteriori (MAP) estimator. 8 / 30

3D-Var MAP estimation The data assimilation problem may be treated via a Bayesian approach to find the maximum-a-posteriori (MAP) estimator. Using the Gaussian probability density functions � � − 1 2 ( z − Hu ) T R − 1 ( z − Hu ) p ( z | u ) ∝ exp � � − 1 2 ( u b − u ) T B − 1 ( u b − u ) p ( u ) ∝ exp 8 / 30

3D-Var MAP estimation The data assimilation problem may be treated via a Bayesian approach to find the maximum-a-posteriori (MAP) estimator. Using the Gaussian probability density functions � � − 1 2 ( z − Hu ) T R − 1 ( z − Hu ) p ( z | u ) ∝ exp � � − 1 2 ( u b − u ) T B − 1 ( u b − u ) p ( u ) ∝ exp Using Bayes formula, p ( u | z ) = p ( z | u ) p ( u ) , the MAP estimator corresponds to p ( z ) the solution of the 3D-Var problem min 1 2 ( z − Hu ) T R − 1 ( z − Hu ) + 1 2 ( u b − u ) T B − 1 ( u b − u ) . 8 / 30

Intuition of 4D-Var 9 / 30

4D-Var Le Dimet and Talagrand (1986) Finite dimensional problem l = 1 [ H ( y ( t i )) − z o ( t i )] T R − 1 � min J ( y , u ) [ H ( y ( t i )) − z o ( t i )] i 2 u i = 0 + 1 2 [ u − u b ] T B − 1 [ u − u b ] subject to: y ( t l ) = M ( y ( t 0 )) (Dynamical system) y ( t 0 ) = u (Initial condition) . 10 / 30

4D-Var Le Dimet and Talagrand (1986) Finite dimensional problem l = 1 [ H ( y ( t i )) − z o ( t i )] T R − 1 � min J ( y , u ) [ H ( y ( t i )) − z o ( t i )] i 2 u i = 0 + 1 2 [ u − u b ] T B − 1 [ u − u b ] subject to: y ( t l ) = M ( y ( t 0 )) (Dynamical system) y ( t 0 ) = u (Initial condition) . Features The dynamic problem incorporates all observations in a given time window; The nonlinear dynamics may be taken into account; The operational use is still a computational challenge. 10 / 30

4D-Var Le Dimet and Talagrand (1986) Finite dimensional problem l = 1 [ H ( y ( t i )) − z o ( t i )] T R − 1 � min J ( y , u ) [ H ( y ( t i )) − z o ( t i )] i 2 u i = 0 + 1 2 [ u − u b ] T B − 1 [ u − u b ] subject to: y ( t l ) = M ( y ( t 0 )) (Dynamical system) y ( t 0 ) = u (Initial condition) . What about the infinite-dimensional problem? 10 / 30

Outline Motivation 1 Analysis of variational data assimilation 2 Optimal placement problem 3 Numerical results 4 Summary of topics 5

Semilinear problem J ( y , u ) = 1 � [ y ( x k , t i ) − z 0 ( x k , t i )] 2 min 2 u k , i + 1 2 � u − u b � 2 B − 1 ∂ y ∂ t + Ay + g ( y ) = 0 in Q = Ω × ] 0 , T [ subject to: y = on Σ = Γ × ] 0 , T [ 0 y ( x , 0 ) = u in Ω , Difficulties Higher regularity of the initial condition is required to get some sort of continuity of the state. Pointwise misfits in the cost leads to right hand sides in M ( Q ) for the adjoint equation. Ill-posedness! 11 / 30

Semilinear problem J ( y , u ) = 1 � [ y ( x k , t i ) − z 0 ( x k , t i )] 2 min 2 u k , i + 1 2 � u − u b � 2 B − 1 ∂ y ∂ t + Ay + g ( y ) = 0 in Q = Ω × ] 0 , T [ subject to: y = on Σ = Γ × ] 0 , T [ 0 y ( x , 0 ) = u in Ω , Difficulties Higher regularity of the initial condition is required to get some sort of continuity of the state. Pointwise misfits in the cost leads to right hand sides in M ( Q ) for the adjoint equation. Ill-posedness! 11 / 30

Semilinear problem T J ( y , u ) = 1 � w k σ i ρ i ( t )[ y ( x k , t ) − z o ( x k , t )] 2 dt � min 2 u k , i 0 B − 1 + ϑ + 1 2 � u − u b � 2 2 �∇ ( u − u b ) � 2 L 2 (Ω) ∂ y ∂ t + Ay + g ( y ) = 0 in Q = Ω × ] 0 , T [ subject to: y = on Σ = Γ × ] 0 , T [ 0 y ( x , 0 ) = u in Ω , Difficulties Higher regularity of the initial condition is required to get some sort of continuity of the state. Pointwise misfits in the cost leads to right hand sides in M ( Q ) for the adjoint equation. Ill-posedness! w and σ are binary vectors, and ρ i ( t ) support functions 11 / 30

Well-posedness Assumption on the nonlinearity g = g ( x , t , y ) : Q × R �→ R satisfies the Carathéodory conditions and is uniformly bounded at the origin, i.e., | g ( x , t , 0 ) | ≤ K , for some K > 0 , g is monotone increasing with respect to y for almost every ( x , t ) ∈ Q , g is twice continuously differentiable with respect to y and | g y ( x , t , y ) | + | g yy ( x , t , y ) | ≤ K , for some K > 0 , for almost all ( x , t ) ∈ Q and any y ∈ R . Theorem If u ∈ H 1 0 (Ω) and the nonlinear term verifies the assumption, then the semilinear equation has a unique solution y ∈ H 2 , 1 ( Q ) ֒ → L 2 ( 0 , T ; C (Ω)) . Moreover, � y � H 2 , 1 ( Q ) ≤ c ( 1 + � u � H 1 0 (Ω) ) , for some c > 0 . 12 / 30

Differentiability Theorem 0 (Ω) → H 2 , 1 ( Q ) , u �→ S ( u ) = y , is Gâteaux The control-to-state mapping S : H 1 differentiable. Its derivative, in direction h ∈ H 1 0 (Ω) , is given by η ∈ H 2 , 1 ( Q ) solution of:  ∂η ∂ t + A η + g ′ ( y ) η = in Q 0  η = on Σ 0 η ( x , 0 ) = in Ω ,  h Adjoint equation − ∂ p ∂ t + A ∗ p + g ′ ( y ) p = µ in Q p = 0 on Σ p ( x , T ) = in Ω , 0 with µ ∈ L 2 ( 0 , T ; M (Ω)) has a unique solution p ∈ L 2 ( 0 , T ; W 1 , r 0 (Ω)) , with m r ∈ [ 1 , m − 1 [ . Casas-Clason-Kunisch 2013, Meyer-Susu 2017 13 / 30