On Instability in Data Assimilation Roland Potthast 1 , 2 , 3 , Boris - PowerPoint PPT Presentation

On Instability in Data Assimilation Roland Potthast 1 , 2 , 3 , Boris Marx 3 , Alexander Moodey 2 , Amos Lawless 2 , Peter Jan van Leeuwen 2 1 German Meteorological Service (DWD), 2 University of Reading, United Kingdom 3 University of G¨ ottingen, Germany Roland Potthast 1/43

Outline Outline Motivation and Introduction. 1 Roland Potthast 2/43

Outline Outline Motivation and Introduction. 1 Variational Data Assimilation 2 Tikhonov and 3dVar 4dVar Spectral Representation Roland Potthast 2/43

Outline Outline Motivation and Introduction. 1 Variational Data Assimilation 2 Tikhonov and 3dVar 4dVar Spectral Representation Convergence Analysis for Cycled Assimilation 1: Range Arguments 3 Setup: Constant System Convergence for Data f ∈ R ( H ) / Divergence for f / ∈ R ( H ) Roland Potthast 2/43

Outline Outline Motivation and Introduction. 1 Variational Data Assimilation 2 Tikhonov and 3dVar 4dVar Spectral Representation Convergence Analysis for Cycled Assimilation 1: Range Arguments 3 Setup: Constant System Convergence for Data f ∈ R ( H ) / Divergence for f / ∈ R ( H ) Convergence Analysis 2: Regularization of Assimilation 4 Setup: High-Frequency Damping Systems Analysis Error Bounds Roland Potthast 2/43

Motivation and Introduction Data Assimilation Motivation I: Numerical Weather Forecast ... Plan Travel Warn and Protect Roland Potthast 3/43

Motivation and Introduction Data Assimilation Motivation II: Numerical Weather Forecast ... Logistics Rivers and Environment Air Control Roland Potthast 4/43

Motivation and Introduction Data Assimilation Data Assimilation at the Deutscher Wetterdienst Roland Potthast 5/43

Motivation and Introduction Data Assimilation Modelling of the Atmosphere: Geometry GME/ICON Resolution 30km COSMO-EU Resolution 7km COSMO-DE Resolution 2.8km Roland Potthast 6/43

Motivation and Introduction Data Assimilation Measurements for State Determination ... Synop, TEMP , Radiosondes, Buoys, Airplanes (AMDAR), Radar, Wind Profiler, Scatterometer, Radiances, GPS/GNSS, Ceilometer, Lidar Roland Potthast 7/43

Motivation and Introduction Data Assimilation Mathematical Setup for Data Assimilation State Space: X state space , containing all state variables in one vector ϕ ϕ state of the atmosphere Roland Potthast 8/43

Motivation and Introduction Data Assimilation Mathematical Setup for Data Assimilation State Space: X state space , containing all state variables in one vector ϕ ϕ state of the atmosphere t k time discretization point ϕ k state at time t k Roland Potthast 8/43

Motivation and Introduction Data Assimilation Mathematical Setup for Data Assimilation State Space: X state space , containing all state variables in one vector ϕ ϕ state of the atmosphere t k time discretization point ϕ k state at time t k M k : X → X model operator at time t k , ϕ k �→ ϕ k + 1 = M ( ϕ k ) Roland Potthast 8/43

Motivation and Introduction Data Assimilation Mathematical Setup for Data Assimilation State Space: X state space , containing all state variables in one vector ϕ ϕ state of the atmosphere t k time discretization point ϕ k state at time t k M k : X → X model operator at time t k , ϕ k �→ ϕ k + 1 = M ( ϕ k ) Observation Space Y k observation space at time t k f k observation vector at time t k H k : X → Y k observation operator x , y points in physical space Roland Potthast 8/43

Motivation and Introduction Data Assimilation Data Assimilation Task Definition (Data Assimilation Task) Given measurements f k at t k for k = 1 , 2 , 3 , ... determine the states ϕ k from the equations H ϕ k = f k , k = 1 , 2 , 3 , ... (1) taking care of the model dynamics given by M k . Roland Potthast 9/43

Motivation and Introduction Data Assimilation Data Assimilation Task Definition (Data Assimilation Task) Given measurements f k at t k for k = 1 , 2 , 3 , ... determine the states ϕ k from the equations H ϕ k = f k , k = 1 , 2 , 3 , ... (1) taking care of the model dynamics given by M k . Usually the measurement space is dynamic, i.e. changing in every time-step. In general H is a non-linear operator, non-injective, ill-posed. The value f k contains significant data error with stochastic components and a dynamic bias. Roland Potthast 9/43

Variational Data Assimilation Tikhonov and 3dVar Outline Motivation and Introduction. 1 Variational Data Assimilation 2 Tikhonov and 3dVar 4dVar Spectral Representation Convergence Analysis for Cycled Assimilation 1: Range Arguments 3 Setup: Constant System Convergence for Data f ∈ R ( H ) / Divergence for f / ∈ R ( H ) Convergence Analysis 2: Regularization of Assimilation 4 Setup: High-Frequency Damping Systems Analysis Error Bounds Roland Potthast 10/43

Variational Data Assimilation Tikhonov and 3dVar Tikhonov Data Assimilation In every assimilation step k ∈ N we solve the variational minimization problem to find the minimum of 2 + || f k + 1 − H ϕ ( b ) 2 , ϕ ∈ X , J ( ϕ ) := α || ϕ − ϕ ( b ) k + 1 || k + 1 || (2) where ϕ ( b ) k + 1 := M ϕ ( a ) k , k = 0 , 1 , 2 , ... (3) For linear operators the minimum is given by the normal equations , which can be reformulated into the update formula � � ϕ ( a ) k + 1 = ϕ ( b ) f k + 1 − H ϕ ( b ) k + 1 + R α (4) k + 1 with R α = ( α I + H ∗ H ) − 1 H ∗ , or in terms of the analysis fields ϕ ( a ) k � � ϕ ( a ) k + 1 = M ϕ ( a ) f k + 1 − HM ϕ ( a ) + R α (5) k k for k = 0 , 1 , 2 , ... . Roland Potthast 11/43

Variational Data Assimilation Tikhonov and 3dVar 3dVAR - 1 Roland Potthast 12/43

Variational Data Assimilation Tikhonov and 3dVar 3dVAR - 1 Start with ϕ ( a ) and for k = 1 , 2 , 3 , ... do: 1 0 Roland Potthast 12/43

Variational Data Assimilation Tikhonov and 3dVar 3dVAR - 1 Start with ϕ ( a ) and for k = 1 , 2 , 3 , ... do: 1 0 Calculate first guess 2 ϕ ( b ) = M k − 1 ϕ ( a ) (6) k k − 1 Roland Potthast 12/43

Variational Data Assimilation Tikhonov and 3dVar 3dVAR - 1 Start with ϕ ( a ) and for k = 1 , 2 , 3 , ... do: 1 0 Calculate first guess 2 ϕ ( b ) = M k − 1 ϕ ( a ) (6) k k − 1 Assimilate data f k at time t k calculating ϕ ( a ) . 3 k Roland Potthast 12/43

Variational Data Assimilation Tikhonov and 3dVar 3dVAR - 2 Functional at time slice 2 J ( ϕ ) = || ϕ − ϕ ( b ) || B − 1 + || f − H ϕ || 2 (7) R − 1 Update Formula + ( B − 1 + H ∗ R − 1 H ) − 1 H ∗ R − 1 ( f k − H ( ϕ ( b ) ϕ ( a ) ϕ ( b ) = k )) k k ϕ ( b ) ( f k − H ( ϕ ( b ) + BH ∗ ( R + HBH ∗ ) − 1 = k ) ) . (8) k � �� � � �� � Kalman gain matrix obs - first guess Roland Potthast 13/43

Variational Data Assimilation Tikhonov and 3dVar 3dVar = Tikhonov Regularization in a weighted space We study a weighted scalar product � � � � � � � � := ϕ, B − 1 ψ := f , R − 1 g ϕ, ψ L 2 , f , g (9) L 2 with some self-adjoint invertible matrices B and R . The adjoint with respect to the weighted scalar product is denoted by H ′ . Roland Potthast 14/43

Variational Data Assimilation Tikhonov and 3dVar 3dVar = Tikhonov Regularization in a weighted space We study a weighted scalar product � � � � � � � � := ϕ, B − 1 ψ := f , R − 1 g ϕ, ψ L 2 , f , g (9) L 2 with some self-adjoint invertible matrices B and R . The adjoint with respect to the weighted scalar product is denoted by H ′ . Then � � � � � � � � f , R − 1 H ϕ R − 1 f , H ϕ H ∗ R − 1 f , ϕ f , H ϕ = L 2 = L 2 = L 2 � � � � � � � � H ∗ R − 1 f , BB − 1 ϕ BH ∗ R − 1 f , B − 1 ϕ BH ∗ R − 1 f , ϕ = = L 2 = = H ′ f , ϕ . Roland Potthast 14/43

Variational Data Assimilation Tikhonov and 3dVar 3dVar = Tikhonov Regularization in a weighted space We study a weighted scalar product � � � � � � � � := ϕ, B − 1 ψ := f , R − 1 g ϕ, ψ L 2 , f , g (9) L 2 with some self-adjoint invertible matrices B and R . The adjoint with respect to the weighted scalar product is denoted by H ′ . Then � � � � � � � � f , R − 1 H ϕ R − 1 f , H ϕ H ∗ R − 1 f , ϕ f , H ϕ = L 2 = L 2 = L 2 � � � � � � � � H ∗ R − 1 f , BB − 1 ϕ BH ∗ R − 1 f , B − 1 ϕ BH ∗ R − 1 f , ϕ = = L 2 = = H ′ f , ϕ . This leads to H ′ = BH ∗ R − 1 (10) Roland Potthast 14/43

Variational Data Assimilation Tikhonov and 3dVar 3dVar = Tikhonov Regularization in a weighted space We study a weighted scalar product � � � � � � � � := ϕ, B − 1 ψ := f , R − 1 g ϕ, ψ L 2 , f , g (9) L 2 with some self-adjoint invertible matrices B and R . The adjoint with respect to the weighted scalar product is denoted by H ′ . Then � � � � � � � � f , R − 1 H ϕ R − 1 f , H ϕ H ∗ R − 1 f , ϕ f , H ϕ = L 2 = L 2 = L 2 � � � � � � � � H ∗ R − 1 f , BB − 1 ϕ BH ∗ R − 1 f , B − 1 ϕ BH ∗ R − 1 f , ϕ = = L 2 = = H ′ f , ϕ . This leads to H ′ = BH ∗ R − 1 (10) and thus BH ∗ ( R + HBH ∗ ) − 1 BH ∗ R − 1 ( I + HBH ∗ R − 1 ) − 1 = = H ′ ( I + HH ′ ) − 1 = ( I + H ′ H ) − 1 H ′ . Roland Potthast 14/43