Variational approach to data assimilation: optimization aspects and - PowerPoint PPT Presentation

Variational approach to data assimilation: optimization aspects and adjoint method Eric Blayo University of Grenoble and INRIA A

Objectives I introduce data assimilation as an optimization problem I discuss the di ff erent forms of the objective functions I discuss their properties w.r.t. optimization I introduce the adjoint technique for the computation of the gradient Link with statistical methods: cf lectures by E. Cosme Variational data assimilation algorithms, tangent and adjoint codes: cf lectures by M. Nodet and A. Vidard E. Blayo - Variational approach to data assimilation

Introduction: model problem Outline Introduction: model problem Definition and minimization of the cost function The adjoint method E. Blayo - Variational approach to data assimilation

Introduction: model problem Model problem Two di ff erent available measurements of a single quantity. Which estimation of its true value ? � ! least squares approach E. Blayo - Variational approach to data assimilation

Introduction: model problem Model problem Two di ff erent available measurements of a single quantity. Which estimation of its true value ? � ! least squares approach 2 obs y 1 = 19 � C and y 2 = 21 � C of the (unknown) Example present temperature x . ⇥ ( x � y 1 ) 2 + ( x � y 2 ) 2 ⇤ I Let J ( x ) = 1 2 x = y 1 + y 2 I Min x J ( x ) = 20 � C � ! ˆ 2 E. Blayo - Variational approach to data assimilation

Introduction: model problem Model problem If 6 = units: y 1 = 66 . 2 � F and y 2 = 69 . 8 � F Observation operator I Let H ( x ) = 9 5 x + 32 I Let J ( x ) = 1 ( H ( x ) � y 1 ) 2 + ( H ( x ) � y 2 ) 2 ⇤ ⇥ 2 x = 20 � C I Min x J ( x ) � ! ˆ E. Blayo - Variational approach to data assimilation

Introduction: model problem Model problem If 6 = units: y 1 = 66 . 2 � F and y 2 = 69 . 8 � F Observation operator I Let H ( x ) = 9 5 x + 32 I Let J ( x ) = 1 ( H ( x ) � y 1 ) 2 + ( H ( x ) � y 2 ) 2 ⇤ ⇥ 2 x = 20 � C I Min x J ( x ) � ! ˆ Drawback # 1: if observation units are inhomogeneous y 1 = 66 . 2 � F and y 2 = 21 � C I J ( x ) = 1 ( H ( x ) � y 1 ) 2 + ( x � y 2 ) 2 ⇤ ⇥ x = 19 . 47 � C !! � ! ˆ 2 E. Blayo - Variational approach to data assimilation

Introduction: model problem Model problem If 6 = units: y 1 = 66 . 2 � F and y 2 = 69 . 8 � F Observation operator I Let H ( x ) = 9 5 x + 32 I Let J ( x ) = 1 ( H ( x ) � y 1 ) 2 + ( H ( x ) � y 2 ) 2 ⇤ ⇥ 2 x = 20 � C I Min x J ( x ) � ! ˆ Drawback # 1: if observation units are inhomogeneous y 1 = 66 . 2 � F and y 2 = 21 � C I J ( x ) = 1 ( H ( x ) � y 1 ) 2 + ( x � y 2 ) 2 ⇤ ⇥ x = 19 . 47 � C !! � ! ˆ 2 Drawback # 2: if observation accuracies are inhomogeneous x = 2 y 1 + y 2 = 19 . 67 � C If y 1 is twice more accurate than y 2 , one should obtain ˆ 3 "✓ x � y 1 ◆ 2 # ◆ 2 ✓ x � y 2 ! J should be J ( x ) = 1 � + 2 1 / 2 1 E. Blayo - Variational approach to data assimilation

Introduction: model problem Model problem General form  ( H 1 ( x ) � y 1 ) 2 � + ( H 2 ( x ) � y 2 ) 2 Minimize J ( x ) = 1 σ 2 σ 2 2 1 2 E. Blayo - Variational approach to data assimilation

Introduction: model problem Model problem General form  ( H 1 ( x ) � y 1 ) 2 � + ( H 2 ( x ) � y 2 ) 2 Minimize J ( x ) = 1 σ 2 σ 2 2 1 2 ( x � y 1 ) 2 ( x � y 2 ) 2 J ( x ) = 1 + 1 If H 1 = H 2 = Id : σ 2 σ 2 2 2 1 2 1 y 1 + 1 y 2 σ 2 σ 2 1 2 which leads to x = ˆ (weighted average) 1 1 + σ 2 σ 2 1 2 E. Blayo - Variational approach to data assimilation

Introduction: model problem Model problem General form  ( H 1 ( x ) � y 1 ) 2 � + ( H 2 ( x ) � y 2 ) 2 Minimize J ( x ) = 1 σ 2 σ 2 2 1 2 ( x � y 1 ) 2 ( x � y 2 ) 2 J ( x ) = 1 + 1 If H 1 = H 2 = Id : σ 2 σ 2 2 2 1 2 1 y 1 + 1 y 2 σ 2 σ 2 1 2 which leads to x = ˆ (weighted average) 1 1 + σ 2 σ 2 1 2 1 1 x )] � 1 Remark: J ”(ˆ x ) = + = [ Var (ˆ (cf BLUE) σ 2 σ 2 | {z } | {z } 1 2 accuracy convexity E. Blayo - Variational approach to data assimilation

Introduction: model problem Model problem Alternative formulation: background + observation If one considers that y 1 is a prior (or background ) estimate x b for x , and y 2 = y is an independent observation, then: ( x � x b ) 2 ( x � y ) 2 J ( x ) = 1 + 1 σ 2 σ 2 2 2 o b | {z } | {z } J o J b and 1 x b + 1 y σ 2 σ 2 σ 2 o b b x = ˆ = x b + ( y � x b ) 1 1 σ 2 b + σ 2 | {z } o + innovation σ 2 σ 2 | {z } o b gain E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Outline Introduction: model problem Definition and minimization of the cost function Least squares problems Linear (time independent) problems The adjoint method E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Outline Introduction: model problem Definition and minimization of the cost function Least squares problems Linear (time independent) problems The adjoint method E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Generalization: arbitrary number of unknowns and observations 0 1 x 1 . R n B C . To be estimated: x = A 2 I . @ x n 0 1 y 1 . R p B C . Observations: y = A 2 I . @ y p R n � R p Observation operator: y ⌘ H ( x ) , with H : I ! I E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Generalization: arbitrary number of unknowns and observations A simple example of observation operator 0 1 x 1 ✓ ◆ x 2 an observation of x 1 + x 2 B C If x = and y = 2 B C x 3 an observation of x 4 @ A x 4 0 1 1 1 0 0 then H ( x ) = Hx with H = 2 2 @ A 0 0 0 1 E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Generalization: arbitrary number of unknowns and observations 0 1 x 1 . R n B . C To be estimated: x = A 2 I . @ x n 0 1 y 1 . R p B C . Observations: y = A 2 I . @ y p R n � R p Observation operator: y ⌘ H ( x ) , with H : I ! I Cost function: J ( x ) = 1 2 k H ( x ) � y k 2 with k . k to be chosen. E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Reminder: norms and scalar products 0 1 u 1 . R n B C . u = A 2 I . @ u n n X ⌘ Euclidian norm: k u k 2 = u T u = u 2 i i = 1 n X Associated scalar product: ( u , v ) = u T v = u i v i i = 1 ⌘ Generalized norm: let M a symmetric positive definite matrix n n X X M -norm: k u k 2 M = u T M u = m ij u i u j i = 1 j = 1 n n X X Associated scalar product: ( u , v ) M = u T M v = m ij u i v j i = 1 j = 1 E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Generalization: arbitrary number of unknowns and observations 0 1 x 1 . R n B . C To be estimated: x = A 2 I . @ x n 0 1 y 1 . R p B . C Observations: y = A 2 I . @ y p R n � R p Observation operator: y ⌘ H ( x ) , with H : I ! I Cost function: J ( x ) = 1 2 k H ( x ) � y k 2 with k . k to be chosen. E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Generalization: arbitrary number of unknowns and observations 0 1 x 1 . R n B . C To be estimated: x = A 2 I . @ x n 0 1 y 1 . R p B . C Observations: y = A 2 I . @ y p R n � R p Observation operator: y ⌘ H ( x ) , with H : I ! I Cost function: J ( x ) = 1 2 k H ( x ) � y k 2 with k . k to be chosen. (Intuitive) necessary (but not su ffi cient) condition for the existence of a unique minimum: p � n E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Formalism “background value + new observations” ◆ ✓ x b � background Y = � new obs y The cost function becomes: 1 1 2 k x � x b k 2 2 k H ( x ) � y k 2 J ( x ) = + o b | {z } | {z } J b J o E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Formalism “background value + new observations” ◆ ✓ x b � background Y = � new obs y The cost function becomes: 1 1 2 k x � x b k 2 2 k H ( x ) � y k 2 J ( x ) = + o b | {z } | {z } J b J o ( x � x b ) T B � 1 ( x � x b ) + ( H ( x ) � y ) T R � 1 ( H ( x ) � y ) = E. Blayo - Variational approach to data assimilation

Definition and minimization of the cost function Least squares problems Formalism “background value + new observations” ◆ ✓ x b � background Y = � new obs y The cost function becomes: 1 1 2 k x � x b k 2 2 k H ( x ) � y k 2 J ( x ) = + o b | {z } | {z } J b J o ( x � x b ) T B � 1 ( x � x b ) + ( H ( x ) � y ) T R � 1 ( H ( x ) � y ) = The necessary condition for the existence of a unique minimum ( p � n ) is automatically fulfilled. E. Blayo - Variational approach to data assimilation