Background-error correlation modelling in variational assimilation - PowerPoint PPT Presentation

. Background-error correlation modelling in variational assimilation using a diffusion equation, with application to oceanography . Anthony Weaver and Isabelle Mirouze CERFACS, Toulouse October 28, 2011 Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Outline . . Data assimilation in oceanography 1 . . Variational data assimilation 2 . . Characteristics of the background-error covariance matrix 3 . . Correlation modelling with a diffusion operator. Part 1: isotropy, 4 boundary conditions, solution algorithm . . Correlation modelling with a diffusion operator. Part 2: anisotropy, 5 inhomogeneity, ensemble estimation . . Concluding remarks 6 Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Outline . . Data assimilation in oceanography 1 . . Variational data assimilation 2 . . Characteristics of the background-error covariance matrix 3 . . Correlation modelling with a diffusion operator. Part 1: isotropy, 4 boundary conditions, solution algorithm . . Correlation modelling with a diffusion operator. Part 2: anisotropy, 5 inhomogeneity, ensemble estimation . . Concluding remarks 6 Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Applications of data assimilation in oceanography Providing initial conditions for climate forecasts. ▶ Monthly, seasonal, multi-annual, decadal. ▶ Global models. ▶ Mainly low resolution ( ∼ 1 ◦ ). Providing initial conditions for ocean forecasts (with a focus on mesoscale eddies). ▶ Days to weeks. ▶ Global, regional and coastal models. ▶ Modest/high resolution ( ∼ 1 / 4 ◦ / ∼ 1 / 12 ◦ +). Reconstructing the history of the ocean (reanalysis). ▶ Mainly global models. ▶ Mainly low/modest resolution for multi-decadal reanalysis. Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

The core components of the global ocean observing system Temp. and salinity from Argo floats Temperature from moorings SST from satellites, ships, buoys SSH from satellite altimeters Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Applications of ocean data assimilation Detecting decadal variability and trends due to global warming in ocean reanalyses. ¡ ( From Balmaseda et al. (2010), European COMBINE project) Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Outline . . Data assimilation in oceanography 1 . . Variational data assimilation 2 . . Characteristics of the background-error covariance matrix 3 . . Correlation modelling with a diffusion operator. Part 1: isotropy, 4 boundary conditions, solution algorithm . . Correlation modelling with a diffusion operator. Part 2: anisotropy, 5 inhomogeneity, ensemble estimation . . Concluding remarks 6 Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Variational data assimilation: I We consider the general 4D-Var assimilation problem: x ∈ R n J [ x ] = 1 + 1 2 ( x − x b ) T B − 1 ( x − x b ) 2 ( G ( x ) − y o ) T R − 1 ( G ( x ) − y o ) min � �� � � �� � J b J o where       . . . . . . . . .       y o =       y o  , G ( x ) = G i ( x )  = H i ( M ( t i , t 0 )( x ))     i . . . . . . . . . Assimilation window t 0 ≤ t i ≤ t N (order of days in the ocean). x = initial condition for temp., salinity, velocity, SSH on 3D grid. dim ( x ) = n ∼ 10 6 − 10 8 . dim ( y o ) = m ∼ 10 5 − 10 6 . Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Variational data assmilation: II We transform the 4D-Var problem into a more convenient form: x ] = 1 x b ) + 1 x b ) T � B − 1 ( � x ) − y o ) T R − 1 ( � x ) − y o ) 2 ( � x ∈ R n J [ � 2 ( � x − � x − � G ( � G ( � min � where     . . . . . .     �     � G ( x ) =  = H i ( M ( t i , t 0 )( K ( � x ))) G i ( x )    . . . . . . x = K − 1 ( x ) is the initial condition transformed into dynamically � decorrelated variables. � B is assumed block diagonal wrt to the transformed variables. The solution is x a = K ( � x a is the minimum of J . x a ) where � Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Variational data assimilation: III We use a solution algorithm based on incremental 4D-Var (Gauss-Newton): x ( k ) ] = 1 x ( k ) − δ � x b , ( k − 1 ) ) T � B − 1 ( δ � x ( k ) − δ � x ( k ) ∈ R n J ( k ) [ δ � x b , ( k − 1 ) ) min 2 ( δ � δ � + 1 x ( k ) − δ y o , ( k − 1 ) ) T R − 1 ( � x ( k ) − δ y o , ( k − 1 ) ) 2 ( � G ( k − 1 ) δ � G k − 1 δ �     where . . . . . .     G ( k − 1 ) = �     H ( k − 1 ) ∂ � M ( k − 1 ) ( t i , t 0 ) K ( k − 1 )  ≈ G i /∂ � x | �    x ( k − 1 ) x = � i . . . . . . x b , ( k − 1 ) = � x b − � x ( k − 1 ) and δ y o , ( k − 1 ) = y o − � x ( k − 1 ) ) . δ � G ( � x ( k ) found approximately using conjugate gradients. Inner problem : δ � Outer problem : very few iterations k are affordable in practice. Setting M ( k − 1 ) ( t i , t 0 ) = I gives a (much cheaper!) 3D-Var algorithm (widely used in ocean DA). Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Outline . . Data assimilation in oceanography 1 . . Variational data assimilation 2 . . Characteristics of the background-error covariance matrix 3 . . Correlation modelling with a diffusion operator. Part 1: isotropy, 4 boundary conditions, solution algorithm . . Correlation modelling with a diffusion operator. Part 2: anisotropy, 5 inhomogeneity, ensemble estimation . . Concluding remarks 6 Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Characteristics of the background-error covariance matrix � B B − 1 is not required when � Specification of � B is used as a preconditioner. � B is an enormous matrix that is difficult to estimate and represent. ▶ Need simplifying assumptions to reduce number of tunable parameters. ▶ Need to account for inhomogeneous and anisotropic structures. ▶ Need computationally efficient operators that can run in parallel. ▶ Need to deal with complex boundaries in the ocean. ▶ Need to apply with complicated grids referenced to the thin-spherical shell geometry ( S 2 × R 1 ) of the Earth. Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Basic properties of � B : I On each CG iteration we need to multiply a vector ψ in state space by � B . B this multiplication in R 3 can be interpreted as For a given block � B i of � ∫ � � B i ( x , x ′ ) ψ i ( x ′ ) d x ′ B i ψ i → R 3 where x = ( x , y , z ) T and d x = d x d y d z . 1 � B i ( x , x ′ ) is a symmetric and positive definite (covariance) function: correlation � �� � � B i ( x , x ′ ) = σ i ( x ) σ i ( x ′ ) C i ( x , x ′ ) with C i ( x , x ) = 1 � �� � std devs Remark: Evaluating the integral equation numerically is in general prohibitively expensive for large-scale problems ! 1 Not to be confused with the state vector x on previous slides! Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Basic properties of � B : II Multiplication by σ i ( x ) is easy. Rescale the variables � ψ i ( x ) = σ i ( x ) ψ i ( x ) . The difficult computation is ∫ C i � R 3 C i ( x , x ′ ) � ψ i ( x ′ ) d x ′ . ψ i → For the ocean (and atmosphere) we need to define a valid correlation operator on the spherical space S 2 ∫ ψ i ( λ ′ , ϕ ′ ) a 2 cos ϕ ′ d λ ′ d ϕ ′ C i � S 2 C i ( λ, ϕ, λ ′ , ϕ ′ ) � ψ i → where a is the Earth’s radius and ( λ, ϕ ) are geographical coordinates. Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011

Outline . . Data assimilation in oceanography 1 . . Variational data assimilation 2 . . Characteristics of the background-error covariance matrix 3 . . Correlation modelling with a diffusion operator. Part 1: isotropy, 4 boundary conditions, solution algorithm . . Correlation modelling with a diffusion operator. Part 2: anisotropy, 5 inhomogeneity, ensemble estimation . . Concluding remarks 6 Large-Scale Inverse Problems and Applications in the Earth Sciences, Linz, Austria, 24-28 October 2011