Didier Auroux University of Nice Sophia Antipolis France auroux@unice.fr Data assimilation for geophysical fluids Modelling and simulation of complex systems : stochastic and deterministic approaches CEMRACS’13 July 22 - August 30, 2013
Talk overview 1. Data and models 2. Variational methods : 4D-VAR 3. Sequential methods : Kalman filter 4. Hybrid scheme : Back and Forth Nudging algorithm 5. Diffusive BFN algorithm CEMRACS’13, CIRM - July 24, 2013 1/57
Motivations Environmental and geophysical studies : forecast the natural evolution � retrieve at best the current state (or initial condition) of the environment. Geophysical fluids (atmosphere, oceans, . . .) : turbulent systems = ⇒ high sensitivity to the initial condition = ⇒ need for a precise identification (much more than observations) Environmental problems (ground pollution, air pollution, hurricanes, . . .) : problems of huge dimension, generally poorly modelized or observed Data assimilation consists in combining in an optimal way the observations of a system and the knowledge of the physical laws which govern it. Main goal : identify the initial condition, or estimate some unknown parame- ters, and obtain reliable forecasts of the system evolution. CEMRACS’13, CIRM - July 24, 2013 2/57
Data assimilation Observations combination model + observations ⇓ Model identification of the initial condition in a geophysical system t Fundamental for a chaotic system (atmosphere, ocean, . . .) Issue : These systems are generally irreversible. Goal : Combine models and data. CEMRACS’13, CIRM - July 24, 2013 3/57
⇒ 1. Data and models 2. Variational methods : 4D-VAR 3. Sequential methods : Kalman filter 4. Hybrid scheme : Back and Forth Nudging algorithm 5. Diffusive BFN algorithm CEMRACS’13, CIRM - July 24, 2013 4/57
Models The equations governing the geophysical flows are derived from the general equations of fluid dynamics. The main variables used to describe the fluids are : – The components of the velocity – Pressure – Temperature – Humidity in the atmosphere, salinity in the ocean – Concentrations for chemical species The constraints applied to these variables are : – Equations of mass conservation. – Momentum equation containing the Coriolis acceleration term, which is essential in the dynamic of flows at extra tropical latitudes. – Equation of energy conservation including law of thermodynamics. – Law of behavior (e.g. Mariotte’s Law). – Equations of chemical kinetics if a pollution type problem is considered. CEMRACS’13, CIRM - July 24, 2013 5/57
Shallow water model τ x ∂ t u − ( f + ζ ) v + ∂ x B = ρ 0 h − ru + ν ∆ u τ y ∂ t v + ( f + ζ ) u + ∂ y B = ρ 0 h − rv + ν ∆ v (1) ∂ t h + ∂ x ( hu ) + ∂ y ( hv ) = 0 CEMRACS’13, CIRM - July 24, 2013 6/57
Quasi-geostrophic ocean model D 1 ( θ 1 (Ψ) + f ) − β ∆ 2 Ψ 1 = F 1 k = 1 : Dt D k ( θ k (Ψ) + f ) − β ∆ 2 Ψ k = 0 2 ≤ k ≤ n − 1 : Dt D n ( θ n (Ψ) + f ) + α ∆Ψ n − β ∆ 2 Ψ n = 0 k = n : Dt CEMRACS’13, CIRM - July 24, 2013 7/57
Primitive equations model ∂U xy ∂p = − [ U. ∇ U ] xy − ρ 0 ∇ xy p + D U ∂z = − ρg ∇ .U = 0 ∂t ∂T ∂S ∂t = −∇ . ( TU ) + D T ∂t = −∇ . ( SU ) + D S ρ = ρ ( T, S, p ) CEMRACS’13, CIRM - July 24, 2013 8/57
Data SYNOP/SHIP data : synoptic networks in red, airport data in blue, ship data in green. CEMRACS’13, CIRM - July 24, 2013 9/57
Data Observations from ten geostationary satellites. CEMRACS’13, CIRM - July 24, 2013 10/57
Data Trajectories of six polar orbiting satellites. CEMRACS’13, CIRM - July 24, 2013 11/57
Data Satellite altimetry (from AVISO web site). CEMRACS’13, CIRM - July 24, 2013 12/57
Data assimilation methods Data assimilation methods : 1. 4D-VAR : optimal control method, based on the minimization of a functional estimating the discrepancy between the model solution and the observations. [Le Dimet-Talagrand (Tellus, vol. 38A, 1986)] 2. Sequential methods : Kalman filtering, extended Kalman and ensemble Kalman filters. [Evensen (Ocean Dynamics, vol. 53, 2003)] 3. Hybrid method : the Back and Forth Nudging. [A.-Blum (Nonlinear Processes in Geophysics, vol. 15, 2008)] CEMRACS’13, CIRM - July 24, 2013 13/57
1. Data and models ⇒ 2. Variational methods : 4D-VAR 3. Sequential methods : Kalman filter 4. Hybrid scheme : Back and Forth Nudging algorithm 5. Diffusive BFN algorithm CEMRACS’13, CIRM - July 24, 2013 14/57
4D-VAR X b Observations X 0 dX dt = F ( X ) , X (0) = X 0 t Y ( t ) : observations of the system, H : observation operator, X b : background, B and R : covariance matrices of background and observation errors respecti- vely. 1 2( X 0 − X b ) T B − 1 ( X 0 − X b ) J ( X 0 ) = � T 1 [ Y ( t ) − H ( X ( t ))] T R − 1 [ Y ( t ) − H ( X ( t ))] dt + 2 0 CEMRACS’13, CIRM - July 24, 2013 15/57
Optimality system Optimization under constraints : � � � T P, dX L ( X 0 , X, P ) = J ( X 0 ) + dt − F ( X ) dt 0 dX dt = F ( X ) Direct model : X (0) = X 0 � ∂F � T − dP P + H T R − 1 [ Y ( t ) − H ( X ( t ))] dt = ∂X Adjoint model : P ( T ) = 0 Gradient of the cost-function : ∂J = B − 1 ( X 0 − X b ) − P (0) ∂X 0 Optimal solution : X 0 = X b + BP (0) [Le Dimet-Talagrand (86)] CEMRACS’13, CIRM - July 24, 2013 16/57
4D-VAR algorithm computation adjoint direct X k ( t ) P k ( t ) X k − → − → 0 ↓ 4D-VAR algorithm : descent X k +1 ∇ J k ∇ J ( X k ← − = 0 ) 0 Each iteration = one integration of the direct model + one integration of the adjoint model. Biggest issue : computation of the adjoint code ! ! ! Big issue : validation of the adjoint code Smaller issue but still big : small number of iterations for operational DA Smaller issue, ... actually really quite small : efficient optimization algo CEMRACS’13, CIRM - July 24, 2013 17/57
Example Quasi-geostrophic ocean model : True solution 4D-VAR identified solution True initial condition (left) and identified initial condition by the 4D-VAR (right), for the upper layer. CEMRACS’13, CIRM - July 24, 2013 18/57
Example True solution 4D-VAR identified solution True initial condition (left) and identified initial condition by the 4D-VAR (right), for the bottom layer. [Luong-Blum-Verron (98), A.-Blum (04)] CEMRACS’13, CIRM - July 24, 2013 19/57
1. Data and models 2. Variational methods : 4D-VAR ⇒ 3. Sequential methods : Kalman filter 4. Hybrid scheme : Back and Forth Nudging algorithm 5. Diffusive BFN algorithm CEMRACS’13, CIRM - July 24, 2013 20/57
Kalman filter The Kalman filter is a recursive filter that estimates the state of a dynamic system from a series of incomplete and noisy measurements. It is an extension to dynamical systems of the optimal interpolation, or BLUE (Best Linear Unbiased Estimator) between the observation and the forecast state. Two steps, repeated for each observation time : prediction and correction. CEMRACS’13, CIRM - July 24, 2013 21/57
! ! Kalman filter Kalman filter : state y o x f ! y o y o t k − 2 k − 1 k CEMRACS’13, CIRM - July 24, 2013 22/57
! ! Kalman filter Kalman filter : state y o x f x a ! y o y o t k − 2 k − 1 k CEMRACS’13, CIRM - July 24, 2013 22/57
! ! Kalman filter Kalman filter : state y o x f x a ! y o y o x f t k − 2 k − 1 k CEMRACS’13, CIRM - July 24, 2013 22/57
! ! Kalman filter Kalman filter : state y o x f x a x a ! y o y o x f t k − 2 k − 1 k CEMRACS’13, CIRM - July 24, 2013 22/57
! ! Kalman filter Kalman filter : state y o x f x f x a x a ! y o y o x f t k − 2 k − 1 k CEMRACS’13, CIRM - July 24, 2013 22/57
! ! Kalman filter Kalman filter : state y o x f x f x a x a x a ! y o y o x f t k − 2 k − 1 k CEMRACS’13, CIRM - July 24, 2013 22/57
Prediction step We denote by X a n the analyzed state at time t n , and M n the system resolvent from time t n to t n +1 . Prediction/forecast step : get a background state at time t n +1 X f n +1 = M n X a n We denote by P f n the covariance matrix of forecast error X f n − X t n and P a n the covariance matrix of analysis error X a n − X t n . � n +1 ) T � P f ( X f n +1 )( X f n +1 − X t n +1 − X t n +1 = E � n − ε n ) T � ( M n X a n − M n X t n − ε n )( M n X a n − M n X t = E P f n +1 = M n P a n M T n + Q n where Q n is the model error covariance matrix at time t n . CEMRACS’13, CIRM - July 24, 2013 23/57
Correction step Analysis step : Correction of the background vector X f n +1 with the innovation vector Y n +1 − H n +1 X f n +1 : n +1 = X f n +1 + K n +1 ( Y n +1 − H n +1 X f X a n +1 ) The new analysis error e a n +1 = X a n +1 − X t n +1 is : n +1 = e f n +1 + K n +1 ( ǫ n +1 − H n +1 e f e a n +1 ) where ǫ is the observation error (of covariance matrix R ) and e f is the forecast error (of covariance matrix P f ). n +1 [ I − K n +1 H n +1 ] T + K n +1 R n +1 K T n +1 = [ I − K n +1 H n +1 ] P f P a n +1 One chooses K n +1 such that the variance of analysis error is minimum : K n +1 = P f n +1 [ H n +1 P f n +1 + R n +1 ] − 1 n +1 H T n +1 H T n +1 = [ I − K n +1 H n +1 ] P f P a Then, n +1 CEMRACS’13, CIRM - July 24, 2013 24/57
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