1. A variational data assimilation system for the global ocean Anthony Weaver CERFACS Toulouse, France Acknowledgements N. Daget, E. Machu, A. Piacentini, S. Ricci, P. Rogel (CERFACS) C. Deltel, J. Vialard (LODYC, Paris) D. Anderson and the ECMWF Seasonal Forecasting Group ENACT project consortium (EC Framework 5)

2. Outline Scientific objectives and development strategy. General formulation and key characteristics of the variational system. Some results from tropical Pacific and global ocean applications. Summary and future directions.

3. Scientific objectives Develop a global ocean data assimilation system that can satisfy two purposes simultaneously: 1.Provide estimates of the ocean state over multi-annual to multi-decadal periods (currently up to 40 years – ERA40). 2.Provide ocean initial conditions for seasonal to multi-annual range forecasts. Much of this work has been coordinated through the EC-FP5 project ENACT (2002-2004).

4. The assimilation method should have a solid theoretical foundation. It must be practical for large-dimensional systems involving GCMs –State vector ~ O(10 6 ) to O(10 7 ) elements. –Non-differentiable parameterizations and algorithms. There should be a clear pathway to more advanced data assimilation systems. Basic considerations in designing a practical data assimilation system

5. Develop a system based on variational data assimilation. Use an “incremental” approach (Courtier et al. 1994, QJRMS). Provide a clear development pathway from 3D-Var 4D-Var: short window, strong model constraint 4D-Var: long window, weak model constraint CERFACS assimilation system development strategy for OPA

6. Development strategy cont. Why 3D-Var? –An effective 3D-Var system provides a solid foundation for 4D- Var (they share most components!). –3D-Var is a simpler and cheaper alternative to 4D-Var. –3D-Var provides a valuable reference for evaluating the cost benefits of 4D-Var. –Some of the flow-dependent features implicit in 4D-Var can be built into 3D-Var. –3D-Var requires significantly less maintenance and development than 4D-Var (the tangent-linear and adjoint of the forecast model are not needed). What can 4D-Var do that 3D-Var can’t? –4D-Var can exploit tendency information in the observations. –4D-Var computes implicitly flow-dependent, time-evolving covariances within the assimilation window.

7. The OPA-VAR assimilation system OPA version 8.2 (Madec et al. 1999) Configurations available for assimilation –Tropical Pacific (TDH): 1 o x 0.5 o at eq., 25 levels (rigid lid) ( Weaver et al. 2003, MWR; Vialard et al. 2003, MWR; Vossepoel et al. 2004, MWR; Ricci et al. 2004, MWR) –Global (ORCA): 2 o x 0.5 o at eq., 31 levels (free surface) Variational assimilation environment –~ 160 Fortran routines (~ 43,000 lines) for the OPA tangent-linear and adjoint models, and associated validation routines. –~ 190 Fortran routines (~ 54,000 lines) for the rest (observation operators, covariance operators, minimization routine,…). –cf. ~ 230 Fortran routines (~76,000 lines) for the global OPA forecast model.

8. General formulation of the variational problem Let denote the vector of prognostic model state variables. Let denote the vector of analysis control variables where Find that minimizes where background term observation term where

9. Incremental formulation Let be an increment to the state Let be an increment to the control where Find that minimizes where background term quadratic obs. term where

10. Choice of analysis control variables In the ocean model = ( T, S, η, u, v) As analysis variables we take = ( T, S u, η u, u u, v u ) and assume these variables are mutually uncorrelated (so is block diagonal). The transformation is a balance constraint (Derber and Bouttier, 1999, Tellus) –strong constraint if S u = η u = u u = v u = 0 –weak constraint if S u ≠ η u ≠ u u ≠ v u ≠ 0 “unbalanced” variables

11. Interpretation of the balance operator If is linear then defines the multivariate covariances in When dim( ) < dim( ), has a null space. E.g., with applied as a strong constraint, the observations will project only onto the “balanced” modes.

12. Choice of balance operator We construct as a lower triangular matrix (and hence easily invertible) transformation using the following constraints: –Linearized local T-S relationships  balanced S (Ricci et al. 2004, MWR) –Dynamic height (baroclinic)  balanced η –Geostrophy, β-plane approx. near eq.  balanced (u, v) We can interpret – S u ≠ S(T)  unbalanced S – η u ≡ barotropic component  unbalanced η –(u u, v u ) ≡ ageostrophic velocity  unbalanced (u, v)

13. Multivariate 3D-Var covariances Ex: covariance relative to a SSH point at (0 o,144 o W) (surface)

14. Choice of linear propagator involves integrating the nonlinear forward model from initial time to the observation times. involves integrating a linear forward model: In 3D-Var (FGAT) persistence In 4D-Var approx. TL model where

15. Linear approximation in the tropical Pacific (from Weaver et al. 2003, MWR) Latitude

16. TL approximation in the tropical Pacific (from Weaver et al. 2003, MWR) TIWs October start date

17. Interpretation of the linear propagator The linear propagator defines how the background error covariances evolve within the assimilation window. E.g., for observations located only at time, the effective background-error covariance matrix at is (cf. Extended Kalman filter)

18. In 3D-Var : In 4D-Var (cf. EKF): 4D-Var (t i =30 days) 3D-Var Diagnosing implicit background temperature error standard deviations ( ) in 4D-Var (Weaver et al. 2003 – MWR)

19. SSH analysis increment Amplitude (cm ) Depth (m ) Impact of a single SSH observation in 4D-Var SSH innovation = 10 cm at (0 o,160 o W) at t = 30 days Temperature analysis increment

20. The minimization is preconditioned via a change of variables so that and where For a single observation, the minimization converges in a single iteration. Preconditioning

21. Specifying background error covariances: general remarks There is not enough information (and never will be) to determine all the elements of (typically > O(10 10 )). must be approximated by a statistical model (e.g., prescribed covariance functions) with a limited number of tunable parameters. In 3D-Var/4D-Var, is implemented as an operator (a matrix-vector product). For the preconditioning transformation we require access to a square-root operator (and its adjoint ). Constructing an effective operator requires substantial development and tuning!

22. We solve a generalized diffusion equation (GDE) to perform the smoothing action of the square-root of the correlation operator ( ). (Weaver and Courtier 2001, QJRMS; Weaver and Ricci 2004, ECMWF Sem. Proc.) Simple parameterizations for the standard deviations of background error ( ) –(Balanced) T: background vertical T-gradient dependent –Unbalanced S: background mixed-layer depth dependent –Unbalanced SSH: function of latitude –Unbalanced (u,v): function of depth Modelling univariate background error covariances

23. Univariate correlation modelling using a diffusion equation (Derber & Rosati 1989 - JPO; Egbert et al. 1994 - JGR; Weaver & Courtier 2001 - QJRMS) A simple 1D example: Consider with constant. on with as Integrate from and with as IC:

24. Solution: This integral solution defines, after normalization, a correlation operator : The kernel of is a Gaussian correlation function where is the length scale. Basic idea : To compute the action of on a discrete grid we can iterate a diffusion operator. This is much cheaper than solving an integral equation directly.

25. Constructing a family of correlation functions on the sphere using a GDE (Weaver & Courtier 2001, QJRMS; Weaver & Ricci 2004 – ECMWF Sem. Procs.) shapespectrum Gaussian L = 500 km Gaussian

26. The full correlation operator is formulated in grid-point space as a sequence of operators is the diffusion operator and is formulated in 3D as a product of a 2D (horizontal) and 1D (vertical) operator. is a diagonal matrix of volume elements, and appears in because of the self-adjointness of. The factor means iterations of the diffusion operator. Some remarks on numerical implementation

27. We can let where is a diffusion tensor that can be used to stretch and/or rotate the coordinates in the correlation model to account for anisotropic or flow-dependent structures. BCs are imposed directly within the discrete expression for using a land-ocean mask. contains normalization factors to ensure the variances of are equal to one. The diffusion approach to correlation modelling has many similarities to spline smoothing (Wahba 1982) and recursive filtering (Purser et al. 2003 - MWR). Some remarks on numerical implementation

28. GDE-generated correlation functions using “time”-implicit scheme Example: T-T correlations at the equator

29. GDE-generated correlation functions Example: flow-dependent correlations (Weaver & Courtier 2001-QJRMS; cf. Riishojgaard 1998-Tellus; Daley & Barker 2001-MWR) Depth 15 o N15 o S 15 o N Background isothermalsT-T correlations

30. Variational formulation: main point The main scientific component of the algorithm is the transformation from control space to observation space in the J o term: Interpolation Ocean model integration Multivariate balance Univariate smoothing

31. Incremental variational formulation And for incremental Var we need the linearized transformation (and its adjoint): Interpolation Linear multivariate balance Univariate smoothing 3D-Var (FGAT) 4D-Var Linear ocean model integration

32. Impact of improved covariances on the mean zonal velocity in the tropical Pacific 1993-96 climatology eastward current bias

33. Impact of in situ T (GTSPP) data assim. on the mean salinity state in the global model 3D-Var univariate (T) Control (no d.a.) Depth (m) Longitude PacificAtlanticIndian PacificAtlanticIndian 500 Equator 0 0

34. Impact of in situ T (GTSPP) data assim. on the mean salinity state in the global model 3D-Var multivariate (T, S, u, v, SSH) Control (no d.a.) PacificAtlanticIndian Longitude PacificAtlanticIndian Equator Depth (m) 500 0 0

35. Global reanalysis set-up and control Experimental set-up for ENACT –Stream 1: 1987 – 2001 –Stream 2: 1962 – 2001 –Daily mean ERA-40 surface fluxes –Weak 3D relaxation to Levitus T and S –Strong relaxation to Reynolds SST (-200 W/m 2 /K) Control: no data assimilation (streams 1 and 2) –Getting a satisfactory control run was not straightforward! Post-correction to ERA-40 precipitation to remove a tropical bias. Stronger relaxation needed at high latitudes to avoid numerical instabilities. Daily correction to global mean E-P to remove sea level drift.

36. Global reanalysis experiments (completed or currently running) 3D-Var (streams 1 and 2) –In situ T data from ENACT QC data-set –10-day window –Multivariate B (with balance) –“Incremental Analysis Updating” (IAU) (Bloom et al. 1994, MWR) 4D-Var (stream 1) –In situ T data from ENACT QC data-set –30-day window –Univariate B (no balance) –Instantaneous update

37. Cycling of 3D-Var and 4D-Var observations Background trajectory Background Analysis “Analysed” trajectory using IAU 10-day window 30-day window

38. ENACT QC historical in situ dataset (Met. Office) 200,000 – 300,000 in situ T observations / month 50,000 – 100,000 in situ S observations / month Example of T data distribution on a 10 day window Jan. 1987Jan. 1995

39. Box regions for ENACT diagnostics

40. Assimilation diagnostics Control 3D-Var Mean ( o C)Standard deviation ( o C) Depth (m) 1987-2001 global temperature statistics

41. Assimilation diagnostics Control 3D-Var Depth (m) Mean ( o C)Standard deviation ( o C) NW extra-trop Pacific 1987-2001 regional temperature statistics

42. Assimilation diagnostics Control 3D-Var Mean ( o C)Standard deviation ( o C) Nino3 Depth (m) 1987-2001 regional temperature statistics

43. Summary 3D-Var (FGAT) and 4D-Var incremental systems developed for a global version of OPA. –Major coding and validation effort required. –Clear development path towards more advanced systems. Substantial effort devoted to developing covariance models and balance operators. –Balance constraints have a significant positive impact in 3D-Var. –And a positive impact in 4D-Var with single observations (but has not yet been evaluated in real-data experiments). Production and assessment of global ocean reanalyses is ongoing (ENACT). –Preliminary results indicate that the assimilation is correcting for a large model bias in the upper ocean. –But assimilation is introducing a bias of its own below the thermocline. –Further improvements to the assimilation system are needed…

44. Future directions Background error modelling and estimation Observation error modelling and quality control Combined in situ T, S, altimeter and SST assimilation Model bias detection/correction Improving the computational efficiency of 4D-Var Ongoing reanalysis production and evaluation