1. FastOpt A road map to an adjoint analysis of the summer 2007 low in Arctic sea-ice area‏ Frank Kauker, Thomas Kaminski, Michael Karcher, Ralf Giering, Rüdiger Gerdes and Michael Vossbeck OASys, FastOpt and AWI http://FastOpt.com http://OASys-Research.de http://awi.de

2. FastOpt Goals ADNOASIM and NAOSIMDAS Construct the adjoint of the coupled sea ice-ocean NAOSIM (ADNAOSIM) by means of the compiler tool TAF (Giering and Kaminski, 1998) NAOSIM was developed at the AWI (Gerdes et al. 2003, Karcher et al. 2005, Kauker et al, 2005)‏ Calculate adjoint sensitivities with ADNOASIM Build a data assimilation system around ADNAOSIM - NAOSIMDAS Perform 4DVAR data assimilation for two time periods: – 1979 to 1981 “high” ice cover, start of remote-sensing data – most recent 2006(7) to 2008(9) “low “ ice cover

3. FastOpt Test configuration Integration time: 1 hour to 1 year Time step: 1 hour 2 x 2 degree horizontal resolution 11 vertical layers Model domain: north of about 50N

4. FastOpt “Wish” configuration Integration time: up to 3 years Time step: 1/2 hour 0.5 x 0.5 degree horizontal resolution 20 vertical layers Model domain: north of about 50N

5. FastOpt Forward integration: Ice extent summer 2007 NSIDC 09/2007NAOSIM 09/2007 NAOSIM 1/12° forced with daily NCEP reanalysis (Gerdes et al., 2008)

6. FastOpt Forward integration: Ice extent summer 2007 NSIDC data includes the Sea of Okhotsk and Bering Sea - model data NOT (Gerdes et al., 2008)

7. FastOpt Forward integration: Ice extent summer 2007 (Gerdes et al., 2008)

8. FastOpt f1f1 f2f2 Statements in code define elementary functions such as +, /, **, sin, exp … fNfN …… Numerical Model f N-1 target quantity (of physical or societal relevance)‏ vector of parameters, initial and boundary conditions Sensitivities via AD

9. FastOpt Sensitivities via AD: Tangent Df 1 Df 2 Df N …… Tangent Linear Model Df N-1 Applies chain rule in forward direction Derivatives of elementary functions are simple, they define local Jacobians Cost of gradient evaluation proportional to length of control vector : One run of TLM per gradient component

10. FastOpt Sensitivities via AD: Adjoint Adjoint Model Df 1 Df 2 Df N …… Df N-1 Applies chain rule in reverse direction Cost of gradient evaluation independent of length of control vector: One run of adjoint for entire gradient But: Reversal of control flow complicates coding

11. FastOpt Compiler Tool TAF TAF (http://FastOpt.com) is a commercial tool for automatic differentiationhttp://FastOpt.com It accepts a model code in Fortran 77-95 and generates a derivative code Can generate tangent linear or adjoint code Can do vector mode (many perturbations at a time)‏ Higher order derivative (e.g., Hessian) code by invoking TAF recursively Command line tool with many options Generated code is structured and well-readable Can handle black box routines (via TAF flow directives)‏ Supports parallelisation (MPI/OpenMP)‏

12. FastOpt Adjoint sensitivity: Ice area summer 2007 target variable y : total ice area A at the end of the integration (t end ) August 2007 input x: all sbc (monthly), model parameter, initial state initialize the model at 1. Mar. 2007 run ADNAOSIM for 6 months output: dA(t end )/dx calculate monthly sbc and initial state anomalies for 2007 rel. to long-term mean gives you the impact of the sbc anomaly on the ice area Results will be presented at the EGU (session OS7) Here adjoint sensitivities for 1979 shown!

13. FastOpt Adjoint sensitivity: Ice area summer 2007 wind stress [10^4 km2/(N/m2)] 2mT [10^4 km2/K] Surface boundary conditions:

14. FastOpt Sensitivities via AD: Test gradient adjoint gradient tangent linear gradient

15. FastOpt Adjoint sensitivity: Ice area summer 2007 Show movie!

16. FastOpt Adjoint sensitivity: Ice area summer 2007 Model parameters:

17. FastOpt Adjoint sensitivity: Ice area summer 2007 Initial state: ocean SST [10^2 km2/K] Temp 60m [10^2 km2/K]

18. FastOpt Adjoint sensitivity: Ice area summer 2007 Initial state: ice ice thickness [10^2 km2/m] ice concentration [10^2 km2]

19. FastOpt Variational Data Assimilation Notation: s : state vector (ocean: u’,v’,s,tpot,ψ ; ice: h,a,age,hsn)‏ t : time d : vector of observations σ : vector observational uncertainties Principle: define vector of control variables x, e.g., initial state (s 0 )‏ forcing/boundary conditions (f)‏ internal model parameters (p)‏ define quality of fit by cost function: minimise J(x) by variation of x d 1 (obs. 1)‏ t s uncertainty for obs. termuncertainty for prior term

20. FastOpt Minimisation Efficient minimisation algorithms use J(x) and the gradient of J(x) in an iterative procedure. Typically the prior value is used as starting point of the iteration. The gradient is helpful as it always points uphill. The adjoint is used to provide the gradient efficiently. Example: Newton algorithm for minimisation Gradient: g(x) = dJ/dx(x)‏ Hessian: H(x) = dg/dx(x) = d 2 J/dx 2 (x)‏ At the minimum, x min : g(x min ) = 0, hence: g(x) = g(x) – g(x min ) ~ H (x) (x-x min )‏ rearranging yields: (x min - x) ~ - H -1 (x) g(x)‏ Smart gradient algorithms use an approximation of H(x)‏ Figure: Tarantola (1987)‏ Figure: Fischer (1996)‏ very high dimensional space

21. FastOpt Next Steps: Adjoint Sensitivities auf summer 2007 ice Speed up adjoint (storing on tape vs. in RAM)‏ Check pointing Make code “smoother” Determine the “assimilation window” First assimilation in coarse resolution of 1979-81 (this summer)‏ Gather data for most recent period

22. FastOpt Next Steps: Data streams recent period: We plan to use almost all DAMOCLES sea ice-ocean data which will be available. AON data are very welcome!

23. FastOpt Next Steps: Data streams recent period: We indent to assimilate ice drift products (met.no) We are gathering ARGO profiles (http://www.coriolis.eu.org) ARGO profile distribution for 2006

24. FastOpt S4D - 4DVar Coupled Ice-Ocean Working Group: The DAMOCLES partners FastOpt and OASys, as well as the US institutes MIT and JPL intend to collaborate on the development of variational data assimilation methods in a working group on polar regions coupled sea ice- ocean model data assimilation and adjoint sensitivities. On the US side, the main variational data assimilation system (built around the MITgcm) is operated by the ECCO consortium. Their experts for the development and maintenance of the assimilation system's coupled sea ice/ocean component are Patrick Heimbach (MIT) and Dimitris Menemenlis (JPL).

25. FastOpt S4D - 4DVar Coupled Ice-Ocean Working Group: Major questions of the working group are: ● What is the range of validity for the linearization of the coupled sea ice-ocean system, i.e., for how large a perturbation is the linearization useful? ● How does this range depend on factors such as the length of the integration period, the spatial and temporal resolution, and details of implementation? ● What are the remedies in an optimization environment (see Koehl and Willebrand, 2000) and for pure for sensitivity calculations? ● Are the adjoint sensitivities robust, i.e., what are the differences between the sensitivities from the MITgcm and those of NAOSIM? ● Which data sets (remotely sensed and in-situ ocean and sea-ice data) are most useful for data assimilation? ● Are XBT ocean data sets usable or do we have to neglect them (see Gouretzki et al. 2007)? ● Is it useful to preprocess data sets? ● How can a quality control be implemented? ● How do we systematically assign weights to the observations and other constraints? ● How do we handle correlated uncertainties? ● How do we quantify representation error? ● Are there common tasks, e.g., in terms of preparing data sets for assimilation, than can be shared between the ECCO and DAMOCLES efforts?

26. FastOpt S4D - 4DVar Coupled Ice-Ocean Working Group: The MIT/JPL/FastOpt/OASys working group is building the collaboration on two events, one one in Boston in autumn 2007 and in Hamburg 2008. ● The Working Visit Boston 2007 focussed on: Presentation of NAOSIM-DAS setup and progress, identification of key sensitivities to be compared ● The Working Visit Hamburg 2008: The WG plans a working session of 2-3 weeks in summer 2008, in Hamburg, Germany. Participants from the US side are Dimitris Menemenlis (JPL) and Patrick Heimbach (MIT). From the EU side, Ralf Giering, Thomas Kaminski, Michael Vossbeck (FastOpt), Michael Karcher, and Frank Kauker (OASys) will participate, The meeting focus will be on comparison and exchange of algorithms for handling of data streams esp. for new data types (glider data, ITP data, high resolution ice- drift), predictability issues

27. FastOpt Fini!