1. ROMS 4-Deimensional Variational (4D-Var) Data Assimilation Algorithms COAWST Modeling System Training WHOI, Woods Hole, MA August 26, 2014 Hernan G. Arango IMCS, Rutgers University Andrew M. Moore University California Santa Cruz

2. Andy Moore – UC Santa Cruz Hernan Arango – Rutgers University Art Miller – Scripps Bruce Cornuelle – Scripps Emanuelle Di Lorenzo – GA Tech Brian Powell – University of Hawaii Javier Zavala-Garay - Rutgers University Julia Levin - Rutgers University John Wilkin - Rutgers University Chris Edwards – UC Santa Cruz Hajoon Song – MIT Anthony Weaver – CERFACS Selime Gürol – CERFACS/ECMWF Polly Smith – University of Reading Emilie Neveu – Savoie University ROMS 4D-Var Team

3. ROMS 4D-Var Data Assimilation b b (t), B b f b (t), B f x b (0), B Model solutions depends on x b (0), f b (t), b b (t), h(t) time x(t) Obs, y x b (t) x a (t)

4. that minimizes the variance given by: Find initial condition increment boundary condition increment surface forcing increment corrections for model error Background error covariance Tangent Linear Model Obs Error Cov. Innovation Data Assimilation

5. K = Kalman Gain Matrix At the minimum of J we have : Model space (control vector) search: (N model x N t ) x (N model x N t ) K K Observation space search: (N obs x N obs ) OR 4D-Variational Data Assimilation (4D-Var)

6. ROMS 4D-Var System Incremental (linearized about a prior) (Courtier et al., 1994) Primal (model grid space search) and dual (observation space search) formulations (Courtier 1997) Primal: Incremental 4D-Var (I4D-Var) Dual: Physical-space Statistical Analysis System, PSAS (4D-PSAS) (Da Silva et al, 1995); (4D-PSAS) T Dual: Indirect Representer (R4D-Var) (Egbert et al, 1994); (R4D-Var) T Strong and weak (dual only) constraint Preconditioned, Lanczos formulation of conjugate gradient (Lorenc, 2003; Tshimanga et al., 2008; Fisher, 1997) Second-level preconditioning for multiple outer-loops Diffusion operator mode for prior covariances (Derber and Bouttier, 1999; Weaver and Courtier, 2001) Multivariate balance operator for prior covariance (Weaver et al., 2001) Background quality control (Andersson and Järvinen, 1999) Physical and ecosystem components Parallel (distributed-memory, MPI) Publications: Moore et al., 2011a, b, c (Progress in Oceanography) WikiROMS Tutorials: https://www.myroms.org/wiki/index.php/4DVar_Tutorial_Introduction

7. ROMS 4D-Var Data Assimilation Systems I4D-Varprimal formulation model grid space search traditional NWP lots of experience strong constraint only (phasing out) R4D-Var dual formulation observations space search formally most correct mathematically rigorous problems with high Rossby numbers strong/weak constraint and 4D-PSAS dual formulation observation space search an excellent compromise more robust for high Rossby numbers formally suboptimal strong/weak constraint and (4D-Var) T

8. I4D-Var Algorithm (Moore et al., 2011a)

9. R4D-Var Algorithm (Moore et al., 2011a)

10. 4D-PSAS Algorithm (Moore et al., 2011a)

11. SST Increments  x(0): California Current I4D-Var 4D-PSAS R4D-Var Model Space Inner-loop 50 Observation Space Observation Space

12. ROMS Obs y, R f b, B f b b, B b x b, B , Q Posterior 4D-Var Priors & Hypotheses Clipped Analyses Ensemble (SV, SO) Hypothesis Tests Forecast dof Adjoint 4D-Var impact Term balance, eigenmodes Uncertainty Analysis error Ensemble 4D-Var ROMS 4D-VAR

13. Primal preconditioned by B has good convergence properties: Dual preconditioned by R -1 has poor convergence properties: Can be partly alleviated using the Minimum Residual Method (El Akkraoui et al., 2008; El Akkraoui and Gauthier, 2010) Restricted Preconditioned Conjugate Gradient (RPCG) ensures that dual 4D-Var converges at same rate as B- preconditioned Primal 4D-Var (Gratton and Tschimanga, 2009; Gürol et al, 2014) Preconditioned Hessian Preconditioned stabilized representer matrix 4D-Var Convergence Issues

14. Conjugate Gradient Convergence Congrad: Lanczos-based Conjugate Gradient algorithm (Fisher, 1998) MINRES:Lanczos-based Minimum Residual (El Akkraoui and Gauthier, 2010) RPCG:Lanczos-based Restricted Preconditioned Conjugate Gradient (Gürol et al, 2014) J min

15. For multiple outer-loops: Augmented Restricted B-Lanczos

16. ROMS 4D-Var Diagnostic Tools Observation impact (Langland and Baker, 2004) Observation sensitivity – adjoint of 4D-Var, (R4D-Var) T, (OSSE) (Gelaro et al., 2004) Singular value decomposition (Barkmeijer et al., 1998) Expected errors (Moore et al., 2012)

17. Observation Sensitivity, 4D-PSAS ADROMS forced by h (a vector corresponding to the velocity grid points that contribute to the transport normal to 37N over the upper 500m) Adjoint of the linearized 4D-Var system, (4D-Var) T WC13 Jan 3-7 Jan 2004, 4D-Var Cycle Based on (4D-Var) T Only available for 4D-PSAS and R4D-Var Quantifies the changes that would result in the circulation estimate, I, as result of changes in the observations or the observation array (Moore et al., 2011c) Observing System Experiments (OSEs): It can be used to predict the changes that will occur in the event of a platform failure/degradation or change in the observation array Adaptive sampling and observation array design Figure show the contribution of the observations from each platform to the total transport increment (red bar) The SSH observations increases the alongshore transport by ~0.55 Sv

18. Observation Impact: 4D-PSAS Jan 3-7 Jan 2004, 4D-Var Cycle WC13 It quantifies the contribution of each observation during a 4D-Var analysis It yields the actual contribution of each observation to the circulation increment Figure show the contribution to the increment from each part of the control vector: initial conditions (IC), surface forcing (SF), and open boundary conditions (BC) Correcting for uncertainties in both IC and SF has the largest impact on the analysis increment The observation sensitivity and impact yield the same total transport increment (I 37N ) However, the contribution of each observation platform is different. This is due to nonlinearity and the approximation to the true gain matrix, K

19. Observations Impact on Alongshore Transport

20. Total number of obs Observation Impact March 2012Dec 2012 March 2012Dec 2012 Ann Kristen Sperrevik (NMO) Observations Impact on Alongshore Transport

21. Impact of HF Radar on 37N Transport

22. Impact of MODIS SST on 37N Transport

23. Regions where ROMS 4D-Var has been used

24. A B C Grid A 10km resolution 380x400x30 Grid B 5km resolution 200x250x42 Grid C 5km resolution 198x156x42 ROMS Grids One of our major objectives is to produce the best ocean state estimate using observations and models (variational data assimilation)

25. Major Straits and Passages ① Mindoro Strait ~420m ② Panay Strait ~570m ③ Sibutu Passage ~320m ④ Dipolog Strait ~504m ⑤ Surigao Strait ~60m ⑥ San Bernadino Strait ~80m ⑦ Tablas Strait ~565m ⑧ Verde Island Passage ~70m

26. CruiseCTDTowed ADCP Moored ADCP GliderAPEX Floater Underway Surface T,S Towed CTD Time Exploratory 2007 Jun 2007 Joint Cruise 2007 Dec 2007 Regional IOP 2008 Jan 2008 Regional IOP 2009 Feb – Mar 2009Observations SST satellite data SSH altimetry HF Radar currents P W X P P P P PPP P PP P X X X X X W WWW X M M Processed for data assimilation Not suitable for data assimilation because of tides Not assimilated Instrument malfunction

27. Satellite-derived SST Products RMSE=0.75 o C

28. Sparse and Incomplete Observations Jun 6–Jul 3, 2007 CTD EM-APEX Gliders UK Met Office EN3 dataset

29. Averaged Sea Surface Temperature June 26 – July 22, 2007 Arango et al., 2011Remarks

30. Averaged Sea Surface Salinity June 26 – July 22, 2007 Arango et al., 2011

31. 2060100140 0 -100 -200 -300 Depth Station Numbers Salinity Observations 2060100140 rms error = 0.17573 Model minus Observations 2060100140 rms error = 0.090601 Model DA minus Observations 0.5 0.25 0 -0.25 -0.5 4DVar Assimilation: Salinity Model Before DA 2060100140 Model After DA 34.9 34.6 34.3 34 33.7 33.4 2060100140 49%

32. 2060100140 0 -100 -200 -300 Depth Station Numbers Temperature Observations 2060100140 rms error = 2.132 Model minus Observations 2060100140 rms error = 1.3227 Model DA minus Observations 4 2 0 -2 -4 4DVar Assimilation: Temperature Model Before DA 2060100140 Model After DA 30 25 20 15 10 5 2060100140 38%

33. Observations used in comparison: ship, glider, and APEX Forecast skill

34. Remarks To our knowledge ROMS is the only ocean community model offering all three 4D-Var systems, (4D-Var) T, and other adjoint-based algorithms ROMS 4D-Var Systems: I4D-Var, R4D-Var, 4D-PSAS Give nearly identical solutions for the same error hypothesis (Courtier, 1997 dual formulation) Fully parallel (MPI) Multivariate Balance Operator: unobserved variables information is extracted from directly observed data using linear balance relationships (Weaver et al., 2006) Efficient Lanczos-based conjugate gradient algorithms Limited-Memory Preconditioners (LMP): Spectral and Ritz (Tshimanga et al., 2008) (4D-Var) T is available for R4D-Var and 4D-PSAS systems used for observation sensitivity, OSEs, adaptive sampling, and posterior error covariance analysis

35. Digital filter – J c to suppress initialization shock (Gauthier and Thépaut, 2001) Non-diagonal R Bias-corrected 4D-Var (Dee, 2005) Time correlations in B Correlations rotated along isopycnals using diffusion tensor (Weaver and Courtier, 2001) Combine 4D-Var and EnKF (hybrid B) TL and AD of parameters Nested 4D-Var Proper Orthogonal Decomposition (POD) for biogeochemistry source and since terms (Pelc, 2013) TL and AD of sea-ice model Planned Developments

36. PhilEX Summary The Philippine Archipelago is very complex and challenging for modeling and predict ROMS forecasts without data assimilation are usually saltier at the surface when compared with the observations. The thermocline is somewhat diffused. The 4D-Var data assimilation corrects these problems: RMSE in temperature is decreased between 35% to 42% RMSE in salinity is decreased between 40% to 49% Excessive salt flux from prescribed lateral boundary conditions for salinity There are large areas in need of sampling in time and space to support and evaluate an ocean prediction system for the Philippine Archipelago

37. Publications Arango, H.G., J.C. Levin, E.N. Curchitser, B. Zhang, A.M. Moore, W. Han, A.L. Gordon, C.M. Lee, and J.B. Girton, 2011: Development of a Hindcast/Forecast Model for the Philippine Archipelago, oceanography, 20(1), 58-69, doi:10.5670/oceanog.2011.04. Fiechter, J., G. Broquet, A.M. Moore, and H.G. Arango, 2011: A data assimilative, coupled physical-biological model for the Coastal Gulf of Alaska, Dyn. Atmos. Ocean, 52, 95-118. Moore, A. M., H. G. Arango, and G. Broquet, 2011: Analysis and forecast error estimates derived from the adjoint of 4D- Var, Mon. Weather Rev., accepted. Moore, A.M., H.G. Arango, G. Broquet, B.S. Powell, A.T. Weaver, and J. Zavala-Garay, 2011a: The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems, Part I: System overview and formulation, Prog. Oceanogr., 91, 34-49, doi:10.1016/j.pocean.2011.05.004. Moore, A.M., H.G. Arango, G. Broquet, C. Edwards, M. Veneziani, B.S. Powell, D. Foley, J. Doyle, D. Costa, and P. Robinson, 2011b: The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems, Part II: Performance and Applications to the California Current System, Prog. Oceanogr., 91, 50-73, doi:10.1016/j.pocean.2011.05.003. Moore, A.M., H.G. Arango, G. Broquet, C. Edwards, M. Veneziani, B.S. Powell, D. Foley, J. Doyle, D. Costa, and P. Robinson, 2011c: The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems, Part III: Observation impact and observation sensitivity in the California Current System, Prog. Oceanogr., 91, 74-94, doi:10.1016/j.pocean.2011.05.005. Zavala-Garay, J., J. L. Wilkin, and H. G. Arango, 2011: Predictability of mesoscale variability in the East Australia Current given strong-constraint data assimilation, J. Phys. Oceanog., accepted. Zhang, W.G., J.L. Wilkin, H.G. Arango, 2010: Toward an integrated observation and modeling system in the New York Bight using variational methods. Part I: 4DVAR data assimilation, Ocean Modeling, 35, 119-133.