ROMS 4-Deimensional Variational (4D-Var) Data Assimilation Algorithms

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

2. ROMS 4D-Var Team • Andy Moore– UCSanta 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 • SelimeGürol– CERFACS/ECMWF • Polly Smith – University of Reading • Emilie Neveu– Savoie University

3. 4D-Var Data Assimilation fb(t), Bf ROMS bb(t), Bb xb(0), B Obs, y xb(t) x(t) xa(t) time Model solutions depends on xb(0), fb(t), bb(t), h(t)

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

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

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 (Anderssonand 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-Vardual 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 Incrementsdx(0): California Current Inner-loop 50 I4D-Var 4D-PSAS R4D-Var Model Space Observation Space Observation Space

12. Ensemble 4D-Var fb, Bf ROMS Obs y, R h, Q bb, Bb xb, B 4D-Var Priors & Hypotheses Hypothesis Tests Posterior Forecast Term balance, eigenmodes Clipped Analyses Ensemble (SV, SO) ROMS 4D-VAR dof Adjoint 4D-Var impact Uncertainty Analysis error

13. 4D-Var Convergence Issues • Primal preconditioned by Bhas good convergence properties: • Dual preconditioned by R-1 has poor convergence properties: • Can be partly alleviated using the Minimum Residual Method (El Akkraouiet 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

14. Conjugate Gradient Convergence Jmin 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)

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

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 • 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 Jan 3-7 Jan 2004, 4D-Var Cycle 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

18. Observation Impact: 4D-PSAS • 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 (𝜹I37N) • However, the contribution of each observation platform is different. This is due to nonlinearity and the approximation to the true gain matrix, K Jan 3-7 Jan 2004, 4D-Var Cycle WC13

19. Observations Impact on Alongshore Transport

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

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

25. Major Straits and Passages Mindoro Strait ~420m Panay Strait ~570m Sibutu Passage ~320m Dipolog Strait ~504m Surigao Strait ~60m San Bernadino Strait ~80m Tablas Strait ~565m Verde Island Passage ~70m

26. Observations • SST satellite data • SSH altimetry • HF Radar currents P X X P P W P X P X P X P P P P M P X W W W P Processed for data assimilation X Not suitable for data assimilation because of tides Not assimilated W Instrument malfunction M

27. Satellite-derived SST Products RMSE=0.75oC

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

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

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

31. 4DVar Assimilation: Salinity Salinity Observations Model Before DA Model After DA 0 34.9 34.6 -100 34.3 Depth 34 -200 33.7 33.4 -300 20 60 100 140 20 60 100 140 20 60 100 140 Station Numbers Model minus Observations Model DA minus Observations 0.5 0.25 0 -0.25 -0.5 20 60 100 140 20 60 100 140 rms error = 0.090601 rms error = 0.17573 49%

32. 4DVar Assimilation: Temperature Temperature Observations Model Before DA Model After DA 0 30 25 -100 20 Depth 15 -200 10 5 -300 20 60 100 140 20 60 100 140 20 60 100 140 Station Numbers Model minus Observations Model DA minus Observations 4 2 0 -2 -4 20 60 100 140 20 60 100 140 rms error = 1.3227 rms error = 2.132 38%

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

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. Planned Developments • Digital filter – Jcto 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

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.