1. Coastal Ocean Observation Lab http://marine.rutgers.edu/cool John Wilkin, Hernan Arango, Julia Levin, Javier Zavala-Garay, Gordon Zhang Regional Ocean Prediction Scott Glenn, Oscar Schofield, Bob Chant Josh Kohut, Hugh Roarty, Josh Graver Coastal Ocean Observation Lab Janice McDonnell Education and Outreach Coastal Observation and Prediction Sponsors: Regional Ocean Prediction http://marine.rutgers.edu/po Education & Outreach http://coolclassroom.org Coastal Ocean Modeling, Observation and Prediction

2. Coastal Ocean Observation Lab http://marine.rutgers.edu/cool/sw06/sw06.htm Integrating Ocean Observing and Modeling Systems for SW06 Analysis and Forecasting Regional Ocean Modeling and Prediction http://marine.rutgers.edu/po/sw06 gliders and CODAR satellite SST, bio-optics high-res regional WRF atmospheric forecast SW06 ship-based obs. ROMS model embedded in NCOM or climatology WRF and NCEP forcing + rivers 2-day cycle IS4DVAR assimilation Real-time data and analysis to ships via ExView and HiSeasNet glider, CODAR, satellite, WRF Daily Bulletin NCOM and ROMS/assimilation 2-day forecasts Model-based re-analysis of submesoscale ocean state ROMS/IS4DVAR assimilation: plus CODAR, Scanfish, moorings, CTDs … high-res nesting in SW06 center ensemble simulations; uncertainty instability, sensitivity analysis, optimal observations Weekly/monthly bulletin ?

3. Regional Ocean Modeling and Prediction for Shallow Water 2006 Rutgers Ocean Modeling and Prediction Group for SW06: –Hernan Arango –John Evans –Naomi Fleming –Gregg Foti –Julia Levin –John Wilkin –Javier Zavala-Garay –Gordon Zhang http://marine.rutgers.edu/po/sw06

4. Outline Strong constraint 4-dimensional variational data assimilation –some math –how it works SW06 configuration –some results Next steps: –SW06 reanalysis Algorithmic tuning, more data, higher resolution –ensemble simulations Forecast and analysis uncertainty and predictability –observing system design

5. Notation ROMS state vector NLROMS equation form: (1) NLROMS propagator form: Observation at time with observation error variance Model equivalent at observation points Unbiased background state with background error covariance

6. Strong constraint 4DVAR Talagrand & Courtier, 1987, QJRMS, 113, 1311-1328 Seek that minimizes subject to equation (1) i.e., the model dynamics are imposed as a ‘strong’ constraint. depends only on “control variables” Cost function as function of control variables J is not quadratic since M is nonlinear.

7. S4DVAR procedure Lagrange function Lagrange multiplier At extrema of, we require: S4DVAR procedure: (1)Choose an (2)Integrate NLROMS and compute J (3)Integrate ADROMS to get (4)Compute (5)Use a descent algorithm to determine a “down gradient” correction to that will yield a smaller value of J (6)Back to (2) until converged. But actually, it doesn’t converge well!

8. Adjoint model integration is forced by the model-data error x b = model state at end of previous cycle, and 1 st guess for the next forecast In 4D-VAR assimilation the adjoint model computes the sensitivity of the initial conditions to mis- matches between model and data A descent algorithm uses this sensitivity to iteratively update the initial conditions, x a, to minimize J b +  (J o ) Observations minus Previous Forecast xx 0 1 2 3 4 time

9. Incremental Strong Constraint 4DVAR (Courtier et al, 1994, QJRMS, 120, 1367-1387 Weaver et al, 2003, MWR, 131, 1360-1378 ) True solution NLROMS solution from Taylor series: ---- TLROMS Propagator Cost function is quadratic now

10. Basic IS4DVAR * procedure * Incremental Strong Constraint 4-Dimensional Variational Assimilation (1)Choose an (2)Integrate NLROMS and save (a) Choose a (b) Integrate TLROMS and compute J (c) Integrate ADROMS to yield (d) Compute (e) Use a descent algorithm to determine a “down gradient” correction to that will yield a smaller value of J (f) Back to (b) until converged (3) Compute new and back to (2) until converged

11. Basic IS4DVAR * procedure * Incremental Strong Constraint 4-Dimensional Variational Assimilation (1)Choose an (2)Integrate NLROMS and save (a) Choose a (b) Integrate TLROMS and compute J (c) Integrate ADROMS to yield (d) Compute (e) Use a descent algorithm to determine a “down gradient” correction to that will yield a smaller value of J (f) Back to (b) until converged (3) Compute new and back to (2) until converged The Devil is in the Details

12. Conjugate Gradient Descent (Long & Thacker, 1989, DAO, 13, 413-440) Expand step (5) in S4DVAR procedure and step (e) in IS4DVAR procedure Two central component: (1) step size determination (2) pre-conditioning (modify the shape of J ) New NLROMS initial condition: ---- step-size (scalar) ---- descent direction Step-size determination: (a) Choose arbitrary step-size and compute new, and (b) For small correction, assume the system is linear, yielded by any step-size is (c) Optimal choice of step-size is the who gives Preconditioning: define use Hessian for preconditioning: is dominant because of sparse obs. Look for minimum J in v space

13. Background Error Covariance Matrix (Weaver & Courtier, 2001, QJRMS, 127, 1815-1846; Derber & Bouttier, 1999, Tellus, 51A, 195-221) Split B into two parts: (1) unbalanced component B u (2) balanced component K b Unbalanced component ---- diagonal matrix of background error standard deviation ---- symmetric matrix of background error correlation for preconditioning, Use diffusion operator to get C 1/2 : assume Gaussian error statistics, error correlation the solution of diffusion equation over the interval with is ---- the solution of diffusion operator ---- matrix of normalization coefficients

14. Adjoint surface temperature states at different time during a three-day period. Initial adjoint forcing area is surrounded by the black frame. Top: southward wind. Bottom: northward wind.

15. The adjoint solution gives sensitivity of SST in the marked area to SST over the a 5-day assimilation interval for steady downwelling and upwelling winds

19. Harvard Box (100kmx100km) ROMS LATTE outer boundary ROMS SW06 outer boundary SW06 Model Domains

20. ROMS SW06 5-km grid for IS4DVAR testing Forcing: NCEP-NAM and WRF USGS Hudson River OTPS tides Open boundaries NCOM and L&G climatology 2-day assimilation cycle 20-km horizontal and 5-m vertical length scales in background error covariance Data: gliders, CTDs, XBTs, Knorr thermosalinograph, daily best-SST composite, AVISO SSH

22. Salt 5mSalt 30mTemp 30m

25. Forecast skill in 2-day interval when initial conditions are adjusted using IS4DVAR Simple forecast: No data assimilation

26. Mesoscale prediction test case: East Australian Current IS4DVAR assimilation daily SST (CSIRO) SSH (AVISO) VOS XBT Tasman Sea Javier Zavala-Garay John Wilkin Hernan Arango Adjoint adjusts all state variables, not just those observed Singular vectors of the tangent linear model give most unstable modes of variability –Optimal perturbations for ensemble simulation –Predictability limits

27. East Australian Current

28. Ensembles of: 1-day forecasts 8-day forecasts 15-day forecasts

29. Assimilating SSH+SST+XBT Assimilating SSH+SST East Australian Current Color: ensemble mean. Contours: individual ensemble members. Black: SSH observations

30. Optimal Perturbation Analysis After assim. SSH+SST+XBT After assim SSH+SST Vertical Structure of SV1Perturbation after 10 daysSingular Vector 1

32. Now what ? SW06 reanalysis of sub-mesoscale ocean state –IS4DVAR algorithmic tuning forecast cycle length; background error covariance (preconditions conjugate gradient search) –More data CODAR, moorings, shipboard ADCP … –Higher resolution –Ensemble simulations forecast skill; quantify predictability; analysis uncertainty MURI COMOP –Observing system design –Physics information

33. Mixing of the Hudson and Raritan Rivers Phytoplankton Absorption Detritus Absorption SeaWiFS chlorophyll Visible RGB SST