Coastal Ocean Observation Lab http://marine.rutgers.edu/cool

1. Coastal Ocean Modeling, Observation and Prediction 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 Regional Ocean Prediction http://marine.rutgers.edu/po Coastal Ocean Observation Lab http://marine.rutgers.edu/cool Education & Outreach http://coolclassroom.org Coastal Observation and Prediction Sponsors:

2. Real-time data and analysis to ships via ExView and HiSeasNet • glider, CODAR, satellite, WRF Daily Bulletin • NCOM and ROMS/assimilation 2-day forecasts Integrating Ocean Observing and Modeling Systems for SW06 Analysis and Forecasting Coastal Ocean Observation Labhttp://marine.rutgers.edu/cool/sw06/sw06.htm Regional Ocean Modeling and Predictionhttp://marine.rutgers.edu/po/sw06 • ROMS model embedded in NCOM or climatology • WRF and NCEP forcing + rivers • 2-day cycle IS4DVAR assimilation • gliders and CODAR • satellite SST, bio-optics • high-res regional WRF atmospheric forecast • SW06 ship-based obs. • 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. 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 Regional Ocean Modeling and Predictionfor Shallow Water 2006

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 4DVARTalagrand & 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: • Choose an • Integrate NLROMS and compute J • Integrate ADROMS to get • Compute • Use a descent algorithm to determine a “down gradient” correction to that will yield a smaller value of J • Back to (2) until converged. But actually, it doesn’t converge well!

8. xb= model state at end of previous cycle, and 1st 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, xa, to minimize Jb+ S(Jo) 0 1 2 3 4 time Observations minus Previous Forecast Adjoint model integration is forced by the model-data error dx

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 • Choose an • 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. The Devil is in the Details Basic IS4DVAR* procedure*Incremental Strong Constraint 4-Dimensional Variational Assimilation • Choose an • 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

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 Bu (2) balanced component Kb • Unbalanced component ---- diagonal matrix of background error standard deviation ---- symmetric matrix of background error correlation • for preconditioning, • Use diffusion operator to get C1/2: 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. SW06 Model Domains ROMS LATTE outer boundary ROMS SW06 outer boundary Harvard Box (100kmx100km)

16. 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

18. Salt 5m Salt 30m Temp 30m

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

22. 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 Mesoscale prediction test case:East Australian Current

23. East Australian Current

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

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

26. Optimal Perturbation Analysis Singular Vector 1 Perturbation after 10 days Vertical Structure of SV1 After assim. SSH+SST+XBT After assim SSH+SST

28. 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

29. SST Mixing of the Hudson and Raritan Rivers Visible RGB Detritus Absorption PhytoplanktonAbsorption SeaWiFS chlorophyll