1. The OR-WA coastal ocean forecast system Initial hindcast assimilation tests 1 Goals for the COMT project: -DA in presence of the Columbia River -Develop the OR-WA forecast model -Try the ensemble-based covariance for the initial conditions (the hybrid Ens- 4DVAR)

2. 8 Jan, 2009 27 June, 2009 8 June, 2009 SST (the plume is relatively colder in winter, warmer in summer) (no DA) 2

3. (1) AVHRR (2) model, no CR (3) model, CR (3) minus (2) SST, May 2009 (no DA)

4. Meridional wind stress (ROMS) Salinity at z=-2 m (obs, model no-CR, model CR) Temperature at z=-2 m Time series, mooring NH10 location (44.65N): (data: courtesy M. Levine, P. M. Kosro, C. Risien) (no DA) 4

5. Initial DA test Available data (assimilated as daily ave.) - AVHRR SST - HF radar surface velocities - Alongtrack SSH (minus mean along the track) 6/22/2009 6/23 6/24 6/25 6/26 6/27 Assimilation window: 22-24 June forecast: 25-27 June

6. → min Day-night composite AVHRR SST HF radar daily ave maps (P. M. Kosro) 4DVAR = dynamically based time- and space- interpolation of data analysis forecast time present - 3 days Along-track altimetry (AVISO) forecast (prior) - Assimilate data in a 3-day interval (TL&ADJ AVRORA) - De-tide the prior - Correct initial conditions in the recent past - Run forecast model (ROMS) with improved initial conditions 6

7. 3-day averaged RMSE prior inverse (std. dev. error in SST: 1.5C) inverse (std. dev. error in SST: 0.5C) analysis forecast

8. Representer method details: correction = linear combination of representers. K= total number of data R=representer matrix (all representers sampled at all data locations) To find b, solve the linear system (or, minimize the quadratic functional f(b)) To obtain a representer: 1 ADJ + 1 TL model run CGM: Solve Pb=h iteratively

9. CGM: search for “P”-orthogonal (conjugate) directions p 1, p 2,… On each iteration, minimize f(b) over a span of {p k } p i Pp j =0 (i  j) Preconditioning: provide a matrix A that is easy to invert, s.t. A -1/2 P A -1/2 is better conditioned than P. CGM only requires A -1 z

10. Egbert’s preconditioning (may be found in Bennett’s textbook): Compute a subset of representers directly Form the preconditoining matrix from the computed representers ADJ TL ADJ TL ADJ TL …  A A ADJ TL CGM (each step requires 1 ADJ and 1 TL run)

11. “Inner loop” convergence for a given choice of model and data error variances: No preconditioning “Re-orthogonalization” of p k Preconditioning w/ A=C d (Carrier et al., 2014) Preconditioning with 280 representers (K=50,191)

12. Model vs/ data errors: The IC error covariance: based on the balanced operator (Weaver et al. (2005)) dSST=0.5C Model error std. dev. Assumed data error std. dev SST (C)0.421.5 SSH (m)0.030.01 u,v (m/s)0.03 for data error std. dev.: SST 1.5C, SSH 0.03 m SST 0.5C, SSH 0.03 m SST 0.5C, SSH 0.01 m RMS (R jj )