Andy Moore, UCSC Hernan Arango, Rutgers Gregoire Broquet, CNRS Chris Edwards & Milena Veneziani, UCSC Brian Powell, U Hawaii Jim Doyle, NRL Monterey Dave Foley, NOAA Pacific Grove A Comprehensive 4D-Var Data Assimilation and Analysis System Applied to the California Current System using ROMS

Acknowledgements Chris Edwards, UCSC Jerome Fiechter, UCSC Gregoire Broquet, UCSC Milena Veneziani, UCSC Javier Zavala, Rutgers Gordon Zhang, Rutgers Julia Levin, Rutgers John Wilkin, Rutgers Brian Powell, U Hawaii Bruce Cornuelle, Scripps Art Miller, Scripps Emanuele Di Lorenzo, Georgia Tech Anthony Weaver, CERFACS Mike Fisher, ECMWF ONR NSF NOPP Dan Costa Patrick Robinson

Adjoint Sensitivity Generalized Stability Analysis 4D-Var Unified ROMS System

Adjoint Sensitivity Generalized Stability Analysis 4D-Var The ROMS Trinity

“Trinity” (The Matrix, 1999)

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 ROMS 4D-Var Ensemble 4D-Var

ROMS 4D-Var Incremental (linearize about a prior) (Courtier et al, 1994) Primal & dual formulations (Courtier 1997) Primal – Incremental 4-Var (I4D-Var) Dual – PSAS (4D-PSAS) & indirect representer (R4D- Var) (Da Silva et al, 1995; Egbert et al, 1994) Strong and weak (dual only) constraint Preconditioned, Lanczos formulation of conjugate gradient (Lorenc, 2003; Tshimanga et al, 2008; Fisher, 1997) Diffusion operator model for prior covariances (Derber & Bouttier, 1999; Weaver & Courtier, 2001) Multivariate balance for prior covariance (Weaver et al, 2005) Physical and ecosystem components Parallel (MPI)

ROMS Obs y, R f b, B f b b, B b x b, B , Q Posterior 4D-Var I4-Var, R4D-Var, 4D-PSAS ROMS 4D-Var Primal & dual formulations

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 Prior Posterior ROMS Prior

Notation & Nomenclature State vector Control vector Observation vector Innovation vector Observation matrix Prior

initial condition increment boundary condition increment forcing increment corrections for model error b b (t), B b f b (t), B f x b (0), B Prior Incremental Formulation & Bayes Theorem Thomas Bayes ( ) Posterior distribution of  z:

initial condition increment boundary condition increment forcing increment corrections for model error b b (t), B b f b (t), B f x b (0), B Prior (background) error covariance Tangent Linear Model sampled at obs points Obs Error Cov. Innovation Prior Incremental Formulation & Bayes Theorem

initial condition increment boundary condition increment forcing increment corrections for model error b b (t), B b f b (t), B f x b (0), B Prior The minimum of J is identified iteratively by searching for ∂J/∂  z=0 Incremental Formulation & Bayes Theorem

The Solution Prior (background) circulation estimate: Observations: Posterior (analysis) circulation estimate: Observation matrix Gain Matrix

z Primal Space y Observation vector z Dual Space Primal vs Dual Formulation Vector of increments

The Priors for ROMS CCS 30km, 10 km & 3 km grids, levels Veneziani et al (2009) Broquet et al (2009) COAMPS forcing ECCO open boundary conditions f b (t), B f b b (t), B b x b (0), B Previous assimilation cycle

Observations (y) CalCOFI & GLOBEC SST & SSH Argo TOPP Elephant Seals Ingleby and Huddleston (2007) Data from Dan Costa

4D-Var Configuration Case studies for a representative case 3-10 March, D-Var applied sequentially, outer-loop, inner-loops 7-14 day assimilation windows Prior D: x L h =50 km, L v =30m,  from clim f L  =300km, L Q =100km,  from COAMPS b L h =100 km, L v =30m,  from clim Super observations formed Obs error R (diagonal): SSH 2 cm SST 0.4 C hydrographic 0.1 C, 0.01psu

Recall the Cost Function The aim of 4D-Var is to find the increments  z corresponding to the minimum variance (maximum likelihood) estimate: initial condition increment boundary condition increment forcing increment corrections for model error The minimum of J is identified iteratively by searching for ∂J/∂  z=0

Primal, strong Dual, strong Dual, weak Jmin 4D-Var Performance 3-10 March, 2003 (10km, 42 levels)

Observations 4D-Var Analysis Posterior Observations 4D-Var Analysis Posterior Observations 4D-Var Analysis Posterior prior Sequential 4D-Var 7-14 days Forecast

I4D-Var (primal) R4D-Var (dual) 4D-PSAS (dual) J initial J final 30km, 1X50, strong Which elements of the control vector exert the largest influence on J?

10km ROMS 30km ROMS I4D-Var, 1 outer, 10 inner Strong constraint R4D-Var, 1 outer, 50 inner Strong constraint Initial log 10 J Final log 10 J

What is most important? i.c. wind stress heat flux freshwater flux open b.c.

ROMS Obs y, R f b, B f b b, B b x b, B , Q Posterior 4D-Var Adjoint 4D-Var observation impact observation sensitivity ROMS 4D-Var

10km ROMS I4D-Var, 1 outer, 10 inner Strong constraint Initial log 10 J Final log 10 J

The California Current

t0t0 t 0 +7t x Analysis Cycle Forecast Cycle Observation Impacts on Forecast Error

Adjoint 4D-Var Observation impacts 7day average transport across 500m isobath upper 14m (Veneziani et al, 2009) 500m Transport Increment = (Posterior-Prior) (Langland & Baker, 2004; Gelaro et al., 2007) initial condition surface forcing boundary conditions Control vector impact Obs impact

rms Analysis Cycle 500m Isobath Transport Offshore Onshore Prior I(x b ) Increment  I = I(x a ) – I(x b )

Analysis Cycle 500m Isobath Transport Satellite SSH Satellite SST T XBT T CTD T Argo T TOPP S CTD S Argo Increment  I = I(x a ) – I(x b )

Correlations Balance Advection Baroclinic waves Barotropic waves Physical processes: Statistics: Average impact of satellite SST

Average impact of satellite SST Advection Horizon Baroclinic Wave Horizon (based on c 1 ~2 ms -1 Chelton et al, 1994) IGW CTW Adjoint CTW (based on v~0.1 ms -1 )

Impact of Argo Salinity Obs

ROMS Obs y, R f b, B f b b, B b x b, B , Q Posterior 4D-Var Adjoint 4D-Var observation impact observation sensitivity ROMS 4D-Var Forecast

10km ROMS I4D-Var, 1 outer, 10 inner Strong constraint R4D-Var, 1 outer, 50 inner Strong constraint Initial log 10 J Final log 10 J

Observations 4D-Var Analysis Posterior Observations 4D-Var Analysis Posterior Observations 4D-Var Analysis Posterior prior Sequential 4D-Var 7-14 days Forecast

t0t0 t 0 +7t x Analysis Cycle Forecast Cycle Overlapping Forecast Cycles Next Analysis Cycle

Forecast Error 14 day forecast of 500m isobath transport at t 0 +14, starting at t 0

t0t0 t 0 +7t x Analysis Cycle Forecast Cycle Overlapping Forecast Cycles Next Analysis Cycle

Forecast Error 14 day forecast of 500m isobath transport at t 0 +14, starting at t 0 7 day forecast of 500m isobath transport at t 0 +14, starting at t 0 +7

t0t0 t 0 +7t x Analysis Cycle Forecast Cycle Overlapping Forecast Cycles Next Analysis Cycle

Forecast Error 14 day forecast of 500m isobath transport at t 0 +14, starting at t 0 7 day forecast of 500m isobath transport at t 0 +14, starting at t 0 +7 Verifying analysis of 500m isobath transport at t 0 +14

t0t0 t 0 +7t x Analysis Cycle Forecast Cycle Overlapping Forecast Cycles Next Analysis Cycle

Forecast Error 14 day forecast of 500m isobath transport at t 0 +14, starting at t 0 7 day forecast of 500m isobath transport at t 0 +14, starting at t 0 +7 Verifying analysis of 500m isobath transport at t day forecast error

t0t0 t 0 +7t x Analysis Cycle Forecast Cycle Overlapping Forecast Cycles Next Analysis Cycle

Forecast Error 14 day forecast of 500m isobath transport at t 0 +14, starting at t 0 7 day forecast of 500m isobath transport at t 0 +14, starting at t 0 +7 Verifying analysis of 500m isobath transport at t day forecast error 7 day forecast error

t0t0 t 0 +7t x Analysis Cycle Forecast Cycle Overlapping Forecast Cycles Next Analysis Cycle

Forecast Error 14 day forecast of 500m isobath transport at t 0 +14, starting at t 0 7 day forecast of 500m isobath transport at t 0 +14, starting at t 0 +7 Verifying analysis of 500m isobath transport at t day forecast error 7 day forecast error Change in e due to assimilation of observations over [t 0 +7,t 0 +14]

Forecast Error 7 day forecast better than 14 day forecast 7 day forecast worse than 14 day forecast

500m Isobath Transport Forecast Error

SST Obs that reduce forecast error SST Obs that increase forecast error RMS Impact of SST Obs on 500m Isobath Transport Forecast Error

ROMS Obs y, R f b, B f b b, B b x b, B , Q Posterior 4D-Var Uncertainty Analysis error ROMS 4D-Var

Expected Posterior Error  b = prior error std  a = posterior error std 3 March, 2003 (10km, 42 levels) Posterior: Posterior covariance: Gain matrix

SSS Zonal wind stress Meridional wind stress Heat Flux

ROMS Obs y, R f b, B f b b, B b x b, B , Q Posterior 4D-Var Priors & Hypotheses Hypothesis Tests ROMS 4D-Var degrees of freedom degrees of reachability array modes

30km Consistency Checks in Obs Space Desroziers et al (2005): Observation matrix

Degrees of Freedom Recall that the optimal increments minimize: No. of dof in obs No. of dof in prior “dof” – degrees of freedom Theoretical min: (Bennett et al, 1993; Cardinali et al, 2004; Desroziers et al., 2009)

Assimilation cycle ( ) Log10(J) dof of obs (30km, 30 level, dual, strong, sequential, 7 day, 200 inner-loops) Less than 10% of all observations provide independent info LOTS OF REDUNDANCY! J b >(J b ) min and indicates over fitting to the obs J≠J min and indicates that prior hypotheses are incorrect

Adjoint Sensitivity Generalized Stability Analysis 4D-Var The ROMS Trinity

Summary and Conclusions

Assimilation impacts on CC No assim Strong Constraint 4D-Var Time mean alongshore flow (37N) (10km, 42 lev) Broquet et al (2009)

Time series of  a -  b (Sv) rms

Correlations Balance Advection Baroclinic waves Barotropic waves Physical processes: Statistics: Average impact of satellite SST (Sv)

Average impact of satellite SST Advection Horizon Baroclinic Wave Horizon (based on c 1 ~2 ms -1 Chelton et al, 1994) IGW CTW Adjoint CTW (Sv)

Average impact CalCOFI & GLOBEC Salinity Obs Advection Horizon Baroclinic Wave Horizon (Sv)