Observations and Ocean State Estimation: Impact, Sensitivity and Predictability Andy Moore University of California Santa Cruz Hernan Arango Rutgers University

Outline State estimation Observation impact Information content Observation sensitivity Forecast error and predictability Examples from the California Current

Ocean State Estimation and Data Assimilation From Bayes’ Theorem: Ocean state vector: PosteriorPriorGainInnovation Obs operator Kalman gain: Prior error cov Obs error cov TL obs operator may include ocean dynamics as in 4D-Var

Practicalities of 4D-Var identified iteratively using 4D-Var Typically done using conjugate gradients (CG) CG is equivalent to Lanczos algorithm: m = number of iterations Typically m<<N obs so: Practical Gain Matrix

Practicalities of 4D-Var identified iteratively using 4D-Var Typically done using conjugate gradients (CG) Practical Gain Matrix Alternatively, view CG and 4D-Var as a function: 4D-Var

4D-Var as a Function Lanczos vectors: 4D-Var Lanczos recursion relation

Two Views of 4D-Var OR Posterior PriorInnovation

The Role of Observations Q: What is the influence of the observations on the analysis? Observation impact: Observation sensitivity: Predictability:

The Role of Observations Q: What is the influence of the observations on the analysis? Observation impact: Observation sensitivity: Predictability:

The Role of Observations Q: What is the influence of the observations on the analysis? Observation impact: Observation sensitivity: Predictability:

Observation Impact Consider a function Q: How does each obs contribute to the analysis? of the circulation:

Observation Impact

Observation Sensitivity Q: How does the analysis change if the observations change? Consider again the function Let

Observation Sensitivity

Observations and Predictability Q: How does each obs contribute to the forecast predictability? Consider now an ensemble of forecast function values for obtained by perturbing priors and obs. Expected forecast error variance: Forecast error covariance

Observations and Predictability Posterior error covariance Tangent Linear 4D-Var Adjoint Linear 4D-Var Forecast error covariance Control priors Tangent linear model where:

The California Current 500m 37N Mesoscale eddies

The California Current 30km, 10 km & 3 km grids, levels Veneziani et al (2009) Broquet et al (2009) Moore et al (2010) COAMPS forcing ECCO open boundary conditions f b (t), B f b b (t), B b x b (0), B x Previous assimilation cycle The Regional Ocean Modeling System (ROMS)

Observations (y) CalCOFI & GLOBEC SST & SSH ARGO Ingleby and Huddleston (2007) Data from Dan Costa ~90% ~10% TOPP Elephant Seals (APB) EN3

Observations Data Assimilation Posterior Observations Data Assimilation Posterior Observations Data Assimilation Posterior prior Sequential Data Assimilation: July 2002-Dec days Forecast

CTD XBT ARGO TOPP All in situ data: July 2002 – Dec 2004

Observation Impact Q: How does each obs contribute to the analysis?

Observation Impacts on Transport 7day average transport Transport increment = (Posterior-Prior) 10km ROMS

Example: 37N Transport No assim With Data Assimilation JAS time mean alongshore Flow (10km, 42 lev) CC CUC CC = California Current CUC = California Under Current

Poleward Equatorward Offshore Onshore Prior alongshore transport (CC+CUC+CJ) Prior cross-shore transport

Analysis Cycle – Observation Impacts Poleward Equatorward Offshore Onshore Alongshore transport Cross-shore transport 10km ROMS rms (Moore et al, 2011c)

Analysis Cycle – Observation Impacts Poleward Equatorward Offshore Onshore Alongshore transport Cross-shore transport 10km ROMS rms (Moore et al, 2011c)

Sv (10 -5 ) Alongshore Transport Impacts IGW CTW (G) Adjoint CTW (G T ) SSH Gyre Circulation

y Observation vector x Degrees of Freedom (dof) “Perfect World” dof ~ N obs y Observation vector x Redundancy dof < N obs

Information Content of Obs Degrees of freedom of obs (dof): 30km Only ~10% of obs contain independent info Tr{KH}/N obs vs assimilation cycle Cardinali et al, 2004; Desroziers et al., 2009 Bennett & McIntosh, 1982 upper & lower bounds (Moore et al, 2011b)

Observation Sensitivity Q: How does the analysis change if the observations change? Q: How does the analysis change if the observation array changes?

Impact vs Sensitivity Single 4D-Var cycle obs Impact on 37N transport Sensitivity of 37N transport to removing observations

Observation Sensitivity: An OSSE 4D-Var change in 37N transport when all SSH removed Change in 37N transport predicted by obs sensitivity Change in 37N transport predicted by obs impact

Observations Data Assimilation Posterior Observations Data Assimilation Posterior Observations Data Assimilation Posterior prior Sequential Data Assimilation: July 2002-Dec days Forecast

Observations and Predictability Q: How does each obs contribute to the forecast predictability?

Forecast Ensembles Small spread Predictable Large spread Unpredictable Expected uncertainty of analysis

Predictability Analysis cycle ending t 0 +7d Analysis cycle ending t 0 t0t0 t 0 +7d t 0 +14dt 0 -7d Forecast cycle ending t Forecast cycle ending t Forecast ensemble

Predictability Alongshore transport Cross-shore transport positive impact of obs on predictability

Summary In situ observations have a large impact on circulation estimates, despite small number. Adjoint operators provide considerable utility for quantifying the impact and value of ocean observations. Routine monitoring of adjoint-based diagnostics → real-time monitoring of observing array. Quantification of the true value of observations.