Representation Error in Ocean Data Assimilation Robert N. Miller James G. Richman College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, OR 97330 Representation Error in Ocean Data Assimilation – p.1/30
Data Assimilation: Assumptions Given • A model: u t − L u = f • Chosen to mimic the “true” state u ( t ) which evolve according to: t − L u ( t ) = f + b ; b r andom u ( t ) • Observations z = H u ( t ) + e obs ; H defines the relation between the state vector and the observed quantities • Question for Today: What, precisely , is u ( t ) ? Representation Error in Ocean Data Assimilation – p.2/30
In Search of the True State • The ocean measured by instruments doesn’t know about physical approximations, coarse resolution or their consequences • It is not subject to the limitations in computing power that restrict models to coarse resolution • Measurements are not subject to the same requirements for approximate physical parameterizations So ask: What quantity in nature is the “true” value of the model state? No specific answers today; Rather a suggestion for what to do while we are waiting. Representation Error in Ocean Data Assimilation – p.3/30
Representation Error • Data assimilation makes use of data misfits, aka innovations : z − H u ( f ) • u ( f ) is the forecast state u t be the “true” ocean, as the instruments • Let ˜ measure it. Representation Error in Ocean Data Assimilation – p.4/30
Representation Error Write the innovation: z − H u ( f ) = z − z ( t ) + z ( t ) − H u ( f ) = ǫ 0 + H (˜ u ( t ) − u ( t ) ) + H ( u ( t ) − u ( f ) ) • ǫ 0 = z − z ( t ) , the instrument error u ( t ) − u ( t ) ) is the representation error • H (˜ • Estimates of its statistics must appear in the terms reserved for instrument error • u ( t ) − u ( f ) is the forecast error Representation Error in Ocean Data Assimilation – p.5/30
Estimating Representation Er- ror Our method for estimating the representation error for SST: 1. Generate a long model run 2. Calculate EOFs of the model run, considered as a matrix whose ( i, j ) element is the value of state element j at time i 3. Determine the number of meaningful degrees of freedom 4. Project the innovations on the meaningful EOFs 5. Subtract the result from the innovations. 6. The difference is an estimate of the representation error Representation Error in Ocean Data Assimilation – p.6/30
Pacific Circulation Model • Parallel Ocean Program (POP) • Domain: • 105 o E to 85 o W • 30 o S to 64 o N • Resolution • 1 o at the Equator, Mercator projection • 0 . 5 o average resolution • 50 vertical levels, 25 in top 500 m • 25 years (1978-2002), forced by NCEP/DOE reanalysis • Initialized from Levitus, 30 year spinup Representation Error in Ocean Data Assimilation – p.7/30
EOFs of SST Anomalies EO F Analy sis ofSea Sur f ace Tem per at ur e Anom alie s Representation Error in Ocean Data Assimilation – p.8/30
Maps of Variance Described by SST EOFs Representation Error in Ocean Data Assimilation – p.9/30
EOF Analysis of HOT SST Anomalies EO F Analy sis ofSea Sur f ace Tem per at ur e Anom alie s ¥ The second EOF of the SST with 4% of the total variance described is dominated by variability in the strength of the subtropical gyre. In the subtropical gyre, this mode describes 30-50% of the SST variance. The SST anomaly at HOT (blue) is dominated by the second mode (red) with little contribution by the other two modes (black) Representation Error in Ocean Data Assimilation – p.10/30
Model and AVHRR Seasonal Anomalies: First EOF Representation Error in Ocean Data Assimilation – p.11/30
The Kalman Filter Given a forecast state u f and observations z , correct according to: u a = u f + K ( z − Hu f ) T + R ) − 1 K = P f H T ( HP f H • P f is the forecast error covariance matrix; K is the Kalman Gain Matrix T + R is an estimate of the covariance of • HP f H the model-data misfit • In the Kalman filter, P f evolves according to model dynamics Representation Error in Ocean Data Assimilation – p.12/30
A Static Reduced State Space Filter 1. Compute multivariate EOFs of the 23 year time series 2. Determine significant degrees of freedom: in this case the Preisendorfer (1988) test indicates 35 significant modes, accounting for 59% of the total variance 3. Estimate P f by fitting the EOFs of the SST misfits with the temperature portions of the multivariate model EOFs 4. Estimate a static Kalman gain and assimilate SST data Representation Error in Ocean Data Assimilation – p.13/30
Results of Data Assimilation Model-data correlations, before and after assimilation: Representation Error in Ocean Data Assimilation – p.14/30
Why Not Better? • The model does pretty well at what it can do. • Most of the signal variability outside of the tropics comes from eddies, and this can’t be usefully assimilated Representation Error in Ocean Data Assimilation – p.15/30
Representation Error • Project model-data misfits on multivariate EOFs. This is the portion of the data that is compatible with the model • Subtract the result from the model-data misfits. This is an estimate of the error of representation • Calculate the EOFs of the error of representation Representation Error in Ocean Data Assimilation – p.16/30
EOFs of SST Error of Represen- tation Representation Error in Ocean Data Assimilation – p.17/30
Next: The NCEP Climate Fore- cast System • Climate models are often run past their forecast horizons • Climate models can only produce forecasts consistent with their internal physics • Representation errors could take on crucial importance • Maybe a statistical characterization of representation errors can be used to devise a stochastic model to simulate them Representation Error in Ocean Data Assimilation – p.18/30
Ocean Component, NCEP CFS • Basically global MOM/POP • Resolution 1 o over most of the ocean, tapering to 0 . 33 o from 30 o N/S to 8 . 5 o • 24 years (1982-2005), 9-month forecasts, from the restart files Representation Error in Ocean Data Assimilation – p.19/30
Spectral Analysis of the CFS Restart Records Temperature EOFs and Time Series of Amplitudes 4 50 11% 2 0 0 −2 −50 −4 100 200 300 1980 1990 2000 2010 4 50 4.3% 2 0 0 −2 −50 −4 100 200 300 1980 1990 2000 2010 5 50 3.5% 0 0 −50 −5 100 200 300 1980 1990 2000 2010 Representation Error in Ocean Data Assimilation – p.20/30
Lead EOF, SSH Lead EOF, SSH 10 60 8 6 40 4 20 2 0 0 −2 −4 −20 −6 −40 −8 −10 −60 −12 50 100 150 200 250 300 350 Representation Error in Ocean Data Assimilation – p.21/30
Second EOF, SSH EOF No. 2, SSH 25 60 20 40 15 10 20 5 0 0 −5 −20 −10 −40 −15 −20 −60 −25 50 100 150 200 250 300 350 Representation Error in Ocean Data Assimilation – p.22/30
Third EOF, SSH EOF No. 3, SSH 10 60 40 5 20 0 0 −5 −20 −40 −10 −60 −15 50 100 150 200 250 300 350 Representation Error in Ocean Data Assimilation – p.23/30
Amplitude of the Lead EOF SOI and First PC 4 2 0 −2 −4 −6 −8 1985 1990 1995 2000 2005 Year Blue=lead PC; Red=SOI. Correlation ≈ 0 . 4 Representation Error in Ocean Data Assimilation – p.24/30
The Preisendorfer Test Spectrum of Model Covariance Matrix 0.12 0.1 Proportion of Total Variance 0.08 0.06 0.04 0.02 0 0 50 100 150 200 250 300 Eigenvalue Number 61 modes ≈ 60% of the total variance Representation Error in Ocean Data Assimilation – p.25/30
EOFs of SST Observations First EOF of Pathfinder SST 60 0.2 40 0 20 -0.2 0 -0.4 -20 -0.6 -40 -0.8 -60 -1 50 100 150 200 250 300 350 Representation Error in Ocean Data Assimilation – p.26/30
Significance Test for SSTA EOFs Preisendorfer Test, AVHRR SSTA 35 30 25 20 15 10 5 0 −5 0 50 100 150 200 250 Representation Error in Ocean Data Assimilation – p.27/30
Amplitude of Lead EOF of SSTA Amplitude of Pathfinder SST EOF 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 1980 1985 1990 1995 2000 2005 2010 Representation Error in Ocean Data Assimilation – p.28/30
Lead EOF of Residuals First EOF of SST Residuals 60 0.2 40 0 20 -0.2 0 -0.4 -20 -0.6 -40 -0.8 -60 -1 50 100 150 200 250 300 350 Representation Error in Ocean Data Assimilation – p.29/30
Amplitude of Lead EOF of Residuals Amplitude of SST Residuals EOF 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 1980 1985 1990 1995 2000 2005 2010 ...This is going to be harder than we thought. Representation Error in Ocean Data Assimilation – p.30/30
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