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Weak and Strong Constraint 4D variational data assimilation: Methods and Applications

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## Weak and Strong Constraint 4D variational data assimilation: Methods and Applications

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**Weak and Strong Constraint 4D variational data**assimilation:Methods and Applications Di Lorenzo, E. Georgia Institute of Technology Arango, H. Rutgers University Moore, A. and B. Powell UC Santa Cruz Cornuelle, B and A.J. Miller Scripps Institution of Oceanography Bennet A. and B. Chua Oregon State University**Short review of 4DVAR theory (an alternative derivation of**the representer method and comparison between different 4DVAR approaches) • Overview of Current applications • Things we struggle with**ASSIMILATION Goal**Initial Guess Best Model Estimate (consistent with observations) (A) (B) WEAK Constraint STRONG Constraint …we want to find the correctionse**Best Model**Estimate Corrections Initial Guess ASSIMILATION Goal (A) (B) WEAK Constraint STRONG Constraint …we want to find the correctionse**Best Model**Estimate Corrections Initial Guess ASSIMILATION Goal**Best Model**Estimate Corrections Initial Guess ASSIMILATION Goal**Best Model**Estimate Corrections Initial Guess ASSIMILATION Goal**Best Model**Estimate Corrections Initial Guess ASSIMILATION Goal Tangent Linear Dynamics**ASSIMILATION Goal**Best Model Estimate Corrections Initial Guess Integral Solution Tangent Linear Propagator**ASSIMILATION Goal**Best Model Estimate Corrections Initial Guess**ASSIMILATION Goal**Best Model Estimate Corrections Initial Guess The Observations**ASSIMILATION Goal**Best Model Estimate Corrections Initial Guess Data misfit from initial guess**ASSIMILATION Goal**is a mapping matrix of dimensions observations X model space def: Data misfit from initial guess**ASSIMILATION Goal**is a mapping matrix of dimensions observations X model space def: Data misfit from initial guess**Quadratic Linear Cost Function for residuals**is a mapping matrix of dimensions observations X model space**Quadratic Linear Cost Function for residuals**2) corrections should not exceed our assumptions about the errors in model initial condition. 1) corrections should reduce misfit within observational error is a mapping matrix of dimensions observations X model space**def:**4DVAR inversion Hessian Matrix**def:**4DVAR inversion Hessian Matrix Representer-based inversion**def:**4DVAR inversion Hessian Matrix Representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix**def:**4DVAR inversion Hessian Matrix Representer-based inversion Representer Coefficients Stabilized Representer Matrix Representer Matrix**An example of Representer Functionsfor the Upwelling System**Computed using the TL-ROMS and AD-ROMS**An example of Representer Functionsfor the Upwelling System**Computed using the TL-ROMS and AD-ROMS**Applications of the ROMS inverse machinery:**• Baroclinic coastal upwelling: synthetic model experiment to test the development • CalCOFI Reanalysis: produce ocean estimates for the CalCOFI cruises from 1984-2006. Di Lorenzo, Miller, Cornuelle and Moisan • Intra-Americas Seas Real-Time DAPowell, Moore, Arango, Di Lorenzo, Milliff et al.**Coastal Baroclinic Upwelling System Model Setup**and Sampling Array section**Applications of inverse ROMS:**• Baroclinic coastal upwelling: synthetic model experiment to test inverse machinery 1) The representer system is able to initialize the forecast extracting dynamical information from the observations. 2) Forecast skill beats persistence 10 day assimilation window 10 day forecast**SKILL of assimilation solution in Coastal**UpwellingComparison with independent observations Weak SKILL Strong Climatology Persistence Assimilation Forecast DAYS Di Lorenzo et al. 2007; Ocean Modeling**Day=0**Day=2 Day=6 Day=10**Assimilation solutions**Day=0 Day=2 Day=6 Day=10**Day=14**Day=18 Day=22 Day=26**Day=14**Day=18 Day=22 Day=26**Forecast**Day=14 Day=14 Day=18 Day=18 Day=22 Day=22 Day=26 Day=26**Intra-Americas Seas Real-Time DAPowell, Moore, Arango, Di**Lorenzo, Milliff et al. www.myroms.org/ias April 3, 2007**CalCOFI Reanlysis: produce ocean estimates for the CalCOFI**cruises from 1984-2006. Di Lorenzo, Miller, Cornuelle and Moisan**…careful**Data Assimilation is NOT a black box**…careful**Data Assimilation is NOT a black box • Typically we do not have sufficient data to constraint the models (e.g. underdetermined systems fitting vs. assimilating data)**…careful**Data Assimilation is NOT a black box • Typically we do not have sufficient data to constraint the models (e.g. underdetermined systems fitting vs. assimilating data) • Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics)**…careful**Data Assimilation is NOT a black box • Typically we do not have sufficient data to constraint the models (e.g. underdetermined systems fitting vs. assimilating data) • Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics) • Coastal data assimilation is STILL a science question (e.g. model biases and Gaussian statistics assumption, inadequate error covariances)**…careful**Data Assimilation is NOT a black box • Typically we do not have sufficient data to constraint the models (e.g. underdetermined systems fitting vs. assimilating data) • Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics) • Coastal data assimilation is STILL a science question (e.g. model biases and Gaussian statistics assumption, inadequate error covariances)**Assimilation of SSTa**True Initial Condition True**Assimilation of SSTa**True Initial Condition Which model has correct dynamics? Model 1 Model 2 True**Wrong Model**Good Model True Initial Condition Model 1 Model 2 True**Time Evolution of solutions after assimilation**Wrong Model DAY 0 Good Model**Time Evolution of solutions after assimilation**Wrong Model DAY 1 Good Model**Time Evolution of solutions after assimilation**Wrong Model DAY 2 Good Model**Time Evolution of solutions after assimilation**Wrong Model DAY 3 Good Model**Time Evolution of solutions after assimilation**Wrong Model DAY 4 Good Model**What if we apply more background constraints?**Wrong Model Good Model True Initial Condition True Model 1 Model 2**Assimilation of data at time**True Initial Condition True Model 1 Model 2