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

STRONG Constraint WEAK Constraint (A)(B) …we want to find the corrections e Best Model Estimate (consistent with observations) Initial Guess ASSIMILATION Goal

STRONG Constraint WEAK Constraint (A)(B) …we want to find the corrections e ASSIMILATION Goal Best Model Estimate Initial Guess Corrections

ASSIMILATION Goal Best Model Estimate Initial Guess Corrections

ASSIMILATION Goal Best Model Estimate Initial Guess Corrections

ASSIMILATION Goal Best Model Estimate Initial Guess Corrections

ASSIMILATION Goal Best Model Estimate Initial Guess Corrections Tangent Linear Dynamics

ASSIMILATION Goal Tangent Linear Propagator Best Model Estimate Initial Guess Corrections Integral Solution

ASSIMILATION Goal Best Model Estimate Initial Guess Corrections

ASSIMILATION Goal Best Model Estimate Initial Guess Corrections The Observations

ASSIMILATION Goal Best Model Estimate Initial Guess Corrections Data misfit from initial guess

ASSIMILATION Goal Data misfit from initial guess def: is a mapping matrix of dimensions observations X model space

ASSIMILATION Goal Data misfit from initial guess def: is a mapping matrix of dimensions observations X model space

Quadratic Linear Cost Function for residuals is a mapping matrix of dimensions observations X model space

2) corrections should not exceed our assumptions about the errors in model initial condition. 1) corrections should reduce misfit within observational error Quadratic Linear Cost Function for residuals is a mapping matrix of dimensions observations X model space

Minimize Linear Cost Function

4DVAR inversion Hessian Matrix def:

4DVAR inversion Representer-based inversion Hessian Matrix def:

4DVAR inversion Hessian Matrix Stabilized Representer Matrix Representer Coefficients Representer Matrix def: Representer-based inversion

4DVAR inversion Hessian Matrix Stabilized Representer Matrix Representer Coefficients Representer Matrix def: Representer-based inversion

An example of Representer Functions for the Upwelling System Computed using the TL-ROMS and AD-ROMS

An example of Representer Functions for 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 Di Lorenzo, Miller, Cornuelle and Moisan  Intra-Americas Seas Real-Time DA Powell, Moore, Arango, Di Lorenzo, Milliff et al.

Coastal Baroclinic Upwelling System Model Setup and Sampling Array section

1) The representer system is able to initialize the forecast extracting dynamical information from the observations. 2) Forecast skill beats persistence Applications of inverse ROMS:  Baroclinic coastal upwelling: synthetic model experiment to test inverse machinery 10 day assimilation window 10 day forecast

SKILL of assimilation solution in Coastal Upwelling Comparison with independent observations SKIL L DAYS Climatology Weak Strong Persistence Assimilation Forecast  Di Lorenzo et al. 2007; Ocean Modeling

Day= 0 Day= 2 Day= 6 Day= 10

Day= 0 Day= 2 Day= 6 Day= 10 Assimilation solutions

Day= 14 Day= 18 Day= 22 Day= 26

Day= 14 Day= 18 Day= 22 Day= 26

Forecast Day= 14 Day= 18 Day= 22 Day= 26 Day= 14 Day= 18 Day= 22 Day= 26

April 3, 2007 Intra-Americas Seas Real-Time DA Powell, Moore, Arango, Di Lorenzo, Milliff et al.

CalCOFI Reanlysis: produce ocean estimates for the CalCOFI cruises from 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) …careful  Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics)

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  Coastal data assimilation is STILL a science question (e.g. model biases and Gaussian statistics assumption, inadequate error covariances)  Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics)

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  Coastal data assimilation is STILL a science question (e.g. model biases and Gaussian statistics assumption, inadequate error covariances)  Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics)

Assimilation of SSTa True True Initial Condition

True True Initial Condition Which model has correct dynamics? Model 1Model 2 Assimilation of SSTa

True True Initial Condition Wrong Model Good Model Model 1Model 2

Time Evolution of solutions after assimilation Wrong Model Good Model DAY 0

Time Evolution of solutions after assimilation Wrong Model Good Model DAY 1

Time Evolution of solutions after assimilation Wrong Model Good Model DAY 2

Time Evolution of solutions after assimilation Wrong Model Good Model DAY 3

Time Evolution of solutions after assimilation Wrong Model Good Model DAY 4

Model 1Model 2 True True Initial Condition Wrong Model Good Model What if we apply more background constraints?

Model 1Model 2 Assimilation of data at time True True Initial Condition

True Gaussian Covariance Explained Variance 24%Explained Variance 83% Explained Variance 99%Explained Variance 89% True Initial Condition Weak Constraint Strong Constraint

True Gaussian Covariance Explained Variance 24%Explained Variance 83% Explained Variance 99%Explained Variance 89% True Initial Condition Weak Constraint Strong Constraint

RMS difference from TRUE Observations Days RMS Less constraint More constraint

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  Coastal data assimilation is STILL a science question (e.g. model biases and Gaussian statistics assumption, inadequate error covariances)  Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics)

A HV =0 A HT =0 A HV =4550 A HT =1000 A HV =4550 A HT =0 INSTABILITY of Linearized model SST [C] Initial Condition Day=5

INSTABILITY of the linearized model (TLM) TLM A HV =4550 A HT =4550 TLM A HV =4550 A HT =1000 TLM A HV =0 A HT =0 Non Linear Model Initial Guess Misfit DAY=5

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  Coastal data assimilation is STILL a science question (e.g. model biases and Gaussian statistics assumption, inadequate error covariances)  Linear sensitivity are not always great! (e.g. Instability of Tangent linear dynamics)

..need research to properly setup a coastal assimilation/forecasting system  Improve model seasonal statistics using surface and open boundary conditions as the only controls.  Predictability of mesoscale flows in the CCS: explore dynamics that control the timescales of predictability. Mosca et al. – (Georgia Tech)

Download: ROMS components Arango H. IOM components Muccino, J. et al. Chua and Bennet (2002) Inverse Ocean Modeling Portal

inverse machinery of ROMS can be applied to regional ocean climate studies …

EXAMPLE: Decadal changes in the CCS upwelling cells Chhak and Di Lorenzo, 2007; GRL

SSTa Composites Observed PDO index Model PDO index Warm PhaseCold Phase Chhak and Di Lorenzo, 2007; GRL

W -130W -120W 30N 40N 50N W -130W -120W 30N 40N 50N COLD PHASE ensemble average WARM PHASE ensemble average April Upwelling Site Pt. Conception Chhak and Di Lorenzo, 2007; GRL Pt. Conception depth [m] Tracking Changes of CCS Upwelling Source Waters during the PDO using adjoint passive tracers enembles

Concentration Anomaly Model PDO PDO lowpassed Surface 0-50 meters (-) meters (-) meters year Changes in depth of Upwelling Cell (Central California) and PDO Index Timeseries Chhak and Di Lorenzo, 2007; GRL Adjoint Tracer

Arango, H., A. M. Moore, E. Di Lorenzo, B. D. Cornuelle, A. J. Miller, and D. J. Neilson, 2003: The ROMS tangent linear and adjoint models: A comprehensive ocean prediction and analysis system. IMCS, Rutgers Tech. Reports. Moore, A. M., H. G. Arango, E. Di Lorenzo, B. D. Cornuelle, A. J. Miller, and D. J. Neilson, 2004: A comprehensive ocean prediction and analysis system based on the tangent linear and adjoint of a regional ocean model. Ocean Modeling, 7, Di Lorenzo, E., Moore, A., H. Arango, Chua, B. D. Cornuelle, A. J. Miller, B. Powell and Bennett A., 2007: Weak and strong constraint data assimilation in the inverse Regional Ocean Modeling System (ROMS): development and application for a baroclinic coastal upwelling system. Ocean Modeling, doi: /j.ocemod References

Data point Assimilation tool Italian constraint New challenges for young coastal oceanographers data assimilators

New challenges for young oceanographers

Model-Data Misfit (vector) Parameters (vector) Model (matrix) Error (vector) e.g. Correction to Initial condition Correction to Boundary or Forcing Biological or Mixing parameters more

Reconstructing the dispersion of a pollutant X km Y km [conc] TIME = 100 Where are the sources? You only know the solution at time=100

Assume you have a quasi perfect model, where you know diffusion K, velocity u and v (1) Least Square Solution Where x (the model parameters) are the unkown, y is the values of the tracers at time=100 (which you know) and E is the linear mapping of the initial condition x into y. Matrix E needs to be computed numerically.

True Solution Reconstruction Initial time Initial time lsq. estimate Final time Final time lsq. estimate

Assume you guess the wrong model. Say you think there is only diffusion (1) Least Square Solution

True Solution Reconstruction Initial time Initial time lsq. estimate Final time Final time lsq. estimate Solution looks good at final time, but initial conditions are completely wrong and the values too high

Limit the size of the model parameters! (which means that the initial condition cannot exceed a certain size) (3) Weighted and Tapered Least Square Solution

True Solution Reconstruction Initial time Initial time lsq. estimate Final time Final time lsq. estimate Solution looks ok, the initial condition is still unable to isolate the source, given that you have a really bad model not including advection. However the initial condition is reasonable with in the diffusion limit, and the size of the initial condition is also within range.

Say you guess the right model however velocities are not quite right is the error in velocity (1) Least Square Solution Let us try again the strait least square estimate

True Solution Reconstruction Initial time Initial time lsq. estimate Final time Final time lsq. estimate Solution looks great, but again the initial condition totally wrong both in the spatial structure and size. So in this case a small error in our model and too much focus on just fitting the data make the lsq solution useless in terms of isolating the source.

Again limit the size of the model parameters! (3) Weighted and Tapered Least Square Solution

True Solution Reconstruction Initial time Initial time lsq. estimate Final time Final time lsq. estimate Solution looks good, the initial condition is able to isolate the sources, the size of the initial condition is within the initial values.

If you do not have the correct model, it is always a good idea to constrain your model parameters, you will fit the data less but will have a smoother inversion. What have we learned?

Weak and Strong Constraint 4D variational data assimilation for coastal/regional applications

Inverse Regional Ocean Modeling System (ROMS) Chua and Bennett (2001) Inverse Ocean Modeling System (IOMs) Moore et al. (2004) NL-ROMS, TL-ROMS, REP-ROMS, AD-ROMS To implement a representer-based generalized inverse method to solve weak constraint data assimilation problems a representer-based 4D-variational data assimilation system for high-resolution basin-wide and coastal oceanic flows Di Lorenzo et al. (2007)

Non Linear Model Tangent Linear Model Representer Model Adjoint Model Sensitivity Analysis Data Assimilation 1) Incremental 4DVAR Strong Constrain 2) Indirect Representer Weak and Strong Constrain 3) PSAS Ensemble Ocean Prediction Stability Analysis Modules ROMS Block Diagram NEW Developments Arango et al Moore et al Di Lorenzo et al. 2007

Adjoint passive tracers ensembles physical circulation independent of

Australia Asia USA Canada Pacific Model Grid SSHa (Feb. 1998) Regional Ocean Modeling System (ROMS)

Model 1Model 2 True True Initial Condition Wrong Model Good Model What if we apply more smoothing?

Model 1Model 2 Assimilation of data at time True True Initial Condition

COLD PHASE ensemble average WARM PHASE ensemble average April Upwelling Site Pt. Conception Chhak and Di Lorenzo, 2007; GRL

What if we really have substantial model errors?

Current application of inverse ROMS in the California Current System (CCS): 1)CalCOFI Reanlysis: produce ocean estimates for the CalCOFI cruises from NASA - Di Lorenzo, Miller, Cornuelle and Moisan 2)Predictability of mesoscale flow in the CCS: explore dynamics that control the timescales of predictability. Mosca and Di Lorenzo 3)Improve model seasonal statistics using surface and open boundary conditions as the only controls.

Comparison of SKILL score of IOM assimilation solutions with independent observations HIRES: High resolution sampling array COARSE: Spatially and temporally aliased sampling array

RP-ROMS with CLIMATOLOGY as BASIC STATE RP-ROMS with TRUE as BASIC STATE RP-ROMS WEAK constraint solution Instability of the Representer Tangent Linear Model (RP-ROMS) SKILL SCORE

TRUE Mesoscale Structure SSH [m] SST [C] ASSIMILATION Setup California Current Sampling: (from CalCOFI program) 5 day cruise 80 km stations spacing Observations: T,S CTD cast 0-500m Currents 0-150m SSH Model Configuration: Open boundary cond. nested in CCS grid 20 km horiz. Resolution 20 vertical layers Forcing NCEP fluxes Climatology initial cond.

SSH [m] WEAK day=5 STRONG day=5 TRUE day=5 ASSIMILATION Results 1 st GUESS day=5

WEAK day=5 STRONG day=5 ASSIMILATION Results ERROR or RESIDUALS SSH [m] 1 st GUESS day=5

WEAK day=0 STRONG day=0 TRUE day=0 Reconstructed Initial Conditions 1 st GUESS day=0

Normalized Observation-Model Misfit Assimilated data: TS 0-500m Free surface Currents 0-150m T S V U  observation number Error Variance Reduction STRONG Case = 92% WEAK Case = 98%