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

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Weak and Strong Constraint 4D variational data
assimilationMethods 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
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• 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

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ASSIMILATION Goal
Initial Guess
Best Model Estimate (consistent with
observations)
(A)
(B)
WEAK Constraint
STRONG Constraint
we want to find the corrections e
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ASSIMILATION Goal
(A)
(B)
WEAK Constraint
STRONG Constraint
we want to find the corrections e
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ASSIMILATION Goal
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ASSIMILATION Goal
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ASSIMILATION Goal
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ASSIMILATION Goal
Tangent Linear Dynamics
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ASSIMILATION Goal
Best Model Estimate
Corrections
Initial Guess
Integral Solution
Tangent Linear Propagator
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ASSIMILATION Goal
Best Model Estimate
Corrections
Initial Guess
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ASSIMILATION Goal
Best Model Estimate
Corrections
Initial Guess
The Observations
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ASSIMILATION Goal
Best Model Estimate
Corrections
Initial Guess
Data misfit from initial guess
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ASSIMILATION Goal
is a mapping matrix of dimensions observations X
model space
def
Data misfit from initial guess
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ASSIMILATION Goal
is a mapping matrix of dimensions observations X
model space
def
Data misfit from initial guess
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Quadratic Linear Cost Function for residuals
is a mapping matrix of dimensions observations X
model space
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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
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Minimize Linear Cost Function
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def
4DVAR inversion
Hessian Matrix
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def
4DVAR inversion
Hessian Matrix
Representer-based inversion
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def
4DVAR inversion
Hessian Matrix
Representer-based inversion
Representer Coefficients
Stabilized Representer Matrix
Representer Matrix
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def
4DVAR inversion
Hessian Matrix
Representer-based inversion
Representer Coefficients
Stabilized Representer Matrix
Representer Matrix
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An example of Representer Functions for the
Upwelling System
Computed using the TL-ROMS and AD-ROMS
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An example of Representer Functions for the
Upwelling System
Computed using the TL-ROMS and AD-ROMS
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• 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.

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Coastal Baroclinic Upwelling System Model
Setup and Sampling Array
section
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• 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
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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
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Day0
Day2
Day6
Day10
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Assimilation solutions
Day0
Day2
Day6
Day10
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Day14
Day18
Day22
Day26
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Day14
Day18
Day22
Day26
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Forecast
Day14
Day14
Day18
Day18
Day22
Day22
Day26
Day26
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Intra-Americas Seas Real-Time DAPowell, Moore,
Arango, Di Lorenzo, Milliff et al.
www.myroms.org/ias
April 3, 2007
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CalCOFI Reanlysis produce ocean estimates for
the CalCOFI cruises from 1984-2006. Di Lorenzo,
Miller, Cornuelle and Moisan
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careful
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careful

• Data Assimilation is NOT a black box

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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)

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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)

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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)

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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)

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Assimilation of SSTa
True Initial Condition
True
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Assimilation of SSTa
True Initial Condition
Which model has correct dynamics?
Model 1
Model 2
True
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Wrong Model
Good Model
True Initial Condition
Model 1
Model 2
True
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Time Evolution of solutions after assimilation
Wrong Model
DAY 0
Good Model
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Time Evolution of solutions after assimilation
Wrong Model
DAY 1
Good Model
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Time Evolution of solutions after assimilation
Wrong Model
DAY 2
Good Model
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Time Evolution of solutions after assimilation
Wrong Model
DAY 3
Good Model
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Time Evolution of solutions after assimilation
Wrong Model
DAY 4
Good Model
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What if we apply more background constraints?
Wrong Model
Good Model
True Initial Condition
True
Model 1
Model 2
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Assimilation of data at time
True Initial Condition
True
Model 1
Model 2
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Strong Constraint
Weak Constraint
True Initial Condition
Explained Variance 24
Explained Variance 83
Gaussian Covariance
Gaussian Covariance
True
Explained Variance 99
Explained Variance 89
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Strong Constraint
Weak Constraint
True Initial Condition
Explained Variance 24
Explained Variance 83
Gaussian Covariance
Gaussian Covariance
True
Explained Variance 99
Explained Variance 89
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RMS difference from TRUE
Less constraint
RMS
More constraint
Days
Observations
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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)

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INSTABILITY of Linearized model
SST C
AHV0 AHT0
Initial Condition
Day5
AHV4550 AHT1000
AHV4550 AHT0
Day5
Day5
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INSTABILITY of the linearized model (TLM)
Misfit DAY5
TLMAHV4550 AHT4550
Non Linear Model Initial Guess
TLMAHV4550 AHT1000
TLMAHV0 AHT0
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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)

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• ..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)

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Inverse Ocean Modeling Portal
Download ROMS components http//myroms.org Arang
o H. IOM componentshttp//iom.asu.edu Muccino,
J. et al. Chua and Bennet (2002)
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inverse machinery of ROMS can be applied to
regional ocean climate studies
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inverse machinery of ROMS can be applied to
regional ocean climate studies
EXAMPLEDecadal changes in the CCS upwelling
cells
Chhak and Di Lorenzo, 2007 GRL
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Observed PDO index Model PDO index
SSTa Composites
Warm Phase
Cold Phase
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2
3
4
Chhak and Di Lorenzo, 2007 GRL
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Tracking Changes of CCS Upwelling Source Waters
during the PDOusing adjoint passive tracers
enembles
WARM PHASEensemble average
COLD PHASEensemble average
April Upwelling Site
Pt. Conception
Pt. Conception
depth m
Chhak and Di Lorenzo, 2007 GRL
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Changes in depth of Upwelling Cell (Central
California) and PDO Index Timeseries
Concentration Anomaly
Adjoint Tracer
year
Model PDO PDO lowpassed Surface 0-50 meters (-)
50-100 meters (-) 150-250 meters
Chhak and Di Lorenzo, 2007 GRL
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References
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,
227-258. 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, doi10.1016/j.ocemod.2006.
08.002.
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New challenges for young coastal oceanographers
data assimilators
Italianconstraint
Assimilation tool
Data point
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New challenges for young oceanographers
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Model-Data Misfit (vector)
Parameters (vector)
Model (matrix)
Error (vector)
e.g. Correction to Initial condition Correction
to Boundary or ForcingBiological or Mixing
parameters more
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Reconstructing the dispersion of a pollutant
TIME 100
conc
Y km
X km
Where are the sources? You only know the
solution at time100
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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 time100 (which
you know) and E is the linear mapping of the
initial condition x into y. Matrix E needs to
be computed numerically.
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True Solution
Initial time
Final time
Initial time lsq. estimate
Final time lsq. estimate
Reconstruction
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Assume you guess the wrong model. Say you
think there is only diffusion
(1) Least Square Solution
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True Solution
Initial time
Final time
Initial time lsq. estimate
Final time lsq. estimate
Reconstruction
Solution looks good at final time, but initial
conditions are completely wrong and the values
too high
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Limit the size of the model parameters! (which
means that the initial condition cannot exceed a
certain size)
(3) Weighted and TaperedLeast Square Solution
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True Solution
Initial time
Final time
Initial time lsq. estimate
Final time lsq. estimate
Reconstruction
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.
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Say you guess the right model however
velocities are not quite right
is the error in velocity
Let us try again the strait least square estimate
(1) Least Square Solution
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True Solution
Initial time
Final time
Initial time lsq. estimate
Final time lsq. estimate
Reconstruction
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.
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Again limit the size of the model parameters!
(3) Weighted and TaperedLeast Square Solution
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True Solution
Initial time
Final time
Initial time lsq. estimate
Final time lsq. estimate
Reconstruction
Solution looks good, the initial condition is
able to isolate the sources, the size of the
initial condition is within the initial values.
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What have we learned?
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.
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Weak and Strong Constraint 4D variational data
assimilationfor coastal/regional applications
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Inverse Ocean Modeling System (IOMs)
Chua and Bennett (2001)
To implement a representer-based generalized
inverse method to solve weak constraint data
assimilation problems
NL-ROMS, TL-ROMS, REP-ROMS, AD-ROMS
Moore et al. (2004)
Inverse Regional Ocean Modeling System (ROMS)
Di Lorenzo et al. (2007)
a representer-based 4D-variational data
assimilation system for high-resolution
basin-wide and coastal oceanic flows
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ROMS Block Diagram NEW Developments
Stability Analysis Modules
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
Arango et al. 2003Moore et al. 2004Di Lorenzo
et al. 2007
Ensemble Ocean Prediction
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Adjoint passive tracers ensembles
physical circulation independent of
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Regional Ocean Modeling System (ROMS)
Pacific Model Grid SSHa (Feb. 1998)
Canada
Asia
USA
Australia
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What if we apply more smoothing?
Wrong Model
Good Model
True Initial Condition
True
Model 1
Model 2
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Assimilation of data at time
True Initial Condition
True
Model 1
Model 2
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Chhak and Di Lorenzo, 2007 GRL
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What if we really have substantial model errors?
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• Current application of inverse ROMS in the
  California Current System (CCS)
• CalCOFI Reanlysis produce ocean estimates for
  the CalCOFI cruises from 1984-2006. NASA - Di
  Lorenzo, Miller, Cornuelle and Moisan
• Predictability of mesoscale flow in the CCS
  explore dynamics that control the timescales of
  predictability. Mosca and Di Lorenzo
• Improve model seasonal statistics using surface
  and open boundary conditions as the only
  controls.

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Comparison of SKILL score of IOM assimilation
solutions with independent observations
HIRES High resolution sampling array
COARSE Spatially and temporally aliased
sampling array
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Instability of the Representer Tangent Linear
Model (RP-ROMS)
SKILL SCORE
RP-ROMS WEAK constraint solution
RP-ROMS with TRUE as BASIC STATE
RP-ROMS with CLIMATOLOGY as BASIC STATE
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ASSIMILATION SetupCalifornia 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. Resolution20 vertical
layersForcing NCEP fluxesClimatology initial
cond.
TRUE Mesoscale Structure
SSH m
SST C
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SSH m
ASSIMILATION Results
STRONG day5
TRUE day5
WEAK day5
1st GUESS day5
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ASSIMILATION Results
SSH m
STRONG day5
ERROR or RESIDUALS
WEAK day5
1st GUESS day5
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Reconstructed Initial Conditions
STRONG day0
TRUE day0
1st GUESS day0
WEAK day0
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Normalized Observation-Model Misfit
?
T
S
U
V
observation number
Assimilated data TS 0-500m Free surface
Currents 0-150m
Error Variance Reduction STRONG Case 92WEAK
Case 98