Budgets and Bias in Data Assimilation

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Budgets and Bias in Data Assimilation Keith
Haines, ESSCDARC, Reading

• Background Marine Informatics
• Assimilation algorithms in Ocean circulation
  models
• Satellite and In Situ data sets
• Physically based covariances simple errors in
  big and Biased models
• Budget diagnostics based on assimilation
• Met Office FOAM, ECMWF Seasonal Forecasting
  collaborations
• DARC-NCOF Fellow Dan Lea based in NCOF group at
  Met Office
• New project (Marine Quest) will look at
  assimilation constraints on Carbon within a
  coupled physics-biochemistry ocean model
• e-Science/Grid Model and Satellite data viewed
  in Google Maps/Earth
• http//lovejoy.nerc-essc.ac.uk8080/Godiva2

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Budgets and Ocean Thermohaline Circulation
Ocean Box-Inverse solution Ganachaud and Wunsch
(2000)
After Broeker

• Closed Budgets of ..
• Heat, Salt, Mass/Volume, Tracers..
• Processes Advection, Surface fluxes, Mixing,
  Data Assimilation

Transport in Sverdrups 1Sv 106 m3 s-1
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Ocean Box-Inverse Assimilation

• Key assumption is for Steady State system
• Therefore can use asynoptic data (different ocean
  sections observed at completely different times)
• Try to correct for known variability eg. Seasonal
  cycle (surface properties and wind induced
  transports)
• Deduce unknown box-exchanges (circulation and
  mixing rates) for closed system
• Often problem underconstrained gt use some Occams
  razor or conditioning assumption (smallest
  consistent flows/mixing rates)

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Transport in Sverdrups 1Sv 106 m3 s-1
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N. Atlantic Water Budgetby density class
(11S-80N)
27.72 28.11
COADS surface fluxes CTD section at 11S Steady
State (cf. Ocean Inverse) gt Mixing
Transformation Flux (Sv)
Speer (1997)
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Walin Budget diagnostics for HadCM3 climate model
(100yr average)
Transformation Flux (Sv)
27.72 28.11
Old and Haines 2006
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Data Assimilation in a time-evolving model?

• Steady state box-inverse models estimate process
  rates or parametrisations like mixing from a 3D
  Variational problem
• Similar Parameter Estimation while matching
  timeevolving data often uses 4DVar Assimilation
• 4DVar very expensive computationally
• The budget within a box concept is subsumed
  into seeking a solution to the temporal model
  equations
• Parameter tuning assumes process representations
  are structurally correct
• Different approach Assimilation corrects for
  model bias so evaluate assimilation as another
  process within Box Budgets
• A posteriori Process Estimation

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Process Estimation v. Parameter Estimation
Parameter estimation 4DVar. Cost function
containing fit to observations, a-priori
info. Tune initial state, sources/sinks, model
parameters (diffusion)..
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Data Assimilation in a time-evolving model?

• Steady state box-inverse models estimate process
  rates or parametrisations like mixing from a 3D
  Variational problem
• Similar Parameter Estimation while matching
  timeevolving data often uses 4DVar Assimilation
• 4DVar very expensive computationally
• The budget within a box concept is subsumed
  into seeking a solution to the temporal model
  equations
• Parameter tuning assumes process representations
  are structurally correct
• Different approach Assimilation corrects for
  model bias so evaluate assimilation as another
  process within Box Budgets
• A posteriori Process Estimation

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OCCAM Assimilation Experiment
Sea Level analysis 28th March 1996
RUN1

• 1993-96
• ECMWF 6hr winds
• Monthly XBT assim.
• 10-day-ly Altimeter assim.
• SST weakly relaxed to Reynolds
• SSS weakly relaxed to Levitus

1/4 x 36 levels Global Ocean Model
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Process Estimation Local Heat Budget Wm-2
Assimilation
Advection

• Bias
• Patterns
• Amplitudes
• Space scales
• Transients

Trend 1993-96
Surface Flux
Mixing
Local Trend Convergence Assimilation
Surface Flux
( Mixing)
(Haines 2003)
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Process Estimation N Atlantic Box Budgets
G Volume Transformation Rate (Sv) (after Walin
1982) Thermodynamically Irreversible Processes
-?G/ ?? dV/dt - ?? G (1) Surface Forcing,
(2) Mixing, (3) Data Assimilation
Fox and Haines (2003) JPO
16Sv
Run1
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Process Estimation in the Ocean

• Locally assimilation corrects for wrong
  Advection eg. Gulf stream overshoots, Eastern
  Pacific thermocline
• Basin average sense assimilation corrects for
  wrong forcing i.e. surface heat flux
• Characteristic of certain processes can help to
  attribute assimilation contributions to
  box-budgets, eg.
• Advection is conservative between regions (no
  sources or sinks)
• Mixing also conservative AND always downgradient

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Relevance to Carbon Budget Modelling and
Assimilation?

• Budget-box representation of terrestrial
  ecosystem
• Conserved quantities Carbon, Nitrogen/Nitrates?..
  ....
• Understand cycling rates in model control
  (seasonal etc.. dependencies)
• Assimilation will try to constrain Amounts of
  conserved properties in each box. Unlikely to
  observe Transformation process rates?
• Success of assimilation may depend on
• Frequency of assimilation
• Rate at which model transformation processes act
• Any feedback between Amounts of property and
  transformation rates
• Generation of unwanted transient processes as
  model adjusts to new data

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Shelf Seas CarbonBiochemistry Modelling
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Bias and Data Assimilation

• Assimilation often correcting for Process Biases
• In OCCAM model
• Locally assimilation corrects for wrong
  Advection eg. mesoscale eddies in the wrong
  location or biased advection eg. Gulf stream
  overshoots
• Basin average sense assimilation corrects for
  wrong forcing i.e. surface heat flux
• Characteristics of certain processes can help to
  attribute assimilation contributions to
  box-budgets, eg.
• Advection is conservative between regions (no
  sources or sinks)
• Mixing also conservative AND always downgradient
• May try to Account for bias when assimilating
  data as it should alter the error weighting
  between model and observations

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Accounting for Bias in Data Assimilation

• Dee (2006) Review in QJRMS
• Variational formulation easiest to understand
  (derivable from Bayesian analysis Drecourt et
  al 2006)
• 2J(x,b,c) (y-b-x)TR-1(y-b-x)
• (x-xfc)TB-1(x-xfc)
• (b-bf)TO-1(b-bf)
• (c-cf)TP-1(c-cf)
• y observation R
  observation error covariance
• x model state B
  model background error covariance
• b observation bias O
  observation bias error covariance
• c model forecast bias P
  model forecast bias error covariance
• Superscript f are forecast values
• Observation operators have been omitted

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Accounting for Bias in Data Assimilation

• Solution (Analysed variables a)
• xa (xf-cf) K (y-bf) (xf-cf) K (BP)
  BPOR-1
• ba bf F (y-bf) (xf-cf) F O
  BPOR-1
• ca cf G (y-bf) (xf-cf) G P
  BPOR-1
• or xa (xf-ca) K1(y-ba) (xf-ca) K1 B
  BR-1
• y observation R
  observation error covariance
• x model state B
  model background error covariance
• b observation bias O
  observation bias error covariance
• c model forecast bias P
  model forecast bias error covariance
• Usual problems are (i) Knowing the Covariance
  errors
• (ii) Sequential 3DVar requires bias models for
  
• bf(t1) Mbba(t) cf(t1) Mcca(t)

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Comments on Bias Modelling

• Known Biases bf (t) cf(t) known a priori eg.
  previous runs
• xa (xf-cf) K (y-bf) (xf-cf) K
  (BP)BPOR-1
• bf (t) 0 cf(t) 0 is particular case
• (BP) total model err cov. (OR) total obs.
  err.
• Persistent Biases bf(t1) ba(t) cf(t1)
  ca(t)
• xa (xf-cf) K (y-bf) (xf-cf) K
  (BP)BPOR-1
• ba bf F (y-bf) (xf-cf) F
  OBPOR-1
• ca cf G (y-bf) (xf-cf) G
  PBPOR-1
• If O,P i.e. F,G are small gt may hope to converge
  to constant b,c
• Simplifications also arise if PaB OßR gt all
  Innovations proportional
• Attribution of Bias When are O,P sufficiently
  different to allow identification of misfits
  (y-bf) (xf-cf) ?
• Should always check misfits are consistent with
  BPOR

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Example Bias Modelling applied toAltimeter Data
Assimilation
Bias Error Covariance O on Mean Sea Level
Mean Sea Level
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Example Bias Modelling applied toAltimeter Data
Assimilation
Mean Sea Level Bias ba
Corrected Mean Sea Level
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CONCLUSIONS

• Biased model parameterisations can be tuned
  through 4DVar but only as far as structural
  errors and computational resources allow
• Alternatively build assimilation increments into
  box-budgets and seek to understand bias as
  process. Retains physically intuitive
  interpretation of Bias and Assimilation
  increments
• Having identified bias it should be accounted for
  during assimilation as it impacts on error
  weighting of model and data. Will need a bias
  model eg. understand its persistence, spatial
  structure, diurnal/seasonal cycling.