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DATA ASSIMILATION WITH IMPERFECT MODELS

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DATA ASSIMILATION WITH IMPERFECT MODELS

Zoltan Toth

and

Malaquias Pena Mendez1

Environmental Modeling Center

NOAA/NWS/NCEP

USA

1SAIC at Environmental Modeling Center, NCEP/NWS

Acknowledgements: Dusanka Zupanski, Guocheng Yuan

http://wwwt.emc.ncep.noaa.gov/gmb/ens/index.html

- ANALYSIS ERRORS
- Observational errors
- Background errors
- Chaotic errors
- Model-related errors
- Stochastic errors
- Systematic errors
- Tendency errors
- State errors
FORECAST DRIFT

- ANALYSIS ERRORS
- Observational errors
- Background errors
- Chaotic errors
- Model-related errors
- Stochastic errors
- Systematic errors
- Tendency errors
- State errors
FORECAST DRIFT

HOW TO REDUCE DRIFT-INDUCED FORECAST ERRORS?

- Mapping initial state on model attractor

- Estimating asymptotic errors

- Reducing model-related errors

- Reducing total forecast errors

- IMPROVED ANALYSES

- GOAL
- Represent nature as truthfully as possible

- USE OBSERVATIONAL DATA
- Incomplete coverage in
- Space
- Time
- Variables

- Noisy
- Assume for this study that observations are unbiased
- Otherwise, de-bias as in Derber & Wu

- Incomplete coverage in
- NEED OTHER (BACKGROUND) INFORMATION TO
- Complete coverage
- Filter out noise
- Choices
- Climatology – Independent of current situation
- Persistence – Dynamics of situation ignored
- Use dynamical short-range forecast - Best choice with caveats
- “Chaotic” forecast errors related to initial uncertainty
- Errors related to use of imperfect model

- COMBINE OBSERVATIONS & BACKGROUND
- Statistical procedure
- Undesirable effects from dynamics point of view
- Observational noise
- Reduced but not eliminated

- Weights on observations & background
- Is truth in between?

- Observational noise

- Undesirable effects from dynamics point of view
- Minimize arbitrariness by
- Relying more on dynamically consistent information
- Eg, ensemble-based background covariance
- Other approaches?

- Relying more on dynamically consistent information

- Statistical procedure
- CRUCIAL ROLE OF BACKGROUND
- How to generate?
- How to use?

- BASICS ABOUT FORECASTING
- Well known facts
- Some assumptions critical
- Will revisit a few

- Some assumptions critical

- Well known facts

- SOURCES OF FORECAST ERRORS
- Initial conditions
- Arise due to initial error (imperfect analysis) and unstable dynamics
- Reasonably well understood

- Arise due to initial error (imperfect analysis) and unstable dynamics
- Model
- Imperfect representation of nature
- Caused, for example, by use of
- Limited domain
- Limited temporal/spatial/physical resolution (truncation)
- Structural errors
- Parametric errors

- Caused, for example, by use of

- Imperfect representation of nature

- Initial conditions
- HOW TO REDUCE FORECAST ERRORS?
- Reduce initial errors
- Make model more similar to nature

- USE OF FORECASTS
- General applications
- In DA cycles

- CHAOTIC ERRORS
- Statistical approach
- “NMC” method (differences between past short-range forecasts verifying at same time)
- Ensemble method (differences between past ensemble forecasts verifying at same time)

- Dynamical approach
- 4DVAR – Norm dependent adjustments
- Ensemble-based DA (large ensemble of current forecasts) – Norm-independent adjustments

- Statistical approach
- MODEL-RELATED ERRORS
- 3 approaches used to cope with model errors in DA:
- Assume model-related errors don’t differ from chaotic errors (Ignore problem)
- Inflation of background errors (ie, move analysis closer to observations)
- Multiply background error covariance matrix in 3/4DVAR
- Increase initial perturbation size in ensemble-based DA

- Inflation of background errors (ie, move analysis closer to observations)
- Assume model-related errors are stochastic with characteristics different from chaotic errors (D. Zupanski et al)
- Introduce additional (model) error covariance term (allow analysis to move closer to obs.)
- How statistics determined?

- Assume errors are systematic (Dee et al)
- Estimate systematic difference between analysis and background
- Before their use, move background by systematic difference closer to analysis

- Assume model-related errors don’t differ from chaotic errors (Ignore problem)

- 3 approaches used to cope with model errors in DA:

Move background toward nature

Move background toward nature

IS THIS THE RIGHT MOVE?

Treat initial and model error the same way?

- TWO COMPONENTS
- Systematic
- Time mean difference

- Stochastic
- What’s left over
- Ignore for now

- What’s left over

- Systematic
- SYSTEMATIC ERROR
- Estimate as
- Climate mean difference
- Regime dependent difference
- Based on most recent data

- Estimate as
- TRADITIONAL PARADIGM FOR ANALYSIS/FORECAST SYSTEM
- Estimate the state of nature as truthfully as possible (analysis);
- Run numerical model forecast from the analysis field;
- Statistically assess the systematic error in the numerical forecast;
- Remove the estimated systematic error from the forecast.

- ASSUMPTION
- Removing systematic error will improve quality of analysis/forecast system
- WILL IT???

- Removing systematic error will improve quality of analysis/forecast system

Attractors of nature & model are different

Nature

Forecast

ORIGIN OF SYSTEMATIC ERROR IN FORECAST

Systematic difference between nature and our model –

Model world is different from reality

- Tendencies are different
- Phenomena evolve differently in time
- Ignore for now

- Phenomena evolve differently in time
- States (ie, realizable, natural states) are different
- Phenomena not (exactly) the same
- Eg, climate mean for nature and model are different
START MODEL FROM STATE OF NATURE

- Eg, climate mean for nature and model are different

- Phenomena not (exactly) the same
- State of nature not compatible with model
- Initial condition near nature is off of model attractor
- Forecast drifts toward model attractor
- Drift-induced errors introduced
REDUCE SYSTEMATIC MODEL-RELATED ERRORS?

- Drift-induced errors introduced

- Tendencies will be imperfect
- Accept that, but

- Can we reduce drift-related errors?

SEARCH FOR BEST INITIAL CONDITIONS FOR IMPERFECT MODELS

How can we reduce drift-induced errors?

What is the best initial condition for an imperfect model?

A state as close to nature as possible (“perfect” initial condition)? - Traditional, “fidelity” paradigm

On/near attractor of nature

Off attractor of model

Forecast drifts form attractor of nature to that of model

Lead-time dependent systematic errors

A state on/near model attractor? – New paradigm?

No forecast drift

“Imperfect” initial conditions?

How to find state on model attractor corresponding to state in nature?

Is there a model trajectory that would “shadow” nature?

Find an initial state on/near a model trajectory that corresponds with observed state

Estimate vector mapping points in nature to points on model attractor

Does such mapping exist?

For now, assume it does

Challenging step

- Estimate the mapping between nature and the model attractors
- Map the observed state of nature into the space of the model attractor
- Move obs. with mapping vector

- Analyze data
- Run the model from the mapped initial condition
- “Remap” the analysis and forecast back to the phase space of nature

New step

Standard procedure

New step

TRADITIONAL PARADIGM

Estimate the state of nature as truthfully as possible (traditional DA)

Run numerical model forecasts from the analysis field

Statistically assess the systematic error in the numerical forecasts

Correct the numerical forecasts for systematic errors

MAPPING PARADIGM

- Estimate the mapping between nature and the model attractors
- Map the observed state of nature into the space of the model attractor
- Move obs. with mapping vector
- Analyze data

- Run the model from the mapped initial condition
- “Remap” the analysis and forecast back to the phase space of nature

Analysis cycle

Analysis cycle

Move analysis toward nature

Move analysis to model attractor

Does mapping exist?

Assumption:

Forecasting would not be possible with imperfect models if mapping did not exist

Not sure, have to try and see

Mapping exists and well estimated if forecast errors with mapping vs. fidelity paradigm reduced

If it exists, how to estimate it?

Don’t need perfect estimate of mapping

Initial state must be closer to model attractor than with fidelity paradigm

Remapping mitigates potential problems with poor mapping vector estimate

The bigger the difference between nature and model, the less likely we can find mapping vector

MAPPING QUESTIONS

- Definition
- Vector that provides best remapped forecast performance

- Estimation
- Difference between long term time means of forecast trajectory & nature
In practice, nature is not known

- Use traditional analyses as proxy:
2.Adaptive technique

- If systematic errors are regime dependent, or climate means are not available
- Details later

- Use traditional analyses as proxy:

- Difference between long term time means of forecast trajectory & nature

- Lorenz (1963) 3-variable model:

“NATURE”

=10

b =8/3

r =28

“MODEL”

=9

z=z+2.5

Runge-Kutta numerical scheme with a time step of 0.01

- Three initialization schemes
- Perfect initial conditions
- Replacement
- All 3 variables observed
- Observational error = 2 (~5% natural variability)

- 3-DVAR:
- 15 time step cycle length (~7 hrs in atmosphere)
- Diagonal R, R=2
- B based on independent forecast errors, empirically tuned variance

Except for very short lead time, mapped forecast beats traditional forecast with or without bias correction

Remapped forecast beats traditional forecast at all leads

PERFECT INITIAL CONDITIONS

67% error reduction

- Drift-induced errors much reduced
- Shadowing period extended 3-fold

REPLACEMENT 3-DVAR

Remapped analysis beats traditional analysis

In the presence of initial errors, error reduction is smaller (~20%)

- Needed when
- Systematic errors are regime dependent or
- Climate means are not available

- Iterative process:
- Based on relatively small amount of data
- Length of iteration period ~15 days
- 10 iterations (~half year)

- Allow first guess fields to drift with each iteration: M = Mprior + MIncr
closer to model attractor

- Based on relatively small amount of data

ALGORITHM

1. Mprior = M = 0

2. Use M in mapping algorithm during next iteration period

3.

4. Update M by M = Mprior + MIncr

Mapping vector estimate

Repeat steps 2-4 for each iteration

Iteration number

Comparison with climate mean mapping

3-DVAR

20+% error reduction

- Mapping vector varies over attractor
- Adaptive mapping captures at least some regime-dependent fluctuations

- Remapped forecasts with adaptive mapping beat those with climate mean mapping
- Differences are
- Small but
- Highly singificant

- Differences are

- “Perfectly” known state is not best initial condition for imperfect models
- Intentionally moving initial condition away from nature, toward model attractor yields superior forecast
- Mapped forecasts used in DA yield superior analysis
- Adaptive, regime dependent mapping vector estimation needs less data and yields improved analysis/forecast performance

Taking a step back brings us closer to reality

- If, like a fly, attracted too close to the light you get burnt;
- By staying back, we can better understand / simulate nature

- Drastic departure from traditional thinking
- Only limited attempts to deviate from fidelity paradigm in literature
- Representativeness error
- Schneider et al ocean anomaly initialization
- M. Clark et al conceptual snow model

- Only limited attempts to deviate from fidelity paradigm in literature
- Concept of mapping proven with simple system
- Works “even” or “only” in simple systems?
- Applicable to more complex systems? Only testing will tell

- Works “even” or “only” in simple systems?
- If it would work in real applications, several implications
- Improved forecasts
- Model used properly, no drift, less arbitrariness

- Improved analyses
- After remapping

- Easier assessment of model errors
- “Asymptotic” model error

- No need for lead-time dependent bias correction
- Significant savings by not having to generate huge reforecast dataset

- Improved forecasts

- ANALYSIS ERRORS
- Observational errors
- Background errors
- Chaotic errors
- Model-related errors
- Stochastic errors
- Systematic errors
- Tendency errors
- State errors
FORECAST DRIFT

SEPARATE INITIAL VALUE AND MODEL-RELATED ERRORS - NEED DIFFERENT ACTIONS

To reduce initial error, draw to “nature”; to reduce model error, draw to model attractor

HOW TO REDUCE DRIFT-INDUCED FORECAST ERRORS?

- Mapping initial state on model attractor

- Estimating asymptotic errors

- Reducing model-related errors

- Reducing total forecast errors

- IMPROVED ANALYSES