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pg 1SAMSI UQ Workshop; Sep. 2011
Ensemble Data Assimilation and Uncertainty Quantification
Jeff AndersonNational Center for Atmospheric Research
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pg 2SAMSI UQ Workshop; Sep. 2011
Observations
…to produce an analysis(best possible estimate).
What is Data Assimilation?
+
Observations combined with a Model forecast…
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pg 3SAMSI UQ Workshop; Sep. 2011
Example: Estimating the Temperature Outside
An observation has a value ( ),
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pg 4SAMSI UQ Workshop; Sep. 2011
Example: Estimating the Temperature Outside
An observation has a value ( ),
and an error distribution (red curve) that is associated with the instrument.
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pg 5SAMSI UQ Workshop; Sep. 2011
Example: Estimating the Temperature Outside
Thermometer outside measures 1C.
Instrument builder says thermometer is unbiased with +/- 0.8C gaussian error.
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pg 6SAMSI UQ Workshop; Sep. 2011
The red plot is , probability of temperature given that To was observed.
Example: Estimating the Temperature Outside
Thermometer outside measures 1C.
€
P T |To( )
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pg 7SAMSI UQ Workshop; Sep. 2011
The green curve is ; probability of temperature given all available prior information .
Example: Estimating the Temperature Outside
We also have a prior estimate of temperature.
€
P T |C( )
€
C
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pg 8SAMSI UQ Workshop; Sep. 2011
Example: Estimating the Temperature Outside
Prior information can include:
1. Observations of things besides T;
2. Model forecast made using observations at earlier times;
3. A priori physical constraints ( T > -273.15C );
4. Climatological constraints ( -30C < T < 40C ).
€
C
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pg 9SAMSI UQ Workshop; Sep. 2011
Combining the Prior Estimate and Observation
€
P T |To,C( ) =P To |T,C( )P T |C( )
Normalization
Bayes Theorem:
Posterior: Probability of T given
observations and Prior. Also called
update or analysis.
Prior
Likelihood: Probability that To is observed if T is true value and given prior information C.
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pg 10SAMSI UQ Workshop; Sep. 2011
Combining the Prior Estimate and Observation
P T |To,C P To |T,C P T |C
normalization
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pg 11SAMSI UQ Workshop; Sep. 2011
Combining the Prior Estimate and Observation
P T |To,C P To |T,C P T |C
normalization
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pg 12SAMSI UQ Workshop; Sep. 2011
Combining the Prior Estimate and Observation
P T |To,C P To |T,C P T |C
normalization
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pg 13SAMSI UQ Workshop; Sep. 2011
Combining the Prior Estimate and Observation
P T |To,C P To |T,C P T |C
normalization
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pg 14SAMSI UQ Workshop; Sep. 2011
Combining the Prior Estimate and Observation
P T |To,C P To |T,C P T |C
normalization
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pg 15SAMSI UQ Workshop; Sep. 2011
Consistent Color Scheme Throughout Tutorial
Green = Prior
Red = Observation
Blue = Posterior
Black = Truth
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pg 16SAMSI UQ Workshop; Sep. 2011
Combining the Prior Estimate and Observation
P T |To,C P To |T,C P T |C normalization
Generally no analytic solution for Posterior.
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pg 17SAMSI UQ Workshop; Sep. 2011
Combining the Prior Estimate and Observation
P T |To,C P To |T,C P T |C normalization
Gaussian Prior and Likelihood -> Gaussian Posterior
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pg 18SAMSI UQ Workshop; Sep. 2011
For Gaussian prior and likelihood…
Prior
Likelihood
Then, Posterior
With
Combining the Prior Estimate and Observation
€
P T |C( ) = Normal Tp,σ p( )
€
P To |T,C( ) = Normal To,σ o( )
€
P T |To,C( ) = Normal Tu,σ u( )
€
σ u = σ p−2 +σ o
−2( )−1
€
Tu =σ u2 σ p
−2Tp +σ o−2To[ ]
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pg 19SAMSI UQ Workshop; Sep. 2011
1. Suppose we have a linear forecast model L
A. If temperature at time t1 = T1, then temperature at t2 = t1 + t is T2 = L(T1)
B. Example: T2 = T1 + tT1
The One-Dimensional Kalman Filter
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pg 20SAMSI UQ Workshop; Sep. 2011
1. Suppose we have a linear forecast model L.
A. If temperature at time t1 = T1, then temperature at t2 = t1 + t is T2 = L(T1).
B. Example: T2 = T1 + tT1 .
A. If posterior estimate at time t1 is Normal(Tu,1, σu,1) then prior at t2 is Normal(Tp,2, σp,2).
Tp,2 = Tu,1,+ tTu,1
σp,2 = (t + 1) σu,1
The One-Dimensional Kalman Filter
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pg 21SAMSI UQ Workshop; Sep. 2011
1. Suppose we have a linear forecast model L.
A. If temperature at time t1 = T1, then temperature at t2 = t1 + t is T2 = L(T1).
B. Example: T2 = T1 + tT1 .
A. If posterior estimate at time t1 is Normal(Tu,1, σu,1) then prior at t2 is Normal(Tp,2, σp,2).
B. Given an observation at t2 with distribution Normal(to, σo)the likelihood is also Normal(to, σo).
The One-Dimensional Kalman Filter
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pg 22SAMSI UQ Workshop; Sep. 2011
1. Suppose we have a linear forecast model L.
A. If temperature at time t1 = T1, then temperature at t2 = t1 + t is T2 = L(T1).
B. Example: T2 = T1 + tT1 .
A. If posterior estimate at time t1 is Normal(Tu,1, σu,1) then prior at t2 is Normal(Tp,2, σp,2).
B. Given an observation at t2 with distribution Normal(to, σo)the likelihood is also Normal(to, σo).
C. The posterior at t2 is Normal(Tu,2, σu,2) where Tu,2 and σu,2 come from page 18.
The One-Dimensional Kalman Filter
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pg 23SAMSI UQ Workshop; Sep. 2011
A One-Dimensional Ensemble Kalman Filter
Represent a prior pdf by a sample (ensemble) of N values:
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pg 24SAMSI UQ Workshop; Sep. 2011
Use sample mean
and sample standard deviation
to determine a corresponding continuous distribution
A One-Dimensional Ensemble Kalman Filter
Represent a prior pdf by a sample (ensemble) of N values:
€
T = Tn Nn=1
N
∑
€
σT = Tn −T ( )2
N −1( )n=1
N
∑
€
Normal T ,σ T( )
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pg 25SAMSI UQ Workshop; Sep. 2011
A One-Dimensional Ensemble Kalman Filter: Model Advance
If posterior ensemble at time t1 is T1,n, n = 1, …, N
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pg 26SAMSI UQ Workshop; Sep. 2011
A One-Dimensional Ensemble Kalman Filter: Model Advance
If posterior ensemble at time t1 is T1,n, n = 1, …, N ,advance each member to time t2 with model, T2,n = L(T1, n) n = 1, …,N .
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pg 27SAMSI UQ Workshop; Sep. 2011
A One-Dimensional Ensemble Kalman Filter: Model Advance
Same as advancing continuous pdf at time t1 …
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pg 28SAMSI UQ Workshop; Sep. 2011
A One-Dimensional Ensemble Kalman Filter: Model Advance
Same as advancing continuous pdf at time t1
to time t2 with model L.
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pg 29SAMSI UQ Workshop; Sep. 2011
A One-Dimensional Ensemble Kalman Filter: Assimilating an Observation
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pg 30SAMSI UQ Workshop; Sep. 2011
Fit a Gaussian to the sample.
A One-Dimensional Ensemble Kalman Filter: Assimilating an Observation
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pg 31SAMSI UQ Workshop; Sep. 2011
Get the observation likelihood.
A One-Dimensional Ensemble Kalman Filter: Assimilating an Observation
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pg 32SAMSI UQ Workshop; Sep. 2011
Compute the continuous posterior PDF.
A One-Dimensional Ensemble Kalman Filter: Assimilating an Observation
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pg 33SAMSI UQ Workshop; Sep. 2011
Use a deterministic algorithm to ‘adjust’ the ensemble.
A One-Dimensional Ensemble Kalman Filter: Assimilating an Observation
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pg 34SAMSI UQ Workshop; Sep. 2011
First, ‘shift’ the ensemble to have the exact mean of the posterior.
A One-Dimensional Ensemble Kalman Filter: Assimilating an Observation
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pg 35SAMSI UQ Workshop; Sep. 2011
First, ‘shift’ the ensemble to have the exact mean of the posterior.Second, linearly contract to have the exact variance of the posterior.
Sample statistics are identical to Kalman filter.
A One-Dimensional Ensemble Kalman Filter: Assimilating an Observation
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pg 36SAMSI UQ Workshop; Sep. 2011
(Ensemble) KF optimal for linear model, gaussian likelihood. In KF, only mean and variance have meaning. Variance defines the uncertainty.
Ensemble allows computation of many other statistics. What do they mean? Not entirely clear.
Example: Kurtosis. Completely constrained by initial ensemble. It is problem specific whether this is even defined!
(See example set 1 in Appendix)
UQ from an Ensemble Kalman Filter
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pg 37SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
Draw 5 values from a real-valued distribution.Call the first 4 ‘ensemble members’.
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pg 38SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
These partition the real line into 5 bins.
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pg 39SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
Call the 5th draw the ‘truth’.1/5 chance that this is in any given bin.
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pg 40SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
Rank histogram shows the frequency of the truth in each bin over many assimilations.
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pg 41SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
Rank histogram shows the frequency of the truth in each bin over many assimilations.
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pg 42SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
Rank histogram shows the frequency of the truth in each bin over many assimilations.
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pg 43SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
Rank histogram shows the frequency of the truth in each bin over many assimilations.
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pg 44SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
Rank histograms for good ensembles should be uniform (caveat sampling noise).Want truth to look like random draw from ensemble.
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pg 45SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
A biased ensemble leads to skewed histograms.
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pg 46SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
An ensemble with too little spread gives a u-shape.This is the most common behavior for geophysics.
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pg 47SAMSI UQ Workshop; Sep. 2011
The Rank Histogram: Evaluating Ensemble Performance
An ensemble with too much spread is peaked in the center.(See example set 1 in appendix)
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pg 48SAMSI UQ Workshop; Sep. 2011
Back to 1-Dimensional (Ensemble) Kalman Filter
All bets are off when model is nonlinear, likelihood nongaussian.
Must assess quality of estimates for each case.
Example: A weakly-nonlinear 1D model:dx/dt = x + 0.4|x|x
(See example set 2 in appendix)
Ensemble statistics can become degenerate.Nonlinear ensemble filters may address this
but beyond scope for today.
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pg 49SAMSI UQ Workshop; Sep. 2011
Ensemble Kalman Filter with Model Error
Welcome to the real world. Models aren’t perfect.
Model (and other) error can corrupt ensemble mean and variance.
(See example set 3 part 1 in appendix)
Adapt to this by increasing uncertainty in prior.
Inflation is common method for ensembles.
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pg 50SAMSI UQ Workshop; Sep. 2011
Dealing with systematic error: Variance Inflation
Observations + physical system => ‘true’ distribution.
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pg 51SAMSI UQ Workshop; Sep. 2011
Dealing with systematic error: Variance Inflation
Observations + physical system => ‘true’ distribution.Model bias (and other errors) can shift actual prior.
Prior ensemble is too certain (needs more spread).
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pg 52SAMSI UQ Workshop; Sep. 2011
Dealing with systematic error: Variance Inflation
Naïve solution: increase the spread in the prior.Give more weight to the observation, less to prior.
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pg 53SAMSI UQ Workshop; Sep. 2011
Ensemble Kalman Filter with Model Error
Errors in model (and other things) result in too much certainty in prior.
Can use inflation to correct for this.Generally tuned on dependent data set.Adaptive methods exist but must be
calibrated.(See example set 3 part 1 in appendix)
Uncertainty errors depend on forecast length.
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pg 54DART-LAB Tutorial -- June 09
Multivariate Ensemble Kalman Filter
So far, we have an observation likelihood for single variable.
Suppose the model prior has additional variables.
Use linear regression to update additional variables.
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pg 55DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Assume that all we know is prior joint distribution.One variable is observed.What should happen to the unobserved variable?
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pg 56DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Assume that all we know is prior joint distribution.One variable is observed.Compute increments for prior ensemble members of observed variable.
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pg 57DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Assume that all we know is prior joint distribution.One variable is observed.Using only increments guarantees that if observation had no impact on observed variable, unobserved variable is unchanged (highly desirable).
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pg 58DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Assume that all we know is prior joint distribution.How should the unobserved variable be impacted?First choice: least squares.Equivalent to linear regression.Same as assuming binormal prior.
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pg 59DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.How should the unobserved variable be impacted?First choice: least squares.Begin by finding least squares fit.
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pg 60DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Next, regress the observed variable increments onto increments for the unobserved variable.Equivalent to first finding image of increment in joint space.
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pg 61DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Next, regress the observed variable increments onto increments for the unobserved variable.Equivalent to first finding image of increment in joint space.
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pg 62DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Next, regress the observed variable increments onto increments for the unobserved variable.Equivalent to first finding image of increment in joint space.
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pg 63DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Next, regress the observed variable increments onto increments for the unobserved variable.Equivalent to first finding image of increment in joint space.
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pg 64DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Next, regress the observed variable increments onto increments for the unobserved variable.Equivalent to first finding image of increment in joint space.
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pg 65DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Regression: Equivalent to first finding image of increment in joint space.Then projecting from joint space onto unobserved priors.
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pg 66DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Regression: Equivalent to first finding image of increment in joint space.Then projecting from joint space onto unobserved priors.
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pg 67DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Regression: Equivalent to first finding image of increment in joint space.Then projecting from joint space onto unobserved priors.
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pg 68DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Regression: Equivalent to first finding image of increment in joint space.Then projecting from joint space onto unobserved priors.
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pg 69DART-LAB Tutorial -- June 09
Ensemble filters: Updating additional prior state variables
Have joint prior distribution of two variables.Regression: Equivalent to first finding image of increment in joint space.Then projecting from joint space onto unobserved priors.
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pg 70DART-LAB Tutorial -- June 09
The Lorenz63 3-Variable ‘Chaotic’ Model
Two lobed attractor.State orbits lobes, switches lobes sporadically.Nonlinear if observations are sparse.Qualitative uncertainty information available from ensemble.Nongaussian distributions are sampled.Quantitative uncertainty requires careful calibration.
(See Example 4 in the appendix)
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pg 71DART-LAB Tutorial -- June 09
Ensemble Kalman Filters with Large Models
Kalman filter is too costly (time and storage) for large models.Ensemble filters can be used for very large applications.Sampling error when computing covariances may become large.Spurious correlations between unrelated observations and state variables contaminate results.Leads to further uncertainty underestimates.Can be compensated by a priori limits on what state variables are impacted by an observation. Called ‘localization’.Requires even more calibration.
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pg 72DART-LAB Tutorial -- June 09
The Lorenz96 40-Variable ‘Chaotic’ Model
Something like weather systems around a latitude circle.Has spurious correlations for small ensembles.Basic implementation of ensemble filter leads to bad uncertainty estimates.Both inflation and localization can improve performance.Quantitative uncertainty requires careful calibration.
(See Example 5 in the appendix)
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pg 73DART-LAB Tutorial -- June 09
Uncertainty and Ensemble Kalman Filters: Conclusions
Ensemble KF variance is exact quantification of uncertainty when things are linear, gaussian, perfect, and ensemble is large enough.
Too little uncertainty nearly universal for geophysical applications.
Uncertainty errors are a function of forecast lead time.
Adaptive algorithms like inflation improve uncertainty estimates.
Quantitative uncertainty estimates require careful calibration.
Out of sample estimates (climate change) are tenuous.
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pg 74DART-LAB Tutorial -- June 09
Appendix: Matlab examples
This is an outline of the live matlab demos from NCAR’s DART system that were used to support this tutorial. The matlab code is available by checking out the DART system from NCAR at:http://www2.image.ucar.edu/forms/dart-software-download
The example scripts can be found in the DART directory:DART_LAB/matlab
Some of the features of the GUI’s for the exercises do not work due to bugs in Matlab releases 2010a and 2010b but should work in earlier and later releases.Many more features of these matlab scripts are described in the tutorial files in the DART directory:DART_LAB/presentationThese presentations are a complement and extension of things presented in this tutorial.
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pg 75DART-LAB Tutorial -- June 09
Appendix: Matlab examples
Example Set 1: Use matlab script oned_modelChange the “Ens. Size” to 10Select “Start Free Run”Observe that the spread curve in the second diagnostic panel quickly converges to have the same prior and posterior values. The rms error varies with time due to observation error, but its time mean should be the same as the spread.Restart this exercise several times to see that the kurtosis curve in the third panel also converges immediately, but to values that change with every randomly selected initial ensemble. Also note the behavior of the rank histograms for this case. They may not be uniform even though the ensemble filter is the optimal solution. In fact, they can be made arbitrarily nonuniform by the selection of the initial conditions.
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pg 76DART-LAB Tutorial -- June 09
Appendix: Matlab examples
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pg 77DART-LAB Tutorial -- June 09
Appendix: Matlab examples
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pg 78DART-LAB Tutorial -- June 09
Appendix: Matlab examples
Example Set 2: Use matlab script oned_modelChange the “Ens. Size” to 10Change “Nonlin. A” to 0.4Select “Start Free Run”
This examines what happens when the model is nonlinear. Observe the evolution of the rms error and spread in the second panel and the kurtosis in the third panel. Also observe the ensemble prior distribution (green) in the first panel and in the separate plot on the control panel (where you changed the ensemble size). In many cases, the ensemble will become degenerate with all but one of the ensemble members collapsing while the other stays separate. Observe the impact on the rank histograms as this evolves.
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pg 79DART-LAB Tutorial -- June 09
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pg 80DART-LAB Tutorial -- June 09
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pg 81DART-LAB Tutorial -- June 09
Appendix: Matlab examples
Example Set 3: Use matlab script oned_modelChange the “Ens. Size” to 10Change “Model Bias” to 1Select “Start Free Run”
This examines what happens when the model used to assimilate observations is biased compared to the model that generated the observations (an imperfect model study). The model for the assimilation adds 1 to what the perfect model would do at each time step. Observe the evolution of the rms error and spread in the second panel and the rank histograms. Now, try adding some uncertainty to the prior ensemble using inflation by setting “Inflation” to 1.5. This should impact all aspects of the assimilation.
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pg 82DART-LAB Tutorial -- June 09
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pg 83DART-LAB Tutorial -- June 09
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pg 84DART-LAB Tutorial -- June 09
Appendix: Matlab examples
Example Set 4: Use matlab script run_lorenz_63Select “Start Free Run”
This run without assimilation shows how uncertainty grows in an ensemble forecast in the 3 variable Lorenz model. After a few trips around the attractor select “Stop Free Run”. Then turn on an ensemble filter assimilation by changing “No Assimilation “ to “EAKF”. Continue the run by selecting “Start Free Run” and observe how the ensemble evolves. You can stop the advance again and use the matlab rotation feature to study the structure of the ensemble.
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pg 85DART-LAB Tutorial -- June 09
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pg 86DART-LAB Tutorial -- June 09
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pg 87DART-LAB Tutorial -- June 09
Appendix: Matlab examples
Example Set 5: Use matlab script run_lorenz_96Select “Start Free Run”
This run without assimilation shows how uncertainty grows in an ensemble forecast in the 40 variable model. After about time=40, select “Stop Free Run” and start an assimilation by changing “No Assimilation” to “EAKF” and selecting “Start Free Run”. Exploring the use of a localization of 0.2 and/or an inflation of 1.1 helps to show how estimates of uncertainty can be improved by modifying the base ensemble Kalman filter.