An Introduction to Ensemble Kalman Filtering · An Introduction to Ensemble Kalman Filtering Jeffrey Anderson NCAR Data Assimilation Research Section ... Environment Canada Operational

6/18/07

ring

tion (DAReS)

7

Anderson: NSF August, 2006 1

An Introduction toEnsemble Kalman Filte

Jeffrey AndersonNCAR Data Assimilation Research Sec

AMS NWP/WAF, 26 June, 200

6/18/07

ng

ith Ensembles


Overview

1. The Data Assimilation Problem

2. A Bayesian View of Ensemble Kalman Filteri

3. Challenges to Ensemble Filters

4. Adaptive Ensemble Filter Algorithms

5. Model and Observing System Development w

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:


The Data Assimilation Problem

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:

_________________________



__________________________________________________

A (Physical) system:Atmosphere, coupled climate system...

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:

_________________________

___________________

odel ‘state vector’.).



__________________________________________________


______________________________________A model of the physical system:

Represents system with discrete vector: the mApproximates time evolution of system (poorly

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:

_________________________

___________________

odel ‘state vector’.).___________________

tion error.ace).licated.



__________________________________________________


______________________________________A model of the physical system:

Represents system with discrete vector: the mApproximates time evolution of system (poorly

______________________________________Observations of the system:

Have a (sometimes poor) estimate of observaCould be sparse and irregular in time (and spRelation to model ‘state vector’ may be compMay have very low information content.

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all three pieces:


Data Assimilation increases information about

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all three pieces:___________


Data Assimilation increases information about______________________________________(Physical) system:

Estimates of state (analyses, posteriors...).Initial conditions for forecasts.Enhanced (physical) understanding.

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___________

l ‘parameters’).




______________________________________Model:

Estimates of model errors.Relative characteristics of different models.Improved model (find good values for mode

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___________

l ‘parameters’).___________

servations.ased information.




______________________________________Model:

Estimates of model errors.Relative characteristics of different models.Improved model (find good values for mode

______________________________________Observations:

Estimates of observation errors.Information content of existing or planned obObserving system designs that provide incre

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Forecasters

ember ENKF


Ensemble Filter Products now Available to

Environment Canada Operational GEM 16 m

6/18/07

Forecasters

KF



University of Washington WRF En

6/18/07

Forecasters

vailable.

d below.

ve additional issues.



1. Many other ensemble forecast products are a

2. Most suffer from the same challenges outline

3. Other ensemble generation methods may ha

6/18/07

ring.

le.

rsion in the valley)

variable.re.


An intr oduction to Ensemble Filte

1: Single variable and observation of that variabLet’s think of it as temperature at SLC.

(Slides are for a mid-winter ski day...inve

2: Single observed variable, single unobservedSLC temperature, Park City temperatu

That’s all there is... (without loss of generality).

6/18/07

, C.

n C and B.

AC) p A C( )x)p x C( )dx

---------------------------------

4 6


Bayes rule:

A: Prior estimate based on all previous informationB: An additional observation.

: Posterior (updated estimate) based o

p A BC( )p B AC( ) p A C( )

p B C( )------------------------------------------

p B(p∫ B(

----------= =

−6 −4 −2 0 20

0.05

0.1

0.15

0.2 Prior PDF

Pro

babi

lity

p A BC( )

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tion, C.

n C and B.


---------------------------------

4 6

Obs. Likelihood


Bayes rule:

A: Prior estimate based on all previous informaB: An additional observation.



p B C( )------------------------------------------

p B(p∫ B(

----------= =

−6 −4 −2 0 20

0.05

0.1

0.15

0.2 Prior PDF

Pro

babi

lity

p A BC( )

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tion, C.

n C and B.


---------------------------------

4 6

Obs. Likelihood


Bayes rule:




p B C( )------------------------------------------

p B(p∫ B(

----------= =

−6 −4 −2 0 20

0.05

0.1

0.15

0.2 Prior PDF

Pro

babi

lity

Product (Numerator)

p A BC( )

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ion, C.

n C and B.


--------------------------------

4 6

Obs. Likelihood

.)


Bayes rule:

A: Prior estimate based on all previous informatB: An additional observation.



p B C( )------------------------------------------

p B(p∫ B(

-----------= =

−6 −4 −2 0 20

0.05

0.1

0.15

0.2 Prior PDF

Pro

babi

lity

Normalization (Denom

p A BC( )

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tion, C.

n C and B.


---------------------------------

4 6

Obs. Likelihood

.)

rior


Bayes rule:




p B C( )------------------------------------------

p B(p∫ B(

----------= =

−6 −4 −2 0 20

0.05

0.1

0.15

0.2 Prior PDF

Pro

babi

lity

Normalization (Denom

Poste

p A BC( )

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ghout


Consistent Color Scheme Throu

Green = Prior

Red = Observation

Blue = Posterior

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s.


---------------------------------

2 4

bs. Likelihood


Bayes rule:

This product is closed for Gaussian distribution


p B C( )------------------------------------------

p B(p∫ B(

----------= =

−4 −2 00

0.2

0.4

0.6

Prior PDF

Pro

babi

lity O

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s.


---------------------------------

2 4

bs. Likelihood


Bayes rule:

This product is closed for Gaussian distribution


p B C( )------------------------------------------

p B(p∫ B(

----------= =

−4 −2 00

0.2

0.4

0.6

Prior PDF

Pro

babi

lity O

Posterior PDF

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dµ2 and


Product of two Gaussians:

Product of d-dimensional normals with meansµ1 ancovariance matricesΣ1 andΣ2 is normal.

N µ1 Σ1,( )N µ2 Σ2,( ) cN µ Σ,( )=

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dµ2 and




Covariance:

Mean:

N µ1 Σ1,( )N µ2 Σ2,( ) cN µ Σ,( )=

Σ Σ11– Σ2

1–+( ) 1–=

µ Σ11– Σ2

1–+( ) 1– Σ11– µ1 Σ2

1– µ2+( )=

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dµ2 and

1 Σ2+ ) 1– µ2 µ1–( )]




Covariance:

Mean:

Weight:

We’ll ignore the weight since we immediatelynormalize products to be PDFs.

N µ1 Σ1,( )N µ2 Σ2,( ) cN µ Σ,( )=

Σ Σ11– Σ2

1–+( ) 1–=

µ Σ11– Σ2

1–+( ) 1– Σ11– µ1 Σ2

1– µ2+( )=

c1

2Π( )d 2⁄ Σ1 Σ2+ 1 2⁄--------------------------------------------------- 12--- µ2 µ1–( )T Σ([–

exp=

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dµ2 and

exponentials.

1 Σ2+ ) 1– µ2 µ1–( )]




Covariance:

Mean:

Weight:

Easy to derive for 1D (d=1); just do products of

N µ1 Σ1,( )N µ2 Σ2,( ) cN µ Σ,( )=

Σ Σ11– Σ2

1–+( ) 1–=

µ Σ11– Σ2

1–+( ) 1– Σ11– µ1 Σ2

1– µ2+( )=

c1

2Π( )d 2⁄ Σ1 Σ2+ 1 2⁄--------------------------------------------------- 12--- µ2 µ1–( )T Σ([–

exp=

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.

le.th’.


---------------------------------

2 4


Bayes rule:

Ensemble filters:Prior is available as finite sample

Don’t know much about properties of this sampMay naively assume it is random draw from ‘tru


p B C( )------------------------------------------

p B(p∫ B(

----------= =

−4 −2 00

0.2

0.4

0.6

Prior Ensemble

Pro

babi

lity

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uous likelihood?

to sample.


---------------------------------

2 4


Bayes rule:

How can we take product of sample with contin

Fit a continuous (Gaussian for now) distribution


p B C( )------------------------------------------

p B(p∫ B(

----------= =

−4 −2 00

0.2

0.4

0.6

Prior Ensemble

Pro

babi

lity

Prior PDF

6/18/07

ly always Gaussian).

e methods below.obs. likelihood.


---------------------------------

2 4

bs. Likelihood


Bayes rule:

Observation likelihood usually continuous (near

If Obs. Likelihood isn’t Gaussian, can generalizFor instance, can fit set of Gaussian kernels to


p B C( )------------------------------------------

p B(p∫ B(

----------= =

−4 −2 00

0.2

0.4

0.6

Prior Ensemble

Pro

babi

lity

Prior PDF

O

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is Gaussian.


---------------------------------

2 4

bs. Likelihood


Bayes rule:

Product of prior Gaussian fit and Obs. likelihood

Computing continuous posterior is simple.BUT, need to have a SAMPLE of this PDF.


p B C( )------------------------------------------

p B(p∫ B(

----------= =

−4 −2 00

0.2

0.4

0.6

Prior Ensemble

Pro

babi

lity

Prior PDF

O

Posterior PDF

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nclear.rform., etc.

2 3


Sampling Posterior PDF:

There are many ways to do this.

Exact properties of different methods may be uTrial and error still best way to see how they peWill interact with properties of prediction models

−2 −1 0 10

0.2

0.4

0.6Posterior PDF

Pro

babi

lity

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2 4


Ensemble Filter Algorithms:

Ensemble Adjustment (Kalman) Filter.

−4 −2 00

0.2

0.4

0.6

Prior Ensemble

Pro

babi

lity

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2 4




Again, fit a Gaussian to sample.−4 −2 00

0.2

0.4

0.6

Prior Ensemble

Pro

babi

lity

Prior PDF

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2 4

bs. Likelihood




Compute posterior PDF.−4 −2 00

0.2

0.4

0.6

Prior Ensemble

Pro

babi

lity

Prior PDF

O

Posterior PDF

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.2 4




Use deterministic algorithm to ‘adjust’ ensemble−4 −2 00

0.2

0.4

0.6

Pro

babi

lity

Posterior PDF

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.f posterior.

2 4




Use deterministic algorithm to ‘adjust’ ensembleFirst, ‘shift’ ensemble to have exact mean o

−4 −2 00

0.2

0.4

0.6

Pro

babi

lity

Posterior PDF

Mean Shifted

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.f posterior.t variance of posterior.

2 4




Use deterministic algorithm to ‘adjust’ ensembleFirst, ‘shift’ ensemble to have exact mean oSecond, use linear contraction to have exac

−4 −2 00

0.2

0.4

0.6

Pro

babi

lity

Posterior PDF

Mean Shifted

Variance Adjusted

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le size.

s ensemble mean,

2 4




i = 1,..., ensemb

p is prior, u is update (posterior), overbar iσ is standard deviation.

−4 −2 00

0.2

0.4

0.6

Pro

babi

lity

Posterior PDF

Mean Shifted

Variance Adjusted

xiu

xip

xp–( ) σu σp⁄( )⋅ xu+=

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sitioned or weighted.

2 4




Bimodality maintained, but not appropriately poNo problem with random outliers.

−4 −2 00

0.2

0.4

0.6

Pro

babi

lity

Posterior PDF

Mean Shifted

Variance Adjusted

6/18/07

rved variablee at Park City.

ngle variable.

nal variable.

ditional variable.

itional variables.


2: Single observed variable, single unobseSLC temperature, temperatur

So far, have known observation likelihood for si

Now, suppose model state vector has an additio

Will examine how ensemble methods update ad

Basic method generalizes to any number of add

Related to Kalman filter in subtle ways.

6/18/07

variables

e that all we knowr joint distribution.

ariable is observed.C temperature)should happen toerved variable?rk City temp.)

like a nastyersion...


Ensemble filters: Updating additional prior state

Assumis prio

One v(SL

What unobs

(Pa

Looks inv

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble

−2 0 2 4Observed Variable

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variables


ariable is observed.

e observedle with one ofus methods.



Assumis prio

One v

Updatvariabprevio

33.5

44.5

5

Uno

bs.


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variables






Assumis prio

One v

Updatvariabprevio

33.5

44.5

5

Uno

bs.


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variables






Assumis prio

One v

Updatvariabprevio

33.5

44.5

5

Uno

bs.


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variables



ute increments fornsemble memberserved variable.



Assumis prio

One v

Compprior eof obs

33.5

44.5

5

Uno

bs.


Increments

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variables






Assumis prio

One v

Compprior eof obs

33.5

44.5

5

Uno

bs.


Increments

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variables






Assumis prio

One v

Compprior eof obs

33.5

44.5

5

Uno

bs.


Increments

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variables






Assumis prio

One v

Compprior eof obs

33.5

44.5

5

Uno

bs.


Increments

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variables






Assumis prio

One v

Compprior eof obs

33.5

44.5

5

Uno

bs.


Increments

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variables



only incrementsntees that ifvation had not on observedle, unobservedle is unchangedy desirable).



Assumis prio

One v

Usingguaraobserimpacvariabvariab(highl

33.5

44.5

5

Uno

bs.


Increments

6/18/07

variables


hould theerved variable beted?

hoice: least squares

alent to linearsion.

as assumingal prior.



Assumis prio

How sunobsimpac

First c

Equivregres

Samebinorm

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

6/18/07

variables

joint priorution of twoles.

hould theerved variable beted?

hoice: least squares

by findingleastes fit.



Have distribvariab

How sunobsimpac

First c

Beginsquar3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

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variables


regress theved variableents ontoents for theerved variable.

alent to first finding of increment inpace.



Have distribvariab

Next, obserincremincremunobs

Equivimagejoint s

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

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variables






Have distribvariab


Equivimagejoint s

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

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variables






Have distribvariab


Equivimagejoint s

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

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variables






Have distribvariab


Equivimagejoint s

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

6/18/07

variables






Have distribvariab


Equivimagejoint s

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

6/18/07

variables


ssion: Equivalent toding image ofent in joint space.

projecting frompace ontoerved priors.

, multiply by priorle correlation.



Have distribvariab

Regrefirst finincrem

Then joint sunobs

Finallysamp

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

6/18/07

variables







Have distribvariab


Then joint sunobs

Finallysamp

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

6/18/07

variables







Have distribvariab


Then joint sunobs

Finallysamp

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

6/18/07

variables







Have distribvariab


Then joint sunobs

Finallysamp

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

6/18/07

variables







Have distribvariab


Then joint sunobs

Finallysamp

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble


Increments

6/18/07

variables

ave an updatedrior) ensemble forobserved variable.



Now h(postethe un

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble

−2 0 2 4Obs.

6/18/07

variables


Gaussians showsean and variancehanged.



Now h(postethe un

Fittingthat mhave c

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble

Prior State Fit

−2 0 2 4Obs.

6/18/07

variables


Gaussians showsean and variancehanged.

features of theistribution mayave changed.



Now h(postethe un

Fittingthat mhave c

Otherprior dalso h

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble

Prior State Fit

Posterior Fit

−2 0 2 4Obs.

6/18/07

variables

CAL POINT:

impact onerved variable is a linearsion, can do thisENDENTLY for

umber oferved variables!

also do many atusing matrixra as in traditionaln Filter.



CRITI

SinceunobssimplyregresINDEPany nunobs

Couldonce algebKalma

3

3.5

4

4.5

5

Uno

bser

ved

Sta

te V

aria

ble

Prior State Fit

Posterior Fit

−2 0 2 4Obs.

6/18/07

nd observations

ference Equation:

(1)

(2)

(nice, not essential).

(3)

(4)

t:

(5)

tk t0≥>


Phase 3: Generalize to geophysical models a

Dynamical system governed by (stochastic) Dif

Observations at discrete times:

Observational error white in time and Gaussian

Complete history of observations is:

Goal: Find probability distribution for state at time

dxt f xt t,( )= G xt t,( )dβt+ t 0≥,

yk h xk tk,( )= vk k;+ 1 2 … tk 1+;, ,=

vk N 0 Rk,( )→

Yτ yl tl τ≤;{ }=

p x t Yt,( )

6/18/07

nd observations

Difference Equation.

(6)

(7)

(8)

ator:

(9)

1)

------

x



State between observation times obtained fromNeed to update state given new observation:

Apply Bayes rule:

Noise is white in time (3) so:

Integrate numerator to get normalizing denomin

p x tk Ytk,( ) p x tk yk Ytk 1–

,,( )=

p x tk Ytk,( )

p yk xk Ytk 1–,( ) p x tk Ytk –

,(

p yk Ytk 1–( )

-----------------------------------------------------------------------=

p yk xk Ytk 1–,( ) p yk xk( )=

p yk Ytk 1– p yk x( ) p x tk Ytk 1–

,( )d∫=

6/18/07

nd observations

(10)

t x and y are vectors.

each observation.

state vector.

1)

)dξ---------



Probability after new observation:

Exactly analogous to earlier derivation except tha

EXCEPT, no guarantee we have prior sample for

SO, let’s make sure we have priors by ‘extending’

p x tk Ytk,

p yk x( ) p x tk Ytk –

,(

p yk ξ( ) p ξ tk Ytk 1–,(∫

---------------------------------------------------------=

6/18/07

nd observations

on vector.

(2)

ues of observations.

mple of state vector x.

observations.

1 tk t0≥>



Extending the state vector to joint state-observati

Recall:

Applying h to x at a given time gives expected val

Get prior sample of obs. by applying h to each sa

Let z = [x, y] be the combined vector of state and

yk h xk tk,( )= vk k;+ 1 2 … tk +;, ,=

6/18/07

nd observations

(10.ext)



NOW, we have a prior for each observation:

p z tk Ytk,

p yk z( ) p z tk Ytk 1–

,( )

p yk ξ( ) p ξ tk Ytk 1–,( )dξ∫

------------------------------------------------------------------=

6/18/07

nd observations

ons in set yk?

of those in set j.

rmalizations.

yk1

yk2 … yk

s, , ,{ }=



One more issue: how to deal with many observati

Let yk be composed of s subsets of observations:

Observational errors for obs. in set i independent

Then:

Can rewrite (10.ext) as series of products and no

yk

p yk z( ) p yki z( )

i 1=

s

∏=

6/18/07

nd observations

ons in set yk?

equentially.

y independent error sequentially.

a.

e obs. error covariance.ated space.

dent errors!



One more issue: how to deal with many observati

Implication: can assimilate observation subsets s

If subsets are scalar (individual obs. have mutualldistributions), can assimilate each observation

If not, have two options:1. Repeat everything above with matrix algebr

2. Do singular value decomposition; diagonalizAssimilate observations sequentially in rotRotate result back to original space.

Good news: Most geophysical obs. have indepen

6/18/07

re)ilable.


Applying an Ensemble Filter.

Ensemble stateestimate after usingprevious observation(analysis).

Ensemble state attime of next obser-vation (prior).

tk tk+1

1. Use model to advanceensemble (3 members heto time at which next observation becomes ava

****

6/18/07

=h(x), byember.

servationsments withed errors canequentially.



2. Get prior ensemble sample of observation, yapplying forward operator h to each ensemble m

Theory: obfrom instruuncorrelatbe done s

y

****

h hh

6/18/07

ibution



3. Getobserved valueandobservational error distrfrom observing system.

y

****

h hh

6/18/07

mbleation errors).

y

ce betweenrs of ensem-imarily increment.



4. Findincrement for each prior observation ense(this is a scalar problem for uncorrelated observ

y

****

h hh Note: Differen

different flavoble filters is probservation in

6/18/07

ariable to linearlyble increments.

y

impact oftion increments onte variable can be sequentially!



5. Use ensemble samples of y and each state vregress observation increments onto state varia

y

****

h hh

Theory:observaeach stahandled

6/18/07

variable are updated,bservation...

y

tk+2



6. When all ensemble members for each state have a new analysis. Integrate to time of next o

y

****

h hh

tk

6/18/07

aotic Model.

on in red.

mble in green.

all three state

r variance = 4.0.

ber ensembles.


Some Fun with the Lorenz-63 3-Variable Ch

Observati

Prior ense

Observingvariables.

Obs. erro

4 20-mem−20

0

20 −20

0

2010

20

30

40

6/18/07

aotic Model.

on in red.

mble in green.



Observati

Prior ense

−20

0

20 −20

0

2010

20

30

40

6/18/07

aotic Model.

on in red.

mble in green.



Observati

Prior ense

−100

10−20

0

2010

20

30

40

6/18/07

aotic Model.

on in red.

mble in green.



Observati

Prior ense

−100

10−20

0

2010

20

30

40

6/18/07

aotic Model.

on in red.

mble in green.

is passing throughble region.



Observati

Prior ense

Ensembleunpredicta

−100

10−20

0

2010

20

30

40

6/18/07

aotic Model.

on in red.

mble in green.

semble heads forthe rest for the



Observati

Prior ense

Part of enone lobe, other.

−100

10−20

0

2010

20

30

40

6/18/07

aotic Model.

on in red.

mble in green.

is not linear here.

regression might be.



Observati

Prior ense

The prior

Standard pretty bad

.

−100

10−20

0

2010

20

30

40

6/18/07

aotic Model.

on in red.

mble in green.

is not linear here.

er hand...

ntrive examples

like this notin real assimilations.



Observati

Prior ense

The prior

On the oth

Hard to cothis bad.

Behavior apparent

−100

10−20

0

2010

20

30

40

6/18/07

o errors.

y

tk+2

pling Error;ian Assumption

Error;inearelation


Basic filter implementation is subject t

y

****

h hh

tk

1. Model Error

2. h errors;Representativeness

4. SamGauss

5. SamplingAssuming LStatistical R

3. ‘Gross’ Obs. Errors

6/18/07

se unobservedariable is known to

related to set ofved variables.

erved variable remainnged.

servations be oftic wind velocity.

variable isity temperature.


Regression sampling error and filter divergence

Suppostate vbe unobser

Unobsshoulduncha

Let obAntarc

State Park C

−3

−2

−1

0

1

2

3

Uno

bser

ved

Sta

te V

aria

ble SD=0.88

MN=0.12

−2 0 2Observed Variable

6/18/07



samples from jointution will haveero correlationcted |corr| = 0.19 samples).

ne observation,. variable mean andhange.




Finitedistribnon-z(expefor 20

After ounobsS.D. c

−3

−2

−1

0

1

2

3

Uno

bser

ved

Sta

te V

aria

ble SD=0.88

MN=0.12

SD=0.82MN=−0.27

After Obs. 1

Sample Correl. = 0.49


6/18/07



erved variableremain unchanged

erved means a random walk asobs. are used.




Unobsshould

Unobsfollowmore

−3

−2

−1

0

1

2

3

Uno

bser

ved

Sta

te V

aria

ble SD=0.88

MN=0.12

SD=0.62MN=0.50

After Obs. 21

Sample Correl. = −0.24


6/18/07




erved standardion is persistentlyased.

ted change in |SD|ative for any non-ample correlation!




Unobsshould

Unobsdeviatdecre

Expecis negzero s

−3

−2

−1

0

1

2

3

Uno

bser

ved

Sta

te V

aria

ble SD=0.88

MN=0.12

SD=0.49MN=0.36

After Obs. 41



6/18/07









Unobsshould

Unobsdeviatdecre

Expecis negzero s

−3

−2

−1

0

1

2

3

Uno

bser

ved

Sta

te V

aria

ble SD=0.88

MN=0.12

SD=0.40MN=0.12

After Obs. 61



6/18/07









Unobsshould

Unobsdeviatdecre

Expecis negzero s

−3

−2

−1

0

1

2

3

Uno

bser

ved

Sta

te V

aria

ble SD=0.88

MN=0.12

SD=0.26MN=0.14

After Obs. 81



6/18/07



ates of unobs.e too confident

rogressively lesst to any meaningfulvations.

sult can be thatingful obs. aretially ignored.




Estimbecom

Give pweighobser

End remeanessen

−3

−2

−1

0

1

2

3

Uno

bser

ved

Sta

te V

aria

ble SD=0.88

MN=0.12

SD=0.19MN=0.03

After Obs. 101

Sample Correl. = −0.29


6/18/07

hows expectedte value of sample

ation vs. trueation.

decrease withle size and for largeorrelations|.



Plot sabsolucorrelcorrel

Errorssamp|real c

0 0.5 10

0.2

0.4

0.6

0.8

1

True Correlation

Exp

ecte

d |S

ampl

e C

orre

latio

n|

10 Members20 Members40 Members80 Members

6/18/07

g error:

is smallce in priors.

n between

r and correct for it.


Ways to deal with regression samplin

1. Ignore it: if number of unrelated observationsand there is some way of maintaining varian

2. Use larger ensembles to limit sampling error.

3. Use additionala priori information about relatioobservations and state variables.

4. Try to determine the amount of sampling erro

6/18/07

g error:

tion between

e from observation.olynomial.

2000?)



3. Use additional a priori information about relaobservations and state variables.

Atmospheric assimilation problems.Weight regression as function of horizontaldistancGaspari-Cohn: 5th order compactly supported p

−2000 −1000 0 10000

0.5

1

Distance from Observation (Km

Reg

ress

ion

Wei

ght

6/18/07

g error:

tion between

variable pairs.

2000?)



3. Use additional a priori information about relaobservations and state variables.

Can use other functions to weight regression.Unclear whatdistance means for some obs./stateReferred to asLOCALIZATION.

−2000 −1000 0 10000

0.5

1

Distance from Observation (Km

Reg

ress

ion

Wei

ght

6/18/07

g error:

r and correct for it:

le correlation.

rge sample correl.

easure sampling error.

with a sample removed.

efficients and weight.



4. Try to determine the amount of sampling erro

A. Could weight regressions based on sampLimited success in tests.For small true correlations, can still get la

B. Do bootstrap with sample correlation to mLimited success.Repeatedly compute sample correlation

C. Use hierarchical Monte Carlo.Have a ‘sample’ of samples.Compute expected error in regression co

6/18/07

g error: ensembles.

N-member ensem-

s. increments for.

s. / state pair:mples of regression

β.y inβ implies staterements should be

regression confi-r,α.


Ways to deal with regression samplin4C. Use hierarchical Monte Carlo: ensemble of

y

****

H HH

y

tk+2

tk

y

****

H HH

y

tk+2

tk

M independentN-memberEnsembles

M groups of bles.

Compute obeach group

For given ob1. Have M sa

coefficient,2. Uncertaint

variable increduced.

3. Compute dence facto

β1

βΜ

RegressionConfidenceFactor,α

6/18/07

g error:

ensembles.

es 4 groups of 20.

efficient,βi.

inimizes:

te increments).



4C. Use hierarchical Monte Carlo: ensemble of

Split ensemble into M independent groups.For instance, 80 ensemble members becom

With M groups get M estimates of regression co

Find regression confidence factorα (weight) that m

Minimizes RMS error in the regression (and sta

αβi β j–[ ]2

i 1 i j≠,=

M∑

j 1=

M∑

6/18/07

g error:

ensembles.

t regression byα.

has repeatedations, cante sample mean or

n statistics forα.

can be used inquent assimilationscalization.

mple mean regression)



4C. Use hierarchical Monte Carlo: ensemble of

Weigh

If one observgeneramedia

Meanαsubseas a lo

α is function of M and (sample SD /sa

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Q: Ratio of sample standard deviation to mean

Re

gre

ssio

n C

on

fide

nce

Fa

cto

r, α

Group Size 2Group Size 4Group Size 8Group Size 16

Q Σβ β⁄=

6/18/07

ssure Obs. at 20N, 60E

60 80 100

an factor level 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 60 80 100

s section at row 18


Localization in GCM can be very complex. Surface Pre

20 40 60 80 100−10

0

10

20

30

40

50

60u mean factor level 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 40 60 80 100−10

0

10

20

30

40

50


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 40−10

0

10

20

30

40

50

60u me

20 40 60 80 100−10

0

10

20

30

40

50


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 40 60 80 100−10

0

10

20

30

40

50


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 40

0.2

0.4

0.6

0.8

1cros

PS to U

6/18/07

rors

independentlyt ongoing.

emble filters...

riance inflation,rior uncertainty

s more impact.

ation’: only letct a set oftate variables.

othly decrease0 as function of


Dealing With Ensemble Filter Er

y

****

h hh

y

tk+2

tk

1. Model Error

2. h errors;Representativeness

4. Sampling Error;Gaussian Assumption

5. Sampling Error;Assuming LinearStatistical Relation

Fix 1, 2, 3HARD bu

Often, ens

1-4: CovaIncrease pto give ob

5. ‘Localizobs. impa‘nearby’ s

Often smoimpact to distance.

3. Gross Obs. Errors

6/18/07

ance Inflation

=> ‘true’ distribution.

−1 0


Model/Filter Error; Filter Divergence and Vari

1. History of observations and physical system

−4 −3 −20

0.5

1

Pro

babi

lity

"TRUE" Prior PDF

6/18/07

ance Inflation

=> ‘true’ distribution.ufficient prior variance.

fident, obs. ignored−1 0



1. History of observations and physical system 2. Sampling error, some model errors lead to ins

3. Can lead to ‘filter divergence’: prior is too con−4 −3 −20

0.5

1

Pro

babi

lity

"TRUE" Prior PDF

Variance Deficient PDF

6/18/07

ance Inflation

=> ‘true’ distribution.ufficient prior variance.

ase spread of prior

.

−1 0

) x+



1. History of observations and physical system 2. Sampling error, some model errors lead to ins

3. Naive solution is Variance inflation: just incre

4. For ensemble member i,

−4 −3 −20

0.5

1

Pro

babi

lity

"TRUE" Prior PDF

Variance Deficient PDF

inflate xi( ) λ xi x–(=

6/18/07

ance Inflation

=> ‘true’ distribution.

2 4



1. History of observations and physical system .

−4 −2 00

0.2

0.4

0.6

0.8

Pro

babi

lity

"TRUE" Prior PDF

6/18/07

ance Inflation

=> ‘true’ distribution.ift in entire distribution.

RTAIN2 4

n (from model)



1. History of observations and physical system 2. Most model errors also lead to erroneous sh

3. Again, prior can be viewed as being TOO CE−4 −2 00

0.2

0.4

0.6

0.8

Pro

babi

lity

"TRUE" Prior PDF Error in Mea

6/18/07

ance Inflation

=> ‘true’ distribution.ift in entire distribution.

RTAIN

or it directly

2 4

n (from model)

ariance Inflated



1. History of observations and physical system 2. Most model errors also lead to erroneous sh

3. Again, prior can be viewed as being TOO CE4. Inflating can ameliorate this5. Obviously, if we knew E(error), we’d correct f

−4 −2 00

0.2

0.4

0.6

0.8

Pro

babi

lity

"TRUE" Prior PDF Error in Mea

V

6/18/07

n

e assimilation

els.

ations can ‘blow up’.

e top of AGCMs.

nd error.


Physical Space Variance Inflatio

Inflate all state variables by same amount befor

Capabilities:

1. Can be very effective for a variety of mod

2. Can maintain linear balances.

3. Stays on local flat manifolds.

4. Simple and inexpensive.

Liabilities:

1. State variables not constrained by observ

For instance unobserved regions near th

2. Magnitude ofλ normally selected by trial a

6/18/07

orenz-63

ence

vation in red.

nsemble in green.


Physical space covariance inflation in L

Observation outside prior: danger of filter diverg

Obser

Prior e−20

0

20 −20

0

2010

20

30

40

6/18/07

orenz-63

divergence avoided

vation in red.

nsemble in green.

semble in magenta.



After inflating, observation is in prior cloud: filter

Obser

Prior e

Inflated en

−20

0

20 −20

0

2010

20

30

40

6/18/07

orenz-63

vation in red.

nsemble in green.



Prior distribution is significantly ‘curved’

Obser

Prior e−20

0

20 −20

0

2010

20

30

40

6/18/07

orenz-63

o be off attractor.to transient off-ehavior or...

w-up’.

vation in red.

nsemble in green.

semble in magenta.



Inflated prior outside attractor. Posterior will alsCan lead attractor b

Model ‘blo

Obser

Prior e

Inflated en

−20

0

20 −20

0

2010

20

30

40

6/18/07

Tolerate Errors.

ystem.th observations., and filter error.


Adjunct Algorithms Developed for DART Can

1. Adaptive Error Tolerant Filters.Automatically detect error in assimilation sAdd uncertainty when model disagrees wiCan deal with LARGE model, observation

6/18/07

rror Tolerant Filter

-observed inconsistency

.

posed to be unbiased.

2 4

hood

S.D.n

obs2


Variance inflation for Observations: An Adaptive E

1. For observed variable, have estimate of prior

2. Expected(prior mean - observation) =

Assumes that prior and observation are supIs it model error or random chance?

−4 −2 00

0.2

0.4

0.6

0.8

Pro

babi

lity

Prior PDF

S.D.

Obs. Likeli

Expected Separatio

Actual 4.714 SDs

σprior2 σ+

6/18/07

rror Tolerant Filter

-observed inconsistency

.

rior and observation.

2 4

hood

S.D.tion

obs2


Variance inflation for Observations: An Adaptive E

1. For observed variable, have estimate of prior

2. Expected(prior mean - observation) =

3. Inflating increases expected separation.Increases ‘apparent’ consistency between p

−4 −2 00

0.2

0.4

0.6

0.8

Pro

babi

lity

Prior PDF Obs. Likeli

Inflated S.D.Expected Separa

Actual 3.698 SDs

σprior2 σ+

6/18/07

Tolerate Errors.

ystem.th observations., and filter error.

emble sampling errors.

problems.


Adjunct Algorithms Developed for DART Can

1. Adaptive Error Tolerant Filters.Automatically detect error in assimilation sAdd uncertainty when model disagrees wiCan deal with LARGE model, observation

2. Hierarchical filters detect and avoid small ensEnsemble of ensembles for tuning period.Limit impact of observations as required.Eliminate unnecessary calculation.

Can apply filters without tuning to large

6/18/07

all three pieces:

l system.

or characteristics.

ion.

ar tangent replacement).


Assimilation increases information about _________________________________________________

1. Get an improved estimate of state of physica

Initial conditions for forecasts.Includes time evolution and ‘balances’.High quality analyses (re-analyses).

_________________________________________________

2. Get better estimates of observing system err

Estimate value of existing or planned observations.Design observing systems that provide increased informat

_________________________________________________

3. Improve model of physical system.

Evaluate model systematic errors.Forward and backward sensitivity analysis (adjoint and lineSelect appropriate values for model parameters.

6/18/07

ary, 2003

ge spread).

inds.

reanalysis.

radiances.


DART/CAM NWP Assimilation: Janu

Model: CAM 3.1 T85L26.

Initialized from a climatological distribution (hu

Observations: Radiosondes, ACARS, Satellite W

Subset of observations used in NCAR/NCEP

Compare to NCEP operational, T254L64, uses

6/18/07

Af ter 6 hours.

CAMstarts withclimatology!Nearly zonal.


NCEP

DART/CAM

Difference.

6/18/07

Af ter 1 day.


NCEP

DART/CAM

Difference.

6/18/07

Af ter 3 days.

CAM gainszonalstructure.


NCEP

DART/CAM

Difference.

6/18/07

Af ter 7 days.

NHconverged.SH poorlyobserved.


NCEP

DART/CAM

Difference.

6/18/07

perature RMS

Hemisphere

NWP system.

1.8 2 2.2 2.4 2.6 2.8MPERATURE K

S Error: Northern Hemisphere

NCEP Opnl.T85 CAM


6-Hour Forecast Observation Space Tem

Tropics Northern

DART/CAM competitive with operational

0.9 1 1.1 1.2 1.3 1.4 1.5

100

200

300

400

500

600

700

800

900

1000

Pre

ssu

re (

hP

a)

TEMPERATURE K

6−Hour Forecast RMS Error: Tropics

NCEP Opnl.T85 CAM

1 1.2 1.4 1.6

100

200

300

400

500

600

700

800

900

1000P

ressu

re (

hP

a)

TE

6−Hour Forecast RM

6/18/07

all three pieces:

l system.

or characteristics.

ion.





Initial conditions for forecasts.Includes time evolution and ‘balances’.

High quality analyses (re-analyses)._________________________________________________



_________________________________________________



6/18/07

-CHEM model.

T CO observations.

ted by Kevin Raeder.


High-quality analysis of CO in Finite Volume CAM

Assimilate standard observations plus MOPIT

Work by Ave Arellano and Peter Hess suppor

6/18/07

6/18/07

all three pieces:

l system.

or characteristics.

ations.ion.






_________________________________________________


Estimate value of existing or planned observDesign observing systems that provide increased informat

_________________________________________________



6/18/07

rvtions in WRF

eric electric field.

low earth satellite.


Assimilating GPS Radio Occultation ObseaAssimilated as refractivity along beam path.Complicated function of T, Q, P and ionosph

Get a sounding as GPS satellite sets relative to

6/18/07

rvtions in WRF

satellite.

300 320

0, 2003


Assimilating GPS Radio Occultation Obsea

Weather Research and Forecasting Model.Regional Weather Prediction model.Configured for CONUS domain, 50 km grid.

Several hundred profiles available from CHAMP

180 200 220 240 260 2800

10

20

30

40

50

60

70

80

Latit

ude

(N)

Longitude (E)

GPS RO locations in CONUS domain, Jan 1−1

6/18/07

rvtions in WRF

close to GPS profiles.

adiosonde profiles.

M, GFDL, GFS...


Assimilating GPS Radio Occultation Obsea

Evaluating Impact of GPS Observations.

Case 1: Assimilate radiosondes EXCEPT thoseCase 2: Also assimilate GPS profiles.

Look at reduction in error from close (unused) r

NOTE: Identical code allows assimilation in CA

6/18/07

Errors in WRFt pair).

0 0.5 1 1.5 mean & rms error (g/kg)

an 1−10, 2003

rad+gpsrad


GPS Radio Occultation Impact on T and QEach plot displays bias (left pair) and RMS (righRed curves include GPS: reduced bias and RMS.

−1 0 1 2 3 4

300

400

500

600

700

800

900

Pre

ssu

re(h

Pa

)

T analysis mean & rms error (K)

Jan 1−10, 2003

−1 −0.5

300

400

500

600

700

800

900

Pre

ssu

re(h

Pa

)

Q analysis

J

6/18/07

all three pieces:

l system.

or characteristics.

ased information.






_________________________________________________


Estimate value of existing or planned observations.

Design observing systems that provide incre_________________________________________________



6/18/07

all three pieces:

l system.

or characteristics.

ion.






_________________________________________________



_________________________________________________



6/18/07

omparisons.perature Bias

.5 0 0.5 1

rth America

_TEMPERATURE bias (C)

guess

analysis

.5 0 0.5 1

rth America

_TEMPERATURE bias (C)

guess

analysis


Example of low-resolution assimilation cCAM spectral vs. FV for January, 2003: Tem

SpectralT21

FiniteVolume2x2.5

−1.5 −1 −0.5 0 0.5 1

100

200

300

400

500

600

700

800

900

1000

Tropics

Pre

ssu

re (

hP

a)

RADIOSONDE_TEMPERATURE bias (C)

guess

analysis

−1.5 −1 −0

100

200

300

400

500

600

700

800

900

1000

No

Pre

ssu

re (

hP

a)

RADIOSONDE

−1.5 −1 −0.5 0 0.5 1

100

200

300

400

500

600

700

800

900

1000

Tropics

Pre

ssu

re (

hP

a)

RADIOSONDE_TEMPERATURE bias (C)

guess

analysis

−1.5 −1 −0

100

200

300

400

500

600

700

800

900

1000

No

Pre

ssu

re (

hP

a)

RADIOSONDE

6/18/07

all three pieces:

l system.

or characteristics.

ion.

near tangent proxy).





_________________________________________________



_________________________________________________


Evaluate model systematic errors.Forward/backward sensitivity analysis (adjoint/liSelect appropriate values for model parameters.

6/18/07

ANY forecast ord analysis quantities

of adjoint andiods.

ns, or functions thereof.

mperature.a temperatures.

semble.


Ensemble Sensitivity Analysis

Can compute correlation (covariance) betweenanalysis quantity and ALL other forecast anor functions thereof at any time lag.

Can get same information as unlimited numberlinear tangent integrations over arbitrary per

Explore relations between variables, observatio

Example 1: Base point is 500 hPa mid-latitude teLook at impact on evolution of 500hP

Similar to linear tangent integration.Significant correlations from 20 member T85 en

6/18/07

t equivalent)

225

K

−1

−0.5

0

0.5

1

00 hPa Temperature


Forward in Time Sensitivity (Linear Tangen

+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

Time lag 00 hours: 500 hPa Temperature to 5

6/18/07

t equivalent)

00 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

t equivalent)

00 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

t equivalent)

00 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

t equivalent)

00 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

t equivalent)

00 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

ANY forecast ord analysis quantities

of adjoint andiods.

ns, or functions thereof.

onal velocity.emperature.


Ensemble Sensitivity Analysis

Can compute correlation (covariance) betweenanalysis quantity and ALL other forecast anor functions thereof.

Can get same information as unlimited numberlinear tangent integrations over arbitrary per

Explore relations between variables, observatio

Example 2: Base point is 500 hPa mid-latitude zLook at impact of previous 500 hPa t

Compare to an adjoint integration.

6/18/07

uivalent)

0 hPa Temperature

225

K

−1

−0.5

0

0.5

1


Backward in Time Sensitivity (Adjoint eq

Time lag -00 hours: 500 hPa Zonal Velocity to 50

+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

uivalent)

0 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

uivalent)

0 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

uivalent)

0 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

uivalent)

0 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

uivalent)

0 hPa Temperature

225

K

−1

−0.5

0

0.5

1




+

degrees east

degr

ees

nort

h

135 18030

40

50

60

70

6/18/07

all three pieces:

l system.

or characteristics.

ion.


ters.





_________________________________________________



_________________________________________________


Evaluate model systematic errors.Forward and backward sensitivity analysis (adjoint and line

Select appropriate values for model parame

6/18/07

Data Assimilation.

AM assimilation of wave drag effi- parameter.

ic values are noised be 0).

ciency< ~4 sug- by modelers.

ar convection.r ‘Wrong Reason’.

LL model problems


Climate Model Parameter Estimation via Ensemble

T21 Cgravityciency

Ocean(shoul

0< effigested

Positive values over NH land expected.Problem: large negative values over tropical land neMay reduce wind bias in tropical troposphere, but fo

Assimilation tries to use free parameter to fix A

0 90 180 270

−45

0

45

−10 −5 0 5 10

6/18/07

DART)

DART.


Data Assimilation Research Testbed (

Software to do everything here (and more) is in

Requires F90 compiler, Matlab.

Available from www.image.ucar.edu/DAReS/.

6/18/07

Documents

An Introduction to Ensemble Kalman Filtering · An Introduction to Ensemble Kalman Filtering Jeffrey Anderson NCAR Data Assimilation Research Section ... Environment Canada Operational