MODEL ERROR ESTIMATION EMPLOYING DATA ASSIMILATION METHODOLOGIES Dusanka Zupanski Cooperative...

MODEL ERROR ESTIMATION MODEL ERROR ESTIMATION EMPLOYING DATA ASSIMILATION EMPLOYING DATA ASSIMILATION

METHODOLOGIESMETHODOLOGIES

Dusanka ZupanskiDusanka ZupanskiCooperative Institute for Research in the AtmosphereCooperative Institute for Research in the Atmosphere

Colorado State UniversityColorado State UniversityFort Collins, CO 80523-1375Fort Collins, CO 80523-1375

ATS Colloquium series29 January 2004

ftp://ftp.cira.colostate.edu/Zupanski/presentations

ftp://ftp.cira.colostate.edu/Zupanski/manuscripts

OUTLINE:OUTLINE:

Data assimilation methodsData assimilation methods

State augmentation approach State augmentation approach

Model error estimation employing variational and Model error estimation employing variational and EnsDA frameworksEnsDA frameworks

Experimental results employing various modelsExperimental results employing various models

Q: What is data assimilation?Q: What is data assimilation?

Conclusions and future workConclusions and future workDusanka Zupanski, CIRA/[email protected]

DATA ASSIMILATION (ESTIMATION THEORY)DATA ASSIMILATION (ESTIMATION THEORY)

Discrete stochastic-dynamic model

Dusanka Zupanski, CIRA/[email protected]

Discrete stochastic observation model

111 )()( : kkkk wxGxMxM

w k-1 – model error (stochastic forcing)

M – non-linear dynamic (NWP) model

G – model (matrix) reflecting the state dependence of model error

kkk xHy )( :D

k – measurement + representativeness error

H – non-linear observation operator (M M D D )

min]([]([2

1][)(][

2

1 11 obs

Tobsb

fTb HHJ yxRyxxxxx ))P

VARIATIONAL APPROACH

(1) State estimate (optimal solution):

)()( 1bobs

TTba xyRPPxxx HHHH

KALMAN FILTER APPROACH

(2) Estimate of the uncertainty of the solution:

TTaf GGQMMPP

Tji

jif MMMM )]()()][()([)( , xpxxpxP ENSEMBLE KALMAN FILTER or EnsDA APPROACH

In EnsDA solution is defined in ensemble subspace (reduced rank problem) !

DATA ASSIMILATION INCLUDES THE FOLLOWING:DATA ASSIMILATION INCLUDES THE FOLLOWING:

KALMAN FILTER APPROACH


State augmentation approach (a model bias example)

1-n1-n

1-n

k1-n

n1-n

n

nn FF

Mw

Φ

x

bΦ

Φx

Φ

xw

)1(

bias model ; conditions initial ; , k0k0k bxbxz

Control variable for the analysis cycle k:

Solve EnKF equations (or EnsDA) equations in terms of control variable z and forecast model F :

TaaTaf ])(][)([ 2/12/1 PFPFFFPP Parameter estimation is a special case of state augmentation approach!

4DVAR framework

Forecast error covariance

Data assimilation(Init. Cond. and Model Error adjust.)

Observations First guess

Init. Cond. and Model Error opt. estimates

Forecast error covariance

Data assimilation(Init. Cond. and Model Error adjust.)

Observations First guess

Init. Cond. and Model Error opt. estimates

Ens. forecasting

Analysis error Covariance

(in ensemble subspace)

EnsDA framework

In EnsDA framework model error does not depend on assumptions regarding forecast error covariance;

data assimilation problem is solved in ensemble subspace


ETA 4DVAR: Surface pressure model error time evolution

(every 2-h over a 12-h data assimilation interval)

From Zupanski et al. 2004 (submitted to MWR)


RAMS 4DVAR: Exner function model error time evolution (lev=5km), every 2-h

From Zupanski et al. 2004 (submitted to MWR)


Eta: Horizontal wind model error time evolution (lev=250hPa), every 2-h


RAMS: Horizontal wind model error time evolution (lev=250hPa), every 2-h


RAMS: Horizontal wind model error, vertical cross-section, every 2-h


EnsDA experiments withKorteweg-de Vries-Burgers (KdVB) model- one-dimensional model- includes non-linear advection, diffusion and dispersion

From Zupanski and Zupanski 2004 (submitted to MWR)

IMPACT OF INCORRECT DIFFUSION(10 Ens, 10 Obs)

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

1 11 21 31 41 51 61 71 81 91

Cycle No.

RM

S e

rro

r

correct_diffusion

incorrect_diffusion

param_estim

IMPACT OF INCORRECT DIFFUSION(10 Ens, 101 Obs)

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

1 11 21 31 41 51 61 71 81 91

Cycle No.

RM

S e

rro

r

correct_diffusion

incorrect_diffusion

param_estim

PARAMETER estimation impact


EnsDA experiments with KdVB model

ESTIM ATION OF DIFFUSION COEFFICIENT 102 ens, 101 obs

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

7.00E-02

8.00E-02

9.00E-02

1.00E-01

1 11 21 31 41 51 61 71 81 91

Cycle No.

dif

fusi

on

co

efic

ien

t va

lue

estimated value

true value

ESTIMATION OF DIFFUSION COEFFICIENT 10 ens, 101 obs

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

1 11 21 31 41 51 61 71 81 91

Cycle No.

dif

fusi

on

co

efic

ien

t va

lue

estimated value

true value

Kalman Filter statistical verification toolsInnovation statistics

)]([][)]([ 12kkf

Tkk HH xyRHHPxy T

2 statistics – can be used to test the stability of an ensemble filter

Innovation vector =obs – first guess

)( kk H xy

Pf – produced by Ensemble Filter algorithm

R – input to Ensemble Filter algorithm

The conditional mean of 2 (normalized by obs) should be equal to one

Innovation histogram – Probability Density Function of normalized innovation vectors

For Gaussian distribution, and with linear observation operator H, the innovation histogram should be equal to standard normal distribution N (0,1)

EnsDA experiments with KdVB model (PARAMETER estimation impact)

Innovation histogram(Parameter etimation 10 ens, 10 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

F

Innovation histogram(Incorrect diffusion, 10 ens, 101 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

F

Innovation histogram(Parameter estimation, 10 ens, 101 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

FInnovation histogram

(Correct diffusion, 10 ens, 101 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

FInnovation histogram

(Correct diffusion 10 ens, 10 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

F

Innovation histogram(Incorrect diffusion, 10 ens, 10 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

F10 obs 101 obs

EnsDA experiments with KdVB model (PARAMETER estimation impact)

INNOVATION 2 TEST(Incorrect diffusion 10 ens, 101 obs)

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

1 11 21 31 41 51 61 71 81 91

Analysis cycleINNOVATION 2 TEST

(Parameter estimation 10 ens, 101 obs)

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

1 11 21 31 41 51 61 71 81 91

Analysis cycleINNOVATION 2 TEST

(Correct diffusion 10 ens, 101 obs)

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

1 11 21 31 41 51 61 71 81 91

Analysis cycle

It would be BEST to have a perfect model, but since this is not the case, it is necessary to estimate model error and use it to correct the model!

EnsDA experiments with KdVB model

Innovation histogram(Model bias estimation 10 ens, 10 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

F

Innovation histogram(Model bias estimation 202 ens, 101 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

FA feasible solution to reduce the number of degrees of freedom is to define bias in terms of small number of parameters.

BIAS estimation results

BIAS estimation may require many observations and large ensemble size !


From Zupanski and Zupanski 2004 (submitted to MWR)

EnsDA experiments with KdVB modelAnalysis error covariance matrix (UNCERTAINTY estimate)

We are employing more and more models!NASA’s GEOS column model

Work in progress in collaboration with:-A. Hou and S. Zhang (NASA/GMAO)-C. Kummerow (CSU/Atmos. Sci.)

Innovation histogram for NASA's GEOS model experiment

(Parameter estimation 10 ens, 110 "REAL"obs)

0.00E+001.00E-012.00E-013.00E-014.00E-015.00E-016.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

F

Preliminary results including parameter estimation:

Innovation histogram for NASA's GEOS model experiment

(Parameter estimation 10 ens, 110 "REAL"obs)

0.00E+001.00E-012.00E-013.00E-014.00E-015.00E-016.00E-01

1 11 21 31 41 51 61 71 81 91 101

Category bins

PD

F

R1/2 = R1/2 = 2

Choice of observation errors directly impacts innovation statistics.Observation error covariance R is the only given input to the system!

Q: What is data assimilation?Q: What is data assimilation?

A:A:Method of defining optimal initial conditions Method of defining optimal initial conditions (classic definition)(classic definition)

Model error estimation methodModel error estimation method

Model development tool (estimate and correct Model development tool (estimate and correct model errors during the model development phase)model errors during the model development phase)

PDF estimationPDF estimation


CONCLUSIONSCONCLUSIONS

To employ full data assimilation power, model error To employ full data assimilation power, model error estimation should be includedestimation should be included

EnsDA approaches are very promising since they can EnsDA approaches are very promising since they can provide not only optimal estimate of the atmospheric provide not only optimal estimate of the atmospheric state, but the state, but the uncertaintyuncertainty of the estimate as well of the estimate as well

FUTURE WORKFUTURE WORKEstimate and correct model errors for various models Estimate and correct model errors for various models (GEOS, RAMS, WRF, etc.) (GEOS, RAMS, WRF, etc.)


An example:

Equivalence between variational and Kalman filter equations

(for linear models and Gaussian statistics)

)()(2

1)()(

2

1 11obs

Tobsb

TbJ yxRyxxxPxx HH

)()( 11obs

Tb

Jyx

HRHxxP

x

HHPx

112

2

RTJ

0)()( 0 11

obsT

bx

JyxRxxP HH

0

)(111

11

obsT

bT

bT

Tb

yRxRxR

xRxxP

HHHHH

HH


An example (continued):

)()()( 111bobs

Tb

Tb xyRxxRxxP HHHH

)()( 1111bobs

TTba xyRRPxxx HHHH

)()( 1bobs

TTba xyRPPxxx HHHH

Important difference: variational methods DO NOT provide forecast

error covariance update (update of P)!

11111 )()( RPPRRP TTTT HHHHHH

Using the matrix equality (e. g., Jaswinski 1970, Appendix 7b:

We obtain the Kalman filter analysis equation:

MODEL ERROR ESTIMATION EMPLOYING DATA ASSIMILATION METHODOLOGIES Dusanka Zupanski Cooperative...

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