Model error estimation employing ensemble data assimilation Dusanka Zupanski and Milija Zupanski CIRA/Colorado State University, Fort Collins, CO, U.S.A.

Model error estimation employing ensemble data assimilation

Dusanka Zupanski and Milija ZupanskiCIRA/Colorado State University, Fort Collins, CO, U.S.A.

EGU General Assembly 2005 NP5.01: Quantifying predictability

24-29 April 2005Vienna, Austria

Dusanka Zupanski, CIRA/[email protected]

OUTLINE

Why do we need to estimate model error?- Data assimilation point of view- General point of view

Methodology- MLEF+State Augmentation

Experimental results- KdVB model (1-d)- CSU-RAMS model (3-d, non-hydrostatic)

Conclusions and future work

Dusanka Zupanski, CIRA/[email protected]

Why do we need to estimate model error?

Goal of CLASSICAL data assimilation methods is to estimate(1) atmospheric state

Goal of ENSEMBLE data assimilation methods is to estimate(1) atmospheric state(2) uncertainty of the estimated state

Data assimilation point of view

Model error influences - adversely - both estimates

ENSEMBLE approaches are more sensitive to model error

Use this opportunity to further improve new methods.

Be happy with the limited benefits of the new methods.

or

Why do we need to estimate model error?

Many additional applications in geophysics would benefit from model error estimation:

Improving current dynamical models Developing new dynamical models Quantifying predictability Quantifying information content of observations Obtaining new knowledge about geophysical processes

General point of view

This presentation is mostly focused on the data assimilation aspect, as a first step towards more general applications.

min]([]([2

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1 11 obs

Tobsb

-f

Tb HHJ yxRyxxxxx ))P

2121 )( CIPfbxx

Change of variable (preconditioning)

x

- control vector in ensemble space of dim Nens

Minimize cost function J

- model state vector of dim Nstate >>Nens

ZZC T

)()( 2121 xRpxRz HH fii

C - information matrix of dim Nens Nens

METHODOLOGY: MLEF approach

fip - columns of

21

fP iz - columns of Z

METHODOLOGY: MLEF + State Augmentation

011,0

1

1,

0

0

00

00

01

nnnn

n

n

1-n

1-nn

nn

n

n

n

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zb

x

I

Ib

x

zn

nΦxx 1-n1-nn,n M

)(bΦΦ Gn0

nn

01-nn

bbbb 01-nn

011, nnnnF zzn

- model state time evolution

- AUGMENTED state time evolution

- serially correlated model error

- model bias

- vector of empirical parameters

Approach applicable to other EnKF methods.

RESULTS: Parameter estimation, KdVB modelESTIMATION OF DIFFUSION COEFFICIENT

(102 ens, 101 obs)

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

1.80E-01

2.00E-01

2.20E-01

2.40E-01

2.60E-01

1 11 21 31 41 51 61 71 81 91

Cycle No.

Dif

fusi

on

co

efic

ien

t va

lue

estim value (0.07)

true value (0.07)

estim value (0.20)

true value (0.20)

Innovation histogram (incorrect diffusion)(neglect_err, 10 ens, 101 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

-5 -4 -3 -2 -1 0 1 2 3 4 5

Category bins

PD

F

Innovation histogram (incorrect diffusion)(param_estim, 10 ens, 101 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

-5 -4 -3 -2 -1 0 1 2 3 4 5

Category bins

PD

F

Innovation histogram (correct diffusion)(correct_model, 10 ens, 101 obs)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

-5 -4 -3 -2 -1 0 1 2 3 4 5

Category bins

PD

F

True parameter recovered. Improved innovation statistics.

INNOVATION 2 TEST (biased model)(neglect_err, 10 ens, 10 obs)

0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01

1 11 21 31 41 51 61 71 81 91

Analysis cycle

INNOVATION 2 TEST (biased model)(bias_estim, 10 ens, 10 obs, bias dim = 101)

0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01

1 11 21 31 41 51 61 71 81 91

Analysis cycle

INNOVATION 2 TEST (biased model)(bias_estim, 10 ens, 10 obs, bias dim = 10)

0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01

1 11 21 31 41 51 61 71 81 91

Analysis cycle

INNOVATION 2 TEST (non-biased model)(correct_model, 10 ens, 10 obs)

0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+01

1 11 21 31 41 51 61 71 81 91

Analysis cycle

NEGLECT BIAS BIAS ESTIMATION (vector size=101)

BIAS ESTIMATION (vector size=10) NON-BIASED MODEL

RESULTS: Bias estimation, KdVB model

It is beneficial to reduce degrees of freedom of the model error.

RESULTS: Bias estimation, KdVB model

Augmented analysis error covariance matrix is updated in each data assimilation cycle. It includes cross-covariance between the initial conditions (IC) error and model error (ME).

An experiment with a simple state dependent model error

Estimate state dependent model error . Define model error components for u, v, T,…,q as:

nun uΦ

nvn vΦ

nqn qΦ

Estimate single parameter

In real atmospheric applications, model errors are commonly more complex, but ARE often STATE DEPENDENT.

EXPERIMENTAL DESIGN

Non-hydrostatic atmospheric model (CSU-RAMS)

- 3d model

- simplified microphysics (level 2)Hurricane Lili case25 1-h DA cycles: 13UTC 1 Oct 2002 – 14 UTC 2 Oct30x20x21 grid points, 15 km grid distance (in the Gulf of Mexico)

- model domain 450km X 300kmControl variable:

- u,v,w,theta,Exner, r_total (initial conditions, dim=54000)

- (dim=1)Model simulated observations with random noise

(7200 obs per DA cycle)Nens=50Iterative minimization of J (1 iteration only)

RESULTS: Parameter estimation, RAMS model

Analysis RMS errors for u-wind(analysis-truth)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 6 11 16 21

Analysis cycle

RM

S u-

win

d (m

s-1)

correct_modelparam_estimneglect_errno_assim

Analysis RMS errors for w-wind(analysis-truth)

0

0.005

0.01

0.015

0.02

0.025

1 6 11 16 21

Analysis cycle

RM

S w

-win

d (m

s-1)

correct_modelparam_estimneglect_errno_assim

U-WIND ANALYSIS ERRORS

W-WIND ANALYSIS ERRORS

Parameter estimation is almost as good as the perfect (correct) model data assimilation experiment.

Both the initial conditions and the parameter are adjusted. Control variable size is 54001.

RESULTS: Parameter estimation, RAMS model

Analysis RMS errors for Exner(analysis-truth)

0

0.5

1

1.5

2

2.5

3

1 6 11 16 21

Analysis cycle

RM

S E

xner

correct_modelparam_estimneglect_errno_assim

Analysis RMS errors for Total Mixing Ratio(analysis-truth)

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

1 6 11 16 21

Analysis cycle

RM

S r

correct_modelparam_estimneglect_errno_assim

Both the initial conditions and the parameter are adjusted. Control variable size is 54001.

EXNER FUNCTION ANALYSIS ERRORS

TOTAL WATER MIXING RATIO ANALYSIS ERRORS

Neglecting model error reduces the benefits of data assimilation.

RESULTS: Parameter estimation, RAMS model

Estimated parameter value is close to the true parameter value.

Empirical parameter

-1.50E-05

-1.00E-05

-5.00E-06

0.00E+00

5.00E-06

1.00E-05

1.50E-05

1 6 11 16 21

Analysis cycle

estimatedtrue

Neglect_err Param_estim

No_assim Correct_model

True

Theta_il

Differences of the order of1.0K-3.0K.

Differences of the order of0.1K.

CONCLUSIONS

Dusanka Zupanski, CIRA/[email protected]

Ensemble based data assimilation methods, if coupled with state augmentation approach, can be effectively used to estimate empirical parameters.

Estimation of model errors can also be effective if number of degrees of freedom of the model error is reduced.

Neglecting model errors leads to degraded data assimilation results.

Capability to update augmented forecast error covariance is an advantage of ensemble based data assimilation approaches.

Sensitivity of ensemble data assimilation approaches to model errors is an OPPORTUNITY for further improvements. This will be further explored in the future.