P1.8 QUANTIFYING AND REDUCING UNCERTAINTY BY EMPLOYING MODEL ERROR ESTIMATION METHODS

Dusanka Zupanski

Cooperative Institute for Research in the AtmosphereColorado State University, Fort Collins, CO

Introduction

Data assimilation methods can be effectively used to estimate errors in dynamical models of the atmosphere and ocean. Variational methods have been successfully used for model error estimation in recent years. New generation data assimilation techniques, referred as Ensemble Kalman fileter-like methods, are also capable of estimating and correcting model errors of dynamical models in terms of biases and empirical parameters. These new methods do not require prescribed background error covariances, since the background error covariances are calculated via ensemble forecasting. Thus, ensemble based data assimilation offers a potential for quantifying and reducing uncertainty of our knowledge about atmospheric and oceanic processes.

Methodology

Maximum Likelihood Ensemble Filter (MLEF, Zupanski 2005; Zupanski and Zupanski 2005)

Developed using ideas fromVariational data assimilation (3DVAR, 4DVAR)Iterated Kalman FiltersEnsemble Transform Kalman Filter (ETKF, Bishop et al. 2001)

Parameter estimation KdVB model

min]([]([2

1][][

2

1 11 obs

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Minimize cost function J

ζCIPfb2121 )( xx

)()( 212121212112

ffff HPRHPRHPRHPC TTT

x

ζ

Change of variable

-augmented control variable of dim Nstate >>Nens

(includes initial conditions, model error, empirical

parameters)

-control variable in ensemble space of dim Nens

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Analysis error covariance

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Forecast error covariance

Experiments

Estimate empirical parameters and model errors of a one-dimensional Korteweg-de Vries-Burgers (KdVB) model

Estimate model errors of a 3-dimensional non-hydrostatic model (CSU-RAMS)

Calculate uncertainties of the estimated model errors in terms of analysis error covariance matrix Pa

“True” diffusion parameters are successfully estimated. Correct innovation statistic (expected to be Gaussian for this experiment) indicates that the estimated Pa is reliable.

Cross-covariance between model state and model error is complex for both models. It would be hard to correctly prescribe this cross-covariance, even for simple dynamical models, such as KdVB. Ensemble based approaches, such as MLEF, can produce complex time evolving covariances for both model state and model error.

AcknowledgementsThis research was partially funded by DoD grant DAAD19-02-2-0005 and NOAA grant NA17RJ1228 .

ReferencesBishop, C. H., B. J. Etherton, and S. Majumjar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part 1: Theoretical aspects. Mon. Wea. Rev., 129, 420–436.Zupanski, D., and M. Zupanski, 2005: Model error estimation employing ensemble data assimilation approach. Submitted to Mon. Wea. Rev. [Available at ftp://ftp.cira.colostate.edu/Zupanski/manuscripts/MLEF_model_err.revised2.pdf]Zupanski, M., 2005: Maximum likelihood ensemble filter: Theoretical aspects. Accepted to Mon. Wea. Rev. [Available at ftp://ftp.cira.colostate.edu/milija/papers/MLEF_MWR.pdf].

M - a non-linear dynamical model

ESTIMATION 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

Bias estimation of KdVB model

Estimated components of the augmented analysis error covariance matrix

Bias estimation of CSU-RAMS model

Estimated components of the augmented analysis error covariance matrix

AMS Ed Lorenz Symposium

- experiments with simulated observations- model error created by using erroneous Coriolis force in the model’s equations