Some Innovative Applications and Approaches Using Nudging Four Dimensional Data Assimilation:

Lili Lei and David R. StaufferDept. of Meteorology, Penn State University

Conference on Applied Inverse ProblemsData Assimilation for Geophysical Problems23 July 2009University of Vienna, Austria

Part II. A Hybrid Nudging-EnKF for Improving Data Assimilation in the Lorenz and Shallow-Water Model

Systems

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Outline

• Motivation

• Methodology

• Experimental design for Lorenz model

• Results of Lorenz model

• Experimental design for shallow-water model

• Results of shallow-water model

• Conclusions

• Future work and acknowledgement

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Motivation

Fujita et al. 2007 Mon. Wea. Rev.

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timetobs

timetobs

timetobs

Nudging:

EnKF:

Hybrid nudging-EnKF:

Methodology for Hybrid Nudging-EnKF

xG w w x xo

s t

d

dt

x x K x xoa b b

xK, w x xo

t

df

dt

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Methodology for Hybrid Nudging-EnKF

Ensemble state

Nudging state

OBS

OBS

time

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Methodology for Hybrid Nudging-EnKF

• The hybrid nudging coefficients:

• The Lorenz model equations:

00 0w w wxx y xzx

dxy x g y y g z z

dtg x x

00 0w w wyx yy zy

dyrx y xz g x x g y y g z z

dt

00 0w w wzzzx zy

dzxy bz g x g z zx g y y

dt

00 0w w wxx y xzx

dxy x g y y g z z

dtg x x

00 0w w wyx yy zy

dyrx y xz g x x g y y g z z

dt

00 0w w wzzzx zy

dzxy bz g x g z zx g y y

dt

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Experimental Design

• Single 3000-step period experiment: From an initial condition, first 1500 time steps of integration are discarded to avoid the effects of transients, and the following 1500-4500 time steps are analyzed.

• 100-sample experiment:100 initial conditions are randomly chosen, and a data assimilation cycle of 1500 time steps is executed following each initial condition. In each data assimilation cycle, the first 500 time steps of integration are discarded, and the following 1000 time steps are used for analysis .

• The observation error variances used to create simulated observations:

2 1.0x 2 1.0y 2 1.0z

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Experimental Design

Exp. Name Exp. Description

TNUDAssimilate observations by traditional nudging with nudging coefficient of 10

EnKFAssimilate observations by ensemble Kalman filter (EnKF)

EnKSAssimilate observations by ensemble Kalman smoother (EnKS)

EnKS_lagAssimilate observations by lagged ensemble Kalman smoother (EnKS) which applies next available observation backward to previous observation time

Hybrid-DAssimilate observations by hybrid nudging-EnKF with diagonal elements only

HybridAssimilate observations by hybrid nudging-EnKF with full matrix

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Experimental Design: Verification

• The root-mean-square (RMS) errors are computed every time step.

• Observation Retention (OR): the average absolute value of the RMS error difference between one time step before the observation time and that at the observation time after the data assimilation.

• Normalized Error and Retention (NER): sum of the average RMS error normalized by that of the EnKS and the OR normalized by that of the EnKS.

EnKS EnKS

RMSE O RNE R = +

RMSE O R

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Comparisons of Hybrid-D and Hybrid

RMSE

Nudging coefficients in Hybrid-D

Nudging coefficients in Hybrid

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RMSE

OR NER

Average parameters in single 3000-step period experiment with ensemble size 100 and perfect model

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RMSE

OR NER

Average parameters in 100-sample experiment with ensemble size 100 and perfect model

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RMSE

OR NER

Average parameters in single 3000-step period experiment with ensemble size 100 and imperfect model

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RMSE

OR NER

Average parameters in 100-sample experiment with ensemble size 100 and imperfect model

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CPU Time Cost (sec)

obsfreq10 obsfreq25 obsfreq50

TNUD 4 5 5

EnKF 16 10 9

EnKS 3414 1385 728

Hybrid 16 10 9

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CPU Time Cost (sec)

obsfreq10 obsfreq25 obsfreq50

TNUD 4 5 5

EnKF 16 10 9

EnKS 3414 1385 728

EnKS_lag 60 55 53

Hybrid 16 10 9

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Comparisons of EnKS_lag and EnKS

RMSE OR

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Summary of Lorenz Model Results

• A hybrid nudging-EnKF approach with potential use for NWP was explored here using the Lorenz three-variable model system.

• The EnKS, which is the golden standard, is more than 100 times more expensive than the EnKF and Hybrid, and it also has large data storage requirements. The EnKS_lag, which is only 4~6 times more expensive than the EnKF and Hybrid, is more practical but has somewhat larger RMS errors and Observation Retention (OR) than the EnKS.

• The hybrid nudging-EnKF with diagonal elements only has larger RMS error than the hybrid nudging-EnKF with full matrix.

• The hybrid nudging-EnKF approach produces somewhat larger / similar average RMS errors than both the EnKF in perfect / imperfect model.

• The hybrid nudging-EnKF has better OR than both the EnKF and the EnKS in general.

• The hybrid nudging-EnKF approach generally produces smaller (better) Normalized Error and Retention (NER, normalized by EnKS) than the EnKF.

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Methodology for Hybrid Nudging-EnKF

• The shallow water model equations:

2 00 0w wwuu t uv t uh t

u u u hu v fv g u g v v g h h

tu

xg

y xu

2 0 00 wwwvu t vv t vh t

v v v hu v fu g v g u u g h h

tg v

x yv

y

2 0 00w w whu t hv t hh t

h h h u vu v h h g u u g v v

t x y x yg h h

0 ,x L 0 y D

2 00 0w wwuu t uv t uh t

u u u hu v fv g u g v v g h h

tu

xg

y xu

2 0 00 wwwvu t vv t vh t

v v v hu v fu g v g u u g h h

tg v

x yv

y

2 0 00w w whu t hv t hh t

h h h u vu v h h g u u g v v

t x y x yg h h

0 ,x L 0 y D

L = 500 km, D = 300 km

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Experimental Design: Initial Conditions

Case I - Wave Case II - Vortex

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Case I Case II

D 300 km 300 km

L 500 km 500 km

f 10-4 s-1 10-4 s-1

g 0.5 ms-2 9.8 ms-2

104 m2s-1 104 m2s-1

dx / dy 10 km 10 km

dt 30 sec 30 sec

B.C.

Periodic B.C in west-east Periodic B.C in west-east

Free-slip rigid wall B.C in south-north

Tendencies of height and wind components = 0.0 in south-north

Inflation factor 1.1 1.1

Localization scale 500 km 100 km

Half-period nudging time window

1 h 1 h

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Experimental Design: Observations

• Simulated 3-hourly observations are generated by finer-scale model simulations. The fine domain has grid spacing of 1 km.

• The observation error variances used to create simulated observations:

• Observation networks:

2 2 20.5u m s 2 2 20.5v m s 2 22.5h m Case I:2 2 22.0u m s 2 2 22.0v m s 2 220.0h m Case II:

OBSN I: 1 OBS

OBSN II: 19 OBS in X direction

OBSN III: 11 OBS in Y direction

OBSN IV: OBSN II + OBSN III

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Experimental Design

Exp. Name Exp. Description

TNUDAssimilate observations by traditional nudging with nudging coefficient of 10-4 s-1

EnKFAssimilate observations by ensemble Kalman filter (EnKF)

EnKS

Assimilate observations by lagged ensemble Kalman smoother (EnKS) which applies next available observation backward to previous observation time every 30 minutes

HybridAssimilate observations by hybrid nudging-EnKF with full matrix

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Experimental Design: Verification

• The verification data is based on the 1-km model simulation and available on every grid point of the 10-km coarse domain.

• The verification data is the average value of surrounding 10*10 1-km grid points from the 1-km “truth” domain.

• The root-mean-square (RMS) errors of height and wind are computed separately every minute.

• Normalized RMS error: the RMS error computed against the “truth” divided by the RMS error of the “truth” computed against its domain-average value.

• Observation Retention (OR): the average absolute value of the RMS error difference between one time step before the observation time and that at the observation time after the data assimilation.

• Normalized Error and Retention (NER): sum of the average RMS error normalized by that of the EnKS and the OR normalized by that of the EnKS.

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Normalized RMS Error of Case I with OBSN II

Height Wind

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RMSE

RMSE – 30min

OR

NER

NER – 30min

Average parameters of height field with different observation frequencies (in hours) in OBSN II

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RMSE

RMSE – 30min

OR

NER

NER – 30min

Average parameters of wind field with different observation frequencies in OBSN II

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RMSE

OR NER

Average parameters of height field with three-hourly observations in different observation networks

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RMSE

OR NER

Average parameters of wind field with three-hourly observations in different observation networks

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Normalized RMS Error of Case II with OBSN II

Height Wind

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RMSE

RMSE – 30min

OR

NER

NER – 30min

Average parameters of height field with different observation frequencies in OBSN II

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RMSE

RMSE – 30min

OR

NER

NER – 30min

Average parameters of wind field with different observation frequencies in OBSN II

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RMSE

OR NER

Average parameters of height field with three-hourly observations in different observation networks

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RMSE

OR NER

Average parameters of wind field with three-hourly observations in different observation networks

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• A hybrid nudging-EnKF data assimilation approach is further investigated using a shallow-water model.

• A quasi-stationary wave (Case I) and a moving vortex (Case II) are used to test the hybrid nudging-EnKF scheme. Three kinds of observation frequencies and four observation networks are applied in the 24-h data assimilation experiments for each case.

• The hybrid EnKF reduces the RMS errors compared to those of the traditional nudging and EnKF applied separately.

• The hybrid EnKF also has the ability to reduce the RMS error as well as or even better than the “gold standard” EnKS, and also to produce better observation retention than the EnKS at a reduced computational cost more similar to that of the EnKF.

Summary of Shallow-Water Model Results

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General Conclusions

• A hybrid nudging-EnKF data assimilation approach is investigated using the Lorenz model and a shallow-water model.

• The hybrid nudging-EnKF retains the spatial (flow-dependent) error correlation weighting function from the EnKF and the gradual corrections of the continuous nudging approach (digital filter unnecessary) to avoid the strong corrections and discontinuities (error spikes) at the analysis steps.

• In the hybrid nudging-EnKF, the model equations assist in the data assimilation process.

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Future Work

• Test the hybrid EnKF in strongly forced / unstable conditions

• Test the hybrid EnKF in forecasting

• …

• Transition hybrid EnkF to WRF

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ACKNOWLEDGEMENTS

• This research is supported by DTRA contract no. HDTRA1-07-C-0076 under the supervision of John Hannan of DTRA.

• The authors would like to thank Aijun Deng, Sue Ellen Haupt, George S. Young and Fuqing Zhang for helpful discussions and comments.