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A multi-scale three-dimensional variational data assimilation scheme

Zhijin Li, , Yi Chao (JPL) James C. McWilliams (UCLA), Kayo Ide (UMD)

The 8th International Workshop on Adjoint Model Applications in Dynamic Meteorology

May 18-22, 2009, Tannersville, PA

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Outline

1. Motivations

2. A multi-scale three-dimensional variational data assimilation (MS-3DVAR) scheme

3. Applications and evaluations

4. Summary

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Data Assimilation and Forecast Cycle

3-day forecast

Aug.100Z

Time

Aug.118Z

Aug.112Z

Aug.106Z

Initialcondition

6-hour forecast

Aug.200Z

xa

xf

6-hour assimilation cycle

xxx fa

Time scales comparable with those of the atmosphere

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Observations and Data assimilation

0

100

200

300

400

500

600

700

800

900

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215

217

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227

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243

245

Year Day

Nu

mb

er o

f C

asts

/Day

<55

<110

<220

<440

<1100

T/S profiles from gliders Ship CTD profiles Aircraft SSTs AUV sections HF radar velocities

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Comparison of Glider-Derived Currents (vertically integrated current)

Black: SIO glider; Red: ROMS

SALT(PSU)

Performance of ROMS3DVAR:AOSN-II, August 2003

(Chao et al., 2009, DSR)

TEMP(C)

Glider temperature/salinity profiles

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Challenge: An Example with SCCOOS

Model domain, the resolution of 1km

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Southern California Coastal Ocean Observing System (SCCOOS)

SIO Glider Tracks

Challenge: How to assimilate sparse vertical profiles along with high resolution observations for a very high resolution model

Aug 2008, simulation without DA

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Data Assimilation Formulation

)()(21

)()(21

min 11 yHxRyHxxxBxxJ TfTf

x

prescribed B optimization algorithm

Variational methods (3Dvar/4Dvar):

Sequential methods (Kalman filter/smoother)

dynamically evolved B analytical solution

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Error Covariance and correlations

CB

C: Correlation matrix : RMSE diagonal matrix

Correlations are the vehicle for spreading out information of observations

Correlation scales are about

15-30km

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Forecast Error Covariance B:Single Observation Experiment

One single observation of SSH

afa xxx (increment)

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Southern California Coastal Ocean Observing System (SCCOOS)

SIO Glider Tracks

Motivation: assimilating sparse vertical profiles along with high resolution observations for a very high resolution model

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Multi-Scale Data Assimilation: Concept

HL

HL

eee

xxx

)()(2

1)()(

2

1min

)()(2

1)()(

2

1min

11

11

HHHHT

HHHfHHH

TfHH

x

LLLLT

LLLfLLL

TfLL

x

yxHRyxHxxBxxJ

yxHRyxHxxBxxJ

S

L

HL

HL yyy

Hxy

Background

Observation

Prerequisites

Multi-scale DA

0

0

THL

THLee

SCCOOS Glider Tracks

(Boer, 1983, MWR)

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Multi-Scale Data Assimilation: Scheme

Low Resolution (LR)

)()(2

1)()(

2

1min 11 yHxRyHxxxBxxJ Taf

LaTaf

Lx

)()(2

1)()(

2

1min 11

LLLLT

LLLfLLL

TfLL

xyxHRyxHxxBxxJ

L

aL

fL

aL xxx

aL

fafL xxx

High Resolution (HR)

Sparse Obs

High Resolution Obs

SCCOOS Glider Tracks

High resolution HF radar

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Work Flowchart of MS-3DVAR

Satellite SST/SSH HF Radar

LR-3DVAR

fLx

Forecast

Smoothed

fx

Start

HR-3DVAR

Increment aLx

afa xxx

aL

fafL xxx

End

Glider/Argo/MooringSmoothed

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Glider Observations vs Analyses

Aug 2008

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CALCOFI Observations

Aug 14-30, 2008 (not assimilated)

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Aug, 2008

Monthly Means

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Analysis vs HF Radar Observations

Taylor diagram (Taylor, 2001, JGR)

19 Surface Currents

Forecasts vs Analyses

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Summary

A multi-scale 3DVAR (MS-3DVAR) scheme has been formulated and developed.

It has been implemented in support of SCCOOS and AOOS-PWS.

The scheme has demonstrated the capability of assimilating sparse and high resolutions observations simultaneously, effectively and reliably.

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3 nested levels: L0 / L1 / L2.

Resolution : 10km / 3.6km / 1.2 km

(L2 : Prince William Sound)

Alaska Ocean Observing System -Prince William Sound:AOOS-PWS

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From USGS

Oil Spill: 1989 Exxon Tanker Wreck Prince William Sound, Alaska

Today

March 24, 1989

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Decomposition of Large and Small Scalesand Estimation of Error Covariance

Generate perturbations: difference between 24h and 48h forecasts, valid at the same time.

Decompose perturbations

Large scales: smoothed fields, with weight of

where L=25km, which is the decorrelation length scale from Russ’s estimation Small scales: the residuals

)2/exp( 22 Lr

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Observational Errors: Representativeness Errors

• Large scale observations: T/S vertical profiles from SIO, mooring and Argo

• Observational errors at the depth of z:

• Tentative values: briefly and empirically

estimated from the RMS profile of the small

scale components

2,0~ zoz Ne

oz

Lz

Lz

Lz zzNe )]/exp(1[,,0~ 2

022

mz 30,6.1 0 mz 50,6.1 0

For temperature

For salinity

CALCOFI Section

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Future Coastal Observing System

(National Research Council, 2003)

Buoy and glider profiles are sparse, with distances between profiles larger than decorrelation scales

Radar and satellite measurements may have very high resolutions, as high as the model resolution

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Glider Temperature and Salinity Profiles

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Surface Tidal Current Comparison (M2)

Corr. RMS Mean Sept. 0.43 3.6 6.6 Oct. 0.44 3.8 6.6 Nov. 0.51 3.7 6.3

Length of Major Axis (cm/s)

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A There-Dimensional Variational Data Assimilation (3DVAR)

• Real-time capability• Implementation with sophisticated and high resolution

model configurations• Flexibility to assimilate various observation

simultaneously

• Development for more advanced scheme

(Li et al., 2006, MWR; Li et al., 2008, JGR, Li et al., 2008, JAOT)

),,,,(

)()(2

1)()(

2

1min 11

vuSTx

yHxRyHxxxBxxJ TfTf

x