A Multi-Scale Three-Dimensional Variational Data

Assimilation Scheme and Its Application to Coastal Oceans

Zhijin Li

Jet Propulsion Laboratory, California Institute of Technology

The 9th Workshop on Adjoint Model Applications in Dynamic Meteorology

Cefalu, Sicily, Italy, 10-14 October 2011

Copyright 2011 California Institute of Technology

Acknowledgements

• Dr Yi Chao and his group (JPL)

• Prof James C McWilliams and his group (UCLA)

• Prof Kayo Ide (UMD)

• We acknowledge the support from NASA Physical

Oceanography Program

3DVAR Data Assimilation and Forecast Cycle

3-day forecast

6-hour forecastxfxxx fa δ+=

Aug.100Z

Time

Aug.118Z

Aug.112Z

Aug.106Z

Initialcondition

forecast

Aug.200Z

xa

x

6-hour assimilation cycle

• Diurnal variation

• Rapid response to wind stresses

• Eddies, fronts, filaments, etc

Autonomous Ocean Sampling Network (AOSN)

Experiment August, 2003

“Bring together sophisticated new robotic vehicles with advanced ocean models to improve our ability to observe and predict the ocean”

www.mbari.org/aosn

9 km 3km 1 km

An Incremental There-Dimensional Variational Data

Assimilation (3DVAR)

f

TT

x

TfTf

x

Hxyy

yxHRyxHxBxxJ

yHxRyHxxxBxxxJ

−=

−−+=

−−+−−=

−−

−−

δ

δδδδδδδ )()(2

1

2

1)(min

)()(2

1)()(

2

1)(min

11

11

1. Real-time capability

2. Implementation with sophisticated and high resolution model

configurations

3. Flexibility to assimilate various observation simultaneously

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

AOSN Intensive Observations

�T/S profiles from gliders

� Ship CTD profiles

� Aircraft SSTs

� AUV sections

� HF radar velocities

0

100

200

300

400

500

600

700

800

900

213

215

217

219

221

223

225

227

229

231

233

235

237

239

241

243

245

Year Day

Num

ber

of C

asts

/Day <55

<110

<220

<440

<1100

� HF radar velocities

T/S Profile Data

Glider and AUV tracks Ships, Aircrafts, and HF radars

Comparison of Glider-Derived Currents (vertically integrated current)

Performance of ROMS3DVARAugust 2003

Black: glider Red: ROMS

(Chao and Li et al., 2009, DSR)

Southern California Coastal Ocean Observing System

(SCCOOS)

SIO Glider Tracks

Challenge: Assimilating sparse vertical profiles along with high

resolution observations for a very high resolution model

Decorrelation length scales: 15-50km

Challenges with 3DVAR

Fourier Series Expansion of Homogenous Errors

dxinxxee

inxexe

n

nn

)exp()(1

)exp()(

−=

=

∫

∑∞

∞

−∞=

dxinxxeen )exp()(2

−= ∫ ∞−π

==

≠==∗

nmc

nmee

nnm ,

,0

Wiener-Khintchine Theorem

)()()( rxexerc += Error Covariance

1

2

)exp()(

)exp()(2

1

nn

n

n

ec

dninrcrc

drinrrcc

=

=

−=

∫

∫∞

∞−

∞

∞−π

power spectral density

Error Covariance in 3DVAR: Smoothing and Spreading

2

2

2)( D

r

erc−

=

−=

2exp

2

22 DnDcn

π

A Multi-Decorrelation Length Scale Scheme for High

Resolution Models?

SL xxx +=

Background Error

SL eee +=

SL

TSL

BBB

ee

+=

= 0

3DVAR with a Background Error Covariance of

Multi-Decorrelation Length Scales

( ) )()(2

1

2

1)(min 11 yxHRyxHxBBxxJ T

SLT

xδδδδδδδ −−++= −−

( )

( ) )()(2

1

2

1)(min

)()(2

1

2

1)(min

11

11

yxHRHHByxHxBxxJ

yxHRHHByxHxBxxJ

ST

LT

SSST

SSx

LT

ST

LLLT

LLx

S

L

δδδδδδδ

δδδδδδδ

δ

δ

−+−+=

−+−+=

−−

−−

(Li et al., 2011, QJRMS, in revision))|(

)|(

yxp

yxp

S

L(Lorenc, 1986)

Multi-Scale Representativeness Errors

to xHye −= δδ

TS HHBR +Observational Error

Covariance for Large Scale

( ) ( ) ( )orS

orom

tS

bS

ttt

tL

oL

eee

xxHyHxyy

xHye

++=

−−−−−=

−= δδ

Measurement error + Representativeness error + Multi-scale representativeness error

Multi-Scale Data Assimilation

hS

hL

h yyy +=

High resolution Observation

( )

( ) )()(2

1

2

1min

)()(2

1

2

1min

11

11

hSSS

ThSSSS

TSS

x

hLLL

ThLLLL

TLL

x

yxHRyxHxBxxJ

yxHRyxHxBxxJ

S

L

δδδδδδδ

δδδδδδδ

δ

δ

−−+=

−−+=

−−

−−

Multi-scale DA

3DVAR Formulations

)()(2

1)()(

2

1min 11 yHxRyHxxxBxxJ TfTf

x−−+−−= −−

( ) )()(11

min11 yxHRHHByxHxBxJ TTT δδδδδδ −+−+=

−−

3DVAR

AB-3DVAR

( )

( ) )()(2

1

2

1min

)()(2

1

2

1min

11

11

yxHRHHByxHxBxJ

yxHRHHByxHxBxJ

ST

LT

SSST

Sx

LT

ST

LLLT

Lx

S

L

δδδδδδ

δδδδδδ

δ

δ

−+−+=

−+−+=

−−

−−

)()(2

1

2

1min

)()(2

1

2

1min

11

11

hSSS

ThSSSS

TS

x

hLLL

ThLLLL

TL

x

yxHRyxHxBxJ

yxHRyxHxBxJ

S

L

δδδδδδ

δδδδδδ

δ

δ

−−+=

−−+=

−−

−−

MS-3DVAR

Experiments with Idealized Problems

cos1

0 N

nkaSx t

k

K

k

tk

tn

+= ∑=

φπ

tkb

tk

K

k

bk

bn

aa

N

nkaSx

β

φπ

=

+= ∑=

cos1

0

True StateBackground/First Guess

5/,200

)1,1(,

1

NKN

ka

N

kktk

tk

k

==

−∈=

=

=

απαφ

γtk

kbk aa β=

ooe

tn

o eaxy +=Observations

Difference between SD-3DVAR, MD-3DVAR,

and MS-3DVAR Solutions

Patchy Observation

( )

−−=

2

22

2exp

D

jibB eij

Analysis Errors

MS-3DVAR Work Flow

SL yy δδ ,

yδObservation innovationForecast fx

fS

fL xx ,

Increment

SS-3DVARLS-3DVAR

IncrementaSxδ

aS

faL

a xxx δ+=aL

ffaL xxx δ+=

aLxδ

SL BB ,

( )

( )

)()(2

1

)()(2

1

2

1min

)()(2

1

)()(2

1

2

1min

1

11

1

11

hSSS

ThSS

ST

LT

SSST

Sx

hLLL

ThLL

LT

ST

LLLT

Lx

yxHRyxH

yxHRHHByxHxBxJ

yxHRyxH

yxHRHHByxHxBxJ

S

L

δδδδ

δδδδδδ

δδδδ

δδδδδδ

δ

δ

−−+

−+−+=

−−+

−+−+=

−

−−

−

−−

Kronecker Product Formulation of

3D Error Correlations

130

100

80

60

Cx(z

a, z

a)

DIS

TA

NC

E F

RO

M S

HO

RE

0.2

0.4

0.6

0.8

1

100

150

200

250D

IST

AN

CE

FR

OM

S. B

OU

ND

AR

Y

Ch(z

a, z

a)

37.50.8

1

ΣΣ= CB

“NMC” Method: 48h-24h Forecast

( ) ( )( )( )T

TT

GGGG

GGGGC

CCC

ηξκηξκ

ηηξκξκξηκ

ηξκξηκ

⊗⊗=

⊗=

⊗=

130 100 80 60 40 20

40

20

DISTANCE FROM SHORE (km)

DIS

TA

NC

E F

RO

M S

HO

RE

0

0.2

50100150200250

50

100

DISTANCE FROM S. BOUNDARY (km)

DIS

TA

NC

E F

RO

M S

. BO

UN

DA

RY

−123.5 −123 −122.5 −122 −121.535

35.5

36

36.5

37

LAT

ITU

DE

(° N

)

LONGITUDE (° W)

−0.2

0

0.2

0.4

0.6

−123.5 −123 −122.5 −122 −121.5LONGITUDE (° W)

Improved Performance with SCCOOS

MS-3DVAR Performance

1989 Exxon Valdez Supertanker Oil Spill

in the Prince William Sound

Struggling sea lion during the tragedy days

Prince William Sound

Blind sea lion present day

Field Experiment 2009 Prediction of

Drifter Trajectories in the Prince William Sound

L0 10km

L1 3.6km

L2 1.2km

Oil Spill: 1989 Exxon Tanker Wreck ,

Prince William Sound, Alaska

Surface CurrentsHF radar observed (Red), ROMS (Black)

Effective Assimilation of High Frequency Radar High

Resolution Velocities during Field Experiment 2009

(Schoch and Chao, 2010, EOS)

Summary

• A multi-scale 3DVAR scheme with partitioned cost functions was

developed

• MS-3dVAR used multi-decorrelation length scales to construct

background error covariancebackground error covariance

• Effectiveness of the assimilation of both sparse and high resolution

observations was improved

� Observation oriented covariance

� Reduced representativeness errors