Predictability of Mesoscale Variability in the East Australian Current given Strong Constraint Data Assimilation John Wilkin Javier Zavala-Garay and Hernan

Predictability of Mesoscale Variability in the East Australian Current given Strong Constraint Data Assimilation

John WilkinJavier Zavala-Garayand Hernan Arango

Institute of Marine and Coastal Sciences Rutgers, The State University of New Jersey

New Brunswick, NJ, USA

Ocean Surface Topography Science Team MeetingHobart, Tasmania, Australia. March 12-15, 2007

http://marine.rutgers.edu

[email protected]

http://marine.rutgers.edu/

http://marine.rutgers.edu/

mailto:[email protected]

East Australia Current ROMS* Model for IS4DVAR

-24

-32

-28

-40

-36

-44

-48145 150 155 160 165

ROMS* Model Configuration

Resolution 0.25 x 0.25 degrees

Grid 64 x 80 x 30

∆x, ∆y ~ 25 km

∆t 1080 sec

Bathymetry 16 to 4895 m

Open boundaries Global NCOM (2001 and 2002)

Forcing Global NOGAPS, daily

De-correlation scale 100 km, 150 m

N outer, N inner loops 10, 3

1/8o resolution version simulates complex EOF “eddy” and “wave” modes of satellite SST and SSH in EAC separation:

* http://myroms.org

Wilkin, J., and W. Zhang, 2006, Modes of mesoscale sea surface height and temperature variability in the East Australian Current J. Geophys. Res. 112, C01013, doi:10.1029/2006JC003590

http://myroms.org/

IS4DVAR*

• Given a first guess (the forward trajectory)…Given a first guess (the forward trajectory)…

* Incremental Strong Constraint 4-Dimensional VARiational data assimilation

( )oR x

ox

IS4DVAR

• Given a first guess (the forward trajectory)…Given a first guess (the forward trajectory)…• and given the available data…and given the available data…

( )oR x

ox

IS4DVAR

• Given a first guess (the forward trajectory)…Given a first guess (the forward trajectory)…• and given the available data…and given the available data…• what change (or increment) to the initial what change (or increment) to the initial

conditions (conditions (ICIC) produces a new forward trajectory ) produces a new forward trajectory that better fits the observations?that better fits the observations?

ox

( )oR x

( )o oR x x

Basic IS4DVAR procedure

(1) Choose an

(2) Integrate NLROMS and save

(a) Choose a

(b) Integrate TLROMS and compute J

(c) Integrate ADROMS to yield

(d) Compute

(e) Use a descent algorithm to determine a “down gradient” correction to that will yield a smaller value of J

(f) Back to (b) until converged

(3) Compute new and back to (2) until converged

(0) (0)bx x

[0, ]t

(0)x

[0, ]t

[ ,0]t (0)(0)oJ

λx

1 (0) (0)(0)J

B x λx

(0)x

(0) (0) (0) x x x

( )tx

Out

er-lo

op

Inne

r-lo

op

NLROMS = Non-linear forward model; TLROMS = Tangent linear; ADROMS = Adjoint

1

1

1

1( )2

12

b

o

Tb b

J

NT

i i i i i ii

J

J x

x x B x x

H x y O H x y

= model-data misfit

The best fit becomes the analysis

assimilation window

The final state becomes the IC for the forecast window

assimilation window forecast

The final state becomes the IC for the The final state becomes the IC for the forecast windowforecast window

assimilation window forecast

verification

Forecast verification is with respect to data not yet assimilated

Days since 1 January 2001, 00:00

XBTs

4DVar Observations and Experiments

7-Day IS4DVAR Experiments

E1: SSH, SSTE2: SSH, SST, XBT

SSH 7-Day Averaged AVISOSST Daily CSIRO HRPT

SSH/SST

SSH/SSTSSH/SST

SSH/SST

Observations ROMS IS4DVAR: SSH/SST

ROMS IS4DVAR: XBT OnlyFirst Guess

Assimilating surface vs. sub-surface observationsEAC IS4DVAR

Observations

E1

E2

E1 – E2

SSHSSH

Temperature along XBT line

Temperature along XBT line

7-Day 4DVar Assimilation cycle

E1: SSH, SST ObservationsE2: SSH, SST, XBT Observations

EAC IS4DVAR

Days since 1 January 2001 00:00

Lag

Fore

cast

Tim

e (w

eeks

)La

g Fo

reca

st T

ime

(wee

ks)

Forecast SSH correlation and RMS error: Experiment E2

SSH Lag Pattern RMS

SSH Lag Pattern Correlation

0.6

RMS error normalized by the expected variance in SSH

lag = -1 week lag = 0 week lag = 1 week lag = 2 weeks lag = 3 weeks lag = 4 weeks

Forecast RMS error:

- typically < 0.5 out to 2 weeks forecast - grows fastest at the open boundaries

The eigenvectors of: RT(t,0) X R(0,t)

…with the largest eigenvalues are the fastest growing perturbations of the Tangent Linear model.

They correspond to the right Singular Vectors of R(0,t) (the ROMS propagator)

and describe perturbations to the initial conditions that lead to the greatest uncertainty in the forecast

ADROMS TLROMS

weights to define norm

Forecast uncertainty: Ensemble predictions using Singular Vectors of the forecast

assimilation forecasts

SV of forecast R()

10

1

, (0,1)i i ii

r a SV a are N

Perturb for ensemble of IC:r scales max|| to 3 cm


Perturbation after 10 daysSingular Vector 1

Afte

r ass

im.

SS

H+S

ST+

XB

TA

fter a

ssim

SS

H+S

ST

Example structures of the singular vectors for Experiments E1 and E2

E1

E2

Vertical structure

E1E1

E2E2

• The optimal perturbations when we include XBTs are more realistic: they tend to be concentrated at the surface, where most of the instability takes place.


Ensemble Prediction: E1

15-day forecast1-day forecast 8-day forecast

White contours: Ensemble setColor: Ensemble meanBlack contour: Observed SSH

Ensemble Prediction: E2

15-day forecast1-day forecast 8-day forecast

White contours: Ensemble setColor: Ensemble meanBlack contour: Observed SSH

• The optimal perturbations when we include XBTs are more realistic: they tend to be concentrated at the surface, where most of the instability takes place.

• When used in an ensemble prediction system the spread When used in an ensemble prediction system the spread of E2 is smaller and verifies better with observations than of E2 is smaller and verifies better with observations than that of E1.that of E1.

• Subsurface XBT data significantly improves the forecastSubsurface XBT data significantly improves the forecast• We have a further source of subsurface We have a further source of subsurface informationinformation

based on surface observations: based on surface observations: synthetic-CTDsynthetic-CTD– a statistically-based proxy deduced from historical EAC data a statistically-based proxy deduced from historical EAC data


Synthetic XBT/CTD example:Statistical projection of satellite SSH and SST

using EOFs of subsurface T(z), S(z)

E1 E3*

E1: SSH, SST*E3: SSH, SST, Syn-CTDSyn-CTD 4-day CSIRO subsurface projection of satellite obs to T(z), S(z)

E1: SSH + SST

Comparison between ROMS temperature analysis (fit) and withheld observations (all available XBTs); the XBT data were not assimilated – they are used here only to evaluate the quality of the reanalysis.

E1: SSH + SST

E3: SSH+SST+Syn-CTD

Comparison between ROMS temperature analysis (fit) and withheld observations (all available XBTs); the XBT data were not assimilated – they are used here only to evaluate the quality of the reanalysis.

0 lag –analysis skill

Comparison between ROMS subsurface temperature predictions and all XBT observations in 2001-2002

correlation RMS error (oC)E3: SSH+SST+Syn-CTD


1 week lag –little loss of skill





2 week lag – forecast beginsto deteriorate






3 week lag – forecast still better than …






3 week lag – forecast still better than …

no assimilation



Conclusions• Skillful ocean state predictions up to 2+ weeks • Assimilation of SST and SSH only is not enough• Assimilation of subsurface information:

– improves estimate of the subsurface – makes forecasts more stable to uncertainty in IC

• Singular Vectors demonstrate ensemble predictions and uncertainty estimation

• Synthetic-CTD subsurface projection adds significant analysis and forecast skill

• The synthetic-CTD is a linear empirical relationship, suggesting a simple dynamical relationship could link surface to subsurface variability– this could be built in to the background error covariance– this idea is not new: Weaver et al 2006, A multivariate balance

operator for variational ocean data assimilation, QJRMS

• Include balance terms in the IS4DVARInclude balance terms in the IS4DVAR• Improve boundary forcingImprove boundary forcing

– better global forecast and/or boundary conditionsbetter global forecast and/or boundary conditions– determine optimal boundary forcing via “weak constraint” data-determine optimal boundary forcing via “weak constraint” data-

assimilation (WS4DVAR)assimilation (WS4DVAR)• Use along-track SSH data instead of gridded multi-Use along-track SSH data instead of gridded multi-

satellite analysissatellite analysis

• Computational effort: 1 week analysis + forecast takes Computational effort: 1 week analysis + forecast takes 4 hours on 8-processors (Opteron 250, PGI, MPICH)4 hours on 8-processors (Opteron 250, PGI, MPICH)

Future Work

• David Griffin (CSIRO) for the SST and Syn-CTDDavid Griffin (CSIRO) for the SST and Syn-CTD• SIO VOS for XBTSIO VOS for XBT• AVISO for SSHAVISO for SSH• John Evans (RU) 4DVAR observation file processingJohn Evans (RU) 4DVAR observation file processing

Thanks to….

(2)

Notation

• ROMS state vector

• NLROMS equation form: (1)

• NLROMS propagator form:

• Observation at time with observation error variance

• Model equivalent at observation points

• Unbiased background state with background error covariance

iy

( ) ( ( )) ( )

(0)

( ) ( )i

t t tt

t t

x N x F

x xx x

( ) Tt x u v T S ζ

it O

( )i i i it H x H x

bx B

1 1( ) ( , )( ( ))i i i it t t t x M x

Strong constraint 4DVAR Talagrand & Courtier, 1987, QJRMS, 113, 1311-1328

• Seek that minimizes

subject to equation (1) i.e., the model dynamics are imposed as a ‘strong’ constraint. depends only on

“control variables” • Cost function as function of control variables

• J is not quadratic since M is nonlinear.

1 1

1

1 1( )2 2

b o

NT T

b b i i i i i ii

J J

J x

x x B x x H x y O H x y

(0), ( ), ( )t tx x F

( )tx

( )tx

1

1

1

1( ) (0) (0) (0) (0)2

1 ( ,0) (0) ( ,0) (0)2

b

o

Tb b

J

NT

i i b i i i b ii

J

J x

t t

x x B x x

H M x y O H M x y

S4DVAR procedureLagrange function Lagrange multiplier

At extrema of , we require:

S4DVAR procedure:

(1) Choose an (2) Integrate NLROMS and compute J(3) Integrate ADROMS to get

(4) Compute

(5) Use a descent algorithm to determine a “down gradient” correction to that will yield a smaller value of J

(6) Back to (2) until converged. But actually, it doesn’t converge well!

1

1

0 ( ) 0

0 0

0 (0) (0) 0 &(0)

0 ( ) 0 . .( )

ii i

i

TTi

i im m mi

b

dL NLROMSdt

dL ADROMSdt

L coupling of NL ADROMS

L i c of ADROMS

x N x Fλ

λ N λ H O Hx yx x

B x x λx

λx

1

( ) ( )N

T ii i i

i

dL J

dt

xx λ N x F ( )i i t F F ( )i i t x x( ) ( )i it i t λ λ λ

L

(0) bx x[0, ]t [ ,0]t (0)λ

1 (0) (0)(0) bJ

B x x λx

(0)x

Adjoint model integration is forced by the model-data error

xb = model state at end of previous cycle, and 1st guess for the next forecast

In 4D-VAR assimilation the adjoint model computes the sensitivity of the initial conditions to mis-matches between model and data

A descent algorithm uses this sensitivity to iteratively update the initial conditions, xa, to minimize Jb+ (Jo)

Observations minus Previous Forecast

x

0 1 2 3 4 time

Incremental Strong Constraint 4DVAR

Courtier et al, 1994, QJRMS, 120, 1367-1387 Weaver et al, 2003, MWR, 131, 1360-1378

• True solution

• NLROMS solution from Taylor series:

---- TLROMS Propagator

• Cost function is quadratic now

b x x x

1

1 1

1 1 1

211 1 1 1

1 ( )

1 1 1 1

( ) ( , ) ( )( , )( ( ) ( ))

( , )( , ) ( ) ( ) ( ( ) )( )

( , ) ( ) ( , ) ( )b i

i i i i

i i b i i

i ii i b i i i

i t

i i b i i i i

t t t tt t t t

t tt t t t O tt

t t t t t t

x

x M xM x x

MM x x xx

M x R x1( , )i it t R

1 1

1

1 1( ) (0) (0) (0) (0)2 2

b o

NTT

i i i ii

J J

J x

x B x G x d O G x d

(0, )i i itG H R

( ,0) (0) ( )i i i i b i i b it t d y H M x y H x

Basic IS4DVAR* procedure*Incremental Strong Constraint 4-Dimensional Variational Assimilation

(1) Choose an


(a) Choose a



(d) Compute




(0) (0)bx x

[0, ]t

(0)x

[0, ]t

[ ,0]t (0)(0)oJ

λx

1 (0) (0)(0)J

B x λx

(0)x

(0) (0) (0) x x x

( )tx

Basic IS4DVAR* procedure*Incremental Strong Constraint 4-Dimensional Variational Assimilation

(1) Choose an


(a) Choose a



(d) Compute




(0) (0)bx x

[0, ]t

(0)x

[0, ]t

[ ,0]t (0)(0)oJ

λx

1 (0) (0)(0)J

B x λx

(0)x

(0) (0) (0) x x x

( )tx

The Devil is in the Details

Conjugate Gradient Descent Long & Thacker, 1989, DAO, 13, 413-440

• Expand step (5) in S4DVAR procedure and step (e) in IS4DVAR procedure

• Two central component: (1) step size determination (2) pre-conditioning (modify the shape of J )

• New NLROMS initial condition: ---- step-size (scalar) ---- descent direction

•

• Step-size determination: (a) Choose arbitrary step-size and compute new , and

(b) For small correction, assume the system is linear, yielded by any step-size is

(c) Optimal choice of step-size is the who gives

• Preconditioning: define

use Hessian for preconditioning: is dominant because of sparse obs.• Look for minimum J in v space

1(0) (0)n n n x x d

nd

1 1(0)n n nn

J

d dx 1

1

( , )(0) (0)n

n n

J Jf

x x

0 (0)x 0J

rJ r0 0( , , )r rJ f J r 0r rJ

1( )2

TJ x x Ex

12v E x

1( )2

TJ z v v

21 1

2TJ

E B H O Hx

B

Background Error Covariance Matrix

Weaver & Courtier, 2001, QJRMS, 127, 1815-1846 Derber & Bouttier, 1999, Tellus, 51A, 195-

221

• Split B into two parts: (1) unbalanced component Bu

(2) balanced component Kb

• Unbalanced component ---- diagonal matrix of background error standard deviation ---- symmetric matrix of background error correlation

• for preconditioning,

• Use diffusion operator to get C1/2: assume Gaussian error statistics, error correlation

the solution of diffusion equation over the interval with is

• ---- the solution of diffusion operator ---- matrix of normalization coefficients

Σ

Tb u bB K B K

u B ΣCΣ

1 2 2TB B B 1 2 1 2bB K ΣC

C

2

2

( )( ) exp2xf x

2

2 02t T x

2

2

1 ( )( ) exp22xx

[0, ]t T 2(0) ( )x

C = ΛLΛΛL

12

1 2 1 2 1 2vC = ΛL L

Documents

Predictability of Mesoscale Variability in the East Australian Current given Strong Constraint Data Assimilation John Wilkin Javier Zavala-Garay and Hernan