Coastal Ocean Observation Labhttp://marine.rutgers.edu/cool

John Wilkin, Hernan Arango, Julia Levin,Javier Zavala-Garay, Gordon Zhang

Regional Ocean Prediction

Scott Glenn, Oscar Schofield, Bob ChantJosh Kohut, Hugh Roarty, Josh Graver

Coastal Ocean Observation Lab

Janice McDonnell Education and Outreach

Coastal Observation and Prediction Sponsors:

Regional Ocean Prediction http://marine.rutgers.edu/po

Education & Outreachhttp://coolclassroom.org

Coastal Ocean Modeling, Observation and Prediction

Coastal Ocean Observation Labhttp://marine.rutgers.edu/cool/sw06/sw06.htm

Integrating Ocean Observing and Modeling Systems for SW06 Analysis and Forecasting

Regional Ocean Modeling and Predictionhttp://marine.rutgers.edu/po/sw06

• gliders and CODAR

• satellite SST, bio-optics

• high-res regional WRF atmospheric forecast

• SW06 ship-based obs.

• ROMS model embedded in NCOM or climatology

• WRF and NCEP forcing + rivers

• 2-day cycle IS4DVAR assimilation

Real-time data and analysis to ships via ExView and HiSeasNet

• glider, CODAR, satellite, WRF Daily Bulletin

• NCOM and ROMS/assimilation 2-day forecasts

Model-based re-analysis of submesoscale ocean state

• ROMS/IS4DVAR assimilation: plus CODAR, Scanfish, moorings, CTDs …

• high-res nesting in SW06 center

• ensemble simulations; uncertainty instability, sensitivity analysis, optimal observations

• Weekly/monthly bulletin ?

Regional Ocean Modeling and Predictionfor Shallow Water 2006

Rutgers Ocean Modeling and Prediction Group for SW06:

– Hernan Arango– John Evans– Naomi Fleming– Gregg Foti– Julia Levin– John Wilkin– Javier Zavala-Garay– Gordon Zhang

http://marine.rutgers.edu/po/sw06

Outline• Strong constraint 4-dimensional variational data

assimilation– some math– how it works

• SW06 configuration– some results

• Next steps:– SW06 reanalysis

• Algorithmic tuning, more data, higher resolution

– ensemble simulations• Forecast and analysis uncertainty and predictability

– observing system design

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

tt t

t

t t

xN x F

x x

x x

( )T

t 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 2b 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

T

b 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 procedure

Lagrange 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

dLNLROMS

dt

dLADROMS

dt

Lcoupling of NL ADROMS

Li c of ADROMS

xN x F

λ

λ Nλ H O Hx y

x 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) b

J

B x x λ

x(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 t

t t t t

t tt t t t O t

t

t t t t t t

x

x M x

M x x

MM x x x

x

M x R x1( , )i it t R

1 1

1

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

2 2b 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

(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 λ

x

(0)x

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

( )tx

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

(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 λ

x

(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 n

n

J

d dx 1

1

( , )(0) (0)n

n n

J Jf

x x

0 (0)x 0JrJ r

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

1( )

2TJ x x Ex

12v E x

1( )

2TJ 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

( )( ) exp

2

xf x

2

20

2t T x

2

2

1 ( )( ) exp

22

xx

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

C = ΛLΛΛL

1

21 2 1 2 1 2

vC = ΛL L

Adjoint surface temperature states at different time during a

three -day period. Initial adjoint forcing area is surrounded by the black frame. Top: southward wind. Bottom: northward wind.

Har

vard

Box

(100

kmx1

00km

)

ROMS LATTE o

uter b

oundary

ROMS SW

06 oute

r boundary

SW06 Model Domains

ROMS SW06

5-km grid for IS4DVAR testing

Forcing:

• NCEP-NAM and WRF USGS Hudson River OTPS tides

• Open boundaries NCOM and L&G climatology

2-day assimilation cycle

20-km horizontal and 5-m vertical length scales in background error covariance

Data:

• gliders, CTDs, XBTs, Knorr thermosalinograph, daily best-SST composite, AVISO SSH

Salt 5m Salt 30m Temp 30m

Forecast skill in 2-day interval when initial conditions are adjusted using IS4DVARSimple forecast: No

data assimilation

Mesoscale prediction test case:East Australian Current

IS4DVAR assimilation• daily SST (CSIRO)• SSH (AVISO)• VOS XBT Tasman

Sea

Javier Zavala-GarayJohn WilkinHernan Arango

• Adjoint adjusts all state variables, not just those observed

• Singular vectors of the tangent linear model give most unstable modes of variability– Optimal perturbations for

ensemble simulation– Predictability limits

East Australian Current

Ensembles of:

1-day forecasts 8-day forecasts 15-day forecasts

Ass

imil

atin

g S

SH

+S

ST

+X

BT

Ass

imil

atin

g S

SH

+S

ST

East Australian Current Color: ensemble mean. Contours: individual ensemble members. Black: SSH observations

Optimal Perturbation AnalysisA

fter

ass

im.

SS

H+

SS

T+

XB

TA

fter

ass

im S

SH

+S

ST

Vertical Structure of SV1Perturbation after 10 daysSingular Vector 1

Now what ?SW06 reanalysis of sub-mesoscale ocean state

– IS4DVAR algorithmic tuning• forecast cycle length; background error covariance

(preconditions conjugate gradient search)

– More data• CODAR, moorings, shipboard ADCP …

– Higher resolution– Ensemble simulations

• forecast skill; quantify predictability; analysis uncertainty

MURI COMOP– Observing system design– Physics information

Mixing of the Hudson and Raritan Rivers

PhytoplanktonAbsorption

Detritus AbsorptionSeaWiFS

chlorophyll

VisibleRGB

SST