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
