predictability of mesoscale variability in the east australian current given strong constraint data...
DESCRIPTION
IS4DVAR * Given a first guess (the forward trajectory)… Given a first guess (the forward trajectory)… * Incremental Strong Constraint 4-Dimensional VARiational data assimilationTRANSCRIPT
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
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
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
Forecast uncertainty: Ensemble predictions using Singular Vectors of the forecast
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.
Forecast uncertainty: Ensemble predictions using Singular Vectors of the forecast
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
Forecast uncertainty: Ensemble predictions using Singular Vectors of the forecast
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
0 lag –analysis skill
1 week lag –little loss of skill
Comparison between ROMS subsurface temperature predictions and all XBT observations in 2001-2002
correlation RMS error (oC)E3: SSH+SST+Syn-CTD
0 lag –analysis skill
1 week lag –little loss of skill
2 week lag – forecast beginsto deteriorate
Comparison between ROMS subsurface temperature predictions and all XBT observations in 2001-2002
correlation RMS error (oC)E3: SSH+SST+Syn-CTD
0 lag –analysis skill
1 week lag –little loss of skill
2 week lag – forecast beginsto deteriorate
3 week lag – forecast still better than …
Comparison between ROMS subsurface temperature predictions and all XBT observations in 2001-2002
correlation RMS error (oC)E3: SSH+SST+Syn-CTD
0 lag –analysis skill
1 week lag –little loss of skill
2 week lag – forecast beginsto deteriorate
3 week lag – forecast still better than …
no assimilation
Comparison between ROMS subsurface temperature predictions and all XBT observations in 2001-2002
correlation RMS error (oC)E3: SSH+SST+Syn-CTD
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
(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
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 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