Post on 05-Aug-2020
1
Lecture 1: Primal 4D-Var
Outline
• ROMS 4D-Var overview• 4D-Var concepts• Primal formulation of 4D-Var• Incremental approach used in ROMS• The ROMS I4D-Var algorithm
ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
η, Q
Posterior
4D-Var
ClippedAnalyses
Ensemble(SV, SO)
HypothesisTests
Forecast
dof Adjoint4D-Var
impact
Term balance,eigenmodes
UncertaintyAnalysiserror
ROMS 4D-VarEnsemble
4D-Var
Priors &Hypotheses
Priors
2
ROMS Obsy, R
fb, Bf
bb, Bb
xb, B
η, Q
Posterior
4D-Var
Priors &Hypotheses
ClippedAnalyses
Ensemble(SV, SO)
HypothesisTests
Forecast
dof Adjoint4D-Var
impact
Term balance,eigenmodes
UncertaintyAnalysiserror
ROMS 4D-VarEnsemble
4D-VarPriors
• Broquet, G., C.A. Edwards, A.M. Moore, B.S. Powell, M. Veneziani and J.D. Doyle, 2009: Application of 4D-variational data assimilation to the California Current System. Dyn. Atmos. Oceans, 48, 69-91.
• Broquet, G., A.M. Moore, H.G. Arango, C.A. Edwards and B.S. Powell, 2009: Ocean state and surface forcing correction using the ROMS-IS4DVAR data assimilation system. Mercator Ocean Quarterly Newsletter, #34. 5-13.
• Broquet, G., A.M. Moore, H.G. Arango, and C.A. Edwards, 2010: Corrections to ocean surface forcing in the California Current System using 4D-variational data assimilation. Ocean Modelling, Under review.
• Di Lorenzo, E., A.M. Moore, H.G. Arango, B.D. Cornuelle, A.J. Miller, B. Powell, B.S. Chua and A.F. Bennett, 2007: Weak and strong constraint data assimilation in the inverse Regional Ocean Modeling System (ROMS): development and application for a baroclinic coastal upwelling system. Ocean Modeling, 16, 160-187.
• Powell, B.S., H.G. Arango, A.M. Moore, E. DiLorenzo, R.F. Milliff and D. Foley, 2008: 4DVAR Data Assimilation in the Intra-Americas Sea with the Regional Ocean Modeling System (ROMS). Ocean Modelling, 23, 130-145.
• Powell, B.S., A.M.Moore, H.G. Arango, E. Di Lorenzo, R.F. Milliff and R.R. Leben, 2009: Near real-time assimilation and prediction in the Intra-Americas Sea with the Regional Ocean Modeling System (ROMS). Dyn. Atmos. Oceans, 48, 46-68.
• Zhang, W.G., J.L. Wilkin, and J.C. Levin, 2010: Towards an integrated observation and modeling system in the New York Bight using variational methods, Part I. Ocean Modelling, Under review.
• Zhang, W.G., J.L. Wilkin, and J.C. Levin, 2010: Towards an integrated observation and modeling system in the New York Bight using variational methods, Part II. Ocean Modelling, Under review
ROMS 4D-Var Applications
,y R
Data Assimilation
bb(t), Bb
fb(t), Bf
xb(0), Bx
Model solutions depends on xb(0), fb(t), bb(t), h(t)
time
x(t)
Obs, yPriorPosterior
ROMS
Prior
3
Notation & Nomenclature
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
TS
x ζuv
(0)( )( )( )
ttt
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
xf
zbη
1
N
y
y
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
y ( )( )H= − bd y x
Statevector
Controlvector
Observationvector
Innovationvector
×
HObservation
operator
Prior
Thomas Bayes(1702-1761)
Bayes Theorem
( | ) ( | ) ( ) ( )p p p p=z y y z z y
( ) ( )( )T 1 exp 1 2 ( ) ( )c H H−= − − −y x R y x
( ) ( )( )T 1 exp 1 2 −× − − −b bz z D z z
Conditional probability:
Maximum likelihood estimate: identify the minimum ofT T1 1( ) ( ) ( ) ( ( )) ( ( ))
2 2NLJ H H− −= − − + − −1 1b bz z z D z z y x R y x
which maximizes p(z|y).
(Wikle and Berliner, 2007)
Posteriordistribution
Prior MarginalDatadistribution
(“likelihood”)
4
Variational Data Assimilation
T T1 1( ) ( ) ( ) ( ( )) ( ( ))2 2NLJ H H− −= − − + − −1 1
b bz z z D z z y x R y x
diag( , , , )= x b fD B B B Q
Background error covariance
( )( | ) exp NLP J∝ −z yConditional Probability:
Problem: Find z=za that minimizes J (i.e. maximizes P)using principles of variational calculus.za is the “maximum likelihood” or “minimum variance”estimate.
JNL is called the “cost” or“penalty” function.
Observation error covariance
( (0), ( ), ( ), ( ))T T T Tt t tδ δ= b fz x ε ε ηinitial
conditionincrement
boundaryconditionincrement
forcingincrement
corrections for model
error
bb(t),Bb
fb(t), Bf
xb(0), Bx
( ) ( )1 11 12 2
TTJ δ δ δ δ− −= + − −z D z G z d R G z d
diag( , , , )= x b fD B B B Q
Prior (background) error covariance
TangentLinear Modelsampled atobs points
ObsErrorCov.
Innovation
( )H= − bd y x
Prior
Incremental Formulation
(Courtier et al., 1994)
(H=linearizedobs operator)
( (0), ( ), ( ), ( ))T T T Tt t tδ δ= b fz x ε ε ηinitial
conditionincrement
boundaryconditionincrement
forcingincrement
corrections for model
error
bb(t),Bb
fb(t), Bf
xb(0), Bx
( ) ( )1 11 12 2
TTJ δ δ δ δ− −= + − −z D z G z d R G z d
Prior
The minimum of J is identified iteratively by searchingfor ∂J/∂δz=0
Incremental Formulation
5
Incremental Formulation
bb(t),Bb
fb(t), Bf
xb(0), Bx
( (0), ( ), ( ), ( ))T T T Tt t tδ δ= b fz x ε ε ηinitial
conditionincrement
boundaryconditionincrement
forcingincrement
corrections for model
error
Assumptions:
(i) δ bz z(ii) (t)= (t)+ (t)δbx x x(iii) (t)δ δx M z(iv) (t) =δ δ δ∗ ∗H x H M z G z
Prior
δ= +bz z zTangent Linear ModelTangent Linear H
==
MH
i i-1 i i(t ) ( (t ), (t ), (t ))=b b b bx x f bM
The Tangent Linear Model(TLROMS)
time
x(t)
Obs, yPrior
0 τPrior is solution of model:
Increment: (t) (t); (t) (t); etcδ δb bx x f fi i-1 i-1
i-1
(t )= ( (t ) (t ), ) ( (t ), )
δδ
++
b
b
b x
x x xx M z
MM
Nonlinearmodel
Tangent linear model
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢
•••••••
⎥⎢ ⎥⎢•
⎥⎢ ⎥⎢ ⎥⎣
•
⎦•
••
⎡ ⎤⎢ ⎥⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢
•••••••
⎥⎢ ⎥⎢•
⎥⎢ ⎥⎢ ⎥⎣
•
⎦•
zPrimalSpace
yObservation
vector
zDual
Space
Primal vs Dual FormulationVector of
increments
“Observationspace”
“Modelspace”
6
The Solution
= +a bz z Kd
T T -1( )= +K DG GDG R
Analysis:
Gain matrix (dual form):
Gain matrix (primal form):1 T 1 1 T 1( )− − − −= +K D G R G G R
Two Spaces
T T -1( )= +K DG GDG RGain (dual):
Gain (primal):1 T 1 1 T 1( )− − − −= +K D G R G G R
obs obsN N×
model modelN N×obs modelN N
Two Spaces
G maps from model spaceto observation space
GT maps from observation spaceto model space
7
Iterative Solution of Primal Formulation(define IS4DVAR, is4dvar_ocean.h)
( ) ( )T1 11 12 2
TJ δ δ δ δ− −= + − −z D z G z d R G z d
Recall the incremental cost function:
At the minimum of J we have J δ∂ ∂ =z 0
( )1 T 1J δ δ δ− −∂ ∂ = + −z D z G R G z d
Given J and ∂J/∂δz, we can identify the δz that minimizes J
bJ oJ
Matrix-less Operations
( )T -1 δ −G R G z dThere are no matrix multiplications!
Zonal shear flow
Matrix-less Operations
( )T -1 δ −G R G z dThere are no matrix multiplications!
Zonal shear flow
Tangent LinearModel
8
Matrix-less Operations
( )T -1 δ −G R G z dThere are no matrix multiplications!
Zonal shear flow
Tangent LinearModel
Matrix-less Operations
( )T -1 δ −G R G z dThere are no matrix multiplications!
Zonal shear flow
Consider a single Observation
Matrix-less Operations
( )T -1 δ −G R G z dThere are no matrix multiplications!
Zonal shear flow
Consider a single Observation
9
Matrix-less Operations
( )T -1 δ −G R G z dThere are no matrix multiplications!
Zonal shear flow
Inverse Obs ErrorCovariance
Matrix-less Operations
( )T -1 δ −G R G z dThere are no matrix multiplications!
Zonal shear flow
Inverse Obs ErrorCovariance
Matrix-less Operations
( )T -1 δ −G R G z dThere are no matrix multiplications!
Zonal shear flow
Adjoint Model
10
Matrix-less Operations
( )T -1 δ −G R G z dThere are no matrix multiplications!
Zonal shear flow
Adjoint Model
Green’s Function
oJ δ∂ ∂ z
Primal 4D-Var Algorithm (I4D-Var)
Choose δz
Run TLROMS
R-1
Run ADROMS
ConjugateGradient
Algorithm
( ), , Jδ δ −G z G z d
( )-1 δ −R G z d
( )T -1 δ −G R G z d1J δ δ−∂ ∂ = +z D z
( )T 1 δ− −G R G z d
δ z
NLROMS, zb xb(t), d
NLROMS, za xa(t)
Choose δz
Run TLROMS
R-1
Run ADROMS
ConjugateGradient
Algorithm
NLROMS, zb
NLROMS, za
Primal 4D-Var Algorithm (I4D-Var)
Outer-loop
Inner-loop
11
Contours of J
Conjugate Gradient (CG) Methods
An Example: ROMS CCS
30km, 10 km & 3 km grids, 30- 42 levelsVeneziani et al (2009)Broquet et al (2009)Moore et al (2010)
COAMPSforcing
ECCO openboundaryconditions
fb(t), Bf
bb(t), Bb
xb(0), Bx
Previous assimilationcycle
Observations (y)
CalCOFI &GLOBEC
SST &SSH
Argo
TOPP Elephant SealsIngleby andHuddleston (2007)
Data from Dan Costa
12
4D-Var Configuration
• Case studies for a representative case3-10 March, 2003.
• 1 outer-loop, 100 inner-loops• 7 day assimilation window• Prior D: x Lh=50 km, Lv=30m, σ from clim
f Lτ=300km, LQ=100km, σ from COAMPS b Lh=100 km, Lv=30m, σ from clim
• Super observations formed• Obs error R (diagonal):
SSH 2 cmSST 0.4 Chydrographic 0.1 C, 0.01psu
Primal, strongDual, strongDual, weakJmin
4D-Var Performance
3-10 March, 2003(10km, 42 levels)
Summary• Strong constraint incremental 4D-Var, primal
formulation: define IS4DVARDrivers/is4dvar_ocean.h
• Matrix-less iterations to identify cost functionminimum using TLROMS and ADROMS
13
References• Broquet, G., C.A. Edwards, A.M. Moore, B.S. Powell, M. Veneziani
and J.D. Doyle, 2009: Application of 4D-variational data assimilation to the California Current System. Dyn. Atmos. Oceans, 48, 69-91.
• Courtier, P., J.-N. Thépaut and A. Hollingsworth, 1994: A strategy for operational implemenation of 4D-Var using an incremental approach. Q. J. R. Meteorol. Soc., 120, 1367-1388.
• Ingleby, B. and M. Huddleston, 2007: Quality control of ocean temperature and salinity profiles - historical and real-time data. J. Mar. Systems, 65, 158-175.
• Moore, A.M., H.G. Arango, G. Broquet, B.S. Powell, J. Zavala-Garayand A.T. Weaver, 2010: The Regional Ocean Modeling System (ROMS) 4-dimensional data assimilation systems: Part I. Ocean Modelling, Submitted.
• Veneziani, M., C.A. Edwards, J.D. Doyle and D. Foley, 2009: A central California coastal ocean modeling study: 1. Forward model and the influence of realistic versus climatological forcing. J. Geophys. Res., 114, C04015, doi:10.1029/2008JC004774.
• Wikle, C.K. and L.M. Berliner, 2007: A Baysian tutorial for data assimilation. Physica D, 230, 1-16.