A variational data assimilation system for the global ocean Anthony Weaver CERFACS Toulouse, France...
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Transcript of A variational data assimilation system for the global ocean Anthony Weaver CERFACS Toulouse, France...
A variational data assimilation system for the global ocean
Anthony WeaverCERFACS
Toulouse, France
Acknowledgements
N. Daget, E. Machu, A. Piacentini, S. Ricci, P. Rogel (CERFACS)
C. Deltel, J. Vialard (LODYC, Paris)
D. Anderson and the ECMWF Seasonal Forecasting Group
ENACT project consortium (EC Framework 5)
Outline
• Scientific objectives and development strategy.
• General formulation and key characteristics of the variational system.
• Some results from tropical Pacific and global ocean applications.
• Summary and future directions.
Scientific objectives
• Develop a global ocean data assimilation system that can satisfy two purposes simultaneously:
1. Provide estimates of the ocean state over multi-annual to multi-decadal periods (currently up to 40 years – ERA40).
2. Provide ocean initial conditions for seasonal to multi-annual range forecasts.
Much of this work has been coordinated through the EC-FP5 project ENACT (2002-2004).
• The assimilation method should have a solid theoretical foundation.
• It must be practical for large-dimensional systems involving GCMs
– State vector ~ O(106) to O(107) elements.– Non-differentiable parameterizations and algorithms.
• There should be a clear pathway to more advanced data assimilation systems.
Basic considerations in designing a practical data assimilation system
• Develop a system based on variational data assimilation.
• Use an “incremental” approach (Courtier et al. 1994, QJRMS).
• Provide a clear development pathway from
3D-Var
4D-Var: short window, strong model constraint
4D-Var: long window, weak model constraint
CERFACS assimilation system development strategy for OPA
Development strategy cont.
• Why 3D-Var?
– An effective 3D-Var system provides a solid foundation for 4D-Var (they share most components!).
– 3D-Var is a simpler and cheaper alternative to 4D-Var.– 3D-Var provides a valuable reference for evaluating the cost
benefits of 4D-Var. – Some of the flow-dependent features implicit in 4D-Var can be
built into 3D-Var.– 3D-Var requires significantly less maintenance and development
than 4D-Var (the tangent-linear and adjoint of the forecast model are not needed).
• What can 4D-Var do that 3D-Var can’t?
– 4D-Var can exploit tendency information in the observations.– 4D-Var computes implicitly flow-dependent, time-evolving
covariances within the assimilation window.
The OPA-VAR assimilation system
• OPA version 8.2 (Madec et al. 1999)
• Configurations available for assimilation
– Tropical Pacific (TDH): 1o x 0.5o at eq., 25 levels (rigid lid)(Weaver et al. 2003, MWR; Vialard et al. 2003, MWR; Vossepoel et al. 2004, MWR; Ricci et al. 2004, MWR)
– Global (ORCA): 2o x 0.5o at eq., 31 levels (free surface)
• Variational assimilation environment
– ~ 160 Fortran routines (~ 43,000 lines) for the OPA tangent-linear and adjoint models, and associated validation routines.
– ~ 190 Fortran routines (~ 54,000 lines) for the rest (observation operators, covariance operators, minimization routine,…).
– cf. ~ 230 Fortran routines (~76,000 lines) for the global OPA forecast model.
General formulation of the variational problem
• Let denote the vector of prognostic model state variables.
• Let denote the vector of analysis control variables where
• Find that minimizes where
background term
observation term
where
)ˆ(xx K
oToo GGJ yxRyx )()( 1
21
)ˆˆ()ˆˆ( 1)ˆ(2
1 bTbbJ xxBxx x
x
ob JJJ )ˆ(x
x̂
x̂
TToi
o ),)(,( yy
Incremental formulation
• Let be an increment to the state
• Let be an increment to the control where
• Find that minimizes where
bxxx
dxGRdxG 121 T
oJ
xBx x ˆˆ 1)ˆ(2
1 TbJ
ob JJJ )ˆ( xx̂
)( bo G xyd
bxxx ˆˆˆ
xKx ˆ
background term
quadratic obs. term
where )ˆ( bb K xx
Choice of analysis control variables
• In the ocean model = ( T, S, η, u, v)
• As analysis variables we take = ( T, Su, ηu, uu, vu)
and assume these variables are mutually uncorrelated
(so is block diagonal).
• The transformation is a balance constraint (Derber and Bouttier, 1999, Tellus)
– strong constraint if Su = ηu = uu = vu = 0
– weak constraint if Su ≠ ηu ≠ uu ≠ vu ≠ 0
x
)ˆ(xx K
)ˆ(xB
x̂
“unbalanced” variables
Interpretation of the balance operator
• If is linear then
• defines the multivariate covariances in
• When dim( ) < dim( ), has a null space.
• E.g., with applied as a strong constraint, the observations will project only onto the “balanced” modes.
KKTKBKB xx )ˆ()(
)(xBK
x̂ x )(xB
K
Choice of balance operator
• We construct as a lower triangular matrix (and hence easily invertible) transformation using the following constraints:– Linearized local T-S relationships balanced S
(Ricci et al. 2004, MWR)
– Dynamic height (baroclinic) balanced η
– Geostrophy, β-plane approx. near eq. balanced (u, v)
• We can interpret– Su ≠ S(T) unbalanced S
– ηu ≡ barotropic component unbalanced η
– (uu, vu) ≡ ageostrophic velocity unbalanced (u, v)
KK
Multivariate 3D-Var covariancesEx: covariance relative to a SSH point at (0o,144oW)
(surface) (surface)
)( )(xB
Choice of linear propagator
• involves integrating the nonlinear forward model from initial time to the observation times .
• involves integrating a linear forward model:
In 3D-Var (FGAT) persistence
In 4D-Var approx. TL model
where
1 ii xx
11),( iiii tt xMx
dxGRdxG 121 T
oJ
)( bo G xyd G
),( 0 ittM
G
xxKx 00 ˆ
0t it
Linear approximation in the tropical Pacific(from Weaver et al. 2003, MWR)
Latitude Latitude
TL approximation in the tropical Pacific (from Weaver et al. 2003, MWR)
TIWs
October start date
Interpretation of the linear propagator
• The linear propagator defines how the background error covariances evolve within the assimilation window .
• E.g., for observations located only at time , the effective background-error covariance matrix at is
(cf. Extended Kalman filter)
Ti
Tii
b ttttt ),(),()( 0)ˆ(0)( MKBKMP xx
],[ 0 itt
itit
Tiii
b ttttt ),(),()( 0)(0)( MBMP xx )()( )( xx BP i
b tIn 3D-Var:
In 4D-Var (cf. EKF):
4D-Var
(ti =30 days)
3D-Var
zzT b |/|
m10z
z
Diagnosing implicit background temperature error standard deviations ( ) in 4D-Var
(Weaver et al. 2003 – MWR)
bT
bT
SSH analysis increment
Am
plit
ud
e (
cm)
De
pth
(m
)
Impact of a single SSH observation in 4D-Var SSH innovation = 10 cm at (0o,160oW) at t = 30 days
Temperature analysis increment
• The minimization is preconditioned via a change of variables
so that and
where
• For a single observation, the minimization converges in a single iteration.
Preconditioning
T)( 2/1)ˆ(
2/1)ˆ()ˆ( xxx BBB
vBxxBv xx2/1)ˆ(
2/1)ˆ( ˆˆ
vvTbJ 2
1 )( 2/1)ˆ( vB xoo JJ
xBx x ˆˆ 1)ˆ(2
1 TbJ
dxKGRdxKG ˆˆ 121 T
oJ
Specifying background error covariances: general remarks
• There is not enough information (and never will be) to determine all the elements of (typically > O(1010)).
• must be approximated by a statistical model (e.g., prescribed covariance functions) with a limited number of tunable parameters.
• In 3D-Var/4D-Var, is implemented as an operator (a matrix-vector product).
• For the preconditioning transformation we require access to a square-root operator (and its adjoint ).
• Constructing an effective operator requires substantial development and tuning!
)ˆ(xB
)ˆ(xB
)ˆ(xB
2/1)ˆ(xB
T)( 2/1)ˆ(xB
)ˆ(xB
• We solve a generalized diffusion equation (GDE) to perform the smoothing action of the square-root of the correlation operator ( ).(Weaver and Courtier 2001, QJRMS; Weaver and Ricci 2004, ECMWF Sem. Proc.)
• Simple parameterizations for the standard deviations of background error ( )– (Balanced) T: background vertical T-gradient dependent– Unbalanced S: background mixed-layer depth dependent– Unbalanced SSH: function of latitude– Unbalanced (u,v): function of depth
Modelling univariate background error covariances
vCΣvBx xxx2/1)ˆ()ˆ(
2/1)ˆ(ˆ
2/1)ˆ(xC
)ˆ(xΣ
Univariate correlation modelling using a diffusion equation
(Derber & Rosati 1989 - JPO; Egbert et al. 1994 - JGR; Weaver & Courtier 2001 - QJRMS)
A simple 1D example:
Consider with constant .
on with as
Integrate from and with as IC:
02
2
zt
0
z 0),( tz z
0t Tt )0,(z
z
zdzeTz Tzz
T)0,(),( 4/)(
4
1 2
Solution:
This integral solution defines, after normalization, a correlation operator :
The kernel of is a Gaussian correlation function
where is the length scale.
Basic idea : To compute the action of on a discrete grid we can iterate a diffusion operator.
This is much cheaper than solving an integral equation directly.
C
C
),(4)0,( TzTz C
2/2 2);( LzeTzf
TL 2
C
z
zdzeTz Tzz
T)0,(),( 4/)(
4
1 2
Constructing a family of correlation functions on the sphere using a GDE
(Weaver & Courtier 2001, QJRMS; Weaver & Ricci 2004 – ECMWF Sem. Procs.)
shape spectrum
Gaussian
L = 500 km
Gaussian
02
1
pP
ppt
• The full correlation operator is formulated in grid-point space as a sequence of operators
• is the diffusion operator and is formulated in 3D as a product of a 2D (horizontal) and 1D (vertical) operator.
• is a diagonal matrix of volume elements, and appears in because of the self-adjointness of .
• The factor means iterations of the diffusion operator.
Some remarks on numerical implementation
2/
2/2/1
2/1
2/12/1
2/12/1
T
T
T
CC
ΛLWWLΛ
ΛLWLΛC
Lvh LLL
W CL
2/1L 2/M
• We can let where is a diffusion tensor that can be used to stretch and/or rotate the coordinates in the correlation model to account for anisotropic or flow-dependent structures.
• BCs are imposed directly within the discrete expression for using a land-ocean mask.
• contains normalization factors to ensure the variances of are equal to one.
• The diffusion approach to correlation modelling has many similarities to spline smoothing (Wahba 1982) and recursive filtering (Purser et al. 2003 - MWR).
Some remarks on numerical implementation
R2
Λ
R
2
C
GDE-generated correlation functionsusing “time”-implicit scheme
Example: T-T correlations at the equator
GDE-generated correlation functions
Example: flow-dependent correlations(Weaver & Courtier 2001-QJRMS; cf. Riishojgaard 1998-Tellus; Daley &
Barker 2001-MWR)
Dep
th
15oN15oS15oS 15oN
Background isothermals T-T correlations
Variational formulation: main point
• The main scientific component of the algorithm is the transformation from control space to observation space in the Jo term:
vBxy x2/1)ˆ(00 ˆ),( b
iii KttMH
Interpolation
Ocean model integration
Multivariate balance
Univariate smoothing
Incremental variational formulation• And for incremental Var we need the linearized transformation (and its adjoint):
vBKMHy x2/1)ˆ(0 ),( ttiii
Interpolation
Linear multivariate balance
Univariate smoothing
IM ),( 0tti 3D-Var (FGAT)
bMtti xxxM /),( 0 4D-Var
Linear ocean model integration
Impact of improved covariances on the mean zonal velocity in the tropical Pacific
1993-96 climatology
eastward current bias
Impact of in situ T (GTSPP) data assim. on the mean salinity state in the global model
3D-Var univariate (T)
Control (no d.a.)
IK
De
pth
(m
)
De
pth
(m
)
Longitude
Longitude
Pacific Atlantic Indian
Pacific Atlantic Indian
500
500
Equator
Equator
0
0
Impact of in situ T (GTSPP) data assim. on the mean salinity state in the global model
3D-Var multivariate (T, S, u, v, SSH)
Control (no d.a.)
IK
Pacific Atlantic Indian
Longitude
Longitude
Pacific Atlantic Indian
Equator
Equator
De
pth
(m
)
De
pth
(m
)500
500
0
0
Global reanalysis set-up and control
• Experimental set-up for ENACT– Stream 1: 1987 – 2001– Stream 2: 1962 – 2001– Daily mean ERA-40 surface fluxes– Weak 3D relaxation to Levitus T and S– Strong relaxation to Reynolds SST (-200 W/m2/K)
• Control: no data assimilation (streams 1 and 2)– Getting a satisfactory control run was not straightforward!
• Post-correction to ERA-40 precipitation to remove a tropical bias.• Stronger relaxation needed at high latitudes to avoid numerical
instabilities.• Daily correction to global mean E-P to remove sea level drift.
Global reanalysis experiments(completed or currently running)
• 3D-Var (streams 1 and 2)– In situ T data from ENACT QC data-set– 10-day window– Multivariate B (with balance)– “Incremental Analysis Updating” (IAU) (Bloom et al. 1994, MWR)
• 4D-Var (stream 1)– In situ T data from ENACT QC data-set– 30-day window– Univariate B (no balance)– Instantaneous update
Cycling of 3D-Var and 4D-Var
observations
Background trajectory
Background
Analysis
“Analysed” trajectory
using IAU
10-day window
30-day window
ENACT QC historical in situ dataset (Met. Office)
• 200,000 – 300,000 in situ T observations / month• 50,000 – 100,000 in situ S observations / month
Example of T data distribution on a 10 day window
Jan. 1987 Jan. 1995
Box regions for ENACT diagnostics
Assimilation diagnostics
)( bo H xy Control
)( bo H xy 3D-Var
)( ao H xy 3D-Var
abo H xHxy )( 3D-Var
Mean (oC) Standard deviation (oC)
De
pth
(m
)
1987-2001 global temperature statistics
Assimilation diagnostics
)( bo H xy Control
)( bo H xy 3D-Var
)( ao H xy 3D-Var
abo H xHxy )( 3D-Var
De
pth
(m
)
Mean (oC) Standard deviation (oC)
NW extra-trop Pacific NW extra-trop Pacific
1987-2001 regional temperature statistics
Assimilation diagnostics
)( bo H xy Control
)( bo H xy 3D-Var
)( ao H xy 3D-Var
abo H xHxy )( 3D-Var
Mean (oC) Standard deviation (oC)
Nino3 Nino3
De
pth
(m
)
1987-2001 regional temperature statistics
Summary
• 3D-Var (FGAT) and 4D-Var incremental systems developed for a global version of OPA.– Major coding and validation effort required.– Clear development path towards more advanced systems.
• Substantial effort devoted to developing covariance models and balance operators.– Balance constraints have a significant positive impact in 3D-Var.– And a positive impact in 4D-Var with single observations (but has
not yet been evaluated in real-data experiments).
• Production and assessment of global ocean reanalyses is ongoing (ENACT). – Preliminary results indicate that the assimilation is correcting for a
large model bias in the upper ocean.– But assimilation is introducing a bias of its own below the
thermocline.– Further improvements to the assimilation system are needed…
Future directions
• Background error modelling and estimation
• Observation error modelling and quality control
• Combined in situ T, S, altimeter and SST assimilation
• Model bias detection/correction
• Improving the computational efficiency of 4D-Var
• Ongoing reanalysis production and evaluation