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