Sequential methods for ocean data assimilation

_________

From theory to practical implementations (I)

P. BRASSEURCNRS/LEGI, Grenoble, FrancePierre.Brasseur@hmg.inpg.fr

J. Ballabrera, L. Berline, F. Birol, J.M. Brankart, G. Broquet, V. Carmillet, F. Castruccio, F. Debost, D. Rozier, Y. Ourmières, T.Penduff, J. Verron

Th. Delcroix, B. Dewitte, Y. duPenhoat, F. Durand, L. Gourdeau

E. Blayo, S. Carme, I. Hoteit, Pham D.T., C. Robert

A. Barth, C. Raick

C.E. Testut, B. Tranchant

L. Parent

OUTLINE

q State-of-the-art1. Kalman filter: fundamentals2. Ocean data assimilation: specific issues3. Error sub-spaces 4. Low rank filters: SEEK and EnKF

q Advanced issues5. Objective validation and evaluation of DA systems6. Error tuning and adaptive schemes7. Improved temporal strategies : FGAT and IAU8. Kalman filtering with inequality constraints

q The MERCATOR Assimilation System

Ocean data assimilation

q Data assimilation involves the optimal combination of measurements with the underlying dynamical principles governing the system under observation.

q Data assimilation can serve several oceanographic objectives:Ø Ocean state estimation in space & time (4D) ;Ø Detection of model errors ;Ø Estimation of budgets & model parameters ;Ø Initialisation, prediction, monitoring ;Ø Optimal design of complex observation systems ;Ø …

q Theories: optimal control (VAR) and optimal estimation (Kalman)

1. Kalman Filter fundamentalsProblem definition

aix

i i+1

yi+1

ai

fi xx M=+1

aix : estimation of the « true » state vector at time i , dimension n

tix

fi 1+x : forecast of the state vector at time i+1, using the linear model M

1+iy : observations available at time i+1 , dimension p

Misfit between forecast and observations should be small :How « small » ?

Notations: Ide et al. (1997)

1+iyai

fi xx M=+1

incomplete data imperfect model

1. Kalman Filter fundamentalsModel-data misfit

1. Kalman Filter fundamentalsUncertainties & PDFs

aix

i i+1

ai

fi xx M=+1

ti

ai

ai xxe −= : error on state estimate at time i ; unknown quantity, but

assume

)(P~)P,( 121

exp01

−→

− ai

ai

Tai

ai

ai N eee

ti

ti 1+−= xx? M : error on model forecast between time i and time i+1 ; assume:

)(Q~Q),( 221

exp0 1

−→ − ??? TN

time

1. Kalman Filter fundamentalsError diagram

aix

tix

fi 1+x

tixM

M

ti 1+x

M

aie

?

fi 1+e

aieM

State space

1. Kalman Filter fundamentalsForecast error

?e?xxxxe −=+−=−= +++f

iti

fi

ti

ai

ai 111 )(MMM

Assuming pdf (1) and (2), model linearity and uncorrelated initial and modelling errors, the forecast error is distributed as :

)(P~)P,( 321

exp0 11

1111

−→ +

−

++++f

if

i

Tfi

fi

fi N eee

The estimation error is amplified by:- unstable model dynamics (M) ; - modelling errors Q .

)(QMMPP

MMM 41

111

+=⇒+=⇒+=

+

+++Ta

if

i

TTTai

ai

Tfi

fi

ai

fi ??eeee?ee

with

oi

tii 111 +++ += exy H : observation available at time i +1

1. Kalman Filter fundamentalsObservations and errors

aix

i i+1

yi+1

ai

fi xx M=+1

A probability distribution for the observation error is assumed:

)(R~R),( 521

exp0 11

11

−→ +

−++

oi

Toi

oi N eee

1. Kalman Filter fundamentalsOptimal estimation

Using Bayes rule at time i+1 :

( ) ( ) ( ) )()(

61

11111

+

+++++

⋅=

i

ti

tii

iti P

PPP

y

xxyyx

given by (5) given by (3)

a factor independent of ti 1+x

The best estimate of is the value of which maxi mize (7), i.e. the minimum of :ti 1+x x

( ) ( )

{ } )()H(R)H()(P)()H(R)H()(P)(

~

721

exp

21

exp21

exp

111

1111

1

111

111

1111

1

111

111

−−+−−−=

−−−⋅

−−−

⋅

++−

++++

−

+++

++−

++++

−

+++

+++

tii

Ttii

ti

fi

fi

Tti

fi

tii

Ttii

ti

fi

fi

Tti

fi

ti

tii PP

xyxyxxxx

xyxyxxxx

xxy

)()H(R)H()(P)()( 811

11

1

11 xyxyxxxxx −−+−−= +−

++

−

++ iT

if

if

iTf

iJ

)()H(RHP)( 90 11

11 xyxxxx −+=⇒= +−

++ iTf

if

iJδ

1. Kalman Filter fundamentalsKalman gain

Equation (9) can be solved for using simple algebra (*), leading to: x

)H(R)H(HPHP fiiTf

iTf

if

i 111

111 ++−

+++ −++= xyxx

Kalman gain = 1+iK

(*) Hint : use matrix equality [ ] [ ] 11211121121212111112112121 −−−−−−− +−=+ XXXXXXXXXXXXX

Note: the forecast and analysis equations can be extended to weakly non-linear models and observation operator .M H

ai

ai

Px ( )

QMMPP +=

=

+

+

Tai

fi

ai

fi M

1

1 xxForecast

1111

−+++ += R)H(HPHPK

Tfi

Tfii

( )( )( ) fiiai

fiii

fi

ai H

111

11111

+++

+++++

−=

−+=

PHKIP

K xyxx

An

alys

is

i i+1

yi+1

1. Kalman Filter fundamentalsAssimilation cycle

i i+1

Sequential assimilation = repeated forecast/analysis cycles

1. Kalman Filter fundamentalsAssimilation sequence

The best estimate at a given time is influenced by all previous observations (Kalman « filter »), and the analysis error covariance reflects the competition between this accumulation of past information and the error growth due to model imperfections .

i+2 i+3

1. Kalman Filter fundamentalsOptimal Interpolation

? Simplification of the Kalman filter: « Optimal Interpolation » To save cost and memory requirements, the KF can be simplified, using time-independent « background » covariance matrix

21211

// DCDBP ==+f

icorrelations

variances

q The forecast error requires 2n model integrations !!!

( ) QMPMQMMPP +=+=+ TaiTaifi 1q The 2n model integrations are useless if model errors are poorly

known (the KF is optimal only when error statistics are perfectly known),

Q

Noting that :

2. Ocean data assimilation : specific issuesModel complexity

(*) ocean circulation model developed at Univ. Miami (RSMAS, E. Chassignet)

q Model variables in HYCOM(*)temperature, salinity, velocity, layer thicknesses, sea-surface height (SSH)

2. Ocean data assimilation : specific issuesState vector dimension

q HYCOM state vector x : 3D grid of the 5 scalar model variables + 2D grid for SSH

q R&D prototype: 1/3° horizontal resolution , 19 hybrid layers

n ~ 5 x 350 x 350 x 19 ~ 1.1 x 107Operational prototype: 1/12° horizontal resolution , 26 hybrid layers

n ~ 5 x 1400 x 1400 x 26 ~ 2.5 x 108

q M operator : dim n x n ~ 6 x 1016 real*8 (i.e. ~ 6000 Earth Simulators !)

ØThe state vector dimension can be huge

ØThe model transition matrix « M » cannot be represented explicitely.Ø Instead, a computer code is used to transition « x » from time i to i+1

2. Ocean data assimilation : specific issuesSpace observations

q Observed variables: - from space: sea-surface height (SSH), sea-surface temperature (SST)

TP/ERS along track data

August 19-26,1993

AVHRR SSTAugust 19-26,1993

2. Ocean data assimilation : specific issuesIn situ observations

q Observed variables: - in situ: T/S profiles (drifting floats, field campaigns, …)

ARGO, July 2004

2. Ocean data assimilation : specific issuesObservations

q Observation vector y : data from various sources, at different time and space resolution

q Radar Altimetry : along-track measurements of SSH anomalies (JASON: 1 obs. / 7 km, ~ 300 km Equatorial tracks separation, repeated every 10 days ;ERS/ENVISAT: ~ 80 km Equatorial tracks separation, repeated every 35 days)

q AVHRR SST : weekly composite images at 4 km resolution (if no clouds)q ARGO flots : 3° x 3° horizontal resolution (targetted), profiles (between

2000 m depth to surface) every 10 days with 1 obs / m along vertical

Ø p = dim y is much smaller than n = dim x : too few observations !Ø The ocean surface relatively well observed by satellites: vertical

extrapolation of data assimilated at the surface into the ocean’s interior has to be consistent with vertical data profiles

Ø The observed variables are closely related to model variables: H is mainly an interpolation operator ( ~ simple)

2. Ocean data assimilation : specific issuesError covariance matrix

q Specification of error covariance matrix ? a0P

0Pq Assume a background state and associated error covariance Consider the analysis step with only one data at a model grid point and the associated observation error .

Ø p = 1 , y is a scalar and H is a vector of the formØ The Kalman gain is then a (n x 1) vector:

Ø The posterior estimate is a correction of the background using the –column of

?

[ ]00100 ,...,,,,...,H =

0x

e

{ } { }ηηηηηηη ε

0021

00 with1

PP)(

R)H(HPHPK =+

=+= − pp

TT

{ } { }ηηηη

ηηηε 000020

with1

xxx =−+

+= )(P)(

ˆp

η0P

2. Ocean data assimilation : specific issuesHorizontal covariance structures

Example: Horizontal covariance relative to a SSH (η) point at (32oN,70oW) MERCATOR Assimilation System - Testut et al.(2004)

Influence function of a single altimeter measurement in the sub-tropical gyre

2. Ocean data assimilation : specific issuesMultivariate covariance structures

Example: covariance relative to a SSH (η) point at (0o,144oW) - Weaver et al.(2003)

Ø The rows/colums of P should be « balanced » dynamically.Ø This requires multivariate covariances

Ø A full-rank representation of P (dim n x n) is still impossible !

2. Ocean data assimilation : specific issuesModel errors Q

q Model variability differs from observed variability

Model Data

Ø Many different model error sources (finite discretizations, representation of bottom topography, atmospheric forcings, etc … ) which cannot be easily quantified in terms of a Q matrix

EKE example :

2. Ocean data assimilation : specific issuesSystematic model errors

q Model mean SSH differs from observed mean SSH

OPA model (Madec et al. 2003) Data (Niiler et al., 2003)

Ø Model biases cannot be properly taken into account with Kalman filters

Mean sea-level difference between Pacific and Atlantic systematically too small in the model