back and forth nudging or seek (bfn, bf-seek)auroux/lefe/doc/29_09_09/auroux.pdf · didier auroux...

Didier Auroux

University of Nice Sophia Antipolis

[email protected]

Work in collaboration with : J. Blum (U. Nice) + work of E. Cosme (U. Grenoble)

Back and Forth Nudging or SEEK

(BFN, BF-SEEK)

INSU-CNRS LEFE-ASSIM BFN/BFN2,

Villefranche-sur-Mer, France. September 29, 2009

Forward nudging

dX

dt= F (X)+K(Xobs − H(X)), 0 < t < T,

X(0) = X0,

where K is the nudging (or gain) matrix.

In the linear case (where F is a matrix), the forward nudging is called

Luenberger or asymptotic observer.

– Meteorology : Hoke-Anthes (1976)

– Oceanography (QG model) : Verron-Holland (1989)

– Atmosphere (meso-scale) : Stauffer-Seaman (1990)

– Optimal determination of the nudging coeffcients :

Zou-Navon-Le Dimet (1992), Stauffer-Bao (1993),

Vidard-Le Dimet-Piacentini (2003)

LEFE/ASSIM 2009, Villefranche-sur-Mer, September 29 2009 1/13

Forward nudging : linear case

Luenberger observer, or asymptotic observer

(Luenberger, 1966)

dX

dt= FX+K(Xobs − HX),

dX

dt= FX, Xobs = HX.

d

dt(X − X) = (F−KH)(X − X)

If F − KH is a Hurwitz matrix, i.e. its spectrum is strictly included in the

half-plane λ ∈ C; Re(λ) < 0, then X → X when t → +∞.


Backward nudging

How to recover the initial state from the final solution ?

Backward model :

dX

dt= F (X), T > t > 0,

X(T ) = XT .

If we apply nudging to this backward model :

dX

dt= F (X)−K(Xobs − HX), T > t > 0,

X(T ) = XT .


BFN : Back and Forth Nudging algorithm

Iterative algorithm (forward and backward resolutions) : [Auroux-Blum (05)]

X0(0) = Xb (first guess)

dXk

dt= F (Xk)+K(Xobs − H(Xk))

Xk(0) = Xk−1(0)

dXk

dt= F (Xk)−K ′(Xobs − H(Xk))

Xk(T ) = Xk(T )

If Xk and Xk converge towards the same limit X, and if K = K ′, then X

satisfies the state equation and fits to the observations.


Choice of the direct nudging matrix K

Implicit discretization of the direct model equation with nudging :

Xn+1 − Xn

∆t= FXn+1 + K(Xobs − HXn+1).

Variational interpretation : direct nudging is a compromise between the mini-

mization of the energy of the system and the quadratic distance to the obser-

vations :

minX

[1

2〈X − Xn, X − Xn〉 −

∆t

2〈FX, X〉 +

∆t

2〈R−1(Xobs − HX), Xobs − HX〉

],

by choosing

K = kHT R−1

where R is the covariance matrix of the errors of observation.

[Auroux-Blum (08)]


Choice of the backward nudging matrix K′

The feedback term has a double role :

• stabilization of the backward resolution of the model (irreversible system)

• feedback to the observations

If the system is observable, i.e. rank[H, HF, . . . , HFN−1] = N , then there

exists a matrix K ′ such that −F −K ′H is a Hurwitz matrix (pole assignment

method).

Simpler solution : one can define K ′ = k′HT R−1, where k′ is e.g. the smallest

value making the backward numerical integration stable.


Shallow water model

∂tu − (f + ζ)v + ∂xB =τx

ρ0h− ru + ν∆u

∂tv + (f + ζ)u + ∂yB =τy

ρ0h− rv + ν∆v

∂th + ∂x(hu) + ∂y(hv) = 0

• ζ = ∂xv − ∂yu is the relative vorticity ;

• B = g∗h +1

2(u2 + v

2) is the Bernoulli potential ;

• g∗ = 0.02 m.s−2 is the reduced gravity ;

• f = f0 + βy is the Coriolis parameter (in the β-plane approximation), with f0 =

7.10−5 s−1 and β = 2.10−11 m−1.s−1 ;

• τ = (τx, τy) is the forcing term of the model (e.g. the wind stress), with a maximum

amplitude of τ0 = 0.05 s−2 ;

• ρ0 = 103 kg.m−3 is the water density ;

• r = 9.10−8 s−1 is the friction coefficient.

• ν = 5 m2.s−1 is the viscosity (or dissipation) coefficient.


Shallow water model

2D shallow water model [Auroux (09)]

State = height h and horizontal velocity (u, v)

Numerical parameters : (run example)

Domain : L = 2000 km × 2000 km ; Rigid boundary and no-slip BC ; Time

step = 1800 s ; Assimilation period : 15 days ; Forecast period : 15 + 45 days

Observations : of h only (∼ satellite obs), every 5 gridpoints in each space

direction, every 24 hours.

Background : true state one month before the beginning of the assimilation

period + white gaussian noise (∼ 10%)

Comparison BFN - 4DVAR : sea height h ; velocity :u and v.


Convergence - perfect obs.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 100 200 300 400 500 600 700

RM

S er

ror

Time steps

h

0

0.05

0.1

0.15

0.2

0.25

0 100 200 300 400 500 600 700

RM

S er

ror

Time steps

u

0

0.05

0.1

0.15

0.2

0.25

0.3

0 100 200 300 400 500 600 700

RM

S er

ror

Time steps

vRelative difference between the BFN

iterates (5 first iterations) and the

true solution versus the time steps,

for h, u and v.


Comparaison - perfect obs.

Relative error h u v

Background state 37.6% 21.5% 30.3%

BFN (5 iterations, converged) 0.44% 1.78% 2.41%

4D-VAR (5 iterations) 0.64% 3.14% 4.47%

4D-VAR (18 iterations, converged) 0.61% 2.43% 3.46%

Relative error of the background state and various identified initial conditions

for the three variables.


Noisy observations

0

0.05

0.1

0.15

0.2

0 500 1000 1500 2000 2500 3000

RM

S er

ror

Time steps

huv

0

0.05

0.1

0.15

0.2

0 500 1000 1500 2000 2500 3000

RM

S er

ror

Time steps

huv

Relative difference bet-

ween the true solution

and the forecast trajec-

tory corresponding to the

BFN (top) and 4D-VAR

(bottom) identified initial

conditions at convergence,

versus time, for the three

variables and in the case

of noisy observations.


BFN-preprocessed 4D-VAR

0

0.01

0.02

0.03

0.04

0.05

0.06

0 500 1000 1500 2000 2500 3000

RM

S er

ror

Time steps

BFN (5 iter., CV)4D-VAR (5 iter.)

BFN + 4D-VAR (2+3 iter.)4D-VAR (16 iter., CV)

Relative difference between the true solution and the forecast trajectory cor-

responding to the BFN (5 iter. CV), 4D-VAR (5 iter., and 16 iter. CV) and

BFN/4D-VAR (2+3 iter., CV in 2+6 iter.) identified initial conditions, versus

time, for the height variable in the case of noisy observations.


Conclusions

• Easy implementation (no linearization, no adjoint state, no minimization

process)

• Very efficient in the first iterations (faster convergence)

• Lower computational and memory costs than other DA methods

• Stabilization of the backward model

• Excellent preconditioner for 4D-VAR (or Kalman filters)

Under investigation :

• Identification of the entire solution of a full primitive model from the

knowledge of the SSH (or SST) only Patrick Bansart

• Efficient resolution of Riccati-like equations for a better backward nudging

matrix


BF-SEEK (Emmanuel Cosme)

Kalman filter :

• Initialization :

xf0 and P

f0 given

• Analysis :

Kn = P fn HT

n [HnP fn HT

n + Rn]−1

xan = xf

n + Kn(xobsn − Hnxfn)

P an = [I − KnHn]P f

n

• Forecast :

xfn+1 = Mn;n+1x

an

Pfn+1 = Mn;n+1P

anMT

n;n+1 + Qn


SEEK filter

Singular Evolutive Extended Kalman filter : [Pham-Verron-Roubaud (98)]

Consider low rank approximations of the covariance matrices P an , while

preserving the stability of the filter : |Sp((I − Kn+1Hn+1)Mn)| < 1.

Initialization of the filter : any covariance matrix P is real and symmetric

diagonalization : P = LDLT , D being diagonal positive.

Initialization :

Pf0 = S

f0 D0S

fT0 = [

√λ1L1, . . . ,

√λrLr] [

√λ1L1, . . . ,

√λrLr]

T

where D0 is the truncated diagonal matrix with the few first eigenvalues.

At any time, the covariance matrices of forecast error can also be decomposed

in a similar way, because their rank is r : P fn = Sf

nSfTn

where Sfn and Sa

n are dim(x0) × r matrices.


Error covariances matrices

Propagation of errors and gain matrices similar to Kalman filter :

P an = Sf

n[I + (HnSfn)T R−1

n (HnSfn)]−1SfT

n = SanSaT

n

(rank(P an ) = r)

As one usually considers 10 to 30 different model modes (number of vectors of

large variability, corresponding to the columns of Sn), it is possible to consider

their time evolution (which was difficult in the full filter). For 1 ≤ j ≤ r :

[Sfn+1]j = M(xa

n + [San]j) − M(xa

n) ≃ M ′[San]j

The last approximation is equivalent to neglect the nonlinear propagation of

errors.


SEEK Filter algorithm

• Initialization :

xf0 and P

f0 given (rank = r, or if not, it will be approximated)

• Analysis :

P fn = Sf

nSfTn

Kn = Sfn[Ir + (HnSf

n)T R−1n (HnSf

n)]−1(HnSfn)T R−1

n

xan = xf

n + Kn[xobsn − Hn(xfn)]

P an = Sf

n[Ir + (HnSfn)T R−1

n (HnSfn)]−1SfT

n

• Forecast :

xfn+1 = Mn;n+1(x

an)

[Sfn+1]j = M(xa

n + [San]j) − M(xa

n), 1 ≤ j ≤ r

Pfn+1 = S

fn+1(S

fn+1)

T


BF-SEEK

Configuration numerique :

• Modele shallow-water (2D), domaine 2000 × 2000 km

• Fenetre d’assimilation de 30 jours, assimilation quotidienne

• Observations de SSH distribuees selon des traces satellitaires Topex/Poseidon

simulees

• 4 cycles aller-retour BF-SEEK

• Pas de propagation dynamique des erreurs ( SEEK ≃ interpolation opti-

male)


BF-SEEK

Exemple d’observations assimilees chaque jour (traces satellitaires simulees)


BF-SEEK

SSH identifiee (gauche) et “exacte” (droite) apres : 0 iter. BF-SEEK


BF-SEEK



BF-SEEK

0 5 10 15 20 25 30Days

20

30

40

50

60

RM

S er

ror

in S

SH (

m)

Erreur sur la SSH en fonction des iterations de BF-SEEK


Conclusions 2

BF-SEEK : seems to work well...


Coming next :

Maelle Nodet

Convergence ou divergence du BFN ?


Motivations

Environmental and geophysical studies : forecast the natural evolution

retrieve at best the current state (or initial condition) of the environment.

Geophysical fluids (atmosphere, oceans, . . .) : turbulent systems =⇒ high

sensitivity to the initial condition =⇒ need for a precise identification (much

more than observations)

Environmental problems (ground pollution, air pollution, hurricanes, . . .) :

problems of huge dimension, generally poorly modelized or observed

Data assimilation consists in combining in an optimal way the observations of

a system and the knowledge of the physical laws which govern it.

Main goal : identify the initial condition, or estimate some unknown parame-

ters, and obtain reliable forecasts of the system evolution.


Data assimilation

t

Observations

Model

combination

model + observations

⇓

identification of the initial condition

of a geophysical system

Fundamental for a chaotic system (atmosphere, ocean, . . .)

Issue : These systems are generally irreversible

Goal : Combine models and data

Typical inverse problem : retrieve the system state from sparse and noisy

observations


4D-VAR

Observations

t

Xb

X0 Model governed by a system of ODE :

dX

dt= F (X), 0 < t < T

X(0) = X0

Xobs(t) : observations of the system, H : observation operator,

Xb : background state (a priori estimation of the initial condition),

B and R : covariance matrices of background and observation errors.

J(X0) =1

2(X0 − Xb)

T B−1(X0 − Xb)

+1

2

∫ T

0

[Xobs(t) − H(X(t))]T

R−1 [Xobs(t) − H(X(t))] dt


Optimality system

Optimization under constraints :

L (X0, X, P ) = J(X0) +

∫ T

0

⟨P ,

dX

dt− F (X)

⟩dt

Direct model :

dX

dt= F (X)

X(0) = X0

Adjoint model :

−dP

dt=

[∂F

∂X

]T

P + HT R−1 [Xobs(t) − H(X(t))]

P (T ) = 0

Gradient of the cost-function :∂J

∂X0= B−1(X0 − Xb) − P (0)

Optimal solution : X0 = Xb + BP (0) [Le Dimet-Talagrand (86)]


Observability conditionLet χ(x) be the time during which the characteristic curve with foot x lies in

the support of K. Then the system is observable if and only if minx

χ(x) > 0.

Partial observations in space : half of the domain is observed.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

T=0.5T=0.3T=0.15T=0.05

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

T=1T=0.5T=0.3T=0.15T=0.05

Decrease rate of the error after one iteration of BFN as a function of the space

variable x, for various final times T .

Linear case (left) : theoretical observability condition = T > 0.5

Nonlinear case (right) : numerical observability condition = T > 1


Quasi-geostrophic ocean model

The oceanic model used in this study is based on the quasi-geostrophic ap-

proximation obtained by writing the conservation of the potential vorticity.

The vertical structure of the ocean is divided into N layers. Each one has a

constant density ρk with a depth Hk (k = 1, . . . , N). We get a coupled system

of N equations :

Dk(θk(Ψ) + f)

Dt+ δk,N C1∆ΨN − C3∆

3Ψk = Fk in Ω×]0, T [,

∀k = 1, . . . , N ;

where :

• Ω ⊂ IR2 is the oceanic basin, and [0, T ] the time interval for the study ;

• Ψk is the stream function in the layer k ;

• C1∆ΨN is the dissipation on the bottom of the ocean ;

• C3∆3Ψk is the parametrization of internal and subgrid dissipation ;


Example

• θk(Ψ) is the potential vorticity in the layer k, given by :

θ1(Ψ)...

θN (Ψ)

= [∆ − [W ]]

Ψ1...ΨN

where [W ] is a N × N tridiagonal matrix ;

• f is the Coriolis force ;

•Dk.

Dtis the Lagrangian derivative in layer k, given by :

Dk.

Dt=

∂.

∂t−

∂Ψk

∂y

∂.

∂x+

∂Ψk

∂x

∂.

∂y=

∂.

∂t+ J(Ψk, .),

where J(., .) is the Jacobian operator J(ϕ, ξ) =∂ϕ

∂x

∂ξ

∂y−

∂ϕ

∂y

∂ξ

∂x;

• Fk is a forcing term. In this model only the tension due to the wind, denoted

τ , is taken into account : F1 = Rotτ and Fk = 0, ∀k ≥ 2.


Example

We consider altimetric measurement of the surface of the ocean given by satel-

lite observations (Topex-Poseidon, Jason). The observed data is the change in

the surface of the ocean. According to the quasi geostrophic approximation it

is proportional to the stream function in the surface layer :

hobs =f0

gΨobs

1

Therefore we will assimilate surface data in order to retrieve the fluid circula-

tion especially in the deep ocean layers.

The control vector is the initial state on the N layers :

u =(Ψk(t = 0)

)

k=1,...,N∈ Uad

The state vector is (Ψk(t)

)

k=1,...,N


QG - BFN convergence

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20

RM

S re

lativ

e di

ffer

ence

BFN iterations

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 5 10 15 20

RM

S re

lativ

e er

ror

BFN iterations

RMS relative difference between two

consecutive BFN iterates (a) and bet-

ween the BFN iterates and the exact

solution (b) versus the number of BFN

iterations.


QG - BFN vs 4D-VAR

10000

1e+06

1e+08

1e+10

1e+12

1e+14

0 5 10 15 20

BFN iterations

J||gradJ1||||gradJ2||||gradJ3||

10000

1e+06

1e+08

1e+10

1e+12

1e+14

0 5 10 15 20

4D-VAR iterations

J||gradJ1||||gradJ2||||gradJ3||

Evolution of the 4D-VAR cost function

and of the 3 gradients of the cost func-

tion in the 3 ocean layers for the BFN

iterates versus the number of BFN ite-

rations (top) and for the 4D-VAR ite-

rates versus the number of 4D-VAR ite-

rations (b).


QG - BFN convergence (noisy obs.)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20

RM

S re

lativ

e di

ffer

ence

BFN iterations

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0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 5 10 15 20

RM

S re

lativ

e er

ror

BFN iterations





iterations in the case of noised observa-

tions.


QG - BFN vs 4D-VAR (noisy obs.)

10000

1e+06

1e+08

1e+10

1e+12

1e+14

0 5 10 15 20

BFN iterations

JgradJ1gradJ2gradJ3

10000

1e+06

1e+08

1e+10

1e+12

1e+14

0 5 10 15 20

4D-VAR iterations

JgradJ1gradJ2gradJ3





rations (top) and for the 4D-VAR ite-

rates versus the number of 4D-VAR ite-

rations (b), in the case of noisy obser-

vations.


QG - BFN vs 4D-VAR (noisy obs.)

0

0.01

0.02

0.03

0.04

0.05

0.06

0 5 10 15 20

RM

S re

lativ

e er

ror

BFN layer 14D-VAR layer 1

0

0.01

0.02

0.03

0.04

0.05

0.06

0 5 10 15 20

RM

S re

lativ

e er

ror

Time (days)


0

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0.02

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0.04

0.05

0.06

0 0 5 10 15 20

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S re

lativ

e er

ror


Evolution in time of the RMS relative

difference between the reference trajec-

tory and the identified trajectories for

the BFN (solid line) and 4D-VAR (dot-

ted line) algorithms, versus time, for each

layer : from surface to bottom.


QG - BFN convergence (model error)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20

RM

S re

lativ

e di

ffer

ence

BFN iterations

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 5 10 15 20

RM

S re

lativ

e er

ror

BFN iterations





iterations in the case of imperfect mo-

del.


QG - BFN vs 4D-VAR (model error)

10000

1e+06

1e+08

1e+10

1e+12

1e+14

0 5 10 15 20

BFN iterations

JgradJ1gradJ2gradJ3

10000

1e+06

1e+08

1e+10

1e+12

1e+14

0 5 10 15 20

4D-VAR iterations

JgradJ1gradJ2gradJ3





rations (a) and for the 4D-VAR iterates

versus the number of 4D-VAR itera-

tions (b) in the case of imperfect model.


Example of convergence results

Viscous linear transport equation :

∂tu − ν∂xxu + a(x)∂xu = −K(u − uobs), u(x, t = 0) = u0(x)

∂tu − ν∂xxu + a(x)∂xu = K ′(u − uobs), u(x, t = T ) = uT (x)

We set w(t) = u(t) − uobs(t) and w(t) = u(t) − uobs(t) the errors.

• If K and K ′ are constant, then ∀t ∈ [0, T ] : w(t) = e(−K−K′)(T−t)w(t)

(still true if the observation period does not cover [0, T ])

• If the domain is not fully observed, then the problem is ill-posed.

Error after k iterations : wk(0) = e−[(K+K′)kT ]w0(0)

exponential decrease of the error, thanks to :

• K + K ′ : infinite feedback to the observations (not physical)

• T : asymptotic observer (Luenberger)

• k : infinite number of iterations (BFN)


back and forth nudging or seek (bfn, bf-seek)auroux/lefe/doc/29_09_09/auroux.pdf · didier auroux...

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