Motion estimation on ocean satellite images by data assimilation in a wavelets reduced model

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Submitted on 30 May 2013

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Motion estimation on ocean satellite images by data assimilation in a wavelets reduced model

Etienne Huot, Giuseppe Papari, Isabelle Herlin, Karim Drifi

To cite this version:

Etienne Huot, Giuseppe Papari, Isabelle Herlin, Karim Drifi. Motion estimation on ocean satellite

images by data assimilation in a wavelets reduced model. EGU - European Geosciences Union General

Assembly, Apr 2013, Vienne, Austria. �hal-00828048�

Motion estimation on ocean satellite images by data assimilation in a wavelets reduced model

E. Huot, G. Papari, I. Herlin, K. Drifi

The presentation concerns the issue of estimating surface motion from satel- lite images, either on rectangular windows or for a whole basin, with complex geographical boundary, such as Black Sea. The spatial domain is in all cases denoted by Ω.

The approach relies on a reduced model that is obtained by Galerkin pro- jection of dynamic equations on subspaces of velocity and image fields. The dynamic model expresses usual assumptions such as the transport of image brightness by motion and advection-diffusion of velocity.

Subspaces of image and motion fields are obtained as finite dimension vecto- rial spaces with scalar and vector wavelets bases, respectiveley denoted ψ l , for l = 1 . . . L and φ k , for k = 1 . . . K. A method has been defined in order to ob- tain the ψ l and φ k elements as solutions of constrained minimization problems described by :



 

 

 

 

(ψ

,...,ψ min

)∈F

n

X

k=1

Q( ψ _k )

B( ψ _k ) = 0, k = 1, ..., n (1)

h ψ _j , ψ _k i = δ j,k

where :

– F is a Hilbert functional space, with given inner product h·, ·i, – Q(ψ) = hL(ψ), ψi is a positively defined quadratic functional,

– B is a linear operator on F, used to express the chosen boundary conditions on image and motion fields,

– and δ j,k is equal to 1 for j = k and to 0 otherwise.

In order to define the scalar basis of the image subspace, the minimization problem is solved with Q(ψ) , R

Ω |∇ψ( r )| ² d ² r , and B(ψ) , n ( r ) ^⊺ ψ( r ), where the vector n ( r ) is zero everywhere apart for the boundary ∂Ω, and it is | n ( r )| = 1, n ( r ) ⊥ ∂Ω, ∀ r ∈ ∂Ω. Thus, we are minimizing the smoothness R

Ω |∇ψ( r )| ² d ² r with the Neumann boundary conditions.

In order to define the vector basis of the motion subspace, the minimization problem (1), is considered with Q(ψ) , R

Ω |∇ψ( r )| ² d ² r , where ψ : Ω 7→ R ² is a planar vector field. As to the operator B(ψ), two possibilities are available :

{B(ψ)}( r ) =

n ( r ) ^⊺ ψ( r ), r ∈ ∂Ω

0, otherwise

1

{B(ψ)}( r ) =

n ( r ) ^⊺ ψ( r ), r ∈ ∂Ω {divψ}( r ), otherwise

In both cases, Neumann boudary conditions are applied, and in the second case, motion is considered as divergence-free.

The sequence of satellite images is projected on the image subspace in order to get a sequence of observation vectors : b(t) = b 1 (t) . . . b L (t) ^T

. These b(t) are then assimilated in the reduced model in order to estimate the values a(t) = a 1 (t) . . . a K (t) T

, a i (t) being the coefficient of the projection of motion w (t) on the basis vector ψ. w (t) is then obtained as

K

X

i=1

a i (t)ψ i . The approach has been used on satellite data acquired by NOAA-AVHRR sensors and on the MyOCean ¹ analysis database for BlackSea.

1. http ://www.myocean.eu.org/

2