Dealing with convergence problems when accounting for ...€¦ · Dealing with convergence problems...

Dealing with convergence problems whenaccounting for correlated observation er-rors in image assimilation

Vincent Chabot1

Maelle Nodet2, Arthur Vidard3

1Meteo-France, RTRA-STAE2Universite de Grenoble

3INRIA Grenoble - Rhone-Alpes

June 2, 2015

Motivation

I Error in dense field, such as satellite images, are correlated in space.

I Model resolutions are increasing. Need to extract finer structure fromobservation.

I Observation error covariance matrices are large and block diagonal.

Hypothesis (in this talk):

I The true R matrix is known.

I The observations are only correlated in space.

Questions:

I How to use this information in a 4D-Var?

I What kind of issue could arise? Why?

1 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard

Outline

1 Modeling R through a change of variable

2 Experiments with an isotropic noise

3 Convergence issue


Technical problems regarding R matrix

Algorithm : 4D-Var with B1/2 preconditioning.Problem : Need to compute R−1(y −Hx) at each iteration.

Constraints:

I R should be invertible,

I the product R−1(y −Hx) should not be too expensive.

For dense field, we can use methods similar to those developed for Bmatrix.

Main differences:

I R needs to be inverted,

I the observation space changes with time.


Representation of spatial correlation in Rthrough a change of variable

There are different ways to represent spatial correlation ([Fisher 2003],[Stewart et al. 2013], [Weaver 2014], ...).In this talk, we use a diagonal matrix after a change of variable (see [Chabot etal. 2014]).

Suppose yo = y t + ε with ε ∼ N (0,R).Then Ayo = Ay t + β with β ∼ N (0,ARAT ).

Aim

Choose A such that DA = diag(ARAT ) ' ARAT .

Here A is an orthonormal wavelet transform.


Spirit of a wavelet transform

Original signal

Approximation

Details

Approx.

Details

Approx.

Details

Approximation Details


Spirit of a wavelet transform

Original signal

Signal at different scalesWavelet

coefficients

= =store

Summary

Use of a ”basis” where each element has some scale, orientation and spatiallocalization properties. Write the cost function as:

(y −H(x))TR−1(y −H(x)) = (y −H(x))TATD−1A A(y −H(x))

Go in wavelet spaceDivide by the varianceReturn in pixel space


Example of covariance matrix : isotropic

and homogeneous case

True Diagonal wavelet modelisation

Orthonormal wavelet transforms do not preserve (in general):

I the variance value (in pixel space),

I the spatial localization,

I the isotropy or the homogeneity of the covariance fields,

but enable to represent (at a cheap cost) some of the error correlations.


Outline



3 Convergence issue


Twin experiment context

Model : Shallow-water ⇒ quantities of interest are (u,v,h)Observations : an image sequence of passive tracer ⇒ H is modelled byan advection–diffusion equation.

Algorithm : 4D-Var with B1/2 preconditioning.B is modeled by diffusion operators [see Weaver and Courtier 2001].Background : (u0, v0, h0) = (0, 0, hmean)

Aim

Control the velocity field via the assimilation of a noisy passive tracer sequence.


Results with homogeneous isotropic noise

Observations : yoti = yt

ti + εiso

Res

idu

al

erro

r(i

n%

)

0.1

1

0 5 10 15 20 25 30 35 40 45 50

Iterations

Pixels diagonal Wavelet diagonal

I Accounting for error correlations leads to a decrease of the residual error.

I There is no convergence issue in this case.


Outline



3 Convergence issue


Convergence issue : best matrix

representation in a wavelet space

The true covariance matrix is used in the wavelet space

yoti = yt

ti + ε with ε = ATD1/2A β β ∼ N (0, I)

A noise realization RMSE with respect to the minimization iterations

Res

idu

al

erro

r(i

n%

)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

Iterations

Pixels diagonalWavelet diagonal

Incorporating the true covariance information leads to some convergence issue.


Using the multiscale aspect of the Wavelet

transform

What happens when discarding information from small scales?

Full image

Discard 1 scale

‖yo ti−

yt t0‖2 R

−1

Time (ti )

Distance between ytt0

and the observations(yo

ti

)i=1,..,240

(Pixel Diagonal)/100

(Wavelet)/100000

(Wavelet without 1 scale)/100

Wavelet without 2 scales


Discarding some information enables to get a better distance notion.



transform


Full image

Discard 2 scales

‖yo ti−

yt t0‖2 R

−1

Time (ti )



ti

)i=1,..,240


(Wavelet)/100000







transform


Full image

Discard 3 scales

‖yo ti−

yt t0‖2 R

−1

Time (ti )



ti

)i=1,..,240


(Wavelet)/100000






Accelerate the convergence rate

Idea

Use only coarsest information at the beginning of the minimization.Along the minimization process, incorporate more and more information on finescale.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

Iterations


Wavelet diagonal: scale by scale incorporation

It accelerates the convergence.


Conclusion and Future work

Conclusion

I It is possible to incorporate spatial error correlations through a change ofvariable.

I This can have some positive impact on the assimilation process.

I This can induced some convergence issues.

I It is possible to overcome this by discarding small scale information at thebeginning of the assimilation process.

Future work

I R formulation in a wavelet space without full image.

I Study the impact of temporal correlation.


Questions ?


Accelerate the convergence rate

Idea

Use only coarsest information at the beginning of the minimization.Along the minimization process, incorporate more and more information on finescale.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

Iterations


Wavelet diagonal: scale by scale incorporationSpectral diagonal

It accelerates the convergence · · · up to a certain point.


Example : inhomogeneous case

True Daubechies: 6 scales

Coiflet: 6 scales Coiflet: 2 scales


Issue with the distance

‖yo ti−

yt t0‖2 R

−1



ti

)i=1,..,240

yot0

yot60

yot125

yot220

ytt0

distance ?

Time (ti )

The order induced by the wavelet distance (which takes into account errorcorrelations) is not the one expected.


Issue with the distance : an homogeneous

isotropic case

‖yo ti−

yt t0‖ R

−1



ti

)i=1,..,240


Variance value in Wavelet Space

Isotropic case: log10(σ2) Inhomogeneous case: log10(σ2)


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