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
1 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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.
2 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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.
3 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Spirit of a wavelet transform
Original signal
Approximation
Details
Approx.
Details
Approx.
Details
Approximation Details
4 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Spirit of a wavelet transform
Original signal
Approximation
Details
Approx.
Details
Approx.
Details
Approximation Details
4 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Spirit of a wavelet transform
Original signal
Approximation
Details
Approx.
Details
Approx.
Details
Approximation Details
4 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Spirit of a wavelet transform
Original signal
Approximation
Details
Approx.
Details
Approx.
Details
Approximation Details
4 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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
5 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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
5 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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.
6 / 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
6 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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.
7 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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.
8 / 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
8 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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.
9 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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
Wavelet without 3 scales
Discarding some information enables to get a better distance notion.
10 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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
Wavelet without 3 scales
Discarding some information enables to get a better distance notion.
10 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Using the multiscale aspect of the Wavelet
transform
What happens when discarding information from small scales?
Full image
Discard 2 scales
‖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
Wavelet without 3 scales
Discarding some information enables to get a better distance notion.
10 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Using the multiscale aspect of the Wavelet
transform
What happens when discarding information from small scales?
Full image
Discard 3 scales
‖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
Wavelet without 3 scales
Discarding some information enables to get a better distance notion.
10 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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
Pixels diagonalWavelet diagonal
Wavelet diagonal: scale by scale incorporation
It accelerates the convergence.
11 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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.
12 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Questions ?
13 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
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
Pixels diagonalWavelet diagonal
Wavelet diagonal: scale by scale incorporationSpectral diagonal
It accelerates the convergence · · · up to a certain point.
13 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Example : inhomogeneous case
True Daubechies: 6 scales
Coiflet: 6 scales Coiflet: 2 scales
13 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Issue with the distance
‖yo ti−
yt t0‖2 R
−1
Distance between ytt0
and the observations(yo
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.
13 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Issue with the distance : an homogeneous
isotropic case
‖yo ti−
yt t0‖ R
−1
Distance between ytt0
and the observations(yo
ti
)i=1,..,240
13 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard
Variance value in Wavelet Space
Isotropic case: log10(σ2) Inhomogeneous case: log10(σ2)
14 / 14Adjoint workshop, June 2, 2015 - Vincent Chabot, Maelle Nodet, Arthur Vidard