Optimized localization and hybridization to lter ensemble-based covariances Benjamin Ménétrier and Tom Auligné NCAR - Boulder - Colorado Roanoke - 06/04/2015

Acknowledgement: AFWA

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Context:

1

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Context: • DA often relies on forecast error covariances.

1

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Context: • DA often relies on forecast error covariances. • This matrix can be sampled from an ensemble of forecasts.

1

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Context: • DA often relies on forecast error covariances. • This matrix can be sampled from an ensemble of forecasts. • Sampling noise arises because of the limited ensemble size.

1

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Context: • DA often relies on forecast error covariances. • This matrix can be sampled from an ensemble of forecasts. • Sampling noise arises because of the limited ensemble size. • Question: how to lter this sampling noise?

1

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Context: • DA often relies on forecast error covariances. • This matrix can be sampled from an ensemble of forecasts. • Sampling noise arises because of the limited ensemble size. • Question: how to lter this sampling noise? Usual methods:

1

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Context: • DA often relies on forecast error covariances. • This matrix can be sampled from an ensemble of forecasts. • Sampling noise arises because of the limited ensemble size. • Question: how to lter this sampling noise? Usual methods: • Covariance localization → tapering with a localization matrix

1

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Context: • DA often relies on forecast error covariances. • This matrix can be sampled from an ensemble of forecasts. • Sampling noise arises because of the limited ensemble size. • Question: how to lter this sampling noise? Usual methods: • Covariance localization → tapering with a localization matrix • Covariance hybridization → linear combination with a static covariance matrix

1

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Questions:

2

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Questions: 1. Can localization and hybridization be considered together?

2

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Questions: 1. Can localization and hybridization be considered together? 2. Is it possible to optimize localization and hybridization coecients objectively and simultaneously?

2

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Questions: 1. Can localization and hybridization be considered together? 2. Is it possible to optimize localization and hybridization coecients objectively and simultaneously? The method should:

2

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Questions: 1. Can localization and hybridization be considered together? 2. Is it possible to optimize localization and hybridization coecients objectively and simultaneously? The method should:

2

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Questions: 1. Can localization and hybridization be considered together? 2. Is it possible to optimize localization and hybridization coecients objectively and simultaneously? The method should: •

use data from the ensemble

only.

2

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Questions: 1. Can localization and hybridization be considered together? 2. Is it possible to optimize localization and hybridization coecients objectively and simultaneously? The method should: •

use data from the ensemble

only.

•

be aordable for high-dimensional

systems.

2

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Introduction Questions: 1. Can localization and hybridization be considered together? 2. Is it possible to optimize localization and hybridization coecients objectively and simultaneously? The method should: •

use data from the ensemble

only.

•

be aordable for high-dimensional

systems.

3. Is hybridization always improving the accuracy of forecast error covariances?

2

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Outline Introduction Linear ltering of sample covariances Joint optimization of localization and hybridization Results Conclusions

3

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Outline Introduction Linear ltering of sample covariances Joint optimization of localization and hybridization Results Conclusions

4

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances

5

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances e: An ensemble of N forecasts {exbp } is used to sample B e B

where:

=

1

N

b δe x N −1 ∑ p=1

b δe x


T

b xb − hexb i and hexb i = δe x =e p p

1 N b e x

p N p∑ =1

5

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances e: An ensemble of N forecasts {exbp } is used to sample B e B

where:

=

1

N

b δe x N −1 ∑ p=1

b δe x


T

b xb − hexb i and hexb i = δe x =e p p

1 N b e x

p N p∑ =1

e →B e? Asymptotic behavior: if N → ∞ , then B

5

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances e: An ensemble of N forecasts {exbp } is used to sample B e B

where:

=

1

N

b δe x N −1 ∑ p=1

b δe x


T

b xb − hexb i and hexb i = δe x =e p p

1 N b e x

p N p∑ =1

e →B e? Asymptotic behavior: if N → ∞ , then B

ee = B e −B e? In practice, N < ∞ ⇒ sampling noise B

5

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances e: An ensemble of N forecasts {exbp } is used to sample B e B

where:

=

N

1

b δe x N −1 ∑ p=1

b δe x


T

b xb − hexb i and hexb i = δe x =e p p

1 N b e x

p N p∑ =1

e →B e? Asymptotic behavior: if N → ∞ , then B

ee = B e −B e? In practice, N < ∞ ⇒ sampling noise B

Theory of sampling error:    2  N (N − 3)  ?2  1 e e e e = E B 2 E Bij − (N − 1)(N − 2) E Bii Bjj ij (N − 1) +

N2

(N − 1) (N − 2) 2

  e E Ξ ijij

5

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances Localization by L (Schur product) Covariance matrix b B

e = L◦B

6

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances Localization by L (Schur product) Covariance matrix b B

e = L◦B

δ xe = √

1

Increment N

b δe x ◦ ∑ p N −1 p=1

L

1/2 α
v

p

6

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances Localization by L (Schur product) Covariance matrix b B

e = L◦B

δ xe = √

1

Increment N

b δe x ◦ ∑ p N −1 p=1

L

1/2 α
v

p

Localization by L + hybridization with B Increment

δ x = β e δ xe + β c

B

1/2

v

c

6

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances Localization by L (Schur product) Covariance matrix b B

δ xe = √

1

Increment N

b δe x ◦ ∑ p N −1

e = L◦B

p=1

L

1/2 α
v

p

Localization by L + hybridization with B Covariance matrix

b h = (β e )2 B

e + (β c )2 L◦B

Increment

B

δ x = β e δ xe + β c

B

1/2

v

c

6

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances Localization by L (Schur product) Covariance matrix b B

δ xe = √

1

Increment N

b δe x ◦ ∑ p N −1

e = L◦B

p=1

L

1/2 α
v

p

Localization by L + hybridization with B Covariance matrix

b h = (β e )2 L ◦ B e + (β c )2 B | {z } Gain Lh

Increment

B

δ x = β e δ xe + β c

B

1/2

v

c

| {z } Offset

6

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Linear ltering of sample covariances Localization by L (Schur product) Covariance matrix b B

δ xe = √

1

Increment N

b δe x ◦ ∑ p N −1

e = L◦B

p=1

L

1/2 α
v

p

Localization by L + hybridization with B Covariance matrix

b h = (β e )2 L ◦ B e + (β c )2 B | {z } Gain Lh

Increment

B

δ x = β e δ xe + β c

B

1/2

v

c

| {z } Offset

e Localization + hybridization = linear ltering of B h c L and β have to be optimized together

6

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Outline Introduction Linear ltering of sample covariances Joint optimization of localization and hybridization Results Conclusions

7

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 1 Step 1: optimizing the localization

only,

without hybridization

8

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 1 Step 1: optimizing the localization

only,

without hybridization

Goal: to minimize the expected quadratic error:

e=E

h e − k L ◦B | {z } e Localized B

e B

?

|{z}

k2

i

(1)

e Asymptotic B

8

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 1 Step 1: optimizing the localization

only,

without hybridization

Goal: to minimize the expected quadratic error:

e=E

h e − k L ◦B | {z } e Localized B

e B

?

|{z}

k2

i

(1)

e Asymptotic B

Light assumptions:

8

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 1 Step 1: optimizing the localization

only,

without hybridization

Goal: to minimize the expected quadratic error:

e=E

h e − k L ◦B | {z } e Localized B

e B

?

k2

i

|{z}

(1)

e Asymptotic B

Light assumptions: ee = B e −B e ? is not correlated • The unbiased sampling noise B e ?. with the asymptotic sample covariance matrix B

8

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 1 Step 1: optimizing the localization

only,

without hybridization

Goal: to minimize the expected quadratic error:

e=E

h e − k L ◦B | {z } e Localized B

e B

?

|{z}

k2

i

(1)

e Asymptotic B

Light assumptions: ee = B e −B e ? is not correlated • The unbiased sampling noise B e ?. with the asymptotic sample covariance matrix B e ? and • The two random processes generating the asymptotic B the sample distribution are independent.

8

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 1 Step 1: optimizing the localization

only,

without hybridization

Goal: to minimize the expected quadratic error:

e=E

h e − k L ◦B | {z } e Localized B

e B

?

k2

i

|{z}

(1)

e Asymptotic B

Light assumptions: ee = B e −B e ? is not correlated • The unbiased sampling noise B e ?. with the asymptotic sample covariance matrix B e ? and • The two random processes generating the asymptotic B the sample distribution are independent. An explicit formula for the optimal localization L is given in Ménétrier et al. 2015 (Montly Weather Review). 8

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 1 This formula of optimal localization L involves: • the ensemble size N e • the sample covariance B e • the sample fourth-order centered moment Ξ

9

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 1 This formula of optimal localization L involves: • the ensemble size N e • the sample covariance B e • the sample fourth-order centered moment Ξ

Lij =

(N − 1)2 N (N − 3)

N

  e E Ξ ijij −  2 e (N − 2)(N − 3) E B ij   e B e E B N −1 ii jj + N (N − 2)(N − 3) EBe 2 

ij

9

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 2 Step 2: optimizing localization and hybridization together

10

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 2 Step 2: optimizing localization and hybridization together Goal: to minimize the expected quadratic error

eh = E

 k

h ◦B e + (β c )2 B

L

|

{z

}

e Localized / hybridized B

−

e B

?

k2



|{z}

e Asymptotic B

10

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 2 Step 2: optimizing localization and hybridization together Goal: to minimize the expected quadratic error

eh = E

 k

h ◦B e + (β c )2 B

L

|

{z

}

e Localized / hybridized B

−

e B

?

k2



|{z}

e Asymptotic B

Same assumptions as before.

10

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Joint optimization: step 2 Step 2: optimizing localization and hybridization together Goal: to minimize the expected quadratic error

eh = E

 k

h ◦B e + (β c )2 B

L

|

{z

}

e Localized / hybridized B

−

e B

?

k2



|{z}

e Asymptotic B

Same assumptions as before. Result of the minimization: a linear system in Lh and (β c )2   e E B Lhij = Lij −  eij2  B ij (β c )2 E Bij
  e B 1 − Lhij E B ∑ ij ij ij c 2 (β ) = 2 ∑ij B ij

(2a) (2b) 10

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Hybridization benets Comparison of: b = L◦B e, • B

with an optimal L minimizing e

b h = Lh ◦ B e + (β c )2 B, • B

with optimal Lh and β c minimizing e h

11

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Hybridization benets Comparison of: b = L◦B e, • B

with an optimal L minimizing e

b h = Lh ◦ B e + (β c )2 B, • B

with optimal Lh and β c minimizing e h

We can show that:

B 2ij Var Beij h c 2 e − e = −(β ) ∑  2 e E B ij ij |

{z

≤0




(3) }

11

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Hybridization benets Comparison of: b = L◦B e, • B

with an optimal L minimizing e

b h = Lh ◦ B e + (β c )2 B, • B

with optimal Lh and β c minimizing e h

We can show that:

B 2ij Var Beij h c 2 e − e = −(β ) ∑  2 e E B ij ij |

{z

≤0




(3) }

With optimal parameters, whatever the static B: Localization + hybridization is better than localization alone 11

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Outline Introduction Linear ltering of sample covariances Joint optimization of localization and hybridization Results Conclusions

12

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Practical implementation An ergodicity assumption is required to estimate the statistical expectations E in practice:

13

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Practical implementation An ergodicity assumption is required to estimate the statistical expectations E in practice: • whole domain average, • local average, • scale dependent average, • etc.

13

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Practical implementation An ergodicity assumption is required to estimate the statistical expectations E in practice: • whole domain average, • local average, • scale dependent average, • etc. → This assumption is independent from earlier theory.

13

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Practical implementation An ergodicity assumption is required to estimate the statistical expectations E in practice: • whole domain average, • local average, • scale dependent average, • etc. → This assumption is independent from earlier theory. Localization Lh and hybridization coecient β c can be computed: • from the ensemble at each assimilation window, • climatologically from an archive of ensembles.

13

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Practical implementation An ergodicity assumption is required to estimate the statistical expectations E in practice: • whole domain average, • local average, • scale dependent average, • etc. → This assumption is independent from earlier theory. Localization Lh and hybridization coecient β c can be computed: • from the ensemble at each assimilation window, • climatologically from an archive of ensembles.

13

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Experimental setup •

WRF-ARW model, large domain, 25 km-resolution, 40 levels

14

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Experimental setup • •

WRF-ARW model, large domain, 25 km-resolution, 40 levels Initial conditions randomized from a homogeneous static B

14

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Experimental setup • • •

WRF-ARW model, large domain, 25 km-resolution, 40 levels Initial conditions randomized from a homogeneous static B Reference and test ensembles (1000 / 100 members)

14

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Experimental setup • • • •

WRF-ARW model, large domain, 25 km-resolution, 40 levels Initial conditions randomized from a homogeneous static B Reference and test ensembles (1000 / 100 members) Forecast ranges: 12, 24, 36 and 48 h

14

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Experimental setup • • • •

WRF-ARW model, large domain, 25 km-resolution, 40 levels Initial conditions randomized from a homogeneous static B Reference and test ensembles (1000 / 100 members) Forecast ranges: 12, 24, 36 and 48 h Temperature at level 7 (∼ 1 km above ground), 48 h-range forecasts

Standard-deviation (K)

Correlations functions

14

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Localization and hybridization •

Optimization of the horizontal localization Lhhor and of the hybridization coecient β c at each vertical level.

15

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Localization and hybridization • •

Optimization of the horizontal localization Lhhor and of the hybridization coecient β c at each vertical level. e Static B = horizontal average of B

15

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Localization and hybridization • • •

Optimization of the horizontal localization Lhhor and of the hybridization coecient β c at each vertical level. e Static B = horizontal average of B Localization length-scale:

15

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Localization and hybridization • • •

Optimization of the horizontal localization Lhhor and of the hybridization coecient β c at each vertical level. e Static B = horizontal average of B Hybridization coecients for zonal wind:

15

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Localization and hybridization • • •

Optimization of the horizontal localization Lhhor and of the hybridization coecient β c at each vertical level. e Static B = horizontal average of B Impact of the hybridization:

15

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Localization and hybridization • • •

Optimization of the horizontal localization Lhhor and of the hybridization coecient β c at each vertical level. e Static B = horizontal average of B Impact of the hybridization: e? • B

is estimated with the reference ensemble

15

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Localization and hybridization • • •

Optimization of the horizontal localization Lhhor and of the hybridization coecient β c at each vertical level. e Static B = horizontal average of B Impact of the hybridization: is estimated with the reference ensemble Expected quadratic errors e and e h are computed

e? • B •

15

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Localization and hybridization • • •

Optimization of the horizontal localization Lhhor and of the hybridization coecient β c at each vertical level. e Static B = horizontal average of B Impact of the hybridization: is estimated with the reference ensemble Expected quadratic errors e and e h are computed Error reduction from e to e h for 25 members

e? • B •

Zonal wind Meridian wind Temperature Specic humidity 4.5 % 4.2 % 3.9 % 1.7 %

15

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Localization and hybridization • • •

Optimization of the horizontal localization Lhhor and of the hybridization coecient β c at each vertical level. e Static B = horizontal average of B Impact of the hybridization: is estimated with the reference ensemble Expected quadratic errors e and e h are computed Error reduction from e to e h for 25 members

e? • B •

Zonal wind Meridian wind Temperature Specic humidity 4.5 % 4.2 % 3.9 % 1.7 % → Hybridization with

B improves the accuracy of the forecast error covariance matrix

15

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Outline Introduction Linear ltering of sample covariances Joint optimization of localization and hybridization Results Conclusions

16

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Conclusions

17

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Conclusions 1. Localization and hybridization are two linear ltering of sample covariances.

joint aspects

of the

17

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Conclusions 1. Localization and hybridization are two linear ltering of sample covariances.

joint aspects

of the

2. We have developed a new objective method to optimize localization and hybridization coecients together:

17

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Conclusions 1. Localization and hybridization are two linear ltering of sample covariances.

joint aspects

of the

2. We have developed a new objective method to optimize localization and hybridization coecients together: •

Based on properties of the ensemble

only

17

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Conclusions 1. Localization and hybridization are two linear ltering of sample covariances.

joint aspects

of the

2. We have developed a new objective method to optimize localization and hybridization coecients together: •

Based on properties of the ensemble

•

Aordable for high-dimensional

only

systems

17

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Conclusions 1. Localization and hybridization are two linear ltering of sample covariances.

joint aspects

of the

2. We have developed a new objective method to optimize localization and hybridization coecients together: •

Based on properties of the ensemble

•

Aordable for high-dimensional

•

Tackling the sampling noise issue only

only

systems

17

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Conclusions 1. Localization and hybridization are two linear ltering of sample covariances.

joint aspects

of the

2. We have developed a new objective method to optimize localization and hybridization coecients together: •

Based on properties of the ensemble

•

Aordable for high-dimensional

•

Tackling the sampling noise issue only

only

systems

3. If done optimally, hybridization always of forecast error covariances.

improves

the accuracy

17

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Conclusions 1. Localization and hybridization are two linear ltering of sample covariances.

joint aspects

of the

2. We have developed a new objective method to optimize localization and hybridization coecients together: •

Based on properties of the ensemble

•

Aordable for high-dimensional

•

Tackling the sampling noise issue only

only

systems

3. If done optimally, hybridization always of forecast error covariances.

improves

the accuracy

Ménétrier, B. and T. Auligné: Optimized Localization and Hybridization to Filter Ensemble-Based Covariances Monthly Weather Review, 2015, accepted 17

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Perspectives Already done in the paper:

18

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Perspectives Already done in the paper: • Extension to vectorial hybridization weights: δ x = β e ◦ δ xe + β c ◦ δ xc

18

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Perspectives Already done in the paper: • Extension to vectorial hybridization weights: δ x = β e ◦ δ xe + β c ◦ δ xc

→ Requires the solution of a nonlinear system A(Lh , β c ) = 0,

performed by a bound-constrained minimization.

18

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Perspectives Already done in the paper: • Extension to vectorial hybridization weights: δ x = β e ◦ δ xe + β c ◦ δ xc

→ Requires the solution of a nonlinear system A(Lh , β c ) = 0, •

performed by a bound-constrained minimization. Heterogeneous optimization: local averages over subdomains

18

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Perspectives Already done in the paper: • Extension to vectorial hybridization weights: δ x = β e ◦ δ xe + β c ◦ δ xc

→ Requires the solution of a nonlinear system A(Lh , β c ) = 0, • •

performed by a bound-constrained minimization. Heterogeneous optimization: local averages over subdomains 3D optimization: joint computation of horizontal and vertical localizations, and hybridization coecients

18

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Perspectives Already done in the paper: • Extension to vectorial hybridization weights: δ x = β e ◦ δ xe + β c ◦ δ xc

→ Requires the solution of a nonlinear system A(Lh , β c ) = 0, • •

performed by a bound-constrained minimization. Heterogeneous optimization: local averages over subdomains 3D optimization: joint computation of horizontal and vertical localizations, and hybridization coecients

To be done:

18

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Perspectives Already done in the paper: • Extension to vectorial hybridization weights: δ x = β e ◦ δ xe + β c ◦ δ xc

→ Requires the solution of a nonlinear system A(Lh , β c ) = 0, • •

performed by a bound-constrained minimization. Heterogeneous optimization: local averages over subdomains 3D optimization: joint computation of horizontal and vertical localizations, and hybridization coecients

To be done: • Tests in a cycled quasi-operational conguration

18

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Perspectives Already done in the paper: • Extension to vectorial hybridization weights: δ x = β e ◦ δ xe + β c ◦ δ xc

→ Requires the solution of a nonlinear system A(Lh , β c ) = 0, • •

performed by a bound-constrained minimization. Heterogeneous optimization: local averages over subdomains 3D optimization: joint computation of horizontal and vertical localizations, and hybridization coecients

To be done: • Tests in a cycled quasi-operational conguration e? • Extension of the theory to account for systematic errors in B (theory is ready, tests are underway...) 18

Introduction

Linear ltering

Joint optimization

Results

Conclusions

Perspectives Thank you for your attention! Any question? Already done in the paper: • Extension to vectorial hybridization weights: δ x = β e ◦ δ xe + β c ◦ δ xc

→ Requires the solution of a nonlinear system A(Lh , β c ) = 0,

• •

performed by a bound-constrained minimization. Heterogeneous optimization: local averages over subdomains 3D optimization: joint computation of horizontal and vertical localizations, and hybridization coecients

To be done: • Tests in a cycled quasi-operational conguration e? • Extension of the theory to account for systematic errors in B (theory is ready, tests are underway...) 18