recent developments in numerical methods for 4d-var · slide 8 ecmwf wavelets on the sphere –...

ECMWFSlide 1

Recent Developments in Numerical Methods for 4d-Var

Mike Fisher

ECMWFSlide 2

Recent Developments

Numerical Methods

4d-Var

ECMWFSlide 3

Outline

Non-orthogonal wavelets on the sphere:

- Motivation: Covariance Modelling

- Definition: Frames

- Application: Wavelet Jb

ECMWFSlide 4

Wavelets on the Sphere – Motivation

0 40 80 120 160

wavenumber n60

50

40

30

20

10

mod

el le

vel

1 1 11

-0.2-0.2

-0.2

-0.2-0.2

-0.1

-0.1

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-0.1

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Global mean vertical correlation for temperature

Vertical Correlationsvary with horizontalspatial scale

~500hPa

~850hPa

~200hPa

ECMWFSlide 5

Wavelets on the Sphere – MotivationVertical Correlationsvary with horizontallocation.

Zonal mean vertical correlation for temperature

~500hPa

~850hPa

~200hPa

ECMWFSlide 6

Wavelets on the Sphere – Motivation

The variation of vertical correlation with location and with horizontal spatial scale are both important features, and both should be included in the covariance model.However, we are severely limited by the enormous size of the covariance matrix (~107×107).Essentially, the covariance matrix must be block-diagonal, with block size NLEVS×NLEVS, and with many identical blocks.Currently, we specify one block per wavenumber, n.

- variation with scale is modelled, variation with location is not.

Alternatively, we could specify one block per gridpoint.- variation with location is modelled, variation with scale is not.

Wavelets provide a way to do both (and still keep things sparse).

ECMWFSlide 7

Wavelets on the Sphere – Frames

It is possible to define orthogonal wavelets on the sphere by first gridding the sphere.Göttelmann (1997, citeseer.ist.psu.edu/227230.html) defined spherical wavelets using splines on a quasi-uniform latitude-longitude grid.Schröder and Sweldens (1995, ACM SIGGRAPH, 161-172) defined them for a triangulation of the sphere.However, these approaches necessarily have special points (poles or vertices). They do not retain rotational symmetry for finite truncations of the wavelet expansion.If we wish to retain rotational symmetry, we must give up on orthogonality.I.e. we must consider frames.

ECMWFSlide 8


Definitions- A family of functions, ; j∈J in a Hilbert space is called a

frame if there exist A>0 and B<∞ such that for all ƒ in the space:

- The condition is sufficient to ensure the existence of a dual frame, ; j∈J with the property:

- A particularly interesting case occurs when A=B. This is called a tight frame. Tight frames are self-dual:

22 2, jj J

A f f B fψ∈

≤ ≤∑

jψ

jψ

1 1, ,j j j jj J j J

f f fA A

ψ ψ ψ ψ∈ ∈

= =∑ ∑

1 , j jj J

f fA

ψ ψ∈

= ∑

ECMWFSlide 9


Tight frames share many of the properties of orthogonal bases. (An orthogonal basis is a tight frame with and A=1).Tight frames define a “transform”, since we may write:

as:

c.f. Fourier series:

1 , j jj J

f fA

ψ ψ∈

= ∑

21jψ =

1, ,j j j jj J

c f f cA

ψ ψ∈

= = ∑

2 ( ) 2 ( )

2 ( )

1ˆ , ( ) d

ˆ( )

bimt b a imt b a

ma

imt b am

f f e f t e tb a

f t f e

π π

π

− − −

∞−

= =−

=

∫

∑m=−∞

ECMWFSlide 10


Example: The Mercedes-Benz Frame- Daubechies, 1992: “Ten Lectures on Wavelets”

Tight frame:

Hence, for any ƒ:

2 23 2 2 2

1

3 1 3 1 3,2 2 2 2 2j y x y x y

jf f f f f f fψ

=

= + − − + − =∑3

1

2 ,3 j j

jf fψ ψ

=

=∑

ψ1= (0 , 1)

ψ3= ((√3)/2 , -1/2)ψ2= (-(√3)/2 , -1/2)

ECMWFSlide 11


Example: The discrete spherical transform.Consider functions where 0≤n≤N, -n≤m≤n.Let be the (m,n)th spherical harmonic, evaluated at the jth gridpoint of the Gaussian grid, and multiplied by sqrt(Gaussian integration weight):

Then:

NB: This is a tight frame, not an orthogonal transform. There are more gridpoints than spectral coefficients.

,ˆ ( , ) ( ) ( , )j j m n j jm n w Yψ φ λ φ=

ˆ ( , )j m nψˆ ( , )f m n

,,

ˆ ˆˆ, ( , ) ( ) ( , ) ( ) ( , )j j m n j j j j j jm n

f f m n w Y w fψ φ λ φ φ λ φ= =∑

,ˆ ˆ ˆ( , ) , ( , ) ( ) ( , ) ( , )j j j j j j m n j j

j jf m n f m n w f Yψ ψ φ λ φ λ φ= =∑ ∑

ECMWFSlide 12

Wavelets on the Sphere – Generalized Frames

The concept of a frame can be generalized to the case where the number of basis functions, is uncountable.The sum in the frame condition becomes an integral:

Most of the properties carry over from the discrete case.In particular, for a tight frame, we have:

- See Kaiser, 1994 “A Friendly Guide to Wavelets”

jψ

22 2, dxD

A f f x B fψ≤ ≤∫

1 , dx xD

f f xA

ψ ψ= ∫

ECMWFSlide 13


We will consider a specific, semi-discrete case.The basis functions, are labelled by 3 indices:

- j (discrete) indicates “scale”- (continuous) is longitude- (continuous) is latitude

We define:

- where is the great-circle distance between and .

, ,j λ φψ

λφ

, , ( , ) ( ( , , , ))j j rλ φψ λ φ λ φ λ φ′ ′ ′ ′= Ψ

( , , , )r λ φ λ φ′ ′ ( , )λ φ′ ′( , )λ φ

ECMWFSlide 14


The inner product is:

Let us write:

Then, substituting for we see that theinner product corresponds to a convolution on the sphere:

, , , ,, ( , ) ( , ) cos( )d dj jf fλ φ λ φψ λ φ ψ λ φ φ λ φΩ

′ ′ ′ ′ ′ ′ ′= ∫

, ,( , ) ,j jf f λ φλ φ ψ=

, , ( , )j λ φψ λ φ′ ′

( , ) ( , ) ( ( , , , )) cos( )d d

I.e.

j j

j j

f f r

f f

λ φ λ φ λ φ λ φ φ λ φΩ

′ ′ ′ ′ ′ ′ ′= Ψ

= ⊗Ψ

∫

ECMWFSlide 15


We seek a tight frame. The condition is:

That is:

I.e.

Evaluating the norms in terms of spherical harmonic coefficients, we have:

2 2, ,, cos( )d dj

jf A fλ φψ φ λ φ

Ω

=∑∫2 2( , ) cos( )d dj

jf A fλ φ φ λ φ

Ω

=∑∫2 2

jj

f A f=∑

2 2

, , ,

ˆ ˆ( , ) ( , )jj m n m n

f m n A f m n=∑ ∑

ECMWFSlide 16


But, remember that , where is a function of great-circle distance.

Hence:

- where are Legendre transform coefficients.

The condition for a tight frame is thus:

2 2

, , ,

ˆ ˆ( , ) ( , )jj m n m n

f m n A f m n=∑ ∑

j jf f= ⊗Ψ jΨ

ˆ ˆ ˆ( , ) ( , ) ( )j jf m n f m n n= Ψ

ˆ ( )j nΨ

2 2

, , ,

ˆ ˆˆ( , ) ( ) ( , )jj m n m n

f m n n A f m nΨ =∑ ∑

ECMWFSlide 17


2 2

, , ,

ˆ ˆˆ( , ) ( ) ( , )jj m n m n

f m n n A f m nΨ =∑ ∑I.e.

For convenience, we will scale the basis functions appropriately, so that A=1:

2 22

, ,

2

ˆ ˆˆ ( ) ( , ) ( , )

ˆ ( )

jm n j m n

jj

n f m n A f m n

n A

Ψ =

⇒ Ψ =

∑ ∑ ∑

∑

2ˆ ( ) 1jj

n nΨ = ∀∑

ECMWFSlide 18


If we have a tight frame, we have the transform property:

But, the right hand side is just another convolution.Hence, the “transform pair” is just:

, , , ,, cos( )d d

( , ) ( ( , , , )) cos( )d d

j jj

j jj

f f

f r

λ φ λ φψ ψ φ λ φ

λ φ λ φ λ φ φ λ φΩ

Ω

′ ′ ′=

′ ′ ′ ′ ′ ′ ′= Ψ

∑∫

∑∫

j j

j jj

f f

f f

= Ψ ⊗

= Ψ ⊗∑

ECMWFSlide 19


The first equation defines .We can easily verify the second equation:

2

ˆˆ( , ) ( ) ( , )

ˆ ( ) ( , )

ˆ ( , )

j j j jj j

jj

f m n n f m n

n f m n

f m n

Ψ ⊗ = Ψ

= Ψ

=

∑ ∑

∑

,j j j jj

f f f f= Ψ ⊗ = Ψ ⊗∑jf

ECMWFSlide 20

Wavelets on the Sphere – Summary

A set of functions of great-circle distance, ; j=0,1,2,... whose Legendre transform coefficients satisfy:

define a tight generalized frame.

The functions define a “transform pair”:

j j

j jj

f f

f f

= Ψ ⊗

= Ψ ⊗∑

2ˆ ( ) 1jj

n nΨ = ∀∑

( )j rΨ

ECMWFSlide 21

Wavelets on the Sphere – Example

The condition:

suggests we define the functions to be B-splines.For example, linear B-splines are triangle functions:

2ˆ ( ) 1jj

n nΨ = ∀∑

2ˆ ( )j nΨ

11

1

121

1

ˆ ( )

0 otherwise

jj j

j j

jj j j

j j

n NN n N

N Nn N

n N n NN N

−−

−

++

+

−< ≤ −

−Ψ = < < −

ECMWFSlide 22


Ψj(n)

ECMWFSlide 23


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ECMWF Analysis VT:Wednesday 1 September 2004 12UTC 850hPa specific humidity

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ECMWFSlide 24


20°E

20°EECMWF Analysis VT:Wednesday 1 September 2004 12UTC 850hPa specific humidity

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ECMWFSlide 25

Spherical Wavelets – Example

T511 Model Orography:

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ECMWFSlide 26

Spherical Wavelets – ExampleConvolve with Ψj:

j jf f= Ψ ⊗

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160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°EECMWF Analysis VT:Wednesday 1 September 2004 12UTC Model Level 1 geopotential height

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ECMWF Analysis VT:Wednesday 1 September 2004 12UTC Model Level 1 geopotential height

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ECMWFSlide 27

Spherical Wavelets – ExampleConvolve again...

j jfΨ ⊗

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ECMWFSlide 28

Spherical Wavelets – Example

... and add, to retrieve the original field: j jj

f f= Ψ ⊗∑

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ECMWFSlide 29

Spherical Wavelets – Other Approaches

The wavelets we have derived are similar to those of Freeden and Windheuser (1996, Adv. Comp. Math. 51-94.

- They are a special case of the very broad range of spherical wavelet decompositions described by Freeden et al. (1998, “Constructive Approximation on the Sphere, OUP).

A different approach, using group theory, is taken by Antoine and Vandergheynst (1999, Appl. and Comput. Harm. Anal. 262-291).

- They define spherical wavelets as coherent states of the productgroup of rotations on the sphere, and dilations on the polar-stereographic tangent plane.

Mhaskar et al. (2000, Adv. Comput. Math.) describe polynomial wavelet frames on the sphere.

- and cite 11 papers, each defining a different approach to the construction of wavelets on the sphere.

ECMWFSlide 30

Wavelet Jb

So, how does all this help us formulate a covariance model? First, let’s review the main idea behind the current Jb.3d/4d-Var determine the analysis by minimizing a cost function:

Usually, we do not minimize directly in terms of x, but formulate the problem as:

NB: L defines the background covariance matrix: B=LLT.

( ) ( ) ( ) ( )T T1 1( ) ( )b b b bJ − −= − − + − − − −x x B x x d H x x R d H x x

( ) ( )TT 1

b

J −= + − −

= +

χ χ d HLχ R d HLχx x Lχ

ECMWFSlide 31

Wavelet Jb

L defines the background covariance matrix: B=LLT.To keep things simple, consider a 2d, univariate model.The current ECMWF covariance model boils down to:

Σ is diagonal in grid space, and corresponds to multiplying each gridpoint by a standard deviation.

- It accounts for the spatial variation of background error.

D is diagonal in spectral space, and corresponds to multiplying each wavenumber, n, by a standard deviation, which is a function of n, only.

- It accounts for the variation of background error with scale.

=L ΣD

ECMWFSlide 32

Wavelet Jb

=L ΣDThe current covariance model separates the spatial and spectral variation of background error.In particular, D corresponds to a convolution.D defines the horizontal correlation of background error for the covariance model.Because D is a convolution, the horizontal correlations are the same everywhere.This is a major shortcoming of the covariance model.

ECMWFSlide 33

Wavelet Jb

To define a covariance model using wavelets, note first that there is no requirement for L to be square.A rectangular matrix L still defines a valid covariance model, B=LLT.We define the control variable as:

where the correspond to different scales.The change of variable matrix is defined by:

1

2=

χχ χ

ˆb j j j

j− = = Ψ ⊗∑x x Lχ Σ χ

jχ

ECMWFSlide 34

Wavelet Jb

The matrices Σj are diagonal in grid space, and account for the spatial variation of background error for each scale.Let us write the convolution with explicitly as a matrix operator , where S is the spherical transform, and is diagonal.Using the symmetry of , and Σj , and the orthogonalityof S, we can write the covariance matrix implied by L as:

We will illustrate the covariance structures generated by wavelet Jb by applying this matrix to delta functions.

Ψ j1 ˆj

−S Ψ S

T 1 2 1j j j

j

− −= =∑B LL S Ψ SΣ S Ψ S

ˆjΨ

b j j jj

ˆjΨ

ˆ− = = Ψ ⊗∑x x Lχ Σ χ

ECMWFSlide 35

Wavelet Jb

Consider the case where there is no spatial variation in the standard deviations: .Then, is diagonal, with elements .

But, , so is a weighted average of

If we choose to be B-splines, then the variation of variance with n is an interpolation between the prescribed .

T 1 2 1j j j

j

− −= =∑B LL S Ψ SΣ S Ψ S

j jσ=Σ I2 1

j j j−Ψ SΣ S Ψ 2 2ˆ ( )j j nσ Ψ

2ˆ ( ) 1jj

nΨ =∑ 2 2ˆ ( )j jj

nσ Ψ∑ 'sjσ

2ˆ ( )j nΨ

'sjσ

ECMWFSlide 36

Wavelet Jb

Suppose we want approximately Gaussian structure functions, with length scale that is a smoothly-varying function of latitude and longitude.Weaver+Courtier (2000) give the following expression for the modal variances corresponding to convolution with a quasi-Gaussian function with lengthscale L:

We simply set:but allow L ( and hence σj) to vary with latitude and longitude.

σ

( )( )

2 22

2 2

exp ( 1) / 2( ; )

(2 1)exp ( 1) / 2n

L n n ab n L

n L n n a

− + = + − + ∑

max( 1) ( ; )j jN b N L= +

ECMWFSlide 37

Wavelet Jb

Desired Length scale (300km – 1300km):

1299998.1299996.

300000. 300007.

80°S80°S

70°S 70°S

60°S60°S

50°S 50°S

40°S40°S

30°S 30°S

20°S20°S

10°S 10°S

0°0°

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40°E 60°E

60°E 80°E

80°E 100°E

100°E 120°E

120°E 140°E

140°E 160°E

160°E

ECMWFSlide 38

Wavelet Jb

Ψ j

0 .0 00 03 4

0 .0 00 03 4

0 .0 00 03 5

0 .0 00 03 5

0 .0 00 03 5

0 .0 00 03 5

0 .0 00 03 5

0 .0 00 03 5

0 .0 00 03 5

0 .0 00 03 5

80°S

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0°0°

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80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°EECMWF Analysis VT:Wednesday 1 September 2004 12UTC 850hPa specific humidity

-0.3 10- 5

-0.2 10- 5

-0.2 10 - 5

-0.2 10 - 5

-0.1 10- 5

-0.8 10- 6

-0.4 10- 6

0.4 10- 6

0.4 10- 6

80°S

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20°S

0°0°

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40°N

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60°N

80°N

80°N


-0 .0 0 0 1

-0 .0 0 0 1

-0 .0 0 0 0 8

-0 .0 0 0 0 8

-0 .0 00 06

-0 .0 00 06

-0 .0 00 04

-0 .0 00 04

-0 .0 00 02

-0 .0 00 02

0 .0 00 02 0 .0 00 02 0 .0 00 02 0 .0 00 02

0.0

0004

0.0

0004

0.0

0004

0.0

0004

80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

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60°N

80°N

80°N


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E


80°S

80°S

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20°S

0°0°

20°N

20°N

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40°N

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60°N

80°N

80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E


80°S

80°S

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60°S

40°S

40°S

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20°S

0°0°

20°N

20°N

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40°N

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60°N

80°N

80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E


80°S

80°S

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20°S

0°0°

20°N

20°N

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80°N


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80°S

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20°S

0°0°

20°N

20°N

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60°N

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80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E

80°S

80°S

60°S

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20°S

0°0°

20°N

20°N

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40°N

60°N

60°N

80°N

80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E

Where x is a set of delta functions

jΨ ⊗ x

ECMWFSlide 39

Wavelet Jb0 .0 25

0 .0 25

0 .0 5

0 .0 5

0 .0 75 0 .0 75

0 .0 75

0 .0 750 .0 75

0 .1

0 .1

0 .1 25

0 .1 250 .1 5

0 .1 5 0 .1 750 .1 750 .1 750 .2

80°S

80°S

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40°S

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80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°EECMWF Analysis VT:Wednesday 1 September 2004 12UTC 850hPa vorticity

-0 .0 05-0 .0 03 75-0 .0 02 5

-0 .0 01 25

-0 .0 01 25

0 .0 01 250 .0 01 25

80°S

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80°N


-0 .2 4-0 .1 6

-0 .1 6

-0 .0 8

-0 .0 8

-0 .0 8

0.08

0 .08

0.0

8

0.0

8

0.1 6

0.1 6

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0°0°

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60°N

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80°N


-0 .0 8-0 .0 8-0 .0 8 -0 .0 8

80°S

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80°N

80°N


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20°N

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60°N

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80°N


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160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E

ECMWF Analysis VT:Wednesday 1 September 2004 12UTC 850hPa vorticity

80°S

80°S

60°S

60°S

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20°S

20°S

0°0°

20°N

20°N

40°N

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80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E


80°S

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0°0°

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80°S

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20°S

0°0°

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80°N


80°S

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20°N

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160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E

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20°N

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80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E

Scale with σj2

at each gridpoint, and for each scale.

ECMWFSlide 40

Wavelet Jb

Convolve again with

0 .0 1 5

0 .0 3

0 .0 4 5

0 .0 6

0 .0 7 5

0 .0 9

0 .1 0 5

0 .1 2

0 .1 3 5

0 .1 5

0 .1 6 5 80°S

80°S

60°S

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40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


-0 .0 02 4

-0 .0 02

-0 .0 016

-0 .0 01 6

-0 .0 01 6

-0 .0 01 2

-0 .0 01 2

-0 .0 00 8

-0 .0 00 8

-0 .0 00 4

-0 .0 00 4

0 .0 00 4

0 .0 00 4

0 .0 0 0 8

0 .0 00 8

0 .0 008

80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


-0 .2-0 .1 5-0 .1-0 .0 5

-0.0 5

-0 .05

-0 .0 5-0 .0 5

0.0

5

0.0

5

0 .05

0.0

5

0 .05

0 .05

0 .1 0 .1

0.1

0 .1

0 .15

0 .150 .1

5

0 .15

80°S

80°S

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20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


-0.08

-0 .0 8 -0 .0 8

-0.0 8

-0.0 8

80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N


80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E

80°S

80°S

60°S

60°S

40°S

40°S

20°S

20°S

0°0°

20°N

20°N

40°N

40°N

60°N

60°N

80°N

80°N

160°W 120°W 80°W 40°W 0° 40°E 80°E 120°E 160°E

ˆ ( )j nΨ

ECMWFSlide 41

Wavelet Jb

Add together to give: 1 2 1j j j

j

− −=∑Bx S Ψ SΣ S Ψ Sx

80°S80°S

70°S 70°S

60°S60°S

50°S 50°S

40°S40°S

30°S 30°S

20°S20°S

10°S 10°S

0°0°

10°N 10°N

20°N20°N

30°N 30°N

40°N40°N

50°N 50°N

60°N60°N

70°N 70°N

80°N80°N

160°W

160°W 140°W

140°W 120°W

120°W 100°W

100°W 80°W

80°W 60°W

60°W 40°W

40°W 20°W

20°W 0°

0° 20°E

20°E 40°E

40°E 60°E

60°E 80°E

80°E 100°E

100°E 120°E

120°E 140°E

140°E 160°E

160°E

-0.081

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

ECMWFSlide 42

Wavelet Jb

E-W cross-sections, and the Gaussians we wanted:

ECMWFSlide 43

Wavelet Jb

The agreement between the Gaussians we wanted, and the functions we got is not perfect!But note:

- No attempt was made to tune the cut-off wavenumbers to improve the accuracy with which the Gaussian structure functions were modelled.

- The implied modal variances are effectively linearly interpolated between those for wavenumbers 0,2,4,8,16,...

- For large lengthscale (1300km), there is little variance beyond about n=10, so there are very few nodes in the interpolation.

- More spectral resolution at large scales may improve the approximation.

- Higher order B-splines would also help.

ECMWFSlide 44

Wavelet Jb

Modal variances.Solid = Gaussians.Dashed = Linear Interpolation.

ECMWFSlide 45

Wavelet Jb

E-W cross-sections for a different choice of spectral bands (0,1,3,6,10,16,25,39,60,91,138,208,313,471,511):

ECMWFSlide 46

Wavelet Jb

We demonstrated Wavelet Jb by producing structure functions of a given analytic form.However, a significant advantage of Wavelet Jb over other approaches to covariance modelling on the sphere (digital filters, diffusion operators, etc.) is that we can calculate the coefficients of the covariance model directly from data.The covariance model is:

The transform property gives us:But has covariance matrix = I, so (in 2d), is simplythe matrix of gridpoint standard deviations ofThis is easily generated, given a sample of bg errors.

ˆb j j j

j− = Ψ ⊗∑x x Σ χ

( )ˆj j j b= Ψ ⊗ −Σ χ x x

jχ jΣ(ˆ

j bΨ ⊗ −x x )

ECMWFSlide 47

Wavelet Jb

Extension of Wavelet Jb to 3 spatial dimensions is straightforward.In 2 dimensions, we have:

where is diagonal. The diagonal elements are standard deviations.In 3 dimensions, becomes block-diagonal, with blocks of dimension NLEVS×NLEVS. The diagonal blocks are symmetric square-roots of vertical covariance matrices.This is not fully general, but is sufficient to capture the variation of vertical correlation with horizontal scale, and with location.

ˆb j j j

j− = Ψ ⊗∑x x Σ χ

jΣ

jΣ

ECMWFSlide 48

Wavelet Jb

Example: Horizontal and vertical Vorticity Correlations.

North America Equatorial Pacific

ECMWFSlide 49

Wavelet Jb – Memory and Cost

At first sight, the memory requirements for Wavelet Jbappear high.However, if then the are strictly band-limited, and may be represented on Gaussian grids of appropriate resolution.

If we arrange for:

then the memory requirement for the control vector is at most p+2 full model grids, where p is the order of the B-splines.Only p+1 full-resolution spherical transforms are required.

ˆ ( ) 0 for j j pn n N +Ψ = > jχ

1

2j

j

NN +≤

ECMWFSlide 50

Wavelet Jb – Memory and Cost

Storage for the vertical covariance matrices is potentially enormous. But, can be reduced to manageable levels by reducing their spatial resolution (e.g. one matrix for every 10 gridpoints).The main CPU requirement is in handling the increased-length control vector.The bottom line is that Wavelet Jb adds about 5% to the cost of a 4d-Var analysis.I think it’s worth it!

ECMWFSlide 51

Summary

Tight frames provide a useful mathematical construct.They share many of the desirable properties of orthogonal bases, but allow considerable flexibility.Using tight frames, a flexible family of wavelets may be defined on the sphere.Unlike grid-based wavelets, there is no pole problem.There is a lot of scope for tuning of the spectral and spatial resolution. (This remains to be explored.)Wavelet Jb is expected to be implemented operationally in the ECMWF 4d-Var system by the end of this year.

recent developments in numerical methods for 4d-var · slide 8 ecmwf wavelets on the sphere –...

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