Recent Developments in Numerical Methods for 4d-Var

Transcription

1 Recent Developments in Numerical Methods for 4d-Var Mike Fisher Slide 1

2 Recent Developments Numerical Methods 4d-Var Slide 2

3 Outline Non-orthogonal wavelets on the sphere: - Motivation: Covariance Modelling - Definition: Frames - Application: Wavelet J b Slide 3

4 -0.1 Wavelets on the Sphere Motivation Global mean vertical correlation for temperature Vertical Correlations vary with horizontal spatial scale model level ~200hPa ~500hPa ~850hPa wavenumber n Slide 4

5 Wavelets on the Sphere Motivation Zonal mean vertical correlation for temperature Vertical Correlations vary with horizontal location. ~200hPa ~500hPa ~850hPa Slide 5

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 (~ ). Essentially, the covariance matrix must be blockdiagonal, 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). Slide 6

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/ html) defined spherical wavelets using splines on a quasiuniform latitude-longitude grid. Schröder and Sweldens (1995, ACM SIGGRAPH, ) 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. Slide 7

8 Wavelets on the Sphere Frames Definitions ψ - A family of functions, { ; J} in a Hilbert space is called a frame if there exist A>0 and B< such that for all ƒ in the space: 2 2 2, ψ J A f f B f - The condition is sufficient to ensure the existence of a dual frame, { ψ ; J} with the property: 1 1 f, ψ ψ = f = f, ψ ψ A A J - A particularly interesting case occurs when A=B. This is called a tight frame. Tight frames are self-dual: f = 1 A J f ψ J ψ, Slide 8

9 Wavelets on the Sphere Frames Tight frames share many of the properties of orthogonal 2 bases. (An orthogonal basis is a tight frame with ψ = 1 and A=1). Tight frames define a transform, since we may write: 1 f = f, A ψ ψ J 1 as: c = f, ψ, f = cψ A c.f. Fourier series: J b ˆ 2 πimt ( b a) 1 2 πimt ( b a) m =, = ( ) d b a a f f e f t e t f() t = m= fˆ e m 2 π imt ( b a) Slide 9

10 Wavelets on the Sphere Frames Example: The Mercedes-Benz Frame - Daubechies, 1992: Ten Lectures on Wavelets ψ 1 = (0, 1) ψ 2 = (-( 3)/2, -1/2) ψ 3 = (( 3)/2, -1/2) Tight frame: = f, ψ = fy + fx fy + fx fy = f Hence, for any ƒ: = 1 f, ψ ψ = f Slide 10

11 Wavelets on the Sphere Frames 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 th gridpoint of the Gaussian grid, and multiplied by sqrt(gaussian integration weight): Then: ψˆ ( mn, ) f ˆ( mn, ) ψˆ ( mn, ) = w( φ ) Y ( λ, φ ) m, n fˆ, ψˆ = fˆ( m, n) w( φ ) Y ( λ, φ ) = w ( φ ) f( λ, φ ) m, n mn, fˆ ( m, n) = f, ψˆ ψˆ ( m, n) = w ( φ ) f( λ, φ ) Y ( λ, φ ) m, n NB: This is a tight frame, not an orthogonal transform. There are more gridpoints than spectral coefficients. Slide 11

12 Wavelets on the Sphere Generalized Frames The concept of a frame can be generalized to the case where the number of basis functions, ψ is uncountable. The sum in the frame condition becomes an integral: A f f, ψ dx B f D x Most of the properties carry over from the discrete case. In particular, for a tight frame, we have: 1 f = f, ψx ψx dx A D - See Kaiser, 1994 A Friendly Guide to Wavelets Slide 12

13 Wavelets on the Sphere Generalized Frames We will consider a specific, semi-discrete case. ψ λ φ The basis functions,,, are labelled by 3 indices: - (discrete) indicates scale λ φ - (continuous) is longitude - (continuous) is latitude We define: ψ, λφ,( λ, φ ) = Ψ ( r( λ, φ, λ, φ)) r( λ, φ, λ, φ) ( λ, φ) - where is the great-circle distance between and. ( λ, φ ) Slide 13

14 Wavelets on the Sphere Generalized Frames The inner product is: f, ψ = f( λ, φ ) ψ ( λ, φ )cos( φ )dλ dφ Let us write:, λφ,, λφ, Ω f ( λφ, ) = f, ψ, λ, φ ψ,,( λ, φ ) Then, substituting for λφ we see that the inner product corresponds to a convolution on the sphere: f ( λ, φ) = f( λ, φ ) Ψ ( r( λ, φ, λ, φ))cos( φ )dλ dφ I.e. f Ω = f Ψ Slide 14

15 Wavelets on the Sphere Generalized Frames We seek a tight frame. The condition is: That is: I.e. Ω Ω f f f, λφ, 2 2, ψ cos( φ)dλdφ = A f 2 2 ( λφ, ) cos( φ)dλdφ= 2 2 = A f A f Evaluating the norms in terms of spherical harmonic coefficients, we have: mn,, mn, 2 2 fˆ ( m, n) = A fˆ( m, n) Slide 15

16 Wavelets on the Sphere Generalized Frames mn,, mn, 2 2 fˆ ( m, n) = A fˆ( m, n) f But, remember that, where is a function of great-circle distance. = f Ψ Ψ Hence: fˆ ( m, n) = fˆ( m, n) Ψˆ ( n) Ψˆ ( n) - where are Legendre transform coefficients. The condition for a tight frame is thus: mn,, mn, 2 2 fˆ( m, n) Ψ ˆ ( n) = A fˆ( m, n) Slide 16

17 Wavelets on the Sphere Generalized Frames I.e. mn,, mn, 2 2 fˆ( m, n) Ψ ˆ ( n) = A fˆ( m, n) Ψ ˆ ( ) ˆ(, ) ˆ n f m n = A f( m, n) mn, mn, ˆ 2 ( n) Ψ = A For convenience, we will scale the basis functions appropriately, so that A=1: ˆ 2 ( n) 1 Ψ = n Slide 17

18 Wavelets on the Sphere Generalized Frames If we have a tight frame, we have the transform property: f =, ψ ψ cos( φ )dλ dφ, λφ,, λφ, = f ( λ, φ ) Ψ ( r( λ, φ, λ, φ))cos( φ )dλ dφ Ω Ω f But, the right hand side is ust another convolution. Hence, the transform pair is ust: f f = Ψ f = Ψ f Slide 18

19 Wavelets on the Sphere Generalized Frames f =Ψ f, f = Ψ f The first equation defines. We can easily verify the second equation: f Ψ (, ) ˆ ( ) ˆ f m n = Ψ n f ( m, n) ˆ 2 = Ψ ( n) f( m, n) = fˆ( m, n ) Slide 19

20 Wavelets on the Sphere Summary A set of functions of great-circle distance, { Ψ () r ; =0,1,2,...} whose Legendre transform coefficients satisfy: ˆ 2 Ψ ( n) = 1 n define a tight generalized frame. The functions define a transform pair : f = Ψ f f = Ψ f Slide 20

21 Wavelets on the Sphere Example The condition: ˆ 2 ( n) 1 Ψ = n ˆ 2 ( ) Ψ n suggests we define the functions to be B-splines. For example, linear B-splines are triangle functions: n N 1 N 1 < n N N N 1 ˆ n N Ψ ( n) = N < n< N 0 otherwise N N + 1 Slide 21

22 Wavelets on the Sphere Example Ψ (n) Slide 22

23 Wavelets on the Sphere Example Analysis VT:Wednesday 1 September UTC 850hPa specific humidity Analysis VT:Wednesday 1 September UTC 850hPa specific humidity Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 2N 2N N N 2N N 2S4S6S8S 2S 2S4S6S8S 2S 2S4S6S8S 2S 4S 4S 4S 6S 6S 6S 8S 8S 8S Analysis VT:Wednesday 1 September UTC 850hPa specific humidity Analysis VT:Wednesday 1 September UTC 850hPa specific humidity Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 2N 2N 2N 2N 2N 2N 2S 4S6S 8S 8S 6S 4S2S 2S 4S6S 8S 8S 6S 4S2S 2S 4S6S 8S 8S 6S 4S2S Slide 23

24 Wavelets on the Sphere Example Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 2E Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 2E Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 2E 2E 2E 2E Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 2E 2E Slide 24

25 Spherical Wavelets Example T511 Model Orography: 16W 14W 12W 10W 8W 6W 4W 2W 2E 4E 6E 8E 10E 12E 14E 16E N 7N N 5N N 3N 2N 2N N 1N 1S 1S 2S 2S S 3S 4S 4S 5S 5S S 6S 7S 7S 8S S 16W 14W 12W 10W 8W 6W 4W 2W 2E 4E 6E 8E 10E 12E 14E 16E Slide 25

26 Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2S 4S6S 8S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2S4S6S8S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2S 4S6S 8S 2N 2S 4S 6S 8S 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Spherical Wavelets Example Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2N 2N 2N Convolve with Ψ : 2S 4S6S 8S 2S 2S 4S6S 8S 2S 4S 4S 6S 6S 8S 8S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height f =Ψ f 2N 2N 2N 2N 2S4S6S8S 2S 2S4S6S8S 2S 4S 4S 6S 6S 8S 8S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2N 2N 2N 2S 4S6S 8S 8S 6S 4S2S 2S 4S6S 8S 8S 6S 4S2S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2N 2N 2N 2S 4S 6S 8S 8S6S4S 2S 2S 4S 6S 8S 8S6S4S 2S Slide 26

27 Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2S 4S6S 8S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2S4S6S8S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2S 4S6S 8S 2N 2S 4S 6S 8S 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Spherical Wavelets Example Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2N 2N 2N Convolve again... 2S 4S6S 8S 2S 2S 4S6S 8S 2S 4S 4S 6S 6S 8S 8S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height Ψ f 2N 2N 2N 2N 2S4S6S8S 2S 2S4S6S8S 2S 4S 4S 6S 6S 8S 8S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2N 2N 2N 2S 4S6S 8S 8S 6S 4S2S 2S 4S6S 8S 8S 6S 4S2S Analysis VT:Wednesday 1 September UTC Model Level 1 geopotential height 2N 2N 2N 2N 2S 4S 6S 8S 8S6S4S 2S 2S 4S 6S 8S 8S6S4S 2S Slide 27

28 Spherical Wavelets Example... and add, to retrieve the original field: 16W 14W 12W 10W 8W 6W 4W 2W 2E 4E 6E 8E f = Ψ 10E 12E 14E 16E f N 7N N 5N 3N 3N 2N 2N 1N 1N 1S 1S 2S 2S S 3S 4S 4S 5S 5S 6S 6S 7S 7S 8S S 16W 14W 12W 10W 8W 6W 4W 2W 2E 4E 6E 8E 10E 12E 14E 16E Slide 28

29 Spherical Wavelets Other Approaches The wavelets we have derived are similar to those of Freeden and Windheuser (1996, Adv. Comp. Math 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 ). - They define spherical wavelets as coherent states of the product group of rotations on the sphere, and dilations on the polarstereographic 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. Slide 29

30 Wavelet J b So, how does all this help us formulate a covariance model? First, let s review the main idea behind the current J b. 3d/4d-Var determine the analysis by minimizing a cost function: J T ( ) 1 T x x B ( x x ) ( d H( x x )) R 1 ( d H( x x )) = + b b b b Usually, we do not minimize directly in terms of x, but formulate the problem as: J T ( ) ( ) χχ d HLχ R d HLχ T 1 = + x= x + Lχ b NB: L defines the background covariance matrix: B=LL T. Slide 30

31 Wavelet J b L defines the background covariance matrix: B=LL T. To keep things simple, consider a 2d, univariate model. The current covariance model boils down to: L = ΣD 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. Slide 31

32 Wavelet J b L = ΣD The 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 maor shortcoming of the covariance model. Slide 32

33 Wavelet J b 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=LL T. We define the control variable as: χ χ χ= χ where the correspond to different scales. The change of variable matrix is defined by: x x = Lχ = Ψˆ Σ χ 1 2 b Slide 33

34 Wavelet J b x x = Lχ = Ψˆ Σ χ b The matrices Σ 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 1 matrix operator S Ψ ˆ S, where S is the spherical transform, and Ψˆ is diagonal. Using the symmetry of Ψˆ, and Σ, and the orthogonality of 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. ˆΨ B LL S Ψ SΣ S Ψ S T Slide 34

35 Wavelet J b B LL S Ψ SΣ S Ψ S = = T Consider the case where there is no spatial variation in the standard deviations: Σ. = σ I Then, Ψ SΣ S Ψ is diagonal, with elements. σ Ψˆ ( ) n But, ˆ 2 ( ) Ψ n =, so σ ˆ Ψ ( n) is a weighted average of σ 's ˆ 2 ( ) Ψ n If we choose to be B-splines, then the variation of variance with n is an interpolation between the prescribed. σ 's Slide 35

36 Wavelet J b 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: b 2 ( n; L) = n ( Lnn ) 2 2 exp ( 1) / 2a + (2n+ 1) exp ( L n( n 1) ) + / 2a 2 2 We simply set: σ = ( N + 1) b( N ; L) max but allow L ( and hence σ ) to vary with latitude and longitude. Slide 36

37 Wavelet J b Desired Length scale (300km 1300km): 16W 14W 12W 10W 8W 6W 4W 2W 2E 4E 6E 8E 10E 12E 14E 16E 7N 7N 5N 5N N 3N 2N 2N 1N 1N 1S 1S 2S 2S 3S 3S 4S S 5S 4S 6S 6S 7S 7S 8S 8S 16W 14W 12W 10W 8W 6W 4W 2W 2E 4E 6E 8E 10E 12E 14E 16E Slide 37

38 2N S 4S6S 8S 2N 2S4S6S8S 2N 2S 4S6S 8S 2N 2S 4S 6S 8S Wavelet J b ˆΨ N S 4S6S 8S Slide 38 2N 2S 4S6S 8S Analysis VT:Wednesday 1 September UTC 850hPa specific humidity Analysis VT:Wednesday 1 September UTC 850hPa specific humidity Analysis VT:Wednesday 1 September UTC 850hPa specific humidity S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 2N 2S4S6S8S 2N 2S4S6S8S Analysis VT:Wednesday 1 September UTC 850hPa specific humidity Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 2N 2S 4S6S 8S 2N 2S 4S6S 8S Analysis VT:Wednesday 1 September UTC 850hPa specific humidity Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 2N 2S 4S 6S 8S 2N 2S 4S 6S 8S Analysis VT:Wednesday 1 September UTC 850hPa specific humidity 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Ψ x Where x is a set of delta functions

39 2N 0.1 2S 4S6S 8S 2N 2S4S6S8S 2N 2S 4S6S 8S 2N 2S 4S 6S 8S Wavelet J b Analysis VT:Wednesday 1 September UTC 850hPa vorticity 2N S 4S6S 8S Slide 39 2N 2S 4S6S 8S Analysis VT:Wednesday 1 September UTC 850hPa vorticity Analysis VT:Wednesday 1 September UTC 850hPa vorticity S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Analysis VT:Wednesday 1 September UTC 850hPa vorticity 2N 2S4S6S8S 2N 2S4S6S8S Analysis VT:Wednesday 1 September UTC 850hPa vorticity Analysis VT:Wednesday 1 September UTC 850hPa vorticity S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Analysis VT:Wednesday 1 September UTC 850hPa vorticity 2N 2S 4S6S 8S 2N 2S 4S6S 8S Analysis VT:Wednesday 1 September UTC 850hPa vorticity Analysis VT:Wednesday 1 September UTC 850hPa vorticity 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 2N 2S 4S 6S 8S 2N 2S 4S 6S 8S Analysis VT:Wednesday 1 September UTC 850hPa vorticity 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Scale with σ 2 at each gridpoint, and for each scale.

40 N S 4S6S 8S N 2S4S6S8S 2N 2S 4S6S 8S 2N 2S 4S 6S 8S Analysis VT:Wednesday 1 September UTC 850hPa vorticity Wavelet J b Analysis VT:Wednesday 1 September UTC 850hPa vorticity 0.03 Convolve again with 0.06 Slide N S 4S6S 8S N 2S 4S6S 8S Analysis VT:Wednesday 1 September UTC 850hPa vorticity S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Analysis VT:Wednesday 1 September UTC 850hPa vorticity 2N 2S4S6S8S 2N 2S4S6S8S Analysis VT:Wednesday 1 September UTC 850hPa vorticity Analysis VT:Wednesday 1 September UTC 850hPa vorticity S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N Analysis VT:Wednesday 1 September UTC 850hPa vorticity 2N 2S 4S6S 8S 2N 2S 4S6S 8S Analysis VT:Wednesday 1 September UTC 850hPa vorticity Analysis VT:Wednesday 1 September UTC 850hPa vorticity 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 2N 2S 4S 6S 8S 2N 2S 4S 6S 8S Analysis VT:Wednesday 1 September UTC 850hPa vorticity 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N 8S 6S 4S 2S 2N ˆ ( ) Ψ n

41 Wavelet J b Add together to give: Bx = S Ψ SΣ S Ψ Sx 16W 14W 12W 10W 8W 6W 4W 2W 2E 4E 6E 8E 10E 12E 14E 16E N 7N 5N 5N N 3N 2N 2N 1N 1N S 1S 2S 2S 3S 3S S 4S 0.2 5S 5S 6S 6S 7S 7S S 8S W 14W 12W 10W 8W 6W 4W 2W 2E 4E 6E 8E 10E 12E 14E 16E Slide 41

42 Wavelet J b E-W cross-sections, and the Gaussians we wanted: Slide 42

43 Wavelet J b 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. Slide 43

44 Wavelet J b Modal variances. Solid = Gaussians. Dashed = Linear Interpolation. Slide 44

45 Wavelet J b 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): Slide 45

46 Wavelet J b We demonstrated Wavelet J b by producing structure functions of a given analytic form. However, a significant advantage of Wavelet J b 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: χ x x = Ψˆ Σχ b Σχ =Ψˆ x x But has covariance matrix = I, so (in 2d), Σ is simply the matrix of gridpoint standard deviations of This is easily generated, given a sample of bg errors. Slide 46 ( ) b ˆ Ψ ( x x ) b

47 Wavelet J b Extension of Wavelet J b to 3 spatial dimensions is straightforward. In 2 dimensions, we have: Σ where is diagonal. The diagonal elements are standard deviations. Σ x x = Ψˆ Σχ b 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. Slide 47

48 Wavelet J b Example: Horizontal and vertical Vorticity Correlations. North America Equatorial Pacific Slide 48

49 Wavelet J b Memory and Cost At first sight, the memory requirements for Wavelet J b appear high. Ψ ˆ ( n) = 0 for n> N + χ However, if p then the are strictly band-limited, and may be represented on Gaussian grids of appropriate resolution. If we arrange for: N N 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. Slide 49

50 Wavelet J b 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 increasedlength control vector. The bottom line is that Wavelet J b adds about 5% to the cost of a 4d-Var analysis. I think it s worth it! Slide 50

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 J b is expected to be implemented operationally in the 4d-Var system by the end of this year. Slide 51