Development of wavelet methodology for weather Data Assimilation

Development of wavelet methodology for weather Data Assimilation Aim´e Fournier

Thomas Aulign´e

Mesoscale and Microscale Meteorology Division National Center for Atmospheric Research

Newton Institute Mathematical and Statistical Approaches to Climate Modelling and Prediction Programme

Fournier & Aulign´ e

Wavelet weather DA

Outline 1

Introduction Purpose: Joint scale & location localization of covariance Background: Wavelet analysis of nonlinear multiscale structures

2

Analysis and modeling methods Horizontal 2D isometric-injective wavelet analysis Background-error covariance model Wavelet estimation of representativeness error

3

Results Variance maps Pseudo-single observation tests Simulated representativeness error

4

Summary Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Joint References localization Background

Purpose: Joint scale & location localization of covariance

Both NWP simulations and observations are increasing in spatio-temporal resolution and complexity, and contain more strongly nonlinear multiscale structures. Data assimilation (DA) must better represent and compute these. Enabling the background covariance to itself be both scale-selective and location-dependent should greatly improve the overall model flow dependence.

Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Joint References localization Background

Other explorations of wavelet methods Research groups Belgian RMI [Deckmyn & Berre 2004] German Weather Service [Rhodin & Anlauf 2007] M´et´eo France and Portuguese Institute of Meteorology [Belo Pereira & Berre 2006; Deckmyn & Berre 2004] UK DARC [Bannister 2007] UK MetOffice [Lorenc 2007, p. 9] Recent meetings ECMWF 2007 Joint SRNWP/Met Office/HIRLAM Workshop 2004

Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Joint References localization Background

Suggestive historical results from wavelet analysis It has long been known that spatial covariance hxm [~rm ], xn [~rn ]i and similar 2-point functions that can be trusted to decay as a function of |~rm − ~rn | no more slowly than a certain estimate, are guaranteed to yield sparse wavelet-matrix representations with a factor O[Na log Na ] (asymptotically O[Na ]) significant entries per rna discretized to Na points [Beylkin et al. 1991, pp. 155–160]. This reduces matrix-multiplication CPU cost from " d # " d # Y Y 3 O Na to O Na a=1

a=1

for Pdd coordinates, but recent methods may reduce this to a=1 O[Na ] [Beylkin & Mohlenkamp 2002]. Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Joint References localization Background

Suggestive covariance results from wavelet analysis

There is an established record for wavelet-based representation of covariance e.g., for a 143-member ensemble of N1 N2 = 128 × 128-pixel face images the “best basis” wavelets [e.g., Fournier 2000] represent 90% of the variance with 2% of the coefficients, and after appending a wavelet-approximated Karhunen-Lo`eve analysis, represent 88% with 0.06% [Wickerhauser 1994].

Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Joint References localization Background

Compression tests on power-law fields wavelet compression rates of random fields ub[x,y] such that |F[ub,kx,ky]| ∝ (1+kb+1)−1

WTPtop 256= 0.39%Wu1 explains 44.18%

u1 on 256×256 grid

b=1 b=2 b=3

−1

10

−2

10

Curves of sorted wavelet-coefficient magnitude |˜ x~ı [s] |2 † xx vs s, for Fourier spectra with random phases and WTP

u on 256×256 grid

−3

10

Wu explains 91.04%

top 256= 0.39%

2

b

2

b

|W[u ]| /||u ||

2

2

X −4

10

u3 on 256×256 grid

~r

~

x[~r ]ei~r •ξ =: xˆξ~ ∝

1 ~ 1+b 1 + |ξ|

WTPtop 256= 0.39%Wu3 explains 99.10%

for b = 1 (blue), 2 (green) and 3 (red).

−5

10

|˜ x~ı [s] | decays more quickly for smoother fields (larger b). −6

10

1

8

64

512 rank

4096 32768

Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods TResults Summary Joint References localization Background T u1 on 256×256 grid W Ptop grid Wu1 explainsW44.18% u2 on 256×256 Ptop 256= 0.39%Wu2 explains 91.04% 256= 0.39% −4 10 om fields ub[x,y] T T +kb+1)−1 u1 on 256×256 grid WTPtop grid Wu1 uexplains u2 on 256×256 W44.18% Ptop grid Wu2 explainsW91.04% on 256×256 P 256= 0.39% 256= 0.39%

Thresholded reconstruction of power-law fields

−2

3

−3

b=1 b=2 b=3

top 25

−5

10

T

T T W Ptop grid Wu1 explains u2 on 256×256 W44.18% Ptop grid Wu2 explainsW91.04% u on 256×256 P Wu explains 99.10% 256=−60.39% 256= 0.39% 3 top 256= 0.39% 3 10 1 8 64 512 4096 32768 rank T u2 on 256×256 grid WTPtop grid Wu2 explainsW91.04% u on 256×256 P Wu explains 99.10% 256= 0.39%

d

−4

3

top 256= 0.39%

3

−5

512 rank

4096 32768 T random-phase power-law fields Maps in physical spaceW91.04% of W Ptop grid Wu2 explains u3 on 256×256 Ptop 256= 0.39%Wu3 explains 99.10% 256= 0.39%

d

−6

1

8

64

T for b = 1 (top left), 2 (top center) and 3 (top right). √ 512 4096 32768 Reconstruction from the largest N1 N2 (0.20%) coefficients represents 44% rank T on 256×256 Ptop99% Wu3 explains 99.10% (bottomu3top), 91%grid (bottom center) Wand (bottom right) of the spatial 256= 0.39% variance †xx.

Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Horizontal References 2D isometric-injective wavelet analysis Background

Isometric-injective wavelet analysis Isometric-injective wavelet analysis in 1D transforms a sequence xn = x[rn ] of any number N of function values at ˜ >N uniform coordinates rn , into a sequence of N scaling-function and wavelet coefficients x˜i :=

N−1 X

N Wn,i xn ,

and back again, xn =

n=0

˜ N−1 X

N ˇ n,i W x˜i ,

i=0

such that

˜ N−1 X i=0

x˜i2 =

N−1 X

xn2 .

n=0

The wavelets used here eliminate the usual severe restriction to a dyadic number Na = 2La ∈ {2, 4, 8, . . .} of points along each coordinate rna . Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Horizontal References 2D isometric-injective wavelet analysis Background

More wavelet-coefficient properties Whereas the index n only describes spatial location, the index i ∈ {¯ı` , . . . ¯ı` +ˇı` − 1} describes both wavenumber band-center and bandwidth ∝ ˇı` and spatial location ∝ (i − ¯ı` )/ˇı` , where i` := b(i`−1 + P)/2c ∈ {iL , . . . i1 }, ˇ ˇı` := b(i`−1 + P)/2c ∈ {ˇıL , . . . ˇı1 } and ¯ı` := iL +

L X

ˇık

k=`+1

˜ := ¯ı0 into L levels, define a dyadic partition of N =: i0 and N ˇ and P  N and P  N are the filter lengths. Furthermore the smoothness of x[r ] is precisely characterized by the collection of x˜i when (i − ¯ı` )/ˇı` corresponds to r . Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Horizontal References 2D isometric-injective wavelet analysis Background

Wavelets of different filter lengths 7C6[25] 7C30[85]

7CP[i] wavelets of filter length P, scale ∝ s ≡ 2−⎣log2i⎦ and location ∝ i/s

0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 32

34

36 XLAT

38

40

42

ˇ = 6 (blue) is less differentiable and has Example wavelets: x˜i = δi,25 , P = P ˇ = 30 (green). fewer vanishing moments than x˜i = δi,85 , P = P Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Horizontal References 2D isometric-injective wavelet analysis Background

Wavelet transform as a hierarchy ˜ ˇ Nx x ˜ ≡ †xWN ∈ †RN and x = W ˜ ∈ RN are computed by a sparse ˜ by post-multiplying the row recursive hierarchy on L levels; e.g., †x † x by †

WN := {C1 {C2 · · · {CL−1 {CL , DL }, DL−1 } · · · , D2 }, D1 } ˜

N N N = {V N } ∈ RN×N , ˜ 1 , . . . V iL , W 1 , . . . W N−i L

N which exhibits the discrete scaling functions V N i and wavelets W i as appropriate products of coarsening (low-pass) and detailing ˇ i`−1 סı` with (high-pass) filters C` ∈ {0, h}i`−1 ×i` and D` ∈ {0, ±h} i`−1 † ˇ ˇ orthogonal rows: {C` , D` } {C` , D` } = I , implying N

ˇ = IN . WN †W Multidimensional functions †x ∈ †RN1 ···Nd are transformed using tensor products W := WN1 ⊗ · · · ⊗ WNd etc. Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Horizontal References 2D isometric-injective wavelet analysis Background

Horizontal control-variable transform The background cost function w.r.t. the difference x between model and background states, or control variable v, is Jb := †xB−1 x ≈ †vv, where B is the background-error covariance. As usual, let x =: Uv, so that B ≈ U †U. This may be ˇ acomplished byqU := P−1 WΣ, where P is an arbitrary factor, ˜ a −1 N 2 h˜ x~ı i and x ˜ := †WPx. Σ = diagia =0 Proof: ˇ 2 ≈ h˜ ˇ †P−1 = hx †xi =: B. U †U = P−1 W(Σ x †x ˜ i) †W Corollary: Jb ≈ †v †U(U †U)−1 Uv = †vv if v ∈ col †U. q a −1 In practice P = diagN p / hx~n2 i works well, where p~n2 is na =0 ~n the cell area at point ~r~n . Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Horizontal References 2D isometric-injective wavelet analysis Background

A criterion to distinguish resolution error Given an observed state y and its model prediction ξ := Hx, let us use their wavelet coefficients y ˜ and ξ˜ to find a “representative” basis of y − ξ. Let y˜~ı1 [s] and ξ˜~ı2 [s] be the s largest-magnitude coefficients, for Q 1 ≤ s ≤ da=1 Na i.e., the s fewest values representing the greatest contribution to the spatial variance. Choose thresholds τ1 = |˜ y~ı1 [s ? ] |, τ2 = |ξ˜~ı2 [s ? ] | by requiring significant logarithmic curve separation v u X ξ˜ u1 s ~ ı [s] 2 ? Yq2 ∀s > s , where Ys := log |Ys | > t . y˜~ı1 [s] s q=1

Reconstruct “representative” y ? and ξ ? by masking all |˜ y~ı| < τ1 and |ξ˜~ı| < τ2 . Fournier & Aulign´ e

Wavelet weather DA

XLA

40 Introduction Analysis and modeling methods Results Summary Variance References maps Pseudo-single observation tests Simulated rep 35°N

New B represents heterogeneity

20 0

"true" Σ2=diag[B≡〈tu′tu′T〉] (

0:

8: 9e+01)

diag[WTdiag[WBWT]W]

diag[ΣWTdiag[WΣ−1BΣ−1WT]WΣ] 80

40°N XLAT

60 40

35°N 20

105°W 100°W XLONG

35°N XLONG

40°N

35°N XLONG

40°N

0

Left: map of “true” unbalanced-temperature variance 1 and latitude r 2 . Σ[~rm ]2 := Bm,m = hTu0 [~rm ]2 i vs longitude rm m Center: map of wavelet Bm,m with P = I. Right: map of wavelet Bm,m with P = diag~rm Σ[~rm ]−1 as above. Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Variance References maps Pseudo-single observation tests Simulated rep

New B represents heterogeneity

New B (center, 7C30 wavelet) represents heterogeneity (left, ensemble), unlike homogeneous recursive filters (right). T -response to a 1 ± 1K observation at {100◦ W, 35◦ N}, η = 0.28, using 30-member N1 N2 = 351 × 451, 5-level dataset. Note the multiscale anisotropic features such as the SW-NE structures over N Texas. Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Variance References maps Pseudo-single observation tests Simulated rep

Represents heterogeneity but still localizes

Response to observation at {100◦ W, 39◦ N}, η = 0.28. Wavelet B costs only O[N1 N2 ] in memory and cpu. Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Variance References maps Pseudo-single observation tests Simulated rep

Also represents isotropy

Response to observation at {104◦ W, 39◦ N}, η = 0.28. The overall horizontal scale is an adjustable parameter len scaling = 0.9 (for RF), nb = L = 7 (for wavelets) and alpha corr scale = 200km (for ensemble), each of which has only been roughly calibrated. Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Variance References maps Pseudo-single observation tests Simulated rep

One more pseudo-single observation test

Response to observation at {104◦ W, 35◦ N}, η = 0.28.

Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Variance References maps Pseudo-single observation tests Simulated rep

Wavelet analysis of simulated satellite radiances

.

Top left: map of full observed outgoing longwave radiation (OLRO, y ). Top center: log10 |˜ y~ı| vs i1 , i2 (log scale). Tick marks are ¯ı` for ` = 0, 1 . . . 10. N1 N2 = 951 × 882. Top right: log10 |˜ y~ı| thresholded as explained earlier (536 = 0.064% of coefficients). & Aulign´ e reconstruction. Wavelet weather DA Bottom left: map Fournier of thresholded

Introduction Analysis and modeling methods Results Summary Variance References maps Pseudo-single observation tests Simulated rep

Thresholding on wavelet-coefficient magnitude

For y (red), ξ (green) and y − ξ (blue), curves of sorted wavelet(solid) and Fourier- (dashed) coefficients. The ∗ marks the threshold rank s ? as explained earlier. Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary Variance References maps Pseudo-single observation tests Simulated rep

Representative differences

Map of y − ξ before (left) and after (right) thresholding.

Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary References

Summary Wavelets have great potential for DA applications, based on their long history of use in related areas, as well as recent applications to DA (see Further Reading, below). At NCAR/MMM we’re using tensor-product wavelets that allow any size of data, and conserve the sum-of-squares to machine precision. We have implemented a new background-error covariance matrix B by incorporating wavelets into the horizontal control-variable transform. PSOTs indicate an advantage in regard to representing heterogeneity and anisotropy. We have also experimented with wavelet thresholding as a tool to address representativeness error.

Fournier & Aulign´ e

Wavelet weather DA

Introduction Analysis and modeling methods Results Summary References

For Further Reading R.N. Bannister, 2007: “Can wavelets improve the representation of forecast error covariances in variational data assimilation?”, Mon. Wea. Rev. 135, 387–408. M. Belo Pereira & L. Berre, 2006: “The use of an ensemble approach to study the background error covariances in a global NWP model”, Mon. Wea. Rev. 134, 2466–2489. G. Beylkin & M.J. Mohlenkamp, 2002: “Numerical operator calculus in higher dimensions”, Proc. Nat. Acad. Sci. 99, 10246–10251. G. Beylkin, R. Coifman & V. Rokhlin, 1991: “Fast wavelet transforms and numerical algorithms I”, Comm. Pure Appl. Math. 44, 141–183. A. Deckmyn & L. Berre, 2004: “Using wavelets for the modelling of LAM background error covariances” [Joint SRNWP/Met Office/HIRLAM Workshop 2004]. ECMWF Workshop on flow-dependent aspects of data assimilation, 2007/6. A. Fournier, 2000: “Introduction to orthonormal wavelet analysis with shift invariance: Application to observed atmospheric blocking spatial structure”, J. Atmos. Sci. 57, 3856–3880. A.K. Gupta & D.K. Nagar, 1999: Matrix Variate Distributions, CRC Press, 367 pp. D.T. Hristopulos, 2002: “New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity”, Stochastic Environmental Research and Risk Assessment 16, 43–62. A.C. Lorenc, 2007: “Ideas for adding flow-dependence to the Met Office VAR system” [ECMWF 2007]. A. Rhodin & H. Anlauf, 2007: “Representation of inhomogeneous, non-separable covariances by sparse wavelet-transformed matrices” [ECMWF 2007]. Joint SRNWP/Met Office/HIRLAM Workshop on Variational Assimilation: Towards 1-4km Resolution, 2004/11. M.V. Wickerhauser, 1994: Adapted wavelet analysis from theory to software, A.K. Peters, 486 pp.

Fournier & Aulign´ e

Wavelet weather DA