w w w .ec.gc.ca Modelling of Ensemble Covariances Meteorological Research Division Environment Canada Mark Buehner BIRS Data Assimilation Workshop February 2008 Background • Canadian NWP centre currently has both a global 4D-Var (for deterministic forecasts) and EnKF (for probabilistic forecasts) • Provides good opportunity to compare two approaches and to evaluate use of flow-dependent ensemble background-error covariances in a variational system • Current approaches for modelling background-error covariances in 4D- Var and EnKF represent two extreme cases: • 4D-Var: horizontally homogeneous, nearly temporally static • EnKF: independently estimated at each grid-point and analysis time • Unlikely that either approach is optimal, best approach probably somewhere in between • Therefore, both systems could be improved with a more general approach to covariance modelling (focus is on correlations in following) 1
Comparison of Covariances used in 3/4D-Var and EnKF 55 o N 0.0 1.2 1.05 0.1 50 o N 0.9 0.2 0.75 • Corrections to T and 0.6 0.3 45 o N 0.45 0.3 0.4 UV in response to a 0.15 40 o N eta 3D-Var 0.5 0 −0.15 0.6 single T obs near the 35 o N −0.3 0.7 −0.45 −0.6 0.8 surface 30 o N −0.75 0.9 −0.9 −1.05 25 o N 1.0 165 o W 155 o W 145 o W 135 o W 165 o W 155 o W 145 o W 135 o W • Black contours show 55 o N 0.0 1.2 1.05 0.1 background T 50 o N 0.9 4D-Var 0.2 0.75 0.6 0.3 45 o N 0.45 0.3 0.4 (obs at • EnKF error 0.15 40 o N eta 0.5 0 −0.15 0.6 covariances from end of 6h 35 o N −0.3 0.7 −0.45 128 ensemble −0.6 0.8 30 o N −0.75 window) 0.9 −0.9 members −1.05 25 o N 1.0 165 o W 155 o W 145 o W 135 o W 165 o W 155 o W 145 o W 135 o W 55 o N 0.0 1.2 1.05 0.1 50 o N 0.9 0.2 0.75 0.6 0.3 45 o N EnKF 0.45 0.3 0.4 0.15 40 o N eta 0.5 0 error cov. −0.15 0.6 35 o N −0.3 0.7 −0.45 −0.6 0.8 30 o N −0.75 −0.9 0.9 −1.05 25 o N 1.0 165 o W 155 o W 145 o W 135 o W 165 o W 155 o W 145 o W 135 o W Outline • Demonstrate complementary effects of spatial and spectral localization applied to ensemble-based error correlations • Implementation issues in realistic NWP variational assimilation systems • Simpler approach to incorporate limited amount of heterogeneity 2
Sampling error in ensemble-based error covariances • Test ability of ensemble-based covariances to reproduce “true” covariances as function of ensemble size and spatial localization • Spatial localization: B grid_loc = B samp ° L grid where L is a simple “correlation” matrix with monotonically decreasing values as a function of separation distance used to do the localization Use operational B matrix as “truth” (homog/isotr. correlations for main • analysis variables), generate ensemble members: e k = B 1/2 ε k where ε k = N (0, I ) • Final value of J o (all operational data) used as simple measure of accuracy of ensemble-based covariances: ability to fit to observations Effect of ensemble size and spatial localization on sampling error Final value of Jo (normalized by value from using “true” B ) as a function of ensemble size and localization radii: Localization radii Localization radii Ensemble size Ensemble size ∞ Horizontal Horizontal Vertical Vertical 32 32 128 128 512 512 (km) (km) (ln(P)) (ln(P)) ∞ ∞ ∞ ∞ 3.15 3.10 2.98 1.00 ∞ ∞ 10 000 10 000 2.72 2.30 1.77 0.96 ∞ ∞ 2 800 2 800 2.09 1.46 1.12 0.84 10 000 10 000 2 2 2.23 1.73 1.31 0.94 2 800 2 1.47 1.11 0.97 0.82 2 800 2 3
Spectral correlation localization • What happens if same type of localization is applied to the correlation matrix in spectral space? – A diagonal correlation matrix in spectral space corresponds with globally homogeneous correlations – Represents an extreme case of spectral correlation localization – More moderate spectral localization should result in correlations with an intermediate amount of heterogeneity • Was shown that localization of correlations in spectral space (multiplication) is equivalent with spatial averaging of correlations in grid-point space (convolution) • Averaging of correlations over a local region should be better than either globally homogeneous or independent for each grid point: – reduced sampling error through averaging, but – still maintain most of spatial/flow dependence of correlations Spectral correlation localization • Localization of correlations in spectral space (multiplication): S B spec_loc S T = ( S B samp S T ) ° L spec where S is spectral transform, L spec is a “correlation” matrix with monotonically decreasing values as a function of the absolute difference in wavenumber • Spatial averaging of correlations in grid-point space (convolution): B spec_loc (x 1 ,x 2 ) = ∫ B samp (x 1 +s,x 2 +s) L spec (s) ds where L spec = (S -1 L spec S -T ) assuming L spec is homogeneous and isotropic in wavenumber space 4
Spatial and spectral correlation localization • Idealized 1-D example using prescribed “true” heterogeneous correlations and estimated correlations from 30 realizations • Spatial localization cannot improve short-range correlations • Spectral localization cannot remove long-range spurious correlations • Combination seems to give best result Original Spatial localization Spectral localization Combined 1 1 1 1 (a) (b) (c) (d) correlation 0.5 0.5 0.5 0.5 0 0 0 0 −0.5 −0.5 −0.5 −0.5 0 50 100 0 50 100 0 50 100 0 50 100 grid−point index grid−point index grid−point index grid−point index Spatial and spectral correlation localization 70 0.3 (a) spatial localization radius 60 0.25 • For this example, a unique optimal combination of 50 0.2 spatial and spectral localization exists (minimum 40 rms error of correlations) 0.15 30 0.1 • Spectral localization dramatically improves local 20 estimate of correlation length scale: (-d 2 C/dx 2 ) -1/2 0.05 10 0 0 • With too much spectral localization, start to loose 0 5 10 15 20 25 heterogeneity (dashed) spectral localization radius Original Spatial localization Spectral localization Combined 10 10 10 10 (a) (b) (c) (d) correlation length scale 8 8 8 8 6 6 6 6 4 4 4 4 2 2 2 2 0 0 0 0 0 50 100 0 50 100 0 50 100 0 50 100 grid−point index grid−point index grid−point index grid−point index 5
Ensemble-based error covariances in 3D-Var • Implementation in preconditioned variational analysis: • no localization: elements of control vector determine global contribution of each ensemble member to the analysis increment: ∆ x = ∑ (e k – <e>) ξ k ( ξ k is a scalar) • spatial localization: elements of control vector determine local contribution of each ensemble member to the analysis increment: ∆ x = ∑ (e k – <e>) o (L 1/2 ξ k ) ( ξ k is a vector) grid • in each case, J b is Euclidean inner product: J b = 1/2 ξ T ξ • can also combine with standard B matrix: ∆ x = β 1/2 ∑ (e k – <e>) o (L 1/2 ξ k ) + (1- β ) 1/2 B 1/2 ξ HI grid Spectral correlation localization Homogeneous Spectral localization Heterogeneous • Apply in variational system, similar 0.0 0.0 0.0 (a) (b) (c) technique as spatial localization 0.1 0.1 0.1 0.2 0.2 0.2 • Elements of control vector 0.3 0.3 0.3 determine local contribution (in 0.4 0.4 0.4 eta 0.5 0.5 0.5 spectral space) to analysis 0.6 0.6 0.6 increment: 0.7 0.7 0.7 0.8 0.8 0.8 ∆ x = S -1 ∑ (S(e k – <e>)) o (L 1/2 ξ k ) 0.9 0.9 0.9 spec 1.0 1.0 1.0 150 o E 170 o E 170 o W 150 o W 150 o E 170 o E 170 o W 150 o W 150 o E 170 o E 170 o W 150 o W Extra tropics 0.0 0.0 0.0 (a) (b) (c) • Spectral correlations forced to zero 0.1 0.1 0.1 beyond total wavenumber 0.2 0.2 0.2 0.3 0.3 0.3 difference of 10 (Gaussian-like 0.4 0.4 0.4 function) eta 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 1.0 1.0 1.0 165 o E 175 o E 175 o W 165 o W 165 o E 175 o E 175 o W 165 o W 165 o E 175 o E 175 o W 165 o W Tropics 6
Spectral correlation localization • Still need to apply spatial localization to damp long-range spurious correlations, however • Current approach may become prohibitively expensive (memory or time) when combining spatial and spectral localization Homogeneous Spectral localization Heterogeneous 45 o N 45 o N 45 o N (a) (b) (c) 30 o N 30 o N 30 o N 15 o N 15 o N 15 o N EQ EQ EQ 15 o S 15 o S 15 o S 30 o S 30 o S 30 o S 45 o S 45 o S 45 o S 135 o E 165 o E 165 o W 135 o W 135 o E 165 o E 165 o W 135 o W 135 o E 165 o E 165 o W 135 o W Combining spatial and spectral correlation localization • transform ensemble members into functions of space and scale: e k (x) = ∑ n e k (n) exp(i2 π nx) = ∑ n e k (x,n) (but too big to store) both control vector ( ξ ) and L depend on space and scale, but L could • be separable: L spec,grid (x 1 ,x 2 ,n 1 ,n 2 ) = L grid (x 1 ,x 2 ) L spec (n 1 ,n 2 ) • follow same approach as before : 1/2 ξ k (x,n) ) ∆ x = ∑ k ∑ n (e k (x,n) – <e(x,n)>) o ( L spec,grid J b = 1/2 ξ T ξ • J b is still the same form : • some similarities with wavelet approach are evident 7
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