Spectral Ensemble Kalman Filters Jan Mandel 12 , Ivan Kasanick y 2 , - PowerPoint PPT Presentation

Spectral Ensemble Kalman Filters Jan Mandel 12 , Ivan Kasanick´ y 2 , Martin Vejmelka 2 , Kryˇ stof Eben 2 , Viktor Fugl´ ık 2 , Marie Turˇ a 2 , Jaroslav Resler 2 , and ciˇ cov´ s 2 Pavel Juruˇ 1 University of Colorado Denver 2 Academy of Sciences of the Czech Republic European Meteorological Society Annual Meeting Prague, October 6, 2014 Supported by the Czech Science Foundation grant GA13–34856S and the U.S. National Science Foundation grant DMS–1216481.

Ensemble Kalman Filter ◮ Incorporate the observation HX ≈ Y in the probability distribution of the state X , represented by the forecast ensemble X f ,1 , . . . , X f , N . ◮ Analysis step: X a , i = X f , i − P N H ∗ ( HP N H ∗ + R ) − 1 ( HX f , i − Y i ) where the superscript ∗ denotes the transpose i = 1 ( X f , i − ¯ X f )( X f , i − ¯ X f ) ∗ is the ensemble N − 1 ∑ N 1 P N = covariance X f = 1 i = 1 X f , i is the ensemble mean ¯ N ∑ N Y i = Y + ǫ i , ǫ i ∼ N ( 0 , R ) are the perturbed data vectors H is the observation operator

The need for localization ◮ Ensemble covariance is low rank N − 1 ⇒ has large numbers far away from the diagonal ◮ X is a random field and Cov ( X ( x ) , X ( y )) ≈ 0 for large distance y − x ◮ Sampling error: P N → P only asymptotically for N → ∞ , but real ensembles are small: N ≈ 20 − few 100 max ◮ Tapering fix: P N → P N ◦ T , multiply term-by-term by a fixed tapering matrix to force small entries when x − y large. But how far away exactly? Depends on N , for large N the convergence P N → P should take over ◮ Expensive/hard to parallelize implementation, banded/sparse matrix operations.

Covariance of random fields ◮ Covariance between the values at two points x , y is the covariance function f x ( x − y ) = Cov ( X ( x ) , X ( y )) ◮ If the covariance function does not change with location , the covariance matrix is diagonal in the Fourier basis u 1 , . . . , u n (sines, cosines or complex exponential)   λ 1 ... Cov ( X ) = F ∗ D = F Cov ( X ) F ∗   F ,   λ n � �� � D ◮ Multiplication by F = [ u 1 , . . . , u n ] ∗ is a discrete Fourier transform ◮ Other orthogonal bases (e.g., wavelets) and frames allow variability with location.

Analysis of spectral diagonal covariance Theorem . Suppose Cov ( X ) after the transformation is diagonal:   λ 1 F Cov ( X ) F ∗ = ...     λ n ◮ Transform the sample covariance P N in the same way, keep only the diagonal part D . ◮ D transformed back is better than P N in the Frobenius norm: n F = 2 E � F ∗ DF − Cov ( X ) � 2 λ 2 ∑ i N i = 1 n F = 2 i + 1 < E � P N − Cov ( X ) � 2 λ 2 ∑ N ∑ λ i λ j N i = 1 i � = j

Spectral EnKF - Simple case: single variable, all observed ◮ Assuming observation operator H = I , data covariance R = I ◮ Compute the diagonal D of the covariance of the transformed � FX f ,1 , . . . , FX f , N � forecast ensemble ◮ Analysis update of the transformed forecast ensemble becomes multiplication by a diagonal matrix: � X f , i − Y i � FX a , i = FX f , i − D ( D + I ) − 1 F ◮ Inverse transform the analysis ensemble: X a , i = F ∗ � FX a , i � , i = 1, . . . , N

Spectral EnKF - More general state and and observation ◮ Low-dimensional and scalar observations: ◮ Use spectral diagonal covariance F ∗ DF in place of the ensemble covariance ◮ Few matrix-vector multiplications to set up ◮ Only need to invert a small matrix or a scalar ◮ Multiple variables on the same grid, one completely observed: ◮ Spectral diagonal crosscovariances between the variables ◮ Multiple variables on different grids, same dimension: ◮ Interpolate all variables to the same grid ◮ Extend the analysis back to the original grids ◮ Both 2D and 3D variables ◮ Treat 2D layers as separate variables

Spectral EnKF - Part of a variable observed OPERA radar rain data Coiflet 5 wavelet ◮ Extend the data to whole domain by zeros ◮ Augment the state by a copy of the variable with 0 outside of the data region ◮ Wavelets, not Fourier, for locality

Shallow water equations ◮ Variables: fluid depth, horizontal velocities ◮ Equations: conservation of mass and horizontal momenta ◮ 64 x 64 grid with step 150 km, depth 10 km ◮ Background σ =100 m, spin-up 15 m, time step 1 s, assimilation cycle 10 seconds

Shallow water equations - Assimilation cycles Mean RMSE from 10 repetitions Water level X momentum Y momentum ensemble size: 16 ensemble size: 16 ensemble size: 16 17 4055 4035 16 4050 4030 15 4045 4025 14 4040 4020 RMSE RMSE RMSE 13 4035 4015 12 4030 4010 No assim. No assim. No assim. 11 4025 EnKF EnKF EnKF DST DST DST 4005 10 4020 DCT DCT DCT DWT DWT DWT 9 4015 4000 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 Assimilation cycle Assimilation cycle Assimilation cycle DST=discrete sine transform, DCT=discrete cosine transform, DWT=wavelet transform, Coiflet 5

Shallow water equations - Convergence with ensemble size Mean RMSE from 10 repetitions Water level height X momentum Y momentum 35 8500 8500 8000 8000 30 7500 7500 7000 7000 25 6500 6500 RMSE RMSE RMSE 6000 6000 20 5500 5500 No assim. No assim. No assim. 5000 5000 15 EnKF EnKF EnKF DST DST DST 4500 4500 DCT DCT DCT DWT DWT DWT 10 4000 4000 5 10 15 5 10 15 5 10 15 Ensemble size Ensemble size Ensemble size DST=discrete sine transform, DCT=discrete cosine transform, DWT=wavelet transform, Coiflet 5

WRF - assimilation setup ◮ WRF 3.6, one domain with resolution 27 x 27 km covering Middle Europe, 39 vertical levels ◮ a common WRF configuration (NOAH Land-surface model, Lin et al. microphysics, Dudhia shortwave radiation, Yonsei PBL) ◮ initial ensemble: perturbations of a deterministic GFS-initialized run ◮ 6 hours spin-up for downscaling ◮ each 2D layer of 3D WRF variables is a separate variable for assimilation ◮ wind interpolated to cell centers for the same dimension ◮ assimilation every hour (6 assimilation cycles)

WRF - assimilation setup ◮ background covariance constructed from a one month simulation of WRF on the same domain, (NMC method, WRFDA routine gen be) ◮ corresponding perturbations by means of WRFDA da wrfvar in randomcv regime V10 y - wind component for two initial ensemble members

WRF - assimilation of 2D potential temperature layer RMSE of 3D WRF variables Potential temperatue Perturbation dry air mass X−wind component Perturbation geopotential 1 70 1.4 55 No assim. No assim. No assim. 1.35 0.95 EnKF EnKF EnKF 65 50 SDEnKF SDEnKF 1.3 SDEnKF 0.9 60 1.25 0.85 45 RMSE [m2/s2] RMSE [ms−1] 1.2 RMSE [Pa] 55 RMSE [K] 0.8 1.15 40 0.75 50 1.1 0.7 35 1.05 45 0.65 1 No assim. 30 40 EnKF 0.6 0.95 SDEnKF 0.55 35 0.9 25 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 Assimilation cycle Assimilation cycle Assimilation cycle Assimilation cycle ◮ 9 ensemble members, 10th member = the truth ◮ observation = the lowest 2D layer of the potential temperature T in the 10th ensemble member, σ = 0.316 K

WRF - assimilation of a station temperature Difference between analysis and forecast mean in the lowest 2D layer of the potential temperature T Standard EnKF Spectral diagonal, sine transform ◮ ensemble with 10 members, observation = T in the middle of the grid, mean forecast + 1 K, σ = 0.2 K

Conclusions ◮ Covariance is diagonal or close in the spectral domain ◮ Fast - FFT or wavelet transform, diagonal matrices ◮ FFT is better in the spatially homogeneous case ◮ Wavelets are better in the spatially nonhomogeneous cases ◮ Important kinds of observation equations supported ◮ Automatic, no tuning of covariance distance ◮ Needs only very small ensembles, about 10 ◮ Preserves convergence in the large ensemble limit Future work: ◮ Better understanding of cross-covariances ◮ Mathematical analysis of more general cases ◮ Frames, better 2D wavelets

References Jonathan D. Beezley, Jan Mandel, and Loren Cobb. Wavelet ensemble Kalman filters. In Proceedings of IEEE IDAACS’2011, Prague, September 2011 . Jan Mandel, Jonathan D. Beezley, and Volodymyr Y. Kondratenko. Fast Fourier transform ensemble Kalman filter with application to a coupled atmosphere-wildland fire model. In Proceedings of MS’10 , World Scientific, 2010. Olivier Pannekoucke, Lo¨ ık Berre, and Gerald Desroziers. Filtering properties of wavelets for local background-error correlations. Q. J. Royal Meteor. Soc. , 133:363–379, 2007. Reinhard Furrer and Thomas Bengtsson. Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. J. Multivariate Anal. , 98:227–255, 2007.

Tom Aulign´ e and Gael Descombes. Background error covariance and GEN BE. 2014 GSI Community Tutorial . Jan Mandel, Loren Cobb, and Jonathan D. Beezley. On the convergence of the ensemble Kalman filter. Appl. Math. , 56:533–541, 2011.