EnKF同化的背景误差协方差矩阵局地化对比研究

doi:10.11867/j.issn.1001-8166.2014.10.1175

在集合数据同化中,背景场误差的协方差估计特别重要。通常有限个成员的集合在估计背景误差协方差矩阵时会引入伪相关,从而造成协方差被低估、滤波发散。虽然协方差膨胀的经验性方法能一定程度缓解协方差被低估的问题,但不能消除协方差的伪相关问题。因此,结合EnKF方案探讨2种消除伪相关的局地化方法(协方差局地化方法和局地分析方法),分析这2种局地化方法对背景误差协方差矩阵、增益矩阵、集合转换矩阵以及同化结果的影响。实验结果表明：局地化方法不仅能消除背景误差协方差矩阵的伪相关,还可以增加背景误差协方差矩阵的秩;在“弱”同化强度下,2种局地化方法的增益矩阵和集合转换矩阵相等;随着同化强度的增大,增益矩阵和集合转换矩阵的差异会变大;在不同的同化强度下,2种局地化方法各具特色,相对而言,协方差局地化方法在更新集合均值和集合扰动上具有较强的鲁棒性。研究结论有助于背景场误差协方差的精细分析和估计。

关键词: EnKF, 协方差局地化, 局地分析, 伪相关

In ensemble data assimilation, the estimate of background error covariance is particuarly important. In general, the use of a finite ensemble size for estimating the background error covariance matrix easily introduces spurious correlations, which leads to the underestimation of covariance and filter divergence. Covariance inflation is an empirical method of correcting the underestimation of background error covariance, but it does not help to solve the problem of long-range spurious correlations. Therefore, based on the EnKF scheme, we explored two localization methods to eliminate the spurious correlations, which were the covariance localization method and the local analysis method. We analyzed their impacts on the background error covariance matrix, gain matrix, ensemble transform matrices and data assimilation results. The experimental results have been obtained. That is, the localization method not only can remove the spurious covariance in the background error covariance matrix, but also can increase the rank of the matrix. In a “weak” assimilation, the gain matrix and ensemble transform matrices of two methods are very close, but the differences of the gain matrix and ensemble transform matrices become more evident with the increase of assimilation strength. Under the different strength of assimilation, two localization methods have their own characteristics, and relatively the covariance localization method has stronger robustness on the update of ensemble mean and ensemble anomalies. This study is very helpful for the fine analysis and estimate of the background error covariance.

Key words: EnKF, Covariance localization, Local analysis, Spurious correlations.

中图分类号:

P237

图1 EnKF同化过程和效果图

Fig.1 The process and result of data assimilation based on EnKF

图2 CL对背景误差协方差矩阵 P^f的影响色度条表示矩阵元素的取值范围

Fig.2 CL’s impact on the background error covariance matrix P^f Colorbar in figure 2 denotes the values range of matrix elements

图 3 舒尔积前后背景误差协方差矩阵 P^f的特征值光谱

Fig.3 The eigenvalues spectrum of the background error covariance matrix P^f before and after Schur Product

图4 LA对背景误差协方差矩阵 P^f的影响( t=50, i=50) 色度条表示矩阵元素的取值范围

Fig.4 LA's impact on the background error covariance matrix P^f ( t = 50, i= 50) colorbar in figure 4 denotes the values range of matrix elements

图5 3种增益矩阵 K对比图( R_{obs_variance}= 10) 色度条表示矩阵元素的取值范围

Fig. 5 The contrast of three kinds of gain matrix K ( R_{obs_variance} = 10) Colorbar in figure 5 denotes the values range of matrix elements

图6 3种增益矩阵 K对比图( R _{obs_variance}= 10 ^-1) 色度条表示矩阵元素的取值范围

Fig.6 The contrast of three kinds of gain matrix K ( R _{obs_variance} = 10 ^-1) Colorbar in figure 6 denotes the values range of matrix elements

图7 3种增益矩阵 K对比图( R _{obs_variance}= 10 ^-4) 色度条表示矩阵元素的取值范围

Fig.7 The contrast of three kinds of gain matrix K ( R _{obs_variance} = 10 ^-4) Colorbar in figure 7 denotes the values range of matrix elements

图8 集合转换矩阵 T对比图( R _{obs_variance}=10) 色度条表示矩阵元素的取值范围

Fig.8 The contrast of the ensemble transformation matrix T( R _{obs_variance}= 10) Colorbar in figure 8 denotes the values range of matrix elements

图9 集合转换矩阵 T对比图( R _{obs_variance}= 10 ^-1) 色度条表示矩阵元素的取值范围

Fig.9 The contrast of the ensemble transformation matrix T( R _{obs_variance}= 10 ^-1) colorbar in figure 9 denotes the values range of matrix elements

图10 集合转换矩阵 T对比图( R _{obs_variance}= 10 ^-4) 色度条表示矩阵元素的取值范围

Fig.10 The contrast of the ensemble transformation matrix T( R _{obs_variance}= 10 ^-4) colorbar in figure 10 denotes the values range of matrix elements

图 11 全局集合转换矩阵 T对比图( R_{obs_variance}= 10和 R_{obs_variance}= 10 ^-4) 色度条表示矩阵元素的取值范围

Fig.11 The contrast of the Global ensemble transformation matrix T( R_{obs_variance}= 10 and R_{obs_variance}= 10 ^-4) colorbar in figure 11 denotes the values range of matrix elements

图12 基于EnKF的同化效果图

Fig.12 The data assimilation effect based on EnKF

图13 基于EnSRF的同化效果图

Fig.13 The data assimilation effect based on EnSRF

图14 基于DEnKF的同化效果图

Fig.14 The data assimilation effect based on DEnKF

[1]	Ma Jianwen, Qin Sixian. The research status review of data assimilation algorithm[J]. Advances in Earth Science, 2012, 27(7): 747-757.
	[马建文,秦思娴.数据同化算法研究现状综述[J].地球科学进展,2012, 27(7): 747-757.]
[2]	Evensen G. Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics[J]. Journal of Geophysical Research: Oceans (1978-2012), 1994, 99(C5): 10143-10162.
[3]	Xiong Chunhui, Zhang Lifeng, Guan Jiping, et al. Development and application of ensemble-variational data assimilation methods[J]. Advances in Earth Science, 2013,28(6):648-656.
	[熊春晖, 张立凤, 关吉平, 等. 集合—变分数据同化方法的发展与应用[J]. 地球科学进展, 2013, 28(6): 648-656.]
[4]	Petrie R. Localization in the Ensemble Kalman Filter[D]. London: University of Reading, 2008.
[5]	Liu Yanhua, Zhang Shuwen, Mao Lu, et al. An evaluation of simulated and estimated land surface states with two different models[J]. Advances in Earth Science, 2013,28(8):913-922.
	[刘彦华, 张述文, 毛璐, 等. 评估两类模式对陆面状态的模拟和估算[J]. 地球科学进展, 2013, 28(8): 913-922.]
[6]	Hamill T M, Whitaker J S, Snyder C. Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter[J]. Monthly Weather Review, 2001, 129(11): 2776-2790.
[7]	Anderson J L, Anderson S L. A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts[J]. Monthly Weather Review, 1999, 127(12): 2741-2758.
[8]	Anderson J L. An ensemble adjustment Kalman filter for data assimilation[J]. Monthly Weather Review, 2001, 129(12): 2884-2903.
[9]	Sakov P, Bertino L. Relation between two common localisation methods for the EnKF[J]. Computational Geosciences, 2011, 15(2): 225-237.
[10]	Oke P R, Sakov P, Corney S P. Impacts of localisation in the EnKF and EnOI: Experiments with a small model[J]. Ocean Dynamics, 2007, 57(1): 32-45.
[11]	Evensen G. Data Assimilation: The Ensemble Kalman Filter (2nd)[M]. Dordrecht: Springer, 2009.
[12]	Schur I. Bemerkungen zur theorie der beschrnkten bilinearformen mit unendlich vielen vernderlichen[J]. Journal für die Reine und Angewante Mathematiek, 1911,140: 1-28.
[13]	Burgers G, Van Leeuwen P J, Evensen G. Analysis scheme in the ensemble Kalman filter[J]. Monthly Weather Review, 1998, 126(6):1719-1724.
[14]	Whitaker J S, Hamill T M. Ensemble data assimilation without perturbed observations[J]. Monthly Weather Review, 2002, 130(7):1913-1924.
[15]	Sakov P, Oke P R. A deterministic formulation of the ensemble Kalman filter: An alternative to ensemble square root filters[J]. Tellus A, 2008, 60(2): 361-371.
[16]	Bishop C H, Etherton B J, Majumdar S J. Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects[J]. Monthly Weather Review, 2001, 129(3): 420-436.
[17]	Horn R A, Johnson C R. Matrix Analysis[M]. New York: Cambridge University Press, 1985.
[18]	Evensen G. Sampling strategies and square root analysis schemes for the EnKF[J]. Ocean Dynamics, 2004, 54(6): 539-560.
[19]	Gaspari G, Cohn S E. Construction of correlation functions in two and three dimensions[J]. Quarterly Journal of the Royal Meteorological Society, 1999, 125(554): 723-757.

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摘要

A Comparative Study of Background Error Covariance Localization in EnKF Data Assimilation