集合背景误差方差中小波阈值去噪方法研究及试验
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摘要
背景误差方差的集合估计值中带有大量采样噪声,在应用之前需进行降噪处理.区别于一般的高斯白噪声,采样噪声具有空间和尺度相关性,部分尺度上的噪声能级远大于平均能级.本文针对背景误差方差中采样噪声的特征,引入小波阈值去噪方法,并根据截断余项的小波系数分布特征发展了一种计算代价很小,能自动修正阈值的算法.一维理想试验结果表明,该方法能滤除大量采样噪声,提高背景误差方差估计值的精度.相对于原来的小波阈值方法,修正阈值后减少了因部分尺度上噪声能级过大导致的残差,去噪后的RMSE减少了13.28%.将该方法应用在实际的集合资料同化系统中,结果表明,小波阈值方法优于谱方法,阈值修正后能在不影响信号的前提下增大小波去噪强度.
A large amount of sampling noise which exists in the ensemble-based background error variance need be reduced effectively before being applied to operational data assimilation system. Unlike the typical Gaussian white noise, the sampling noise is scaled and space-dependent, thus making its energy level on some scales much larger than the average.Although previous denoising methods such as spectral filtering or wavelet thresholding have been successfully used for denoising Gaussian white noise, they are no longer applicable for dealing with this kind of sampling noise. One can use a different threshold for each scale, but it will bring a big error especially on larger scales. Another modified method is to use a global multiplicative factor, α, to adjust the filtering strength based on the optimization of trade-off between removal of the noise and averaging of the useful signal. However, tuning α is not so easy, especially in real operational numerical weather prediction context. It motivates us to develop a new nearly cost-free filter whose threshold can be automatically calculated.According to the characteristics of sampling noise in background error variance, a heterogeneous filtering method similar to wavelet threshold technology is employed. The threshold, TA, determined by iterative algorithm is used to estimate the truncated remainder whose norm is smaller than TA. The standard deviation of truncated remainder term is regard as first guess of sampling noise. Non-Guassian term of sampling noise, whose coefficient modulus is above TA, is regarded as a small probability event. In order to incorporate such a coefficient into the domain of [-T, T ], a semi-empirical formula is used to calculate and approach the ideal threshold.Investigations are first conducted in a one-dimensional(1D) framework: several methods such as spectral filter,primal wavelet filter, optimal wavelet filter, and proposed filter are used to denoise the scale and space-dependent sampling noise in variance estimations. Their validity and performance are compared and examined with different ensemble sizes.Results show that the proposed method can efficiently filter out a large amount of sampling noise efficiently and improve the estimation accuracy of background error variance. Compared with previous filters, the modified threshold can help to reduce some residual noise arising from the scales where the noise level is above the average level, and the filtering performance of the new method is improved by almost 13.28%. Application to the real ensemble four-dimensional variational data assimilation system is then considered. Results show that the wavelet threshold method outperforms the spectral method. Modified threshold can enhance denosing without deforming the signal of interest.A new nearly cost-free filter is proposed to reduce the scale and space-dependent sampling noise in ensemble-based background error variance. It is able to remove most of the sampling noises, while extracting the signal of interest.Compared with those of primal wavelet filter and spectral filter, the performance and efficiency of proposed method are improved in 1D framework and real data assimilation system experiments. Further work should focus on the sphere wavelets, which is appropriate for analysing and reconstructing the signals on the sphere in global spectral models.
A large amount of sampling noise which exists in the ensemble-based background error variance need be reduced effectively before being applied to operational data assimilation system. Unlike the typical Gaussian white noise, the sampling noise is scaled and space-dependent, thus making its energy level on some scales much larger than the average.Although previous denoising methods such as spectral filtering or wavelet thresholding have been successfully used for denoising Gaussian white noise, they are no longer applicable for dealing with this kind of sampling noise. One can use a different threshold for each scale, but it will bring a big error especially on larger scales. Another modified method is to use a global multiplicative factor, α, to adjust the filtering strength based on the optimization of trade-off between removal of the noise and averaging of the useful signal. However, tuning α is not so easy, especially in real operational numerical weather prediction context. It motivates us to develop a new nearly cost-free filter whose threshold can be automatically calculated.According to the characteristics of sampling noise in background error variance, a heterogeneous filtering method similar to wavelet threshold technology is employed. The threshold, TA, determined by iterative algorithm is used to estimate the truncated remainder whose norm is smaller than TA. The standard deviation of truncated remainder term is regard as first guess of sampling noise. Non-Guassian term of sampling noise, whose coefficient modulus is above TA, is regarded as a small probability event. In order to incorporate such a coefficient into the domain of [-T, T ], a semi-empirical formula is used to calculate and approach the ideal threshold.Investigations are first conducted in a one-dimensional(1D) framework: several methods such as spectral filter,primal wavelet filter, optimal wavelet filter, and proposed filter are used to denoise the scale and space-dependent sampling noise in variance estimations. Their validity and performance are compared and examined with different ensemble sizes.Results show that the proposed method can efficiently filter out a large amount of sampling noise efficiently and improve the estimation accuracy of background error variance. Compared with previous filters, the modified threshold can help to reduce some residual noise arising from the scales where the noise level is above the average level, and the filtering performance of the new method is improved by almost 13.28%. Application to the real ensemble four-dimensional variational data assimilation system is then considered. Results show that the wavelet threshold method outperforms the spectral method. Modified threshold can enhance denosing without deforming the signal of interest.A new nearly cost-free filter is proposed to reduce the scale and space-dependent sampling noise in ensemble-based background error variance. It is able to remove most of the sampling noises, while extracting the signal of interest.Compared with those of primal wavelet filter and spectral filter, the performance and efficiency of proposed method are improved in 1D framework and real data assimilation system experiments. Further work should focus on the sphere wavelets, which is appropriate for analysing and reconstructing the signals on the sphere in global spectral models.
引文
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[9]Berre L,Varella H,Desroziers G 2015 Q.J.R.Meteorol.Soc.141 2803
[10]Raynaud L,Berre L,Desroziers G 2009 Q.J.R.Meteorol.Soc.135 1177
[11]Pereira M B,Berre L 2006 Mon.Weather Rev.134 2466
[12]Raynaud L,Berre L,Desroziers G 2008 Q.J.R.Meteorol.Soc.134 1003
[13]Wiener N 1949 Extrapolation,Interpolation,and Smoothing of Stationary Time Series(Cambridge:Massachusetts Institute of Technology)pp86–90
[14]Bonavita M,Isaksen L,Hólm E 2012 Q.J.R.Meteorol.Soc.138 1540
[15]Liu B N,Zhang W M,Cao X Q,Zhao Y L,Huang Q B2015 China J.Geophys.58 1526(in Chinese)[刘柏年,张卫民,曹小群,赵延来,皇群博,罗雨2015地球物理学报58 1526]
[16]Donoho D L,Johnstone J M 1994 Biometrical 81 425
[17]Parrish D F,Derber J C 1992 Mon.Weather Rev.1201747
[18]Fisher M 2003 Proceedings ECMWF Seminar on“Recent Developments in Data Assimilation for Atmosphere and Ocean”Reading,September 8–12,2003 p45
[19]Isaksen L,Fisher M,Berner J 2006 ECMWF Tech.Memo.492
[20]Moore S,Wood S,Davies P 1998 Annals of Statistics26 1
[21]Daley R 1993 Atmospheric Data Analysis1993(Cambridge:Cambridge University Press)pp46–50
[22]Azzalini A,Farge M,Schneider K 2005 Appl.Comput.Harmon.Anal.18 177
[23]Yen R N V,Farge M,Schneider K 2012 Physica D Nonlinear Phenomena 241 186
[24]Pannekoucke O,Raynaud L,Farge M 2014 Q.J.R.Meteorol.Soc.140 316
[25]Zhang W M,Liu B N,Cao X Q,Zhao Y L,Zhu M B,Zhao W J 2016 Acta Meteorol Sin.74 410(in Chinese)[张卫民,刘柏年,曹小群,赵延来,朱孟斌,赵文静2016气象学报74 410]
[2]Wang S C,LI Y,Zhang W M,Zhao J,Cao X Q 2011Acta Phys.Sin.60 099203(in Chinese)[王舒畅,李毅,张卫民,赵军,曹小群2011物理学报60 099203]
[3]Laroche S,Gauthier G 1998 Tellus Ser.A 50 557
[4]Derber J,Bouttier F 1999 Tellus Ser.A 51 195
[5]Buizza R,Houtekamer P L,Pellerin G,Toth Z,Zhu Y J,Wei M Z 2005 Mon.Weather Rev.133 1076
[6]Houtekamer P L,Mitchell H L 1998 Mon.Weather Rev.126 3
[7]Evensen G 1994 J.Geophys.Res.99 10143
[8]Bonavita M,Raynaud L,Isaksen L 2011 Q.J.R.Meteorol.Soc.137 423
[9]Berre L,Varella H,Desroziers G 2015 Q.J.R.Meteorol.Soc.141 2803
[10]Raynaud L,Berre L,Desroziers G 2009 Q.J.R.Meteorol.Soc.135 1177
[11]Pereira M B,Berre L 2006 Mon.Weather Rev.134 2466
[12]Raynaud L,Berre L,Desroziers G 2008 Q.J.R.Meteorol.Soc.134 1003
[13]Wiener N 1949 Extrapolation,Interpolation,and Smoothing of Stationary Time Series(Cambridge:Massachusetts Institute of Technology)pp86–90
[14]Bonavita M,Isaksen L,Hólm E 2012 Q.J.R.Meteorol.Soc.138 1540
[15]Liu B N,Zhang W M,Cao X Q,Zhao Y L,Huang Q B2015 China J.Geophys.58 1526(in Chinese)[刘柏年,张卫民,曹小群,赵延来,皇群博,罗雨2015地球物理学报58 1526]
[16]Donoho D L,Johnstone J M 1994 Biometrical 81 425
[17]Parrish D F,Derber J C 1992 Mon.Weather Rev.1201747
[18]Fisher M 2003 Proceedings ECMWF Seminar on“Recent Developments in Data Assimilation for Atmosphere and Ocean”Reading,September 8–12,2003 p45
[19]Isaksen L,Fisher M,Berner J 2006 ECMWF Tech.Memo.492
[20]Moore S,Wood S,Davies P 1998 Annals of Statistics26 1
[21]Daley R 1993 Atmospheric Data Analysis1993(Cambridge:Cambridge University Press)pp46–50
[22]Azzalini A,Farge M,Schneider K 2005 Appl.Comput.Harmon.Anal.18 177
[23]Yen R N V,Farge M,Schneider K 2012 Physica D Nonlinear Phenomena 241 186
[24]Pannekoucke O,Raynaud L,Farge M 2014 Q.J.R.Meteorol.Soc.140 316
[25]Zhang W M,Liu B N,Cao X Q,Zhao Y L,Zhu M B,Zhao W J 2016 Acta Meteorol Sin.74 410(in Chinese)[张卫民,刘柏年,曹小群,赵延来,朱孟斌,赵文静2016气象学报74 410]