不同矩阵分解方法对海洋数据同化的影响
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摘要
在海洋数据同化领域,集合最优插值方法中,矩阵求逆过程所使用的奇异值分解(singular value decomposition,SVD)十分耗时。对集合最优插值中逆矩阵的求逆过程进行优化,分别使用LU分解、Choleskey分解、QR分解来替代SVD分解。首先,通过LU分解(Choleskey分解或QR分解)得到相应的三角矩阵(或正交矩阵);然后,利用分解后的矩阵来实现相关逆矩阵的计算。由于LU分解、Choleskey分解、QR分解的算法复杂度都远小于SVD分解,因此改进后的同化程序能得到大幅度的性能提升。数值结果表明,所采用的三种矩阵分解方法相比于SVD分解,都能将集合最优插值的计算效率提升至少两倍以上。值得一提的是,在四种矩阵分解中Choleskey分解使得整个同化程序的性能达到了最优。
In the field of ocean data assimilation, the SVD(singular value decomposition) method used in the matrix inversion process of ensemble optimal interpolation is very time-consuming. This paper uses matrix LU decomposition,Choleskey decomposition and QR decomposition respectively to replace singular value decomposition to optimize the matrix inversion process in the ensemble optimal interpolation. Firstly, LU(Choleskey or QR) decomposition breaks down some matrices into two products of triangular matrices(or orthogonal matrices). Secondly, the matrices obtained by decomposition process, are used rapidly to calculate the inverse of matrix. As the algorithm complexity of LU(Choleskey or QR) decomposition is much smaller than that of the singular value decomposition, the performance of the assimilation program is improved greatly. The numerical experiments show that the three kinds of matrix decomposition methods used in this paper can improve the computational efficiency of the ensemble at least two times compared with the traditional implementation based on singular value decomposition. Furthermore, it should be noted that the Choleskey decomposition makes the entire assimilation program achieve the best performance.
In the field of ocean data assimilation, the SVD(singular value decomposition) method used in the matrix inversion process of ensemble optimal interpolation is very time-consuming. This paper uses matrix LU decomposition,Choleskey decomposition and QR decomposition respectively to replace singular value decomposition to optimize the matrix inversion process in the ensemble optimal interpolation. Firstly, LU(Choleskey or QR) decomposition breaks down some matrices into two products of triangular matrices(or orthogonal matrices). Secondly, the matrices obtained by decomposition process, are used rapidly to calculate the inverse of matrix. As the algorithm complexity of LU(Choleskey or QR) decomposition is much smaller than that of the singular value decomposition, the performance of the assimilation program is improved greatly. The numerical experiments show that the three kinds of matrix decomposition methods used in this paper can improve the computational efficiency of the ensemble at least two times compared with the traditional implementation based on singular value decomposition. Furthermore, it should be noted that the Choleskey decomposition makes the entire assimilation program achieve the best performance.
引文
[1]Natvik L J,Evensen G.Assimilation of ocean colour data into a biochemical model of the North Atlantic:part 1.Data assimilation experiments[J].Journal of Marine Systems,2003,40/41:127-153.
[2]Wan L Y,Zhu J,Laurent B,et al.Initial ensemble generation and validation for ocean data assimilation using HYCOMin the Pacific[J].Ocean Dynamics,2008,58(2):81-99.
[3]Ma Z P,Jing A Q.Data assimilation method applied in marine science-its significance,system configuration and development situation[J].Coastal Engineering,2005,24(4):83-99.
[4]Miller R N,Ehret L L.Ensemble generation for models of multimodal systems[J].Monthly Weather Review,2002,130(130):2313-2333.
[5]Behringer D,Xue Y.Ocean data assimilation for seasonal to interannual prediction[R/OL].The ARGO,2006.http://www.Argo.ucsd.edu/.
[6]Natvik L J,Evensen G.Assimilation of ocean colour data into a biochemical model of the North Atlantic:part 2.Statistical analysis[J].Journal of Marine Systems,2003,40/41:155-169.
[7]Wang H,Liu G M,Wan L Y.Review on the data assimilation into marine ecosystem model[J].Advances in Earth Science,2007,22(10):989-996.
[8]Bleck R.An oceanic general circulation model framed in hybrid isopycnic-cartesian coordinates[J].Ocean Modelling,2002,4(1):55-88.
[9]Wu X R,Wang X D,Li W,et al.Review of the application of ocean data assimilation on data fusion techniques[J].Journal of Ocean Technology,2015,34(3):97-103.
[10]Keppenne C L,Rienecker M M.Assimilation of temperature into an isopycnal ocean general circulation model using a parallel ensemble Kalman fiter[J].Journal of Marine Systems,2003,40/41:363-380.
[11]Talagrand O,Courtier P.Variational assimilation of meteorological observations with the adjoint vorticity equation.I:theory[J].Quarterly Journal of the Royal Meteorological,2010,113(478):1311-1328.
[12]Haugen E J,Evensen G.Assimilation of SLA and SST data into an OGCM for the Indian Ocean[J].Ocean Dynamics,2002,52(3):133-151.
[13]Cai Y H,Cheng P F,Wen H J,et al.Study on global sound speed field of seawater using ARGO profile data[J].Remote Sensing Information,2014,29(5):13-19.
[14]Yan C X,Xie J P,Zhu J.An analysis system for rapid estimate of three-dimensional ocean temperature,salinity,and current fields and its application in the Gulf of Aden[J].Climatic and Environmental Research,2011,16(4):419-428.
[15]Xiao X J,He N,Zhang Z Q,et al.Variation assimilation using satellite data of sea surface temperature and altimeter[J].Journal of Tropical Oceanography,2011,30(3):1-8.
[16]Zhang R H,Zhu J,Xu J P,et al.Argo global ocean data assimilation and its applications in short-term climate prediction and oceanic analysis[J].Chinese Journal of Atmospheric Sciences,2013,37(2):411-424.
[17]Evensen G.The ensemble Kalman filter:theoretical formulation and practical implementation[J].Ocean Dynamics,2003,53(4):343-367.
[18]Hamill T M,Snyder C.A hybrid ensemble Kalman filter 3Dvariational analysis scheme[J].Monthly Weather Review,2000,128(8):2905-2919.
[19]Zhuang Z R.Design of the ensemble Kalman filter data assimilation system application in the ensemble prediction[D].Beijing:Graduate University of Chinese Academy of Sciences,2007.
[20]Reichle R H,Mclaughlin D B,Entekhabi D.Hydrologic data assimilation with the ensemble Kalman filter[J].Monthly Weather Review,2002,130(1):103-114.
[21]Lv G K.Application and improvement of ensemble optimal interpolatiom on HYCOM and ROMS[D].Qingdao:Ocean University of China,2013.
[22]Whitaker J S,Hamill T M.Ensemble data assimilation without perturbed observations[J].Monthly Weather Review,2002,130(7):1913-1924.
[23]Toth Z,Kalnay E.Ensemble forecasting at NMC:the generation of perturbations[J].Bulletin of the American Meteorological Society,1993,74(12):2317-2330.
[24]Mao K C,Sun F P.Performance comparison between extended Kalman filtering and unscented Kalman filtering[J].Hydrographic Surveying and Charting,2006,26(5):4-6.
[25]Yan C X,Zhu J.Choice of ensemble members for ensemble optimal interpolation[J].Climatic and Environmental Research,2011,16(4):452-458.
[26]Verlaan M,Heemink A W.Nonlinearity in data assimilation applications:a practical method for analysis[J].Monthly Weather Review,2001,129(6):1578-1589.
[27]Yin F L,Yang Y Y,Wang C D,et al.Research on the advances of singular value decomposition and its application in the high dimensional data mining[J].Mathematics in Practice and Theory,2011,41(15):171-177.
[28]Niu S Z,Niu X X,Yang Y X.Digital watermarking algorithm based on LU decomposition[J].Journal of Electronics&Information Technology,2004,26(10):1620-1625.
[29]Wei C J,Zhang C S,Liu J.A new method of solving inverse matrix based on Cholesky matrix[J].Electronic Design Engineering,2014,22(1):159-161.
[30]Lin X L,Jiang Y L.QR decomposition and algorithm for unitary symmetric matrix[J].Chinese Journal of Computers,2005,28(5):817-822.
[3]马寨璞,井爱芹.海洋科学中的数据同化方法--意义、结构与发展现状[J].海岸工程,2005,24(4):83-99.
[9]吴新荣,王喜冬,李威,等.海洋数据同化与数据融合技术应用综述[J].海洋技术学报,2015,34(3):97-103.
[13]蔡艳辉,程鹏飞,文汉江,等.基于ARGO浮标数据的全球海水声速场研究[J].遥感信息,2014,29(5):13-19.
[14]闫长香,谢基平,朱江.一个快速海洋三维温盐流分析系统及在亚丁湾临近海域的应用[J].气候与环境研究,2011,16(4):419-428.
[15]肖贤俊,何娜,张祖强,等.卫星遥感海表温度资料和高度计资料的变分同化[J].热带海洋学报,2011,30(3):1-8.
[16]张人禾,朱江,许建平,等.Argo大洋观测资料的同化及其在短期气候预测和海洋分析中的应用[J].大气科学,2013,37(2):411-424.
[19]庄照荣.集合卡尔曼滤波资料同化系统的设计及其在集合预报中的应用[D].北京:中国科学院研究生院,2007.
[21]吕国坤.集合最优插值同化方法在HYCOM及ROMS模式中应用及改进[D].青岛:中国海洋大学,2013.
[24]毛克诚,孙付平.扩展卡尔曼滤波与采样卡尔曼滤波性能比较[J].海洋测绘,2006,26(5):4-6.
[25]闫长香,朱江.集合最优插值中的样本选取[J].气候与环境研究,2011,16(4):452-458.
[27]尹芳黎,杨雁莹,王传栋,等.矩阵奇异值分解及其在高维数据处理中的应用[J].数学的实践与认识,2011,41(15):171-177.
[28]牛少彰,钮心忻,杨义先.基于矩阵LU分解的数字水印算法[J].电子与信息学报,2004,26(10):1620-1625.
[29]魏婵娟,张春水,刘健.一种基于Cholesky分解的快速矩阵求逆方法设计[J].电子设计工程,2014,22(1):159-161.
[30]蔺小林,蒋耀林.酉对称矩阵的QR分解及其算法[J].计算机学报,2005,28(5):817-822.
[2]Wan L Y,Zhu J,Laurent B,et al.Initial ensemble generation and validation for ocean data assimilation using HYCOMin the Pacific[J].Ocean Dynamics,2008,58(2):81-99.
[3]Ma Z P,Jing A Q.Data assimilation method applied in marine science-its significance,system configuration and development situation[J].Coastal Engineering,2005,24(4):83-99.
[4]Miller R N,Ehret L L.Ensemble generation for models of multimodal systems[J].Monthly Weather Review,2002,130(130):2313-2333.
[5]Behringer D,Xue Y.Ocean data assimilation for seasonal to interannual prediction[R/OL].The ARGO,2006.http://www.Argo.ucsd.edu/.
[6]Natvik L J,Evensen G.Assimilation of ocean colour data into a biochemical model of the North Atlantic:part 2.Statistical analysis[J].Journal of Marine Systems,2003,40/41:155-169.
[7]Wang H,Liu G M,Wan L Y.Review on the data assimilation into marine ecosystem model[J].Advances in Earth Science,2007,22(10):989-996.
[8]Bleck R.An oceanic general circulation model framed in hybrid isopycnic-cartesian coordinates[J].Ocean Modelling,2002,4(1):55-88.
[9]Wu X R,Wang X D,Li W,et al.Review of the application of ocean data assimilation on data fusion techniques[J].Journal of Ocean Technology,2015,34(3):97-103.
[10]Keppenne C L,Rienecker M M.Assimilation of temperature into an isopycnal ocean general circulation model using a parallel ensemble Kalman fiter[J].Journal of Marine Systems,2003,40/41:363-380.
[11]Talagrand O,Courtier P.Variational assimilation of meteorological observations with the adjoint vorticity equation.I:theory[J].Quarterly Journal of the Royal Meteorological,2010,113(478):1311-1328.
[12]Haugen E J,Evensen G.Assimilation of SLA and SST data into an OGCM for the Indian Ocean[J].Ocean Dynamics,2002,52(3):133-151.
[13]Cai Y H,Cheng P F,Wen H J,et al.Study on global sound speed field of seawater using ARGO profile data[J].Remote Sensing Information,2014,29(5):13-19.
[14]Yan C X,Xie J P,Zhu J.An analysis system for rapid estimate of three-dimensional ocean temperature,salinity,and current fields and its application in the Gulf of Aden[J].Climatic and Environmental Research,2011,16(4):419-428.
[15]Xiao X J,He N,Zhang Z Q,et al.Variation assimilation using satellite data of sea surface temperature and altimeter[J].Journal of Tropical Oceanography,2011,30(3):1-8.
[16]Zhang R H,Zhu J,Xu J P,et al.Argo global ocean data assimilation and its applications in short-term climate prediction and oceanic analysis[J].Chinese Journal of Atmospheric Sciences,2013,37(2):411-424.
[17]Evensen G.The ensemble Kalman filter:theoretical formulation and practical implementation[J].Ocean Dynamics,2003,53(4):343-367.
[18]Hamill T M,Snyder C.A hybrid ensemble Kalman filter 3Dvariational analysis scheme[J].Monthly Weather Review,2000,128(8):2905-2919.
[19]Zhuang Z R.Design of the ensemble Kalman filter data assimilation system application in the ensemble prediction[D].Beijing:Graduate University of Chinese Academy of Sciences,2007.
[20]Reichle R H,Mclaughlin D B,Entekhabi D.Hydrologic data assimilation with the ensemble Kalman filter[J].Monthly Weather Review,2002,130(1):103-114.
[21]Lv G K.Application and improvement of ensemble optimal interpolatiom on HYCOM and ROMS[D].Qingdao:Ocean University of China,2013.
[22]Whitaker J S,Hamill T M.Ensemble data assimilation without perturbed observations[J].Monthly Weather Review,2002,130(7):1913-1924.
[23]Toth Z,Kalnay E.Ensemble forecasting at NMC:the generation of perturbations[J].Bulletin of the American Meteorological Society,1993,74(12):2317-2330.
[24]Mao K C,Sun F P.Performance comparison between extended Kalman filtering and unscented Kalman filtering[J].Hydrographic Surveying and Charting,2006,26(5):4-6.
[25]Yan C X,Zhu J.Choice of ensemble members for ensemble optimal interpolation[J].Climatic and Environmental Research,2011,16(4):452-458.
[26]Verlaan M,Heemink A W.Nonlinearity in data assimilation applications:a practical method for analysis[J].Monthly Weather Review,2001,129(6):1578-1589.
[27]Yin F L,Yang Y Y,Wang C D,et al.Research on the advances of singular value decomposition and its application in the high dimensional data mining[J].Mathematics in Practice and Theory,2011,41(15):171-177.
[28]Niu S Z,Niu X X,Yang Y X.Digital watermarking algorithm based on LU decomposition[J].Journal of Electronics&Information Technology,2004,26(10):1620-1625.
[29]Wei C J,Zhang C S,Liu J.A new method of solving inverse matrix based on Cholesky matrix[J].Electronic Design Engineering,2014,22(1):159-161.
[30]Lin X L,Jiang Y L.QR decomposition and algorithm for unitary symmetric matrix[J].Chinese Journal of Computers,2005,28(5):817-822.
[3]马寨璞,井爱芹.海洋科学中的数据同化方法--意义、结构与发展现状[J].海岸工程,2005,24(4):83-99.
[9]吴新荣,王喜冬,李威,等.海洋数据同化与数据融合技术应用综述[J].海洋技术学报,2015,34(3):97-103.
[13]蔡艳辉,程鹏飞,文汉江,等.基于ARGO浮标数据的全球海水声速场研究[J].遥感信息,2014,29(5):13-19.
[14]闫长香,谢基平,朱江.一个快速海洋三维温盐流分析系统及在亚丁湾临近海域的应用[J].气候与环境研究,2011,16(4):419-428.
[15]肖贤俊,何娜,张祖强,等.卫星遥感海表温度资料和高度计资料的变分同化[J].热带海洋学报,2011,30(3):1-8.
[16]张人禾,朱江,许建平,等.Argo大洋观测资料的同化及其在短期气候预测和海洋分析中的应用[J].大气科学,2013,37(2):411-424.
[19]庄照荣.集合卡尔曼滤波资料同化系统的设计及其在集合预报中的应用[D].北京:中国科学院研究生院,2007.
[21]吕国坤.集合最优插值同化方法在HYCOM及ROMS模式中应用及改进[D].青岛:中国海洋大学,2013.
[24]毛克诚,孙付平.扩展卡尔曼滤波与采样卡尔曼滤波性能比较[J].海洋测绘,2006,26(5):4-6.
[25]闫长香,朱江.集合最优插值中的样本选取[J].气候与环境研究,2011,16(4):452-458.
[27]尹芳黎,杨雁莹,王传栋,等.矩阵奇异值分解及其在高维数据处理中的应用[J].数学的实践与认识,2011,41(15):171-177.
[28]牛少彰,钮心忻,杨义先.基于矩阵LU分解的数字水印算法[J].电子与信息学报,2004,26(10):1620-1625.
[29]魏婵娟,张春水,刘健.一种基于Cholesky分解的快速矩阵求逆方法设计[J].电子设计工程,2014,22(1):159-161.
[30]蔺小林,蒋耀林.酉对称矩阵的QR分解及其算法[J].计算机学报,2005,28(5):817-822.