A Method of Improving Error Covariances in EnKF and Its Application to Data Assimilation
-
摘要: 集合卡尔曼滤波 (the Ensemble Kalman Filter,简称EnKF) 中将预报集合的统计协方差作为预报误差协方差,但该估计可能严重偏离真实的预报误差协方差,影响同化精度。基于极大似然估计理论,发展了一种优化预报误差协方差矩阵的实时膨胀方法,即MLE (the Maximum Likelihood Estimation) 方法。利用蒙古国基准站Delgertsgot (简称DGS站) 观测资料,基于EnKF方法和MLE方法,在通用陆面模式 (the Common Land Model,简称CoLM) 中同化了地表温度和10 cm土壤温度观测资料,建立了土壤温度同化系统。结果表明:MLE方法对地表温度和各层土壤温度 (尤其深层土壤温度) 的估计比EnKF方法准确。考虑到浅层和深层土壤温度的差别,在实施MLE方法时对浅层和深层土壤温度采用了不同的膨胀因子。对比膨胀因子为单一标量时的结果,多因子膨胀能缓解深层土壤温度的不合理膨胀,改善同化效果。Abstract:
In the ensemble Kalman filter (EnKF), the forecast error covariance matrix is estimated as the sampling covariance matrix of the forecast ensemble. However, previous studies suggest that the sampling error resulting from finite-size ensembles may make such estimations far from the true forecast error covariance, and finally degrade the performance of EnKF. A common way to address this problem is covariance inflation with a time-constant inflation factor. A time-dependent infiation approach on forecast error covariance matrix (i.e., MLE method) is developed based on the maximum likelihood estimation theory, so as to improve estimates of forecast error covariances. At Delgertsgot (DGS) Station in the Mongolian Plateau reference site, point observations of ground temperature and soil temperature at the depth of 10 cm are assimilated into the Common Land Model (CoLM) with a frequency of every 12 hours, using two assimilation algorithms (EnKF method and MLE method), in order to test the effectivity of MLE in practical assimilation. In this way, a soil temperature assimilation system is constructed on the point scale.Results indicate that MLE method performs better than EnKF method for ground temperature and soil temperatures at most depths (especially for soil temperatures at deeper depths). Moreover, considering differences between soil temperatures at shallower depths and those at deeper depths, different inflation factors are adopted to them when implementing MLE method. Compared to results of MLE using a single scalar inflation factor, it shows that multiple-factor inflation is able to alleviate the unreasonable inflation of soil temperatures at deeper depths and therefore get better assimilation results. In addition, the leaf area index (LAI) in the CoLM is updated dynamically by MODIS LAI products, and results derived using MODIS LAI are compared to those derived using LAI computed by experiential formula, so as to study the effect of the LAI accuracy on simulated and assimilated soil temperatures. It shows that using MODIS LAI can get better simulation of soil temperature at depths of 0 cm and 3 cm, as well as more accurate assimilation of soil temperature at most depths.The inflation factor is set to be variable in time, but constant in space. However, variables such as soil temperature and soil moisture behave quite differently at shallow surfaces and deep depths, and observations may be unevenly distributed in space in regional assimilation researches. Therefore, it is necessary to adopt different inflation factors to different variables or to the same variable at different locations. In the future, it is necessary to develop a time-and-space dependent inflation method and test its capability in real applications.
-
Key words:
- data assimilation;
- EnKF;
- error covariance inflation
-
表 1 2003年9月1—30日土壤温度模拟场和同化场平均的均方根误差 (单位:K)
Table 1 Root mean square error of the simulated and assimilated soil temperature from 1 Sep to 30 Sep in 2003(unit:K)
方法 土壤温度 0 cm 3 cm 10 cm 40 cm 100 cm COLM模拟 3.705 3.704 1.068 3.618 12.907 EnKF 3.655 3.485 0.698 1.836 4.217 MLE1 3.334 3.215 0.752 2.410 5.574 MLE2 3.148 3.168 0.652 1.289 3.493 -
[1] Evensen G.Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte-Carlo methods to forecast error statistics.J Geophys Res, 1994, 99(C5):10143-10162. doi: 10.1029/94JC00572 [2] Burgers G, Leeuwen P J V, Evensen G.Analysis scheme in the Ensemble Kalman Filter.Mon Wea Rev, 1998, 126:1719-1724. doi: 10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2 [3] Evensen G.The ensemble Kalman filter:Theoretical formulation and practical implementation.Ocean Dynam, 2003, 53(4):343-367. doi: 10.1007/s10236-003-0036-9 [4] Senegas J, Wackernagel H H, Rosenthal W, et al.Error covariance modeling in sequential data assimilation.Stoch Env Res Risk A, 2001, 15:65-86. doi: 10.1007/PL00009788 [5] Daley R.Atmospheric Data Analysis.Cambridge:Cambridge University Press, 1991. [6] Kalnay E.Atmospheric Modeling, Data Assimilation, and Predictability.Cambridge:Cambridge University Press, 2002. [7] Anderson J L, Anderson S L.A Monte Carlo implementation of the non-linear filtering problem to produce ensemble assimilations and forecasts.Mon Wea Rev, 1999, 127:2741-2758. doi: 10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2 [8] Constantinescu E M, Sandu A, Chai T, et al.Ensemble-based chemical data assimilation.I:General approach.Q J R Meteorol Soc, 2007, 133:1229-1243. doi: 10.1002/(ISSN)1477-870X [9] Dee D P.On-line estimation of error covariance parameters for atmospheric data assimilation.Mon Wea Rev, 1995, 123(4):1128-1145. doi: 10.1175/1520-0493(1995)123<1128:OLEOEC>2.0.CO;2 [10] Dee D P, da Silva A M.Maximum-likelihood estimation of forecast and observation error covariance parameters.Part Ⅰ:Methodology.Mon Wea Rev, 1999, 127:1822-1834. doi: 10.1175/1520-0493(1999)127<1822:MLEOFA>2.0.CO;2 [11] Dee D P, Gaspari G, Redder C, et al.Maximum-likelihood estimation of forecast and observation error covariance parameters.Part Ⅱ:Applications.Mon Wea Rev, 1999, 127:1835-1849. doi: 10.1175/1520-0493(1999)127<1835:MLEOFA>2.0.CO;2 [12] Wang X, Bishop C H.A comparison of breeding and ensemble transform Kalman filter ensemble forecast schemes.J Atmos Sci, 2003, 60:1140-1158. doi: 10.1175/1520-0469(2003)060<1140:ACOBAE>2.0.CO;2 [13] Li H, Kalnay E, Miyoshi T.Simultaneous estimation of covariance inflation and observation errors within an ensemble Kalman filter.Q J R Meteorol Soc, 2009, 135(639):523-533. doi: 10.1002/qj.v135:639 [14] Miyoshi T.The gaussian approach to adaptive covariance inflation and its implementation with the Local Ensemble Transform Kalman Filter.Mon Wea Rev, 2011, 139:1519-1535. doi: 10.1175/2010MWR3570.1 [15] Zheng X G.An adaptive estimation of forecast error covariance parameters for Kalman filtering data assimilation.Adv Atmos Sci, 2009, 26:154-160. doi: 10.1007/s00376-009-0154-5 [16] Liang X, Zheng X G, Zhang S P, et al.Maximum likelihood estimation of inflation factors on error covariance matrices for ensemble Kalman filter assimilation.Q J R Meteorol Soc, 2011, 138:263-273, doi: 10.1002/qj.912. [17] Huang C L, Li X, Lu L.Retrieving soil temperature profile by assimilating MODIS LST products with ensemble Kalman filter.Rem Sens Environ, 2008, 112:1320-1336. doi: 10.1016/j.rse.2007.03.028 [18] Yang K, Koike T, Kaihotsu I, et al.Validation of a dual-pass microwave land data assimilation system for estimating surface soil moisture in semiarid regions.J Hydrometeorology, 2009, 10(3):780-793. doi: 10.1175/2008JHM1065.1 [19] 杨晓峰, 陆其峰, 杨忠东.基于AMSR-E土壤湿度产品的LIS同化试验.应用气象学报, 2013, 24(4):435-445. doi: 10.11898/1001-7313.20130406 [20] 吴统文, 宋连春, 刘向文, 等.国家气候中心短期气候预测模式系统业务化进展.应用气象学报, 2013, 24(5):533-543. doi: 10.11898/1001-7313.20130503 [21] 王莉, 黄嘉佑.Kalman滤波的试验应用研究.应用气象学报, 1999, 10(3):276-282. http://qikan.camscma.cn/jams/ch/reader/view_abstract.aspx?file_no=19990370&flag=1 [22] 赵晓琳, 朱国富, 李泽椿.基于TIGGE资料识别适应性观测敏感区的应用研究.应用气象学报, 2010, 21(4):405-415. doi: 10.11898/1001-7313.20100403 [23] Whitaker J S, Hamill T H.Ensemble data assimilation without perturbed observations.Mon Wea Rev, 2002, 130:1913-1924. doi: 10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2 [24] Dai Y J, Zeng X B, Dickinson R E, et al.The common land model.Bull Amer Meteor Soc, 2003, 84:1013-1023. doi: 10.1175/BAMS-84-8-1013 [25] 孟春雷, 张朝林.路面气象数值预报模型及性能检验.应用气象学报, 2012, 23(4):451-458. doi: 10.11898/1001-7313.20120408 [26] Koike T.Coordinated Enhanced Observing Period (CEOP)-An initial step for integrated global water cycle observation.World Meteorological Organization Bulletin, 2004, 53(2):115-121. [27] 龚建东, 赵刚.全球资料同化中误差协方差三维结构的准确估计与应用:背景误差协方差调整与数值试验分析.气象学报, 2006, 64(6):669-682. doi: 10.11676/qxxb2006.065 [28] 曹小群, 黄思训, 张卫民, 等.区域三维变分同化中背景误差协方差的模拟.气象科学, 2008, 28(1):8-14. http://www.cnki.com.cn/Article/CJFDTOTAL-QXKX200801004.htm [29] 马旭林, 庄照荣, 薛纪善, 等.GRAPES非静力数值预报模式的三维变分资料同化系统的发展.气象学报, 2009, 67(1):50-60. doi: 10.11676/qxxb2009.006 [30] 王曼, 李华宏, 段旭, 等.WRF模式三维变分中背景误差协方差估计.应用气象学报, 2011, 22(4):482-492. doi: 10.11898/1001-7313.20110411 [31] Jin R, Li X.Improving the estimation of hydrothermal state variables in the active layer of frozen ground by assimilating in situ observations and SSM/I data.Sci China Ser D:Earth Sci, 2009, 39(9):1220-1231. http://en.cnki.com.cn/Article_en/CJFDTOTAL-JDXG200911007.htm [32] Anderson J L.Spatially and temporally varying adaptive covariance inflation for ensemble filters.Tellus, 2009, 61:72-83. doi: 10.1111/tea.2008.61.issue-1