-
摘要:
集合卡尔曼滤波（ensemble Kalman filter, EnKF）是最流行的数据同化方法之一。然而，在处理非高斯问题时，EnKF存在局限性。为了解决非高斯问题并准确描述含水介质连通性，将正态分数变换（normal-score transformation, NST）与多重数据同化集合平滑器（ensemble smoother with multiple data assimilation, ES-MDA）相结合，提出NS-ES-MDA方法。通过对比实验，验证了NS-ES-MDA方法估计非高斯分布含水层渗透系数场的有效性。相较于重启正态分数集合卡尔曼滤波器（restart normal-score ensemble Kalman filter, rNS-EnKF）方法，NS-ES-MDA在吸收相同数据后，参数估计精度提升约34%，计算效率提升约35%。此外，NS-ES-MDA方法受“异参同效”现象的影响较小，具有较强的更新能力，能够保障得到较准确的参数估计值。研究可为非高斯分布含水层参数估计提供一种有效的求解方法。
Abstract:The ensemble Kalman filter (EnKF) is one of the most widely used data assimilation methods. However, it exhibits limitations in handling non-Gaussian problems. To effectively address such issues and accurately describe the connectivity of aquifers, a novel approach named NS-ES-MDA is developed in this study. The proposed NS-ES-MDA synergistically combines the normal-score transformation (NST) with ensemble smoother with multiple data assimilation (ES-MDA). Through comparative experiments, the efficacy of NS-ES-MDA in estimating the hydraulic conductivity of non-Gaussian distributed aquifers is demonstrated. By assimilating the same dataset, NS-ES-MDA exhibits approximately 34% improvement in parameter estimation accuracy and about 35% enhancement in computational efficiency compared to the restart normal-score ensemble Kalman filter (rNS-EnKF). Furthermore, the NS-ES-MDA shows case robustness against the “equifinality” and displays remarkable updating capabilities, which leads to more precise parameter estimates. This study provides an effective solution for parameter estimation in non-Gaussian distributed aquifers.
-
-
表 1 二维算例对数渗透系数参考场参数设置
Table 1. Parameters of the reference lnK
岩性 变异函数
类型均值
/（m·d−1）标准差
/（m·d−1）x方向
变程/my方向
变程/m砂土 指数型 2.0 0.5 200 200 黏土 指数型 −1.5 0.5 200 200 
下载: 导出CSV
表 2 二维算例含水层渗透系数场估计的情景设置及估计结果
Table 2. Estimation of the two-dimensional aquifer hydraulic conductivity field in different scenarios
情景 方法
（迭代次数）IRMSE（lnK）
/（m·d−1）IES（lnK）
/（m·d−1）运行
时间/s与S0的
比值/%S0 rNS-EnKF（） 1.38 1.19 1680 100% S1 NS-ES-MDA（1） 1.40 1.31 254 15% S2 NS-ES-MDA（2） 1.28 1.18 354 21% S3 NS-ES-MDA（4） 1.05 0.95 628 37% S4 NS-ES-MDA（6） 0.92 0.82 901 54% S5 NS-ES-MDA（8） 0.91 0.76 1084 65% S6 NS-ES-MDA（10） 0.92 0.74 1285 76% 注：rNS-EnKF不是迭代算法，没有迭代次数；IRMSE（lnK）代表集合均值与真实lnK值的均方根误差；IES（lnK）是后验参数集合的集合扩散；运行时间指在CPU主频为3.80 GHz的计算机上运行所用时间；与S0的比值为不同情景运行时间与S0运行时间的比值。
下载: 导出CSV
表 3 水头控制点处评价指标
Table 3. Metrics at head control points
评价指标 初始集合 S0 S1 S2 S3 S4 S5 S6 IRMSE（Head #1）/m 12.10 1.07 38.00 5.82 1.03 0.70 0.24 1.22 IES（Head #1）/m 27.88 4.47 50.17 18.28 8.53 6.27 5.51 5.03 INSE（Head #1） 0.03 0.99 −8.59 0.78 0.99 1.00 1.00 0.99 IRMSE（Head #2）/m 57.96 3.14 230.50 41.99 5.30 6.14 3.00 1.42 IES（Head #2）/m 50.92 12.54 202.68 76.99 26.19 14.53 9.68 8.84 INSE（Head #2） 0.72 1.00 −3.40 0.85 1.00 1.00 1.00 1.00 IRMSE（Head #3）/m 123.40 9.81 416.68 72.18 19.15 24.69 11.03 3.89 IES（Head #3）/m 100.99 17.04 255.07 98.46 38.17 21.65 13.06 12.09 INSE（Head #3） 0.70 1.00 −2.42 0.90 0.99 0.99 1.00 1.00
下载: 导出CSV
表 4 浓度控制点处的评价指标
Table 4. Metrics at concentration control points
评价指标 初始集合 S0 S1 S2 S3 S4 S5 S6 IRMSE（Conc #4）/（g·m−3） 4.57 3.80 3.28 2.41 1.25 0.89 0.84 0.71 IES（Conc #4）/（g·m−3） 6.58 2.69 4.72 3.19 1.29 0.84 0.79 0.75 INSE（Conc #4） −0.21 0.42 0.57 0.75 0.92 0.97 0.96 0.97 IRMSE（Conc #5）/（g·m−3） 1.86 2.74 3.09 2.54 0.80 0.28 0.24 0.25 IES（Conc #5）/（g·m−3） 2.61 1.64 2.18 1.84 0.99 0.55 0.38 0.33 INSE（Conc #5） 0.34 0.79 0.67 0.72 0.95 0.99 0.99 0.99 IRMSE（Conc #6）/（g·m−3） 25.12 26.63 28.40 28.07 27.79 18.57 9.99 6.84 IES（Conc #6）/（g·m−3） 13.19 8.82 9.77 8.61 7.96 6.76 6.22 5.58 INSE（Conc #6） −0.21 −0.83 −1.06 −0.98 −1.06 0.57 0.99 1.00
下载: 导出CSV
-
[1] 张江江. 地下水污染源解析的贝叶斯监测设计与参数反演方法[D]. 杭州:浙江大学,2017. [ZHANG Jiangjiang. Bayesian monitoring design and parameter inversion for groundwater contaminant source identification[D]. Hangzhou:Zhejiang University,2017. (in Chinese with English abstract)]
ZHANG Jiangjiang. Bayesian monitoring design and parameter inversion for groundwater contaminant source identification[D]. Hangzhou: Zhejiang University, 2017. （in Chinese with English abstract）
[2] 薛佩佩,文章,梁杏. 地质统计学在含水层参数空间变异研究中的应用进展与发展趋势[J]. 地质科技通报,2022,41(1):209 − 222. [XUE Peipei,WEN Zhang,LIANG Xing. Application and development trend of geostatistics in the research of spatial variation of aquifer parameters[J]. Bulletin of Geological Science and Technology,2022,41(1):209 − 222. (in Chinese with English abstract)]
XUE Peipei, WEN Zhang, LIANG Xing. Application and development trend of geostatistics in the research of spatial variation of aquifer parameters[J]. Bulletin of Geological Science and Technology, 2022, 41（1）: 209 − 222. （in Chinese with English abstract）
[3] 曾献奎. 地下水流数值模拟不确定性分析与评价[D]. 南京:南京大学,2012. [ZENG Xiankui. The uncertainty analysis and assessment of groundwater flow numerical simulation[D]. Nanjing:Nanjing University,2012. (in Chinese with English abstract)]
ZENG Xiankui. The uncertainty analysis and assessment of groundwater flow numerical simulation[D]. Nanjing: Nanjing University, 2012. （in Chinese with English abstract）
[4] CHEN Yan,ZHANG Dongxiao. Data assimilation for transient flow in geologic formations via ensemble Kalman filter[J]. Advances in Water Resources,2006,29(8):1107 − 1122. doi: 10.1016/j.advwatres.2005.09.007
[5] HUANG Chunlin,HU B X,LI Xin,et al. Using data assimilation method to calibrate a heterogeneous conductivity field and improve solute transport prediction with an unknown contamination source[J]. Stochastic Environmental Research and Risk Assessment,2009,23(8):1155 − 1167. doi: 10.1007/s00477-008-0289-4
[6] 胡旭冉. 基于集合卡尔曼滤波融合的西北太平洋海表面温度时空分布特征研究[D]. 上海:上海海洋大学,2018. [HU Xuran. Spatial and temporal distribution characteristics of sea surface temperature based on ensemble Kalman filtering fusion in the northwest Pacific Ocean[D]. Shanghai:Shanghai Ocean University,2018. (in Chinese with English abstract)]
HU Xuran. Spatial and temporal distribution characteristics of sea surface temperature based on ensemble Kalman filtering fusion in the northwest Pacific Ocean[D]. Shanghai: Shanghai Ocean University, 2018. （in Chinese with English abstract）
[7] 马超群. 基于集合卡尔曼滤波的大气化学资料同化技术研究[D]. 南京:南京大学,2020. [MA Chaoqun. Study of atmospheric chemistry data assimilation based on ensemble Kalman filter[D]. Nanjing:Nanjing University,2020. (in Chinese with English abstract)]
MA Chaoqun. Study of atmospheric chemistry data assimilation based on ensemble Kalman filter[D]. Nanjing: Nanjing University, 2020. （in Chinese with English abstract）
[8] 杨运,吴吉春,骆乾坤,等. 考虑预报偏差的迭代式集合卡尔曼滤波在地下水水流数据同化中的应用[J]. 水文地质工程地质,2022,49(6):13 − 23. [YANG Yun,WU Jichun,LUO Qiankun,et al. Application of the bias aware Ensemble Kalman Filter with Confirming Option(Bias-CEnKF) in groundwater flow data assimilation[J]. Hydrogeology & Engineering Geology,2022,49(6):13 − 23. (in Chinese with English abstract)]
YANG Yun, WU Jichun, LUO Qiankun, et al. Application of the bias aware Ensemble Kalman Filter with Confirming Option（Bias-CEnKF） in groundwater flow data assimilation[J]. Hydrogeology & Engineering Geology, 2022, 49（6）: 13 − 23. （in Chinese with English abstract）
[9] 高善露. 基于集合卡尔曼滤波的油藏参数反演方法研究[D]. 合肥:合肥工业大学,2019. [GAO Shanlu. Research of the ensemble Kalman filter for reservoir parameter inversion method[D]. Hefei:Hefei University of Technology,2019. (in Chinese with English abstract)]
GAO Shanlu. Research of the ensemble Kalman filter for reservoir parameter inversion method[D]. Hefei: Hefei University of Technology, 2019. （in Chinese with English abstract）
[10] GODOY V A,NAPA-GARCÍA G F,GÓMEZ-HERNÁNDEZ J J. Ensemble random forest filter:An alternative to the ensemble Kalman filter for inverse modeling[J]. Journal of Hydrology,2022,615:128642. doi: 10.1016/j.jhydrol.2022.128642
[11] ZOVI F,CAMPORESE M,HENDRICKS FRANSSEN H J,et al. Identification of high-permeability subsurface structures with multiple point geostatistics and normal score ensemble Kalman filter[J]. Journal of Hydrology,2017,548:208 − 224. doi: 10.1016/j.jhydrol.2017.02.056
[12] MORADKHANI H,HSU K L,GUPTA H,et al. Uncertainty assessment of hydrologic model states and parameters:Sequential data assimilation using the particle filter[J]. Water Resources Research,2005,41(5):W05012.
[13] ZHANG Jiangjiang,ZHENG Qiang,WU Laosheng,et al. Using deep learning to improve ensemble smoother:Applications to subsurface characterization[J]. Water Resources Research,2020,56(12):e2020WR027399. doi: 10.1029/2020WR027399
[14] CAO Zhendan,LI Liangping,CHEN Kang. Bridging iterative Ensemble Smoother and multiple-point geostatistics for better flow and transport modeling[J]. Journal of Hydrology,2018,565:411 − 421. doi: 10.1016/j.jhydrol.2018.08.023
[15] ZHOU Haiyan,GÓMEZ-HERNÁNDEZ J J,HENDRICKS FRANSSEN H J,et al. An approach to handling non-Gaussianity of parameters and state variables in ensemble Kalman filtering[J]. Advances in Water Resources,2011,34(7):844 − 864. doi: 10.1016/j.advwatres.2011.04.014
[16] XU Teng,JAIME GÓMEZ-HERNÁNDEZ J,ZHOU Haiyan,et al. The power of transient piezometric head data in inverse modeling:An application of the localized normal-score EnKF with covariance inflation in a heterogenous bimodal hydraulic conductivity field[J]. Advances in Water Resources,2013,54:100 − 118. doi: 10.1016/j.advwatres.2013.01.006
[17] TANG Q,KURTZ W,BRUNNER P,et al. Characterisation of river-aquifer exchange fluxes:The role of spatial patterns of riverbed hydraulic conductivities[J]. Journal of Hydrology,2015,531:111 − 123. doi: 10.1016/j.jhydrol.2015.08.019
[18] XU Teng,GÓMEZ-HERNÁNDEZ J J. Simultaneous identification of a contaminant source and hydraulic conductivity via the restart normal-score ensemble Kalman filter[J]. Advances in Water Resources,2018,112:106 − 123. doi: 10.1016/j.advwatres.2017.12.011
[19] SKJERVHEIM J A,EVENSEN G,HOVE J,et al. An Ensemble Smoother for assisted History Matching[C]// SPE Reservoir Simulation Symposium All Days. Woodlands:Society of Petroleum Engineers,2011:SPE-141929-MS.
[20] EMERICK A A,REYNOLDS A C. Ensemble smoother with multiple data assimilation[J]. Computers & Geosciences,2013,55:3 − 15.
[21] EVENSEN G. Analysis of iterative ensemble smoothers for solving inverse problems[J]. Computational Geosciences,2018,22(3):885 − 908. doi: 10.1007/s10596-018-9731-y
[22] 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. doi: 10.1175/1520-0493(2001)129<2776:DDFOBE>2.0.CO;2
[23] ANDERSON J L. Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter[J]. Physica D:Nonlinear Phenomena,2007,230(1/2):99 − 111.
[24] CHEN Yan,OLIVER D S. Cross-covariances and localization for EnKF in multiphase flow data assimilation[J]. Computational Geosciences,2010,14(4):579 − 601. doi: 10.1007/s10596-009-9174-6
[25] 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. doi: 10.1002/qj.49712555417
[26] BAKKER M,POST V,LANGEVIN C D,et al. Scripting MODFLOW model development using python and FloPy[J]. Ground Water,2016,54(5):733 − 739. doi: 10.1111/gwat.12413
[27] CHEN Zi,XU Teng,GÓMEZ-HERNÁNDEZ J J,et al. Contaminant spill in a sandbox with non-gaussian conductivities:Simultaneous identification by the restart normal-score ensemble Kalman filter[J]. Mathematical Geosciences,2021,53(7):1587 − 1615. doi: 10.1007/s11004-021-09928-y
[28] NASH J E,SUTCLIFFE J V. River flow forecasting through conceptual models part I—a discussion of principles[J]. Journal of Hydrology,1970,10(3):282 − 290. doi: 10.1016/0022-1694(70)90255-6
[29] GRAVEY M,MARIETHOZ G. QuickSampling v1.0:A robust and simplified pixel-based multiple-point simulation approach[J]. Geoscientific Model Development,2020,13(6):2611 − 2630. doi: 10.5194/gmd-13-2611-2020
[30] STREBELLE S. Multiple-point statistics simulation models:Pretty pictures or decision-making tools?[J]. Mathematical Geosciences,2021,53(2):267 − 278. doi: 10.1007/s11004-020-09908-8
[31] 崔凯鹏. 集合卡尔曼滤波在地下水流及溶质运移数据同化中的应用探讨[D]. 南京:南京大学,2013. [CUI Kaipeng. The application and discussion of ensemble Kalman filter method in groundwater flow and solute transport data assimilation[D]. Nanjing:Nanjing University,2013. (in Chinese with English abstract)]
CUI Kaipeng. The application and discussion of ensemble Kalman filter method in groundwater flow and solute transport data assimilation[D]. Nanjing: Nanjing University, 2013. （in Chinese with English abstract）
[32] WEN Xianhuan,GÓMEZ-HERNÁNDEZ J J. Numerical modeling of macrodispersion in heterogeneous media:A comparison of multi-Gaussian and non-multi-Gaussian models[J]. Journal of Contaminant Hydrology,1998,30(1/2):129 − 156.
-
图(4)
表(4)
计量
- 文章访问数: 431
- PDF下载数: 65
- 施引文献: 0