Estimation of conductivity fields by using a correlation-based localization scheme of iterative ensemble smoother
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摘要:
在地下水流和溶质运移问题中，有较多研究基于物理距离局域化集合同化方法反演水文地质参数。当反演参数与观测信息之间不存在物理距离时，这种方法不适用。为了克服这个局限，通过渗透系数与水头信息之间的相关性计算局域化方法的阻滞因子，构建基于相关性的局域化迭代集合平滑方法。为了方便比较，将该方法和一种基于物理距离的局域化迭代集合平滑一同用于同化水头信息反演二维孔隙承压含水层的渗透系数场。算例中考虑了不同集合大小、观测误差及观测数量等因子的组合，便于分析其对渗透系数反演精度的影响。研究结果显示：（1）在所有算例中新方法得到的渗透系数均方根误差范围为[0.8307, 0.9590]，都小于基于物理距离方法的均方根误差，范围为[0.8394, 1.0000]；（2）基于物理距离方法得到的渗透系数场空间上存在不连续性，而新方法的结果不存在此现象。文章提出了一种新的基于相关性局域化迭代平滑方法，该方法不需要依赖参数与观测信息之间的物理距离且参数反演精度高于基于物理距离的方法，可作为参数反演的科学工具。
Abstract:In the studies of groundwater flow and solute transport, many efforts have been made to estimate hydrogeological parameters through physical-distance based localization schemes of ensemble assimilation approaches. However, these methods are unavailable when there are no physical distances between parameters and observations. To avoid this limitation, we calculate the tapering factors in terms of the correlation coefficients between parameters and observations and develop a novel correlation-based localization scheme of iterative ensemble smoother. For the purpose of comparison, a physical distance-based scheme of iterative ensemble smoother together with the new approach are used to assimilate hydraulic head information and estimate the hydraulic conductivity field of a 2D confined aquifer. Among the test cases, we consider different configurations of ensemble size, observation error and number of observations, and their impacts on the accuracy of conductivity estimation can be well explored. The results show that (1) the root mean square error of hydraulic conductivity, RMSE, obtained through the new approach for each test case of interest, is lower than its counterpart through the physical distance-based approach, with the RMSE ranges of [0.8307, 0.9590] and [0.8394, 1.0000] for all test cases through the two approaches, respectively. (2) The estimated conductivity field has discontinuities when using the physical distance-based approach, but this does not happen when using the new approach. In this study, we develop a novel correlation-based localization scheme of iterative ensemble smoother, which is free of the definitions of physical distances between parameters and observations, yields higher accuracy of parameter estimation in comparison with the physical distance-based approach, and can be a useful tool for estimating hydrogeological parameters.
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图 3 当N = 50，100，500时，通过DL_iES和CL_iES在第20次迭代后获得的Y估计场和Y方差(
$\sigma _{\text{Y}}^2$ )估计场Figure 3.
图 4 在Nmon = 16，48，168时，通过DL_iES和CL_iES在第20次迭代后获得的Y估计场和Y方差(
$\sigma _{\text{Y}}^2$ )估计场Figure 4.
图 5 当N = 50，100，500和100000的
$\left| {{\rho _{k1}}} \right|$ 及${r_{k1}}\left| {{\rho _{k1}}} \right|$ 分布图Figure 5.
表 1 第1组第20次迭代后DL_iES和CL_iES得到的RMSE、SY和Eh值
Table 1. Final values of RMSE, SY and Eh through DL_iES and CL_iES for Group 1 at the 20th iteration
参数 模型 集合大小 50 100 500 RMSE DL_iES 1.0000 0.9779 0.8394 CL_iES 0.9590 0.9151 0.8307 SY DL_iES 0.9901 0.9689 0.7887 CL_iES 0.9359 0.9016 0.8424 Eh DL_iES 0.1132 0.1115 0.0829 CL_iES 0.1059 0.0978 0.0858 
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表 2 第2组第20次迭代后DL_iES和CL_iES得到的RMSE、SY和Eh值
Table 2. Final values of RMSE, SY and Eh through DL_iES and CL_iES for Group 2 at the 20th iteration
参数 模型 观测误差 0.1 0.01 0.001 RMSE DL_iES 0.9744 0.9779 0.9801 CL_iES 0.9187 0.9151 0.9162 SY DL_iES 0.9691 0.9689 0.9690 CL_iES 0.9029 0.9016 0.9015 Eh DL_iES 0.2728 0.1115 0.0789 CL_iES 0.2677 0.0978 0.0571
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表 3 第3组第20次迭代后DL_iES和CL_iES得到的RMSE、SY和Eh值
Table 3. Final values of RMSE, SY and Eh through DL_iES and CL_iES for Group 3 at the 20th iteration
参数 模型 观测数量 16 48 168 RMSE DL_iES 0.9999 0.9779 0.9575 CL_iES 0.9537 0.9151 0.8974 SY DL_iES 0.9862 0.9689 0.9543 CL_iES 0.9281 0.9016 0.8850 Eh DL_iES 0.1023 0.1115 0.1090 CL_iES 0.0948 0.0978 0.0932
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表 4 第4组第20次迭代后CL_iES得到的RMSE、SY和Eh值
Table 4. Final values of RMSE, SY and Eh through CL_iES for Group 4 at the 20th iteration
参数 相关系数高斯白噪音标准差的倍数 1.0 1.5 2.0 2.5 3.0 RMSE 0.8782 0.8920 0.9151 0.9443 0.9706 SY 0.8367 0.8715 0.9016 0.9273 0.9500 Eh 0.0901 0.0936 0.0978 0.1027 0.1071
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