Application of tidal harmonic analysis based on iteratively reweighted least squares method in the Qiantang River
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摘要: 基于普通最小二乘法的传统调和分析方法对噪声敏感，易受测量误差和强非潮过程干扰的影响。基于迭代权重最小二乘法的调和分析通过赋予异常值较小权重降低其干扰，相比普通最小二乘法可有效提高精度和稳定性，但目前仍没有研究对这两种方法在感潮河段潮位分析中精度差异进行系统性比较。本研究基于浙江钱塘江实测潮位数据，通过理想实验和实际实验，系统对比了两种方法。结果表明：（1）在短时序（<3个月）数据情况下，迭代权重最小二乘法结果精度优于普通最小二乘法，平均矢量差可减小超过2 cm；时序延长（>3个月）后，两者差异缩小。（2）在钱塘江下游，两种方法差异不大；但在中上游（如仓前至桐庐段）等受到强烈河流径流影响的河段，迭代权重最小二乘法对调和分析结果有所改善，尤其对长周期分潮的改善效果更为显著。（3）迭代权重最小二乘法通过有效抑制高噪声和异常值干扰，显著提升了钱塘江潮位站调和分析结果的稳定性和准确性。因此，在数据质量不佳或背景噪声较强的区域（例如感潮河段），基于迭代权重最小二乘法的调和分析方法具有重要应用价值。Abstract: Traditional harmonic analysis based on the ordinary least squares (OLS) method is sensitive to noise and susceptible to contamination by measurement errors and strong non-tidal processes. Harmonic analysis utilizing the iteratively reweighted least squares (IRLS) method reduces the influence of outliers by assigning them smaller weights, thereby effectively improving accuracy and stability compared to the OLS method. However, a systematic comparison of the precision of these two methods in tidal-level analysis within tidal rivers is still lacking. This study systematically compares the two methods using measured water level data from the Qiantang River in Zhejiang Province, China, through both idealized and practical experiments. The results indicate that: (1) For short time series (<3 months), the IRLS method yields more accurate results than OLS, with the mean vector difference reduced by over 2 cm, while the difference between the two methods diminishes as the time series lengthens (>3 months). (2) In the lower reaches of the Qiantang River, the difference between the two methods is minimal. However, in the middle to upper reaches (e.g., from Cangqian to Tonglu), where the river is strongly influenced by freshwater runoff, the IRLS method improves the harmonic analysis results, particularly for long-period constituents. (3) The IRLS method significantly enhances the stability and accuracy of harmonic analysis results for tidal stations along the Qiantang River by effectively suppressing high-level noise and outlier interference. Therefore, the IRLS-based harmonic analysis method holds significant application value in regions with poor data quality or high background noise, such as tidal rivers.
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图 2 2011年钱塘江水位时间序列（a）和富春江电站流量过程（b）
Fig. 2 Time series of water level in the Qiantang River (a) and discharge of Fuchun River Power Station (b) in 2011
图 4 对构造潮不同数据长度采用不同求解方法得到的分潮平均矢量差
Fig. 4 Mean vector differences of tidal constituents derived from different solution methods for varying lengths of synthetic tide data
图 6 钱塘江沿程潮位站M2分潮特征变化及两种方法比较
（子图a是采用OLS方法得到的调和常数；子图b是分别采用OLS方法和IRLS方法得到的两组调和常数的差异的绝对值；子图c是两种方法得到的调和常数的矢量差相对变化；子图d是两种方法得到的振幅差的相对变化；子图b、c、d中的本底值是OLS方法得到的结果，即子图a中的数据）
Fig. 6 Along-channel changes in M2 tidal characteristics and a comparison of two analysis methods
a. Harmonic constants from the OLS method, b. absolute differences between harmonic constants derived from the OLS and IRLS methods, c. relative vector differences, d. relative amplitude differences. The results from the OLS method (subplot a) serve as the baseline for subplot b, c, and d
图 9 钱塘江沿程潮位站MSf分潮特征变化及两种方法比较（子图说明类似图6）
Fig. 9 Along-channel changes in MSf tidal characteristics and a comparison of two analysis methods (subplot descriptions are similar to those in Fig. 6)
图 7 钱塘江沿程潮位站K1分潮特征变化及两种方法比较（子图说明类似图6）
Fig. 7 Along-channel changes in K1 tidal characteristics and a comparison of two analysis methods (subplot descriptions are similar to those in Fig. 6)
图 8 钱塘江沿程潮位站M4分潮特征变化及两种方法比较（子图说明类似图6）
Fig. 8 Along-channel changes in M4 tidal characteristics and a comparison of two analysis methods (subplot descriptions are similar to those in Fig. 6)
图 10 迭代权重最小二乘法在乍浦站和桐庐站分别采用的权重值
Fig. 10 Weights applied by the IRLS method at the Zhapu and Tonglu stations
图 11 OLS与IRLS方法回报水位的差异比较
a. 不同站位水位差值的时间序列；b. 总水位及分潮组分差异最大值的沿程分布
Fig. 11 Comparison of water level differences between OLS and IRLS hindcasts
a. Time series of water level differences at different stations, b. along-channel distribution of the maximum differences in total water level and tidal constituents
B1 钱塘江沿程潮位站S2分潮特征变化及两种方法比较（子图说明类似图6）
B1 Along-channel changes in S2 tidal characteristics and a comparison of two analysis methods (subplot descriptions are similar to those in Fig. 6)
B2 钱塘江沿程潮位站O1分潮特征变化及两种方法比较（子图说明类似图6）
B2 Along-channel changes in O1 tidal characteristics and a comparison of two analysis methods (subplot descriptions are similar to those in Fig. 6)
B3 钱塘江沿程潮位站MS4分潮特征变化及两种方法比较（子图说明类似图6）
B3 Along-channel changes in MS4 tidal characteristics and a comparison of two analysis methods (subplot descriptions are similar to those in Fig. 6)
B4 钱塘江沿程潮位站Mf分潮特征变化及两种方法比较（子图说明类似图6）
B4 Along-channel changes in Mf tidal characteristics and a comparison of two analysis methods (subplot descriptions are similar to those in Fig. 6)
B5 钱塘江沿程潮位站Sa分潮特征变化及两种方法比较（子图说明类似图6）
B5 Along-channel changes in Sa tidal characteristics and a comparison of two analysis methods (subplot descriptions are similar to those in Fig. 6)
表 1 乍浦站各分潮调和常数
Tab. 1 Harmonic constants of the tidal constituents at Zhapu
分潮类型 名称 振幅/cm 迟角/（°） 信噪比 半日潮 M2 228.5 3.0 40447.0 S2 85.3 65.9 5578.9 N2 39.1 345.5 1193.5 K2 22.2 57.9 361.6 全日潮 K1 33.5 220.1 860.6 O1 19.7 171.8 295.2 P1 8.3 226.8 53.1 Q1 3.2 149.4 7.8 浅水分潮 M4 13.3 276.2 138.2 MS4 12.2 334.6 114.6 MN4 4.6 260.1 16.9 M6 8.3 130.6 55.0 注：表中迟角基准时间为东八区（UTC+8），下同。 
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表 2 采用不同权重函数求解得到的分潮平均矢量差统计（单位：cm）
Tab. 2 Mean vector difference statistics of tidal constituents for different weighting functions (unit: cm)
方法 矢量差 1个月 2个月 3个月 4个月 5个月 6个月 9个月 12个月 普通最小二乘法 10.8 3.3 0.6 0.4 0.3 0.3 0.2 0.1 迭代权重
最小二
乘法Cauchy 9.7 1.2 0.5 0.4 0.3 0.2 0.2 0.1 Andrews 9.6 1.2 0.5 0.4 0.3 0.2 0.2 0.1 Bisquare 9.6 1.2 0.5 0.4 0.3 0.2 0.2 0.1 Fair 10.0 1.3 0.5 0.4 0.3 0.2 0.2 0.1 Huber 9.3 1.2 0.5 0.4 0.3 0.2 0.2 0.1 Logistic 9.8 1.2 0.5 0.4 0.3 0.2 0.2 0.1 Talwar 9.5 1.7 1.2 0.6 0.4 0.3 0.2 0.1 Welsch 9.6 1.1 0.5 0.4 0.3 0.2 0.2 0.1
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表 3 两种方法得到的结果比较
Tab. 3 Comparison of the results from the two methods
调和常数 M2 K1 振幅/cm 迟角/（°） 振幅/cm 迟角/（°） 天文潮 228.5 3.0 33.5 220.1 普通最小二乘法 223.5 2.9 41.8 233.5 差值 −5.0 −0.1 8.3 13.4 迭代权重最小二乘法（Huber） 228.7 3.1 33.6 220.5 差值 0.2 0.1 0.1 0.4 迭代权重最小二乘法（Cauchy） 228.5 3.0 33.5 220.1 差值 0.0 0.0 0.0 0.0
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表 4 两种方法回报水位差异（绝对值）比较（单位：cm）
Tab. 4 Comparison of hindcasted water level differences (absolute value) between the OLS and IRLS methods (unit: cm)
乍浦 澉浦 仓前 七堡 闸口 富阳 桐庐 总水位 最大值 2.9 10.2 12.2 20.1 27.7 47.8 78.6 平均值 0.7 2.4 3.0 6.2 9.0 15.4 24.4 长周期分潮组分 最大值 1.2 5.6 7.4 16.6 25.1 45.2 73.7 平均值 0.4 1.7 2.6 5.9 8.9 15.4 24.3 全日潮组分 最大值 0.5 1.0 1.8 1.5 1.4 0.9 4.5 平均值 0.2 0.3 0.6 0.5 0.5 0.3 1.5 半日潮组分 最大值 1.5 4.4 3.6 4.2 2.6 2.2 3.6 平均值 0.5 1.6 1.2 1.4 0.9 0.7 1.4 1/4分潮组分 最大值 1.0 2.0 2.7 4.2 2.4 0.7 1.6 平均值 0.3 0.6 0.9 1.3 0.8 0.2 0.5
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表 A1 IRLS方法不同权重函数及特点
Tab. A1 Different weighting functions and their characteristics for the IRLS method
分类 方法 权重函数 说明 连续衰减型 Cauchy ${ \omega =\dfrac{1}{1+{r}^{2}}} $ 对异常值的惩罚相对“温和”但持续，对中等残差的数据点就有明显的降权效果，但对极大残差不会像某些函数那样急剧降权。特点是衰减曲线平滑，重尾分布，对极端值不敏感。 Logistic ${ \omega =\dfrac{\mathrm{tanh} (r)}{r} }$ 与Cauchy类似，也是一种平滑的S形曲线，但通常从1衰减到0的速度和形状略有不同，提供另一种连续衰减的选择。特点是平滑衰减，有明确的上下渐近线。 Fair ${ \omega =\dfrac{1}{1+\left| r\right| } }$ 提供了一个非常平滑且简单的衰减过程，计算效率高。特点是计算简单，权重随|r|线性倒数衰减，对异常值的处理相对温和。 Welsch $ {\omega ={{\mathrm{e}}}^{-{{r}^{2}}}} $ 对小幅残差给予近乎全额的权重，但当残差超过一定阈值后，权重会非常迅速地衰减至零。特点是初期衰减缓慢，后期（大残差时）衰减极为迅速，对显著异常值非常“强硬”。 二次衰减型 Bisquare ${ \omega =\begin{cases} {(1-{{r}^{2}})}^{2},\left| r\right| < 1\\ 0,\left| r\right| \geqslant 1\\ \end{cases}} $ 是一种“硬”阈值方法，将超过阈值的点完全排除在计算之外。特点是对异常值的惩罚非常“严厉”且明确，能完全忽略极端异常值。 线性衰减与
混合范数型Huber ${ \omega =\dfrac{1}{\mathrm{max} (1,\left| r\right| )}} $ 保留小残差数据的高效率，同时对大残差施加更稳健的线性惩罚。特点是兼顾效率与稳健性，是广泛使用的基准函数。 Andrews $ {\omega =\begin{cases} \dfrac{\mathrm{sin} \left(r\right)}{r},\left| r\right| < {\text{π}} \\ 0,\left| r\right| \geqslant {\text{π}} \\ \end{cases}} $ 在阈值内提供了一种非线性的衰减方式，之后直接截断。特点是基于正弦函数，在阈值内提供独特的非线性衰减。 硬截断型 Talwar ${ \omega =\begin{cases} 1,\left| r\right| < 1\\ 0,\left| r\right| \geqslant 1\\ \end{cases}} $ 非常简单粗暴，将所有数据点分为“完全接受”和“完全拒绝”两类。特点是最为严格，计算简单，但可能因忽略过多信息而导致效率低下。
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表 A2 桐庐站和澉浦站采用OLS和IRLS方法得到的调和常数
Tab. A2 Harmonic constants derived from the OLS and IRLS methods at Tonglu and Ganpu
分潮名称 桐庐站振幅(cm)/迟角(°)/信噪比 澉浦站振幅(cm)/迟角(°)/信噪比 OLS方法 IRLS方法 OLS方法 IRLS方法 M2 22.2/ 220.7/80.0 24.3/221.2/76.0 262.0/18.9/ 100000.0 263.9/18.6/ 98000.0 S2 14.6/296.8/34.0 13.7/298.0/24.0 100.0/81.3/ 14000.0 99.1/81.2/ 14000.0 N2 5.0/189.3/4.0 4.8/187.0/3.0 42.7/356.2/ 2600.0 43.0/356.7/ 2600.0 K2 6.4/290.3/6.4 6.3/295.3/4.9 25.0/76.2/860.0 24.4/77.3/800.0 K1 7.5/12.2/9.0 8.6/19.1/9.5 33.6/225.2/ 1600.0 33.7/224.8/ 1600.0 O1 8.8/278.0/12.0 8.1/282.7/8.3 19.7/175.0/550.0 19.5/174.1/520.0 P1 10.7/353.0/18.0 11.4/0.8/17.0 8.4/230.6/100.0 8.1/229.2/92.0 Q1 1.7/235.0/0.4 2.2/244.3/0.6 3.0/153.4/13.0 2.9/154.1/11.0 M4 6.9/17.4/7.8 7.3/19.4/6.9 24.6/343.7/890.0 24.5/341.9/850.0 MS4 7.5/84.4/9.3 7.9/88.7/8.0 21.7/41.3/680.0 21.5/41.9/650.0 MN4 2.3/359.0/0.8 2.5/4.3/0.8 8.7/320.8/110.0 8.7/323.8/110.0 M6 1.8/150.9/0.5 1.8/154.1/0.4 6.8/215.8/68.0 7.2/212.6/74.0 MSf 67.0/77.8/730.0 57.2/79.7/420.0 4.1/86.5/25.0 2.7/84.9/10.0 MSm 26.5/298.8/110.0 5.2/328.7/3.3 1.8/52.4/4.6 2.1/68.6/5.8 Mf 9.1/65.6/13.0 17.1/61.5/37.0 3.2/359.8/15.0 2.6/13.6/9.6 Sa 53.8/185.3/470.0 25.5/217.3/83.0 23.8/229.5/810.0 21.8/233.9/660.0
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