﻿ 基于正则化与恒星日滤波的BDS多路径误差削减方法
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 大地测量与地球动力学  2024, Vol. 44 Issue (2): 116-121  DOI: 10.14075/j.jgg.2023.04.180

### 引用本文

LI Xinzhong, XIONG Yongliang, XU Shaoguang. BDS Multipath Reduction Method Based on Regularization and Sidereal Filtering[J]. Journal of Geodesy and Geodynamics, 2024, 44(2): 116-121.

### Foundation support

National Natural Science Foundation of China, No.41674028, No.41274044; Science and Technology Program of Sichuan Province, No. 2022YFG0169.

### Corresponding author

XIONG Yongliang, PhD, professor, PhD supervisor, majors in the theories and applications of high precision GNSS, E-mail: ylxiong@sina.com.

### 第一作者简介

LI Xinzhong, PhD candidate, majors in the theories and applications of high precision GNSS, E-mail: xinz_li@126.com.

### 文章历史

1. 西南交通大学地球科学与环境工程学院，成都市犀安路999号，611756

1 理论模型 1.1 单差残差重构

 $\nabla \Delta \mathit{\Phi}_{\mathrm{rb}}^{\mathrm{jk}}=\nabla \Delta \rho_{\mathrm{rb}}^{\mathrm{jk}}+\lambda \nabla \Delta N_{\mathrm{rb}}^{\mathrm{jk}}+\nabla \Delta m+\nabla \Delta \varepsilon$ (1)

 $\mathit{\boldsymbol{Ds}}{{ = }}\mathit{\boldsymbol{d}}$ (2)

 $\begin{gathered} {\left[\begin{array}{ccccc} \omega_1 & \omega_2 & \omega_3 & \cdots & \omega_n \\ 1 & -1 & 0 & \cdots & 0 \\ 1 & 0 & -1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 0 & 0 & \cdots & -1 \end{array}\right]\left[\begin{array}{c} s_{\mathrm{rb}}^1 \\ s_{\mathrm{rb}}^2 \\ s_{\mathrm{rb}}^3 \\ \vdots \\ s_{\mathrm{rb}}^n \end{array}\right]=} \\ {\left[\begin{array}{c} \sum \omega_i s_{\mathrm{rb}}^i \\ s_{\mathrm{rb}}^1-s_{\mathrm{rb}}^2 \\ s_{\mathrm{rb}}^1-s_{\mathrm{rb}}^3 \\ \vdots \\ s_{\mathrm{rb}}^1-s_{\mathrm{rb}}^n \end{array}\right]=\left[\begin{array}{c} 0 \\ d_{\mathrm{rb}}^{12} \\ d_{\mathrm{rb}}^{13} \\ \vdots \\ d_{\mathrm{rb}}^{1 n} \end{array}\right]} \end{gathered}$ (3)

1.2 Tikhonov正则化建模 1.2.1 Tikhonov正则化建模提取多路径

 $\varphi_i=m_i+\varepsilon_i, i=1, 2, \cdots, n$ (4)

 $\begin{gathered} J\left(m_i\right)= \\ \sum\limits_{i=1}^n \omega_i\left(\varphi_i-m_i\right)^2+\alpha \sum\limits_{i=1}^{n-1}\left(m_{i+1}-m_i\right)^2 \end{gathered}$ (5)

 $\boldsymbol{J}=(\boldsymbol{\varphi}-\boldsymbol{m})^{\mathrm{T}} \boldsymbol{W}(\boldsymbol{\varphi}-\boldsymbol{m})+\alpha \boldsymbol{m}^{\mathrm{T}} \boldsymbol{R} \boldsymbol{m}$ (6)

 $(\boldsymbol{W}+\alpha \boldsymbol{R}) \hat{\boldsymbol{m}}=\boldsymbol{W} \boldsymbol{\varphi}$ (7)

1.2.2 正则化矩阵R的确定

 $\boldsymbol{m}^{\mathrm{T}} \boldsymbol{R} \boldsymbol{m}=\sum\limits_{i=1}^{n-1}\left(m_{i+1}-m_i\right)^2$ (8)

 $\boldsymbol{R}=\boldsymbol{G}^{\mathrm{T}} \boldsymbol{G}$ (9)

1.2.3 正则化参数α的确定

 $\eta_i=\omega_i\left(\varphi_i-\hat{m}_i\right), i=1, 2, \cdots, n$ (10)

 $\hat{m}(\alpha)=\frac{1}{B+1} \sum\limits_{b=0}^B \hat{m}_b(\alpha)$ (11)

b=0时，φ为原始载波相位单差残差。定义误差统计方法为：

 \begin{aligned} E(\alpha)= & \frac{1}{n B} \sum\limits_{b=0}^B\left[\hat{m}_b(\alpha)-\hat{m}(\alpha)\right]^{\mathrm{T}} \\ & {\left[\hat{m}_b(\alpha)-\hat{m}(\alpha)\right] } \end{aligned} (12)

 $\hat{\alpha}=\operatorname{argmin} E(\alpha), \alpha \in\{0.1, 1, 10, 50, 100, \cdots\}$ (13)

1) 采用自举法选取正则化参数$\hat{\alpha}$

2) 代入公式(7)，计算对应的多路径误差$\hat{m}$

3) 循环计算以0.1$\hat{\alpha}$为步长的区间$\left[{\hat \alpha - m\hat \alpha, \hat \alpha + \widehat {n\alpha }} \right]$内每一个αi值。mn根据经验选取，通过反复实验，当m=0.5、n=1时，在保证得到最优正则化参数的同时，计算效率处于可接受的范围内。

4) 重复步骤2)，计算每个αi值对应的多路径误差$\hat{m}_i$和模型误差E(αi)。

5) 输出min{E(αi)}对应的正则化参数αopt

2 数据处理与分析

 图 1 部分卫星在前后2个周期(doy237~238、doy231~238)的载波相位单差残差 Fig. 1 Residual of carrier phase single difference at doy237-238 and doy231-238 for partial satellites

 图 2 所有卫星对应的平均和分段多路径误差重复时间 Fig. 2 Mean MRT estimates and hourly MRT estimates for all satellites

 图 3 C01、C07于doy237，C28于doy231利用TB、TC和小波滤波方法提取的多路径误差 Fig. 3 Multipath error extracted by C01, C07 at doy237, C28 at doy231 using TB, TC and wavelet filtering method

 图 4 C04、C06和C19于doy238应用TB、TC和小波滤波方法恒星日滤波前后的载波相位SD残差 Fig. 4 Carrier phase SD residual of C04, C06 and C19 before and after sidereal filtering by applying TB, TC and wavelet filtering method on doy238

doy238恒星日滤波缓解多路径误差前后载波相位单差残差的均方根和改进量如图 5所示，可以看出，所有可见卫星对TB、TC和小波滤波的平均改进百分比分别为33.9%、40.5%和32.6%。鉴于BDS独特的星座特性，分别统计每类卫星的改进量：对于GEO卫星，采用TB、TC和小波滤波方法，平均改进量分别为37.9%、45.9%、36.7%；对于IGSO卫星，采用3种方法的平均改进量依次为30.5%、38.2%、26.4%；对于MEO卫星，采用3种方法平均改进33.3%、37.5%、34.5%。由此可以得出，采用TC方法应用恒星日滤波后，改进效果最佳。另外，对于GEO卫星的多路径改进的效果较其他两类卫星更为显著。

 图 5 doy238恒星日滤波前后载波相位单差残差的均方根和改进量 Fig. 5 RMS and improvement of carrier phase SD residual before and after sidereal filtering on doy238

doy238多路径误差改正前后均方根如表 2所示，可以看出，NU方向定位精度较E方向高，定位结果较优。3种方法改正效果都非常显著，经过TB、TC和小波滤波后，ENU方向改进分别为23%、26.3%、41.7%，24.8%、26.3%、42.7%和22.4%、23.7%、41.7%。总体上TC方法的多路径误差改正效果更优。

3 结语

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BDS Multipath Reduction Method Based on Regularization and Sidereal Filtering
LI Xinzhong1     XIONG Yongliang1     XU Shaoguang1
1. Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University, 999 Xi'an Road, Chengdu 611756, China
Abstract: On the basis of the reconstructed single difference residual of carrier phase, we estimate the multipath repetition time of Beidou satellite by using the segmentation idea; we then extract the multipath of single difference residual of carrier phase by regularization method and classical wavelet filtering method respectively to obtain a "clean" single difference residual sequence. The experimental results show that Tikhonov regularization is feasible to correctly extract the multipath signal, the multipath signal is smoother than the original residual measurement, and the estimation method of the regularization parameter is further optimized. After using the optimized Tikhonov regularization method and sidereal filtering, the single difference residual of carrier phase is improved by 40.5% on average. The coordinate residual in E, N and U directions are improved by 24.8%, 26.3% and 42.7%, respectively. The optimized Tikhonov regularization is superior to the traditional wavelet filtering method in both the observation and coordinate domains.
Key words: BDS; Tikhonov regularization; sidereal filtering; multipath error