﻿ 基于改进Cao算法的奇异谱分析法及其在北斗多路径去噪中的应用
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 大地测量与地球动力学  2019, Vol. 39 Issue (1): 25-30  DOI: 10.14075/j.jgg.2019.01.005

引用本文

YU Bin, YANG Shaomin. Singular Spectrum Analysis Based on Improved Cao Algorithm and Its Application in Beidou Multipath Filtering[J]. Journal of Geodesy and Geodynamics, 2019, 39(1): 25-30.

Foundation support

National Natural Science Foundation of China, No. 41574017.

Corresponding author

YANG Shaomin, PhD, researcher, majors in geodesy and geodynamics, E-mail:269047966@qq.com.

第一作者简介

YU Bin, postgraduate, majors in high precision GNSS data processing, E-mail: 719117834@qq.com.

文章历史

1. 中国地震局地震研究所地震大地测量重点实验室，武汉市洪山侧路40号，430071;
2. 中国地震局地壳应力研究所武汉科技创新基地，武汉市洪山侧路40号，430071

1 SSA基本原理 1.1 SSA算法

1) 计算轨迹矩阵。

 $\begin{array}{l} \mathit{\boldsymbol{X}}{\rm{ = }}\left[{{X_1}, {X_2}, \cdots , {X_K}} \right] = \left( {{x_{ij}}} \right)_{i, j = 1}^{L, K} = \\ \;\;\;\;\left[\begin{array}{l} {x_1}\;\;\;\;{x_2}\;\;\; \cdots \;\;\;{x_{N - L + 1}}\\ {x_2}\;\;\;\;{x_3}\;\;\; \cdots \;\;\;{x_{N - L + 2}}\\ \; \vdots \;\;\;\;\;\; \vdots \;\;\;\;\; \ddots \;\;\;\;\;\;\; \vdots \\ {x_L}\;\;\;{x_{L + 1}}\;\; \cdots \;\;\;\;\;\;{x_N} \end{array} \right] \end{array}$ (1)

2) 奇异值分解。

3) 分组。

4) 重构。

 ${y_k} = \left\{ \begin{array}{l} \frac{1}{k}\sum\limits_{p = 1}^k {y_{p, k - p + 1}^*, 1 \le k \le {L^*}} \\ \frac{1}{{{L^*}}}\sum\limits_{p = 1}^{{L^*}} {y_{p, k - p + 1}^*, {L^*} \le k \le {K^*}} \\ \frac{1}{{N - k + 1}}\sum\limits_{p = k - {K^*} + 1}^{N - {K^*} + 1} {y_{p, k - p + 1}^*, {K^*} \le k \le N} \end{array} \right.$ (2)

 ${x_n} = \sum\limits_{k = 1}^p {\tilde x_n^k, n = 1, 2, \cdots , N}$ (3)
1.2 Cao算法确定嵌入维数

 $\begin{array}{l} d\left( {i, M} \right) = \frac{{\left\| {{\mathit{\boldsymbol{X}}_i}\left( {M + 1} \right) - {\mathit{\boldsymbol{X}}_{n\left( {i, M} \right)}}\left( {M + 1} \right)} \right\|}}{{\left\| {{\mathit{\boldsymbol{X}}_i}\left( M \right) - {\mathit{\boldsymbol{X}}_{n\left( {i, M} \right)}}\left( M \right)} \right\|}}, \\ \;\;\;\;\;\;\;\;i = 1, 2, \cdots , N - M \end{array}$ (4)

 $E\left( M \right) = \frac{1}{{N - M}}\sum\limits_{i = 1}^{N - M} {d\left( {i, M} \right)}$ (5)

 ${E_1}\left( M \right) = \frac{{E\left( {M + 1} \right)}}{{E\left( M \right)}}$ (6)

MM0时，E1(M)停止变化或者波动较小，则最小嵌入维数为M0+1。

1.3 改进Cao算法

Cao算法确定嵌入维数L是以E1(M)停止变化为标准。然而对于实际时间序列来说，E1(M)起伏不定，很难达到严格的稳定状态，因此实际中通过主观判断确定最佳嵌入维数L有一定的困难。针对此问题，本文提出一种嵌入维数的稳定性判定准则用于最佳嵌入维数的确定，计算过程如下：

1) 定义Δi=$\left| {\frac{{{E_1}\left( {i + 1} \right)}}{{{E_1}\left( i \right)}} - 1} \right|$, 1≤iN-1；

2) 根据E1序列的波动设定阈值ε1，遍历E1序列得到第一个Δiε1对应的下标j

3) 令ε2=${\mathit{\bar \Delta }_i}$, jiN-1，${\mathit{\bar \Delta }_i}$表示取平均值；

4) 若Δi+2Δi+1Δiε2, jiN-2，记录此时Δi的值，并找到最小Δi对应的下标k，则嵌入维数L=k+1。

2 算例分析

 图 1 连续3 d坐标残差序列 Fig. 1 Residual sequence for three consecutive days

 图 2 嵌入维度L和重构阶次P的选取 Fig. 2 The selection of embedded dimension and reconstitution order

 图 3 SSA、db10小波重构多路径误差序列 Fig. 3 The sequence of multipath error reconstituted by SSA and db10 wavelet

3 北斗多路径误差周期计算

3.1 多路径误差时间序列相关性计算

3.2 北斗卫星轨道重复周期计算

 $\left\{ \begin{array}{l} n = \sqrt {GM} /{a^{\frac{3}{2}}} + \Delta n\\ {T_0} = 2\pi /n\\ {T_R} = 2{T_0} \end{array} \right.$ (9)

 图 4 广播星历计算得到的北斗卫星轨道重复周期 Fig. 4 Beidou satellite orbital repetition periods from broadcast ephemeris
4 基于北斗多路径误差周期的恒星日滤波

 图 5 doy175多路径误差改正后坐标序列 Fig. 5 The coordinate series of doy 175 after multipath error correction was applied

 图 6 doy176多路径误差改正后坐标序列 Fig. 6 The coordinate series of doy 176 after multipath error correction was applied

5 结语

1) 本文提出的嵌入维数的稳定性判定准则大大降低了Cao算法的主观性，将其运用到SSA中能显著提高SSA算法的准确性和计算效率。

2) 北斗系统3类卫星有着不同的轨道运动特性，不同类型的卫星轨道重复周期变化趋势不同，其中GEO卫星轨道重复周期变化幅度最大，IGSO卫星次之，MEO卫星轨道重复周期的变化幅度最小。北斗系统多路径误差周期约为86 160 s。

3) 基于SSA的北斗恒星日滤波算法相比基于小波的恒星日滤波算法能更有效地消除多路径误差的影响，经SSA方法滤波后的坐标序列精度优于小波滤波结果，利用该滤波方法可显著提高北斗动态变形监测的精度。

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Singular Spectrum Analysis Based on Improved Cao Algorithm and Its Application in Beidou Multipath Filtering
YU Bin1,2     YANG Shaomin1,2
1. Key Laboratory of Earthquake Geodesy, Institute of Seismology, CEA, 40 Hongshance Road, Wuhan 430071, China;
2. Wuhan Base of Institute of Crustal Dynamics, CEA, 40 Hongshance Road, Wuhan 430071, China
Abstract: In this paper, we study Beidou sidereal filtering algorithm based on improved SSA. The embedding dimension is determined by phase space reconstitution Cao algorithm, and we propose an improvement for the shortcomings of Cao algorithm. Thus, the computational efficiency and accuracy of the SSA method are improved. The repetitive cycle characteristics of different constellation satellites of Beidou system are analyzed, and the repeat cycle of the Beidou multipath error is estimated to be 86 160 s. The results of Beidou short baseline solution are processed by sidereal filtering based on singular spectrum analysis method and traditional wavelet method. The results indicate that the effect of SSA multipath filtering method are better than the wavelet method, and the multipath error in original coordinate sequence can be eliminated well.
Key words: singular spectrum analysis; improved Cao algorithm; repeat cycle; sidereal filtering