﻿ 基于MSSA的区域GPS站点季节性信号提取
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 大地测量与地球动力学  2019, Vol. 39 Issue (5): 516-520, 543  DOI: 10.14075/j.jgg.2019.05.015

### 引用本文

WANG Hao, YUE Jianping, XIANG Yunfei, et al. Seasonal Signals Extraction of Regional GPS Stations Based on MSSA[J]. Journal of Geodesy and Geodynamics, 2019, 39(5): 516-520, 543.

### About the first author

WANG Hao, postgraduate, majors in GNSS coordinate time series analysis, E-mail:austwanghao@163.com.

### 文章历史

1. 河海大学地球科学与工程学院，南京市佛城西路8号，211100

1 多通道奇异谱分析

MSSA的研究对象为多维中心化时间序列xi(l)，上标l表示通道序号，下标i为时间序号，l=0, 1, 2, …, Li=0, 1, 2, …, N。分别将第l通道时间序列xi(l)按照嵌入维数M、时滞1排列成M行、NM+1列的时滞矩阵，则：

 $\mathit{\boldsymbol{X}}{\rm{ }} = \left[ {\begin{array}{*{20}{c}} {{\rm{ }}x_1^{\left( 1 \right)}}&{x_2^{\left( 1 \right)}}&{ \ldots {\rm{ }}}&{x_{i + 1}^{\left( 1 \right)}}&{ \ldots {\rm{ }}}&{{\rm{ }}x_{N - M + 1}^{\left( 1 \right)}}\\ \vdots &{ \vdots }& \ddots &{ \vdots }& \ddots &{ \vdots }\\ {x_M^{\left( 1 \right)}}&{x_{M + 1}^{\left( 1 \right)}}&{ \ldots {\rm{ }}}&{x_{i + M}^{\left( 1 \right)}}&{ \ldots {\rm{ }}}&{x_N^{\left( 1 \right)}}\\ {x_1^{\left( 2 \right)}}&{x_2^{\left( 2 \right)}}&{ \ldots {\rm{ }}}&{x_{i + 1}^{\left( 2 \right)}}&{ \ldots {\rm{ }}}&{x_{N - M + 1}^{\left( 2 \right)}}\\ { \vdots }&{ \vdots }& \ddots &{ \vdots }& \ddots &{ \vdots }\\ {x_M^{\left( 2 \right)}}&{x_{M + 1}^{\left( 2 \right)}}&{ \ldots {\rm{ }}}&{x_{i + M}^{\left( 2 \right)}}&{ \ldots {\rm{ }}}&{x_N^{\left( 2 \right)}}\\ { \vdots }&{ \vdots }& \ddots &{ \vdots }& \ddots &{ \vdots }\\ {x_1^{\left( L \right)}}&{x_2^{\left( L \right)}}&{ \ldots {\rm{ }}}&{x_{i + 1}^{\left( L \right)}}&{ \ldots {\rm{ }}}&{x_{N - M + 1}^{\left( L \right)}}\\ { \vdots }&{ \vdots }& \ddots &{ \vdots }& \ddots &{ \vdots }\\ {x_M^{\left( L \right)}}&{x_{M + 1}^{\left( L \right)}}&{ \ldots {\rm{ }}}&{x_{i + M}^{\left( L \right)}}&{ \ldots {\rm{ }}}&{x_N^{\left( L \right)}} \end{array}} \right]$ (1)

 $\mathit{\boldsymbol{C}}{_X} = \left[ {\begin{array}{*{20}{c}} {{\rm{ }}\mathit{\boldsymbol{C}}{_{11}}}&{\mathit{\boldsymbol{C}}{_{12}}}&{ \ldots {\rm{ }}}&{\mathit{\boldsymbol{C}}{_{1L}}}\\ {\mathit{\boldsymbol{C}}{_{21}}}&{\mathit{\boldsymbol{C}}{_{22}}}&{ \ldots {\rm{ }}}&{\mathit{\boldsymbol{C}}{_{2L}}}\\ \vdots & \vdots & \ddots & \vdots \\ {\mathit{\boldsymbol{C}}{_{1L}}}&{\mathit{\boldsymbol{C}}{_{L2}}}&{ \ldots {\rm{ }}}&{\mathit{\boldsymbol{C}}{_{LL}}} \end{array}} \right]$ (2)

 ${(\mathit{\boldsymbol{C}}{_{ll\prime }})_{j, j\prime }} = \frac{1}{{{\rm{ }}N - \left| {j - j\prime } \right|}}\sum\limits_{i = 1{\rm{ }}}^{N - \left| {j - j\prime } \right|} {} {\rm{ }}x_{i + j}^{(l)}x_{i + j - j\prime }^{(l\prime )}$ (3)

 ${a_{i, k}} = {\rm{ }}{\mathit{\boldsymbol{X}}_i}{\mathit{\boldsymbol{P}}_k} = \sum\limits_{l = 1}^L {} \sum\limits_{j = 1}^M {} x_{i + j}^{(l)}P_{j, k}^{(l)}$ (4)

 $\begin{array}{l} x_{i, k}^{(l)} = \\ \left\{ \begin{array}{l} \frac{1}{{i{\rm{ }}}}\sum\limits_{j = 1}^i {} {\rm{ }}{a_{i - j, k}}P_{j, k}^{(l)}, 1 \le i \le M - 1\\ \frac{1}{{M{\rm{ }}}}\sum\limits_{j = 1}^M {} {\rm{ }}{a_{i - j, k}}P_{j, k}^{(l)}, {\rm{ }}M \le i \le N - M + 1\\ \frac{1}{{N - i + 1}}\sum\limits_{j = i - N + M}^M {} {\rm{ }}{a_{i - j, k}}P_{j, k}^{(l)}, N - M + 2 \le i \le N \end{array} \right. \end{array}$ (5)

L×M个RC线性相加应与原始序列xi(l)相同。为了达到去除噪声的效果，可选取前K项重建成分RC重构原始序列主要信号：

 $\hat x_i^{\left( l \right)} = \sum\limits_{k = 1}^K {} {\rm{ }}x_{j, k}^{(l)}$ (6)

2 实验与分析 2.1 实验数据

2.2 季节性信号提取

 图 1 BJFS与TAIN两站ST-EOF 1~6仿真结果 Fig. 1 Simulation results of BJFS and TAIN two stations ST-EOF 1~6

 图 2 BJFS与TAIN两站重建成分RC 1~6的仿真结果 Fig. 2 Simulation results of BJFS and TAIN two stations reconstruction components RC 1~6

 图 3 采用MSSA提取4个站点时间序列的1 a项和0.5 a项 Fig. 3 Use MSSA to extract annual and semi-annual items of four stations time series

2.3 提取效果对比

 图 4 采用SSA提取4站时间序列的1 a项和0.5 a项 Fig. 4 Use SSA to extract annual and semi-annual items of four stations time series

 图 5 SSA/MSSA提取季节信号时奇异谱方差所占百分比 Fig. 5 The percentage of singular spectral variance of seasonal signal extractionby SSA or MSSA

 图 6 SSA、MSSA处理BJFS与TAIN站点时间序列时前10个重建成分重构信号对比 Fig. 6 Comparison of the top ten reconstruction components reconstructed signals of BJFS and TAIN stations time series by SSA or MSSA

 图 7 BJFS与TAIN两站坐标时间序列功率谱分析 Fig. 7 Power spectrum analysis of BJFS and TAIN two stations coordinate time series
3 结语

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Seasonal Signals Extraction of Regional GPS Stations Based on MSSA
WANG Hao1     YUE Jianping1     XIANG Yunfei1     ZHANG Chengcai1
1. School of Earth Sciences and Engineering, Hohai University, 8 West-Focheng Road, Nanjing 211100, China
Abstract: In this paper, MSSA is used to simultaneously extract seasonal signals from multiple GPS stations' elevation time series, mainly including annual and semi-annual items. The experiment verifies that MSSA can identify and extract seasonal signals in time series of GPS stations. At the same time, MSSA can use the reconstruction components(RC) to reconstruct the signals which can extract the useful signals from the original time series and remove interference noise. By comparing the percentages of the top 10 singular spectral variance and the reconstructed signals obtained from MSSA and SSA, it is shown that the MSSA reconstruction curve only contains the time-varying and common seasonal signals of multiple stations, effectively eliminating the influence of single site-specific noise and local phenomenon.
Key words: MSSA; seasonal signals extraction; SSA; reconstructed signal