# Potential Contributors to Common Mode Error in Array GPS Displacement Fields in Taiwan Island

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Data Processing

#### 2.1. GNSS Data Processing

#### 2.2. Principal Component Analysis

#### 2.3. Independent Component Analysis

- Centralize and whiten the observed data.
- Choose an initial weight vector of unit norm $w$.
- Update ${w}^{+}$ through ${w}^{+}=E\left[Zg\left({w}^{\mathrm{T}}Z\right)\right]-E\left[{g}^{\prime}\left({w}^{\mathrm{T}}Z\right)\right]w$.
- Normalize $w$ by $w={w}^{+}/{w}^{+}$.
- Return to step 3 if $w$ is not converged.

## 3. Results

#### 3.1. Common Mode Error Extraction Using PCA/ICA

#### 3.2. Noise Analysis of GNSS Coordinate Time Series

## 4. Discussion

#### 4.1. Potential Geophysical Interpretation of the CME

#### 4.2. Effect of Removing the CME

## 5. Conclusions

- Both PCA and ICA can effectively remove the CME. The average RMS of PCA and ICA in the U direction shifted from 6.47 mm to 5.58 mm and 5.40 mm, respectively, a decreased by approximately 14% and 17%. However, the CMEs of the two approaches reveal notable differences in their temporal and spatial characteristics. Figure 6 shows that the PCA separates only one CME and the ICA separates two CMEs. We therefore believe that PCA may eliminate the original site information, whereas ICA retains more original site information.
- There are notable differences in the ICA filtering effect between the east and west of Taiwan. The RMS reduction rate in the west is relatively large, whereas that in the east small, which is partially due to topography. There are many mountains on the eastern side of Taiwan and the stations are sparsely distributed, whereas the western side is relatively flat and the stations are relatively dense. Another explanation is the orogenic processes, as there is a topographic uplift in the eastern region and sinking in the southwest region due to groundwater extraction.
- To explore the possible geophysical sources of ICA’s CMEs, we compare the CMEs of ICA with the predicted average loading displacements caused by changes in the atmospheric and hydrological loadings. It is found that GPS_IC1 and ATML, and GPS_IC2 and LWS are consistent in terms of amplitude and characteristics. The correlation between GPS_IC1 and ATML is 0.55, and the correlation coefficient between GPS_IC2 and LWS is 0.40. This indicates that seasonal changes in Taiwan are related to the movement of water in addition to atmospheric pressure.
- We used Hector software to analyze the noise characteristics of the time series of all stations prior to filtering. The average spectral index shifted from -0.72 to -0.92, which indicates that the most suitable noise model in Taiwan is W + FN. Filtering can also effectively reduce PL noise. After filtering, PL noise is reduced by an average of approximately 28%. The average annual cycle item is also significantly reduced by approximately 60%. ICA filtration is more advantageous than PCA filtration. The noise sequence filtered by ICA and PCA at the GS39 station was analyzed to verify the above conclusions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Taiwan continuous global positioning system (CGPS) network. The black, green, blue, and purple circles represent CGPS sites operated by the Central Weather Bureau (CWB), the Institute of Earth Sciences (IES), the Central Geological Survey (CGS) and the Water Resources Agency (WRA) and other research units. Among them, the sites represented by red triangles are used in this ICA/PCA study.

**Figure 3.**Original GPS time series (

**left**) and preprocessing sequence of the CHKU, JUNA, S101, SFON, and SHAN stations (

**right**).

**Figure 6.**Spatial response of five PCs (first row) and ICs (second row) at each site in Taiwan. The red arrow represents the positive spatial response, and the blue arrow represents the negative spatial response.

**Figure 8.**U component power spectrum of the GS39 site noise sequence. The black line represents the power spectrum, and the red line represents the slope.

**Figure 9.**Comparison of the GPS_ICs (calculated with IC1 and IC2) and mass loading deformation signals (ATML and LWS). The upper figure shows GPS_IC1 and the ATML, and the lower figure shows GPS_IC2 and the LWS.

**Figure 10.**Comparison of the power spectrum of the GS39 site noise sequence and noise sequence filtered by PCA and ICA.

Order of PCs | Individual Contribution Rate (%) | Cumulative Contribution Rate (%) |
---|---|---|

1 | 24.7 | 24.7 |

2 | 6.5 | 31.2 |

3 | 4.6 | 35.8 |

4 | 3.7 | 39.5 |

5 | 3.6 | 43.1 |

6 | 3.0 | 46.1 |

7 | 2.7 | 48.8 |

8 | 2.5 | 51.3 |

9 | 2.4 | 53.7 |

10 | 2.3 | 56.0 |

Method | RMS/mm | |
---|---|---|

Before | 6.47 | |

After | PCA | 5.58 |

ICA | 5.40 |

**Table 3.**Estimated parameters using Hector software before and after spatiotemporal filtering (ICA).

Sites | Annual Amplitude (mm) | Semi-Annual (mm) | PL Amplitude | WN Amplitude | $\mathbf{Spectral}\text{}\mathbf{Index}\text{}\mathit{\kappa}$ | |||||
---|---|---|---|---|---|---|---|---|---|---|

Before | After | Before | After | Before | After | Before | After | Before | After | |

‘ANKN’ | 1.23 ± 0.33 | 0.70 ± 0.25 | 0.55 ± 0.25 | 0.50 ± 0.21 | 10.32 | 8.07 | 4.08 | 4.13 | −0.47 | −0.37 |

‘CHKU’ | 2.41 ± 0.31 | 0.40 ± 0.19 | 0.60 ± 0.23 | 0.39 ± 0.15 | 8.63 | 4.94 | 3.50 | 4.00 | −0.67 | −0.93 |

‘CHNT’ | 3.80 ± 0.83 | 2.80 ± 0.80 | 0.88 ± 0.45 | 0.77 ± 0.40 | 20.32 | 19.12 | 6.17 | 6.40 | −0.98 | −1.05 |

‘CLAN’ | 4.30 ± 1.00 | 2.39 ± 0.94 | 1.58 ± 0.65 | 1.35 ± 0.60 | 24.21 | 23.04 | 6.09 | 6.23 | −1.01 | −1.10 |

‘CTOU’ | 2.54 ± 0.45 | 0.67 ± 0.31 | 0.53 ± 0.26 | 0.59 ± 0.24 | 11.60 | 8.14 | 4.90 | 5.24 | −0.85 | −1.02 |

‘CWEN’ | 3.41 ± 0.37 | 0.69 ± 0.20 | 0.55 ± 0.25 | 0.44 ± 0.16 | 9.99 | 5.37 | 3.88 | 3.75 | −0.73 | −0.65 |

‘DAHU’ | 2.63 ± 0.45 | 1.58 ± 0.38 | 0.48 ± 0.25 | 0.50 ± 0.24 | 12.19 | 9.52 | 4.24 | 4.73 | −0.75 | −0.89 |

‘DOSH’ | 3.90 ± 0.38 | 0.36 ± 0.18 | 0.40 ± 0.21 | 0.35 ± 0.15 | 10.94 | 5.36 | 3.80 | 4.52 | −0.64 | −0.68 |

‘FLON’ | 2.76 ± 0.46 | 1.64 ± 0.42 | 1.58 ± 0.39 | 1.33 ± 0.36 | 15.33 | 14.31 | 0.00 | 0.01 | −0.45 | −0.43 |

‘FNGU’ | 2.76 ± 0.37 | 0.44 ± 0.21 | 0.85 ± 0.27 | 0.64 ± 0.17 | 9.52 | 4.60 | 4.26 | 4.40 | −0.82 | −1.25 |

‘GS15’ | 2.90 ± 0.29 | 0.91 ± 0.20 | 0.40 ± 0.19 | 0.31 ± 0.13 | 8.37 | 3.90 | 3.54 | 4.32 | −0.60 | −1.19 |

‘GS16’ | 2.18 ± 0.45 | 0.67 ± 0.31 | 0.63 ± 0.30 | 0.42 ± 0.21 | 12.38 | 9.58 | 5.43 | 5.57 | −0.73 | −0.70 |

‘GS21’ | 2.03 ± 0.36 | 0.82 ± 0.25 | 0.58 ± 0.24 | 0.37 ± 0.15 | 8.48 | 4.36 | 3.95 | 3.84 | −0.94 | −1.47 |

‘GS22’ | 2.63 ± 0.43 | 1.72 ± 0.33 | 0.49 ± 0.25 | 0.43 ± 0.20 | 10.89 | 7.58 | 3.73 | 4.00 | −0.85 | −0.99 |

‘GS31’ | 1.80 ± 0.40 | 2.04 ± 0.33 | 0.45 ± 0.23 | 0.30 ± 0.16 | 9.54 | 6.58 | 3.82 | 3.78 | −0.99 | −1.24 |

‘GS33’ | 2.86 ± 0.36 | 0.49 ± 0.23 | 0.58 ± 0.24 | 0.50 ± 0.17 | 8.59 | 4.68 | 4.44 | 4.42 | −0.93 | −1.37 |

‘GS39’ | 3.94 ± 0.33 | 1.32 ± 0.24 | 0.58 ± 0.24 | 0.42 ± 0.16 | 9.13 | 4.81 | 3.02 | 3.75 | −0.67 | −1.17 |

‘HUAL’ | 4.06 ± 0.57 | 1.38 ± 0.45 | 0.89 ± 0.38 | 0.51 ± 0.25 | 14.22 | 10.14 | 5.96 | 5.68 | −0.89 | −1.11 |

‘ILAN’ | 4.10 ± 0.70 | 2.35 ± 0.43 | 0.92 ± 0.41 | 0.47 ± 0.23 | 14.38 | 9.43 | 6.60 | 5.59 | −1.27 | −1.09 |

‘JHCI’ | 3.72 ± 0.32 | 0.63 ± 0.18 | 0.56 ± 0.24 | 0.57 ± 0.16 | 10.23 | 5.50 | 2.79 | 3.83 | −0.45 | −0.37 |

‘JONP’ | 3.17 ± 0.34 | 0.58 ± 0.18 | 0.51 ± 0.23 | 0.28 ± 0.13 | 9.75 | 4.45 | 3.83 | 4.26 | −0.63 | −0.76 |

‘JPEI’ | 3.05 ± 0.46 | 1.70 ± 0.43 | 0.60 ± 0.29 | 0.68 ± 0.29 | 12.07 | 10.31 | 6.14 | 6.43 | −0.78 | −0.94 |

‘JULI’ | 2.20 ± 0.32 | 0.83 ± 0.26 | 0.63 ± 0.25 | 0.57 ± 0.20 | 9.33 | 5.70 | 4.98 | 6.01 | −0.51 | −0.95 |

‘JUNA’ | 3.16 ± 0.41 | 0.83 ± 0.27 | 0.46 ± 0.23 | 0.32 ± 0.16 | 10.38 | 5.75 | 4.56 | 4.66 | −0.89 | −1.19 |

‘PAOL’ | 2.84 ± 0.47 | 0.65 ± 0.31 | 0.52 ± 0.27 | 0.50 ± 0.24 | 12.30 | 9.80 | 5.68 | 5.58 | −0.80 | −0.78 |

‘S101’ | 3.50 ± 0.37 | 1.78 ± 0.29 | 0.72 ± 0.28 | 0.57 ± 0.23 | 11.76 | 8.98 | 2.63 | 3.96 | −0.50 | −0.51 |

‘S106’ | 2.90 ± 0.36 | 0.50 ± 0.22 | 0.35 ± 0.18 | 0.29 ± 0.14 | 9.21 | 5.70 | 4.42 | 4.36 | −0.82 | −0.87 |

‘S170’ | 2.14 ± 0.33 | 0.43 ± 0.20 | 0.96 ± 0.26 | 0.66 ± 0.17 | 9.22 | 5.00 | 3.94 | 4.27 | −0.71 | −1.00 |

‘SFON’ | 3.58 ± 0.48 | 1.19 ± 0.38 | 0.52 ± 0.26 | 0.42 ± 0.21 | 11.43 | 8.74 | 5.12 | 5.10 | −0.92 | −1.00 |

‘SHAN’ | 0.79 ± 0.40 | 1.87 ± 0.44 | 0.84 ± 0.37 | 0.63 ± 0.29 | 14.82 | 11.21 | 4.70 | 5.28 | −0.75 | −0.85 |

‘SHJU’ | 2.97 ± 0.45 | 1.20 ± 0.30 | 0.84 ± 0.31 | 0.76 ± 0.21 | 11.27 | 6.04 | 4.56 | 4.80 | −0.92 | −1.26 |

‘SHMN’ | 1.67 ± 0.34 | 0.36 ± 0.19 | 0.85 ± 0.28 | 0.71 ± 0.24 | 11.85 | 10.08 | 0.03 | 0.02 | −0.36 | −0.28 |

‘SINY’ | 3.82 ± 0.49 | 0.50 ± 0.26 | 1.01 ± 0.36 | 0.76 ± 0.27 | 13.29 | 9.31 | 5.64 | 5.57 | −0.73 | −0.73 |

‘TACH’ | 2.85 ± 0.49 | 0.68 ± 0.33 | 0.44 ± 0.23 | 0.36 ± 0.19 | 11.92 | 8.04 | 4.06 | 4.15 | −0.98 | −1.34 |

‘TOFN’ | 2.51 ± 0.46 | 0.5 ± 0.26 | 0.59 ± 0.28 | 0.62 ± 0.24 | 11.58 | 8.50 | 4.07 | 4.15 | −0.91 | −0.98 |

‘TSIO’ | 1.85 ± 0.45 | 0.96 ± 0.39 | 1.11 ± 0.33 | 0.96 ± 0.29 | 11.60 | 9.61 | 4.26 | 4.60 | −0.86 | −1.04 |

‘VR01’ | 1.60 ± 0.42 | 0.86 ± 0.33 | 0.46 ± 0.23 | 0.31 ± 0.16 | 10.31 | 6.85 | 4.46 | 4.40 | −0.97 | −1.33 |

‘WANS’ | 3.33 ± 0.49 | 2.05 ± 0.40 | 0.77 ± 0.32 | 0.83 ± 0.29 | 11.98 | 9.71 | 5.33 | 5.12 | −0.95 | −0.96 |

‘WARO’ | 2.01 ± 0.44 | 0.96 ± 0.32 | 0.81 ± 0.33 | 0.38 ± 0.20 | 13.44 | 10.57 | 5.08 | 4.06 | −0.55 | −0.47 |

‘WUFN’ | 2.61 ± 0.28 | 0.95 ± 0.18 | 0.51 ± 0.21 | 0.25 ± 0.12 | 9.00 | 4.00 | 2.43 | 4.46 | −0.42 | −0.82 |

‘WUKU’ | 2.71 ± 1.07 | 2.65 ± 1.05 | 1.99 ± 0.78 | 1.95 ± 0.76 | 29.23 | 28.39 | 0.05 | 1.59 | −0.92 | −0.95 |

‘YENL’ | 2.31 ± 0.36 | 0.73 ± 0.29 | 0.42 ± 0.21 | 0.31 ± 0.16 | 9.71 | 6.73 | 5.21 | 5.62 | −0.70 | −1.05 |

‘YM03’ | 1.61 ± 0.51 | 1.80 ± 0.40 | 0.78 ± 0.35 | 0.94 ± 0.32 | 13.74 | 11.38 | 5.84 | 5.23 | −0.80 | −0.64 |

‘YM05’ | 2.68 ± 0.47 | 0.77 ± 0.34 | 0.85 ± 0.34 | 0.59 ± 0.26 | 12.98 | 9.91 | 3.95 | 4.51 | −0.72 | −0.83 |

Mean | 2.77 ± 0.45 | 1.12 ± 0.34 | 0.72 ± 0.30 | 0.59 ± 0.23 | 12.08 | 8.72 | 4.21 | 4.46 | −0.77 | −0.92 |

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## Share and Cite

**MDPI and ACS Style**

Ma, X.; Liu, B.; Dai, W.; Kuang, C.; Xing, X.
Potential Contributors to Common Mode Error in Array GPS Displacement Fields in Taiwan Island. *Remote Sens.* **2021**, *13*, 4221.
https://doi.org/10.3390/rs13214221

**AMA Style**

Ma X, Liu B, Dai W, Kuang C, Xing X.
Potential Contributors to Common Mode Error in Array GPS Displacement Fields in Taiwan Island. *Remote Sensing*. 2021; 13(21):4221.
https://doi.org/10.3390/rs13214221

**Chicago/Turabian Style**

Ma, Xiaojun, Bin Liu, Wujiao Dai, Cuilin Kuang, and Xuemin Xing.
2021. "Potential Contributors to Common Mode Error in Array GPS Displacement Fields in Taiwan Island" *Remote Sensing* 13, no. 21: 4221.
https://doi.org/10.3390/rs13214221