# Hybrid Wavelet and Principal Component Analyses Approach for Extracting Dynamic Motion Characteristics from Displacement Series Derived from Multipath-Affected High-Rate GNSS Observations

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

_{j}is j decomposition level of short-period movement components; and n is the number of decomposition levels. Figure 2 illustrates the development model and processing. To evaluate the performance of MSPCAW in extracting the dynamic component, their results were compared with LVDT observations.

## 3. Test Description

#### 3.1. Data Collection

#### 3.2. Multipath

#### 3.3. GNSS Data Processing

## 4. Results and Discussion

#### 4.1. Denoising and Noise Evaluation

#### 4.2. Motions’ Characteristics

## 5. Conclusions

- -
- The multipath error and measurement noise have affected the amplitude accuracy of the displacement obtained from GNSS results. The multipath contaminates the semi-static component and can be removed using a low-pass filter. The background and dynamic noises contaminate the GNSS results, and the Gaussian assumption is adequate to describe the distribution of these noises at a probability of 99.7%.
- -
- The comparison between the short-period and dynamic component wavelet energy in the current study shows that the low energy wavelet affects the accuracy of the dynamic properties of movements by 90.5% and 86.54% for the relative GNSS positioning and PPP methods, respectively. The amplitude and the frequency contents of GNSS-measured displacement are equal and are similar to the frequencies that are estimated by the LVDT. Furthermore, the number of observations in the time series is an essential factor for the extraction of the wavelet energy impact in the GNSS results.
- -
- Evaluation of the four events shows that the MSPCAW model can be used to estimate an accurate performance of GNSS-derived displacements in terms of the time and frequency domains. The evaluation of motion amplitudes for these events assessed by LVDT and extracted by MSPCAW from GNSS results show that the average accuracy of extracting the amplitude of movements is 80.94% and 79.77% for the relative GNSS positioning and PPP techniques, respectively. Meanwhile, the spectrum analysis of extracted displacement from GNSS results shows that the average variation of errors in computing the amplitude of the movements, relative to LVDT measurements, is 0.54 mm and 0.62 mm for the relative GNSS positioning and PPP, respectively. Results also show that the extracted dominant frequency of movements by both GNSS methods and LVDT is equal. From these results, it is concluded that the proposed hybrid method can be used to extract the accurate dynamic characteristic of engineering structures, such as high-rise buildings and suspended bridges, where the GNSS observations are subject to harsh environments.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Scale and wavelet functions of Sym6, (

**b**) two decomposition level of 10 Hz Global Navigation Satellite System (GNSS) results.

**Figure 2.**Multiscale principal component analysis and wavelet transform (MSPCAW) model diagram and study process.

**Figure 6.**L1 Multipath superimposed time series of Satellite G16 for rover (black) and base station (red).

**Figure 7.**Four events amplitude (

**a**) and frequency (

**b**) for LVDT and amplitude (

**c**) and frequency (

**d**) for both GNSS solutions.

**Figure 8.**MSPCAW evaluation in amplitude, (

**a**) apparent displacement, (

**b**) relative measurement filter, (

**c**) short-period component, and (

**d**) dynamic component.

**Figure 10.**MSPCAW evaluation in frequency, (

**a**) measurement frequency, (

**b**) semi-static frequency, (

**c**) short-period frequency, (

**d**) dynamic frequency.

**Figure 11.**Dynamic movements of seismic movement of events: (

**a**) event 2, (

**b**) event 3, and (

**c**) event 4.

**Figure 12.**Residual Q-Q plot of events 1 (

**a**), 2(

**b**), 3 (

**c**), and 4 (

**d**) and box-plots of events 2 (

**e**) and 4 (

**f**).

**Figure 13.**Spectrum diagram of the four events: (

**a**) event 1, (

**b**) event 2, (

**c**) event 3, and (

**d**) event 4.

Technique | Apparent Displacement | Short-Period Displacement | Dynamic Displacement | ||||||
---|---|---|---|---|---|---|---|---|---|

Mean | Std | Range | Mean | Std | Range | Mean | Std | Range | |

LVDT | 9.2 | 4.7 | 29.5 | --- | --- | --- | 9.2 | 4.7 | 29.5 |

Relative-GNSS | 9.6 | 5.7 | 52.8 | 9.3 | 4.9 | 42.6 | 9.2 | 4.8 | 32.4 |

PPP-GNSS | 8.8 | 5.3 | 46.0 | 8.5 | 4.4 | 37.3 | 8.5 | 4.4 | 33.5 |

**Table 2.**The correlation coefficient of dynamic movements extracted from GNSS with LVDT measurements.

Method | Event1 | Event 2 | Event 3 | Event 4 |
---|---|---|---|---|

Relative | 0.99 | 0.94 | 0.98 | 0.95 |

PPP | 0.99 | 0.94 | 0.98 | 0.94 |

Event | Technique | Min (mm) | Max (mm) | Mean (mm) | Std (mm) | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

1 | Relative | −4.4 | 5.7 | −0.03 | 1.6 | 0.101 | 2.640 |

PPP | −4.4 | 5.3 | −0.03 | 1.7 | 0.141 | 2.527 | |

2 | Relative | −3.2 | 3.0 | 0.00 | 1.1 | 0.175 | 2.864 |

PPP | −4.2 | 3.5 | 0.00 | 1.1 | 0.181 | 3.261 | |

3 | Relative | −4.3 | 3.7 | −0.00 | 1.3 | −0.116 | 2.863 |

PPP | −4.3 | 3.7 | −0.01 | 1.4 | −0.100 | 2.906 | |

4 | Relative | −7.5 | 6.9 | 0.00 | 3.2 | −0.008 | 1.957 |

PPP | −8.4 | 7.5 | 0.00 | 3.3 | −0.026 | 2.041 |

Events | LVDT | Relative-GNSS | PPP-GNSS | |||
---|---|---|---|---|---|---|

Amplitude (mm) | Frequency (Hz) | Amplitude (mm) | Frequency (Hz) | Amplitude (mm) | Frequency (Hz) | |

1 | 13.3 | 0.50 | 12.4 | 0.50 | 12.5 | 0.50 |

2 | 4.2 | 1.00 | 4.1 | 1.00 | 4.1 | 1.00 |

3 | 8.5 | 1.00 | 8.1 | 1.00 | 7.8 | 1.00 |

4 | 12.7 | 1.00 | 11.9 | 1.00 | 11.8 | 1.00 |

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**MDPI and ACS Style**

Kaloop, M.R.; Yigit, C.O.; El-Mowafy, A.; Dindar, A.A.; Bezcioglu, M.; Hu, J.W.
Hybrid Wavelet and Principal Component Analyses Approach for Extracting Dynamic Motion Characteristics from Displacement Series Derived from Multipath-Affected High-Rate GNSS Observations. *Remote Sens.* **2020**, *12*, 79.
https://doi.org/10.3390/rs12010079

**AMA Style**

Kaloop MR, Yigit CO, El-Mowafy A, Dindar AA, Bezcioglu M, Hu JW.
Hybrid Wavelet and Principal Component Analyses Approach for Extracting Dynamic Motion Characteristics from Displacement Series Derived from Multipath-Affected High-Rate GNSS Observations. *Remote Sensing*. 2020; 12(1):79.
https://doi.org/10.3390/rs12010079

**Chicago/Turabian Style**

Kaloop, Mosbeh R., Cemal O. Yigit, Ahmed El-Mowafy, Ahmet A. Dindar, Mert Bezcioglu, and Jong Wan Hu.
2020. "Hybrid Wavelet and Principal Component Analyses Approach for Extracting Dynamic Motion Characteristics from Displacement Series Derived from Multipath-Affected High-Rate GNSS Observations" *Remote Sensing* 12, no. 1: 79.
https://doi.org/10.3390/rs12010079