# A Novel Method of Generating Deformation Time-Series Using Interferometric Synthetic Aperture Radar and Its Application in Mexico City

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Data Preprocessing

#### 2.2. Mathematical Model

## 3. Study Area and Datasets

#### 3.1. Geological Background

#### 3.2. Data Processing

^{5}m; the center incidence angle in the study area $\theta =44.5\xb0$; the interferometric perpendicular baseline ${B}_{\perp}<120\text{}\mathrm{m}$; and the maximum relative error of the Shuttle Radar Topography Mission (SRTM) DEM is 10 m. If we set $\Delta z=10\text{}\mathrm{m}$, the computed $\mathrm{max}\left({d}_{\Delta z}\right)=1.8\text{}\mathrm{mm}$. Due to the real annual rate being larger than 10 $\mathrm{cm}$, the impact of elevation error can be ignored in the latter solution.

## 4. Experimental Results and Analysis

#### 4.1. Annual Velocity

#### 4.2. Mathematical Model

#### 4.3. Time-Series Deformation

#### 4.4. Atmospheric and Nonlinear Results

## 5. Discussion

#### 5.1. Mathematical Models

#### 5.2. Spatio-Temporal Changes of the Deformation in Mexico City

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Framework of interferogram unwrapping, error detection and correction. The boxes represent the detailed operations, the parallelograms indicate the processed results and the arrows point to the next process.

**Figure 2.**The coverage of Mexico City soil composition (plot according to [31]) and study area. The black box in the upper right figure indicates the scope of the Sentinel-1 dataset and the blue rectangle is the study area. In the main figure, the purple and red lines represent the scope of the transitional zone and the lacustrine clay border and the blue box denotes the study area. The lacustrine clay border defines the scope of clayed layer overlaying sand and gravel aquifer, transitional zone distinguishes the different geotechnical units.

**Figure 3.**The baseline distribution of the interferograms, where the blue triangles show the times of the SLC image acquisitions and the purple lines display the formed interferograms.

**Figure 4.**Subsidence velocity derived by the stacking method (unit: cm/year), where the black rectangles represent stable areas, the purple star indicates the reference point, the blue triangles are the feature points for the time-series deformation analysis and the blue box is the study area, the grey line indicates the administrative boundaries.

**Figure 5.**Time-series statistical chart in the stable areas by different functional models and weighting functions. Unit weight and adp weight represent equal and adaptive weight distributions, respectively. The left column shows the results of area a and the right column shows the results of area b. The figures in the three lines show the time-series results of the Usai method, the Berardino method and the proposed method, respectively.

**Figure 6.**Displacement time-series along LOS direction with unit weights by Method (unit: cm). The negative value represents the increased distance between ground target and satellite antenna direction.

**Figure 7.**Time-series results of points P1, P2 and P3 (see Figure 4) between 2015/04/13 and 2016/09/10, where the horizontal axis represents time and the vertical axis indicates the deformation in the LOS direction.

**Figure 9.**The atmospheric and nonlinear phase analysis of interferograms 20150823_20151115, 20160326_20160817, 20160501_20160513 and 20160805_20160817. The first column marked IFG represents the interferometric phase removed linear displacement; NONL and ATM in the second and third columns are the solved nonlinear phase and atmospheric results with Method III; IFG-NONL and IFG-ATM are the outcomes of subtracting the nonlinear and atmospheric contributions from the interferometric phase, respectively; IFG-ATM-NONL is the residual phase, which has removed nonlinear and atmospheric components from the interferometric phase. Unit: cm.

Method | Observation $\mathit{L}$ | Unknown Parameter $\mathit{X}$ | Coefficient Matrix $\mathit{A}$ | Calculation Criteria | Random Model $\mathit{P}$ | Atmosphere | Representative Literature |
---|---|---|---|---|---|---|---|

I | $\delta {\phi}_{j},j=1,2\cdots ,N$ | ${d}_{j},j=1,2\cdots ,M$ | $N\times M$ | LS | I | ignore | Usai (2001) |

II | $\delta {\phi}_{j},j=1,2\cdots ,N$ | $\overline{\mathrm{v}},\text{}\overline{\mathrm{a}},\text{}\Delta \overline{\mathrm{a}}$ | $N\times 3$ | LS | I | filter | Berardino et al. (2002) |

III | $\delta {\phi}_{j},j=1,2\cdots ,N$ | $\overline{v},{S}_{j},j=1,2\cdots ,M$ | $N\times \left(M+1\right)$ | SVD | $\mathbf{I},\mathbf{P}$ | Parameter estimation, filter |

**Table 2.**Mean and standard deviation of the interferometric phase, nonlinear displacement and atmospheric phase in stable areas a and b.

IFG | NONL | ATM | IFG-NONL | IFG-ATM | IFG-ATM-NONL | |||
---|---|---|---|---|---|---|---|---|

20150823_20151115 ${\mathrm{B}}_{\perp}=2\mathrm{m}$ $\Delta \mathrm{t}=84\text{}\mathrm{day}$ | Area a | mean | 2.28 | 0.65 | 1.73 | 1.63 | 0.55 | −0.10 |

std | 0.47 | 0.12 | 0.32 | 0.45 | 0.33 | 0.31 | ||

Area b | mean | 0.11 | 0.01 | 0.09 | 0.10 | 0.03 | 0.02 | |

std | 0.49 | 0.15 | 0.34 | 0.39 | 0.29 | 0.26 | ||

20160326_20160817 ${\mathrm{B}}_{\perp}=-25\mathrm{m}$ $\Delta \mathrm{t}=144\text{}\mathrm{day}$ | Area a | mean | 0.16 | 0.09 | −0.83 | 0.07 | 1.00 | 0.90 |

std | 0.67 | 0.25 | 0.42 | 0.57 | 0.55 | 0.46 | ||

Area b | mean | −1.89 | −0.30 | −1.19 | −1.59 | −0.69 | −0.40 | |

std | 0.82 | 0.22 | 0.50 | 0.65 | 0.38 | 0.23 | ||

20160501_20160513 ${\mathrm{B}}_{\perp}=-9\mathrm{m}$ $\Delta \mathrm{t}=\text{}12\text{}\mathrm{day}$ | Area a | mean | 2.32 | 0.16 | 1.40 | 2.16 | 0.92 | 0.76 |

std | 0.52 | 0.07 | 0.49 | 0.54 | 0.12 | 0.12 | ||

Area b | mean | −0.21 | 0.10 | −0.07 | −0.31 | −0.14 | −0.24 | |

std | 0.55 | 0.15 | 0.45 | 0.45 | 0.16 | 0.15 | ||

20160805_20160817 ${\mathrm{B}}_{\perp}=-43\mathrm{m}$ $\Delta \mathrm{t}=\text{}12\text{}\mathrm{day}$ | Area a | mean | −2.10 | −0.16 | −1.93 | −1.94 | −0.18 | −0.02 |

std | 0.28 | 0.12 | 0.34 | 0.28 | 0.31 | 0.28 | ||

Area b | mean | -0.40 | -0.12 | -0.12 | -0.28 | -0.28 | -0.16 | |

std | 0.33 | 0.14 | 0.24 | 0.31 | 0.22 | 0.16 |

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**MDPI and ACS Style**

Wang, X.; Zhang, Q.; Zhao, C.; Qu, F.; Zhang, J.
A Novel Method of Generating Deformation Time-Series Using Interferometric Synthetic Aperture Radar and Its Application in Mexico City. *Remote Sens.* **2018**, *10*, 1741.
https://doi.org/10.3390/rs10111741

**AMA Style**

Wang X, Zhang Q, Zhao C, Qu F, Zhang J.
A Novel Method of Generating Deformation Time-Series Using Interferometric Synthetic Aperture Radar and Its Application in Mexico City. *Remote Sensing*. 2018; 10(11):1741.
https://doi.org/10.3390/rs10111741

**Chicago/Turabian Style**

Wang, Xiaying, Qin Zhang, Chaoying Zhao, Feifei Qu, and Juqing Zhang.
2018. "A Novel Method of Generating Deformation Time-Series Using Interferometric Synthetic Aperture Radar and Its Application in Mexico City" *Remote Sensing* 10, no. 11: 1741.
https://doi.org/10.3390/rs10111741