# Implementation of Non-Linear Non-Parametric Persistent Scatterer Interferometry and Its Robustness for Displacement Monitoring

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Review of NN-PSI

_{n}in any pixel at the n

^{th}acquisition that is expressed by the baseline distance b

_{n}, the reference range distance r, the elevation direction s, and the displacement in the line of sight d(s, t

_{n}).

_{0}can be selected using the statistics of the temporal coherence in the given EV spectrum. Once the height is decided, the scattering distribution is turned into the spectral distribution ${\gamma}_{t}\left({s}_{0},v\right)$, which is the profile of the temporal coherence at the selected height. The displacement is reconstructed by Equation (2):

#### 2.2. The Range and Resolution of EV Spectrum

#### 2.3. Observation Intervals

## 3. Proposed Method

_{0}for the displacement estimation was decided by the maximum temporal coherence in the given EV spectrum. This method works properly for many types of non-linear displacements. However, some periodical displacements cannot be reconstructed due to the wrong height selection. With the periodical displacements, the maximum temporal coherence in the EV spectrum does not always show the height that should be used for the displacement estimation. Instead of searching for the maximum coherence for the correct height detection, the following method is proposed:

- (1)
- Decide the range and sampling resolution of EV spectrum by the ambiguity based on the observation conditions.
- (2)
- Select the height by the minimum average value of the total coherence values at each height in the velocity direction.
- (3)
- Extract coherence profile along the velocity direction at the selected height, ${\gamma}_{t}\left({s}_{0},v\right)$ in Step 2.
- (4)
- Estimate the displacement with the coherence profile by Equation (2).

## 4. Simulations and Verification

#### 4.1. Simulation Method

#### 4.1.1. Displacement Types

#### Step Displacement

#### Exponential Displacement

#### Sinusoidal Displacement

#### 4.1.2. Observation Conditions

#### 4.1.3. Observation Intervals

#### 4.2. Simulation Results

#### 4.2.1. Step Displacement

#### 4.2.2. Exponential Displacement

#### 4.2.3. Sinusoidal Displacement

#### 4.2.4. Observation Interval

## 5. Real Data Processing

#### 5.1. Study Area

^{2}, so that the difference between the reference displacement and the measurement displacements would be large enough. The study area is shown in Figure 9.

#### 5.2. Method and Materials

#### 5.3. Results of the Validation with GEONET

#### 5.4. Periodical Displacement

## 6. Discussion

#### 6.1. Evaluation of the Proposed Method

#### 6.2. Periodical Displacements

#### 6.3. Other Improvement by NN-PSI

#### 6.4. Atomospheric Correction

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**System geometry in the plane orthogonal to the orbit direction. Each satellite shows the positions of the acquisition antennas over repeated passes [25].

**Figure 4.**The distribution of spatial baselines related with the acquisition times. (

**a**) shows dense acquisitions for multi-baseline model. (

**b**) shows sparse baselines and acquisitions [27]. The red point shows the reference observation in the multi-baseline model, the black point shows other observations, and the gray line shows the connections of the pairs.

**Figure 5.**The simulation results with the step function. (

**a**–

**d**) The simulation outputs: EV spectrum, profile of the average coherence at each height along the velocity direction, profile of the coherence at the selected height along the velocity, and the original displacement in black and the estimated displacement by non-linear non-parametric persistent scatterer interferometry (NN-PSI) and conventional PSI (ConvPSI) in red and blue, respectively. The gray line in (

**b**) indicates the selected height.

**Figure 6.**The simulation results with the exponential function. (

**a**–

**d**) The simulation outputs: EV spectrum, profile of the average coherence at each height along the velocity direction, profile of the coherence at the selected height along the velocity, and the original displacement in black, the estimated displacement by ConvPSI in blue, and NN-PSI in red, respectively. The gray line in (

**b**) indicates the selected height.

**Figure 7.**The simulation results with the sinusoidal function. (

**a**–

**d**) The simulation outputs: EV spectrum, profile of the average coherence at each height along the velocity direction, profile of the coherence at the selected height along the velocity, and the original displacement in black, the estimated displacement by ConvPSI in blue, and NN-PSI in red. The gray line in (

**b**) indicates the selected height.

**Figure 8.**The vertical axis shows the amount of the displacement divided by the wavelength. The horizontal axes on the bottom and top show the velocity ambiguity of the X-band and C-band, respectively.

**Figure 9.**The average intensity image of Sentinel-1 in the study area. Pt1, Pt2, Pt3, and Pt4 are the location of the GNSS Earth Observation Network System (GEONET) stations. A1 and A2 are the evaluation areas, and Pt5 and Pt6 are the evaluation points. The green star is the location of the reference point for NN-PSI and ConvPSI.

**Figure 10.**The combination of the interferometric pairs used in the PSI approaches. The yellow square shows the master acquisition, and the black squares are the slave acquisitions. The horizontal axis shows the date of the acquisition, the vertical axis shows the baseline length in meters, and the gray lines show the combination of the interferometric pairs.

**Figure 11.**The black dot shows the GEONET displacements, and the red line shows the displacements estimated by NN-PSI. (

**a**,

**b**) the displacements at Pt1 and Pt2, respectively.

**Figure 12.**(

**a**,

**c**,

**e**) The location of the displacement, EV spectrum, and NN-PSI in red and ConvPSI in blue at Pt5. (

**b**,

**d**,

**f**) The location of the displacement, EV spectrum, NN-PSI in red, and ConvPSI in blue at Pt6. (

**g**,

**h**) The daily average temperature measured by the Japan Metrological Agency at the closest stations from Pt5 and Pt6. (

**i**,

**j**) GEONET measurement in black and NN-PSI in red at Pt3 and Pt4. The satellite imagery in (

**a**,

**b**) is distributed in the GSImap [29].

**Figure 13.**The red and blue lines show the resulting displacement of NN-PSI and ConvPSI, respectively. (

**a**) The representative displacement in A1, and (

**b**) in A2.

**Figure 14.**(

**a**) The mean velocity calculated with GEONET data of T1, and (

**b**) that of T2. The white lines show the boundaries of the prefectures in Japan.

Displacement Type | Total Amount of the Simulated Displacement in 500 Days |
---|---|

Step | −0.25$\lambda $ |

Exponential | −0.5$\lambda $ |

Sinusoidal ^{1} | 0.5$\lambda $ |

^{1}The amount of the displacement indicates the peak to peak value.

Items | Values |
---|---|

Slant range distance | 700 km |

Wavelength | 1 $\lambda $ |

Incidence angle | 45° |

Baseline variance ^{1} | ±6% |

Backscatter coefficient | 5 dB |

Number of observations | 51 |

Observation interval | 10 days |

Total observation period | 500 days |

Simulated height | 0 m |

Height ambiguity | 306 m |

Height step | 1 m |

Velocity ambiguity | 566 mm/year |

Velocity step | 1 mm/year |

^{1}The value in this column is the ratio of the baseline and the critical baseline.

Parameters | Values |
---|---|

Satellite sensor | Sentinel-1 |

Monitoring period | January 2017–December 2018 |

Number of acquisitions | 59 |

Time interval of the acquisitions | 12 days |

Date of the master acquisition | 16 October 2017 |

Incidence angle | 39.0° |

Wavelength ($\lambda $) | 55.5 mm |

Average baseline distance | 39.1 m |

Average temporal baseline | 12.6 days |

Height resolution | 113.8 m |

Height ambiguity | 585.6 m |

Velocity resolution | 14.1 mm/year |

Velocity ambiguity | 802.1 mm/year |

Scheme | RMSE (mm) | Temporal Coherence |
---|---|---|

Pt1 | 2.1 | 0.69 |

Pt2 | 3.5 | 0.77 |

Method | T1 (mm/year) | T2 (mm/year) |
---|---|---|

A1 | ||

NN-PSI | −17.1 | −19.0 |

ConvPSI | −10.0 | 28.8 |

A2 | ||

NN-PSI | −14.1 | −17.9 |

ConvPSI | −12.8 | 49.4 |

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**MDPI and ACS Style**

Ogushi, F.; Matsuoka, M.; Defilippi, M.; Pasquali, P.
Implementation of Non-Linear Non-Parametric Persistent Scatterer Interferometry and Its Robustness for Displacement Monitoring. *Sensors* **2021**, *21*, 1004.
https://doi.org/10.3390/s21031004

**AMA Style**

Ogushi F, Matsuoka M, Defilippi M, Pasquali P.
Implementation of Non-Linear Non-Parametric Persistent Scatterer Interferometry and Its Robustness for Displacement Monitoring. *Sensors*. 2021; 21(3):1004.
https://doi.org/10.3390/s21031004

**Chicago/Turabian Style**

Ogushi, Fumitaka, Masashi Matsuoka, Marco Defilippi, and Paolo Pasquali.
2021. "Implementation of Non-Linear Non-Parametric Persistent Scatterer Interferometry and Its Robustness for Displacement Monitoring" *Sensors* 21, no. 3: 1004.
https://doi.org/10.3390/s21031004