# Improvement of Persistent Scatterer Interferometry to Detect Large Non-Linear Displacements with the 2π Ambiguity by a Non-Parametric Approach

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## Abstract

**:**

## 1. Introduction

## 2. Basic Concept and Methodology

#### 2.1. Multi-Baseline Model

_{n}in any azimuth–range pixel (x, r) at the n

^{th}(n = 0, …, N) acquisition is expressed with the baseline distance b

_{n}, the reference range distance r, and the elevation direction s, and each acquisitions is supposed to have a displacement in the line of sight d(s, t

_{n}) in Equation (1):

#### 2.2. Calculation Procedure

#### 2.3. Scattering Distribution

#### 2.4. ConvPSI

#### 2.5. NN-PSI

## 3. Simulation

- The applicable displacements of NN-PSI were investigated by changing the magnitude and period of the displacements (simulation-1);
- How the velocity ranges that are used in the generation of SDM affect the resulting displacement with NN-PSI was investigated (simulation-2).

_{max}) and CDF variance. As shown in Figure 4b, the period of the displacement becomes shorter as the CDF variance is lowered.

## 4. Simulation Results and Discussions

#### 4.1. Simulation-1

_{max}increases. Figure 4a shows that a displacement with a D

_{max}value of less than 0.25λ is correctly reconstructed by the ConvPSI technique, because the value of RMSE is less than 0.1λ, but a displacement of more than 0.25λ could not be reconstructed, and RMSE increases as D

_{max}increases. It is also clear that the results obtained using the ConvPSI technique are not affected by the CDF variance, because RMSE remains the same in the CDF variance direction. These results prove that ConvPSI works only with a small displacement, with a D

_{max}value of less than 0.25λ.

_{max}value of more than 0.25λ can be reconstructed, and these RMSEs are less than 0.1λ. When the CDF variance is 20, the displacement with a D

_{max}value of λ can be reconstructed correctly. When the CDF variance is lowered, the RMSEs tend to be larger, with the same values of D

_{max}. When the CDF variance is lowered to one, a D

_{max}value with a minimum RMSE of about 0.25λ is the same as that resulting from the use of ConvPSI. This result indicates that the NN-PSI is dependent on the period of the displacement, and RMSE becomes larger when the CDF variance is smaller. However, it is clear that NN-PSI can adapt to larger displacements that are not handled correctly by ConvPSI.

_{max}is increased. The maximum value of the coherence in the SDM in Figure 5a is highest with a CDF variance of 15 and a D

_{max}value of 0.25λ, and becomes lower as D

_{max}increases, as shown in Figure 5c,e. The area of the high coherence values in Figure 6a has a rounded shape, but for the displacement with D

_{max}values of 0.75λ and 1.5λ, the rounded shape becomes smaller and distorted. This indicates that the displacement that is close to the linear trend shows a rounded shape with high coherence values in the SDM, whereas the non-linear displacement does not strongly show it.

_{max}value of 0.25λ. With a CDF variance of 15 and a D

_{max}value of 0.75λ in this simulation, ConvPSI cannot reconstruct the displacement because of the phase jump incurred by the 2λ displacement ambiguity, but the displacement of NN-PSI does not have a phase jump, and the displacement is correctly estimated, as shown in Figure 6d. Neither of the methods can reconstruct the displacement with a CDF variance of 15 and a D

_{max}value of 1.5λ, as shown in Figure 6f. There are big differences between the resulting displacements and the original one. The result of ConvPSI has several phase jumps, and the result of NN-PSI underestimates the displacement. However, there are no phase jumps in the NN-PSI result.

#### 4.2. Simulation-2

_{max}, and Figure 7 shows results from both ConvPSI and NN-PSI. In Figure 7a, the displacement with a D

_{max}value of 0.25$\lambda $ can be reconstructed with an RMSE of 0.1$\lambda $. As D

_{max}is increased, RMSE increases, and RMSE reaches more than 0.5$\lambda $ when the D

_{max}is more than 1.75$\lambda $. This result is the same as that from the first simulation and indicates that the velocity range does not influence the results of ConvPSI. In Figure 7b, the displacement with a D

_{max}value of less than $2.5\lambda $ can be reconstructed with an RMSE of 0.1$\lambda $, when the velocity range is set to be wider than a range from −250 to 250 mm/year. In simulation-1, the largest displacement that can be correctly reconstructed is generated with a D

_{max}value of $\lambda ,$ when the velocity range is set to be from −70 to 70 mm/year (Figure 4b). Thus, increasing the velocity range expands the magnitude of the displacement, of which NN-PSI enables a correct estimation. The maximum D

_{max}value of a displacement that NN-PSI can reconstruct with a low RMSE is around 2.5$\lambda $ with larger velocity ranges, but beyond this range, RMSE becomes higher with NN-PSI, although the velocity ranges are increased.

_{max}values of 1.5λ and 2.5λ in Figure 8a,c,e, and the shape becomes a horizontal circle as the D

_{max}is increased in Figure 8g. The right side (positive velocity) of the coherence values at the selected height of 0 are clearly low in the SDMs with a D

_{max}value of 1.5λ and λ in Figure 8a,c,e, but this pattern is not clear in the SDM with a D

_{max}value of λ in Figure 8g. This result indicates that a positive mean velocity does not exist in the selected elevation in the SDM, and the resulting displacement becomes a negative direction (subsidence).

_{max}values of 1.5λ and 2.5λ by NN-PSI agree well with the original ones in Figure 8b,d,f. With a D

_{max}value of 4λ, NN-PSI cannot estimate the displacement, and the result of NN-PSI shows an uplift that is opposite to the original one in Figure 8h. In this case, the estimation is less accurate than that of ConvPSI due to the strong uplift estimated by NN-PSI.

_{max}value of $2.5\lambda $ can be estimated by NN-PSI by increasing the velocity range, but the estimation of the displacement becomes less accurate than that of ConvPSI, once the displacement is beyond the capability of NN-PSI. One further advantage of using the wider velocity range in NN-PSI is that the fitting of the resulting displacement is better than with the smaller velocity range. Compared to the time evolutions of simulation-1 and simulation-2, the resulting displacement by NN-PSI, shown by the red line in Figure 6d, has some gaps between the resulting and the original displacements, whereas the output displacements, shown by the red lines in Figure 8d,f, do not have any gaps between the resulting and original displacements.

## 5. Experiment with Actual Observation Data

#### 5.1. ConvPSI and NN-PSI

_{max}value of 0.5$\lambda $ and a CDF variance of 30, and these are allowable parameters for NN-PSI to reconstruct the displacement but not for ConvPSI due to the displacement magnitude larger than a D

_{max}value of 0.25$\lambda $. As explained by the simulation, the difference between ConvPSI and NN-PSI at Pt3 should come from the computation limitation of ConvPSI, and NN-PSI is still able to measure the displacement, despite the ambiguity.

#### 5.2. Comparison with SBAS Results

## 6. Conclusions

_{max}) and a period (CDF variance) of the displacement. The results show that NN-PSI is able to reconstruct the displacement with a D

_{max}value of three-quarters of the wavelength (0.75$\lambda $) and a CDF variance of 15, whereas ConvPSI cannot reconstruct the same displacement due to the 2$\pi $ displacement ambiguity. As for the actual numbers, the displacements of roughly 2, 4, and 18 cm with X-, C-, and L-bands, respectively, over 80 days should be reconstructed by NN-PSI. In the second simulation, the influence of the velocity range was investigated, and it was confirmed that the displacement with a D

_{max}value of 2.5$\lambda $, which is roughly equivalent to 8 cm for X-band, 14 cm for C-band, and 60 cm for L-band, can be reconstructed with a velocity range from −250 to 250 mm/year. The wider the velocity range, the larger the displacement that can be reconstructed by NN-PSI. The allowable D

_{max}value is raised by up to three times that in simulation-1, with a wider velocity range.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**System geometry in the plane orthogonal to the orbit direction. Each satellite shows the positions of the acquisition antennas over repeated passes [26].

**Figure 2.**Process flows of the conventional persistent scatterer interferometry (PSI) and the proposed method.

**Figure 3.**The single scattering point is selected, where the scattering distribution (temporal coherence) is the maximum. Conventional PSI (ConvPSI) uses a single elevation ${s}_{0}$ and a mean velocity ${v}_{0}$, and non-parametric non-linear PSI (NN-PSI) uses a single elevation and the profile (v

_{min}through v

_{max}) of the temporal coherence for the displacement reconstruction.

**Figure 4.**(

**a**) The displacement model in the simulation. (

**b**) The displacements, with the different cumulative distribution function (CDF) variances in the CDF period shown in (

**a**).

**Figure 5.**(

**a**) The root mean square error (RMSE) distribution with the ConvPSI; and (

**b**) the RMSE distribution with the NN-PSI. The vertical axis shows the CDF variance, and the horizontal axis shows the D

_{max}.

**Figure 6.**(

**a**) The scattering distribution map (SDM) with a D

_{max}value of 0.25$\lambda $ and a CDF variance of 15. The horizontal dashed white line shows the selected height, and the vertical dashed white line shows the selected velocity in the ConvPSI. (

**b**) The resulting displacement of (

**a**). (

**c**) The SDM with a D

_{max}value of 0.75$\lambda $ and a CDF variance of 15. (

**d**) The resulting displacement of (

**c**). (

**e**) The SDM with a D

_{max}value of 1.5$\lambda $ and a CDF variance of 15. (

**f**) The resulting displacement of (

**e**). In the time plots, the red, blue, and black lines show the NN-PSI, ConvPSI, and modeled displacements, respectively.

**Figure 7.**The results of the simulation for the velocity range: (

**a**) the RMSE distribution with ConvPSI; and (

**b**) the RMSE distribution of NN-PSI.

**Figure 8.**(

**a**) The SDM with a CDF variance of 1.5$\lambda $ and a velocity range from −150 to 150 mm/year. (

**b**) The original and resulting displacements of (

**a**). (

**c**) The SDM with a CDF variance of 1.5$\lambda $ and a velocity range from −250 to 250 mm/year. (

**d**) The resulting displacements of (

**c**). (

**e**) The SDM with a CDF variance of 2.5$\lambda $ and a velocity range from −250 to 250 mm/year. (

**f**) The resulting displacements of (

**e**). (

**g**) The SDM with a CDF variance of 4$\lambda $ and a velocity range from −300 to 300 mm/year. (

**h**) The resulting displacements of (

**g**). In the time plots, the red, blue, and black lines show NN-PSI, ConvPSI, and modeled displacements, respectively.

**Figure 9.**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 10.**The distribution of the resulting points of PSI around Szent Gellért Station. The color of the points represents the mean velocity of ConvPSI. Pt1, Pt2, and Pt3 are the selected points, and the time evolutions were investigated. The background map was obtained from OpenStreetMap (OSM).

**Figure 11.**(

**a**–

**c**) The time evolutions at Pt1, Pt2, and Pt3. In the time plots, the red, blue, and purple lines show NN-PSI, ConvPSI, and Small BAseline Subset (SBAS) displacements, respectively.

Items | Simulation-1 | Simulation-2 |
---|---|---|

Slant range distance | 700 km | |

Incidence angle | 45° | |

Baseline variance ^{1} | ±4% | |

Backscatter coefficient | 5 dB | |

Number of observations | 41 | |

Observation interval | 10 days | |

Total period | 410 days | |

No-displacement period | 200 days | |

CDF period | 210 days | |

Search velocity range | ±70 mm/year | 0–±300 mm/year |

CDF variance | 1–20 | 20 |

D_{max} (λ: wavelength) | 0–3λ | 0–4λ |

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

Satellite sensor | TerraSAR-X |

Monitoring period | 24 October 2008–16 April 2010 |

Number of acquisitions | 42 |

Time interval of the acquisitions | 11 days |

Date of the master acquisition | 4 July 2009 |

Incidence angle | 44.5° |

Wavelength ($\lambda $) | 31.066 mm |

Method | Pt1 | Pt2 | Pt3 |
---|---|---|---|

ConvPSI | 0.31 | −0.14 | −0.66 |

NN-PSI | 0.35 | 0.88 | 0.95 |

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## Share and Cite

**MDPI and ACS Style**

Ogushi, F.; Matsuoka, M.; Defilippi, M.; Pasquali, P. Improvement of Persistent Scatterer Interferometry to Detect Large Non-Linear Displacements with the 2*π* Ambiguity by a Non-Parametric Approach. *Remote Sens.* **2019**, *11*, 2467.
https://doi.org/10.3390/rs11212467

**AMA Style**

Ogushi F, Matsuoka M, Defilippi M, Pasquali P. Improvement of Persistent Scatterer Interferometry to Detect Large Non-Linear Displacements with the 2*π* Ambiguity by a Non-Parametric Approach. *Remote Sensing*. 2019; 11(21):2467.
https://doi.org/10.3390/rs11212467

**Chicago/Turabian Style**

Ogushi, Fumitaka, Masashi Matsuoka, Marco Defilippi, and Paolo Pasquali. 2019. "Improvement of Persistent Scatterer Interferometry to Detect Large Non-Linear Displacements with the 2*π* Ambiguity by a Non-Parametric Approach" *Remote Sensing* 11, no. 21: 2467.
https://doi.org/10.3390/rs11212467