# An Approach to Persistent Scatterer Interferometry

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The PSIG Procedure

- Candidate Cousin PS (CPS) selection. In this step, a set of PSs with phases characterized by a moderate spatial variation is sought. This is accomplished by using at least a seed PS and searching for its “cousins”, i.e., PSs with similar characteristics (the details are described in the next section). An iterative process is used to ensure an appropriate CPS coverage and density;
- Phase unwrapping consistency check. This check is based on a LS estimation, followed by the analysis of the so-called residuals (the details are described in the next section). The final set of CPSs is selected at this stage;
- Estimation of deformation velocity and RTE. The deformation velocity and RTE are computed over a dense set of PSs (much denser than the selected CPSs), from the M wrapped APS-free interferograms, using the method of the periodogram [1]. Optionally, an extension of the two-parameter model can be used to account for the thermal expansion [24];
- RTE removal. The RTE phase component is removed from the wrapped APS-free interferograms. The linear deformation component can optionally be removed and then, in a later stage, added back to the deformation time series. The same procedure can be done with the thermal expansion component;
- 2+1D phase unwrapping. A 2+1D phase unwrapping is executed on the set of M APS- and RTE-free interferograms to obtain the final deformation phase time series, a quality index for each time series and other parameters related to the detection and correction of unwrapping errors (detailed information is provided in Section 4.2).

## 3. First Processing Block

- Candidate CPS selection;
- 2D phase unwrapping on the selected CPSs;
- Phase unwrapping consistency check.

#### 3.1. Candidate CPS Selection

_{ij}:

_{Defo_i,j}is the phase difference due to deformation, Δφ

_{Ther_i,j}is the phase difference due to thermal expansion, Δφ

_{RTE_i,j}is the phase difference due to the RTE, Δφ

_{Atmo_i,j}is the phase difference due to the atmospheric component and Δφ

_{Noise_i,j}is the difference of the phase noise components. The procedure includes the following steps:

- (1)
- At least one seed PS is chosen with the following characteristics: φ
_{Defo_SEED}= 0, φ_{Ther_SEED}= 0, φ_{RTE_SEED}= 0 and φ_{Noise_SEED}= 0 small. In other words, a PS located on the ground, with no deformation or thermal expansion and characterized by small noise is sought. - (2)
- The candidate CPSs are those PSs that satisfy the following condition:$${\left|\mathrm{\Delta}{\phi}_{i,\mathit{SEED}}^{unw}\right|}_{90\%}<\mathit{Thr}$$
_{Atmo_i,j}. It is worth noting that the above condition is more restrictive than the ones based on the classical two-parameter model (mean deformation velocity and RTE), see [32,33]. However, the proposed procedure is computationally lighter than the one based on the two-parameter model. - (3)
- This operation is repeated recursively, using a given candidate CPS as seed, until the area of interest is sufficiently covered by CPSs. Additional seeds might be required to cover isolated areas or to increase the CPS density in a given area.

#### 3.2. Phase Unwrapping Consistency Check

_{MS}is the unwrapped interferometric phase (the observation), S and M are the slave and master images (the unknowns). The lth image contains the following phase components:

_{Defo_l}is the deformation component, φ

_{Ther_l}is the thermal expansion component, φ

_{Atmo_l}is the atmospheric component, φ

_{RTE_l}is the RTE component and φ

_{Noise_l}is the phase noise. M equations with N − 1 unknowns written for each pixel: the phase of the first image φ

_{0}is set to zero because Equation (3) has a datum defect: it is translation-invariant. The system of observation equations can be written as:

_{0l}, …, Δφ

_{ij}, …, Δφ

_{m(N − 1)})

^{T}is the M-dimensional observation vector, A is the design matrix that expresses the set of scalar Equations (3) in matrix form and x = (φ

_{1}, …, φ

_{i}, …, φ

_{j}, …, φ

_{N − 1})

^{T}is the N-1-dimensional vector of unknowns. A stochastic model is associated with the functional model in Equation (5), which is represented by weight matrix P. Usually, the observations are equally weighted. The LS solution is given by:

_{res}is the vector of residuals.

#### 3.3. Candidate CPS Selection: Main Parameters

^{2}), the Burgos (Spain) dataset, which covers 113 km

^{2}and is based on TerraSAR-X Stripmap images, and the Seattle dataset, which consists of CosmoSkyMed Stripmap images and covers an area of 365 km

^{2}around the city of Seattle (USA). The three datasets were processed using Win = 400 pixels and Thr defined as a function of the distance from the seed (D), which was computed using these values: 0.8 rad with D = 0 m, 1.1 rad with D = 100 m, 1.35 rad with D = 200 m, 1.52 rad with D = 300 m. To the authors’ experience, working with X-band data, a maximum Thr of 0.8 rad, with D = 0 m, should be used. The increase of Thr with the distance takes into account the term Δφ

_{Atmo_i,j}from Equation (1): its values were derived by analysing the average autocorrelation function of the atmospheric component of the Stripmap SAR images of the Barcelona dataset. In order to speed up the computation, the CPS selection was restricted to all PSs with DA smaller than 0.3.

^{2}with a density of 600 candidate CPS/km

^{2}, using nine seeds; 87% of the candidate CPSs satisfied the filter from Equation (7) and became selected CPSs. In the Burgos dataset, 82,841 candidate CPSs were found over an area of 113 km

^{2}, with a density of 733 candidate CPS/km

^{2}, from one seed. The higher density is mainly attributable to the smaller number of images of this dataset. Seventy-four percent of the candidate CPSs were selected as CPSs. Finally, in the Seattle dataset, 34,594 candidate CPSs were found in an area of approximately 365 km

^{2}, with a density of 95 candidate CPS/km

^{2}, from five seeds. The remarkably reduced density is not due to the different sensor (CosmoSkyMed): it is caused by the wide perpendicular baseline distribution of this dataset, [−800; 600 m] with respect to the distribution of the Barcelona dataset [−170, 150 m]. In this case, 82% of the candidate CPSs were selected as CPSs. It is worth noting that an alternative approach to CPS selection could be used, by explicitly modelling the phase component due to RTE. This would allow us to obtain more CPSs. However, this approach would require more complex, multi-baseline model assisted, phase unwrapping algorithms.

## 4. Third Processing Block

- Estimating the deformation velocity and RTE and, optionally, the thermal expansion;
- Removing the RTE component from the APS-free interferograms. The deformation velocity and the thermal expansion components are optionally removed;
- Performing the 2+1D phase unwrapping.

#### 4.1. 2+1D Phase Unwrapping

_{res}associated with a given observation:

_{M}and φ̆

_{S}are the a posteriori LS estimated image phases. The second parameter is the redundancy of the network of interferograms and images. For each image, it is important to consider the number of interferograms that are directly tied to it, i.e., the number of interferograms where the given image appears either as master or slave. The main steps of the 1D phase unwrapping algorithm include:

- (1)
- Performing the first LS estimation, computing the residuals (Equation (6));
- (2)
- Temporally removing the highest absolute residual (outlier candidate) from the network, and performing a new LS estimation;
- (3)
- Checking the residual of the outlier candidate: if it is a multiple of 2π (within a given tolerance), the observation is corrected and reaccepted. In this way, it is possible to correct the unwrapping errors. Otherwise, the decision of re-entering or rejecting the outlier candidate is based on the comparison of its old and new residuals;
- (4)
- Performing a new LS estimation, computing the residuals and restarting the procedure from point (2). The procedure is executed iteratively from points (2)–(4) until there are no remaining outlier candidates, i.e., there is no residual above a given threshold. Then, the correction of the unwrapping-related errors is extended to all the residuals that, within a given tolerance, are multiples of 2π.

_{hh}is the hth diagonal element of R, which is called the local redundancy. The procedure to correct the unwrapping errors explained above is performed on the corrected residuals. If there is sufficient system redundancy, the LS spreading of errors is mitigated and the unwrapping errors are properly identified. However, it is worth emphasising that this requires that the majority of the interferograms connected to a given image are correctly unwrapped. The algorithm checks the available local redundancy at each iteration and leaves a given outlier candidate untouched if its redundancy is too low. If this occurs, the corresponding parts of the network have to be checked off-line after concluding the automatic analysis. That is, the 1D phase unwrapping only works over the redundant parts of the analysed network. This is often a limitation in the case of ERS and Envisat images where, due to the dispersion of baselines, it is often difficult to obtain a unique and hyper-connected set of interferograms. By contrast, this is not a problem with X-band data and, more importantly, it will not be a problem with the Sentinel-1 data.

#### 4.2. 2+1D Phase Unwrapping: Description of the Outputs

## 5. Analysis of the Barcelona Dataset

^{2}. Only the major deformation phenomena are visible in this image: the airport and the port of Barcelona and other subsidence and uplift areas located at the top right part of the image. However, more than 30 major deformation phenomena were found over the entire frame. Although not all of them have been analysed yet, there are several examples of deformation due to soil compaction, water abstraction, landslides, underground construction works (metro line and metro stations), etc. These results represent a valuable source of information for the public and private entities in charge of the maintenance of indispensable assets of the Barcelona metropolitan area.

## 6. Conclusions

^{2}has been covered with more than 5.4 million PSs, with an average density of 5300 PS/km

^{2}. For each PS, besides the deformation velocity and deformation time series, key information (a quality index and the number of correction for each image of a given time series) has been generated. Even if it has been only tested using X-band data, it is expected that the same procedure can be straightforwardly extended to other types of PSI data.

## Acknowledgments

_{2}storage experiments, MTH1486”, of the same Program. The CosmoSkyMed Seattle dataset mentioned in this work was acquired by an internal project with the company Soldata (Nanterre, France).

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Flow chart showing the main processing steps of the PSIG chain. The dashed lines indicate the three main processing blocks.Notes:

^{(*)}The thermal dilation component can optionally be estimated;

^{(+)}The deformation velocity and the thermal dilation components can optionally be removed.

**Figure 2.**Example of plot of Least Squares (LS) residuals at the first iteration (

**top**) and final iteration (

**middle**). This example refers to a dataset of 236 interferograms. The two plots display the residuals of 79 PSs. The plot shown at the bottom is a profile along a–a (in blue) and b–b (in purple), which displays the residuals of a single PS.

**Figure 3.**Examples of time series (diamonds) and the associated corrections per each image (squares). “Good” time series (

**top**), “Fair” time series (

**middle**) and “Warning” time series (

**bottom**) are shown. These examples were derived from the Barcelona dataset with a network.

**Figure 4.**SAR mean amplitude of the Barcelona dataset and distribution of the nine seed Persistent Scatterers (PSs) used for the candidate CPS selection.

**Figure 6.**Deformation velocity map covering the airport and port of Barcelona (

**top**), which corresponds to the blue frame shown in Figure 5. Quality index map of the same area (

**bottom**), which displays three classes: “Good” (green), “Fair” (yellow) and “Warning” (red).

Dataset | # Images | # Seed PSs | Area (km^{2}) | Candidate CPSs | Candidate CPS Density (PS/km^{2}) | Selected CPSs (%) |
---|---|---|---|---|---|---|

Barcelona | 28 | 9 | 1019 | 611,813 | 600 | 87 |

Burgos | 11 | 1 | 113 | 82,841 | 733 | 74 |

Seattle | 30 | 5 | 365 | 34,594 | 95 | 82 |

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**MDPI and ACS Style**

Devanthéry, N.; Crosetto, M.; Monserrat, O.; Cuevas-González, M.; Crippa, B.
An Approach to Persistent Scatterer Interferometry. *Remote Sens.* **2014**, *6*, 6662-6679.
https://doi.org/10.3390/rs6076662

**AMA Style**

Devanthéry N, Crosetto M, Monserrat O, Cuevas-González M, Crippa B.
An Approach to Persistent Scatterer Interferometry. *Remote Sensing*. 2014; 6(7):6662-6679.
https://doi.org/10.3390/rs6076662

**Chicago/Turabian Style**

Devanthéry, Núria, Michele Crosetto, Oriol Monserrat, María Cuevas-González, and Bruno Crippa.
2014. "An Approach to Persistent Scatterer Interferometry" *Remote Sensing* 6, no. 7: 6662-6679.
https://doi.org/10.3390/rs6076662