# Statistically-Based Trend Analysis of MTInSAR Displacement Time Series

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. F Test

_{0}; i.e., that the difference of the residuals between the measured and approximated data using the two polynomials is not significant. In more detail, the null hypothesis is not rejected with a level of significance $\alpha $ if

_{0}is not satisfied by a certain degree n, we could therefore increase by 1 the degree and perform the test again, until the condition is reached in which an additional degree does not violate the H

_{0}hypothesis, which would correspond to the sought minimum degree.

#### 2.2. Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC)

- $\mathsf{\Delta}\mathrm{AIC}\left({n}_{p}\right)\le 2$: both models have substantial support;
- $4\le \mathsf{\Delta}\mathrm{AIC}\left({n}_{p}\right)\le 7$: $P\left({n}_{p}\right)$ has considerably less support than $P\left({n}_{m}\right)$;
- $\mathsf{\Delta}\mathrm{AIC}\left({n}_{p}\right)>10$: $P\left({n}_{p}\right)$ has no support.

#### 2.3. ${F}_{A}$ Index

## 3. Performance Analysis

- ${\overline{n}}_{p}\left(\mathrm{RMSE}\right)=\mathrm{min}\left(\left\{{n}_{p}|\mathrm{RMSE}\left({n}_{p}\right)\le {\mathrm{Th}}_{\mathrm{RMSE}}\right\}\right){\mathrm{with}\mathrm{Th}}_{\mathrm{RMSE}}=0.5\mathrm{cm}$;
- ${\overline{n}}_{p}\left({R}^{2}\right)=\mathrm{min}\left(\left\{{n}_{p}|{R}^{2}\left({n}_{p}\right)\ge {\mathrm{Th}}_{{R}^{2}}\right\}\right){\mathrm{with}\mathrm{Th}}_{{R}^{2}}=0.8$;
- ${\overline{n}}_{p}\left({\gamma}_{t}\right)=\mathrm{min}\left(\left\{{n}_{p}|{\gamma}_{t}\left({n}_{p}\right)\ge {\mathrm{Th}}_{{\gamma}_{t}}\right\}\right){\mathrm{with}\mathrm{Th}}_{{\gamma}_{t}}=0.8$;
- ${\overline{n}}_{p}\left({F}_{A}\right)=\mathrm{min}\left(\left\{{n}_{p}|{F}_{A}\left({n}_{p}\right)\le {F}_{A\alpha}\right\}\right)\mathrm{with}{F}_{A\alpha}=4$;
- ${\overline{n}}_{p}\left(F\right)=\mathrm{min}\left(\left\{{n}_{p}|F\left({n}_{p}\right)\le {F}_{\mathsf{\alpha}}\right\}\right)\mathrm{with}{F}_{\mathsf{\alpha}}=4$;
- ${\overline{n}}_{p}\left(\mathsf{\Delta}\mathrm{AIC}\right)=\mathrm{min}\left(\left\{{n}_{p}|\mathsf{\Delta}\mathrm{AIC}\left({n}_{p}\right)\le {\mathsf{\Delta}}_{A}\right\}\right){\mathrm{with}\mathsf{\Delta}}_{A}=2$.

## 4. Results and Discussion

#### Results from Real Data

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

^{®}platform. The Sentinel-1 data were provided through the Copernicus Program of the European Union. Finally, the authors thank M. Mottola for her support.

## Conflicts of Interest

## References

- Crosetto, M.; Monserrat, O.; Cuevas-González, M.; Devanthéry, N.; Crippa, B. Persistent Scatterer Interferometry: A review. ISPRS J. Photogramm. Remote Sens.
**2015**, 115, 78–89. [Google Scholar] [CrossRef] [Green Version] - Ho Tong Minh, D.; Hanssen, R.; Rocca, F. Radar Interferometry: 20 Years of Development in Time Series Techniques and Future Perspectives. Remote Sens.
**2020**, 12, 1364. [Google Scholar] [CrossRef] - Caro Cuenca, M.; Hanssen, R.F.; Hooper, A.J.; Arikan, M. Surface deformation of the whole Netherlands after PSI analysis. In Proceedings of the Fringe 2011 Workshop, Frascati, Italy, 19–23 September 2011; pp. 1–27. [Google Scholar]
- Costantini, M.; Ferretti, A.; Minati, F.; Falco, S.; Trillo, F.; Colombo, D.; Novali, F.; Malvarosa, F.; Mammone, C.; Vecchioli, F.; et al. Analysis of surface deformations over the whole Italian territory by interferometric processing of ERS, Envisat and COSMO-SkyMed radar data. Remote Sens. Environ.
**2017**, 202, 250–275. [Google Scholar] [CrossRef] - Dehls, J.F.; Larsen, Y.; Marinkovic, P.; Lauknes, T.R.; Stodle, D.; Moldestad, D.A. INSAR.No: A National Insar Deformation Mapping/Monitoring Service in Norway—From Concept To Operations. In Proceedings of the IGARSS 2019—2019 IEEE International Geoscience and Remote Sensing Symposium, Yokohama, Japan, 28 July–2 August 2019; pp. 5461–5464. [Google Scholar]
- EU-GMS Task Force. European Ground Motion Service (EU-GMS). A Proposed Copernicus Service Element. 2017. Available online: https://land.copernicus.eu/user-corner/technical-library/egms-white-paper (accessed on 11 June 2021).
- Ferretti, A.; Prati, C.; Rocca, F. Nonlinear subsidence rate estimation using permanent scatterers in differential SAR interferometry. IEEE Trans. Geosci. Remote Sens.
**2000**, 38, 2202–2212. [Google Scholar] [CrossRef] [Green Version] - Cohen-Waeber, J.; Bürgmann, R.; Chaussard, E.; Giannico, C.; Ferretti, A. Spatiotemporal Patterns of Precipitation-Modulated Landslide Deformation From Independent Component Analysis of InSAR Time Series. Geophys. Res. Lett.
**2018**, 45, 1878–1887. [Google Scholar] [CrossRef] - Berti, M.; Corsini, A.; Franceschini, S.; Iannacone, J.P. Automated classification of Persistent Scatterers Interferometry time series. Nat. Hazards Earth Syst. Sci.
**2013**, 13, 1945–1958. [Google Scholar] [CrossRef] [Green Version] - Chang, L.; Hanssen, R.F. A Probabilistic Approach for InSAR Time-Series Postprocessing. IEEE Trans. Geosci. Remote Sens.
**2016**, 54, 421–430. [Google Scholar] [CrossRef] [Green Version] - Refice, A.; Pasquariello, G.; Bovenga, F. Model-Free Characterization of SAR MTI Time Series. IEEE Geosci. Remote Sens. Lett.
**2020**, 1–5. [Google Scholar] [CrossRef] - Aminikhanghahi, S.; Cook, D.J. A survey of methods for time series change point detection. Knowl. Inf. Syst.
**2017**, 51, 339–367. [Google Scholar] [CrossRef] [Green Version] - Hussain, E.; Novellino, A.; Jordan, C.; Bateson, L. Offline-Online Change Detection for Sentinel-1 InSAR Time Series. Remote Sens.
**2021**, 13, 1656. [Google Scholar] [CrossRef] - Muggeo, V.M.R. Estimating regression models with unknown break-points. Stat. Med.
**2003**, 22, 3055–3071. [Google Scholar] [CrossRef] - Lee, T.S. Change-point problems: Bibliography and review. J. Stat. Theory Pract.
**2010**. [Google Scholar] [CrossRef] - Fisher, R.R.A. Statistical Methods for Research Workers; Oliver & Boyd: Edinburgh, UK, 1932; ISBN 0050021702. [Google Scholar]
- Akaike, H. A new look at the statistical model identification. IEEE Trans. Automat. Contr.
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Schwarz, G. Estimating the Dimension of a Model. Ann. Stat.
**1978**, 6, 461–464. [Google Scholar] [CrossRef] - Colesanti, C.; Ferretti, A.; Novali, F.; Prati, C.; Rocca, F. SAR monitoring of progressive and seasonal ground deformation using the permanent scatterers technique. IEEE Trans. Geosci. Remote Sens.
**2003**, 41, 1685–1701. [Google Scholar] [CrossRef] [Green Version] - Bovenga, F.; Nutricato, R.; Refice, A.; Guerriero, L.; Chiaradia, M.T. SPINUA: A flexible processing chain for ERS/ENVISAT long term interferometry. In Proceedings of the 2004 Envisat & ERS Symposium, Salzburg, Austria, 6–10 September 2004; pp. 473–478. [Google Scholar]
- Crosetto, M.; Monserrat, O.; Cuevas-González, M.; Devanthéry, N.; Luzi, G.; Crippa, B. Measuring thermal expansion using X-band persistent scatterer interferometry. ISPRS J. Photogramm. Remote Sens.
**2015**, 100, 84–91. [Google Scholar] [CrossRef] [Green Version] - Morishita, Y.; Hanssen, R.F. Deformation Parameter Estimation in Low Coherence Areas Using a Multisatellite InSAR Approach. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 4275–4283. [Google Scholar] [CrossRef] - Draper, N.R.; Smith, H. Applied Regression Analysis; Wiley Series in Probability and Statistics; John Wiley & Sons, Inc.: New York, NY, USA, 1998; ISBN 9780471170822. [Google Scholar]
- Motulsky, H.; Christopoulos, A. Fitting Models to Biological Data Using Linear and Nonlinear Regression. A Practical Guide to Curve Fitting; GraphPad Software: San Diego, CA, USA, 2003; ISBN 0198038348/9780198038344. [Google Scholar]
- Forbes, C.; Evans, M.; Hastings, N.; Peacock, B. Statistical Distributions, 4th ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2011; ISBN 978-0-470-39063-4. [Google Scholar]
- Burnham, K.P.; Anderson, D.R. Multimodel Inference. Sociol. Methods Res.
**2004**, 33, 261–304. [Google Scholar] [CrossRef] - Dai, K.; Peng, J.; Zhang, Q.; Wang, Z.; Qu, T.; He, C.; Li, D.; Liu, J.; Li, Z.; Xu, Q.; et al. Entering the Era of Earth Observation-Based Landslide Warning Systems: A Novel and Exciting Framework. IEEE Geosci. Remote Sens. Mag.
**2020**, 8, 136–153. [Google Scholar] [CrossRef] [Green Version] - Moretto, S.; Bozzano, F.; Esposito, C.; Mazzanti, P.; Rocca, A. Assessment of landslide pre-failure monitoring and forecasting using satellite SAR interferometry. Geosciences
**2017**, 7, 36. [Google Scholar] [CrossRef] [Green Version] - Carlà, T.; Intrieri, E.; Raspini, F.; Bardi, F.; Farina, P.; Ferretti, A.; Colombo, D.; Novali, F.; Casagli, N. Perspectives on the prediction of catastrophic slope failures from satellite InSAR. Sci. Rep.
**2019**, 9. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Xu, Q.; Peng, D.; Zhang, S.; Zhu, X.; He, C.; Qi, X.; Zhao, K.; Xiu, D.; Ju, N. Successful implementations of a real-time and intelligent early warning system for loess landslides on the Heifangtai terrace, China. Eng. Geol.
**2020**, 278, 105817. [Google Scholar] [CrossRef] - Crosetto, M.; Solari, L.; Mróz, M.; Balasis-Levinsen, J.; Casagli, N.; Frei, M.; Oyen, A.; Moldestad, D.A.; Bateson, L.; Guerrieri, L.; et al. The evolution of wide-area DInSAR: From regional and national services to the European ground motion service. Remote Sens.
**2020**, 12, 2043. [Google Scholar] [CrossRef] - Bischoff, C.A.; Ferretti, A.; Novali, F.; Uttini, A.; Giannico, C.; Meloni, F. Nationwide deformation monitoring with SqueeSAR
^{®}using Sentinel-1 data. Proc. Int. Assoc. Hydrol. Sci.**2020**, 382, 31–37. [Google Scholar] [CrossRef] [Green Version] - Lanari, R.; Bonano, M.; Casu, F.; De Luca, C.; Manunta, M.; Manzo, M.; Onorato, G.; Zinno, I. Automatic generation of Sentinel-1 continental scale DInSAR deformation time series through an extended P-SBAS processing pipeline in a cloud computing environment. Remote Sens.
**2020**, 12, 2961. [Google Scholar] [CrossRef] - Schlögl, M.; Widhalm, B.; Avian, M. Comprehensive time-series analysis of bridge deformation using differential satellite radar interferometry based on Sentinel-1. ISPRS J. Photogramm. Remote Sens.
**2021**, 172, 132–146. [Google Scholar] [CrossRef]

**Figure 1.**Quantile values ${F}_{\alpha}$ of the Fisher distribution $f\left(1,N\right)$, computed for $N={N}_{t}-{n}_{p}$ between 10 and 120, and for confidence levels p between 0.90 and 0.99. For ${N}_{t}\ge 100$ and ${n}_{p}\le 4$, $N\approx {N}_{t}$.

**Figure 2.**(

**a**) Simulated MTInSAR displacement time series (black circles) covering a timespan of T = 600 days, sampled at dt = 6 days, constructed with a piecewise linear trend (blue dashed line) with velocities ${v}_{1}$ = 0 and ${v}_{2}$ = −3 cm/year, with transition at time ${t}_{1}=T/2$, and by an additive noise signal (red dots) equivalent to a coherence ${\gamma}_{\mathrm{IN}}=0.87$; (

**b**) trends derived by modeling the simulated displacements (black crosses) by polynomial models of degree ${n}_{p}$ ranging from 1 to 4. The legend reports the values of the quality indices computed for each polynomial model.

**Figure 3.**Trends of $\mathrm{RMSE},{R}^{2},{\gamma}_{t},{F}_{A},F$, and AIC as a function of the velocity change $dv={v}_{2}-{v}_{1}$ in cm/year, computed for fitting polynomials of order ${n}_{p}$ ranging from 1 to 4. Plots along each row refer to different noise levels, decreasing as ${\gamma}_{\mathrm{IN}}$ increases from 0.50 to 0.90. The results refer to simulated piecewise linear trends as in Figure 2, with a total timespan of T = 600 days, sampled at dt = 6 days, with the discontinuity at time ${t}_{1}=T/2$, and white Gaussian added noise.

**Figure 4.**Best approximating polynomial degrees (labeled as P1, P2, P3, and P4, respectively, on the y axes) as a function of the time series velocity change $dv$ in cm/year (on the x axes), selected by using $\mathrm{RMSE},{R}^{2},{\gamma}_{t},{F}_{C},F$, and ΔAIC, respectively (from the top to bottom rows of the figure). Plots along each row refer to different noise levels, decreasing as ${\gamma}_{\mathrm{IN}}$ increases from 0.50 to 0.90. Results refer again to simulated piecewise linear trends, as described for the plots in Figure 3.

**Figure 5.**Flow chart of the processing scheme proposed for the analysis of the MTInSAR displacement time series.

**Figure 6.**Examples of results derived by running the proposed trend analysis on simulated time series with 100 samples at a time spacing of 6 days. Legends at the bottom of each plot show the values of the quality indices computed for each polynomial model. Upper-right insets report the trend analysis outputs (${\overline{n}}_{p},{F}_{A},{\gamma}_{t})$. The plot in (

**a**) refers to a relatively noisy linear trend. Plots (

**b**,

**c**) refer to piecewise linear trends with velocities ${v}_{1}$ = 0 and ${v}_{2}$= −3 cm/year with a transition time ${t}_{1}=T/2$, and noise levels corresponding to temporal coherences of 0.9 (low noise) and 0.57 (strong noise), respectively. Plot (

**d**) refers to a piecewise linear trend with velocities ${v}_{1}$= 0 and ${v}_{2}$= −3 cm/year and a transition time ${t}_{1}=0.8\xb7T$, and a noise level corresponding to a temporal coherence of 0.75.

**Figure 7.**(

**a**) Scatterplot of ${\gamma}_{t,\mathrm{OUT}}$ versus ${\gamma}_{t,\mathrm{IN}}$; (

**b**) ${\gamma}_{t,\mathrm{OUT}}$ histograms corresponding to polynomial degrees ranging from 1 (linear) to 4; (

**c**–

**e**) examples of nonlinear displacement time series selected according to the output of the proposed trend analysis. The continuous colored lines and black crosses represent the polynomial functions and the MTInSAR displacement values in the input, respectively.

**Table 1.**Values explored for each of the parameters involved in the simulation scheme. The timespan of the simulated MTInSAR time series is $T={N}_{t}\xb7dt$.

Input Parameter | Values |
---|---|

$\lambda $ | 5.6 cm |

${N}_{t}$ | 100 |

$dt$ | 6 days |

${t}_{1}$ | (0.5, 0.6, 0.7, 0.8, 0.9)⋅T |

${v}_{1}$ | 0 |

${v}_{2}$ | (0, −1, −2, −3, −4, −5) cm/year |

${\gamma}_{\mathrm{IN}}$ | (0.5, 0.6, 0.7, 0.8, 0.9) |

${n}_{p}$ | (1, 2, 3, 4) |

${N}_{r}$ | 100 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bovenga, F.; Pasquariello, G.; Refice, A.
Statistically-Based Trend Analysis of MTInSAR Displacement Time Series. *Remote Sens.* **2021**, *13*, 2302.
https://doi.org/10.3390/rs13122302

**AMA Style**

Bovenga F, Pasquariello G, Refice A.
Statistically-Based Trend Analysis of MTInSAR Displacement Time Series. *Remote Sensing*. 2021; 13(12):2302.
https://doi.org/10.3390/rs13122302

**Chicago/Turabian Style**

Bovenga, Fabio, Guido Pasquariello, and Alberto Refice.
2021. "Statistically-Based Trend Analysis of MTInSAR Displacement Time Series" *Remote Sensing* 13, no. 12: 2302.
https://doi.org/10.3390/rs13122302