# Multi-Temporal Small Baseline Interferometric SAR Algorithms: Error Budget and Theoretical Performance

## Abstract

**:**

## 1. Introduction

## 2. The SB InSAR Framework

## 3. Relative Error Bounds of Ground Deformation Measurements via the SB Methods

## 4. How Do the Relative Error Bounds Depend on the Perpendicular Baseline Threshold of the SB Interferograms?

## 5. How Does the Temporal Coherence Get Valuable Information on the SBAS-InSar Products Error?

## 6. Experimental Results

## 7. Discussion

## 8. Conclusions and Future Perspectives

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Massonnet, D.; Rossi, M.; Carmona, C.; Adragna, F.; Peltzer, G.; Feigl, K.; Rabaute, T. The Displacement Field of the Landers Earthquake Mapped by Radar Interferometry. Nature
**1993**, 364, 138–142. [Google Scholar] [CrossRef] - Bürgmann, R.; Rosen, P.A.; Fielding, E.J. Synthetic Aperture Radar Interferometry to Measure Earth’s Surface Topography and Its Deformation. Annu. Rev. Earth Planet. Sci.
**2000**, 28, 169–209. [Google Scholar] [CrossRef] - Rosen, P.A.; Hensley, S.; Joughin, I.R.; Li, F.K.; Madsen, S.N.; Rodriguez, E.; Goldstein, R.M. Synthetic Aperture Radar Interferometry. Proc. IEEE
**2000**, 88, 333–382. [Google Scholar] [CrossRef] - Li, F.K.; Goldstein, R.M. Studies of Multibaseline Spaceborne Interferometric Synthetic Aperture Radars. IEEE Trans. Geosci. Remote Sens.
**1990**, 28, 88–97. [Google Scholar] [CrossRef] - Fialko, Y.; Simons, M.; Agnew, D. The Complete (3-D) Surface Displacement Field in the Epicentral Area of the 1999 MW7.1 Hector Mine Earthquake, California, from Space Geodetic Observations. Geophys. Res. Lett.
**2001**, 28, 3063–3066. [Google Scholar] [CrossRef] [Green Version] - Diao, F.; Walter, T.R.; Wang, R. Continued Fault Locking near Istanbul: Evidence of High Earthquake Potential from InSAR Observation. In Proceedings of the EGU General Assembly 2015, Vienna, Austria, 12–17 April 2015; Volume 17, p. 15247. [Google Scholar]
- Chaussard, E.; Wdowinski, S.; Cabral-Cano, E.; Amelung, F. Land Subsidence in Central Mexico Detected by ALOS InSAR Time-Series. Remote Sens. Environ.
**2014**, 140, 94–106. [Google Scholar] [CrossRef] - Hussain, E.; Wright, T.J.; Walters, R.J.; Bekaert, D.; Hooper, A.; Houseman, G.A. Geodetic Observations of Postseismic Creep in the Decade after the 1999 Izmit Earthquake, Turkey: Implications for a Shallow Slip Deficit. J. Geophys. Res. Solid Earth
**2016**, 121, 2980–3001. [Google Scholar] [CrossRef] [Green Version] - Ruch, J.; Pepe, S.; Casu, F.; Acocella, V.; Neri, M.; Solaro, G.; Sansosti, E. How Do Volcanic Rift Zones Relate to Flank Instability? Evidence from Collapsing Rifts at Etna. Geophys. Res. Lett.
**2012**, 39. [Google Scholar] [CrossRef] - Del Negro, C.; Currenti, G.; Solaro, G.; Greco, F.; Pepe, A.; Napoli, R.; Pepe, S.; Casu, F.; Sansosti, E. Capturing the Fingerprint of Etna Volcano Activity in Gravity and Satellite Radar Data. Sci. Rep.
**2013**, 3, 3089. [Google Scholar] [CrossRef] [Green Version] - Jiang, L.; Lin, H.; Cheng, S. Monitoring and Assessing Reclamation Settlement in Coastal Areas with Advanced InSAR Techniques: Macao City (China) Case Study. Int. J. Remote Sens.
**2011**, 32, 3565–3588. [Google Scholar] [CrossRef] - Wang, J. Geodetic and Remote-Sensing Sensors for Dam Deformation Monitoring. Sensors
**2018**, 18, 3682. [Google Scholar] - Milillo, P.; Giardina, G.; DeJong, M.J.; Perissin, D.; Milillo, G. Multi-Temporal InSAR Structural Damage Assessment: The London Crossrail Case Study. Remote Sens.
**2018**, 10, 287. [Google Scholar] [CrossRef] [Green Version] - Sousa, J.J.; Bastos, L. Multi-Temporal SAR Interferometry Reveals Acceleration of Bridge Sinking before Collapse. Nat. Hazards Earth Syst. Sci.
**2013**, 13, 659–667. [Google Scholar] [CrossRef] - Noviello, C.; Verde, S.; Zamparelli, V.; Fornaro, G.; Pauciullo, A.; Reale, D.; Nicodemo, G.; Ferlisi, S.; Gulla, G.; Peduto, D. Monitoring Buildings at Landslide Risk With SAR: A Methodology Based on the Use of Multipass Interferometric Data. IEEE Geosci. Remote Sens. Mag.
**2020**, 8, 91–119. [Google Scholar] [CrossRef] - Blanco-Sànchez, P.; Mallorquí, J.J.; Duque, S.; Monells, D. The Coherent Pixels Technique (CPT): An Advanced DInSAR Technique for Nonlinear Deformation Monitoring. In Earth Sciences and Mathematics; Camacho, A.G., Díaz, J.I., Fernández, J., Eds.; Pageoph Topical Volumes: Birkhäuser, Basel, Switzerland, 2008; Volume 1, pp. 1167–1193. ISBN 978-3-7643-8907-9. [Google Scholar]
- Pepe, A.; Solaro, G.; Calò, F.; Dema, C. A Minimum Acceleration Approach for the Retrieval of Multiplatform InSAR Deformation Time Series. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2016**, 9, 3883–3898. [Google Scholar] [CrossRef] - Cao, N.; Lee, H.; Jung, H.C. A Phase-Decomposition-Based PSInSAR Processing Method. IEEE Trans. Geosci. Remote Sens.
**2016**, 54, 1074–1090. [Google Scholar] [CrossRef] - 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] - 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] - Ferretti, A.; Prati, C.; Rocca, F. Permanent Scatterers in SAR Interferometry. IEEE Trans. Geosci. Remote Sens.
**2001**, 39, 8–20. [Google Scholar] [CrossRef] - Ferretti, A.; Fumagalli, A.; Novali, F.; Prati, C.; Rocca, F.; Rucci, A. A New Algorithm for Processing Interferometric Data-Stacks: SqueeSAR. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 3460–3470. [Google Scholar] [CrossRef] - Fornaro, G.; Verde, S.; Reale, D.; Pauciullo, A. CAESAR: An Approach Based on Covariance Matrix Decomposition to Improve Multibaseline–Multitemporal Interferometric SAR Processing. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 2050–2065. [Google Scholar] [CrossRef] - Goel, K.; Adam, N. A Distributed Scatterer Interferometry Approach for Precision Monitoring of Known Surface Deformation Phenomena. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 5454–5468. [Google Scholar] [CrossRef] - Berardino, P.; Fornaro, G.; Lanari, R.; Sansosti, E. A New Algorithm for Surface Deformation Monitoring Based on Small Baseline Differential SAR Interferograms. IEEE Trans. Geosci. Remote Sens.
**2002**, 40, 2375–2383. [Google Scholar] [CrossRef] [Green Version] - Hetland, E.A.; Musé, P.; Simons, M.; Lin, Y.N.; Agram, P.S.; DiCaprio, C.J. Multiscale InSAR Time Series (MInTS) Analysis of Surface Deformation. J. Geophys. Res. Solid Earth
**2012**, 117. [Google Scholar] [CrossRef] [Green Version] - Mora, O.; Mallorqui, J.J.; Broquetas, A. Linear and Nonlinear Terrain Deformation Maps from a Reduced Set of Interferometric SAR Images. IEEE Trans. Geosci. Remote Sens.
**2003**, 41, 2243–2253. [Google Scholar] [CrossRef] - Samsonov, S.; d’Oreye, N. Multidimensional Time-Series Analysis of Ground Deformation from Multiple InSAR Data Sets Applied to Virunga Volcanic Province. Geophys. J. Int.
**2012**, 191, 1095–1108. [Google Scholar] [CrossRef] [Green Version] - Samiei-Esfahany, S.; Martins, J.E.; van Leijen, F.; Hanssen, R.F. Phase Estimation for Distributed Scatterers in InSAR Stacks Using Integer Least Squares Estimation. IEEE Trans. Geosci. Remote Sens.
**2016**, 54, 5671–5687. [Google Scholar] [CrossRef] [Green Version] - Schmitt, M.; Stilla, U. Maximum-Likelihood Estimation for Multi-Aspect Multi-Baseline SAR Interferometry of Urban Areas. ISPRS J. Photogramm. Remote Sens.
**2014**, 87, 68–77. [Google Scholar] [CrossRef] - Sowter, A.; Bateson, L.; Strange, P.; Ambrose, K.; Syafiudin, M.F. DInSAR Estimation of Land Motion Using Intermittent Coherence with Application to the South Derbyshire and Leicestershire Coalfields. Remote Sens. Lett.
**2013**, 4, 979–987. [Google Scholar] [CrossRef] [Green Version] - Pepe, A.; Yang, Y.; Manzo, M.; Lanari, R. Improved EMCF-SBAS Processing Chain Based on Advanced Techniques for the Noise-Filtering and Selection of Small Baseline Multi-Look DInSAR Interferograms. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 4394–4417. [Google Scholar] [CrossRef] - Usai, S. A Least Squares Database Approach for SAR Interferometric Data. IEEE Trans. Geosci. Remote Sens.
**2003**, 41, 753–760. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Zhu, X.X. Robust Estimators for Multipass SAR Interferometry. IEEE Trans. Geosci. Remote Sens.
**2016**, 54, 968–980. [Google Scholar] [CrossRef] - Zhang, L.; Ding, X.; Lu, Z. Modeling PSInSAR Time Series Without Phase Unwrapping. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 547–556. [Google Scholar] [CrossRef] - Werner, C.; Wegmuller, U.; Strozzi, T.; Wiesmann, A. Interferometric Point Target Analysis for Deformation Mapping. In Proceedings of the IGARSS 2003 IEEE International Geoscience and Remote Sensing Symposium, Proceedings (IEEE Cat. No.03CH37477), Toulouse, France, 21–25 July 2003; Volume 7, pp. 4362–4364. [Google Scholar]
- Doin, M.-P.; Lodge, F.; Guillaso, S.; Jolivet, R.; Lasserre, C.; Ducret, G.; Grandin, R.; Pathier, E.; Pinel, V. Presentation of the Small Baseline NSBAS Processing Chain on a Case Example: The Etna Deformation Monitoring from 2003 to 2010 Using Envisat Data. In Proceedings of the Fringe Symposium, Frascati, Italy, 20 September 2011. [Google Scholar]
- Falabella, F.; Serio, C.; Zeni, G.; Pepe, A. On the Use of Weighted Least-Squares Approaches for Differential Interferometric SAR Analyses: The Weighted Adaptive Variable-LEngth (WAVE) Technique. Sensors
**2020**, 20, 1103. [Google Scholar] [CrossRef] [Green Version] - Crosetto, M.; Tscherning, C.C.; Crippa, B.; Castillo, M. Subsidence Monitoring Using SAR Interferometry: Reduction of the Atmospheric Effects Using Stochastic Filtering. Geophys. Res. Lett.
**2002**, 29, 26-1–26-4. [Google Scholar] [CrossRef] - Zebker, H.A.; Villasenor, J. Decorrelation in Interferometric Radar Echoes. IEEE Trans. Geosci. Remote Sens.
**1992**, 30, 950–959. [Google Scholar] [CrossRef] [Green Version] - Rodriguez, E.; Martin, J.M. Theory and Design of Interferometric Synthetic Aperture Radars. IEE Proc. F Radar Signal Process.
**1992**, 139, 147–159. [Google Scholar] [CrossRef] - Pinel-Puyssegur, B.; Michel, R.; Avouac, J. Multi-Link InSAR Time Series: Enhancement of a Wrapped Interferometric Database. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2012**, 5, 784–794. [Google Scholar] [CrossRef] - Raucoules, D.; Bourgine, B.; de Michele, M.; Le Cozannet, G.; Closset, L.; Bremmer, C.; Veldkamp, H.; Tragheim, D.; Bateson, L.; Crosetto, M.; et al. Validation and Intercomparison of Persistent Scatterers Interferometry: PSIC4 Project Results. J. Appl. Geophys.
**2009**, 68, 335–347. [Google Scholar] [CrossRef] [Green Version] - Gong, W.; Thiele, A.; Hinz, S.; Meyer, F.; Hooper, A.; Agram, P. Comparison of Small Baseline Interferometric SAR Processors for Estimating Ground Deformation. Remote Sens.
**2016**, 8, 330. [Google Scholar] [CrossRef] [Green Version] - Shanker, P.; Casu, F.; Zebker, H.A.; Lanari, R. Comparison of Persistent Scatterers and Small Baseline Time-Series InSAR Results: A Case Study of the San Francisco Bay Area. IEEE Geosci. Remote Sens. Lett.
**2011**, 8, 592–596. [Google Scholar] [CrossRef] - Shamshiri, R.; Nahavandchi, H.; Motagh, M.; Hooper, A. Efficient Ground Surface Displacement Monitoring Using Sentinel-1 Data: Integrating Distributed Scatterers (DS) Identified Using Two-Sample t-Test with Persistent Scatterers (PS). Remote Sens.
**2018**, 10, 794. [Google Scholar] [CrossRef] [Green Version] - Zhao, Q.; Pepe, A.; Gao, W.; Lu, Z.; Bonano, M.; He, M.L.; Wang, J.; Tang, X. A DInSAR Investigation of the Ground Settlement Time Evolution of Ocean-Reclaimed Lands in Shanghai. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2015**, 8, 1763–1781. [Google Scholar] [CrossRef] - Desnos, Y.L.; Foumelis, M.; Engdahl, M. Sentinel-1 Mission Scientific Exploitation Activities. IEEE Int. Geosci. Remote Sens. Symp.
**2017**, 19, 19364. [Google Scholar] - Zan, F.D.; Guarnieri, A.M. TOPSAR: Terrain Observation by Progressive Scans. IEEE Trans. Geosci. Remote Sens.
**2006**, 44, 2352–2360. [Google Scholar] [CrossRef] - Scheiber, R.; Jäger, M.; Prats-Iraola, P.; Zan, F.D.; Geudtner, D. Speckle Tracking and Interferometric Processing of TerraSAR-X TOPS Data for Mapping Nonstationary Scenarios. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2015**, 8, 1709–1720. [Google Scholar] [CrossRef] [Green Version] - Yague-Martinez, N.; Prats-Iraola, P.; Zan, F.D. Coregistration of Interferometric Stacks of Sentinel-1A TOPS Data. In Proceedings of the EUSAR 2016: 11th European Conference on Synthetic Aperture Radar, Hamburg, Germany, June 2016; pp. 1–6. [Google Scholar]
- Fattahi, H.; Agram, P.; Simons, M. A Network-Based Enhanced Spectral Diversity Approach for TOPS Time-Series Analysis. IEEE Trans. Geosci. Remote Sens.
**2017**, 55, 777–786. [Google Scholar] [CrossRef] [Green Version] - Zan, F.D.; Zonno, M.; López-Dekker, P. Phase Inconsistencies and Multiple Scattering in SAR Interferometry. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 6608–6616. [Google Scholar] [CrossRef] [Green Version] - Eshqi Molan, Y.; Lu, Z. Modeling InSAR Phase and SAR Intensity Changes Induced by Soil Moisture. IEEE Trans. Geosci. Remote Sens.
**2020**, 58, 4967–4975. [Google Scholar] [CrossRef] - Brancato, V.; Hajnsek, I. Separating the Influence of Vegetation Changes in Polarimetric Differential SAR Interferometry. IEEE Trans. Geosci. Remote Sens.
**2018**, 56, 6871–6883. [Google Scholar] [CrossRef] - De Zan, F.; Parizzi, A.; Prats-Iraola, P.; López-Dekker, P. A SAR Interferometric Model for Soil Moisture. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 418–425. [Google Scholar] [CrossRef] [Green Version] - Pepe, A.; Lanari, R. On the Extension of the Minimum Cost Flow Algorithm for Phase Unwrapping of Multitemporal Differential SAR Interferograms. IEEE Trans. Geosci. Remote Sens.
**2006**, 44, 2374–2383. [Google Scholar] [CrossRef] - Papoulis, A.; Pillai, S.U. Probability, Random Variables and Stochastic Processes with Errata Sheet, 4th ed.; McGraw-Hill Education: Boston, MA, USA, 2015; ISBN 978-0-07-122661-5. [Google Scholar]
- Chandrasekaran, S.; Ipsen, I.C.F. On the Sensitivity of Solution Components in Linear Systems of Equations. Siam J. Matrix Anal. Appl.
**1995**, 16, 93–112. [Google Scholar] [CrossRef] [Green Version] - Vaccaro, R.; Kot, A. A Perturbation Theory for the Analysis of SVD-Based Algorithms. In Proceedings of the ICASSP ’87 IEEE International Conference on Acoustics, Speech, and Signal Processing, Dallas, TX, USA, 6–9 April 1987; Volume 12, pp. 1613–1616. [Google Scholar]
- Wei, M. The Perturbation of Consistent Least Squares Problems. Linear Algebra Its Appl.
**1989**, 112, 231–245. [Google Scholar] [CrossRef] [Green Version] - Demmel, J.W. Applied Numerical Linear Algebra; SIAM: Philadelphia, PA, USA, 1997; ISBN 978-0-89871-389-3. [Google Scholar]
- Hanssen, R.F. Radar Interferometry: Data Interpretation and Error Analysis; Springer Science & Business Media: Berlin, Germany, 2001; ISBN 978-0-7923-6945-5. [Google Scholar]
- Parizzi, A.; Brcic, R. Adaptive InSAR Stack Multilooking Exploiting Amplitude Statistics: A Comparison Between Different Techniques and Practical Results. IEEE Geosci. Remote Sens. Lett.
**2011**, 8, 441–445. [Google Scholar] [CrossRef] [Green Version] - Pepe, A.; Mastro, P.; Jones, C.E. Adaptive Multilooking of Multitemporal Differential SAR Interferometric Data Stack Using Directional Statistics. IEEE Trans. Geosci. Remote Sens.
**2020**, 1–16. [Google Scholar] [CrossRef] - Jiang, M.; Ding, X.; Hanssen, R.F.; Malhotra, R.; Chang, L. Fast Statistically Homogeneous Pixel Selection for Covariance Matrix Estimation for Multitemporal InSAR. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 1213–1224. [Google Scholar] [CrossRef] - Bamler, R.; Hartl, P. Synthetic Aperture Radar Interferometry. Inverse Probl.
**1998**, R1–R54. [Google Scholar] [CrossRef] - Strang, G. Linear Algebra and Its Applications, 4th ed.; Cengage Learning: Belmont, CA, USA, 2006; ISBN 978-0-03-010567-8. [Google Scholar]
- Golub, G.H.; Reinsch, C. Singular Value Decomposition and Least Squares Solutions. In Handbook for Automatic Computation: Volume II: Linear Algebra; Wilkinson, J.H., Reinsch, C., Bauer, F.L., Householder, A.S., Olver, F.W.J., Rutishauser, H., Samelson, K., Stiefel, E., Eds.; Springer: Berlin/Heidelberg, Germany, 1971; pp. 134–151. ISBN 978-3-642-86940-2. [Google Scholar]
- Agram, P.S.; Simons, M. A Noise Model for InSAR Time Series. J. Geophys. Res. Solid Earth
**2015**, 120, 2752–2771. [Google Scholar] [CrossRef] - Yu, C.; Li, Z.; Penna, N.T. Interferometric Synthetic Aperture Radar Atmospheric Correction Using a GPS-Based Iterative Tropospheric Decomposition Model. Remote Sens. Environ.
**2018**, 204, 109–121. [Google Scholar] [CrossRef] - Goldstein, R. Atmospheric Limitations to Repeat-Track Radar Interferometry. Geophys. Res. Lett.
**1995**, 22, 2517–2520. [Google Scholar] [CrossRef] [Green Version] - Pepe, A.; Calò, F. A Review of Interferometric Synthetic Aperture RADAR (InSAR) Multi-Track Approaches for the Retrieval of Earth’s Surface Displacements. Appl. Sci.
**2017**, 7, 1264. [Google Scholar] [CrossRef] [Green Version] - Zwieback, S.; Hensley, S.; Hajnsek, I. Soil Moisture Estimation Using Differential Radar Interferometry: Toward Separating Soil Moisture and Displacements. IEEE Trans. Geosci. Remote Sens.
**2017**, 55, 5069–5083. [Google Scholar] [CrossRef] - Michaelides, R.J.; Zebker, H.A.; Zheng, Y. An Algorithm for Estimating and Correcting Decorrelation Phase From InSAR Data Using Closure Phase Triplets. IEEE Trans. Geosci. Remote Sens.
**2019**, 57, 10390–10397. [Google Scholar] [CrossRef] - Casu, F.; Manzo, M.; Lanari, R. A Quantitative Assessment of the SBAS Algorithm Performance for Surface Deformation Retrieval from DInSAR Data. Remote Sens. Environ.
**2006**, 102, 195–210. [Google Scholar] [CrossRef] - Casu, F.; Elefante, S.; Imperatore, P.; Zinno, I.; Manunta, M.; Luca, C.D.; Lanari, R. SBAS-DInSAR Parallel Processing for Deformation Time-Series Computation. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2014**, 7, 3285–3296. [Google Scholar] [CrossRef] - Manunta, M.; Luca, C.D.; Zinno, I.; Casu, F.; Manzo, M.; Bonano, M.; Fusco, A.; Pepe, A.; Onorato, G.; Berardino, P.; et al. The Parallel SBAS Approach for Sentinel-1 Interferometric Wide Swath Deformation Time-Series Generation: Algorithm Description and Products Quality Assessment. IEEE Trans. Geosci. Remote Sens.
**2019**, 57, 6259–6281. [Google Scholar] [CrossRef] - 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] - Rocca, F. Modeling Interferogram Stacks. IEEE Trans. Geosci. Remote Sens.
**2007**, 45, 3289–3299. [Google Scholar] [CrossRef] - Pepe, A. Theory and Statistical Description of the Enhanced Multi-Temporal InSAR (E-MTInSAR) Noise-Filtering Algorithm. Remote Sens.
**2019**, 11, 363. [Google Scholar] [CrossRef] [Green Version] - Jr, F.J.M. The Kolmogorov-Smirnov Test for Goodness of Fit. J. Am. Stat. Assoc.
**1951**, 46, 68–78. [Google Scholar] [CrossRef] - Martinez-Espla, J.J.; Martinez-Marin, T.; Lopez-Sanchez, J.M. A Particle Filter Approach for InSAR Phase Filtering and Unwrapping. IEEE Trans. Geosci. Remote Sens.
**2009**, 47, 1197–1211. [Google Scholar] [CrossRef] - Ferraioli, G.; Shabou, A.; Tupin, F.; Pascazio, V. Multichannel Phase Unwrapping With Graph Cuts. IEEE Geosci. Remote Sens. Lett.
**2009**, 6, 562–566. [Google Scholar] [CrossRef] - Fornaro, G.; Pauciullo, A.; Reale, D. A Null-Space Method for the Phase Unwrapping of Multitemporal SAR Interferometric Stacks. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 2323–2334. [Google Scholar] [CrossRef] - Lu, Y.; Wang, X.; Zhang, X. Weighted Least-Squares Phase Unwrapping Algorithm Based on Derivative Variance Correlation Map. Optik
**2007**, 118, 62–66. [Google Scholar] [CrossRef] - Zhang, K.; Ge, L.; Hu, Z.; Ng, A.H.; Li, X.; Rizos, C. Phase Unwrapping for Very Large Interferometric Data Sets. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 4048–4061. [Google Scholar] [CrossRef] - Mardia, K.V.; Jupp, P.E. Directional Statistics; John Wiley & Sons: Hoboken, NJ, USA, 2009; ISBN 978-0-470-31781-5. [Google Scholar]
- Goldstein, R.M.; Werner, C.L. Radar Interferogram Filtering for Geophysical Applications. Geophys. Res. Lett.
**1998**, 25, 4035–4038. [Google Scholar] [CrossRef] [Green Version] - Costantini, M.; Rosen, P.A. A Generalized Phase Unwrapping Approach for Sparse Data. In Proceedings of the IEEE 1999 International Geoscience and Remote Sensing Symposium, IGARSS’99 (Cat. No.99CH36293), Hamburg, Germany, 28 June–2 July 1999; Volume 1, pp. 267–269. [Google Scholar]
- Torres, R.; Løkås, S.; Geudtner, D.; Rosich, B. Sentinel-1A LEOP and Commissioning. In Proceedings of the 2014 IEEE Geoscience and Remote Sensing Symposium, Quebec City, QC, Canada, 13–18 July 2014; pp. 1469–1472. [Google Scholar]

**Figure 1.**Examples of SB InSAR data pair distributions in the temporal/perpendicular baseline domain. (

**a**) SAR data collected by the ERS/ENVISAT sensors in Yellowstone, U.S. (track 41, frame 2709) region. The applied thresholds on the maximum geometrical and temporal baselines are 800 m and three years, respectively. (

**b**) A triangular-shaped network of SB interferograms related to a SAR data set collected by the ENVISAT sensor (ascending orbit, VV polarization) in the area of Pearl River Delta, China, from 2006 to 2010.

**Figure 2.**Pictorial representation of the lack of temporal phase consistency among a set of three images A, B, and C. The terms $\mathsf{\Delta}{n}_{i}^{uncor}$ and ${K}_{i}^{uncorr}$ are the time-uncorrelated phase alterations and the time-uncorrelated phase unwrapping mistakes, respectively.

**Figure 3.**Vectorial representation of the least-squares problem discussed in Equation (4). The grey area identifies the co-domain of the linear transformation given by Equation (4).

**Figure 4.**Study of the perpendicular baseline distribution of sets of SAR images collected by the first-generation SAR sensors. (

**a**–

**e**) Representation in the temporal/perpendicular baseline plane of relevant sets of SAR data acquired by the ERS and ENVISAT sensors on the five selected test-site areas of Afar, Ethiopia, Abruzzi, Central Italy, San Andreas Fault region, U.S., the Mt. Etna zone, Sicily Island, and the metropolitan area of the city of Naples in Italy. The data span the time interval between 1992 and 2010. (

**f**) Theoretical density probability function (pdf) (red line) and the empirical (black line) density probability function of the whole potential set of interferometric perpendicular baselines. Note that the theoretical pdf is drawn using Equation (21). In contrast, the empirical pdf is computed by the data by merely drawing the (normalized) histogram of the SAR dataset interferometric perpendicular baselines, see [58] for details on the probability distribution function of a random variable.

**Figure 5.**Theoretical Distribution of the perpendicular baselines of the InSAR data. (

**A**) Normal distribution of the perpendicular baseline of InSAR data pairs. (

**B**) Distribution of the perpendicular baselines of the InSAR data pairs below the critical baseline boundary. (

**C**) Distribution of the perpendicular baseline of the selected small baseline interferograms obtained by imposing a maximum allowed absolute perpendicular baseline equal to Bmax.

**Figure 6.**The plot of the term $\mathrm{\Pi}-1$ in Equation (23) vs. the maximum perpendicular baseline of the selected SB interferograms and various values of the standard deviation of the distribution of the perpendicular baseline of the whole possible InSAR data pairs.

**Figure 7.**The plot of the unwrapped phase’s relative errors vs. the selected maximum allowed perpendicular baseline of the SB interferograms, for different values of the ground mean displacement rate of the imaged scenes.

**Figure 8.**Upper bounds of the relative error of the measurement model parameters vs. the maximum perpendicular baseline of the SB interferogram, considering different mean ground displacement values of the observed SAR pixel on the terrain.

**Figure 9.**Sets of SB interferograms relevant to South California’s area obtained by processing a group of ENVISAT/ASAR images. The SB InSAR data pairs are identified by imposing a constraint on the maximum perpendicular baseline of (

**a**) 1100 m, (

**b**) 800 m, (

**c**) 600 m, (

**d**) 400 m, and (

**e**) 200 m. Dates shown in this Figure are expressed with the format (day, month, year).

**Figure 10.**2003–2010 Mean deformation velocity map of South California obtained using the SBAS method to the group of SB interferograms identified in Figure 9e and corresponding to a maximum value of the InSAR perpendicular baselines equal to 200 m. The map is in radar coordinates, and the ground deformation is superimposed on an amplitude SAR image of the area. Shown deformation values are saturated between +/− 10 mm/year.

**Figure 11.**Experimental SBAS Results. (

**a**) Comparing the temporal coherence values of the five performed runs of the SBAS inversion obtained by progressively relaxing the constraint on the maximum allowed perpendicular baseline of the interferograms, from 1100 m to 200 m; (

**b**) zoomed view of (

**a**) the interval of very high temporal coherence values (higher than 0.9).

**Figure 12.**Map of the Relative Errors of the SB unwrapped phases relevant to the test-case with SB interferograms with a maximum perpendicular baseline of 200 m.

**Figure 13.**(

**a**) Temporal coherence map of South California area (test-site case with the maximum perpendicular baseline of 200 m), (

**b**) map of the absolute error (upper bound) of the ground mean displacement rate after SBAS inversion; the map does not take into account the effects of APS and the residual topography of the area.

**Figure 14.**The plot of the ground displacement rate absolute error of the SBAS measurements vs. the temporal coherence.

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**MDPI and ACS Style**

Pepe, A.
Multi-Temporal Small Baseline Interferometric SAR Algorithms: Error Budget and Theoretical Performance. *Remote Sens.* **2021**, *13*, 557.
https://doi.org/10.3390/rs13040557

**AMA Style**

Pepe A.
Multi-Temporal Small Baseline Interferometric SAR Algorithms: Error Budget and Theoretical Performance. *Remote Sensing*. 2021; 13(4):557.
https://doi.org/10.3390/rs13040557

**Chicago/Turabian Style**

Pepe, Antonio.
2021. "Multi-Temporal Small Baseline Interferometric SAR Algorithms: Error Budget and Theoretical Performance" *Remote Sensing* 13, no. 4: 557.
https://doi.org/10.3390/rs13040557