# Resolving Three-Dimensional Surface Motion with InSAR: Constraints from Multi-Geometry Data Fusion

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## Abstract

**:**

## 1. Introduction

- What magnitude of error is made when projecting LOS to vertical and neglecting the possibility of horizontal motion?
- How can we rigorously estimate vertical and horizontal surface motions from the InSAR LOS geometry?
- Can we solve for the N–S component of motion using a multi-geometry data fusion of InSAR LOS measurements (without using the less precise pixel offset tracking or multi-aperture interferometry methods, independent geodetic data sources or data from right- and left-looking geometries)?

## 2. Methods

#### 2.1. InSAR Line of Sight Viewing Geometry

#### 2.2. Multi-Geometry Data Fusion

**A**for the case of the Envisat SAR database used in Section 4 is:

**A**) is close to 1 is considered well-conditioned; a linear system for which cond(

**A**) >> 1 is considered ill-conditioned (see [69], p. 281). The condition number of a non-square design matrix

**A**can be obtained as the ratio of maximum and minimum values of a singular value decomposition (svd) of

**A**:

**A**is erased. The condition number of

**A**is then 1.9 for the example given in in Equation (5), proving that the linear system is much better conditioned (by a factor of 10) compared to the case when estimating all three components. When the N component is omitted, Equation (3) can be solved when velocity data is available from only one ascending and one descending geometry, which is a common observation scenario for InSAR studies:

## 3. Application to Simulated Data

#### 3.1. Simulated Deformation Using a Mogi Model

^{3}/yr at a depth of 500 m. The resulting subsidence rate (downward motion) is strongest at the location directly above the source and decreases with distance away from that location. Maximum horizontal velocities are offset from the source location, with the offset magnitude depending on the source depth. In this example the maximum vertical velocity is −9.55 mm/yr. The maximum horizontal velocity is ±3.68 mm/yr and located at a horizontal distance of 350 m from the source location.

#### 3.2. Fusion of Simulated Displacement Data

## 4. Application to Observed Data

#### 4.1. Envisat SAR Database in the Sydney Region

- Calculation of a small baseline subset (SBAS) network of interferograms using the GAMMA software [81]. The SBAS network is generated based on coherence and the requirement for a minimum and maximum number of connections for each image. The results of this step are interferometric phase images with major orbital and topographic contributions to the phase signal removed.
- Quality check and outlier filtering of the resulting subset of image pixels, including the exclusion of pixels with phase unwrapping errors. Furthermore, a stable area (zero-mean and low variation of displacement values), common to all seven independent InSAR analyses is defined. The stable area then serves as a spatial reference for all InSAR-derived data sets.

#### 4.2. Fusion of Envisat Velocity Data

^{3}/yr) of the model are applicable to this deforming zone. However, as previously discussed, the horizontal velocities produced by this real-world deformation phenomenon exceed those of the Mogi model. This indicates that in this case the deformation source cannot be described by a single point source within an elastic half-space. Both projected U components (ascending and descending) can be seen to deviate significantly from the combined vertical and modelled vertical components. The maximum subsidence values of both projections exceed the combined ones and are shifted by several hundred metres to the west (ascending) or to the east (descending). The ascending track displayed in Figure 10 (track 467) has a relatively shallow incidence angle ($\theta =44$°), which further compounds the error effect on the projected vertical component (when compared to the descending track 359, with a steeper incidence angle, $\theta =34$°). Both ascending and descending projected U velocity estimates indicate uplift (ascending: east of the subsidence bowl, descending: west of the subsidence bowl), whereas the combined U component as well as the Mogi model indicate only subsidence. This example from a real-world data set demonstrates that projecting LOS measurements to vertical may result in misleading interpretation of InSAR-derived motions, particularly when a strong horizontal signal component is present.

## 5. Discussion

## 6. Conclusions

- Projection of InSAR LOS measurements into the vertical direction without considering the impact of horizontal motion is generally not recommended. Instead, data from ascending and descending geometries should be combined whenever available in the area of interest. The error in the projected vertical measurement depends on the amount of horizontal motion as well as on the incidence angle $\theta $ of the LOS geometry. As a rule of thumb (assuming Mogi-like deformation), the maximum error introduced into the projected vertical measurement can be approximated by $0.385\xb7tan\theta $ of the maximum vertical motion. Vertical measurements resulting from LOS projection can be wrong in terms of magnitude, direction and location.
- Multi-geometry data fusion of LOS InSAR measurements from several viewing geometries allows for robust estimation of horizontal and vertical motions. Weighted LSA can be applied when data from three or more viewing geometries are available with at least one ascending and one descending geometry. The matrix formulation for data fusion on a pixel-by-pixel basis allows for inclusion of data from different sensors (X-band, C-band, L-band, etc.) as well as independent data sources such as GNSS. Using exact incidence angles for each pixel is recommended, since large variations of incidence angle across range exist for most InSAR imaging geometries (17° over the full 250 km extent of a Sentinel-1 Interferometric Wide Swath image).
- The N–S component of motion resulting from multi-geometry data fusion of LOS InSAR measurements derived from past or current space-borne SAR sensors is poorly constrained, even if several different geometries are available (InSAR data from seven different tracks). If the N–S component is expected to be at a similar magnitude as the E–W component or smaller, less error is introduced into the combined results by omitting the N–S component from the LSA and only solving for the vertical and E–W components of motion.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- 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] - Ferretti, A.; Prati, C.; Rocca, F. Permanent scatterers in SAR interferometry. IEEE Trans. Geosci. Remote Sens.
**2001**, 39, 8–20. [Google Scholar] [CrossRef] [Green Version] - Adam, N.; Kampes, B.; Eineder, M.; Worawattanamateekul, J.; Kircher, M. The development of a scientific permanent scatterer system. In Proceedings of the ISPRS Workshop High Resolution Mapping from Space, Hannover, Germany, 23–25 December 2003. [Google Scholar]
- Hooper, A.; Zebker, H.; Segall, P.; Kampes, B. A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers. Geophys. Res. Lett.
**2004**, 31. [Google Scholar] [CrossRef] [Green Version] - Kampes, B.M. Displacement Parameter Estimation Using Permanent Scatterer Interferometry. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 2005. [Google Scholar]
- Hanssen, R.F. Radar Interferometry: Data Interpretation and Error Analysis; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2001. [Google Scholar]
- Teatini, P.; Tosi, L.; Strozzi, T.; Carbognin, L.; Wegmüller, U.; Rizzetto, F. Mapping regional land displacements in the Venice coastland by an integrated monitoring system. Remote Sens. Environ.
**2005**, 98, 403–413. [Google Scholar] [CrossRef] - Stramondo, S.; Bozzano, F.; Marra, F.; Wegmuller, U.; Cinti, F.R.; Moro, M.; Saroli, M. Subsidence induced by urbanisation in the city of Rome detected by advanced InSAR technique and geotechnical investigations. Remote Sens. Environ.
**2008**, 112, 3160–3172. [Google Scholar] [CrossRef] - Solari, L.; Ciampalini, A.; Raspini, F.; Bianchini, S.; Moretti, S. PSInSAR Analysis in the Pisa Urban Area (Italy): A Case Study of Subsidence Related to Stratigraphical Factors and Urbanization. Remote Sens.
**2016**, 8, 120. [Google Scholar] [CrossRef] - Yang, C.-S.; Zhang, Q.; Xu, Q.; Zhao, C.-Y.; Peng, J.-B.; Ji, L.-Y. Complex Deformation Monitoring over the Linfen–Yuncheng Basin (China) with Time Series InSAR Technology. Remote Sens.
**2016**, 8, 284. [Google Scholar] [CrossRef] - Zhou, L.; Guo, J.; Hu, J.; Li, J.; Xu, Y.; Pan, Y.; Shi, M. Wuhan Surface Subsidence Analysis in 2015–2016 Based on Sentinel-1A Data by SBAS-InSAR. Remote Sens.
**2017**, 9, 982. [Google Scholar] [CrossRef] - Zheng, M.; Deng, K.; Fan, H.; Du, S. Monitoring and Analysis of Surface Deformation in Mining Area Based on InSAR and GRACE. Remote Sens.
**2018**, 10, 1392. [Google Scholar] [CrossRef] - Galloway, D.L.; Hudnut, K.W.; Ingebritsen, S.E.; Phillips, S.P.; Peltzer, G.; Rogez, F.; Rosen, P.A. Detection of aquifer system compaction and land subsidence using interferometric synthetic aperture radar, Antelope Valley, Mojave Desert, California. Water Resour. Res.
**1998**, 34, 2573–2585. [Google Scholar] [CrossRef] [Green Version] - Amelung, F.C.; Galloway, D.L.; Bell, J.W.; Zebker, H.A.; Laczniak, R.J. Sensing the ups and downs of Las Vegas. Geology
**1999**, 27, 483–486. [Google Scholar] [CrossRef] - Schmidt, D.A.; Bürgmann, R. Time-dependent land uplift and subsidence in the Santa Clara valley, California, from a large interferometric synthetic aperture radar data set. J. Geophys. Res. Solid Earth
**2003**, 108. [Google Scholar] [CrossRef] [Green Version] - Wisely, B.A.; Schmidt, D. Deciphering vertical deformation and poroelastic parameters in a tectonically active fault-bound aquifer using InSAR and well level data, San Bernardino basin, California. Geophys. J. Int.
**2010**, 181, 1185–1200. [Google Scholar] [CrossRef] [Green Version] - Hung, W.-C.; Hwang, C.; Chen, Y.-A.; Chang, C.-P.; Yen, J.-Y.; Hooper, A.; Yang, C.-Y. Surface deformation from persistent scatterers SAR interferometry and fusion with leveling data: A case study over the Choushui River Alluvial Fan, Taiwan. Remote Sens. Environ.
**2011**, 115, 957–967. [Google Scholar] [CrossRef] - Wang, H.; Wright, T.J.; Yu, Y.; Lin, H.; Jiang, L.; Li, C.; Qiu, G. InSAR reveals coastal subsidence in the Pearl River Delta, China. Geophys. J. Int.
**2012**, 191, 1119–1128. [Google Scholar] [CrossRef] - Chaussard, E.; Amelung, F.; Abidin, H.; Hong, S.-H. Sinking cities in Indonesia: ALOS PALSAR detects rapid subsidence due to groundwater and gas extraction. Remote Sens. Environ.
**2013**, 128, 150–161. [Google Scholar] [CrossRef] - Ge, L.; Ng, A.H.-M.; Li, X.; Abidin, H.Z.; Gumilar, I. Land subsidence characteristics of Bandung Basin as revealed by ENVISAT ASAR and ALOS PALSAR interferometry. Remote Sens. Environ.
**2014**, 154, 46–60. [Google Scholar] [CrossRef] - Qu, F.; Lu, Z.; Zhang, Q.; Bawden, G.W.; Kim, J.-W.; Zhao, C.; Qu, W. Mapping ground deformation over Houston-Galveston, Texas using multi-temporal InSAR. Remote Sens. Environ.
**2015**, 169, 290–306. [Google Scholar] [CrossRef] - Du, Z.; Ge, L.; Li, X.; Ng, A. Subsidence Monitoring over the Southern Coalfield, Australia Using both L-Band and C-Band SAR Time Series Analysis. Remote Sens.
**2016**, 8, 543. [Google Scholar] [CrossRef] - Xu, B.; Feng, G.; Li, Z.; Wang, Q.; Wang, C.; Xie, R. Coastal Subsidence Monitoring Associated with Land Reclamation Using the Point Target Based SBAS-InSAR Method: A Case Study of Shenzhen, China. Remote Sens.
**2016**, 8, 652. [Google Scholar] [CrossRef] - Zhang, Y.; Wu, H.a.; Kang, Y.; Zhu, C. Ground Subsidence in the Beijing-Tianjin-Hebei Region from 1992 to 2014 Revealed by Multiple SAR Stacks. Remote Sens.
**2016**, 8, 675. [Google Scholar] [CrossRef] - Parker, A.; Filmer, M.; Featherstone, W. First Results from Sentinel-1A InSAR over Australia: Application to the Perth Basin. Remote Sens.
**2017**, 9, 299. [Google Scholar] [CrossRef] - Alshammari, L.; Large, D.; Boyd, D.; Sowter, A.; Anderson, R.; Andersen, R.; Marsh, S. Long-Term Peatland Condition Assessment via Surface Motion Monitoring Using the ISBAS DInSAR Technique over the Flow Country, Scotland. Remote Sens.
**2018**, 10, 1103. [Google Scholar] [CrossRef] - Gao, M.; Gong, H.; Chen, B.; Li, X.; Zhou, C.; Shi, M.; Si, Y.; Chen, Z.; Duan, G. Regional Land Subsidence Analysis in Eastern Beijing Plain by InSAR Time Series and Wavelet Transforms. Remote Sens.
**2018**, 10, 365. [Google Scholar] [CrossRef] - Hung, W.-C.; Hwang, C.; Chen, Y.-A.; Zhang, L.; Chen, K.-H.; Wei, S.-H.; Huang, D.-R.; Lin, S.-H. Land Subsidence in Chiayi, Taiwan, from Compaction Well, Leveling and ALOS/PALSAR: Aquaculture-Induced Relative Sea Level Rise. Remote Sens.
**2018**, 10, 40. [Google Scholar] [CrossRef] - Ng, A.; Wang, H.; Dai, Y.; Pagli, C.; Chen, W.; Ge, L.; Du, Z.; Zhang, K. InSAR Reveals Land Deformation at Guangzhou and Foshan, China between 2011 and 2017 with COSMO-SkyMed Data. Remote Sens.
**2018**, 10, 813. [Google Scholar] [CrossRef] - Tang, W.; Yuan, P.; Liao, M.; Balz, T. Investigation of Ground Deformation in Taiyuan Basin, China from 2003 to 2010, with Atmosphere-Corrected Time Series InSAR. Remote Sens.
**2018**, 10, 1499. [Google Scholar] [CrossRef] - Yang, M.; Yang, T.; Zhang, L.; Lin, J.; Qin, X.; Liao, M. Spatio-Temporal Characterization of a Reclamation Settlement in the Shanghai Coastal Area with Time Series Analyses of X-, C-, and L-Band SAR Datasets. Remote Sens.
**2018**, 10, 329. [Google Scholar] [CrossRef] - Raucoules, D.; Maisons, C.; Carnec, C.; Le Mouelic, S.; King, C.; Hosford, S. Monitoring of slow ground deformation by ERS radar interferometry on the Vauvert salt mine (France): Comparison with ground-based measurement. Remote Sens. Environ.
**2003**, 88, 468–478. [Google Scholar] [CrossRef] - Short, N.; LeBlanc, A.-M.; Sladen, W.; Oldenborger, G.; Mathon-Dufour, V.; Brisco, B. RADARSAT-2 D-InSAR for ground displacement in permafrost terrain, validation from Iqaluit Airport, Baffin Island, Canada. Remote Sens. Environ.
**2014**, 141, 40–51. [Google Scholar] [CrossRef] - Sun, H.; Zhang, Q.; Zhao, C.; Yang, C.; Sun, Q.; Chen, W. Monitoring land subsidence in the southern part of the lower Liaohe plain, China with a multi-track PS-InSAR technique. Remote Sens. Environ.
**2017**, 188, 73–84. [Google Scholar] [CrossRef] - Zhou, W.; Li, S.; Zhou, Z.; Chang, X. Remote Sensing of Deformation of a High Concrete-Faced Rockfill Dam Using InSAR: A Study of the Shuibuya Dam, China. Remote Sens.
**2016**, 8, 255. [Google Scholar] [CrossRef] - Zhou, C.; Gong, H.; Chen, B.; Li, J.; Gao, M.; Zhu, F.; Chen, W.; Liang, Y. InSAR Time-Series Analysis of Land Subsidence under Different Land Use Types in the Eastern Beijing Plain, China. Remote Sens.
**2017**, 9, 380. [Google Scholar] [CrossRef] - Chen, G.; Zhang, Y.; Zeng, R.; Yang, Z.; Chen, X.; Zhao, F.; Meng, X. Detection of Land Subsidence Associated with Land Creation and Rapid Urbanization in the Chinese Loess Plateau Using Time Series InSAR: A Case Study of Lanzhou New District. Remote Sens.
**2018**, 10, 270. [Google Scholar] [CrossRef] - Milillo, P.; Giardina, G.; DeJong, M.; Perissin, D.; Milillo, G. Multi-Temporal InSAR Structural Damage Assessment: The London Crossrail Case Study. Remote Sens.
**2018**, 10, 287. [Google Scholar] [CrossRef] - Wright, T.J.; Parsons, B.E.; Lu, Z. Toward mapping surface deformation in three dimensions using InSAR. Geophys. Res. Lett.
**2004**, 31. [Google Scholar] [CrossRef] [Green Version] - Teatini, P.; Castelletto, N.; Ferronato, M.; Gambolati, G.; Janna, C.; Cairo, E.; Marzorati, D.; Colombo, D.; Ferretti, A.; Bagliani, A.; et al. Geomechanical response to seasonal gas storage in depleted reservoirs: A case study in the Po River basin, Italy. J. Geophys. Res. Earth Surf.
**2011**, 116. [Google Scholar] [CrossRef] [Green Version] - Ferretti, A. Satellite InSAR data: Reservoir Monitoring from Space; EAGE Publications: Houten, The Netherlands, 2014. [Google Scholar]
- Lundgren, P.; Casu, F.; Manzo, M.; Pepe, A.; Berardino, P.; Sansosti, E.; Lanari, R. Gravity and magma induced spreading of Mount Etna volcano revealed by satellite radar interferometry. Geophys. Res. Lett.
**2004**, 31. [Google Scholar] [CrossRef] [Green Version] - Borgia, A.; Tizzani, P.; Solaro, G.; Manzo, M.; Casu, F.; Luongo, G.; Pepe, A.; Berardino, P.; Fornaro, G.; Sansosti, E.; et al. Volcanic spreading of Vesuvius, a new paradigm for interpreting its volcanic activity. Geophys. Res. Lett.
**2005**, 32. [Google Scholar] [CrossRef] [Green Version] - Manzo, M.; Ricciardi, G.P.; Casu, F.; Ventura, G.; Zeni, G.; Borgström, S.; Berardino, P.; Del Gaudio, C.; Lanari, R. Surface deformation analysis in the Ischia Island (Italy) based on spaceborne radar interferometry. J. Volcanol. Geotherm. Res.
**2006**, 151, 399–416. [Google Scholar] [CrossRef] - Lanari, R.; Casu, F.; Manzo, M.; Zeni, G.; Berardino, P.; Manunta, M.; Pepe, A. An Overview of the Small BAseline Subset Algorithm: A DInSAR Technique for Surface Deformation Analysis. In Deformation and Gravity Change: Indicators of Isostasy, Tectonics, Volcanism, and Climate Change; Birkhäuser: Basel, Switherlands, 2007; pp. 637–661. [Google Scholar]
- 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] - de Michele, M.; Raucoules, D.; de Sigoyer, J.; Pubellier, M.; Chamot-Rooke, N. Three-dimensional surface displacement of the 2008 May 12 Sichuan earthquake (China) derived from Synthetic Aperture Radar: Evidence for rupture on a blind thrust. Geophys. J. Int.
**2010**, 183, 1097–1103. [Google Scholar] [CrossRef] - Raucoules, D.; de Michele, M.; Malet, J.P.; Ulrich, P. Time-variable 3D ground displacements from high-resolution synthetic aperture radar (SAR). application to La Valette landslide (South French Alps). Remote Sens. Environ.
**2013**, 139, 198–204. [Google Scholar] [CrossRef] [Green Version] - Wang, X.; Liu, G.; Yu, B.; Dai, K.; Zhang, R.; Ma, D.; Li, Z. An integrated method based on DInSAR, MAI and displacement gradient tensor for mapping the 3D coseismic deformation field related to the 2011 Tarlay earthquake (Myanmar). Remote Sens. Environ.
**2015**, 170, 388–404. [Google Scholar] [CrossRef] - Jo, M.-J.; Jung, H.-S.; Won, J.-S.; Lundgren, P. Measurement of three-dimensional surface deformation by Cosmo-SkyMed X-band radar interferometry: Application to the March 2011 Kamoamoa fissure eruption, Kīlauea Volcano, Hawai’i. Remote Sens. Environ.
**2015**, 169, 176–191. [Google Scholar] [CrossRef] - Jo, M.-J.; Jung, H.-S.; Won, J.-S. Measurement of precise three-dimensional volcanic deformations via TerraSAR-X synthetic aperture radar interferometry. Remote Sens. Environ.
**2017**, 192, 228–237. [Google Scholar] [CrossRef] - Hu, J.; Li, Z.W.; Ding, X.L.; Zhu, J.J.; Zhang, L.; Sun, Q. Resolving three-dimensional surface displacements from InSAR measurements: A review. Earth-Sci. Rev.
**2014**, 133, 1–17. [Google Scholar] [CrossRef] - Burgmann, R.; Hilley, G.; Ferretti, A.; Novali, F. Resolving vertical tectonics in the San Francisco Bay Area from permanent scatterer InSAR and GPS analysis. Geology
**2006**, 34, 221–224. [Google Scholar] [CrossRef] - Gourmelen, N.; Amelung, F.; Lanari, R. Interferometric synthetic aperture radar—GPS integration: Interseismic strain accumulation across the Hunter Mountain fault in the eastern California shear zone. J. Geophys. Res. Solid Earth
**2010**, 115. [Google Scholar] [CrossRef] - Hammond, W.C.; Blewitt, G.; Li, Z.; Plag, H.-P.; Kreemer, C. Contemporary uplift of the Sierra Nevada, western United States, from GPS and InSAR measurements. Geology
**2012**, 40, 667–670. [Google Scholar] [CrossRef] - Hussain, E.; Wright, T.J.; Walters, R.J.; Bekaert, D.P.S.; Lloyd, R.; Hooper, A. Constant strain accumulation rate between major earthquakes on the North Anatolian Fault. Nat. Commun.
**2018**, 9, 1392. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gudmundsson, S.; Gudmundsson, M.T.; Björnsson, H.; Sigmundsson, F.; Rott, H.; Carstensen, J.M. Three-dimensional glacier surface motion maps at the Gjálp eruption site, Iceland, inferred from combining InSAR and other ice-displacement data. Ann. Glaciol.
**2002**, 34, 315–322. [Google Scholar] [CrossRef] - Samsonov, S.; Tiampo, K. Analytical optimization of a DInSAR and GPS dataset for derivation of three-dimensional surface motion. IEEE Geosci. Remote Sens. Lett.
**2006**, 3, 107–111. [Google Scholar] [CrossRef] - Catalao, J.; Nico, G.; Hanssen, R.; Catita, C. Merging GPS and Atmospherically Corrected InSAR Data to Map 3-D Terrain Displacement Velocity. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 2354–2360. [Google Scholar] [CrossRef] [Green Version] - Hu, J.; Zhu, J.J.; Li, Z.W.; Ding, X.L.; Wang, C.C.; Sun, Q. Robust Estimating Three-Dimensional Ground Motions from Fusion of InSAR and GPS Measurements. In Proceedings of the 2011 International Symposium on Image and Data Fusion, Tengchong, China, 9–11 August 2011; pp. 1–4. [Google Scholar]
- Caro Cuenca, M.; Hanssen, R.F.; Hooper, A.; 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. 1–8. [Google Scholar]
- Fuhrmann, T.; Caro Cuenca, M.; Knöpfler, A.; van Leijen, F.J.; Mayer, M.; Westerhaus, M.; Hanssen, R.F.; Heck, B. Estimation of small surface displacements in the Upper Rhine Graben area from a combined analysis of PS-InSAR, levelling and GNSS data. Geophys. J. Int.
**2015**, 203, 614–631. [Google Scholar] [CrossRef] - Mogi, K. Relations between the eruptions of various volcanoes and the deformations of the ground surfaces around them. Bull. Earthq. Res. Inst.
**1958**, 36, 99–134. [Google Scholar] - Van Leijen, F.J. Persistent Scatterer Interferometry Based on Geodetic Estimation Theory. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 2014. [Google Scholar]
- ESA. EnviSat ASAR Product Handbook, Issue 2.2. 2007, p. 564. Available online: https://earth.esa.int/pub/ESA_DOC/ENVISAT/ASAR/asar.ProductHandbook.2_2.pdf (accessed on 24 January 2019).
- ESA. Sentinel-1 SAR User Guide—Interferometric Wide Swath. 2018. Available online: https://sentinels.copernicus.eu/web/sentinel/user-guides/sentinel-1-sar/acquisition-modes/interferometric-wide-swath (accessed on 24 January 2019).
- Li, J.; Heap, A.D. A Review of Spatial Interpolation Methods for Environmental Scientists; Record 2008/23 Geoscience Australia: Canberra, Australia, 2008.
- Dai, K.; Liu, G.; Li, Z.; Li, T.; Yu, B.; Wang, X.; Singleton, A. Extracting Vertical Displacement Rates in Shanghai (China) with Multi-Platform SAR Images. Remote Sens.
**2015**, 7, 9542. [Google Scholar] [CrossRef] - Tarantola, A. Inverse Problem Theory and Methods for Model Parameter Estimation; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2005; p. 348. [Google Scholar]
- Mossop, A.; Segall, P. Subsidence at The Geysers Geothermal Field, N. California from a comparison of GPS and leveling surveys. Geophys. Res. Lett.
**1997**, 24, 1839–1842. [Google Scholar] [CrossRef] [Green Version] - Carnec, C.; Fabriol, H. Monitoring and modeling land subsidence at the Cerro Prieto Geothermal Field, Baja California, Mexico, using SAR interferometry. Geophys. Res. Lett.
**1999**, 26, 1211–1214. [Google Scholar] [CrossRef] - Lisowski, M. Analytical volcano deformation source models. In Volcano Deformation: Geodetic Monitoring Techniques; Springer: Berlin/Heidelberg, Germany, 2006; pp. 279–304. [Google Scholar] [CrossRef]
- Segall, P. Earthquake and Volcano Deformation; Princeton University Press: Princeton, NJ, USA, 2010. [Google Scholar]
- Dawson, J.; Tregoning, P. Uncertainty analysis of earthquake source parameters determined from InSAR: A simulation study. J. Geophys. Res. Solid Earth
**2007**, 112. [Google Scholar] [CrossRef] [Green Version] - Rocca, F. Modeling Interferogram Stacks. IEEE Trans. Geosci. Remote Sens.
**2007**, 45, 3289–3299. [Google Scholar] [CrossRef] - Gourmelen, N.; Amelung, F.; Casu, F.; Manzo, M.; Lanari, R. Mining-related ground deformation in Crescent Valley, Nevada: Implications for sparse GPS networks. Geophys. Res. Lett.
**2007**, 34. [Google Scholar] [CrossRef] [Green Version] - Fuhrmann, T. Surface Displacements from Fusion of Geodetic Measurement Techniques Applied to the Upper Rhine Graben Area. Ph.D. Thesis, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, 2016. [Google Scholar]
- Samieie-Esfahany, S.; Hanssen, R.F.; van Thienen-Visser, K.; Muntendam-Bos, A. On the effect of horizontal deformation on InSAR subsidence estimates. In Proceedings of the Fringe 2009 Workshop, Frascati, Italy, 30 November–4 December 2009; pp. 1–7. [Google Scholar]
- Morishita, Y.; Kobayashi, T.; Yarai, H. Three-dimensional deformation mapping of a dike intrusion event in Sakurajima in 2015 by exploiting the right- and left-looking ALOS-2 InSAR. Geophys. Res. Lett.
**2016**, 43, 4197–4204. [Google Scholar] [CrossRef] - Adam, N.; Kampes, B.; Eineder, M. Development of a Scientific Permanent Scatterer System: Modifications for Mixed ERS/ENVISAT Time Series. In Proceedings of the Envisat Symposium 2004, Salzburg, Austria, 6–10 September 2004. [Google Scholar]
- Wegmüller, U.; Werner, C. GAMMA SAR processor and interferometry software. In Proceedings of the Third ERS Symposium on Space at the Service of Our Environment, Florence, Italy, 14–21 March 1997. [Google Scholar]
- 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] - Hooper, A.; Segall, P.; Zebker, H. Persistent scatterer interferometric synthetic aperture radar for crustal deformation analysis, with application to Volcán Alcedo, Galápagos. J. Geophys. Res. Solid Earth
**2007**, 112. [Google Scholar] [CrossRef] [Green Version] - Hooper, A. A multi-temporal InSAR method incorporating both persistent scatterer and small baseline approaches. Geophys. Res. Lett.
**2008**, 35. [Google Scholar] [CrossRef] [Green Version] - Hooper, A.; Bekaert, D.; Spaans, K.; Arıkan, M. Recent advances in SAR interferometry time series analysis for measuring crustal deformation. Tectonophysics
**2012**, 514–517, 1–13. [Google Scholar] [CrossRef] - Fuhrmann, T.; Garthwaite, M.; Lawrie, S.; Brown, N. Combination of GNSS and InSAR for Future Australian Datums. In Proceedings of the IGNSS Symposium 2018, Sydney, Australia, 7–9 February 2018; pp. 1–13. [Google Scholar]
- Torres, R.; Snoeij, P.; Geudtner, D.; Bibby, D.; Davidson, M.; Attema, E.; Potin, P.; Rommen, B.; Floury, N.; Brown, M.; et al. GMES Sentinel-1 mission. Remote Sens. Environ.
**2012**, 120, 9–24. [Google Scholar] [CrossRef] - 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] - De Luca, C.; Zinno, I.; Manunta, M.; Lanari, R.; Casu, F. Large areas surface deformation analysis through a cloud computing P-SBAS approach for massive processing of DInSAR time series. Remote Sens. Environ.
**2017**, 202, 3–17. [Google Scholar] [CrossRef] - Kalia, A.C.; Frei, M.; Lege, T. A Copernicus downstream-service for the nationwide monitoring of surface displacements in Germany. Remote Sens. Environ.
**2017**, 202, 234–249. [Google Scholar] [CrossRef] - Novellino, A.; Cigna, F.; Brahmi, M.; Sowter, A.; Bateson, L.; Marsh, S. Assessing the Feasibility of a National InSAR Ground Deformation Map of Great Britain with Sentinel-1. Geosciences
**2017**, 7, 19. [Google Scholar] [CrossRef]

**Figure 1.**Schematic view of the interferometric synthetic aperture radar (InSAR) viewing geometry for line of sight (LOS) measurements on ascending and descending satellite passes.

**Figure 2.**Simulated velocity fields resulting from a Mogi deformation source with a volume change of 10,000 m

^{3}/yr at 500 m depth: (

**a**) vertical surface velocity (subsidence rate) resulting from a negative rate of volume change, (

**b**) E–W component of velocity resulting from the same rate of volume change. Top-view images of (

**a**) and (

**b**) are displayed in Figure 4

**b**and

**a**, respectively.

**Figure 3.**Maximum vertical and horizontal displacement rates resulting from Mogi model simulations with varying source depths and volume changes. Displacement rate is expressed as velocity in mm/yr assuming a steady volume change resulting in a constant displacement rate. Three discrete rates of volume change are compared for vertical and horizontal motion in (

**a**). Contours of maximum vertical (

**b**) and horizontal (

**c**) velocities in mm/yr plotted as a function of depth and volume change of the source.

**Figure 4.**Simulated velocities from a Mogi deformation model: (

**a**) E component, (

**b**) U component, (

**c**) ascending LOS with −16.0° heading and 33.9° incidence angle, (

**d**) descending LOS with −165.5° heading and 18.9° incidence angle. Note that the colour scale saturates at ±5 mm/yr. The N–S component is equivalent to the E–W component due to the symmetry of the Mogi model, but rotated anti-clockwise by 90°.

**Figure 5.**Simulated velocities at Gaussian noise level 0.5 mm/yr (one realisation from 1000 runs): (

**a**) ascending LOS with −16.0° heading and 33.9° incidence angle, (

**b**) estimated U component from fusion of seven viewing geometries, (

**c**) U component from projection of LOS velocities, (

**d**) difference between (

**b**) and (

**c**). Note that the colour scale saturates at ±5 mm/yr.

**Figure 6.**Mean absolute difference (MAD) between estimated and modelled velocity components at different noise levels and for different combinations, i.e., all seven viewing geometries (solid lines), four viewing geometries (dashed lines), and two viewing geometries (dotted lines): (

**a**) least-squares adjustment (LSA) of all three velocity components, (

**b**) LSA of E and U velocity components only. Note that the y-scale is different in (

**a**) and (

**b**) to allow for visualisation of the smaller MAD values in the (

**b**) case.

**Figure 7.**Overview of the Envisat SAR database used within this paper: (

**a**) footprint of ascending and descending Envisat SAR images in the Sydney region, and position of Envisat satellite at time of image acquisition from the seven orbital tracks in E–U plane (

**b**) and in N–U plane (

**c**).

**Figure 8.**Multi-geometry data fusion of Envisat InSAR data: (

**a**) ascending LOS velocities (track 152), (

**b**) descending LOS velocities (track 359), (

**c**) estimated E velocities, (

**d**) estimated U velocities from data fusion of seven geometries. The circle indicates the defined stable area common to all seven InSAR analyses and used as spatial reference. Coordinate axes: UTM, zone 56.

**Figure 9.**Velocities observed at the deforming zone southwest of Picton: (

**a**) descending LOS (track 359), (

**b**) estimated U component from fusion of seven tracks, (

**c**) U component from projection of LOS velocities, (

**d**) difference between (

**b**) and (

**c**). Dashed black lines denote the envelope of points used to derive the E–W profiles shown in Figure 10. Coordinate axes: UTM, zone 56.

**Figure 10.**Profile through the deforming zone shown in Figure 9. E and U velocity components from Mogi model (dotted lines) and from multi-geometry data fusion (solid lines), projected U components from ascending (track 467) and descending (track 359) LOS velocities (dashed lines). Data gaps are reduced for the combined and projected signals by averaging the velocities of pixels falling within an envelope of 1.8 km width on either side of the profile line (see dashed black lines in Figure 9) and at intervals of 100 m along profile, and subsequently smoothing the profiles in the east direction.

**Table 1.**Mean absolute difference (MAD) and maximum difference (MAX) between projected vertical velocities and simulated velocities (both without adding noise) for different incidence angles. The maximum vertical velocity of this simulation is −9.55 mm/yr.

Incidence Angle | MAD (mm/yr) | MAX (mm/yr) |
---|---|---|

15° | 0.15 | 0.98 |

20° | 0.20 | 1.34 |

25° | 0.26 | 1.71 |

30° | 0.32 | 2.12 |

35° | 0.38 | 2.57 |

40° | 0.46 | 3.08 |

45° | 0.55 | 3.68 |

50° | 0.65 | 4.38 |

**Table 2.**Difference of estimated velocities to original model values for Gaussian noise level of 0.5 mm/yr, mean absolute difference (MAD) and maximum difference (MAX).

Type of Fusion | E (mm/yr) | N (mm/yr) | U (mm/yr) | |||
---|---|---|---|---|---|---|

MAD | MAX | MAD | MAX | MAD | MAX | |

all three components | ||||||

Seven tracks | 0.37 | 2.12 | 3.21 | 18.41 | 0.48 | 2.73 |

Four tracks | 0.45 | 2.61 | 4.60 | 26.32 | 0.63 | 3.63 |

Two tracks | - | - | - | - | - | - |

E and U only | ||||||

Seven tracks | 0.33 | 1.84 | - | - | 0.20 | 1.23 |

Four tracks | 0.45 | 2.51 | - | - | 0.24 | 1.43 |

Two tracks | 0.53 | 3.02 | - | - | 0.36 | 2.14 |

**Table 3.**Overview of the seven Envisat data sets including information on the LOS geometries. A: ascending, D: descending.

Orbital Track | Pass Direction | Start Date | End Date | Number of Images Used | Number of Interferograms | Satellite Heading | Incidence Angle | Envisat Image Mode |
---|---|---|---|---|---|---|---|---|

338 | A | 15 June 2006 | 2 September 2010 | 42 | 109 | −14.5° | 19.0° | IS1 |

381 | A | 18 June 2006 | 5 September 2010 | 43 | 117 | −15.5° | 28.9° | IS3 |

152 | A | 2 June 2006 | 24 September 2010 | 36 | 97 | −16.0° | 33.9° | IS4 |

467 | A | 31 March 2007 | 31 October 2009 | 26 | 83 | −17.3° | 44.1° | IS7 |

173 | D | 8 July 2006 | 25 September 2010 | 31 | 81 | −165.5° | 18.9° | IS1 |

402 | D | 19 June 2006 | 6 September 2010 | 42 | 130 | −165.1° | 22.9° | IS2 |

359 | D | 16 June 2006 | 3 September 2010 | 41 | 129 | −164.1° | 33.8° | IS4 |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fuhrmann, T.; Garthwaite, M.C.
Resolving Three-Dimensional Surface Motion with InSAR: Constraints from Multi-Geometry Data Fusion. *Remote Sens.* **2019**, *11*, 241.
https://doi.org/10.3390/rs11030241

**AMA Style**

Fuhrmann T, Garthwaite MC.
Resolving Three-Dimensional Surface Motion with InSAR: Constraints from Multi-Geometry Data Fusion. *Remote Sensing*. 2019; 11(3):241.
https://doi.org/10.3390/rs11030241

**Chicago/Turabian Style**

Fuhrmann, Thomas, and Matthew C. Garthwaite.
2019. "Resolving Three-Dimensional Surface Motion with InSAR: Constraints from Multi-Geometry Data Fusion" *Remote Sensing* 11, no. 3: 241.
https://doi.org/10.3390/rs11030241