{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:47:21Z","timestamp":1760240841298,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2019,10,1]],"date-time":"2019-10-01T00:00:00Z","timestamp":1569888000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>This paper analyses the critical phase unwrapping step in a differential interferometric phase (D-InSAR) stack where both the solving of conventional methods and alternative approaches are discussed. It can be shown that including the temporal relationship between interferograms in the phase unwrapping step improves the results. This leads to the three-dimensional extended minimum cost flow algorithm. To unwrap the phase in a multitemporal way a motion model has to be considered. The estimation of these parameters is an important step. By default, the parameters are estimated in an iterative search process, where in each step, a linear program has to be solved. The best parameters are defined by the minimal costs. Often the choice of this search space is not straightforward. Furthermore, with this discrete optimization function, the solution is often not unique. This paper presents an alternative way to estimate the motion model parameters by maximizing a continuous function, the ensemble phase coherence. With the help of a closed-loop simulation and real data, both methods, the standard and the alternative way, are numerically compared and analyzed. Consequently, it is shown that maximizing the ensemble phase coherence is a good alternative to the established iterative procedure. It offers the advantage that the run time can be reduced considerably and is thus well suited in the processing of large data sets.<\/jats:p>","DOI":"10.3390\/rs11192295","type":"journal-article","created":{"date-parts":[[2019,10,1]],"date-time":"2019-10-01T11:11:16Z","timestamp":1569928276000},"page":"2295","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["On the Analysis of the Phase Unwrapping Process in a D-InSAR Stack with Special Focus on the Estimation of a Motion Model"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9901-1866","authenticated-orcid":false,"given":"Christina","family":"Esch","sequence":"first","affiliation":[{"name":"Institute of Geodesy and Geoinformation, University of Bonn, 53115 Bonn, Germany"}]},{"given":"Jo\u00ebl","family":"K\u00f6hler","sequence":"additional","affiliation":[{"name":"Institute of Geodesy and Geoinformation, University of Bonn, 53115 Bonn, Germany"}]},{"given":"Karlheinz","family":"Gutjahr","sequence":"additional","affiliation":[{"name":"Joanneum Research, DIGITAL Institute for Information and Communication Technologies, 8010 Graz, Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3522-3146","authenticated-orcid":false,"given":"Wolf-Dieter","family":"Schuh","sequence":"additional","affiliation":[{"name":"Institute of Geodesy and Geoinformation, University of Bonn, 53115 Bonn, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1109\/36.898661","article-title":"Permanent Scatterers in SAR Interferometry","volume":"39","author":"Ferretti","year":"2001","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2375","DOI":"10.1109\/TGRS.2002.803792","article-title":"A New Algorithm for Surface Deformation Monitoring Based on Small Baseline Differential SAR Interferograms","volume":"40","author":"Berardino","year":"2002","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3460","DOI":"10.1109\/TGRS.2011.2124465","article-title":"A New Algorithm for Processing Interferometric Data-Stacks: SqueeSAR","volume":"49","author":"Ferretti","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_4","first-page":"33","article-title":"Leitnivellements in Bodenbewegungsgebieten des Bergbaus-Eine Langj\u00e4hrige Kernaufgabe der Landesvermessung in Nordrhein-Westfalen","volume":"3","author":"Boje","year":"2008","journal-title":"N\u00d6V NRW"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2374","DOI":"10.1109\/TGRS.2006.873207","article-title":"On the Extension of the Minimum Cost Flow Algorithm for Phase Unwrapping of Multitemporal Differential SAR Interferograms","volume":"44","author":"Pepe","year":"2006","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1109\/36.499752","article-title":"Phase Unwrapping by Means of Multigrid Techniques for Interferometric SAR","volume":"34","author":"Pritt","year":"1996","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2692","DOI":"10.1364\/JOSAA.14.002692","article-title":"Two-Dimensional Phase Unwrapping with Minimum Weighted Discontinuity","volume":"14","author":"Flynn","year":"1997","journal-title":"JOSA A"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Eineder, M., Hubig, M., and Milcke, B. (1998, January 6\u201310). Unwrapping Large Interferograms Using the Minimum Cost Flow Algorithm. Proceedings of the IGARSS \u201998, Sensing and Managing the Environment, 1998 IEEE International Geoscience and Remote Sensing, Symposium Proceedings, (Cat. No.98CH36174), Seattle, WA, USA.","DOI":"10.1109\/IGARSS.1998.702806"},{"key":"ref_9","unstructured":"Ghiglia, D.C., and Pritt, M.D. (1998). Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software, WileyBlackwell."},{"key":"ref_10","unstructured":"Ferretti, A., Monti-Guarnieri, A., Prati, C., Rocca, F., and Massonnet, D. (2007). InSAR Principles: Guidelines for SAR Interferometry Processing and Interpretation, Part C: InSAR processing: A mathematical approach, ESA."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2323","DOI":"10.1109\/TGRS.2010.2102767","article-title":"A Null-Space Method for the Phase Unwrapping of Multitemporal SAR Interferometric Stacks","volume":"49","author":"Fornaro","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1109\/TASSP.1977.1162923","article-title":"A New Phase Unwrapping Algorithm","volume":"25","author":"Tribolet","year":"1977","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Goldstein, R.M., Zebker, H.A., and Werner, C.L. (1988). Satellite Radar Interferometry: Two-Dimensional Phase Unwrapping. Radio Sci., 23.","DOI":"10.1029\/RS023i004p00713"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1364\/JOSAA.15.000586","article-title":"Phase Unwrapping Algorithms for Radar Interferometry: Residue-Cut, Least-Squares, and Synthesis Algorithms","volume":"15","author":"Zebker","year":"1998","journal-title":"JOSA A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1364\/JOSAA.11.000107","article-title":"Robust Two Dimensional Weighted and Unweighted Phase Unwrapping That Uses Fast Transforms and Iterative Methods","volume":"11","author":"Ghiglia","year":"1994","journal-title":"JOSA A"},{"key":"ref_16","first-page":"261","article-title":"A Phase Unwrapping Method Based on Network Programming","volume":"Volume 406","author":"Costantini","year":"1997","journal-title":"ERS SAR Interferometry"},{"key":"ref_17","unstructured":"Costantini, M., and Rosen, P. (July, January 28). Generalized Phase Unwrapping Approach for Sparse Data. Proceedings of the IEEE 1999 International Geoscience and Remote Sensing Symposium, IGARSS\u201999 (Cat. No.99CH36293), Hamburg, Germany."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2737","DOI":"10.1364\/JOSAA.24.002737","article-title":"Phase unwrapping in three dimensions with application to InSAR time series","volume":"24","author":"Hooper","year":"2007","journal-title":"JOSA A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1364\/JOSAA.27.000605","article-title":"Edgelist Phase Unwrapping Algorithm for Time Series InSAR Analysis","volume":"27","author":"Shanker","year":"2010","journal-title":"JOSA A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"758","DOI":"10.1109\/TGRS.2011.2162630","article-title":"A General Formulation for Redundant Integration of Finite Differences and Phase Unwrapping on a Sparse Multidimensional Domain","volume":"50","author":"Costantini","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_21","unstructured":"Schrijver, A. (1986). Theory of Linear and Integer Programming, John Wiley & Sons, Inc."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Dantzig, G. (1963). Linear Programming and Extensions, Princeton University Press.","DOI":"10.7249\/R366"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/BF02579150","article-title":"A New Polynomial-Time Algorithm for Linear Programming","volume":"4","author":"Karmarkar","year":"1984","journal-title":"Combinatorica"},{"key":"ref_24","unstructured":"Wilson, R.J. (1986). Introduction to Graph Theory, John Wiley & Sons, Inc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1137\/0109002","article-title":"An Out-of-Kilter Method for Minimal-Cost Flow Problem","volume":"9","author":"Fulkerson","year":"1961","journal-title":"J. Soc. Ind. Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1287\/opre.36.1.93","article-title":"Relaxation Methods for Minimum Cost Ordinary and Generalized Network Flow Problems","volume":"36","author":"Bertsekas","year":"1988","journal-title":"Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4394","DOI":"10.1109\/TGRS.2015.2396875","article-title":"Improved EMCF-SBAS Processing Chain Based on Advanced Techniques for the Noise-Filtering and Selection of Small Baseline Multi-Look DInSAR Interferograms","volume":"53","author":"Pepe","year":"2015","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1109\/36.312890","article-title":"Intensity and Phase Statistics of Multilook Polarimetric and Interferometric SAR Imagery","volume":"32","author":"Lee","year":"1994","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.isprsjprs.2014.08.002","article-title":"A Very Fast Phase Inversion Approach for Small Baseline Style Interferogram Stacks","volume":"97","author":"Zhang","year":"2014","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1093\/comjnl\/7.4.308","article-title":"A Simplex Method for Function Minimization","volume":"7","author":"Nelder","year":"1965","journal-title":"Comput. J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by Simulated Annealing","volume":"220","author":"Kirkpatrick","year":"1983","journal-title":"Science"},{"key":"ref_32","first-page":"288","article-title":"Ausgleichung von H\u00f6hennetzen aus mehreren Epochen unter Ber\u00fccksichtigung von Bodenbewegungen","volume":"4","author":"Halsig","year":"2013","journal-title":"Z. F\u00fcr Geod. Geoinf. Landmanag."},{"key":"ref_33","first-page":"173","article-title":"25 Jahre Bodenbewegungen in der Niederrheinischen Bucht\u2013Ein kombinierter Ansatz aus D-InSAR und amtlichen Leitnivellements","volume":"3","author":"Esch","year":"2019","journal-title":"Z. F\u00fcr Geod. Geoinf. Landmanag."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/11\/19\/2295\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:26:50Z","timestamp":1760189210000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/11\/19\/2295"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,1]]},"references-count":33,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2019,10]]}},"alternative-id":["rs11192295"],"URL":"https:\/\/doi.org\/10.3390\/rs11192295","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2019,10,1]]}}}