Abstract

Phase unwrapping is a key procedure in interferometric synthetic aperture radar studies, translating ambiguous phase observations to topography, and surface deformation estimates. Some unwrapping algorithms are conducted along specific paths based on different selection criteria. In this study, we analyze six unwrapping paths: line scan, maximum coherence, phase derivative variance, phase derivative variance with branch-cut, second-derivative reliability, and the Fisher distance. The latter is a new path algorithm based on Fisher information theory, which combines the phase derivative with the expected variance to get a more robust path, potentially performing better than others in the case of low image quality. In order to compare only the performance of the paths, the same unwrapping function (phase derivative integral) is used. Results indicate that the Fisher distance algorithm gives better results in most cases.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Zebker and P. Rosen, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” in Geoscience and Remote Sensing Symposium, 1994. IGARSS ’94. Surface and Atmospheric Remote Sensing: Technologies, Data Analysis and Interpretation, International, vol.  1, (IEEE, 1994), pp. 286–288.
  2. M. Simons, Y. Fialko, and L. Rivera, “Coseismic deformation from the 1999 Mw 7.1 Hector Mine, California, earthquake as inferred from InSAR and GPS observations,” Bull. Seismol. Soc. Am. 92, 1390–1402 (2002).
    [CrossRef]
  3. R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
    [CrossRef]
  4. R. Lanari, P. Lundgren, and E. Sansosti, “Dynamic deformation of Etna volcano observed by satellite radar interferometry,” Geophys. Res. Lett. 25, 1541–1544 (1998).
    [CrossRef]
  5. T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
    [CrossRef] [PubMed]
  6. F. Amelung, D. L. Galloway, J. W. Bell, H. A. Zebker, and R. J. Laczniak, “Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation,” Geology 27, 483–486 (1999).
    [CrossRef]
  7. G. Bawden, W. Thatcher, R. Stein, K. Hudnut, and G. Peltzer, “Tectonic contraction across Los Angeles after removal of groundwater pumping effects,” Nature 412, 812–815(2001).
    [CrossRef] [PubMed]
  8. Z. Perski, “Applicability of ERS-1 and ERS-2 InSAR for land subsidence monitoring in the Silesian coal mining region, Poland,” International Archives of Photogrammetry and Remote Sensing 32, 555–558 (1998).
  9. N. Gourmelen, F. Amelung, F. Casu, M. Manzo, and R. Lanari, “Mining-related ground deformation in Crescent Valley, Nevada: implications for sparse GPS networks,” Geophys. Res. Lett. 34, L09309 (2007).
    [CrossRef]
  10. R. M. Goldstein, H. Engelhardt, B. Kamb, and R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion—application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
    [CrossRef] [PubMed]
  11. S. Wdowinski, F. Amelung, F. Miralles-Wilhelm, T. Dixon, and R. Carande, “Space-based measurements of sheet-flow characteristics in the Everglades wetland, Florida,” Geophys. Res. Lett. 31, L15503 (2004).
    [CrossRef]
  12. S. Wdowinski, S.-W. Kim, F. Amelung, T. Dixon, F. Miralles-Wilhelm, and R. Sonenshein, “Space-based detection of wetlands surface water level changes from L-band SAR interferometry,” Rem. Sens. Environ. 112, 681–696 (2008).
    [CrossRef]
  13. M. I. Skolnik, Introduction to Radar Systems, 2nd ed. (McGraw Hill, 1980).
  14. R. F. Hanssen, Radar Interferometry: Data Interpretation and Error Analysis (Kluwer, 2001).
  15. H. A. Zebker and J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
    [CrossRef]
  16. J. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. 25, 170–177 (1977).
    [CrossRef]
  17. D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [CrossRef]
  18. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470–2470 (1982).
    [CrossRef] [PubMed]
  19. H. A. Zebker and R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
    [CrossRef]
  20. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  21. R. Bamler and P. Hartl, “Synthetic aperture radar interferometry,” Inverse Probl. 14, R1–R54 (1998).
    [CrossRef]
  22. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632(1991).
    [CrossRef] [PubMed]
  23. J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868–882 (1998).
    [CrossRef]
  24. M. Kim and H. Griffiths, “Phase unwrapping of multibaseline interferometry using Kalman filtering,” presented at the 7th International Conference on Image Processing and its Applications , Manchester, UK (13–15 July 1999).
  25. O. Loffeld, H. Nies, S. Knedlik, and Y. Wang, “Phase unwrapping for SAR interferometry: a data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Remote Sens. 46, 47–58 (2008).
    [CrossRef]
  26. J. Martinez-Espla, T. Martinez-Marin, and J. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47, 1197–1211 (2009).
    [CrossRef]
  27. W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
    [CrossRef]
  28. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  29. M. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
    [CrossRef]
  30. M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
    [CrossRef]
  31. C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401–414 (2000).
    [CrossRef]
  32. C. W. Chen and H. A. Zebker, “Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimization,” J. Opt. Soc. Am. A 18, 338–351(2001).
    [CrossRef]
  33. C. W. Chen and H. A. Zebker, “Phase unwrapping for large SAR interferograms: statistical segmentation and generalized network models,” IEEE Trans. Geosci. Remote Sens. 40, 1709–1719 (2002).
    [CrossRef]
  34. J. M. Bioucas-Dias and J. Leitao, “InSAR phase unwrapping: a Bayesian approach,” in Geoscience and Remote Sensing Symposium, 2001 (IEEE, 2001), pp. 396–400.
  35. A. Ferretti, C. Prati, and F. Rocca, “Permanent scatterers in SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 39, 8–20 (2001).
    [CrossRef]
  36. P. Berardino, G. Fornaro, R. Lanari, and E. Sansosti, “A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms,” IEEE Trans. Geosci. Remote Sens. 40, 2375–2383 (2002).
    [CrossRef]
  37. A. Hooper, “Persistent scatter radar interferometry for crustal deformation studies and modeling of volcanic deformation,” Ph.D. dissertation (Stanford University, 2006).
  38. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).
  39. R. Krämer, “Auf Kalman-Filtern basierende Phasen- und Parameterestimation zur Lösung der Phasenvieldeutigkeitsproblematik bei der Höhenmodellerstellung aus SAR-Interferogrammen,” Ph.D. dissertation (Universitat-GH Siegen, 1998).
  40. M. Herráez, D. Burton, M. Lalor, and M. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
    [CrossRef] [PubMed]
  41. H. Abdul-Rahman, M. Gdeisat, D. Burton, M. Lalor, F. Lilley, and C. Moore, “Fast and robust three-dimensional best path phase unwrapping algorithm,” Appl. Opt. 46, 6623–6635(2007).
    [CrossRef] [PubMed]
  42. E. Moore, “Machine models of self-reproduction,” in Mathematical Problems in the Biological Sciences, R.Bellman, ed. (American Mathematical Society, 1962), pp. 17–33.
  43. G. Peano, “On a curve which entirely fills a plane domain,” Math. Ann. 36–157160 (1890).
    [CrossRef]
  44. H. Sagan and J. Holbrook, Space-Filling Curves (Springer-Verlag, 1994).
    [CrossRef]
  45. C. A. Pickover, The Math Book (Sterling, 2009).
  46. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge University, 1992).
  47. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am 69, 393–399(1979).
    [CrossRef]
  48. B. Spottiswoode, “2D phase unwrapping algorithms,” http://www.mathworks.com/matlabcentral/fileexchange/22504.
  49. J. Buckland, J. Huntley, and S. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
    [CrossRef] [PubMed]
  50. J. Quiroga, A. González-Cano, and E. Bernabeu, “Stable-marriages algorithm for preprocessing phase maps with discontinuity sources,” Appl. Opt. 34, 5029–5038 (1995).
    [CrossRef] [PubMed]
  51. R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
    [CrossRef] [PubMed]
  52. B. Gutmann and H. Weber, “Phase unwrapping with the branch-cut method: role of phase-field direction,” Appl. Opt. 39, 4802–4816 (2000).
    [CrossRef]
  53. S. Karout, M. Gdeisat, D. Burton, and M. Lalor, “Residue vector, an approach to branch-cut placement in phase unwrapping: theoretical study,” Appl. Opt. 46, 4712–4727 (2007).
    [CrossRef] [PubMed]
  54. R. Pawula, S. Rice, and J. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun. 30, 1828–1841 (1982).
    [CrossRef]
  55. F. Edgeworth, “On the probable errors of frequency-constants (contd.),” J. Roy. Statist. Soc. 71, 651–678 (1908).
    [CrossRef]
  56. T. Cover, J. Thomas, and J. Wiley, Elements of Information Theory (Wiley, 1991).
    [CrossRef]
  57. B. Frieden, Physics from Fisher Information: a Unification (Cambridge University, 1998).
    [CrossRef]
  58. D. Just and R. Bamler, “Phase statistics of interferograms with applications to synthetic aperture radar,” Appl. Opt. 33, 4361–4368 (1994).
    [CrossRef] [PubMed]
  59. J. Lee, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
    [CrossRef]
  60. A. Ferretti, C. Prati, and F. Rocca, “Multibaseline InSAR DEM reconstruction: the wavelet approach,” IEEE Trans. Geosci. Remote Sens. 37, 705–715 (1999).
    [CrossRef]
  61. F. Lombardini, F. Gini, and P. Matteucci, “Application of array processing techniques to multibaseline InSAR for layover solution,” in Proceedings of the 2001 IEEE Radar Conference, 2001 (IEEE, 2002), pp. 210–215.
  62. A. Ferretti, A. Monti-Guarnieri, C. Prati, F. Rocca, and D. Massonet, InSAR Principles-Guidelines for SAR Interferometry Processing and Interpretation, (ESA, 2007), Vol.  19, Chap. Part B.
  63. J. W. Eaton, D. Bateman, and S. Hauberg, GNU Octave Manual (Network Theory, 2008).
  64. B. Kampes, “MATLAB toolbox for InSAR,” http://enterprise.lr.tudelft.nl/doris/software/insarmatlab.tar.gz.
  65. B. Kampes and S. Usai, “Doris: the Delft object-oriented radar interferometric software,” presented at the International Symposium on Operationalization of Remote Sensing, Enschede, The Netherlands, 16–20 August 1999.
  66. T. Strozzi and U. Wegmüller, “Land subsidence in Mexico City mapped by ERS differential SAR interferometry,” in Proceedings of the IEEE 1999 International Geoscience and Remote Sensing Symposium (IEEE, 1999), pp. 1940–1942.
  67. E. Cabral-Cano, T. H. Dixon, F. Miralles-Wilhelm, O. Diaz-Molina, O. Sanchez-Zamora, and R. E. Carande, “Space geodetic imaging of rapid ground subsidence in Mexico City,” Geol. Soc. Am. Bull. 120, 1556–1566 (2008).
    [CrossRef]
  68. P. López-Quiroz, M. Doin, F. Tupin, P. Briole, and J. Nicolas, “Time series analysis of Mexico City subsidence constrained by radar interferometry,” J. Appl. Geophys. 69, 1–15(2009).
    [CrossRef]
  69. H. Friis, “Noise figures of radio receivers,” Proc. IREE Aust. 32, 419–422 (1944).
    [CrossRef]
  70. J. W. Tukey, Exploratory Data Analysis, Behavioral Science: Quantitative Methods (Addison-Wesley, 1977).
  71. E. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).

2009

J. Martinez-Espla, T. Martinez-Marin, and J. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47, 1197–1211 (2009).
[CrossRef]

P. López-Quiroz, M. Doin, F. Tupin, P. Briole, and J. Nicolas, “Time series analysis of Mexico City subsidence constrained by radar interferometry,” J. Appl. Geophys. 69, 1–15(2009).
[CrossRef]

2008

S. Wdowinski, S.-W. Kim, F. Amelung, T. Dixon, F. Miralles-Wilhelm, and R. Sonenshein, “Space-based detection of wetlands surface water level changes from L-band SAR interferometry,” Rem. Sens. Environ. 112, 681–696 (2008).
[CrossRef]

O. Loffeld, H. Nies, S. Knedlik, and Y. Wang, “Phase unwrapping for SAR interferometry: a data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Remote Sens. 46, 47–58 (2008).
[CrossRef]

E. Cabral-Cano, T. H. Dixon, F. Miralles-Wilhelm, O. Diaz-Molina, O. Sanchez-Zamora, and R. E. Carande, “Space geodetic imaging of rapid ground subsidence in Mexico City,” Geol. Soc. Am. Bull. 120, 1556–1566 (2008).
[CrossRef]

2007

2006

T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
[CrossRef] [PubMed]

2004

S. Wdowinski, F. Amelung, F. Miralles-Wilhelm, T. Dixon, and R. Carande, “Space-based measurements of sheet-flow characteristics in the Everglades wetland, Florida,” Geophys. Res. Lett. 31, L15503 (2004).
[CrossRef]

2002

M. Simons, Y. Fialko, and L. Rivera, “Coseismic deformation from the 1999 Mw 7.1 Hector Mine, California, earthquake as inferred from InSAR and GPS observations,” Bull. Seismol. Soc. Am. 92, 1390–1402 (2002).
[CrossRef]

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

C. W. Chen and H. A. Zebker, “Phase unwrapping for large SAR interferograms: statistical segmentation and generalized network models,” IEEE Trans. Geosci. Remote Sens. 40, 1709–1719 (2002).
[CrossRef]

P. Berardino, G. Fornaro, R. Lanari, and E. Sansosti, “A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms,” IEEE Trans. Geosci. Remote Sens. 40, 2375–2383 (2002).
[CrossRef]

M. Herráez, D. Burton, M. Lalor, and M. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
[CrossRef] [PubMed]

2001

C. W. Chen and H. A. Zebker, “Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimization,” J. Opt. Soc. Am. A 18, 338–351(2001).
[CrossRef]

A. Ferretti, C. Prati, and F. Rocca, “Permanent scatterers in SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 39, 8–20 (2001).
[CrossRef]

G. Bawden, W. Thatcher, R. Stein, K. Hudnut, and G. Peltzer, “Tectonic contraction across Los Angeles after removal of groundwater pumping effects,” Nature 412, 812–815(2001).
[CrossRef] [PubMed]

2000

1999

F. Amelung, D. L. Galloway, J. W. Bell, H. A. Zebker, and R. J. Laczniak, “Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation,” Geology 27, 483–486 (1999).
[CrossRef]

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

A. Ferretti, C. Prati, and F. Rocca, “Multibaseline InSAR DEM reconstruction: the wavelet approach,” IEEE Trans. Geosci. Remote Sens. 37, 705–715 (1999).
[CrossRef]

1998

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

R. Lanari, P. Lundgren, and E. Sansosti, “Dynamic deformation of Etna volcano observed by satellite radar interferometry,” Geophys. Res. Lett. 25, 1541–1544 (1998).
[CrossRef]

Z. Perski, “Applicability of ERS-1 and ERS-2 InSAR for land subsidence monitoring in the Silesian coal mining region, Poland,” International Archives of Photogrammetry and Remote Sensing 32, 555–558 (1998).

R. Bamler and P. Hartl, “Synthetic aperture radar interferometry,” Inverse Probl. 14, R1–R54 (1998).
[CrossRef]

J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868–882 (1998).
[CrossRef]

1996

M. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[CrossRef]

1995

1994

1993

R. M. Goldstein, H. Engelhardt, B. Kamb, and R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion—application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

1992

H. A. Zebker and J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
[CrossRef]

1991

1988

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

1987

1986

H. A. Zebker and R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

1982

K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470–2470 (1982).
[CrossRef] [PubMed]

R. Pawula, S. Rice, and J. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun. 30, 1828–1841 (1982).
[CrossRef]

1979

B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am 69, 393–399(1979).
[CrossRef]

1977

J. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. 25, 170–177 (1977).
[CrossRef]

1944

H. Friis, “Noise figures of radio receivers,” Proc. IREE Aust. 32, 419–422 (1944).
[CrossRef]

1908

F. Edgeworth, “On the probable errors of frequency-constants (contd.),” J. Roy. Statist. Soc. 71, 651–678 (1908).
[CrossRef]

1890

G. Peano, “On a curve which entirely fills a plane domain,” Math. Ann. 36–157160 (1890).
[CrossRef]

Abdul-Rahman, H.

Aktug, B.

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Amelung, F.

S. Wdowinski, S.-W. Kim, F. Amelung, T. Dixon, F. Miralles-Wilhelm, and R. Sonenshein, “Space-based detection of wetlands surface water level changes from L-band SAR interferometry,” Rem. Sens. Environ. 112, 681–696 (2008).
[CrossRef]

N. Gourmelen, F. Amelung, F. Casu, M. Manzo, and R. Lanari, “Mining-related ground deformation in Crescent Valley, Nevada: implications for sparse GPS networks,” Geophys. Res. Lett. 34, L09309 (2007).
[CrossRef]

S. Wdowinski, F. Amelung, F. Miralles-Wilhelm, T. Dixon, and R. Carande, “Space-based measurements of sheet-flow characteristics in the Everglades wetland, Florida,” Geophys. Res. Lett. 31, L15503 (2004).
[CrossRef]

F. Amelung, D. L. Galloway, J. W. Bell, H. A. Zebker, and R. J. Laczniak, “Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation,” Geology 27, 483–486 (1999).
[CrossRef]

Ayele, A.

T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
[CrossRef] [PubMed]

Ayhan, M.

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Bamler, R.

Bateman, D.

J. W. Eaton, D. Bateman, and S. Hauberg, GNU Octave Manual (Network Theory, 2008).

Bawden, G.

G. Bawden, W. Thatcher, R. Stein, K. Hudnut, and G. Peltzer, “Tectonic contraction across Los Angeles after removal of groundwater pumping effects,” Nature 412, 812–815(2001).
[CrossRef] [PubMed]

Bell, J. W.

F. Amelung, D. L. Galloway, J. W. Bell, H. A. Zebker, and R. J. Laczniak, “Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation,” Geology 27, 483–486 (1999).
[CrossRef]

Berardino, P.

P. Berardino, G. Fornaro, R. Lanari, and E. Sansosti, “A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms,” IEEE Trans. Geosci. Remote Sens. 40, 2375–2383 (2002).
[CrossRef]

Bernabeu, E.

Biggs, J.

T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
[CrossRef] [PubMed]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and J. Leitao, “InSAR phase unwrapping: a Bayesian approach,” in Geoscience and Remote Sensing Symposium, 2001 (IEEE, 2001), pp. 396–400.

Bone, D. J.

Briole, P.

P. López-Quiroz, M. Doin, F. Tupin, P. Briole, and J. Nicolas, “Time series analysis of Mexico City subsidence constrained by radar interferometry,” J. Appl. Geophys. 69, 1–15(2009).
[CrossRef]

Buckland, J.

Bürgmann, R.

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Burton, D.

Cabral-Cano, E.

E. Cabral-Cano, T. H. Dixon, F. Miralles-Wilhelm, O. Diaz-Molina, O. Sanchez-Zamora, and R. E. Carande, “Space geodetic imaging of rapid ground subsidence in Mexico City,” Geol. Soc. Am. Bull. 120, 1556–1566 (2008).
[CrossRef]

Carande, R.

S. Wdowinski, F. Amelung, F. Miralles-Wilhelm, T. Dixon, and R. Carande, “Space-based measurements of sheet-flow characteristics in the Everglades wetland, Florida,” Geophys. Res. Lett. 31, L15503 (2004).
[CrossRef]

Carande, R. E.

E. Cabral-Cano, T. H. Dixon, F. Miralles-Wilhelm, O. Diaz-Molina, O. Sanchez-Zamora, and R. E. Carande, “Space geodetic imaging of rapid ground subsidence in Mexico City,” Geol. Soc. Am. Bull. 120, 1556–1566 (2008).
[CrossRef]

Casella, G.

E. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).

Casu, F.

N. Gourmelen, F. Amelung, F. Casu, M. Manzo, and R. Lanari, “Mining-related ground deformation in Crescent Valley, Nevada: implications for sparse GPS networks,” Geophys. Res. Lett. 34, L09309 (2007).
[CrossRef]

Chen, C. W.

Costantini, M.

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

Cover, T.

T. Cover, J. Thomas, and J. Wiley, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Cumming, I.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

Cusack, R.

Demir, C.

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Diaz-Molina, O.

E. Cabral-Cano, T. H. Dixon, F. Miralles-Wilhelm, O. Diaz-Molina, O. Sanchez-Zamora, and R. E. Carande, “Space geodetic imaging of rapid ground subsidence in Mexico City,” Geol. Soc. Am. Bull. 120, 1556–1566 (2008).
[CrossRef]

Dixon, T.

S. Wdowinski, S.-W. Kim, F. Amelung, T. Dixon, F. Miralles-Wilhelm, and R. Sonenshein, “Space-based detection of wetlands surface water level changes from L-band SAR interferometry,” Rem. Sens. Environ. 112, 681–696 (2008).
[CrossRef]

S. Wdowinski, F. Amelung, F. Miralles-Wilhelm, T. Dixon, and R. Carande, “Space-based measurements of sheet-flow characteristics in the Everglades wetland, Florida,” Geophys. Res. Lett. 31, L15503 (2004).
[CrossRef]

Dixon, T. H.

E. Cabral-Cano, T. H. Dixon, F. Miralles-Wilhelm, O. Diaz-Molina, O. Sanchez-Zamora, and R. E. Carande, “Space geodetic imaging of rapid ground subsidence in Mexico City,” Geol. Soc. Am. Bull. 120, 1556–1566 (2008).
[CrossRef]

Doin, M.

P. López-Quiroz, M. Doin, F. Tupin, P. Briole, and J. Nicolas, “Time series analysis of Mexico City subsidence constrained by radar interferometry,” J. Appl. Geophys. 69, 1–15(2009).
[CrossRef]

Eaton, J. W.

J. W. Eaton, D. Bateman, and S. Hauberg, GNU Octave Manual (Network Theory, 2008).

Ebinger, C.

T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
[CrossRef] [PubMed]

Edgeworth, F.

F. Edgeworth, “On the probable errors of frequency-constants (contd.),” J. Roy. Statist. Soc. 71, 651–678 (1908).
[CrossRef]

Engelhardt, H.

R. M. Goldstein, H. Engelhardt, B. Kamb, and R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion—application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

Ferretti, A.

A. Ferretti, C. Prati, and F. Rocca, “Permanent scatterers in SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 39, 8–20 (2001).
[CrossRef]

A. Ferretti, C. Prati, and F. Rocca, “Multibaseline InSAR DEM reconstruction: the wavelet approach,” IEEE Trans. Geosci. Remote Sens. 37, 705–715 (1999).
[CrossRef]

A. Ferretti, A. Monti-Guarnieri, C. Prati, F. Rocca, and D. Massonet, InSAR Principles-Guidelines for SAR Interferometry Processing and Interpretation, (ESA, 2007), Vol.  19, Chap. Part B.

Fialko, Y.

M. Simons, Y. Fialko, and L. Rivera, “Coseismic deformation from the 1999 Mw 7.1 Hector Mine, California, earthquake as inferred from InSAR and GPS observations,” Bull. Seismol. Soc. Am. 92, 1390–1402 (2002).
[CrossRef]

Fielding, E.

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Figueiredo, M. A. T.

J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868–882 (1998).
[CrossRef]

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge University, 1992).

Fornaro, G.

P. Berardino, G. Fornaro, R. Lanari, and E. Sansosti, “A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms,” IEEE Trans. Geosci. Remote Sens. 40, 2375–2383 (2002).
[CrossRef]

Frieden, B.

B. Frieden, Physics from Fisher Information: a Unification (Cambridge University, 1998).
[CrossRef]

Friis, H.

H. Friis, “Noise figures of radio receivers,” Proc. IREE Aust. 32, 419–422 (1944).
[CrossRef]

Frolich, R. M.

R. M. Goldstein, H. Engelhardt, B. Kamb, and R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion—application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

Galloway, D. L.

F. Amelung, D. L. Galloway, J. W. Bell, H. A. Zebker, and R. J. Laczniak, “Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation,” Geology 27, 483–486 (1999).
[CrossRef]

Gdeisat, M.

Ghiglia, D. C.

Gini, F.

F. Lombardini, F. Gini, and P. Matteucci, “Application of array processing techniques to multibaseline InSAR for layover solution,” in Proceedings of the 2001 IEEE Radar Conference, 2001 (IEEE, 2002), pp. 210–215.

Goldrein, H. T.

Goldstein, R. M.

R. M. Goldstein, H. Engelhardt, B. Kamb, and R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion—application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

H. A. Zebker and R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

González-Cano, A.

Gourmelen, N.

N. Gourmelen, F. Amelung, F. Casu, M. Manzo, and R. Lanari, “Mining-related ground deformation in Crescent Valley, Nevada: implications for sparse GPS networks,” Geophys. Res. Lett. 34, L09309 (2007).
[CrossRef]

Griffiths, H.

M. Kim and H. Griffiths, “Phase unwrapping of multibaseline interferometry using Kalman filtering,” presented at the 7th International Conference on Image Processing and its Applications , Manchester, UK (13–15 July 1999).

Gutmann, B.

Hanssen, R. F.

R. F. Hanssen, Radar Interferometry: Data Interpretation and Error Analysis (Kluwer, 2001).

Hartl, P.

R. Bamler and P. Hartl, “Synthetic aperture radar interferometry,” Inverse Probl. 14, R1–R54 (1998).
[CrossRef]

Hauberg, S.

J. W. Eaton, D. Bateman, and S. Hauberg, GNU Octave Manual (Network Theory, 2008).

Herráez, M.

Holbrook, J.

H. Sagan and J. Holbrook, Space-Filling Curves (Springer-Verlag, 1994).
[CrossRef]

Hooper, A.

A. Hooper, “Persistent scatter radar interferometry for crustal deformation studies and modeling of volcanic deformation,” Ph.D. dissertation (Stanford University, 2006).

Hoppel, K.

J. Lee, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

Hudnut, K.

G. Bawden, W. Thatcher, R. Stein, K. Hudnut, and G. Peltzer, “Tectonic contraction across Los Angeles after removal of groundwater pumping effects,” Nature 412, 812–815(2001).
[CrossRef] [PubMed]

Hunt, B. R.

B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am 69, 393–399(1979).
[CrossRef]

Huntley, J.

Huntley, J. M.

Itoh, K.

Just, D.

Kamb, B.

R. M. Goldstein, H. Engelhardt, B. Kamb, and R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion—application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

Kampes, B.

B. Kampes and S. Usai, “Doris: the Delft object-oriented radar interferometric software,” presented at the International Symposium on Operationalization of Remote Sensing, Enschede, The Netherlands, 16–20 August 1999.

B. Kampes, “MATLAB toolbox for InSAR,” http://enterprise.lr.tudelft.nl/doris/software/insarmatlab.tar.gz.

Karout, S.

Keir, D.

T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
[CrossRef] [PubMed]

Kim, M.

M. Kim and H. Griffiths, “Phase unwrapping of multibaseline interferometry using Kalman filtering,” presented at the 7th International Conference on Image Processing and its Applications , Manchester, UK (13–15 July 1999).

Kim, S.-W.

S. Wdowinski, S.-W. Kim, F. Amelung, T. Dixon, F. Miralles-Wilhelm, and R. Sonenshein, “Space-based detection of wetlands surface water level changes from L-band SAR interferometry,” Rem. Sens. Environ. 112, 681–696 (2008).
[CrossRef]

Knedlik, S.

O. Loffeld, H. Nies, S. Knedlik, and Y. Wang, “Phase unwrapping for SAR interferometry: a data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Remote Sens. 46, 47–58 (2008).
[CrossRef]

Krämer, R.

R. Krämer, “Auf Kalman-Filtern basierende Phasen- und Parameterestimation zur Lösung der Phasenvieldeutigkeitsproblematik bei der Höhenmodellerstellung aus SAR-Interferogrammen,” Ph.D. dissertation (Universitat-GH Siegen, 1998).

Laczniak, R. J.

F. Amelung, D. L. Galloway, J. W. Bell, H. A. Zebker, and R. J. Laczniak, “Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation,” Geology 27, 483–486 (1999).
[CrossRef]

Lalor, M.

Lanari, R.

N. Gourmelen, F. Amelung, F. Casu, M. Manzo, and R. Lanari, “Mining-related ground deformation in Crescent Valley, Nevada: implications for sparse GPS networks,” Geophys. Res. Lett. 34, L09309 (2007).
[CrossRef]

P. Berardino, G. Fornaro, R. Lanari, and E. Sansosti, “A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms,” IEEE Trans. Geosci. Remote Sens. 40, 2375–2383 (2002).
[CrossRef]

R. Lanari, P. Lundgren, and E. Sansosti, “Dynamic deformation of Etna volcano observed by satellite radar interferometry,” Geophys. Res. Lett. 25, 1541–1544 (1998).
[CrossRef]

Lee, J.

J. Lee, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

Lehmann, E.

E. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).

Leitao, J.

J. M. Bioucas-Dias and J. Leitao, “InSAR phase unwrapping: a Bayesian approach,” in Geoscience and Remote Sensing Symposium, 2001 (IEEE, 2001), pp. 396–400.

Leitao, J. M. N.

J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868–882 (1998).
[CrossRef]

Lenk, O.

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Lilley, F.

Loffeld, O.

O. Loffeld, H. Nies, S. Knedlik, and Y. Wang, “Phase unwrapping for SAR interferometry: a data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Remote Sens. 46, 47–58 (2008).
[CrossRef]

Lombardini, F.

F. Lombardini, F. Gini, and P. Matteucci, “Application of array processing techniques to multibaseline InSAR for layover solution,” in Proceedings of the 2001 IEEE Radar Conference, 2001 (IEEE, 2002), pp. 210–215.

López-Quiroz, P.

P. López-Quiroz, M. Doin, F. Tupin, P. Briole, and J. Nicolas, “Time series analysis of Mexico City subsidence constrained by radar interferometry,” J. Appl. Geophys. 69, 1–15(2009).
[CrossRef]

Lopez-Sanchez, J.

J. Martinez-Espla, T. Martinez-Marin, and J. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47, 1197–1211 (2009).
[CrossRef]

Lundgren, P.

R. Lanari, P. Lundgren, and E. Sansosti, “Dynamic deformation of Etna volcano observed by satellite radar interferometry,” Geophys. Res. Lett. 25, 1541–1544 (1998).
[CrossRef]

Mango, S.

J. Lee, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

Manzo, M.

N. Gourmelen, F. Amelung, F. Casu, M. Manzo, and R. Lanari, “Mining-related ground deformation in Crescent Valley, Nevada: implications for sparse GPS networks,” Geophys. Res. Lett. 34, L09309 (2007).
[CrossRef]

Martinez-Espla, J.

J. Martinez-Espla, T. Martinez-Marin, and J. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47, 1197–1211 (2009).
[CrossRef]

Martinez-Marin, T.

J. Martinez-Espla, T. Martinez-Marin, and J. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47, 1197–1211 (2009).
[CrossRef]

Massonet, D.

A. Ferretti, A. Monti-Guarnieri, C. Prati, F. Rocca, and D. Massonet, InSAR Principles-Guidelines for SAR Interferometry Processing and Interpretation, (ESA, 2007), Vol.  19, Chap. Part B.

Mastin, G. A.

Matteucci, P.

F. Lombardini, F. Gini, and P. Matteucci, “Application of array processing techniques to multibaseline InSAR for layover solution,” in Proceedings of the 2001 IEEE Radar Conference, 2001 (IEEE, 2002), pp. 210–215.

McClusky, S.

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Miller, A.

J. Lee, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

Miralles-Wilhelm, F.

S. Wdowinski, S.-W. Kim, F. Amelung, T. Dixon, F. Miralles-Wilhelm, and R. Sonenshein, “Space-based detection of wetlands surface water level changes from L-band SAR interferometry,” Rem. Sens. Environ. 112, 681–696 (2008).
[CrossRef]

E. Cabral-Cano, T. H. Dixon, F. Miralles-Wilhelm, O. Diaz-Molina, O. Sanchez-Zamora, and R. E. Carande, “Space geodetic imaging of rapid ground subsidence in Mexico City,” Geol. Soc. Am. Bull. 120, 1556–1566 (2008).
[CrossRef]

S. Wdowinski, F. Amelung, F. Miralles-Wilhelm, T. Dixon, and R. Carande, “Space-based measurements of sheet-flow characteristics in the Everglades wetland, Florida,” Geophys. Res. Lett. 31, L15503 (2004).
[CrossRef]

Monti-Guarnieri, A.

A. Ferretti, A. Monti-Guarnieri, C. Prati, F. Rocca, and D. Massonet, InSAR Principles-Guidelines for SAR Interferometry Processing and Interpretation, (ESA, 2007), Vol.  19, Chap. Part B.

Moore, C.

Moore, E.

E. Moore, “Machine models of self-reproduction,” in Mathematical Problems in the Biological Sciences, R.Bellman, ed. (American Mathematical Society, 1962), pp. 17–33.

Nicolas, J.

P. López-Quiroz, M. Doin, F. Tupin, P. Briole, and J. Nicolas, “Time series analysis of Mexico City subsidence constrained by radar interferometry,” J. Appl. Geophys. 69, 1–15(2009).
[CrossRef]

Nies, H.

O. Loffeld, H. Nies, S. Knedlik, and Y. Wang, “Phase unwrapping for SAR interferometry: a data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Remote Sens. 46, 47–58 (2008).
[CrossRef]

Pawula, R.

R. Pawula, S. Rice, and J. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun. 30, 1828–1841 (1982).
[CrossRef]

Peano, G.

G. Peano, “On a curve which entirely fills a plane domain,” Math. Ann. 36–157160 (1890).
[CrossRef]

Peltzer, G.

G. Bawden, W. Thatcher, R. Stein, K. Hudnut, and G. Peltzer, “Tectonic contraction across Los Angeles after removal of groundwater pumping effects,” Nature 412, 812–815(2001).
[CrossRef] [PubMed]

Perski, Z.

Z. Perski, “Applicability of ERS-1 and ERS-2 InSAR for land subsidence monitoring in the Silesian coal mining region, Poland,” International Archives of Photogrammetry and Remote Sensing 32, 555–558 (1998).

Pickover, C. A.

C. A. Pickover, The Math Book (Sterling, 2009).

Prati, C.

A. Ferretti, C. Prati, and F. Rocca, “Permanent scatterers in SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 39, 8–20 (2001).
[CrossRef]

A. Ferretti, C. Prati, and F. Rocca, “Multibaseline InSAR DEM reconstruction: the wavelet approach,” IEEE Trans. Geosci. Remote Sens. 37, 705–715 (1999).
[CrossRef]

A. Ferretti, A. Monti-Guarnieri, C. Prati, F. Rocca, and D. Massonet, InSAR Principles-Guidelines for SAR Interferometry Processing and Interpretation, (ESA, 2007), Vol.  19, Chap. Part B.

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge University, 1992).

Pritt, M.

M. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

Quiroga, J.

Rice, S.

R. Pawula, S. Rice, and J. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun. 30, 1828–1841 (1982).
[CrossRef]

Rivera, L.

M. Simons, Y. Fialko, and L. Rivera, “Coseismic deformation from the 1999 Mw 7.1 Hector Mine, California, earthquake as inferred from InSAR and GPS observations,” Bull. Seismol. Soc. Am. 92, 1390–1402 (2002).
[CrossRef]

Roberts, J.

R. Pawula, S. Rice, and J. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun. 30, 1828–1841 (1982).
[CrossRef]

Rocca, F.

A. Ferretti, C. Prati, and F. Rocca, “Permanent scatterers in SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 39, 8–20 (2001).
[CrossRef]

A. Ferretti, C. Prati, and F. Rocca, “Multibaseline InSAR DEM reconstruction: the wavelet approach,” IEEE Trans. Geosci. Remote Sens. 37, 705–715 (1999).
[CrossRef]

A. Ferretti, A. Monti-Guarnieri, C. Prati, F. Rocca, and D. Massonet, InSAR Principles-Guidelines for SAR Interferometry Processing and Interpretation, (ESA, 2007), Vol.  19, Chap. Part B.

Romero, L. A.

Rosen, P.

H. A. Zebker and P. Rosen, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” in Geoscience and Remote Sensing Symposium, 1994. IGARSS ’94. Surface and Atmospheric Remote Sensing: Technologies, Data Analysis and Interpretation, International, vol.  1, (IEEE, 1994), pp. 286–288.

Sagan, H.

H. Sagan and J. Holbrook, Space-Filling Curves (Springer-Verlag, 1994).
[CrossRef]

Sanchez-Zamora, O.

E. Cabral-Cano, T. H. Dixon, F. Miralles-Wilhelm, O. Diaz-Molina, O. Sanchez-Zamora, and R. E. Carande, “Space geodetic imaging of rapid ground subsidence in Mexico City,” Geol. Soc. Am. Bull. 120, 1556–1566 (2008).
[CrossRef]

Sansosti, E.

P. Berardino, G. Fornaro, R. Lanari, and E. Sansosti, “A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms,” IEEE Trans. Geosci. Remote Sens. 40, 2375–2383 (2002).
[CrossRef]

R. Lanari, P. Lundgren, and E. Sansosti, “Dynamic deformation of Etna volcano observed by satellite radar interferometry,” Geophys. Res. Lett. 25, 1541–1544 (1998).
[CrossRef]

Simons, M.

M. Simons, Y. Fialko, and L. Rivera, “Coseismic deformation from the 1999 Mw 7.1 Hector Mine, California, earthquake as inferred from InSAR and GPS observations,” Bull. Seismol. Soc. Am. 92, 1390–1402 (2002).
[CrossRef]

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar Systems, 2nd ed. (McGraw Hill, 1980).

Sonenshein, R.

S. Wdowinski, S.-W. Kim, F. Amelung, T. Dixon, F. Miralles-Wilhelm, and R. Sonenshein, “Space-based detection of wetlands surface water level changes from L-band SAR interferometry,” Rem. Sens. Environ. 112, 681–696 (2008).
[CrossRef]

Spottiswoode, B.

B. Spottiswoode, “2D phase unwrapping algorithms,” http://www.mathworks.com/matlabcentral/fileexchange/22504.

Stein, R.

G. Bawden, W. Thatcher, R. Stein, K. Hudnut, and G. Peltzer, “Tectonic contraction across Los Angeles after removal of groundwater pumping effects,” Nature 412, 812–815(2001).
[CrossRef] [PubMed]

Stork, A.

T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
[CrossRef] [PubMed]

Strozzi, T.

T. Strozzi and U. Wegmüller, “Land subsidence in Mexico City mapped by ERS differential SAR interferometry,” in Proceedings of the IEEE 1999 International Geoscience and Remote Sensing Symposium (IEEE, 1999), pp. 1940–1942.

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge University, 1992).

Thatcher, W.

G. Bawden, W. Thatcher, R. Stein, K. Hudnut, and G. Peltzer, “Tectonic contraction across Los Angeles after removal of groundwater pumping effects,” Nature 412, 812–815(2001).
[CrossRef] [PubMed]

Thomas, J.

T. Cover, J. Thomas, and J. Wiley, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Tribolet, J.

J. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. 25, 170–177 (1977).
[CrossRef]

Tukey, J. W.

J. W. Tukey, Exploratory Data Analysis, Behavioral Science: Quantitative Methods (Addison-Wesley, 1977).

Tupin, F.

P. López-Quiroz, M. Doin, F. Tupin, P. Briole, and J. Nicolas, “Time series analysis of Mexico City subsidence constrained by radar interferometry,” J. Appl. Geophys. 69, 1–15(2009).
[CrossRef]

Turkezer, A.

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Turner, S.

Usai, S.

B. Kampes and S. Usai, “Doris: the Delft object-oriented radar interferometric software,” presented at the International Symposium on Operationalization of Remote Sensing, Enschede, The Netherlands, 16–20 August 1999.

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge University, 1992).

Villasenor, J.

H. A. Zebker and J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
[CrossRef]

Wang, Y.

O. Loffeld, H. Nies, S. Knedlik, and Y. Wang, “Phase unwrapping for SAR interferometry: a data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Remote Sens. 46, 47–58 (2008).
[CrossRef]

Wdowinski, S.

S. Wdowinski, S.-W. Kim, F. Amelung, T. Dixon, F. Miralles-Wilhelm, and R. Sonenshein, “Space-based detection of wetlands surface water level changes from L-band SAR interferometry,” Rem. Sens. Environ. 112, 681–696 (2008).
[CrossRef]

S. Wdowinski, F. Amelung, F. Miralles-Wilhelm, T. Dixon, and R. Carande, “Space-based measurements of sheet-flow characteristics in the Everglades wetland, Florida,” Geophys. Res. Lett. 31, L15503 (2004).
[CrossRef]

Weber, H.

Wegmüller, U.

T. Strozzi and U. Wegmüller, “Land subsidence in Mexico City mapped by ERS differential SAR interferometry,” in Proceedings of the IEEE 1999 International Geoscience and Remote Sensing Symposium (IEEE, 1999), pp. 1940–1942.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Wiley, J.

T. Cover, J. Thomas, and J. Wiley, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Wright, T.

T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
[CrossRef] [PubMed]

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Xu, W.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

Yirgu, G.

T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
[CrossRef] [PubMed]

Zebker, H. A.

C. W. Chen and H. A. Zebker, “Phase unwrapping for large SAR interferograms: statistical segmentation and generalized network models,” IEEE Trans. Geosci. Remote Sens. 40, 1709–1719 (2002).
[CrossRef]

C. W. Chen and H. A. Zebker, “Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimization,” J. Opt. Soc. Am. A 18, 338–351(2001).
[CrossRef]

C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401–414 (2000).
[CrossRef]

F. Amelung, D. L. Galloway, J. W. Bell, H. A. Zebker, and R. J. Laczniak, “Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation,” Geology 27, 483–486 (1999).
[CrossRef]

H. A. Zebker and J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
[CrossRef]

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

H. A. Zebker and R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

H. A. Zebker and P. Rosen, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” in Geoscience and Remote Sensing Symposium, 1994. IGARSS ’94. Surface and Atmospheric Remote Sensing: Technologies, Data Analysis and Interpretation, International, vol.  1, (IEEE, 1994), pp. 286–288.

Appl. Opt.

K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470–2470 (1982).
[CrossRef] [PubMed]

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632(1991).
[CrossRef] [PubMed]

D. Just and R. Bamler, “Phase statistics of interferograms with applications to synthetic aperture radar,” Appl. Opt. 33, 4361–4368 (1994).
[CrossRef] [PubMed]

R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
[CrossRef] [PubMed]

J. Quiroga, A. González-Cano, and E. Bernabeu, “Stable-marriages algorithm for preprocessing phase maps with discontinuity sources,” Appl. Opt. 34, 5029–5038 (1995).
[CrossRef] [PubMed]

J. Buckland, J. Huntley, and S. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
[CrossRef] [PubMed]

B. Gutmann and H. Weber, “Phase unwrapping with the branch-cut method: role of phase-field direction,” Appl. Opt. 39, 4802–4816 (2000).
[CrossRef]

M. Herráez, D. Burton, M. Lalor, and M. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
[CrossRef] [PubMed]

S. Karout, M. Gdeisat, D. Burton, and M. Lalor, “Residue vector, an approach to branch-cut placement in phase unwrapping: theoretical study,” Appl. Opt. 46, 4712–4727 (2007).
[CrossRef] [PubMed]

H. Abdul-Rahman, M. Gdeisat, D. Burton, M. Lalor, F. Lilley, and C. Moore, “Fast and robust three-dimensional best path phase unwrapping algorithm,” Appl. Opt. 46, 6623–6635(2007).
[CrossRef] [PubMed]

Bull. Seismol. Soc. Am.

M. Simons, Y. Fialko, and L. Rivera, “Coseismic deformation from the 1999 Mw 7.1 Hector Mine, California, earthquake as inferred from InSAR and GPS observations,” Bull. Seismol. Soc. Am. 92, 1390–1402 (2002).
[CrossRef]

R. Bürgmann, M. Ayhan, E. Fielding, T. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, and A. Turkezer, “Deformation during the 12 November 1999 Duzce, Turkey, earthquake, from GPS and InSAR data,” Bull. Seismol. Soc. Am. 92, 161–171 (2002).
[CrossRef]

Geol. Soc. Am. Bull.

E. Cabral-Cano, T. H. Dixon, F. Miralles-Wilhelm, O. Diaz-Molina, O. Sanchez-Zamora, and R. E. Carande, “Space geodetic imaging of rapid ground subsidence in Mexico City,” Geol. Soc. Am. Bull. 120, 1556–1566 (2008).
[CrossRef]

Geology

F. Amelung, D. L. Galloway, J. W. Bell, H. A. Zebker, and R. J. Laczniak, “Sensing the ups and downs of Las Vegas: InSAR reveals structural control of land subsidence and aquifer-system deformation,” Geology 27, 483–486 (1999).
[CrossRef]

Geophys. Res. Lett.

R. Lanari, P. Lundgren, and E. Sansosti, “Dynamic deformation of Etna volcano observed by satellite radar interferometry,” Geophys. Res. Lett. 25, 1541–1544 (1998).
[CrossRef]

S. Wdowinski, F. Amelung, F. Miralles-Wilhelm, T. Dixon, and R. Carande, “Space-based measurements of sheet-flow characteristics in the Everglades wetland, Florida,” Geophys. Res. Lett. 31, L15503 (2004).
[CrossRef]

N. Gourmelen, F. Amelung, F. Casu, M. Manzo, and R. Lanari, “Mining-related ground deformation in Crescent Valley, Nevada: implications for sparse GPS networks,” Geophys. Res. Lett. 34, L09309 (2007).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

J. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. 25, 170–177 (1977).
[CrossRef]

IEEE Trans. Commun.

R. Pawula, S. Rice, and J. Roberts, “Distribution of the phase angle between two vectors perturbed by Gaussian noise,” IEEE Trans. Commun. 30, 1828–1841 (1982).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

M. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[CrossRef]

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

J. Lee, K. Hoppel, S. Mango, and A. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

A. Ferretti, C. Prati, and F. Rocca, “Multibaseline InSAR DEM reconstruction: the wavelet approach,” IEEE Trans. Geosci. Remote Sens. 37, 705–715 (1999).
[CrossRef]

H. A. Zebker and J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
[CrossRef]

O. Loffeld, H. Nies, S. Knedlik, and Y. Wang, “Phase unwrapping for SAR interferometry: a data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Remote Sens. 46, 47–58 (2008).
[CrossRef]

J. Martinez-Espla, T. Martinez-Marin, and J. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47, 1197–1211 (2009).
[CrossRef]

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

C. W. Chen and H. A. Zebker, “Phase unwrapping for large SAR interferograms: statistical segmentation and generalized network models,” IEEE Trans. Geosci. Remote Sens. 40, 1709–1719 (2002).
[CrossRef]

A. Ferretti, C. Prati, and F. Rocca, “Permanent scatterers in SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 39, 8–20 (2001).
[CrossRef]

P. Berardino, G. Fornaro, R. Lanari, and E. Sansosti, “A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms,” IEEE Trans. Geosci. Remote Sens. 40, 2375–2383 (2002).
[CrossRef]

IEEE Trans. Image Process.

J. M. N. Leitao and M. A. T. Figueiredo, “Absolute phase image reconstruction: a stochastic nonlinear filtering approach,” IEEE Trans. Image Process. 7, 868–882 (1998).
[CrossRef]

International Archives of Photogrammetry and Remote Sensing

Z. Perski, “Applicability of ERS-1 and ERS-2 InSAR for land subsidence monitoring in the Silesian coal mining region, Poland,” International Archives of Photogrammetry and Remote Sensing 32, 555–558 (1998).

Inverse Probl.

R. Bamler and P. Hartl, “Synthetic aperture radar interferometry,” Inverse Probl. 14, R1–R54 (1998).
[CrossRef]

J. Appl. Geophys.

P. López-Quiroz, M. Doin, F. Tupin, P. Briole, and J. Nicolas, “Time series analysis of Mexico City subsidence constrained by radar interferometry,” J. Appl. Geophys. 69, 1–15(2009).
[CrossRef]

J. Geophys. Res.

H. A. Zebker and R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

J. Opt. Soc. Am

B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am 69, 393–399(1979).
[CrossRef]

J. Opt. Soc. Am. A

J. Roy. Statist. Soc.

F. Edgeworth, “On the probable errors of frequency-constants (contd.),” J. Roy. Statist. Soc. 71, 651–678 (1908).
[CrossRef]

Math. Ann.

G. Peano, “On a curve which entirely fills a plane domain,” Math. Ann. 36–157160 (1890).
[CrossRef]

Nature

G. Bawden, W. Thatcher, R. Stein, K. Hudnut, and G. Peltzer, “Tectonic contraction across Los Angeles after removal of groundwater pumping effects,” Nature 412, 812–815(2001).
[CrossRef] [PubMed]

T. Wright, C. Ebinger, J. Biggs, A. Ayele, G. Yirgu, D. Keir, and A. Stork, “Magma-maintained rift segmentation at continental rupture in the 2005 Afar dyking episode,” Nature 442, 291–294 (2006).
[CrossRef] [PubMed]

Proc. IREE Aust.

H. Friis, “Noise figures of radio receivers,” Proc. IREE Aust. 32, 419–422 (1944).
[CrossRef]

Radio Sci.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Rem. Sens. Environ.

S. Wdowinski, S.-W. Kim, F. Amelung, T. Dixon, F. Miralles-Wilhelm, and R. Sonenshein, “Space-based detection of wetlands surface water level changes from L-band SAR interferometry,” Rem. Sens. Environ. 112, 681–696 (2008).
[CrossRef]

Science

R. M. Goldstein, H. Engelhardt, B. Kamb, and R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion—application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

Other

M. I. Skolnik, Introduction to Radar Systems, 2nd ed. (McGraw Hill, 1980).

R. F. Hanssen, Radar Interferometry: Data Interpretation and Error Analysis (Kluwer, 2001).

H. A. Zebker and P. Rosen, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” in Geoscience and Remote Sensing Symposium, 1994. IGARSS ’94. Surface and Atmospheric Remote Sensing: Technologies, Data Analysis and Interpretation, International, vol.  1, (IEEE, 1994), pp. 286–288.

H. Sagan and J. Holbrook, Space-Filling Curves (Springer-Verlag, 1994).
[CrossRef]

C. A. Pickover, The Math Book (Sterling, 2009).

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C (Cambridge University, 1992).

A. Hooper, “Persistent scatter radar interferometry for crustal deformation studies and modeling of volcanic deformation,” Ph.D. dissertation (Stanford University, 2006).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

R. Krämer, “Auf Kalman-Filtern basierende Phasen- und Parameterestimation zur Lösung der Phasenvieldeutigkeitsproblematik bei der Höhenmodellerstellung aus SAR-Interferogrammen,” Ph.D. dissertation (Universitat-GH Siegen, 1998).

E. Moore, “Machine models of self-reproduction,” in Mathematical Problems in the Biological Sciences, R.Bellman, ed. (American Mathematical Society, 1962), pp. 17–33.

M. Kim and H. Griffiths, “Phase unwrapping of multibaseline interferometry using Kalman filtering,” presented at the 7th International Conference on Image Processing and its Applications , Manchester, UK (13–15 July 1999).

J. M. Bioucas-Dias and J. Leitao, “InSAR phase unwrapping: a Bayesian approach,” in Geoscience and Remote Sensing Symposium, 2001 (IEEE, 2001), pp. 396–400.

J. W. Tukey, Exploratory Data Analysis, Behavioral Science: Quantitative Methods (Addison-Wesley, 1977).

E. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).

T. Cover, J. Thomas, and J. Wiley, Elements of Information Theory (Wiley, 1991).
[CrossRef]

B. Frieden, Physics from Fisher Information: a Unification (Cambridge University, 1998).
[CrossRef]

B. Spottiswoode, “2D phase unwrapping algorithms,” http://www.mathworks.com/matlabcentral/fileexchange/22504.

F. Lombardini, F. Gini, and P. Matteucci, “Application of array processing techniques to multibaseline InSAR for layover solution,” in Proceedings of the 2001 IEEE Radar Conference, 2001 (IEEE, 2002), pp. 210–215.

A. Ferretti, A. Monti-Guarnieri, C. Prati, F. Rocca, and D. Massonet, InSAR Principles-Guidelines for SAR Interferometry Processing and Interpretation, (ESA, 2007), Vol.  19, Chap. Part B.

J. W. Eaton, D. Bateman, and S. Hauberg, GNU Octave Manual (Network Theory, 2008).

B. Kampes, “MATLAB toolbox for InSAR,” http://enterprise.lr.tudelft.nl/doris/software/insarmatlab.tar.gz.

B. Kampes and S. Usai, “Doris: the Delft object-oriented radar interferometric software,” presented at the International Symposium on Operationalization of Remote Sensing, Enschede, The Netherlands, 16–20 August 1999.

T. Strozzi and U. Wegmüller, “Land subsidence in Mexico City mapped by ERS differential SAR interferometry,” in Proceedings of the IEEE 1999 International Geoscience and Remote Sensing Symposium (IEEE, 1999), pp. 1940–1942.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Example of a fixed-integration unwrapping path. The dot indicates the start of the path and the arrow indicates the direction. Path is shown as the black stroke. The path can be drawn from start to end without removing the tip of the pen.

Fig. 2
Fig. 2

Example of a region-growth algorithm. In the top row, the gray-scale background indicates a measure of signal quality, where the lighter colors indicate higher quality. The bottom row shows the unwrapping mask, where solved points are marked with S and neighboring points are marked with N. During initialization, a reference point is selected. Pixels in the 4-pixel neighborhood are marked as ready to solve. Starting from the reference point, unwrapping continues with the highest quality neighbors.

Fig. 3
Fig. 3

DEM generated for the synthetic dataset using Eq. (16). The dynamic range of the DEM is about 1450 m . The DEM is 256 × 256 pixels and each pixel has the dimensions 80 m × 80 m .

Fig. 4
Fig. 4

(a)–(b) The DEM phase, (c)–(d) interferogram phase, and (e)–(f) coherence are shown for 100 and 150 m baselines. The DEM phase shows the unwrapped, noise-free interferogram. Note the increased fringe rate in (d) compared to (c), as well as reduced coherence in (f) compared to (e).

Fig. 5
Fig. 5

(a)–(c) The DEM phase, (d)–(f) interferogram phase, (g)–(i) coherence, and (j)–(l) residual phase for the Envisat interferograms 14,337–14,838, 17,343–18,345, and 18,345–19,347 over Mexico City. The DEM phase images are calculated using the SRTM DEM and baseline configurations from the interferograms. Note the reduced coherence in nonurban areas for interferograms 17,343–18,345, and 18,345–19,347 (h)–(i) compared to 14,337–14,838 (g). Also note the residuals are showing inconsistencies between the simulated DEM and interferogram phase (j)–(l).

Fig. 6
Fig. 6

Average of ten solutions of synthetic dataset for 100 and 150 m perpendicular baselines with different Gaussian noise, for the compared paths. The scale is in radians and each figure is normalized with the number shown in the bottom right corner. The white areas in the PDV-BC solutions indicate inaccessible areas by branch cuts. Residual figures show the propagation of errors for each path. Note the small footprint of residuals for FD solutions.

Fig. 7
Fig. 7

Solutions for the Envisat interferograms are shown in the same way as Fig. 6. Except the LS and PDV-BC, all paths return decent solutions for 14,337–14,838. Visual comparison of the results and residuals show that the FD returns reasonable results for interferograms 17,343–18,345 and 18,345–19,347 where others returned large residuals.

Fig. 8
Fig. 8

Misfit values along various unwrapping paths for synthetic interferograms with 100 and 150 m perpendicular baseline. The misfit (y axis) is in the log scale to better show the high range of values. The curvature of the misfit curves indicate how the algorithms perform along the path, such that rapid increase in misfit indicates propagated errors. (a) Note that the MC achieves very low misfit values. (b) Note that the MC starts propagating errors early on and does not recover. Also note that the FD starts propagating the errors the latest.

Fig. 9
Fig. 9

Paths followed by the path algorithms shown as a color-coded map. Blue indicates the beginning of the path and red indicates the end. (a) Note that the unreachable areas in the PDV-BC are larger with the 150 m baseline. Also note the differences between the PDV and FD paths. The PDV algorithm unwraps the areas shown in the white boxes earlier in the 150 m baseline case. The PDV-BC algorithm masks out those areas, and the FD algorithm delays the unwrapping of the areas marked with the white boxes until the end. (b) Note that the paths are quite different for the MC, PDV, PDV-BC, SDR and FD algorithms for different interferograms. This is due to the differences in general coherence level of the interferograms. The LS path changes depending on the different starting points for each interferogram.

Fig. 10
Fig. 10

Misfit values along various unwrapping paths for the Envisat interferograms. The misfit (y axis) is shown in log scale to better represent the high range of values. (a) Note the last rise in the PDV-BC path. This indicates that the PDV-BC successfully left a bad area until the end, and the unwrapping function failed to unwrap that area. The PDV, PDV-BC, SDR and FD have similar misfits, indicating similar paths, with differences at the end. (b) The PDV-BC path ends early on. The LS performs better than the MC, PDV, and SDR. (c) The MC achieves better misfit than the LS, PDV, and SDR. The PDV, PDV-BC, and FD start the same. The FD achieves the lowest misfit.

Fig. 11
Fig. 11

Histograms show the combined phase derivative errors in both directions (azimuth and range) for each unwrapping path. The y range of all the plots are between 0 and 250,000, whereas the x range is between π and π. Red and blue guidelines are drawn to aid comparison. The red line marks the highest peak in the histograms, and the blue line indicates the highest peak of cycle error.

Tables (3)

Tables Icon

Table 1 Baseline Information for the Envisat Interferograms

Tables Icon

Table 2 Misfit and Average Error for Synthetic Data

Tables Icon

Table 3 Misfits for the Envisat Dataset

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

ϕ I = ϕ M ϕ S = 4 π ( R M R S ) λ ,
γ total = γ baseline × γ temporal × γ Doppler × γ volume × γ thermal × γ processing × γ orbit × γ atmosphere .
arg [ X ( e j ω ) ] = 0 ω arg [ X ( e j η ) ] d η arg [ X ( e j 0 ) ] = 0 ,
ϕ ( m ) = ϕ ( 0 ) + n = 1 m W 2 Δ W 1 [ ϕ ( n ) ] ,
r ( i , k ) = × ϕ ( i , k ) = W δ i ϕ ( i , k ) + W δ k ϕ ( i + 1 , k ) W δ i ϕ ( i , k + 1 ) W δ k ϕ ( i , k ) ,
ϕ ^ 0 = k = 1 n arg ( ϕ 0 ) + arg ( ϕ k × conj ( ϕ 0 ) ) n ,
ϕ ^ 0 = k = 1 n Q ( k ) × [ arg ( ϕ 0 ) + arg ( ϕ k × conj ( ϕ 0 ) ) ] k = 1 n Q ( k ) ,
χ 2 = 1 N × k = 0 N [ ϕ ^ ( k ) ϕ ( k ) ] 2 σ 2 ( k ) ,
F = k = 0 N k * Q ( k )
γ = E [ M S * ] E [ M M * ] E [ S S * ] ,
γ ^ = 1 N i = 0 N M i S i * 1 N i = 0 N M i M i * 1 N i = 0 N S i S i * ,
PDV k = n = 0 N σ x 2 ( n ) + σ y 2 ( n ) ,
I ( ϕ ) = E { [ ϕ log L ( ϕ ; X ) ] 2 | ϕ } ,
I 0 , 1 ( ϕ ) = ( ϕ 1 ϕ 0 ) 2 2 σ 0 2 + log 2 π σ 0 2 ,
I | 01 | = 0.5 × ( I 1 , 0 + I 0 , 1 ) .
DEM ( x , y ) = 3 × ( 1 x ) 2 × e x 2 ( y + 1 ) 2 10 × ( x / 5 x 3 y 5 ) × e x 2 y 2 1 / 3 × e ( x + 1 ) 2 y 2 ,
γ = γ baseline × γ noise .
F = F 1 + F 2 1 G 1 + F 3 1 G 1 G 2 + + F K 1 G 1 G 2 G K 1 ,
F K = SNR IN SNR OUT N ,

Metrics