Abstract

The problem of phase unwrapping in two dimensions has been studied extensively in the past two decades, but the three-dimensional (3D) problem has so far received relatively little attention. We develop here a theoretical framework for 3D phase unwrapping and also describe two algorithms for implementation, both of which can be applied to synthetic aperture radar interferometry (InSAR) time series. We test the algorithms on simulated data and find both give more accurate results than a two-dimensional algorithm. When applied to actual InSAR time series, we find good agreement both between the algorithms and with ground truth.

© 2007 Optical Society of America

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  1. R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry: two-dimensional phase unwrapping," Radio Sci. 23, 713-720 (1988).
    [CrossRef]
  2. D. C. Ghiglia and L. A. Romero, "Minimum Lp-norm two-dimensional phase unwrapping," J. Opt. Soc. Am. A 13, 1999-2013 (1996).
    [CrossRef]
  3. M. Costantini, "A novel phase unwrapping method based on network programming," IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
    [CrossRef]
  4. H. A. Zebker and Y. P. Lu, "Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms, J. Opt. Soc. Am. A 15, 586-598 (1998).
    [CrossRef]
  5. 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]
  6. A. Ferretti, C. Prati, and F. Rocca, "Permanent scatterers in SAR interferometry," IEEE Trans. Geosci. Remote Sens. 39, 8-20 (2001).
    [CrossRef]
  7. A. Hooper, H. Zebker, P. Segall, and B. Kampes, "A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers," Geophys. Res. Lett. 31, 611-615 (2004).
    [CrossRef]
  8. A. Hooper, P. Segall, and H. Zebker, "Persistent scatterer InSAR for crustal deformation analysis, with application to Volcán Alcedo, Galápagos," J. Geophys. Res. 112, B07407 (2007).
    [CrossRef]
  9. B. M. Kampes, "Displacement parameter estimation using permanent scatterer interferometry," Ph.D. dissertation (Delft University of Technology, 2005).
  10. 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]
  11. D. A. Schmidt and R. Bürgmann, "Time-dependent land uplift and subsidence in the Santa Clara valley, California, from a large interferometric synthetic aperture radar data set," J. Geophys. Res. 108, 2416-2428 (2003).
    [CrossRef]
  12. J. M. Huntley, "Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901-3908 (2001).
    [CrossRef]
  13. R. Cusack and N. Papadakis, "New robust 3D phase unwrapping algorithm: Application to magnetic field mapping and undistorting echo-planar images," Neuroimage 16, 754-764 (2002).
    [CrossRef] [PubMed]
  14. O. Marklund, J. M. Huntley, and R. Cusack, "Robust unwrapping algorithm for 3-D phase volumes of arbitrary shape containing knotted phase singularity loops," research report (Luleå University of Technology, 2005).
  15. M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. Beauregard, "Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping," Appl. Opt. 45, 2711-2722 (2006).
    [CrossRef] [PubMed]
  16. M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, "Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc., Interface 3, 415-427 (2006).
    [CrossRef]
  17. M. Jenkinson, "Fast, automated N-dimensional phase-unwrapping algorithm," Magn. Reson. Med. 49, 193-197 (2003).
    [CrossRef] [PubMed]
  18. 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]
  19. C. W. Chen, "Statistical-cost network-flow approaches to two-dimensional phase unwrapping for radar interferometry," Ph.D. dissertation (Stanford University, 2001).
  20. R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
    [CrossRef]
  21. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).
  22. M. Costantini and P. A. Rosen, "Generalized phase unwrapping approach for sparse data," in Proceedings of the IEEE 1999 International Geoscience and Remote Sensing Symposium (IGARSS) (IEEE1999), Vol. 1, pp. 267-269.
  23. K. Brakke, "The surface evolver," Exp. Math. 1, 141-165 (1992).
  24. R. F. Hanssen, Radar Interferometry Data Interpretation and Error Analysis (Springer, 2001).
  25. R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
    [CrossRef]
  26. K. Mogi, "Relations between the eruptions of various volcanoes and the deformations of the ground surfaces around them," Bull. Earthquake Res. Inst., Univ. Tokyo 36, 111-123 (1958).
  27. M. Battaglia, P. Segall, J. Murray, P. Cervell, and J. Langbein, "The mechanics of unrest at Long Valley caldera, California: 1. Modeling the geometry of the source using GPS, leveling and two-color EDM data," J. Volcanol. Geotherm. Res. 127, 195-217 (2003).
    [CrossRef]
  28. Y. Fialko, M. Simons, and Y. Khazan, "Finite source modelling of magmatic unrest in Socorro, New Mexico, and Long Valley, California," Geophys. J. Int. 146, 191-200 (2001).
    [CrossRef]

2007

A. Hooper, P. Segall, and H. Zebker, "Persistent scatterer InSAR for crustal deformation analysis, with application to Volcán Alcedo, Galápagos," J. Geophys. Res. 112, B07407 (2007).
[CrossRef]

2006

M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. Beauregard, "Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping," Appl. Opt. 45, 2711-2722 (2006).
[CrossRef] [PubMed]

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, "Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc., Interface 3, 415-427 (2006).
[CrossRef]

2004

A. Hooper, H. Zebker, P. Segall, and B. Kampes, "A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers," Geophys. Res. Lett. 31, 611-615 (2004).
[CrossRef]

2003

D. A. Schmidt and R. Bürgmann, "Time-dependent land uplift and subsidence in the Santa Clara valley, California, from a large interferometric synthetic aperture radar data set," J. Geophys. Res. 108, 2416-2428 (2003).
[CrossRef]

M. Jenkinson, "Fast, automated N-dimensional phase-unwrapping algorithm," Magn. Reson. Med. 49, 193-197 (2003).
[CrossRef] [PubMed]

M. Battaglia, P. Segall, J. Murray, P. Cervell, and J. Langbein, "The mechanics of unrest at Long Valley caldera, California: 1. Modeling the geometry of the source using GPS, leveling and two-color EDM data," J. Volcanol. Geotherm. Res. 127, 195-217 (2003).
[CrossRef]

2002

R. Cusack and N. Papadakis, "New robust 3D phase unwrapping algorithm: Application to magnetic field mapping and undistorting echo-planar images," Neuroimage 16, 754-764 (2002).
[CrossRef] [PubMed]

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]

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]

J. M. Huntley, "Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901-3908 (2001).
[CrossRef]

Y. Fialko, M. Simons, and Y. Khazan, "Finite source modelling of magmatic unrest in Socorro, New Mexico, and Long Valley, California," Geophys. J. Int. 146, 191-200 (2001).
[CrossRef]

2000

1998

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

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

H. A. Zebker and Y. P. Lu, "Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms, J. Opt. Soc. Am. A 15, 586-598 (1998).
[CrossRef]

R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
[CrossRef]

1996

1992

K. Brakke, "The surface evolver," Exp. Math. 1, 141-165 (1992).

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]

1958

K. Mogi, "Relations between the eruptions of various volcanoes and the deformations of the ground surfaces around them," Bull. Earthquake Res. Inst., Univ. Tokyo 36, 111-123 (1958).

Adam, N.

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

Bamler, R.

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

Battaglia, M.

M. Battaglia, P. Segall, J. Murray, P. Cervell, and J. Langbein, "The mechanics of unrest at Long Valley caldera, California: 1. Modeling the geometry of the source using GPS, leveling and two-color EDM data," J. Volcanol. Geotherm. Res. 127, 195-217 (2003).
[CrossRef]

Beauregard, D. A.

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, "Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc., Interface 3, 415-427 (2006).
[CrossRef]

M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. Beauregard, "Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping," Appl. Opt. 45, 2711-2722 (2006).
[CrossRef] [PubMed]

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]

Brakke, K.

K. Brakke, "The surface evolver," Exp. Math. 1, 141-165 (1992).

Bürgmann, R.

D. A. Schmidt and R. Bürgmann, "Time-dependent land uplift and subsidence in the Santa Clara valley, California, from a large interferometric synthetic aperture radar data set," J. Geophys. Res. 108, 2416-2428 (2003).
[CrossRef]

Cervell, P.

M. Battaglia, P. Segall, J. Murray, P. Cervell, and J. Langbein, "The mechanics of unrest at Long Valley caldera, California: 1. Modeling the geometry of the source using GPS, leveling and two-color EDM data," J. Volcanol. Geotherm. Res. 127, 195-217 (2003).
[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]

M. Costantini and P. A. Rosen, "Generalized phase unwrapping approach for sparse data," in Proceedings of the IEEE 1999 International Geoscience and Remote Sensing Symposium (IGARSS) (IEEE1999), Vol. 1, pp. 267-269.

Cusack, R.

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, "Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc., Interface 3, 415-427 (2006).
[CrossRef]

M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. Beauregard, "Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping," Appl. Opt. 45, 2711-2722 (2006).
[CrossRef] [PubMed]

R. Cusack and N. Papadakis, "New robust 3D phase unwrapping algorithm: Application to magnetic field mapping and undistorting echo-planar images," Neuroimage 16, 754-764 (2002).
[CrossRef] [PubMed]

O. Marklund, J. M. Huntley, and R. Cusack, "Robust unwrapping algorithm for 3-D phase volumes of arbitrary shape containing knotted phase singularity loops," research report (Luleå University of Technology, 2005).

Davidson, G. W.

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

Ferretti, A.

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

Fialko, Y.

Y. Fialko, M. Simons, and Y. Khazan, "Finite source modelling of magmatic unrest in Socorro, New Mexico, and Long Valley, California," Geophys. J. Int. 146, 191-200 (2001).
[CrossRef]

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]

Ghiglia, D. C.

D. C. Ghiglia and L. A. Romero, "Minimum Lp-norm two-dimensional phase unwrapping," J. Opt. Soc. Am. A 13, 1999-2013 (1996).
[CrossRef]

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

Goldstein, R. M.

R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
[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]

Graves, M. J.

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, "Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc., Interface 3, 415-427 (2006).
[CrossRef]

M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. Beauregard, "Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping," Appl. Opt. 45, 2711-2722 (2006).
[CrossRef] [PubMed]

Hanssen, R. F.

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

Hooper, A.

A. Hooper, P. Segall, and H. Zebker, "Persistent scatterer InSAR for crustal deformation analysis, with application to Volcán Alcedo, Galápagos," J. Geophys. Res. 112, B07407 (2007).
[CrossRef]

A. Hooper, H. Zebker, P. Segall, and B. Kampes, "A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers," Geophys. Res. Lett. 31, 611-615 (2004).
[CrossRef]

Huntley, J. M.

M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. Beauregard, "Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping," Appl. Opt. 45, 2711-2722 (2006).
[CrossRef] [PubMed]

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, "Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc., Interface 3, 415-427 (2006).
[CrossRef]

J. M. Huntley, "Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901-3908 (2001).
[CrossRef]

O. Marklund, J. M. Huntley, and R. Cusack, "Robust unwrapping algorithm for 3-D phase volumes of arbitrary shape containing knotted phase singularity loops," research report (Luleå University of Technology, 2005).

Jenkinson, M.

M. Jenkinson, "Fast, automated N-dimensional phase-unwrapping algorithm," Magn. Reson. Med. 49, 193-197 (2003).
[CrossRef] [PubMed]

Just, D.

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

Kampes, B.

A. Hooper, H. Zebker, P. Segall, and B. Kampes, "A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers," Geophys. Res. Lett. 31, 611-615 (2004).
[CrossRef]

Kampes, B. M.

B. M. Kampes, "Displacement parameter estimation using permanent scatterer interferometry," Ph.D. dissertation (Delft University of Technology, 2005).

Khazan, Y.

Y. Fialko, M. Simons, and Y. Khazan, "Finite source modelling of magmatic unrest in Socorro, New Mexico, and Long Valley, California," Geophys. J. Int. 146, 191-200 (2001).
[CrossRef]

Lanari, R.

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]

Langbein, J.

M. Battaglia, P. Segall, J. Murray, P. Cervell, and J. Langbein, "The mechanics of unrest at Long Valley caldera, California: 1. Modeling the geometry of the source using GPS, leveling and two-color EDM data," J. Volcanol. Geotherm. Res. 127, 195-217 (2003).
[CrossRef]

Lu, Y. P.

Marklund, O.

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, "Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc., Interface 3, 415-427 (2006).
[CrossRef]

O. Marklund, J. M. Huntley, and R. Cusack, "Robust unwrapping algorithm for 3-D phase volumes of arbitrary shape containing knotted phase singularity loops," research report (Luleå University of Technology, 2005).

Mogi, K.

K. Mogi, "Relations between the eruptions of various volcanoes and the deformations of the ground surfaces around them," Bull. Earthquake Res. Inst., Univ. Tokyo 36, 111-123 (1958).

Murray, J.

M. Battaglia, P. Segall, J. Murray, P. Cervell, and J. Langbein, "The mechanics of unrest at Long Valley caldera, California: 1. Modeling the geometry of the source using GPS, leveling and two-color EDM data," J. Volcanol. Geotherm. Res. 127, 195-217 (2003).
[CrossRef]

Papadakis, N.

R. Cusack and N. Papadakis, "New robust 3D phase unwrapping algorithm: Application to magnetic field mapping and undistorting echo-planar images," Neuroimage 16, 754-764 (2002).
[CrossRef] [PubMed]

Prati, C.

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

Pritt, M. D.

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

Rocca, F.

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

Romero, L. A.

Rosen, P. A.

M. Costantini and P. A. Rosen, "Generalized phase unwrapping approach for sparse data," in Proceedings of the IEEE 1999 International Geoscience and Remote Sensing Symposium (IGARSS) (IEEE1999), Vol. 1, pp. 267-269.

Ruiz, P. D.

Salfity, M. F.

M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. Beauregard, "Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping," Appl. Opt. 45, 2711-2722 (2006).
[CrossRef] [PubMed]

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, "Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc., Interface 3, 415-427 (2006).
[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]

Schmidt, D. A.

D. A. Schmidt and R. Bürgmann, "Time-dependent land uplift and subsidence in the Santa Clara valley, California, from a large interferometric synthetic aperture radar data set," J. Geophys. Res. 108, 2416-2428 (2003).
[CrossRef]

Segall, P.

A. Hooper, P. Segall, and H. Zebker, "Persistent scatterer InSAR for crustal deformation analysis, with application to Volcán Alcedo, Galápagos," J. Geophys. Res. 112, B07407 (2007).
[CrossRef]

A. Hooper, H. Zebker, P. Segall, and B. Kampes, "A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers," Geophys. Res. Lett. 31, 611-615 (2004).
[CrossRef]

M. Battaglia, P. Segall, J. Murray, P. Cervell, and J. Langbein, "The mechanics of unrest at Long Valley caldera, California: 1. Modeling the geometry of the source using GPS, leveling and two-color EDM data," J. Volcanol. Geotherm. Res. 127, 195-217 (2003).
[CrossRef]

Simons, M.

Y. Fialko, M. Simons, and Y. Khazan, "Finite source modelling of magmatic unrest in Socorro, New Mexico, and Long Valley, California," Geophys. J. Int. 146, 191-200 (2001).
[CrossRef]

Werner, C. L.

R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
[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]

Zebker, H.

A. Hooper, P. Segall, and H. Zebker, "Persistent scatterer InSAR for crustal deformation analysis, with application to Volcán Alcedo, Galápagos," J. Geophys. Res. 112, B07407 (2007).
[CrossRef]

A. Hooper, H. Zebker, P. Segall, and B. Kampes, "A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers," Geophys. Res. Lett. 31, 611-615 (2004).
[CrossRef]

Zebker, H. A.

Appl. Opt.

Bull. Earthquake Res. Inst., Univ. Tokyo

K. Mogi, "Relations between the eruptions of various volcanoes and the deformations of the ground surfaces around them," Bull. Earthquake Res. Inst., Univ. Tokyo 36, 111-123 (1958).

Exp. Math.

K. Brakke, "The surface evolver," Exp. Math. 1, 141-165 (1992).

Geophys. J. Int.

Y. Fialko, M. Simons, and Y. Khazan, "Finite source modelling of magmatic unrest in Socorro, New Mexico, and Long Valley, California," Geophys. J. Int. 146, 191-200 (2001).
[CrossRef]

Geophys. Res. Lett.

R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
[CrossRef]

A. Hooper, H. Zebker, P. Segall, and B. Kampes, "A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers," Geophys. Res. Lett. 31, 611-615 (2004).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

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

R. Bamler, N. Adam, G. W. Davidson, and D. Just, "Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry," IEEE Trans. Geosci. Remote Sens. 36, 913-921 (1998).
[CrossRef]

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

J. Geophys. Res.

D. A. Schmidt and R. Bürgmann, "Time-dependent land uplift and subsidence in the Santa Clara valley, California, from a large interferometric synthetic aperture radar data set," J. Geophys. Res. 108, 2416-2428 (2003).
[CrossRef]

A. Hooper, P. Segall, and H. Zebker, "Persistent scatterer InSAR for crustal deformation analysis, with application to Volcán Alcedo, Galápagos," J. Geophys. Res. 112, B07407 (2007).
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J. R. Soc., Interface

M. F. Salfity, J. M. Huntley, M. J. Graves, O. Marklund, R. Cusack, and D. A. Beauregard, "Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms," J. R. Soc., Interface 3, 415-427 (2006).
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J. Volcanol. Geotherm. Res.

M. Battaglia, P. Segall, J. Murray, P. Cervell, and J. Langbein, "The mechanics of unrest at Long Valley caldera, California: 1. Modeling the geometry of the source using GPS, leveling and two-color EDM data," J. Volcanol. Geotherm. Res. 127, 195-217 (2003).
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M. Jenkinson, "Fast, automated N-dimensional phase-unwrapping algorithm," Magn. Reson. Med. 49, 193-197 (2003).
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Figures (15)

Fig. 1
Fig. 1

Simple phase discontinuity surface intersecting a 2D data set. The surface is bounded by a residue loop. Where the surface intersects the 2D data set results in a discontinuity line, which is bounded by a positive and a negative residue, arising where the residue loop intersects the data set.

Fig. 2
Fig. 2

Single-cycle branch in (a) 2D and (b) 3D. The numbers in (a) represent unwrapped phase values in cycles, the lines represent single-cycle discontinuities, and the dot represents a residue at the branch point. In (b), the surfaces represent single-cycle discontinuity surfaces and the lines represent residue loops, with the dashed part being the branch line.

Fig. 3
Fig. 3

Multiple-cycle discontinuity surface. The loops represent residue loops, the right loop bounding the edge of the surface and the left loop separating the 2 π phase discontinuity region of the surface from the 4 π discontinuity region. Note that an L -norm solution would place a minimal surface within each residue loop.

Fig. 4
Fig. 4

Residue identification and connection. Each apex represents a data point and the connecting lines represent the arcs along which phase differences are calculated. Residues, indicated by a + or −, are identified by integrating arc phase differences around each triangular or rectangular face. (a) shows an element with residues on two faces. A residue loop enters the element through one face and exits through the other. (b) shows an element with residues on four faces. The four residues could be linked to form either two separate loops (as shown) or one twisted figure-eight loop.

Fig. 5
Fig. 5

Residue loop that is truncated at the data volume boundaries. The continuous parts of the loop are contained within the data volume, while the dashed parts lie outside. The loop is truncated four times, once at the top boundary and three times at the right boundary.

Fig. 6
Fig. 6

Relationship between the length of the arcs connecting data points, L a r c , and the fraction of arcs bounding residues, N r e s N a r c . These data are from the Lost Hills example (Subsection 5B). The model we fit assumes a logarithmic relationship.

Fig. 7
Fig. 7

Optimizing the connection of truncated residue loops. In (a), the connection between truncated loop ends A and B is broken. In (b) the connection between end C and D is broken, and A is reconnected to C. The probability of proceeding from (a) to (b) is given by P = A B ( A B + A C ) where A B is the distance between A and B, and A C is the distance between A and C.

Fig. 8
Fig. 8

Simulated change in volume of a center of dilation that is used to calculate the deformation phase contribution to the simulated phase data in Fig. 9.

Fig. 9
Fig. 9

Simulated phase data in radians, (a) unwrapped and (b) wrapped, including deformation phase, atmospheric phase delay, and noise. The unwrapped phase is referenced to the top-right corner to enable comparison with Fig. 10.

Fig. 10
Fig. 10

Simulated phase data in radians unwrapped using (a) the quasi- L 3D algorithm, (b) the stepwise 3D algorithm, and (c) an iterative least-squares 2D algorithm. In all cases the phase is referenced to the top-right corner.

Fig. 11
Fig. 11

Difference in cycles between the simulated unwrapped phase and that estimated by the three algorithms, (a) the quasi- L 3D algorithm, (b) the stepwise 3D algorithm, and (c) an iterative least-squares 2D algorithm.

Fig. 12
Fig. 12

Comparison of unwrapping accuracy for the simulated data for the three different algorithms.

Fig. 13
Fig. 13

Lost Hills region persistent scatterer interferograms in radians, (a) wrapped phase, (b) phase unwrapped using the quasi- L -norm 3D algorithm, and (c) phase unwrapped using the stepwise 3D algorithm. The date of the master acquisition is February 22, 2003 and only every ninth interferogram is shown. The background image in gray is the mean SAR amplitude of all 28 passes and the points represent PS pixels, with color indicating the relative phase difference with respect to the northeast corner.

Fig. 14
Fig. 14

Long Valley persistent scatterer interferograms in radians, (a) wrapped phase and (b) phase unwrapped using the stepwise 3D algorithm. The date of the master acquisition is June 22, 1997 and only every seventh interferogram is shown. The background image in gray is topography in shaded relief and the points represent PS pixels, with color indicating the relative phase difference with respect to the northwest corner.

Fig. 15
Fig. 15

Comparison of relative vertical motion between benchmarks 23EG and G916 (see Fig. 14 for locations) from leveling and GPS, to that calculated from unwrapped PS phase, (a) assuming phase changes are due to vertical displacement only and (b) assuming there is also a horizontal component of displacement proportional to the vertical. The error bars represent 68% confidence bounds. Also shown is the scaled line length change between CASA and KRAK as measured by EDM, which is a proxy for vertical deformation.

Equations (5)

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i , j w i , j ( x ) Δ ϕ i , j ( x ) Δ ψ i , j ( x ) p + i , j w i , j ( y ) Δ ϕ i , j ( y ) Δ ψ i , j ( y ) p ,
i , j , k w i , j , k ( x ) Δ ϕ i , j , k ( x ) Δ ψ i , j , k ( x ) p + i , j , k w i , j , k ( y ) Δ ϕ i , j , k ( y ) Δ ψ i , j , k ( y ) p + i , j , k w i , j , k ( z ) Δ ϕ i , j , k ( z ) Δ ψ i , j , k ( z ) p ,
i , j w i , j ( s ) Δ ϕ i , j ( s ) Δ ψ i , j ( s ) p + i , j w i , j ( t ) Δ ϕ i , j ( t ) Δ ψ i , j ( t ) p ,
N r e s N edge = k log L edge + c ,
P = A B A B + A C ,

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