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doi: 10.3934/ipi.2021043
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A mathematical perspective on radar interferometry

 Department of Mathematics, North Carolina State University, Campus Box 8205, Raleigh, NC 27695, USA

* Corresponding author: Mikhail Gilman

Received  July 2020 Revised  April 2021 Early access July 2021

Radar interferometry is an advanced remote sensing technology that utilizes complex phases of two or more radar images of the same target taken at slightly different imaging conditions and/or different times. Its goal is to derive additional information about the target, such as elevation. While this kind of task requires centimeter-level accuracy, the interaction of radar signals with the target, as well as the lack of precision in antenna position and other disturbances, generate ambiguities in the image phase that are orders of magnitude larger than the effect of interest.

Yet the common exposition of radar interferometry in the literature often skips such topics. This may lead to unrealistic requirements for the accuracy of determining the parameters of imaging geometry, unachievable precision of image co-registration, etc. To address these deficiencies, in the current work we analyze the problem of interferometric height reconstruction and provide a careful and detailed account of all the assumptions and requirements to the imaging geometry and data processing needed for a successful extraction of height information from the radar data. We employ two most popular scattering models for radar targets: an isolated point scatterer and delta-correlated extended scatterer, and highlight the similarities and differences between them.

Citation: Mikhail Gilman, Semyon Tsynkov. A mathematical perspective on radar interferometry. Inverse Problems & Imaging, doi: 10.3934/ipi.2021043
References:
 [1] R. Bamler and P. Hartl, Synthetic aperture radar interferometry, Inverse Problems, 14 (1998), R1–R54. doi: 10.1088/0266-5611/14/4/001.  Google Scholar [2] F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces, Translated and Edited by Carol B. Vesecky and John F. Vesecky. International Series in Natural Philosophy, Pergamon Press, Oxford-New York, 1979.   Google Scholar [3] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon Press, New York, 1963.   Google Scholar [4] F. Bovenga, D. Derauw, F. M. Rana, C. Barbier, A. Refice, N. Veneziani and R. Vitulli, Multi-chromatic analysis of SAR images for coherent target detection, Remote Sensing, 6 (2014), 8822-8843.   Google Scholar [5] G. Brigot, M. Simard, E. Colin-Koeniguer and A. Boulch, Retrieval of forest vertical structure from PolInSAR data by machine learning using LIDAR-derived features, Remote Sensing, 11 (2019), 381. Google Scholar [6] M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), 301-312.  doi: 10.1137/S0036144500368859.  Google Scholar [7] M. Cheney and B. Borden, Fundamentals of radar imaging, CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 79 (2009).  Google Scholar [8] S. Cloude, Polarisation: Applications in Remote Sensing, Oxford University Press, 2010.   Google Scholar [9] M. Crosetto, O. Monserrat, M. Cuevas-González, N. Devanthéry and B. Crippa, Persistent scatterer interferometry: A review, ISPRS Journal of Photogrammetry and Remote Sensing, 115 (2016), 78-89.   Google Scholar [10] I. G. Cumming and F. H. Wong, Digital Processing of Synthetic Aperture Radar Data. Algorithms and Implementation, Artech House, Boston, 2005. Google Scholar [11] L. J. Cutrona, Synthetic Aperture Radar, 2$^{nd}$ edition, M. Skolnik, editor, Radar Handbook, 21, McGraw-Hill, New-York, 1990. Google Scholar [12] D. Derauw, A. Orban and C. Barbier, Wide band SAR sub-band splitting and inter-band coherence measurements, Remote Sensing Letters, 1 (2010), 133-140.   Google Scholar [13] M. Eineder, C. Minet, P. Steigenberger, X. Cong and T. Fritz, Imaging geodesy — toward centimeter-level ranging accuracy with TerraSAR-X, IEEE Transactions on Geoscience and Remote Sensing, 49 (2011), 661-671.   Google Scholar [14] A. Ferretti, C. Prati and F. Rocca, Permanent scatterers in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 39 (2001), 8-20.   Google Scholar [15] H. Foroosh, J. B. Zerubia and M. Berthod, Extension of phase correlation to subpixel registration, IEEE Transactions on Image Processing, 11 (2002), 188-200.   Google Scholar [16] G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing, Electronic Engineering Systems Series. CRC Press, Boca Raton, FL, 1999.   Google Scholar [17] G. Franceschetti and D. Riccio, Scattering, Natural Surfaces, and Fractals, Elsevier, 2006. Google Scholar [18] F. Gatelli, A. Monti Guamieri, F. Parizzi, P. Pasquali, C. Prati and F. Rocca, The wavenumber shift in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 32 (1994), 855-865.   Google Scholar [19] M. Gilman, E. Smith and S. Tsynkov, Transionospheric Synthetic Aperture Imaging, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Cham, Switzerland, 2017.  Google Scholar [20] M. Gilman and S. Tsynkov, A mathematical model for SAR imaging beyond the first Born approximation, SIAM J. Imaging Sci., 8 (2015), 186-225.  doi: 10.1137/140973025.  Google Scholar [21] R. M. Goldstein, H. A. Zebker and C. L. Werner, Satellite radar interferometry: Two-dimensional phase unwrapping, Radio science, 23 (1988), 713-720.   Google Scholar [22] J. W. Goodman, Statistical properties of laser speckle patterns, in Laser Speckle and Related Phenomena, (1984), 9–75. Google Scholar [23] M. Guizar-Sicairos, S. T. Thurman and J. R. Fienup, Efficient subpixel image registration algorithms, Opt. Lett., 33 (2008), 156-158.   Google Scholar [24] R. F. Hanssen, Radar Interferometry: Data Interpretation and Error Analysis Remote Sensing and Digital Image Processing, Kluwer Academic Publishers, New York, 2001. Google Scholar [25] E. W. Hoen and H. A. Zebker, Penetration depths inferred from interferometric volume decorrelation observed over the Greenland ice sheet, IEEE Transactions on Geoscience and Remote Sensing, 38 (2000), 2571-2583.   Google Scholar [26] J. A. Jackson, B. D. Rigling and R. L. Moses, Canonical scattering feature models for 3D and bistatic SAR, IEEE Transactions on Aerospace and Electronic Systems, 46 (2010), 525-541.   Google Scholar [27] C. V. Jakowatz, Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia and P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach, Springer, 1996. Google Scholar [28] D. Just and R. Bamler, Phase statistics of interferograms with applications to synthetic aperture radar, Applied Optics, 33 (1994), 4361-4368.   Google Scholar [29] J.-S. Lee and E. Pottier, Polarimetric Radar Imaging from Basics to Applications, Optical Science and Engineering. CRC Press, Boca Raton, 2009.   Google Scholar [30] G. D. Martino, A. Iodice, D. Riccio and G. Ruello, Equivalent number of scatterers for SAR speckle modeling, IEEE Transactions on Geoscience and Remote Sensing, 52 (2014), 2555-2564.   Google Scholar [31] D. Massonnet and J. -Claude Souyris, Imaging with Synthetic Aperture Radar, Engineering Sciences: Electrical Engineering. EFPL Press. Distributed by CRC Press, Lausanne, Switzerland, 2008.   Google Scholar [32] A. Moreira, P. Prats-Iraola, M. Younis, G. Krieger, I. Hajnsek and K. P. Papathanassiou, A tutorial on synthetic aperture radar, IEEE Geoscience and Remote Sensing Magazine, 1 (2013), 6-43.   Google Scholar [33] C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images, Artech House Remote Sensing Library. Artech House, Boston, 1998. Google Scholar [34] P. A. Rosen, S. Hensley, I. R. Joughin, F. K. Li, S. N. Madsen, E. Rodriguez and R. M. Goldstein, Synthetic aperture radar interferometry, Proceedings of the IEEE, 88 (2000), 333-382.   Google Scholar [35] H. S. Stone, M. Orchard, E. Chang and S. Martucci, A fast direct Fourier-based algorithm for subpixel registration of images, IEEE Transactions on Geoscience and Remote Sensing, 39 (2001), 2235-2243.   Google Scholar [36] Q. Tian and M. N. Huhns, Algorithms for subpixel registration, Computer Vision, Graphics, and Image Processing, 35 (1986), 220-233.   Google Scholar [37] A. Voronovich, Wave Scattering from Rough Surfaces, Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1999.  Google Scholar [38] M. Wermuth, A. Hauschild, O. Montenbruck and R. Kahle, TerraSAR-Xprecise orbit determination with real-time GPS ephemerides, Advances in Space Research, 50 (2012), 549-559.   Google Scholar [39] B. Yazıcı, I. Son and H. Cagri Yanik, Doppler synthetic aperture radar interferometry: A novel SAR interferometry for height mapping using ultra-narrowband waveforms, Inverse Problems, 34 (2018), 055003. doi: 10.1088/1361-6420/aab24c.  Google Scholar [40] B. Yonel, I. Son and B. Yazici, Exact multistatic interferometric imaging via generalized ıirtinger flow, IEEE Transactions on Computational Imaging, 6 (2020), 711-726.   Google Scholar [41] Y. T. Yoon, M. Eineder, N. Yague-Martinez and O. Montenbruck, TerraSAR-Xprecise trajectory estimation and quality assessment, IEEE Transactions on Geoscience and Remote Sensing, 47 (2009), 1859-1868.   Google Scholar [42] H. A. Zebker and J. Villasenor, Decorrelation in interferometric radar echoes, IEEE Transactions on Geoscience and Remote Sensing, 30 (1992), 950-959.   Google Scholar

show all references

References:
 [1] R. Bamler and P. Hartl, Synthetic aperture radar interferometry, Inverse Problems, 14 (1998), R1–R54. doi: 10.1088/0266-5611/14/4/001.  Google Scholar [2] F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces, Translated and Edited by Carol B. Vesecky and John F. Vesecky. International Series in Natural Philosophy, Pergamon Press, Oxford-New York, 1979.   Google Scholar [3] P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon Press, New York, 1963.   Google Scholar [4] F. Bovenga, D. Derauw, F. M. Rana, C. Barbier, A. Refice, N. Veneziani and R. Vitulli, Multi-chromatic analysis of SAR images for coherent target detection, Remote Sensing, 6 (2014), 8822-8843.   Google Scholar [5] G. Brigot, M. Simard, E. Colin-Koeniguer and A. Boulch, Retrieval of forest vertical structure from PolInSAR data by machine learning using LIDAR-derived features, Remote Sensing, 11 (2019), 381. Google Scholar [6] M. Cheney, A mathematical tutorial on synthetic aperture radar, SIAM Rev., 43 (2001), 301-312.  doi: 10.1137/S0036144500368859.  Google Scholar [7] M. Cheney and B. Borden, Fundamentals of radar imaging, CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 79 (2009).  Google Scholar [8] S. Cloude, Polarisation: Applications in Remote Sensing, Oxford University Press, 2010.   Google Scholar [9] M. Crosetto, O. Monserrat, M. Cuevas-González, N. Devanthéry and B. Crippa, Persistent scatterer interferometry: A review, ISPRS Journal of Photogrammetry and Remote Sensing, 115 (2016), 78-89.   Google Scholar [10] I. G. Cumming and F. H. Wong, Digital Processing of Synthetic Aperture Radar Data. Algorithms and Implementation, Artech House, Boston, 2005. Google Scholar [11] L. J. Cutrona, Synthetic Aperture Radar, 2$^{nd}$ edition, M. Skolnik, editor, Radar Handbook, 21, McGraw-Hill, New-York, 1990. Google Scholar [12] D. Derauw, A. Orban and C. Barbier, Wide band SAR sub-band splitting and inter-band coherence measurements, Remote Sensing Letters, 1 (2010), 133-140.   Google Scholar [13] M. Eineder, C. Minet, P. Steigenberger, X. Cong and T. Fritz, Imaging geodesy — toward centimeter-level ranging accuracy with TerraSAR-X, IEEE Transactions on Geoscience and Remote Sensing, 49 (2011), 661-671.   Google Scholar [14] A. Ferretti, C. Prati and F. Rocca, Permanent scatterers in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 39 (2001), 8-20.   Google Scholar [15] H. Foroosh, J. B. Zerubia and M. Berthod, Extension of phase correlation to subpixel registration, IEEE Transactions on Image Processing, 11 (2002), 188-200.   Google Scholar [16] G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing, Electronic Engineering Systems Series. CRC Press, Boca Raton, FL, 1999.   Google Scholar [17] G. Franceschetti and D. Riccio, Scattering, Natural Surfaces, and Fractals, Elsevier, 2006. Google Scholar [18] F. Gatelli, A. Monti Guamieri, F. Parizzi, P. Pasquali, C. Prati and F. Rocca, The wavenumber shift in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 32 (1994), 855-865.   Google Scholar [19] M. Gilman, E. Smith and S. Tsynkov, Transionospheric Synthetic Aperture Imaging, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Cham, Switzerland, 2017.  Google Scholar [20] M. Gilman and S. Tsynkov, A mathematical model for SAR imaging beyond the first Born approximation, SIAM J. Imaging Sci., 8 (2015), 186-225.  doi: 10.1137/140973025.  Google Scholar [21] R. M. Goldstein, H. A. Zebker and C. L. Werner, Satellite radar interferometry: Two-dimensional phase unwrapping, Radio science, 23 (1988), 713-720.   Google Scholar [22] J. W. Goodman, Statistical properties of laser speckle patterns, in Laser Speckle and Related Phenomena, (1984), 9–75. Google Scholar [23] M. Guizar-Sicairos, S. T. Thurman and J. R. Fienup, Efficient subpixel image registration algorithms, Opt. Lett., 33 (2008), 156-158.   Google Scholar [24] R. F. Hanssen, Radar Interferometry: Data Interpretation and Error Analysis Remote Sensing and Digital Image Processing, Kluwer Academic Publishers, New York, 2001. Google Scholar [25] E. W. Hoen and H. A. Zebker, Penetration depths inferred from interferometric volume decorrelation observed over the Greenland ice sheet, IEEE Transactions on Geoscience and Remote Sensing, 38 (2000), 2571-2583.   Google Scholar [26] J. A. Jackson, B. D. Rigling and R. L. Moses, Canonical scattering feature models for 3D and bistatic SAR, IEEE Transactions on Aerospace and Electronic Systems, 46 (2010), 525-541.   Google Scholar [27] C. V. Jakowatz, Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia and P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach, Springer, 1996. Google Scholar [28] D. Just and R. Bamler, Phase statistics of interferograms with applications to synthetic aperture radar, Applied Optics, 33 (1994), 4361-4368.   Google Scholar [29] J.-S. Lee and E. Pottier, Polarimetric Radar Imaging from Basics to Applications, Optical Science and Engineering. CRC Press, Boca Raton, 2009.   Google Scholar [30] G. D. Martino, A. Iodice, D. Riccio and G. Ruello, Equivalent number of scatterers for SAR speckle modeling, IEEE Transactions on Geoscience and Remote Sensing, 52 (2014), 2555-2564.   Google Scholar [31] D. Massonnet and J. -Claude Souyris, Imaging with Synthetic Aperture Radar, Engineering Sciences: Electrical Engineering. EFPL Press. Distributed by CRC Press, Lausanne, Switzerland, 2008.   Google Scholar [32] A. Moreira, P. Prats-Iraola, M. Younis, G. Krieger, I. Hajnsek and K. P. Papathanassiou, A tutorial on synthetic aperture radar, IEEE Geoscience and Remote Sensing Magazine, 1 (2013), 6-43.   Google Scholar [33] C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images, Artech House Remote Sensing Library. Artech House, Boston, 1998. Google Scholar [34] P. A. Rosen, S. Hensley, I. R. Joughin, F. K. Li, S. N. Madsen, E. Rodriguez and R. M. Goldstein, Synthetic aperture radar interferometry, Proceedings of the IEEE, 88 (2000), 333-382.   Google Scholar [35] H. S. Stone, M. Orchard, E. Chang and S. Martucci, A fast direct Fourier-based algorithm for subpixel registration of images, IEEE Transactions on Geoscience and Remote Sensing, 39 (2001), 2235-2243.   Google Scholar [36] Q. Tian and M. N. Huhns, Algorithms for subpixel registration, Computer Vision, Graphics, and Image Processing, 35 (1986), 220-233.   Google Scholar [37] A. Voronovich, Wave Scattering from Rough Surfaces, Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1999.  Google Scholar [38] M. Wermuth, A. Hauschild, O. Montenbruck and R. Kahle, TerraSAR-Xprecise orbit determination with real-time GPS ephemerides, Advances in Space Research, 50 (2012), 549-559.   Google Scholar [39] B. Yazıcı, I. Son and H. Cagri Yanik, Doppler synthetic aperture radar interferometry: A novel SAR interferometry for height mapping using ultra-narrowband waveforms, Inverse Problems, 34 (2018), 055003. doi: 10.1088/1361-6420/aab24c.  Google Scholar [40] B. Yonel, I. Son and B. Yazici, Exact multistatic interferometric imaging via generalized ıirtinger flow, IEEE Transactions on Computational Imaging, 6 (2020), 711-726.   Google Scholar [41] Y. T. Yoon, M. Eineder, N. Yague-Martinez and O. Montenbruck, TerraSAR-Xprecise trajectory estimation and quality assessment, IEEE Transactions on Geoscience and Remote Sensing, 47 (2009), 1859-1868.   Google Scholar [42] H. A. Zebker and J. Villasenor, Decorrelation in interferometric radar echoes, IEEE Transactions on Geoscience and Remote Sensing, 30 (1992), 950-959.   Google Scholar
Traditional presentation of geometry for radar interferometry in the vertical cross-track plane. Points ${\mathit{\boldsymbol{z}}}$ and ${\mathit{\boldsymbol{T}}}$ are on the same circle centered at ${\mathit{\boldsymbol{x}}}^ {(0)}$ (dashed line)
Calculation of the flat Earth phase in cross-track radar interferometry. The points ${\mathit{\boldsymbol{z}}}'$ and ${\mathit{\boldsymbol{T}}}$ have zero elevation, and both circles are centered at ${\mathit{\boldsymbol{x}}}^ {(0)}$
The annulus in the vertical cross-track plane due to the main lobe of the $\mathop{\mathrm{{sinc}}}\nolimits$ in $(49)$. It is centered at ${\mathit{\boldsymbol{x}}}$ and has central radius $R_ {\mathit{\boldsymbol{z}}}$. Its thickness $2\Delta_ {\rm{R}}$ is defined by the system bandwidth $B$, see formula (50). Vertical localization of radar targets can be performed using either interferometry or external information about the elevation
Radar interferometry with two coequal antennas. All coordinates are specified in the slant reference frame $(u,v)$. To illustrate formula (88), note that $l_1+l_2 = z_ {vb}-z_ {va}$
)">Figure 5.  Interferometry of an extended vertically stratified scatterer. In the analysis of (106), $V(z'_u) = \tau \mathop{\mathrm{{sinc}}}\nolimits (Bz'_u/c)$ is considered as a function of $z_u = y_u+z'_u$ (cf. Fig. 3)
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