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Formulating a Vector Wave Expression for Polarimetric GNSS Surface Scattering

By Robert N. Treuhaft, Stephen T. Lowe, and Estel Cardellach
Progress In Electromagnetics Research B, Vol. 33, 257-276, 2011


This paper formulates a simple vector integral expression for electromagnetic waves received after scattering from a surface. The derived expression is an alternative to the Stratton-Chu equation frequently used for polarimetric surface scattering. It is intended for use in polarimetric Global Navigation Satellite System (GNSS) ocean remote sensing, or any type of polarimetric remote sensing from surfaces, when the surface roughness pattern is known from simulation or data. This paper is intended to present a complete accounting of the steps leading to the simpler vector integral expression. It therefore starts with the scalar case, using Maxwell's equations and Green's theorem. It principally treats the case of a transmitter within the integration volume, but discusses how the formalism changes if the transmitter is outside of the integration volume, as with plane waves. It then shows how the scalar expression can be extended to a vector expression for the component of the electric field in an arbitrary receive-polarization direction due to scattering from a rough surface of an incident wave with an arbitrary transmit polarization. It uses the Kirchhoff, or tangent-plane, approximation in which each facet on the ocean is considered to specularly reflect the incoming signal. The derived vector expression is very similar to that for a scalar wave, but it includes all vector properties of the scattering. Equivalence is demonstrated between the Stratton-Chu equation and the derived, simpler expression, which is operationally easier to code than the Stratton-Chu equation in many modeling applications.


Robert N. Treuhaft, Stephen T. Lowe, and Estel Cardellach, "Formulating a Vector Wave Expression for Polarimetric GNSS Surface Scattering," Progress In Electromagnetics Research B, Vol. 33, 257-276, 2011.


    1. Garrison, J. L. and S. J. Katzberg, "Effect of sea roughness on bistatically scattered range coded signals from the global positioning system ," Geophysical Research Letters, Vol. 25, No. 13, 2257-2260, 1998.

    2. Zavorotny, V. U. and A. G. Voronovich, "Scattering of GPS signals from the ocean with wind remote sensing application," IEEE Trans. on Geosci. and Rem. Sens., Vol. 38, No. 2, 951-964, 2000.

    3. Martin-Neira, M., M. Caparrini, J. Font-Rossello, S. Lannelongue, and C. S. Vallmitjana, "The PARIS concept: An experimental demonstration of sea surface altimetry using GPS reflected signals ," IEEE Trans. on Geosci. and Rem. Sens., Vol. 39, No. 1, 142-150, 2001.

    4. Treuhaft, R. N., S. T. Lowe, C. Zuffada, and Y. Chao, "2-cm GPS altimetry over crater lake," Geophysical Research Letters, Vol. 22, No. 23, 4343-4346, 2001.

    5. Lowe, S. T., C. Zuffada, Y. Chao, P. Kroger, L. E. Young, and J. L. LaBrecque, "5-cm precision aircraft ocean altimetry using GPS reflections," Geophysical Research Letters, Vol. 29, No. 10, 1375, 2002.

    6. Ruffini, G., M. Soulat, M. Caparrini, O. Germain, and M. Martin-Neira-, "The eddy experiment: Accurate GNSS-R ocean altimetry from low altitude aircraft," Geophysical Research Letters, Vol. 31, L12306-L12309, 2001.

    7. Cardellach, E. and A. Rius, "A new technique to sense nonGaussian features of the sea surface from L-band bi-static GNSS reflections," Remote Sensing Environment, Vol. 112, No. 6, 2927-2937, 2008.

    8. Yang, D. K., Y. Q. Zhang, Y. Lu, and Q. S. Zhang, "GPS reflections for sea surface wind speed measurement," IEEE Trans. on Geosci. and Rem. Sens., Vol. 5, No. 4, 569-572, 2008.

    9. Semmling, A. M., G. Beyerle, R. Stosius, G. Dick, J. Wickert, F. Fabra, E. Cardellach, S. Ribo, A. Rius, S. B. Yudanov, and S. d'Addio, "Detection of arctic ocean tides using interferometric GNSS-R signals ," Geophysical Research Letters, Vol. 38, L04103, 2011, doi: 10.1029/2010GL046005.

    10. Zuffada, C., A. Fung, J. Parker, M. Okolicanyi, and E. Huang, "Polarization properties of the GPS signal scattered off a wind-driven ocean," IEEE Transactions on Antennas and Propagation, Vol. 52, 172-188, 2004.

    11. Jackson, J. D., Classical Electrodynamics, Vol. 41, 428, Wiley and Sons, New York, 1975.

    12. Elfouhaily, T., B. Chapron, and K. Katsaros, "A unified directional spectrum for long and short wind-driven waves," Journal of Geophysical Research, Vol. 102, No. C7, 15781-15796, 1997.

    13. Beckmann, P. and A. Spizzichino, "The Scattering of Electromagnetic Waves from Rough Surfaces," Ch. 3, 178-180, Macmillan, New York, 1963.

    14. Stratton, J. A. and L. J. Chu, "Diffraction theory of electromagnetic waves," Physical Review, Vol. 56, 99-107, 1939.

    15. Fung, A. K., Z. Li, and K. S. Chen, "Backscattering from a randomly rough dielectric surface," IEEE Trans. on Geosci. and Rem. Sens., Vol. 30, No. 2, 356-369, 1992.

    16. Jin, Y. Q., "Electromagnetic Scattering and Modelling for Quantitative Remote Sensing,", World Scientific, Singapore, 1993.

    17. Tsang, L. and J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topics, John Wiley and Songs, Inc., New York, 2001.

    18. Silver, S., Microwave Antenna Theory and Design, 108, McGraw-Hill, New York, 1949.

    19. Bass, F. G. and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces, Pergamon Press, Oxford, 1979.