volume | PIER Journals

Abstract

Energy conservation is an important consideration in wave scattering and transmission from random rough surfaces and is particularly important in passive microwave remote sensing. In this paper, we study energy conservation in scattering from layered random rough surfaces using the second order small perturbation method (SPM2). SPM2 includes both first order incoherent scattering and a second order correction to the coherent fields. They are combined to compute the total reflected and transmitted powers, as a sum of integrations over wavenumber k_x, in which each integration includes the surface power spectra of a rough interface weighted by an emission kernel function (assuming the roughness of each interface is uncorrelated). We calculate the corresponding kernel functions which are the power spectral densities for one-dimensional (1D) surfaces in 2D scattering problems and examine numerical results for the cases of 2 rough interfaces and 51 rough interfaces. Because it is known that the SPM when evaluated to second order conserves energy, and it can be applied to second order for arbitrary surface power spectra, energy conservation can be shown to be satisfied for each value of k_x in the kernel functions. The numerical examples show that energy conservation is obeyed for any dielectric contrast, any layer configuration and interface, and for arbitrary roughness spectra. The values of reflected or transmitted powers predicted, however, are accurate only to second order in surface roughness.

1. Tsang, L., J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves, Vol. 1: Theory and Applications, 426, Wiley Interscience, 2000.
doi:10.1002/0471224286

2. Tsang, L. and J. A. Kong, Scattering of Electromagnetic Waves, Vol. 3: Advanced Topics, 413, Wiley Interscience, 2001.
doi:10.1002/0471224278

3. Tsang, L., X. Gu, and H. Braunisch, "Effects of random rough surface on absorption by conductors at microwave frequencies," IEEE Microwave and Wireless Components Letters, Vol. 16, No. 4, 221-223, 2006.
doi:10.1109/LMWC.2006.872109

4. Bagley, J. Q., L. Tsang, K. H. Ding, and A. Ishimaru, "Optical transmission through a plasmon film lens with small roughness: Enhanced spatial resolution of images of single source and multiple sources," Journal of the Optical Society of America B, Vol. 28, No. 7, 1766-1777, 2011.
doi:10.1364/JOSAB.28.001766

5. Tsang, L., K. H. Ding, X. Li, P. N. Duvelle, J. H. Vella, J. Goldsmith, C. L. H. Devlin, and N. I. Limberopoulos, "Studies of the influence of deep subwavelength surface roughness on fields of plasmonic thin film based on Lippmann-Schwinger equation in the spectral domain," Journal of the Optical Society of America B, Vol. 32, No. 5, 878-891, 2015.
doi:10.1364/JOSAB.32.000878

6. Tabatabaeenejad, A. and M. Moghaddam, "Bistatic scattering from three-dimensional layered rough surfaces," IEEE Transactions on Geoscience and Remote Sensing, Vol. 44, No. 8, 2102-2114, 2006.
doi:10.1109/TGRS.2006.872140

7. Afifi, S. and R. Dusseaux, "On the co-polarized scattered intensity ratio of rough layered surfaces: The probability law derived from the small perturbation method," IEEE Transactions on Antennas Propagation, Vol. 60, No. 4, 2133-2138, 2012.
doi:10.1109/TAP.2012.2186258

8. Johnson, J. T., "Third-order small-perturbation method for scattering from dielectric rough surfaces," Journal of the Optical Society of America A, Vol. 16, No. 11, 2720-2736, 1999.
doi:10.1364/JOSAA.16.002720

9. Chiu, T. C. and K. Sarabandi, "Electromagnetic scattering interaction between a dielectric cylinder and a slightly rough surface," IEEE Transaction on Antennas and Propagation, Vol. 47, No. 5, 902-913, 1999.
doi:10.1109/8.774155

10. Soubret, A., G. Berginc, and C. Bourrely, "“Backscattering enhancement of an electromagnetic wave scattered by two-dimensional rough layers," Journal of the Optical Society of America A, Vol. 18, No. 11, 2778-2788, 2001.
doi:10.1364/JOSAA.18.002778

11. Demir, M. A. and J. T. Johnson, "Fourth and higher-order small perturbation solution for scattering from dielectric rough surfaces," Journal of the Optical Society of America A, Vol. 20, No. 12, 2330-2337, 2003.
doi:10.1364/JOSAA.20.002330

12. Soubert, A., G. Berginc, and C. Bourrely, "Application of reduced Rayleigh equations to electromagnetic wave scattering by two-dimensional randomly rough surfaces," Physical Review B, Vol. 63, No. 24, 245411, 2011.
doi:10.1103/PhysRevB.63.245411

13. Zamani, H., A. Tavakoli, and M. Dehmollaian, "Second-order perturbative solution of scattering from two rough surfaces with arbitrary dielectric profiles," IEEE Transactions on Antennas and Propagation, Vol. 63, No. 12, 5767-5776, 2015.
doi:10.1109/TAP.2015.2484387

14. Zamani, H., A. Tavakoli, and M. Dehmollaian, "Second-order perturbative solution of crosspolarized scattering from multilayered rough surfaces," IEEE Transactions on Antennas and Propagation, Vol. 64, No. 5, 1877-1890, 2016.
doi:10.1109/TAP.2016.2535503

15. Demir, M. A., Perturbation theory of electromagnetic scattering from layered media with rough interfaces, Ph.D. Dissertation, Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH, 2007.

16. Imperatore, P., A. Iodice, and D. Riccio, "Electromagnetic wave scattering from layered structures with an arbitrary number of rough interfaces," IEEE Transactions on Geoscience and Remote Sensing, Vol. 47, No. 4, 1056-1072, 2009.
doi:10.1109/TGRS.2008.2007804

17. Wu, C. and X. Zhang, "Second-order perturbative solutions for 3-D electromagnetic radiation and propagation in a layered structure with multilayer rough interfaces," IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, Vol. 8, No. 1, 180-194, 2015.
doi:10.1109/JSTARS.2014.2320506

18. Zamani, H., A. Tavakoli, and M. Dehmollaian, "Scattering from layered rough surfaces: Analytical and numerical investigations," IEEE Transactions on Geoscience and Remote Sensing, Vol. 54, No. 6, 3685-3696, 2016.
doi:10.1109/TGRS.2016.2524639

19. Demir, M. A., J. T. Johnson, and T. J. Zajdel, "A study of the fourth-order small perturbation method for scattering from two-layer rough surfaces," IEEE Transactions on Geoscience and Remote Sensing, Vol. 50, No. 9, 3374-3382, 2012.
doi:10.1109/TGRS.2011.2182614

20. Pinel, N., J. T. Johnson, and C. Bourlier, "A geometrical optics model of three dimensional scattering from a rough layer with two rough surfaces," IEEE Transactions on Antennas and Propagation, Vol. 58, No. 3, 809-816, 2010.
doi:10.1109/TAP.2009.2039306

21. Tabatabaeenejad, A., X. Duan, and M. Moghaddam, "Coherent scattering of electromagnetic waves from two-layer rough surfaces within the Kirchhoff regime," IEEE Transactions on Geoscience and Remote Sensing, Vol. 51, No. 7, 3943-3953, 2013.
doi:10.1109/TGRS.2012.2229391

22. Huang, S. and L. Tsang, "Electromagnetic scattering of randomly rough soil surfaces based on numerical solutions of Maxwell equations in three-dimensional simulations using a hybrid UV/PBTG/SMCG method," IEEE Transactions on Geoscience and Remote Sensing, Vol. 50, No. 10, 4025-4035, 2012.
doi:10.1109/TGRS.2012.2189776

23. Chen, K. S., T. D. Wu, L. Tsang, Q. Li, J. Shi, and A. K. Fung, "Emission of rough surfaces calculated by the integral equation method with comparison to three-dimensional moment method simulations," IEEE Transactions on Geoscience and Remote Sensing, Vol. 41, No. 1, 90-101, 2003.
doi:10.1109/TGRS.2002.807587

24. Du, Y., J. A. Kong, Z. Y. Wang, W. Z. Yan, and L. Peng, "A statistical integral equation model for shadow-corrected EM scattering from a Gaussian rough surface," IEEE Transactions on Antennas and Propagation, Vol. 55, No. 6, 1843-1855, 2007.
doi:10.1109/TAP.2007.898503

25. Kuo, C. H. and M. Moghaddam, "Scattering from multilayer rough surfaces based on the extended boundary condition method and truncated singular value decomposition," IEEE Transactions on Antennas and Propagation, Vol. 54, No. 10, 2917-2929, 2006.
doi:10.1109/TAP.2006.882160

26. Kuo, C. H. and M. Moghaddam, "Electromagnetic scattering from multilayer rough surfaces with arbitrary dielectric profiles for remote sensing of subsurface soil moisture," IEEE Transactions on Geoscience and Remote Sensing, Vol. 45, No. 2, 349-366, 2007.
doi:10.1109/TGRS.2006.887164

27. Duan, X. and M. Moghaddam, "3-D vector electromagnetic scattering from arbitrary random rough surfaces using stabilized extended boundary condition method for remote sensing of soil moisture," IEEE Transactions on Geoscience and Remote Sensing, Vol. 50, No. 1, 87-103, 2011.
doi:10.1109/TGRS.2011.2160549

28. Duan, X. and M. Moghaddam, "Bistatic vector 3-D scattering from layered rough surfaces using stabilized extended boundary condition method," IEEE Transactions on Geoscience and Remote Sensing, Vol. 51, No. 5, 2722-2733, 2013.
doi:10.1109/TGRS.2012.2215614

29. Jezek, K. C., J. T. Johnson, M. R. Drinkwater, G. Macelloni, L. Tsang, M. Aksoy, and M. Durand, "Radiometric approach for estimating relative changes in intraglacier average temperature," IEEE Transactions on Geoscience and Remote Sensing, Vol. 53, No. 1, 134-143, 2015.
doi:10.1109/TGRS.2014.2319265

30. Tan, S., M. Aksoy, M. Brogioni, G. Macelloni, M. Durand, K. C. Jezek, T. L. Wang, L. Tsang, J. T. Johnson, M. R. Drinkwater, and L. Brucker, "Physical models of layered polar firn brightness temperatures from 0.5 to 2 GHz," IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, Vol. 8, No. 7, 3681-3691, 2015.
doi:10.1109/JSTARS.2015.2403286

31. Brogioni, M., G. Macelloni, F.Montomoli, and K. C.Jezek, "Simulating multifrequency groundbased radiometric measurements at Dome C-Antarctica," IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, Vol. 8, No. 9, 4405-4417, 2015.
doi:10.1109/JSTARS.2015.2427512

32. Tsang, L., J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol. 2: Numerical Simulations, 705, Wiley Interscience, 2001.
doi:10.1002/0471224308

33. Johnson, J. T. and M. Zhang, "Theoretical study of the small slope approximation for ocean polarimetric thermal emission," IEEE Transactions on Geoscience and Remote Sensing, Vol. 37, No. 5, 2305-2316, 1999.
doi:10.1109/36.789627

Dr. Tianlin Wang

Professor Leung Tsang

Professor Joel Johnson

Professor Shurun Tan