4

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RANDOM ROUGH SURFACE SIMULATIONS

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Perfect Electric Conductor (Non-Penetrable Surface)

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1.1 1.2 1.3 1.4

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Integral Equation Matrix Equation: Dirichlet Boundary Condition (EFIE for TE Case) Tapering of Incident Waves and Calculation of Scattered Waves Random Rough Surface Generation

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1.4.1 Gaussian Rough Surface 1.4.2 Fractal Rough Surface

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Neumann Boundary Condition (MFIE for TM Case)

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Two-Media Problem

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2.1 2.2 2.3 2.4

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TE and TM Waves Absorptivity, Emissivity and Reflectivity Impedance Matrix Elements: Numerical Integrations Simulation Results

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2.4.1 Gaussian Surface and Comparisons with Analytical Methods 2.4.2 Dirichlet Case of Gaussian Surface with Ocean Spectrum and Fractal Surface 2.4.3 Bistatic Scattering for Two Media Problem with Ocean Spectrum

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Topics of Numerical Simulations

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Periodic Boundary Condition MFIE for TE Case of PEC Impedance Boundary Condition

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Microwave Emission of Rough Ocean Surfaces

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- 111-

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4 RANDOM ROUGH SURFACE SIMULATIONS

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Waves Scattering from Real-Life Rough Surface Profiles 166

5.1 5.2 5.3

Introduction Rough Surface Generated by Three Methods Numerical Results of the Three Methods

References and Additional Readings

166 167 169

4 RANDOM ROUGH SURFACE SIMULATIONS

In this chapter we study random rough surface simulations of onedimensional surface for two-dimensional scattering problem. The simulations of rough surface scattering started in the late 1970's and continue to the present day [Axline and Fung, 1978; Thorsos, 1988; Thorsos and Jackson, 1991; Maystre et al. 1991; Devayya and Wingham, 1992; Thorsos and Jackson, 1989; Thorsos and Broschat, 1995; Maradudin et al. 1990; Michel and O'Donnell, 1992; McGurn and Maradudin, 1993; Chan et al. 1991; NietoVesperinas and Soto-Crespo, 1987]. The main purposes of the early simulations were to validate analytic scattering theory and to investigate backscattering enhancement. The numerical method in this chapter is based on the formulation of integral equations and converting the integral equations into matrix equations using the method of moments. We discuss the Dirichlet problem and Neumann problem and illustrate the results using Gaussian surfaces, surfaces with ocean spectrum, and fractal surfaces. Next, we discuss dielectric surface and the calculation of emissivity for applications in passive remote sensing. In particular, we address the accuracy issue in the calculation of emissivity. The accurate calculation of emissivity distinguishes the emphasis of rough surface simulations in this book. Most researchers on rough surface simulations emphasize on bistatic scattering and backscattering, which are usually measured and plotted in dB scale. For such simulations, an accuracy of 25% or 1 dB is acceptable. However, for passive remote sensing calculations, the physics is based on energy conservation. The key result in passive remote sensing is the difference of emissivity between a rough surface and a flat surface. The difference is small and can be a few percent to less than 1%. For ocean remote sensing, that difference is particularly small, e.g., 0.003 or 0.3%. This corresponds to a brightness temperature difference of less than a Kelvin between a rough surface and a flat surface. The ability to distinguish that small difference actually forms the basis of passive remote sensing of ocean wind. This means that for passive remote sensing numerical simulations, energy conservation has to be within 0.3%. Such a stringent requirement is not needed in active remote sensing simulations, where an energy conservation of 96% is deemed to be good. Thus numerical methods of simulations for active remote sensing can be different from passive remote sensing because of the large difference in accuracy requirements. One key implication for calculations in passive remote sensing is that the rough surface needs to have a fine discretization. In this chapter, we will also introduce examples of real life surface profiles measured for rocky surfaces, soil surfaces, and snow surfaces.

4 RANDOM ROUGH SURFACE SIMULATIONS

Perfect Electric Conductor (Non-Penetrable Surface)

1.1 Integral Equation

Consider an incident wave 1f;inc(r) impinging upon a random surface (Fig. 4.1.1) with height profile z = f (x). In two-dimensional scattering problem r = xx + ZZ, the wavefunction '!/J(r) is (4.1.1) where '!/Js(r) is the scattered wave. The wavefunction obeys the equation

(\72

+ k2) '!/J = 0

(4.1.2)

The two-dimensional Green's function obeys the equation (4.1.3) and

g(r,r')

~H61)(klr-r'i)

(4.1.4)

o (Va)

Let the spaces above and below the rough surface be denoted by region and region 1 (VI)' We use the Green's theorem to get