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J Quant Spectrosc Radiat Transfer, 109, pp. 2195–2206. 10 Models for Scattering from Rough Surfaces F. Ticconi 1 , L. Pulvirenti 2 and N. Pierdicca 2 1 Dept. of Information Engineering and Computer Science, University of Trento, 2 Dept. Information Engineering, Electronics and Telecommunications - “Sapienza University of Rome, Italy 1. Introduction Models for scattering of electromagnetic waves from random rough surfaces have been developed during the last two centuries and the scientific interest in the problem remains strong also today due to the importance of this phenomenon in diverse areas of science, such as measurements in optics, geophysics, communications and remote sensing of the Earth. Such models can be categorised into empirical models, analytical models and a combination of the two. Though very simple, empirical models are greatly dependent on the experimental conditions. In spite of their complexity, only theoretical models can yield a significant understanding of the interaction between the electromagnetic waves and the Earth’s surface, although an exact solution of equations governing this interaction may not always be available and approximate methods have to be used. The semi-empirical models, which are based on both physical considerations and experimental observations, can be set between these two kinds of models and can be easily inverted. In this survey, we will focus on the analytical models and we study more in detail the Kirchhoff Approximation (KA), the Small Perturbation Method (SPM) and the Integral Equation Method (IEM). The Kirchhoff Approximation and the Small Perturbation Methods represent early approaches to scattering which are still much used, whereas the Integral Equation Method represents a newer approach which has a larger domain of validity. These methods have been found to be the most common in the literature and many of the other methods are based or have much in common with these approaches. In section 2, we begin by giving a brief presentation of the scattering problem and introduce some concepts and results from the theory of electromagnetic fields which are often used in this context. We will also define the bistatic scattering coefficient, due to the importance of this type of measurement in many remote sensing applications, and in particular in the retrieval of soil moisture content. In section 3, we give a brief presentation on the Kirchhoff Approximation and its close variants, the Physical Optics (PO) and the Geometrical Optics (GO). In section 4, we give a brief presentation of the Small Perturbation Method and in section 5 we will present the Integral Equation Model. 2. Some concepts of the electromagnetic theory and surface parameters In this section we will give a brief presentation of some concepts on theories of electromagnetism and statistical characterisation of surfaces, which are often used for Electromagnetic Waves 204 modelling scattering of electromagnetic waves from random rough surfaces. We will also define the bistatic scattering coefficient due to the importance of this type of measurement in many remote sensing applications. 2.1 The Maxwell’s equations and the wave equation The basic laws of the electromagnetism are given by the Maxwell’s equations which, for linear, homogeneous, isotropic, stationary and not dispersive media, can be written as (Balanis, 1989): t     B E (2.1.1) ci t      D HJJ (2.1.2)   D (2.1.3) 0  B (2.1.4) where E is the electric field vector, D is the electric flux density, H is the magnetic field vector, B is the magnetic flux density, J is the conduction electric current density, J i is the impressed electric current density and  is the electric charge density. Maxwell’s equations together with the boundary conditions, give a complete description of the field vectors at any points (including discontinuities) and at any time. In rough surface scattering, the surface enters in the boundary conditions (see equations (2.2.1)-(2.2.4)), which have to be also supplied at infinity. If we consider time-harmonic variation of the electromagnetic field, the instantaneous field vectors can be related to their complex forms. Thus the Maxwell’s equations can be written in a much simpler form: j    EH (2.1.5)   ici jj       HEJEJ (2.1.6)    E (2.1.7) 0   H (2.1.8) where we assumed the region characterised by permeability  , permittivity  and conductivity  (lossy medium). To obtain the governing equation for the electric field, we take the curl of (2.1.5) and then replace (2.1.6). Thus, 2 ci j      EEJ (2.1.9) which is known as the inhomogeneous Helmholtz vector wave equation. In a free-source region, 0  E and (2.1.9) simplifies to: Models for Scattering from Rough Surfaces 205 22 0 c   EE (2.1.10) In rectangular coordinates, a simple solution to (2.1.10) has the form:   0 j e    kr Er E (2.1.11) where E 0 is a constant complex vector which determines the polarisation characteristics and the complex propagation vector, k , is defined as: ˆ ˆˆ x y z kkk  kx y z (2.1.12) with the components satisfying 222 2 2 xyz c kkk k     (2.1.13) Equation (2.1.11) represents a plane wave and k is the propagation constant. Most analytical methods for scattering from rough surfaces assume this kind of incident wave, which if linearly polarised can be rewritten as:   0 ˆˆ i j ii Ee E   kr Er p p (2.1.14) where ˆ ii kkk, ˆ p is the unit polarisation vector and E 0 is the amplitude. The associated magnetic field is given by:     ˆ ii i  Hr k Er (2.1.15) where c    is the wave impendence in the medium. 2.2 Integral theorems and other results used in scattering models We will present some results for electromagnetic fields which are often used as a starting point in the analytical models for scattering from rough surfaces. These equations are approximated and simplified using different methods and assumptions in the analytical solutions for scattering from rough surfaces. We will not show how the equations in this section are derived, but derivation can be found in the references. Consider an electromagnetic plane wave incident on a rough surface as shown in figure 2.2.1. Fig. 2.2.1. Scattering of electromagnetic field on surface separating two media. Electromagnetic Waves 206 Across any surface interface, the electromagnetic field should satisfy continuity conditions given by (Balanis, 1989):   1 ˆ 0  nEE (2.2.1)   1 ˆ s  nHH J (2.2.2)   11 ˆ s    nE E (2.2.3)   11 ˆ 0   nH H (2.2.4) where ˆ n is the unit normal vector of the rough surface (pointing in the region 0). The electric surface current density, J s , and the charge surface density,  s , at the rough interface are zero unless the scattering surface (or one of the media) is a perfect conductor. Using the fact that the fields satisfy the Helmholtz wave equation (2.1.9), it can be shown that in the region 0, the electromagnetic fields E and H, satisfy Huygens’ principle and the radiation boundary condition at infinity and E is given by (Ulaby et al, 1982; Tsang et al, 2000):         1 ˆˆ ,, i S jds         Er E r Grr n Hr Grr n Er (2.2.5) where G is the dyadic Green function (to the vector Helmholtz equation) which is represented by:   2 ,,g k        Grr I rr (2.2.6) Here I is the unit dyadic and   ,g  rr is the Green function that satisfies the scalar wave equation. It assumes the following expression:  , 4 jk e g        rr rr rr (2.2.7) In (2.2.5) the first term on the right-hand side represents the field generated by a current source in an unbounded medium with permittivity  and permeability  and corresponds to the incident field. Hence, the electromagnetic field in the region 0 is expressed as the sum of two contributions: one is given by the incident field   i Er ; the other contribution is given by the surface integrals that involve the tangential components t E and t H of the fields at the boundary S 1 (note that ˆˆ t    nEnE and ˆˆ t    nHnH) and represents the scattered field due to the presence of surface. The equation (2.2.5) constitutes the mathematical basis of Huygens’ principle in vector form. According to this principle, the electromagnetic field in a source-free region ( 0J ) is uniquely determined once its tangential components are assigned on the boundary of the region. However, since in the region 0, the existence of the impressed current J has been Models for Scattering from Rough Surfaces 207 assumed, the total electric field can be expressed as the sum of two terms, the incident and scattering ones:       is Er E r E r (2.2.8) Thus, the scattered field can be written as:        1 ˆˆ ,, s S jds          Er Grr nHr Grr n Er (2.2.9) If the observation point is in the far field region, the Green function in (2.2.9) can be simplified and the scattering field can be written as (Ulaby et al,1982; Tsang et al, 2000):           1 ˆˆˆ j s S Keds            kr Er r n Er r n Hr (2.2.10) where 4 jkr K j ke r    and ˆ r is the unit vector pointing in the direction of observation. The tangential surface fields ˆ  nE and ˆ  nH can be also expressed as (Poggio & Miller, 1973): 2 ˆˆ ˆ 2 4 i ndsnE nE p ¢ ´= ´ - ´ ò e (2.2.11) 2 ˆˆ ˆ 2 4 i ds        nH nH n  (2.2.12) and 2 ˆˆ 4 ttt ds      nE n e (2.2.13) 2 ˆˆ 4 ttt ds      nH n  (2.2.14) where       111 ˆˆ ˆ j kG G G    nH nE nE e (2.2.15)    111 ˆˆ ˆ jk GGG    nE nH nH  (2.2.16)         22 2 2 2 ˆˆ ˆ 1 t r jk G G G                   nH nE nE e (2.2.17)     2 222 2 ˆˆ ˆ 1 t r jk GGG                      nE nH nH  (2.2.18) and ˆ n , ˆ  n , ˆ t n , ˆ t  n are the unit normal vectors to the surface and ˆˆ t  nn , ˆˆ t  nn , ˆ nE and ˆ nH are the total tangential fields on the rough surface in the medium above the Electromagnetic Waves 208 separating interface; G 1 and G 2 are the Green’s functions in medium above and below the interface, respectively, and 21r    , 21r    , 222    and 222 k    . 2.3 The nature of surface scattering When an electromagnetic wave impinges the surface boundary between two semi-infinitive media, the scattering process takes place only at the surface boundary if the two media can be assumed homogeneous. Under such supposition, the problem at issue is indicated as surface scattering problem. On the other hand, if the lower medium is inhomogeneous or is a mixture of materials of different dielectric properties, then a portion of the transmitted wave scattered backward by the inhomogeneities may cross the boundary surface into the upper medium. In this case scattering takes place within the volume of the lower medium and it is referred to as volume scattering. In most cases both the scattering processes are involved, although only one of them can be dominant. In the case of bare soil, which will be assumed to be a homogeneous body, surface scattering is the only process taken into consideration. When the surface boundary separating the two semi-infinitive media is perfectly smooth the reflection is in the specular direction and is described by the Fresnel reflection laws. On the other hand, when the surface boundary becomes rough, the incident wave is partly reflected in the specular direction and partly scattered in all directions. Qualitatively, the relationship between surface roughness and surface scattering can be illustrated through the example shown in Figure 2.3.1. For the specular surface, the angular radiation pattern of the reflected wave is a delta function centred about the specular direction as shown in Figure 2.3.1 (a). For the slightly rough surface (Figure 2.3.1 (b)), the angular radiation pattern consists of two components: a reflected component and a scattered component. The reflected component is again in the specular direction, but the magnitude of its power is smaller than that for smooth surface. This specular component is often referred to as the coherent scattering component. The scattered component, also known as the diffuse or incoherent component, consists of power scattered in all directions, but its magnitude is smaller than that of the coherent component. As the surface becomes rougher, the coherent component becomes negligible. Note that the specular component represents also the mean scattered field (in statistical sense), whereas the diffuse component has a stochastic behaviour, associated to the randomness of the surface roughness. (a) (b) (c) Fig. 2.3.1. Relative contributions of coherent and diffuse scattering components for different surface-roughness conditions: (a) specular, (b) slightly rough, (c) very rough. 2.3.1 Characterisation of soil roughness A rough surface can be described by a height function   ,zx y   . There are basically two categories of methods which are being used to measure surface roughness. The roughness  i  s  s  i  s  i [...]... 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Radio and Electronic Engineer, vol 45, pp 241-2 47 Deryck, L (1 978 ) Natural propagation of electromagnetic waves in tunnels, IEEE Transactions on Vehicular Technology, vol VT- 27, pp 145-150 Damoso, E & Padova, S (1 976 ) Propagation and radiation of VHF radio signals in motorway tunnels, IEEE Trans on Vehicular Technology, vol VT-25, pp 39-45 232 Electromagnetic Waves Mandich, D & Carpine, E An experimental... 200.0 300.0 231 Electromagnetic Wave Propagation in Circular Tunnels 500 450  =10-2 r = 4.08 Attenuation in dB/100m 400 TM01 350 300 250 200 150  =10-1 100 50 300.0 200.0 150.0 120.0 100.0 85 .7 75.0 66.6 60.0 54.5 50.0 46.2 42.9 40.0 37. 5 35.3 33.3 31.6 30.0 28.6 0 Frequency in MHz Fig 4 Frequency and attenuation in a circular tunnel 4 Conclusion The propagation of electromagnetic waves in a circular... vol 4 172 , pp 169- 176 Fung A K (1994) Microwave Scattering and Emission Models and Their Application, Boston, MA, Arthech House Poggio A J., Miller E K (1 973 ) “Integral Equation solution of Three Dimensional Scattering Problems”, Computed Techniques for Electromagnetics, Pergamon, New York Saillard M., Sentenac A (2001) “Rigorous Solutions for Electromagnetic Scattering from Rough Surfaces”, Waves in... 0 22 2 0 230 Electromagnetic Waves 450 Attenuation in dB/100m 400 TE01  =10-2 r = 4.08 350 300 250 200 150 100  =10-1 50 0 46.2 50.0 54.5 60.0 66.6 75 .0 85 .7 100.0 120.0 150.0 200.0 300.0 Frequency in MHz Fig 2 Frequency and attenuation in a circular tunnel 500 450  =10-2 r = 4.08 Attenuation in dB/100m 400 350 TM11 300 250 200 150 100  =10-1 50 0 46.2 50.0 54.5 60.0 66.6 75 .0 85 .7 Frequency in... R & Strong, P.(1 975 ) Theory of the propagation of UHF radio waves in cool mine tunnels, IEEE Trans Antennas Propagat AP-23, pp 192-205 Delogne, P (1982) Leaky feeders and subsurface radio communication, Peter Peregrinus, Stevegage, U.K Delogne,P & Safak, M (1 975 ) Electromagnetic Theory of the leaky, Coaxial Cable, The Radio and Electronic Engineer, vol 45, pp 233-240 Deryck, L (1 975 ) Control of mode... (Fung, 1994)): 226 Electromagnetic Waves 2  oqp  ( ksz )n Fqp (  kx ,  ky )  ( kz )n Fqp (  ksx ,  ksy ) 2 2 k 2  2 ( kz2  ksz )   2 n e W ( n ) ( ksx  kx , ksy  ky )  n ! ( kz  ksz )n fqp e kz ksz  2 2 n 1 (5.1.22) 6 Conclusions We have presented the results from a literature search of models for scattering of electromagnetic waves from random rough surfaces In particular we have... “Rigorous Solutions for Electromagnetic Scattering from Rough Surfaces”, Waves in Random Media, vol 11, pp 103-1 37 Tsang L., Kong J A., Ding K.-H (2000) Electromagnetic Waves, theories and applications, Jon Wiley and Sons, New York Tsang L., Kong J A (2001) Scattering of Electromagnetic Waves, Advanced Topics, Jon Wiley and Sons, New York Thorsos E I., Jackson D (1989) “The Validity of the Perturbation... Electromagnetic Waves and Applications, 20 (6), pp .70 7 -71 5 Abo-Seida, O (2002) Propagation of electromagnetic waves in a rectangular tunnel , ACES Journal, vol 17, pp 170 - 175 Marcuvitz, N (1951) Waveguide Handbook , Mc Graw-Hill, New York and London, pp 5 572 Part 4 Analysis and Applications of Periodic Structures and Waveguide Components ... scattering coefficient was obtained 220 Electromagnetic Waves 4 The small perturbation method The Small Perturbation Method (SPM) belongs to a large family of perturbation expansion solutions to the wave equation The approach is based on formulating the scattering as a partial differential equation boundary value problem The basic idea is to find a solution in terms of plane waves that matches the surface boundary . effects at scattering by randomly oriented cluster of spherical particles. J. Quant. Spectrosc. Radiat. Transfer, 61, pp. 76 7 77 3. Tishkovets, V., Litvinov, P., Petrova, E., Jockers, K., &. (1980) Light scattering by irregularly shaped particles. New York, Plenum Press. Effects of Interaction of Electromagnetic Waves in Complex Particles 201 Shen, Y., Draine, B. T., &. pp. 371 -381 Voshchinnikov, N. V., Il'in, V. B., Henning, Th., & Dubkova, D. N. (2006) Dust extinction and absorption: the challenge of porous grains, Astron. Astrophys., 445, pp.1 67- 177 .