© 2005 by CRC Press 157 6 Wave Scattering by Rough Surfaces 6.1 STATISTICAL CHARACTERISTICS OF A SURFACE Many natural surfaces, such as the soil surface or water surface of the ocean, can be regarded as smooth only in certain circumstances. In general, these surfaces should be considered to be rough, and their interaction with radiowaves should be seen as a scattering process. Whether or not we assume that the surface is rough generally depends on the problem formulation and, particularly, on the ratio of the roughness scales and the wavelength. The nature of the roughness varies depending on the type of surface. Sea surface roughness is a result of the interaction of the wind with the water surface. This interaction has a nonlinear character. A great number of waves with different frequencies and wave numbers are generated as a result, and their mixture leads to oscillations of the sea surface height according to the stochastic function of coordinate and time. However, the velocity of the sea surface movement is small compared to the velocity of light, so time dependence cannot be taken into account in the first approximation. Soil roughness can form as a result of wind erosion, urban activity, and other causes. The soil roughness is also a random function of coordinates. Again, a dependence on time is not considered in the beginning and we are dealing generally with wave scattering by random surfaces. The specific surface will be described by a random function of the elevation The average value so it is assumed that the average surface is given by the plane The function is supposed to be statistically homogeneous. It simplifies the problem substantially, as the statistical homogeneity of the real natural rough surfaces take place in the restricted cases. The correlation function: (6.1) depends, in this case, on the coordinate difference of the points involved. In many cases, the surface may be assumed to be statistically isotropic. Then, the correlation function depends only on the module (i.e., on the distance between the points of correlation). The correlation function has significant value within the correlation radius which is often defined by the relation: (6.2) ζ(), { }.sswhere x,y= ζ() ,s = 0 z0.= ζ()s ˆ K ζ ζζ ′ − ′′ () = ′ () ′′ () ss s s ′ − ′′ ss ld d 2 2 2 2 0 12 = () = ∫∫ ∞ ζ π ζ ζζ KKss (s)s s. TF1710_book.fm Page 157 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 158 Radio Propagation and Remote Sensing of the Environment It is necessary to represent the correlation scales along the main axis of the anisotropy ellipse in the case of statistical anisotropy. If we suppose that roughness occupies the bounded surface Σ for whose measure is much larger than the radius of correlation, then the following Fourier expansion is correct: . (6.3) The spatial spectrum: (6.4) is a random function with zero mean. Its correlation function is written as: By introducing the “gravity center” coordinate and the difference , the last integral can be written in the form: Let us represent the elevation fluctuation spectrum of the examined surface as: (6.5) according to the Wiener–Chintchin theorem. The integral in this expression can be spread over infinite limits because the size of the chosen surface was set much larger than the correlation scale. Thus, (6.6) It follows from here, in particular, that at : ζζ() ()sq= ⋅ ∫  ed iqs q 2  ζ π ζ() ()qs= −⋅ ∫ 1 4 2 2 ed iqs s Σ  ζζ π ζ ′ () ′′ () = ′ − ′′ () ∗ − ′ ⋅ ′ − ∫∫ qq ss qs 1 16 2 ˆ K Σ e i ′′ ′ ⋅ ′′ () ′′′ qs ssdd 22 . ′ + ′′ =ss S2 ′ − ′′ =ss s  ζζ π ζ ′ () ′′ () = − ′ + ′′ ⋅    ∗ qq qq s 1 16 2 2 ˆ ()expKs i    − ′ − ′′ () ⋅ ∫∫ de d i 22 sS qq S . ΣΣ  Kq K ζζ π () ˆ () .= −⋅ ∫ 1 4 2 2 ss qs ed i   ζζ π ζ ′ () ′′ () = ′ + ′′       − ′ − ′′ qq qq qq 1 4 2 2 K e i (() ⋅ ∫ S Sd 2 . Σ ′ = ′′ =qq q TF1710_book.fm Page 158 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Wave Scattering by Rough Surfaces 159 (6.7) where we have simplified the expression by not distinguishing between the value of the surface and its square. If then the integral in Equation (6.6) can be also taken to infinite limits by maintaining the condition . As a result, (6.8) The correlation function of the scalar product is: (6.9) In particular, (6.10) Let us now turn to the case of a statistically isotropic surface. We can assume that the main area of integration is situated in the interval therefore, the following estimation is valid: On the other hand, it is easy to determine from Equation (6.2) that As a result, we have: (6.11) Further, we will encounter the correlation vector : (6.12)   ζ π ζ () () ,q Kq 2 2 4 = Σ ′ ≠ ′′ qq, ′ − ′′ >>qq Σ 1   ζζ δ ζ ′ () ′′ () = ′ + ′′       ′ − ′′ () qq qq qqK 2 . ∇ ′ () ⋅∇ = −∇ == ′ ⋅ ζζ ζζ sqss qs () () () ,sKsKq 22 q 2  ed i −− ′′ ∫ s . ∇ () = () ∫ ζ ζ 2 2 q 2  K qqd . 0 ≤≤q2π l; ∇ () ≅ = ∫ ζπ π ζ π ζ 2 0 2 3 4 20 8 0  KK() ().qq 3 d l l l 22 2 40= () 〈〉πζ ζ  K . ∇ () ≅ζπ ζ 2 3 2 2 2 l . = () ⋅ ′ − ′′ ∫ ied i qq q qs s  K ζ () . 2 TF1710_book.fm Page 159 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 160 Radio Propagation and Remote Sensing of the Environment And, finally, we can calculate the value: The last expression is easily transformed to the form: (6.13) The previous discussion was concerned with smooth and differentiable surfaces. It is often convenient to eliminate the requirement of differentiability when describ- ing natural surfaces (sea, soil, etc.). To illustrate, we will analyze the structure function of properties of the surface: (6.14) assuming statistical isotropy of the surface. It is usual to suppose in the case of a smooth surface that the first derivative of the correlation function at zero is equal to zero ( ). Then, the expansion: (6.15) for the structure function is valid close to zero. Such a surface has slopes, which means that the angle is equal to zero and the dispersion (the mean-square slope) is defined as: . (6.16) The differentiability of the surface is understood in such sense. Many natural surfaces have a fractal character. 78 Their structure function satisfies the equation: (6.17) The index α is connected with fractal dimension β (Hausdorff–Besicovitch dimen- sion) by the relation: . (6.18) ζζ π ζζ π ζ () ( ) () ˆ sq s K  = ′ () ′ ≅ −⋅ ′ 1 4 1 4 2 2 2 ss s qs ed i −− ′ () ′ −⋅ ′ ∫∫ ss qs ed i 2 . Σ ζζ ζ () ( ) () .sq Kq   = −⋅ e iqs D ζ ζζ ′ − ′′ () = ′ () − ′′ ()     ss s s 2 ˆ () ′ =K ζ 00 D() ˆ ()sK s 2 = ′′ ζ 0 ∇ () = −∇ = − ′′ = → ζ ζζ 2 2 020 1 2 ˆ () ˆ () lim ) KK (s s 0 2 s D DB(s s) = α αβ= − () 22 TF1710_book.fm Page 160 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Wave Scattering by Rough Surfaces 161 The value obtained lies within the range , which leads to an index interval of in Equation (6.17). It would not be correct, in this case, to address the differentiability of the surface. The maximum value of α leads to Equation (6.15), which is typical for smooth surfaces. The value , which follows from the theory of Brownian motion, corresponds to the Brownian fractal. Let us now turn our attention to the properties of rough surfaces. Generally, we cannot expect to develop an exact technique to solve the problem of radiowave diffraction on stochastic surfaces; instead, we must rely on approximation methods that, as a rule, are found effective for asymptotic cases. In our case, the roughness is small compared to the wavelength, which is the opposite of the case of large inhomogeneities. The method of small perturbation is effective in our case, and the Kirchhoff approach is best suited for the second case. Recently, some attempts have been made to find analytical solutions of the latter problem on the basis of an integral equation model (IEM) for electromagnetic fields 79 ; however, only some refinement of results have been reported, and the IEM is undergoing improvement. 80 6.2 RADIOWAVE SCATTERING BY SMALL INHOMOGENEITIES AND CONSEQUENT APPROXIMATION SERIES Let us assume that the described surface separates into two media. The upper medium has permittivity equal to unity. The permittivity of the lower medium is equal to ε . We assume that a plane wave of single amplitude is incident from the side of the z-coordinate positive values. To find the scattered field it is necessary to solve Maxwell’s equations for both media while maintaining the boundary conditions: (6.19) on the surface. The numbers 1 and 2 indicate, respectively, the fields over and under the examined surface. If e z is the single normal to the average plane z = 0, then the single normal to the surface ζ ( s ) satisfies the equation: (6.20) Let us assume that the roughness is small; that is, . (6.21) 12 <<β 02<<α βα==15 1.( ) Eg r er ii ik e i == ⋅ ,{}x,y,z nEE nE E HH× −     = ⋅− () = − () =(),( ), 12 1 2 1 2 000 ζζζ ε ns e () = −∇ + ∇ () z ζ ζ1 2 . k ζ 2 1<< TF1710_book.fm Page 161 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 162 Radio Propagation and Remote Sensing of the Environment This inequality means that the probability is extremely low that deviations of the roughness from the average plane can be more than the wavelength. We also assume that the considered rough surface slopes are small, or << 1. Equation (6.11) allows us to assume that the correlation radius of such a surface is much larger than the height; that is, We can now apply the method of sequential approximations. Let us represent the unknown fields in the form of series: (6.22) Here, j = 1, 2, and the sum terms represent expansion over the growing degree of Naturally, the sum terms of the same small size satisfy Maxwell’s equations. The fields, however, are expanded into a Taylor series of the form: Taking into account the approximation , the border conditions in Equation (6.19) are transferred from surface ζ ( s ) on the plane z = 0. Further, the corresponding expansions are continued until the second order of small- ness is obtained. Let us set the term of the same order of smallness equal to zero to obtain the boundary conditions for fields of a different order. These boundary conditions for the field of the zero order have the form: (6.23) From here on, we will omit the subscript z = 0. The boundary conditions for the electromagnetic field of the first order can be written as: (6.24) ()∇ζ 2 l >> 〈〉ζ 2 . EE E E HH H H jj j j j j j j =+++⋅⋅⋅ =++ () () () () () ( , 012 0 1 22) .+ ⋅⋅⋅ ζζand .∇ EE EE () () .ζ ∂ ∂ ζ ∂ ∂ ζ=+ + +⋅⋅⋅0 1 2 2 2 z z 2 ne e= −∇ − ∇ + ⋅⋅⋅ zz ζζ()/ 2 2 eE E eE E z zz × −     = ⋅− = (),( ) () () () () 1 0 2 0 0 1 0 2 0 0 ε (() = − () = ==zz0 1 0 2 0 0 00,. () () HH eEE EE e z × − ()     = ∇ × − ()     − 1 1 2 1 1 0 2 0() () ( ) ( ) ζ zz z I,× − ()       () − = − ∂ ∂ ζEE HH 1 0 2 0 1 1 2 1 () () () () ζζ ∂ ∂ ε z II , z HH eE E 1 0 2 0 1 1 2 1 () () () () − () () × − ()     = ∇ × − ()     − × − ( ζε ∂ ∂ EE e EE 1 0 2 0 1 0 2 0() () () () z z ))       () ζ III . TF1710_book.fm Page 162 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Wave Scattering by Rough Surfaces 163 Analogous expressions can be obtained for the second-order fields. We will not provide them here but refer the reader to Armand, 52 where the second approach is analyzed in detail. Further actions deal with using Maxwell’s equations to solve for every approx- imation at given boundary conditions. In doing so, the incident plane wave (more exactly, the source of the radiated wave) is the source for the field of the zero approximation. The fields of subsequent approaches are excited by the preceding fields. So, a system of constrained fields is obtained for which it is necessary to solve a succession of Maxwell’s equations for fields of different orders. We should bear in mind when doing so that the boundary conditions for the normal to the average plane field components are odd in some sense, but we must keep them in order to minimize calculations when they are indirectly presented. Note that these boundary conditions are analogous to Equations (1.93) and (1.94), which means that our problem is reduced to a problem of fields excited by surface currents. The formal solution of this problem is Equation (1.111) which will be used from here on. Let us begin with the zero-arch approximation. It does not require any special consideration as it is reduced to a problem of plane wave reflection by the plane the formulae in a more convenient form. In particular, the electric field on the surface is written as: (6.25) Here, w i is the wave vector of the incident wave on the plane z = 0, and Having found the field component of the zero approximation, let us now compute the fields of the first approximation of the perturbation theory. The first step is to rewrite the equivalent surface currents and charges of Equation (6.23) with a more compact right side. We refer the reader to Bass 48 and Armand 52 for more details on the procedure, as we provide only the results here. The first expression of Equation (6.23) can be expressed as: (6.26) where the surface magnetic current is: (6.27) The expression obtained for the magnetic current is not only compact but also convenient as it is expressed with regard to only the field over the interface. EEE g e e 1 0 0 00 00 00 2 1 () =+= + + − + + +  ir ii a ab ab ab ε ε z      ⋅ ()         ⋅ eg ws z i i e i . w ii k= sin .θ eEE K zm c × −     = −() , () () () 1 1 2 11 4π KeeE mzz 4 () . 1 1 0 1 = − × ∇     ⋅ () c π ε ε ζ TF1710_book.fm Page 163 Thursday, September 30, 2004 1:43 PM interface. This problem has been examined in Chapter 3, but here we will rewrite © 2005 by CRC Press 164 Radio Propagation and Remote Sensing of the Environment The second boundary condition is rewritten as: (6.28) where the surface electric current is: (6.29) Note that the expression for the surface electric charge has the form: (6.30) Now we can use Equation (6.25) to establish that: (6.31) where (6.32) The electric current is represented as: (6.33) where vector i is defined as: (6.34) Now, let us compute the Fourier transform of the introduced surface currents. For the magnetic current: (6.35) eHH K ze c × − ()     = 1 0 2 01 4π () , KeeE e ikc () . 1 1 0 4 = − () ××         ε π ζ 1 zz δ ε π ζ ∂ ∂ ζ e () . 1 1 0 1 0 1 4 = − ⋅ () −∇⋅ ()       z z eE E Ke ws mz () , 1 = ℑ × ∇     ⋅ ζe i i ℑ= − () + () ⋅ () – z ε πε 1 2 0 00 a ab i eg. Kir ws e i ik e i (1) = ζ() i eg ee= − () ⋅ () + − − +      ca ab ab i i ε πε ε 1 z z 0 00 00 2 1  − +           g i ab 00 .  Kes ws ws mz () . 1 2 2 4 = ℑ × ∇     () −⋅ ⋅ ∫ π ζeed i i i Σ TF1710_book.fm Page 164 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Wave Scattering by Rough Surfaces 165 Let us take into account that The inte- gral of the first term is transformed into such over the boundary of the surface Σ , and we can set it equal to zero, as the roughness is zero, or the incident field is small on the border of the radiating antenna footprint. The integral of the second summand is transformed to: (6.36) according to Equation (6.4). Similarly, for the electric current: (6.37) Now, according to Equation (1.111), the Fourier image of the scattered field is represented in the first approach as: (6.38) where: (6.39) The field itself is expressed via the integral: (6.40) As the integral extends toward infinite limits, the scattered field satisfies Gaussian probability law. Given a sufficient distance from the surface, the integral can be calculated using the stationary phase method. Let us designate a = cos θ and and introduce the vector: (6.41) and we obtain: (6.42) ee e ie i i ii i ii −⋅ ⋅ − () ⋅− () ∇ = ∇ + ws ws wws ww w()( )ζζ ζ ⋅⋅ s .   Kewww mz ()1 = ℑ ×     − () i i ζ   Kiww ei ik () . 1 = − () ζ   Ew qF ww 1 1 2 1 4 () , () = + () ×     − () ik c i π εγ γ ζ 1 Fw eq qi ei()= ℑ ×     + ×     + − () + × εε γγ z z 11 12 1 kk k     . Er qF ww ws 1 1 1 12 4 () () = ×     − () + ⋅ + ik c e i ii π ζ εγ γ  γγ 1 2z d w. ∫ b = −εθsin 2 pe e eθφ θ φ θ φ θ,sincossinsincos, () =++ xyz Er eFe 1 1 22 8 () () cos = × ()     + πθ ε k c e ab ik s r ss ss r  ζ ww s − () i . TF1710_book.fm Page 165 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 166 Radio Propagation and Remote Sensing of the Environment Here, the subscript s indicates the direction of scattering, and . (6.43) Computation of the field scattered into the lower medium is done in the same way. The formulae for scattered fields allow us to compute easily the scattering amplitudes into upper and into lower semispaces and to determine accordingly the cross section of the scattering. It is necessary to take into account that the scattering amplitude is a stochastic value with a mean value equal to zero. The squared module of the scattering amplitude is also an occasional quantity; therefore, an important definition of the cross section is It is appropriate to take into account Equation (6.7), which leads us to the conclusion that the scattering section is proportional to the square of the illuminated surface. So, it is reasonable to introduce a definition of the scattering cross section per unit of surface σ 0 , the so-called reflectivity. It is a dimensionless value that characterizes the scattering properties of the surface regardless of size: (6.44) Thus, the intensity of the wave scattering is proportional in the first approximation to the power of the surface spatial component with wave vector . Its absolute value is: (6.45) The subscript i represents values related to the incident wave. The result obtained indicates that the incident wave interacts most effectively with only one of the spatial harmonics of the surface. This effect is said about to be a resonance one. The spatial spectrum of the surface is a rather acute function of angles, so the angle dependence of the scattering intensity (scattering indicatrix) is generally defined by the properties of the function Particularly, we can assert that maximum scattering occurs in the direction of the specular reflection when We must pay special attention to the very important case of backscattering. Particular interest is raised here by the fact that the radar images are formed against a background of wave backscattering. In this case, , and the following expression is obtained for the backscattering reflectivity: Fe e p p i e i sz () = ℑ ×+ ×+ − + ×ε ε 1 ab σ = 〈〉||.f 2 σ πθ ε ζ 0 24 2 2 16 e eFe s s 2 ss ss () = × ()     + k c ab cos  K www s − () i . kw w= − s i k sss =+−−k iii sin sin sin sin cos( ). 22 2θθθθϕϕ  K ζ k () . ϕϕ θθ ss and== ii . ee s = − i TF1710_book.fm Page 166 Thursday, September 30, 2004 1:43 PM [...]... dζ (6. 67) −∞ is the characteristic function of the surface As sizes of the area Σ are much larger than the wavelength, the integral on the right-hand side is the delta-function, and we have: f = −iπ k ( B n ⋅ez ) ( ) ( ) Pζ 2ka i δ  k u i − u s    It is also necessary to consider the smallness of the slope of the surface by averaging vector B (n ⋅ e z ); therefore, we may be restricted by the. .. ( 0 ) (6. 134) TF1710_book.fm Page 190 Thursday, September 30, 2004 1:43 PM 190 Radio Propagation and Remote Sensing of the Environment which represents the section of the antenna pattern by the mean plane of the scattering surface We neglected the difference from unity of the first factor of Equation (1.114) by assuming sufficient sharpness of the antenna pattern As a result, we now have a rather obvious... roughness and slopes are found to be weakly correlated for any separation of the correlation points We have reason, then, to assume a complete absence of the mentioned correlation between roughness and slopes Moreover, due to the tendency toward the small-scale part of the spectrum, the correlation radius of the slopes is on the order of the inner scale, while the large scales dominates the mechanism of scattering... analogous to Equations (6. 23) and (6. 24), where the fields in the medium are assumed to be equal to zero We can show that the field on the plane z = 0 is: E (1) z= 0 () = 2ikζ s g i (6. 54) Then, the field of the first order can be expressed by the integral: ∫ ( ) E (1) (r) = 2ikg i ζ w e iw ⋅ s+iz k 2 −w 2 d 2w (6. 55) in any point of the space Here, the transversal coordinates of the point of observation are... impulse expands and the duration given by Equation (6. 113) should be taken into account instead of the initial pulse duration The physical sense of impulse expansion is clear The pulses reflected from the peaks and hollows of rough surfaces come to ζ 2 c , which the receiving point with a mutual delay on the order of 2 cos θi changes the duration of the summary impulse The effect is not revealed at the condition... analyze the components of different moments The first moment (coherent part of the field) is equal to: E j = E (j0 ) + E (j2) , © 2005 by CRC Press H j = H (j0 ) + H (j2) TF1710_book.fm Page 168 Thursday, September 30, 2004 1:43 PM 168 Radio Propagation and Remote Sensing of the Environment One can see that the second approximation changes the coherent component of the field This change proceeds at the second... (6. 135) e reflecting the fact that the power of the field in the receiving point consists of field powers scattered by infinitely small, in the physical sense, elements of the surface The values of these powers are proportional to the antenna footprint value at the point of scattering We can refer to Equation (6. 135) for integration over the plane that is perpendicular to vector L0 and passes through the. .. of the slopes and to integrate over the entire region of the ∇ζ change Due to the delta-function properties, we can obtain the value of the distribution function by the argument: ∇ζ = ui − us ai + as (6. 75) The points where the gradient has this value correspond to the surface areas where specular reflection takes place (so-called specular or bright points) The scattering vector es coincides with the. .. K ζ w ∂z ∫ ( ) k 2 − w 2 d 2w (6. 60) TF1710_book.fm Page 170 Thursday, September 30, 2004 1:43 PM 170 Radio Propagation and Remote Sensing of the Environment This expression represents the amplitude of the plane wave describing, on average, the reflected field of the second order When calculating the interference term, we also must take into account that the real part of Poynting’s vector is important,... 178 Radio Propagation and Remote Sensing of the Environment In the marginal case when α = 2 (the differentiable surface), we have Equation (6. 81), if we assume: E= π3 ζ2 2l 2 ( )  B −e i   n ⋅ez  , () 2  4F 2 0  = cos 2 θ i   (6. 88) Equation (6. 88) is correct in all cases independent of the fractal dimension value Thus, we must always consider wave scattering due to the large-scale part of the . CRC Press 160 Radio Propagation and Remote Sensing of the Environment And, finally, we can calculate the value: The last expression is easily transformed to the form: (6. 13) The previous. Page 165 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 166 Radio Propagation and Remote Sensing of the Environment Here, the subscript s indicates the direction of scattering, and Press 158 Radio Propagation and Remote Sensing of the Environment It is necessary to represent the correlation scales along the main axis of the anisotropy ellipse in the case of statistical