© 2005 by CRC Press 111 5 Radiowave Scattering 5.1 CROSS SECTION OF SCATTERING We have already considered radiowave propagation in inhomogeneous media; how- ever, the rate of change of media parameters was so small in our examples that the geometrical optics technique was rather workable in the first approximation. In this chapter, we will consider situations when the spatial variation of the medium can be sharp on the wavelength scale including the jump changes. Examples include drop formation of clouds and rain, areas of vegetation, hailstones, etc. The primary phenomena with regard to radiowave interaction with such inhomogeneities are diffraction and the associated wave scattering. Thus, certainly, wave absorption in the inhomogeneities does take place. Both processes — scattering and absorption — lead to attenuation of the incident wave power flow, and a phenomenon known as extinction occurs. The waves field can be represented, in the cases discussed here, as the sum of the incident and scattered waves; that is, E = E i + E s and H = H i + H s . The field of scattered waves is described by: (5.1) Here, ε ( r ) is the permittivity of the scattering waves of the homogeneities. It is assumed, then, that the permittivity of the medium is equal to unity in the absence of inhomogeneities. We may consider the second term in Equation (5.1) to be a current with density: (5.2) Then, the scattered field can be expressed in the form of an integral from the inserted equivalent current. The integral must be extended, in this case, to the volume taken up by all of the scattering particles (inhomogeneities). We shall first pay attention to the simple case of one particle and will consider the field at the wave zone. We must use Equations (1.37) and (1.38) to obtain: (5.3) Here, integration is performed over the volume of the scattering inhomogeneities. ∇ ×= ∇ ×=−− −     EH H E E ss s s ik ik ik,().ε r 1 jrErE e ik i () () () .r = −−     = −−     c 4π ε ω π ε1 4 1 E r E rr s r rr r =××             − () − ke ik ik2 4 1 π ε exp ⋅⋅ ′       ′ =×       ∫ r rH r E rr ss d V 3 ,. TF1710_book.fm Page 111 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 112 Radio Propagation and Remote Sensing of the Environment From this point forward, we will assume as a rule that the sources of radiation are so far away that the incident field within the scattering volume (or the particle size) does not differ significantly from the plane wave. As we have already seen, it is necessary for the size of the Fresnel zone to exceed the scale of the scattering inhomo- geneities. Let us assume that the incident wave has the form E i = g i exp and H i = Here, single vector g i is the vector of linear polarization, and e i is the single vector of the propagation direction of the incident wave. The incident wave amplitude is assumed to be equal to unity. The vector e s = r /r determines the direction of the wave scattering. The scattering characteristics of a particle (inhomogeneity) are defined by the vector: (5.4) called the scattering amplitude. The scattered wave is described in these notations by the expression: (5.5) Equation (5.5) has a general nature and is not automatically a consequence of Equation (5.3); in particular, a value of ε = ∞ (a metallic particle) in Equation (5.3) makes the current definition, Equation (5.2), meaningless. Equation (5.5), however, keeps its meaning in such a case. Equation (5.3) itself is, in essence, an integral form of the initial Maxwell equations. Field E under the integral is not generally known beforehand and should be found by solving the problem of radiowave dif- fraction on the inhomogeneity being considered. It can be determined only in very few cases from any previous findings, usually with the same approach. The amplitude of scattering is a function of the incident wave direction and its polarization, as well as the direction of scattering. The scattering is, in the general case, accompanied by a change of polarization. The dependence of the scattering amplitude on the incident wave propagation direction and its polarization is obvious; therefore, as a rule we will omit the corresponding vectors from our list of scattering amplitudes and arguments and, for brevity, will keep only the dependence on vector e s . The power flow density of the incident wave is S i = c /8 π . Poynting’s vector of scattered wave is: (5.6) ik i er⋅ ()     eE eg er ii i i i ik×     =×     ⋅ ()     exp . fe e g e E e e isi k ik,, exp () =×× ()     − () −⋅ 2 4 1 π ε ss s ′′ ()     ′ ∫ rrd V 3 , Efeeg H ef ss r ss r rr = () =×     i ik ik i ee ,, , . Se e ss s 2 Sf r = () i 2 . TF1710_book.fm Page 112 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radiowave Scattering 113 The following value is the differential cross section of scattering: . (5.7) It has the dimensions of a square and corresponds to the power flow density scattered in direction e s within a solid angular unit. It should be pointed out that in radar science a rather different definition of the scattering differential cross section is used and is connected with the one introduced here by the simple equality . The total flow of the power scattered in all directions is characterized by the cross section of scattering: (5.8) where the integration is provided with respect to solid angle Ω . Let us now turn our attention to the absorbed power. As we already know, the density of the absorbed power is determined by Equation (1.20). Integrating over the scattered volume gives us the power value absorbed by a particle. The normal- ization of this power per incident wave power flow density determines the cross section of absorption: (5.9) The cross section of absorption is small if the scattering inhomogeneity is almost transparent ( ε′′ << 1) or the particle has very high conductivity (E → 0). The physical difference between σ s and σ a is that the cross section of scattering characterizes the spatial redistribution of the incident wave power flow, and the cross section of absorption defines the efficiency of transfer of this energy to heat. The summed value is the total cross section: (5.10) which determines wave attenuation. If the density of the particles is not very high, then we can assume in the first approximation that they scatter independently, which means that the field of scattering of adjacent particles is much smaller than the field of the incident wave; that is, the scattering of fields already scattered does not play a particular role. According to these assumptions, the power of the field scattered by the particle assembly is equal to the sum of the powers scattered by every particle separately, which applies to irregular distribution of particles inside the scattering volume. In fact, it is believed that particles scatter incoherently. Let us consider the σ di i i i eeg eeg,, ,, ss f () = () 2 σπσ d r d () = 4 σσ ππ sds feg eeg ii i i dd,,, , () = () = ∫∫ ΩΩ 2 44 σε a V kd= ′′ ′ () ′ () ′ ∫ rr rE 2 3 . σσσ ts =+ a TF1710_book.fm Page 113 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 114 Radio Propagation and Remote Sensing of the Environment volume of a medium having a single section and length ds along the direction of wave propagation. The power flow density at the input of the considered element is S and at the output it is S + d S. The difference between these densities is determined by wave scattering and absorption by particles that are inside the volume; that is, d S = –S N σ t ds , where N is the particle density in the volume. The corresponding transfer equation is: , (5.11) for which the solution (at σ t , N = const ) is: (5.12) where the value: (5.13) is the coefficient of extinction , which defines the degree of wave attenuation in the scattering medium. The value: (5.14) is the scattering albedo and determines the role of scattering in the general balance of propagating wave energy losses. 5.2 SCATTERING BY FREE ELECTRONS The scattering of electromagnetic waves by electrons is one of the most vivid examples of the scattering process. Due to the incident field, electrons take on an oscillatory motion that is followed by radiation. This reradiation field is the field of electron scattering. Let us, first of all, formulate the equation of electron motion in the field of the incident wave: (5.15) Here, ν is the frequency of electron collisions (so, in this case, the electrons are not absolutely free, because introducing the frequency of collision takes into account their interaction with other particles), and d ds N S S t = −σ S=S 0 e s−Γ , Γ = Nσ t ˆ A = σ σ s t d dt d dt d dt e m 2 2 3 3 rr r E+ − = −νγ . TF1710_book.fm Page 114 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radiowave Scattering 115 (5.16) The term with the third derivative describes the strength of the radiation friction. 36 It is assumed that the processes are not relativistic, so the influence of the wave magnetic field on the electrons is neglected. In the case of harmonic time dependence, the solution to the movement equation is: (5.17) The inducted dipole moment of an electron is: (5.18) The reradiation power is calculated with the help of Equation (1.44), and, being relative to the incident wave power flow density, gives us the following cross section of scattering: (5.19) Here, (5.20) is the classical radius of electron. Usually, ν / ω , ωγ << 1; the corresponding members can be neglected; and we come to the classical Thomson’s formula: (5.21) The cross section of absorption can be calculated based on the following discussion. The work performed by the field at the electron is the criterion of the wave energy absorption. The work performed in a unit of time is equal to the product γ ==⋅ − 2 3 625 10 2 20 e mc 3 .sec. r E = ++ ()     e miωω ν ωγ 2 . pr E = − = − ++ ()     e e mi 2 2 ωω ν ωγ . σ π ν ω ωγ se = ++ () 8 3 1 1 2 2 a . a e mc e 2 cm==⋅ − 2 13 282 10. σ π s 2 = 8 3 a e . TF1710_book.fm Page 115 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 116 Radio Propagation and Remote Sensing of the Environment –1/2Re( e v · E ), where v is the electron velocity. Dividing this product by the incident wave power flow density gives us the absorption cross section: (5.22) We have already said that taking into account electron collisions assumes that the electrons are in an environment of other particles, including charged ones (i.e., plasma). The interaction between charged particles of plasma leads to collective effects. Generally speaking, it is not correct to consider electron motion without taking into account movement in the closest environment. Also, reradiation processes of waves by electrons without partially coherent radiation of the electromagnetic energy by its neighbors cannot be considered. So, the result obtained has an approx- imate character. The conditions of its validity and the role of collective effects will be further discussed. 5.3 OPTICAL THEOREM The value of the total cross section is connected with the amplitude of scattering forward in the direction of the incident wave. To show this, 10 we will surround the particle by a sphere of large radius that tends to infinity. Let us consider the energy balance in the volume surrounded by this sphere. Equation (1.17), in which external currents are not considered, will serve as our base. Let us perform the integration over the volume and transform the volume integral from ∇ · S into a surface one: (5.23) Because the field is the sum of incident and scattered waves, the Poynting vector of the summary wave is S = S i + S s + S i s , where the interference term can be written as: The integral of S i is equal to zero, and the integral of S s gives us the flow of the scattered power, equal to S i σ s . Thus, we obtain: . (5.24) σ πνωγ ωνωγ πν ω a e e ac ac = + () ++ () ≅ 4 4 2 22 22 . Se⋅ () = − ∫ s Srd ia 2 4 Ωσ π . SEHEH ii isss c 8 =×     +×     {} π Re . σ π tss S = −⋅ () ∫ 1 2 4 i i rdSe Ω TF1710_book.fm Page 116 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radiowave Scattering 117 The expanded view is: In essence, the integration is performed with respect to spherical angles defining the direction of vector e s . Because kr >> 1, the stationary phase method can be used to calculate the integral. The points of the stationary phase are related to the directions e s = e i and e s = – e i . The corresponding asymptotic integration leads to the result: (5.25) One should bear in mind, in this case, that the contribution of the stationary point, e s = – e i , is missed because its real part is equal to zero. One should also take into account the product e s · f ( e s ) = 0, due to transversality of the scattering field. The result is referred to as the optical theorem , and it demonstrates that extinction is determined by the forward scattering intensity. Note that polarization of the forward scattered radiation cannot be orthogonal to the incident wave polarization, which applies to particles of isotropic material. Another definition of optical theorem is based on Equation (5.4): (5.26) The optical theorem allows us to calculate easily the maximum value of the total cross section for nontransparent bodies whose sizes are large compared to the wavelength. The Kirchhoff approximation may be used, in this case, to describe the scattered field. According to the Babinet principle, the scattered field in the fare zone is: From here, the amplitude of the forward scattering: (5.27) σ tss r1 = −⋅ () −⋅ () ⋅ ()     ∗∗ −−⋅ re ii ik i Re gf ef ge eee ee gf eg ef s ss ()     + { + ⋅ () ⋅ () −⋅ () ⋅ () ∫ 4π ii ii e iikr i d 1−⋅ ()           } ee s Ω. σ π t = () () 4 k ii Im .gf e σε t = − () ⋅ () −⋅ ′ ()     ′ ∫ kikd ii V Im exp .1 3 gE er r E g ee er r s r ss r = ⋅ () −⋅ ′ ()     ′ ik e ik d i ik i 2 2 π exp . AA ∫ fe g er r i i A ik ik d () = −⋅ ′ ()     ′ ∫ 2 2 π exp . s TF1710_book.fm Page 117 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 118 Radio Propagation and Remote Sensing of the Environment Now, the mutual orthogonality of vectors e i and r ′ should be taken into consideration. So, (5.28) Here, A is the body section transversal to the direction of the incident wave propa- gation. Thus, the total cross section of scattering by a large-size body is equal to double the square of its section. It would be reasonable to ask at what ratio of body size to wavelength we can use Equation (5.28) with reasonable accuracy. It is not possible to answer that question in general terms, as it is necessary to solve the wave diffraction problem and the answers may differ for bodies with different shapes and different physical properties. Analysis of calculation results (e.g., see King 77 for a metallic sphere and a metallic disc with radius a ) shows that Equation (5.28) leads to satisfactory results for wavelengths λ = a and shorter. 5.4 SCATTERING FROM A THIN SHEET A thin sheet with thickness d and square A is an example of when the scattering problem can be solved without actually determining an exact solution. If transverse sizes of the sheet are much larger than the wavelength, we can use the solution of Naturally, such an approach is based on the smallness of the edge effect. Use of the term thin sheet indicates observance of the condition kd << 1, and the value can be large. A thin sheet can be a vegetable leaves model. As the scattering takes place at rather low frequencies (corresponding to wavelengths measured in centi- meters), the sheet can be semitransparent, and at high frequencies (waves measured in millimeters) it practically does not transmit radiowaves. Let us use the results of Section 3.5 assuming that ε 1 = ε 3 = 1 and ε 2 = ε . The calculation can be performed using the Kirchhoff approximation and the theory of diffraction. For the case of horizontally polarized waves (H-waves), we can use Equation (1.75), which is represented in this case as the sum: Note that: σ t = 2A. kd ε EE Er r ii ikR AA i e R d T +=− ′ () ′ − () ∞ − ∫ s z 1 22 2 π ∂ ∂ θ π ∂ ∂zz Er r i ikR A e R d ′ () ′ ∫ 2 . − () ′ =+ () ′ ∞ − ∫∫ 1 2 1 2 22 ππ dd AA i A rE r. TF1710_book.fm Page 118 Thursday, September 30, 2004 1:43 PM wave reflection by a homogeneous layer (see Section 3.3) to calculate the field. © 2005 by CRC Press Radiowave Scattering 119 So, (5.29) Defining a scattered field in such a way is a generalization of the Babinet principle for semitransparent screens. Equation (5.29) can be rewritten as: (5.30) Correspondingly, the amplitude of scattering is: (5.31) At last, using the expression for the incident wave and Equation (5.25), we obtain: (5.32) Note that Equation (5.28) applies in the case of an opaque sheet. In the case of vertical polarization (E-waves), an equation similar to Equation (5.30) must be written for the magnetic field. If the magnetic field is now expressed through an electrical one, then it is easy to show that we again obtain Equation (5.32), where only the coefficient of transmission for vertically polarized waves should be used. The cross section of absorption is calculated on the basis of simple energetic considerations. The wave power flow incident at the sheet is equal to S i A cos θ i . The flow value of the reflected power is . The flow of the transmitted power is determined by the value . The difference between the power flow of the incident wave and the reflected plus transmitted powers gives us the energy absorbed inside the sheet in a unit of time. This, in turn, determines the cross section of absorption: (5.33) EErr s z = − () ′ () ′ ∫ 1 2 2 T e R d i i ikR A θ π ∂ ∂ . EeeEr e szs r r s = − ()     ⋅ () ′ () −⋅ ik T e e i ik i ik 1 2 θ π ′′ () ′ ∫ r rd A 2 . fe e e E r er szs s () = − ()     ⋅ () ′ () −⋅ ′ ik T e i i ik 1 2 θ π (() ′ ∫ d A 2 r . σθ θ t = − ()     21AT ii cos Re . S iii AFcos θθ () 2 S iii ATcos ( )θθ 2 σ θ θθθ a i iii AFT AFT i i = −− () = − () − () S S cos cos 1 1 22 222       . TF1710_book.fm Page 119 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 120 Radio Propagation and Remote Sensing of the Environment Now, it is not difficult to calculate the value of the cross section of scattering: (5.34) Next, it is important to determine the value of the backscattering differential cross section. By the same reasoning used earlier, we can introduce the amplitude of the backscattered field in the form: (5.35) for the case of H-waves. Because we are considering the case of large plates compared to the wavelength, the main scattering is concentrated in the direction of the specula reflection For this reason, the reflection coeffi- cient in Equation (5.35) is outside the integral sign at an angle equal to the incident one. For the same reason, the integral in this formula is an “acute” function of angles. The cross section of the scattering in the back semisphere is described by the formula: , (5.36) where the function Ψ ( Ω ) is determined by Equation (1.117), except that in this case it is called the indicatrix of diffraction . Its form is defined by the geometrical shape of the sheet. We can say, generally, that the indicatrix angle spread has a value on the order . In the case of E-waves, the reflection coefficient for vertically polarized waves and vector instead of vector g i must be used in Equation (5.35). 5.5 WAVE SCATTERING BY SMALL BODIES We often come across cases when natural objects are smaller than the wavelength, including drops of rain, clouds, practically the entire microwave region, vegetation cover, etc. The spatial structure of the electrical field inside these objects is the same as it would be when scatterers (i.e., particles) are placed in the electrostatic field. As we are dealing with particles smaller than the wavelength, then the exponent index in Equation (5.4) approaches zero, and we can write: σσσ θ θ θ saii i AF T= − = () + − ()       t cos . 22 1 fe g ee r ee r szs s () = ⋅ ()() ′ − () ⋅ ∫ ik Fe d i i ik A i 2 2 π θ ee eee szz = −⋅ () () ii 2 σ π θ di kA Fe s () = () 22 2 2 4 ΨΩ() ∆Ω ≅ λ 2 A geeeeg iiii −⋅ () ×× ()   2 zz TF1710_book.fm Page 120 Thursday, September 30, 2004 1:43 PM [...]... ′ S (5. 88) on the basis of Equations (1.83) and (1.84) Thus, the integral is divided into two to cover both the illuminated and shady parts of the surface According to the Kirchhoff approach, the field on the illuminated part is approximately equal to the one that follows from the laws of wave reflection by the plane interface of two media The field of the incident wave substitutes for the field of the. .. incident wave of single amplitude In the case of the spherical particle, the internal field is parallel to the incident one, and the amplitude of the considered induced moment is equal to the polarizability (or perceptivity) of the particle: α= ε −1 3 a ε+2 (5. 57) In general, for arbitrarily shaped particles, the relation between the components of the induced dipole moment and the components of the incident... (5. 87) TF1710_book.fm Page 133 Thursday, September 30, 2004 1:43 PM Radiowave Scattering 133 Equation (5. 75) is a particular case of the second of these equations for small-sized particles Generally, the resonant members of these sums will be the largest, but the sum of the other members will not necessarily be negligible Because of this fact, the dependence, for example, of the cross sections on the. .. j (5. 124) t j that is, the total cross section of the particle assembly, σ (t Σ ) , is equal to the sum of the cross sections of the particles themselves If the scattering particles have a complex shape and rotate, then σ (t Σ ) is also found to be a random value because the cross sections σ (tj ) are the same Therefore, the average value of σ (t Σ ) is taken to be a measure of the scattering by the. .. scattering by the volume of particles in view of the body of the corresponding shape The last term is becoming more clear due to our introducing a definition of the effective permittivity The fluctuation intensity of the amplitude of scattering of the particle assembly is: ∆f Σ 2 = fΣ 2 − fΣ 2 (5. 137) The first summand is determined by Equation (5. 133) and the second one is calculated from Equation (5. 127),... e (s ) , g (i ) r − rj (5. 122) Here, rj is the radius vector of the jth particle, E(rj) is the complex amplitude of the summary wave incident on it, e i( j ) and g i( j ) are the vectors of direction and polarization of this wave at the point of the jth particle location, and e ( j ) is the scattering s vector of the chosen particle Let us regard an assembly of particles of low density, for which... ε+2 (5. 46) Let us point out some important considerations The polarization of waves scattered forward and backward by the sphere coincides with the polarization of the incident wave The cross section of scattering is much smaller than the cross section of absorption due to the smallness of the product ka Accordingly, the total cross section calculated on the basis of the optical theorem is equal to the. .. calculations the smallness of the imaginary part of the permittivity is assumed; therefore, no sign of the module is seen in the formulae Nevertheless, the size of the sphere is set so large that the wave inside the sphere is completely absorbed even in cases of little imaginary part of the permittivity So, at large values of permittivity:  16 4 ln ε  σ ← = πa 2  1 − + s ε    3 ε (5. 110) The cross... the shady side of the surface according to the Babinet principle Thus, ( ) ( ) ( ) f e i , e s , g i = f1 e i , e s , g i + f 2 e i , e s , g i , (5. 89) where the first summand corresponds to the integral covering the illuminated side and the second one to the shady side of the surface It is obvious that the first item describes the wave scattering by the illuminated part of the surface and the second one... describes the diffraction phenomenon and the forward scattering connected with it In the process of integration with respect to the illuminated part of the surface, © 20 05 by CRC Press TF1710_book.fm Page 134 Thursday, September 30, 2004 1:43 PM 134 Radio Propagation and Remote Sensing of the Environment 0 we should take into consideration the relation H 0 =  e r ⋅ E r  between the magnetic r   and electric . 20 05 by CRC Press 114 Radio Propagation and Remote Sensing of the Environment volume of a medium having a single section and length ds along the direction of wave propagation. The. PM © 20 05 by CRC Press 124 Radio Propagation and Remote Sensing of the Environment The ranges of change of Euler’s angles are given by the intervals and It is easy to establish that the internal. 20 05 by CRC Press 130 Radio Propagation and Remote Sensing of the Environment the drop; however, first we will neglect the absorption and assume that ε′′ = 0, and we conclude that: on the basis of