© 2005 by CRC Press 211 7 Radiowave Propagation in a Turbulent Medium 7.1 PARABOLIC EQUATION FOR THE FIELD IN A STOCHASTIC MEDIUM We have already discussed radiowave propagation in a turbulent medium using the small fluctuations of wave parameters (amplitude and phase) that, because of an accumulative effect, do not always occur even at small fluctuations of the medium permittivity. Moreover, the geometrical optics approach is restricted to the value of the distance traveled by the wave in a turbulent medium, as it does not take into account the effects of wave diffraction on the inhomogeneities. It is possible to use the geometrical optics approximation in cases where the Fresnel zone is smaller than the inhomogeneity scale (i.e., when where is some scale between the outer and inner scales of the turbulence). For these reasons, we must develop a mathematical instrument to describe in greater detail the processes of wave propa- gation in a turbulent medium. One such instrument is the parabolic equation method. The parabolic equation method is based on the following approach. Because we are dealing with a medium with little fluctuation of dielectric permittivity, we can neglect the polarization effects and describe the field by the scalar wave equation written for the radiated component of the electromagnetic field. It is necessary to take into account that turbulent pulsation of the medium generally has dimensions that are significantly greater than the wavelength which means that forward wave can apply the small angle approximation and use a parabolic equation of the type shown in Equation (1.81) for the fluctuating field description. If we insert Equation (1.80) into the wave equation then we can obtain the equation for the fluctuating field: (7.1) which is similar to Equation (1.81). Thus, we can suppose, as before, that permittivity differs little from unity. Substitution of the wave equation with the parabolic one means that we are neglecting waves scattered backward. Note that if we neglect transverse derivatives of the field, then the typical geometrical optics approach follows from Equation (7.1): l L >> λ , l 20 22 ik U UkU ∂ ∂ µ z + ∇ += ⊥ TF1710_book.fm Page 211 Thursday, September 30, 2004 1:43 PM geometrical optics approximation. The results obtained in Chapter 3 are limited to scattering by turbulent inhomogeneities is of primary important (see Chapter 5). We © 2005 by CRC Press 212 Radio Propagation and Remote Sensing of the Environment (7.2) Here s is the vector orthogonal to the main wave propagation direction (z-axis). The solution of the obtained parabolic equation is a random function. The first step is to determine the mean field or its coherent component. Let us designate , then average Equation (7.1) to obtain: (7.3) To solve this equation is rather difficult because the explicit form of the last term is unknown. The situation is improved when the permittivity fluctuation can be considered Gaussian. The Novikov–Furutzu formula can be applied in this case, 53 and we obtain: (7.4) Here, we deliberately separated the transverse and longitudinal coordinates. The reason for this separation will become clear later. Using the Novikov–Furutzu for- mula does not improve the situation without implementing further simplifications related to the necessity to have an explicit view of the functional derivative We can estimate its value using Equation (7.2), and the func- tional derivative is: According to the rules of the functional analysis: , (7.5) UU ik zdz go z zss,exp ,. () = ()         ∫ 0 0 2 µ UU= 20 22 ik U Uk U ∂ ∂ µ z + ∇ += ⊥ . µ δ δµ ε ss ss s s ,, , , , zz zz z z ()() = − ′ − ′ () () ′′ () U U dK 22 ′′ ∫∫ s dz. δδµU ss,,.zz () ′′ () δ δµ δµ δµ U ik U go go z z z zs s s s s , , , , , () ′′ () = () () ′′ 2 zz z z () ∫ d . 0 δµ δµ δ s s ss , , z z zz () ′ () = − ′ () − ′ () Θ TF1710_book.fm Page 212 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radiowave Propagation in a Turbulent Medium 213 where the step function is: (7.6) As a result, we can write: (7.7) and it is easy to find that: (7.8) To solve this expression it is necessary to take into account that the integral in Equation (7.2) is subject to Gaussian statistics. If the distance traveled by the wave in the turbulent medium sufficiently exceeds the turbulence scale, then, as we have already shown, the integral in the last formula is estimated by the value of the order l z. It follows, then, that the scale of the mean field change along the wave propagation direction is estimated by the value Consequently, the scale of the functional derivative change is the same. This scale is much larger than the turbulence scale as it is assumed that the phase fluctuation shift in the one-inhomogeneity frame is The integration by z ′ in Equation (7.4) is concentrated closely to z in the interval of the order l . It means that the functional derivative is not changed within the essential interval of integration. Therefore, we can substitute z for z ′ in the functional derivative, Equation (7.4). Then, we can prove that in our case: (7.9) From Equation (7.4) we have: (7.10) Θ zz z z for z >z , 1 2 for z =z , 1forz − ′ () = − ′ () = ′ ′ ′ δ zd 0 <<z. z        ∫ 0 δ δµ δ U ik U go go z z zzz s s sss , , , () ′′ () = () − ′ () − ′ ( 2 Θ )) , UU k dd go 0 zz zzzzzs,exp , () = − ′ − ′′ () ′′′ ∫∫ 0 2 0 8 0K ε         . µ 2 1 22 klµ . klµ 2 1<< . δ δµ δ U ik U s s sss , , ,. z z z () ′ () = () − ′ () 4 µ ss s,, ,()zz z ()() = () U ik UA 4 0 TF1710_book.fm Page 213 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 214 Radio Propagation and Remote Sensing of the Environment and the function (see Equations (4.78) and (4.79)): (7.11) It should be pointed out that it is not difficult to achieve the same result for the delta-correlation of the permittivity fluctuations in the wave propagation direction, having assumed that: (7.12) This rather artificial assumption reflects the fact that the scale of the field change in the wave propagation direction is much larger than the turbulence scale. This also emphasizes the need to use the Markovian approximation for the field computation. Using this approximation, the field is defined in each point by the previous values of permittivity over the z-axis. As a result, the mean field is described by the equation: (7.13) assuming Gaussian statistics of permittivity and a Markovian approximation. Let V represent the solution of the parabolic equation at A (0) = 0. Then, the general solution, Equation (7.13), can be written down in the form: (7.14) The solution for function V is represented by Equation (1.78). Also, the role of turbulent pulsation of the medium becomes apparent when extinction is considered to calculate the mean (coherent) field. The corresponding extinction coefficient is equal to: (7.15) The last formula in this series is obtained by assuming turbulence isotropy. If we compare it with Equation (5.178), then it becomes a simple matter to determine that which corresponds to the theory of scattering. We recall that is the wave scattering by unit volume of the turbulent inhomogeneities. It follows, from this comparison, that the extinction coefficient is approximately equal to Ad ed i () , , .ss q q qs = () = () ⊥ ⋅ ⊥ −∞ ∞ ⊥ ∫∫ KK εε πzz2 0 2  K ε δ(, () ( .ssz) z)= A 2 4 00 2 3 ik U U ik AU ∂ ∂z + ∇ += ⊥ () UV k Ass,,exp().zz z () = () −       2 8 0 Γ == () = () ⊥⊥ ⊥⊥ k A k dk 22 222 4 0 2 00() , , π π εε  KKqq qqdd 2 0 q ⊥ ∞ ∫∫ . Γ = σ s 0 , σ s 0 µ 22 kl TF1710_book.fm Page 214 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radiowave Propagation in a Turbulent Medium 215 (a more specific value is given, for example, by Equation (5.179)). Let us point out in this context that the estimation presented by us, a priori , of the rapidity of field attenuation with distance does not differ from the one defined by the extinction coefficient. 7.2 THE FUNCTION OF MUTUAL COHERENCE In this part, we will derive an equation for the second moment of the field, according to the same assumption as before. First of all, we will consider the function of mutual coherence: (7.16) This function corresponds to the definition provided earlier by Equation (1.124), and we derived Equation (7.16) in much the same way as Equation (1.125). As a result, we now have: (7.17) Here, we have introduced subscripts representing variables with respect to which differentiation in the transverse Laplacian is performed. Further, the reasoning used to derive the equation for the coherent field component can also be applied in this case. Doing so gives us: (7.18) The equation for the other second moment is derived in the same way: (7.19) and can be written as: (7.20) In order to solve Equation (7.17), let us utilize the method used to solve Equation (1.125): Γ U UUss s s 12 1 2 1 2 ,, , , .zzz () = ()() ∗ 2 2 12 ik k U U ∂ ∂ µµ Γ Γ z zz 22 + ∇−∇ () + () − ()    ss 12 ss,,  ()() = ∗ UUss 12 0,,.zz 2 2 0 12 22 3 12 ik ik AA U U ∂ ∂ Γ Γ z + ∇−∇ () + () −− ()    ss ss  =Γ U 0.  Γ U UUss s s 12 ,, , ,zzz () = ()() 12 2 2 0 22 3 12 2 ik z ik AAss U U ∂ ∂   Γ Γ+ ∇ + ∇ () − + −  ss 1 () ( )    =  Γ U 0. TF1710_book.fm Page 215 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 216 Radio Propagation and Remote Sensing of the Environment (7.21) instead of Equation (1.128), represented here, for short, as: (7.22) We deliberately separated out in this definition the z variable to emphasize the possible dependence of turbulence parameters on the longitudinal coordinate (e.g., altitude in the atmosphere of Earth). Furthermore, we will make the following changes to the variables: (the first of these two equations is the equation of characteristics). As a result of these changes, the ordinary differential equation now can be written as: with the solution: or (7.23) The function is determined from the boundary condition: (7.24) from which it follows that: (7.25) k k H UU ∂ ∂ ∂ ∂  ΓΓ z z,+       + () =W s s 3 4 0 HAA e d i () () () ,z, z,ss qq qs = − = − () () ⊥ ⋅ ⊥ 021 0 2 π ε  K ⊥⊥ ∫ . sW s− = ′ =zzkz, d dz k H z k U  Γ = − ′ +       2 4 z,s W  Γ U z M k Hz z k dz= ′ () − ′′ + ′       ′   ∫ Ws s W ,exp , 2 0 4        Γ U M z k k Hz k zz= −       − ′ −− ′ ()     Ws W s W ,exp , 2 4   ′           ∫ dz z 0 . M Ws, () ΓΓ U (,,) (,),Ss Ss0 0 = Med i (,) (,) .Ws Ss S WS = − ∫ 1 4 2 0 2 π Γ TF1710_book.fm Page 216 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radiowave Propagation in a Turbulent Medium 217 As a result, we have the following common solution: (7.26) or after substituting we obtain: (7.27) The solution of the equation for the function unfortunately cannot be found by any analytical form; therefore, it is necessary to resort to approximate methods for its solution. 7.3 PROPERTIES OF THE FUNCTION H Before turning to particular examples, let us discuss in detail some properties of the function , which mainly determines the behavior of the coherence function (see Equation (7.22)). Let us assume isotropic turbulence for simplicity. This allows us to obtain: (7.28) by integration in the polar coordinate system. Let us point out once more that, in the case of turbulence isotropy, the function H depends only on the distance between the points of correlation. Naturally, this property is retained in the coherence func- tion. To be specific in further calculations, we can use Equation (4.80), and we obtain: (7.29) ΓΓ U z k i k H = ′ −       − ′ () − { − ∫∫ 1 4 4 2 0 2 π Ss W WS S,exp zz z k dz d d z , s W SW−            ′ ∫ 0 22 Wss= − ′ () k z ΓΓ U k i k = ′′ () − ′ () − ′ () −    ∫∫ 2 2 0 4π z z 2 Ss S S s s,exp −−−− ′ ()            ′′ ∫ k H z dz d d 2 0 22 4 z z z ,.sss Ss  Γ U H z,s () HJ d() ,z,s q s z,q q q= − ()     () ⊥⊥⊥⊥ ∞ ∫ 41 0 2 0 0 π ε  K HC J ed m (z,s) z q s qq q q 2 = () − ()     ⊥ − ⊥⊥ ⊥ 41 1 22 0 2 π ε ++       ⊥ ∞ ∫ q q 2 0 2 ν . 0 TF1710_book.fm Page 217 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 218 Radio Propagation and Remote Sensing of the Environment In this case, we can neglect the role of small-scale turbulence. Then, it is convenient to use the relation: (7.30) known from the theory of Bessel functions. 44 Here, is the Macdonald func- tion, 44 and (7.31) Further, we are particularly interested in the case when Let us point out, however, that because we have neglected the role of small-scale turbulence it is also necessary to observe the inequality The following approximation is appli- cable for MacDonald functions at small values of argument: (7.32) This relationship is true when index p is not an integer number; however, the expansion has a rather different form when the subscripts are integers (which will not be considered here). We can see that the last term of Equation (7.32) should be kept at p < 1, and it is acceptable to have the second term in the opposite case. So, for the small separation, we have greater accuracy in those cases for which the correlation distance is much smaller than the turbulence scale: (7.33) At large values of argument, the MacDonald function tends exponentially to zero, and function H is equal to the first factor of Equation (7.31). Studying the problem in general, we can confirm that the spatial coherence of a wave propagating in a turbulent medium becomes small under the following the condition: . (7.34) Jb a d ab q Kab p p q pq q q qp xx x x () + () = + () + + − − 1 22 1 21Γ (() ∞ ∫ , 0 K p ()x H C K= − − − ()() () − − − 2 1 1 2 22 1 2 π ν ν ν ε ν ν ν qz) 1 qs 0 2 0 ( Γ 11 qs 0 ()         . qs<<1. 0 qs>>1. m K p p p p p p ()x x x ≅ () − − ()       − − () + − 2 1 1 1 2 1 1 1 2 ΓΓ Γ pp p ()               x 2 2 . H C ≅ () −−       + − () () 2 1 1 2 2 22 2 π νν ν ν ε qz qs 2 0 2 0 Γ Γ qqs 2 0               −22ν . k Hz z dz 2 0 4 1,s z ss z −− ′ ()       ≥ ∫ TF1710_book.fm Page 218 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radiowave Propagation in a Turbulent Medium 219 7.4 THE COHERENCE FUNCTION OF A PLANE WAVE Let us now consider a plane wave as an example of calculating the coherence function. In this case, the excitement sources are uniform, in the plane . So, for the plane wave, where is the density of the power source. It is easy to see in this case that integration with respect to gives us , and, as a result: (7.35) We shall assume hereafter that the turbulence spectral index is less than 2. We can only keep the second summand from Equation (7.33). We can now write, assuming constant turbulence parameters along the wave propagation direction, (7.36) Let us now determine the coherence scale by equating the exponent index to unity. The scale represented in such a way is defined from the equation: (7.37) At the same time, we must maintain to make Equation (7.36) valid. Obviously, it is possible if the wave has traveled such a large distance that: (7.38) Let us emphasize that the coherence scale is smaller than the turbulence scale, although it can be greater in those cases when a weak inequality substitutes for a strong one; also, in all of the cases considered here, extinction is appreciable and the field coherent component is small. For the Kolmogorov spectrum, , and the equation for the coherence scale can be formulated in the simple form: (7.39) where the distance traveled (z) was replaced with L . z0= Γ 00 2 2Ss, () = u u 0 2 ′ S 4 2 πδW () Γ U uk Hz dz(, exp , .ssz) z = − ()         ∫ 0 22 0 24 Γ Γ U u k (, exps z) qs z 0 = − − ()() − − 0 2 22 22 21 2 2 2 πν µ ν ν ν −− () − ()           132Γν q 0 . qs q 0cog 0 () = − () − () − () − − 22 21 22 2132 2 ν ν νν πνµ Γ Γ k zz . qs 0co g << 1 µµ 22 22 kklzq z>>1 0 ≅ . ν = 11 6 s cog l klL       = 5 3 22 107. , µ TF1710_book.fm Page 219 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 220 Radio Propagation and Remote Sensing of the Environment 7.5 THE COHERENCE FUNCTION OF A SPHERICAL WAVE The sources function for the spherical wave is given as the product: (7.40) The integration in Equation (7.27) is easily performed to give us: (7.41) after a simple transform. Here, the dependence on vector S appears distinct from the plane wave. In some instances, this dependence is not valid, such as when a spherical wave can be seen in the approach of a plane wave. This dependence disappears when we consider a field on a sphere of constant radius. So, further, we shall restrict ourselves to the case of S = 0. Instead of (7.36), we now have the following: (7.42) The following estimation is true for a spherical wave rather than Equation (7.39): . (7.43) The coherence radius for the spherical wave is approximately twice as large as that for the plane wave. Γ 0 2 0 2 2 2 22 ′′ () = ′ + ′       ′ − ′      Ss S s S s , π δδ u k  . Γ U u i k k HSs Ss s,, exp ,z z z z 4 z1 2 () = () −− ()   0 22 2 ζζ             ∫ dζ 0 1 Γ Γ U u k 0 2 2 0 2 22 2 2 2 ,, exps z z qs z 2 2 0 () = − − ()() − πν µ ν 22-1 0 q ν νν ν− () − () − ()           12 1 32Γ . s cog l klL       = 5 3 22 285. µ TF1710_book.fm Page 220 Thursday, September 30, 2004 1:43 PM . is of primary important (see Chapter 5). We © 2005 by CRC Press 212 Radio Propagation and Remote Sensing of the Environment (7. 2) Here s is the vector orthogonal to the main wave propagation. TF 171 0_book.fm Page 213 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 214 Radio Propagation and Remote Sensing of the Environment and the function (see Equations (4 .78 ) and (4 .79 )): (7. 11) It. TF 171 0_book.fm Page 2 17 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 218 Radio Propagation and Remote Sensing of the Environment In this case, we can neglect the role of small-scale