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FIGURE 4.16. Geometrical presentation of a quasi-smooth terrain described by Kirchhoff s approximation.
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ELECTROMAGNETIC ASPECTS OF WAVE PROPAGATION OVER TERRAIN
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is a superposition of the incident eld E0 and the eld E re ected from the plane LL0 . This plane is tangential to the surface S at point r, as shown in Figure 4.16. The scattered electromagnetic wave at the observation point R can be represented by the values of E and H on the surface S and by using the well-known Green s function presentation for the point source, G expfikRg, that is, R Ei r 2 S E0 expfikR1 g ; R1 Hi r 2 S H0 expfikR1 g R1 4:64
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The nal expression for the scattered led is E R ik eik R1 R2 f n H H0 n E E0 rr R2 R1 R2 4p
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rr R2 n E E0 gds Here, as follows from Figure 4.17, R1 and R2 are the distances from the current point r x; y; z 0 at the at surface z 0 to the source point O and the observation point e e R; R1 and R2 are the distances from the current point r x; y; z at the surface S over which the integration in (4.65) takes place; & r is the height of the surface S at the arbitrary point r x; y; z . If the source and observations point are located in the fare e eld zone relative to surface S, that is, k R1 ) 1 and k R2 ) 1, the integral in (4.65) for the scattered eld in the direction ks can be rewritten as E k; ks ik eik R10 R20 4p R10 R20 f n H H0 n E E0 rr R2
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Here R10 and R20 are the distances between the arbitrary point r x; y; z 0 on the surface S0 , which is the projection of the rough surface S at the plane z 0, and
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FIGURE 4.17. Re ection from a quasi-smooth terrain.
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the source O and observed point R, respectively. For future analysis of the integral in e e (4.66), it is convenient to present the distances R1 and R2 through the vector r x; y; z 0 that lies on the at surface z 0 and the value of surface height & r at this current point (see Figure 4.17): q r 2 z0 & 2 % R1 az & q e R2 d r 2 z & 2 % R2 bz & q p where R1 r 2 z2 ; R2 d r 2 z2 ; e R1
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z az R01 and bz Rz2 are the z-components of vectors a rr R1 and b rr R2 (i.e., the projections of these vectors at the z-axis). We analyze the expression (4.66) that describes the scattered eld for two cases that are useful in practice for overthe-terrain propagation by introducing some new variables according to References [9 14]: q ks k, k ka krr R1 , ks kb krr R2 . In the case of a perfectly conducting Earth s surface, the expression (4.66) can be simpli ed taking into account that the electric and magnetic components of the electromagnetic eld are mutually perpendicular, H0 k E0 =k; and that we concentrate only on the short-wave approximation qz & ) 1 , that is,
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4:68
After statistical averaging of integral (4.68), the average scattered eld can be presented as hE k; ks i E 0 k; ks Gf c where E 0 k; ks ik eik R10 R20 b E0 q 2p R10 R20 qz expf iq rgdr 4:70 4:69
is the eld re ected from area S0 of the plane z 0, and Gf c is the effective re ection coef cient from the rough terrain, which for the surface S with a Gaussian distribution, can be presented as [21 27] Gf c % expf 2k2 & 2 sin2 cg 4:71
Here, c is the slip angle (see Fig. 4.17). One can see that the effective re ection coef cient decreases exponentially with an increase of roughness height & r .