ANALYSIS OF ELECTROMAGNETIC WAVE PROPAGATION IN FOREST ENVIRONMENT ALONG MULTIPLE PATHS

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1 Progress In Electromagnetics Research, PIER 3, , 1999 ANALYSIS OF ELECTROMAGNETIC WAVE PROPAGATION IN FOREST ENVIRONMENT ALONG MULTIPLE PATHS L. W. Li, T. S. Yeo, P. S. Kooi, M. S. Leong, and J. H. Koh Centre for Microwave and RF Department of Electrical Engineering National University of Singapore 10 Kent Ridge Crescent, Singapore Introduction. Formulation of the Problem 3. Analytical Electric Fields in Closed Form 3.1 Direct and Multiple Reflected Waves 3. Lateral Waves 3..1 Along the air-canopy interface 3.. Along the canopy-trunk interface 3..3 Along the trunk-ground interface 4. Results and Discussion 5. Conclusions References Appendix. Formulas Used in the Analysis 1. INTRODUCTION The presence of forest foliage causes attenuation of radio waves along a radio path and reduces the communication range of the radio equipment. In planning a communication link, quantitative knowledge of the excess transmission loss suffered by the radio waves due to the presence of the foliage is essential. Radio wave propagation in vegetated environment has been the subject of intensive studies. Those earlier studies usually considered radio propagation in forests modeled as a homogeneous and isotropic dielectric layer placed over a conducting

2 138 Li et al. earth 1 5. In that analysis, the Hertz potential theory was used, and satisfactory results were obtained. Recently, a four-layered model has been widely adopted and commonly used in the analysis of propagation mechanism in forest environments 6 8. Two lossy dielectric layers placed over a semi-infinite ground plane are used to represent the canopy layer and trunk layer of the forest, respectively. To solve the problem of the electromagnetic wave propagation in this model, eigenfunction expansion of the dyadic Green s function is used in this paper. The use of the dyadic Green s function for the analysis of the electromagnetic wave propagation in semi-infinite media was described by Tai 9 and a generalization of these functions for the case of an N-layered medium 10, and expression for the coefficients of the scattered dyadic Green s functions in multi-layered medium have been recently obtained 11, 1. The similar ideas developed in 13 will be also implemented inside the current paper. In 13, the analysis has been conducted only on the z - components of the electromagnetic fields due to its length restriction, so that the overall effects of the forest and the global contributions due to all the electromagnetic field components cannot be seen clearly. In this paper, the dyadic Green s functions for the four-layered planar geometry are first determined. From these functions, all the the electric field components of the electromagnetic waves radiated from an inclined dipole antenna located in the trunk layer were obtained in Section, and then were evaluated in Section 3, using the saddle point technique which accounts for the direct and multiple reflected waves, and the branch-cut integration technique which gives contributions of lateral waves along various interfaces 14. The transmission losses for vertical, horizontal, and 45 inclined dipoles were finally calculated numerically in Section 4 for typical forests. Also, an error occurring in the publication 5 has been pointed out and corrected in this paper.. FORMULATION OF THE PROBLEM The geometry that will be considered is shown in Fig. 1. The layer II represents a medium of tree crowns (canopy) with dielectric parameters ɛ,µ and σ, while layer III represents a medium of tree trunks (trunk) with dielectric parameters ɛ 3,µ 3 and σ 3. The height of layer II measured along the z-direction is H 1, while the height of layer III is H. Throughout the analysis, the air (layer I) and ground (layer IV) are assumed to extend to infinity, with dielectric parameters given

3 Analysis of electromagnetic wave propagation in forest 139 Figure 1. Geometry for the forest model of four-layers. by ɛ 0 and µ 0, and ɛ 4,µ 4 and σ 4, respectively. An electric dipole with an inclination angle α with respect to the boundary plane between layers and an observation point are both located in layer III. The source that feeds the electric dipole has a harmonic time dependence given by exp( jωt ). The dielectric permittivities of the four layers are expressed in their complex form, i.e., ɛ l = ɛ rl ɛ o = ɛ l (1 jσ l/ωɛ l ) where ɛ rl ( l =1,, 3, 4 ) is the relative complex dielectric constant, σ l the conductivity of medium l, and ɛ o the free space permittivity. The wavenumbers for the four layers are k 1 in the air semi-space, k in the canopy layer, k 3 in the trunk layer, and k 4 in the semi-space ground, where k n = ω (µ o ɛ n ). The magnetic permeability µ l in every layer is taken equal to that µ 0 of the free space. The current density of an electric dipole with an inclination α with respect to the boundary plane between layers, as shown in Fig. 1, may be expressed as J 3 (r )=(P x x P z ẑ)δ(x 0)δ(y 0)δ(z z o ) =P x (cos φ ρ sin φ φ)pz ẑ δ(ρ )δ(z z o ) πρ, (1) where z 0 = z is the height of the dipole, and P x and P z are the horizontal and vertical dipole moments, respectively.

4 140 Li et al. Using the dyadic Green s function for a four-layered planar geometry given by Li et al. 1, and following a procedure similar to that described by Cavalcante et al. 5, one obtains for the region H z z : E 3 (r) = ωµ 3 4π 0 dλ h 3 Px { (1 A 33 M)M o1λ (h 3 )C 33 M M o1λ (h 3 ) e jh3z B 33 M M o1λ (h 3 )D 33 MM o1λ (h 3 ) e jh3z jh 3 k 3 (1 A 33 N )N e1λ (h 3 )C 33 N N e1λ (h 3 ) e jh3z jh 3 k 3 B 33 N N e1λ (h 3 )D 33 N N e1λ (h 3 ) e jh3z } P { zλ (1A 33 N k 3 ) Ne0λ (h 3 )C 33 N N e0λ (h 3 ) e jh3z B 33 N N e0λ (h 3 )D 33 N N e0λ (h 3 ) e jh3z }, () and the the region 0 z z : E < 3 (r) = ωµ 3 4π where h l = 0 dλ { A 33 P x h MM o1λ (h 3 )CM 33 M o1λ (h 3 ) e jh3z 3 B 33 M M o1λ (h 3 )(1D 33 M)M o1λ (h 3 ) e jh3z jh 3 k 3 A 33 N N e1λ (h 3 )C 33 N N e1λ (h 3 ) e jh3z jh 3 B 33 N N e1λ (h 3 ) ( 1DN 33 k 3 ) Ne1λ (h 3 ) e jh3z } P { zλ A 33 N N e0λ (h 3 )CN 33 N e0λ (h 3 ) e jh3z k 3 BN 33 N e0λ (h 3 ) ( 1DN 33 ) Ne0λ (h 3 ) e jh3z }, (3) BM,N 33, C33 M,N, D33 M,N k l λ ( l =1,, 3 and 4), the coefficients A 33 M,N, of scattered dyadic Green s functions are expressed by Li et al. 1 and are not given here to save space. Using the expressions for Me o nλ(h) and Ne o nλ(h) (see Tai 9 and Li et al. 1), expressions in cylindrical coordinates for the field com-

5 Analysis of electromagnetic wave propagation in forest 141 ponents of () and (3) may be obtained as E 3z E 3z < = jωµ o λ { dλ Φ 4πk3 0 D3 V P x J 1 (λρ) cos φ Φ < jp } zλ Φ J 0 (λρ) h 3 Φ <, (4a) E E < = ωµ { { o dλ J1 (λρ) 1 P x cos φ 4π 0 h 3 ρ D3 H Φ ( ) 1ρ h3 dj 1 (λρ) 1 Φ } Φ < ρ 1ρ k 3 dρ D3 V Φ < ρ ± jp zλh 3 dj 0 (λρ) 1 Φ } k3 dρ D3 V Φ <, (4b) E 3φ E 3φ < = ωµ { o dλ dj1 (λρ) 1 Φ P x sin φ 1φ 4π 0 h 3 dρ D3 H Φ < 1φ ( ) h3 J 1 (λρ) 1 Φ } φ, (4c) ρ k 3 D V 3 Φ < φ < where the intermediates Φ± and Φ < (1,,3)(ρ,φ) are derived as Φ ± = 1 ± R3 V e jh3z { R V R1 V e jhh1 e jh3(hz) } 1R1 V R V e jhh1 e jh3(zz ), Φ < ± = 1 R3 V e jh3z{ R V R1 V e jhh1 e jh3(hz ) } ± 1R1 V R V e jhh1 e jh3(z z), Φ 1ρ = 1 R3 H e jh3z { e jh3(zz ) 1R1 H R H e jhh1 R H R1 H e jhh1 e jh3(hz)}, Φ < 1ρ = 1 R3 H e jh3z { e jh3(zz ) 1R1 H R H e jhh1 R } H R1 H e jhh1 e jh3(hz ), Φ ρ = 1R3 V e jh3z { e jh3(zz ) 1R1 V R V e jhh1 R V R1 V e jhh1 e jh3(hz)}, (5a) (5b) (5c) (5d) (5e)

6 14 Li et al. Φ < ρ = 1R3 V e jh3z { e jh3(zz ) 1R1 V R V e jhh1 R } V R1 V e jhh1 e jh3(hz ), (5f) Φ = 1 R3 V e jh3z { e jh3(zz ) 1R1 V R V e jhh1 R V R1 V e jhh1 e jh3(hz)}, (5g) Φ < = 1R3 V e jh3z { e jh3(zz ) 1R1 V R V e jhh1 R } V R1 V e jhh1 e jh3(hz ), (5h) Φ 1φ = 1 R3 H e jh3z { e jh3(zz ) 1R1 H R H e jhh1 R H R1 H e jhh1 e jh3(hz)}, (5i) Φ < 1φ = 1 R3 H e jh3z { e jh3(zz ) 1R1 H R H e jhh1 R } H R1 H e jhh1 e jh3(hz ), (5j) Φ φ = 1R3 V e jh3z { e jh3(zz ) 1R1 V R V e jhh1 R V R1 V e jhh1 e jh3(hz)}, (5k) Φ < φ = 1R3 V e jh3z { e jh3(zz ) 1R1 V R V e jhh1 R } V R1 V e jhh1 e jh3(hz ) ; (5l) the reflection coefficients R H,V i ( i =1, and 3 ) for TE (H) mode and TM (V) mode are given respectively by R H f R V f = µ f h f1 µ f1 h f µ f h f1 µ f1 h f, (6a) = µ f h f kf1 µ f1h f1 kf µ f h f kf1 µ f1h f1 kf ; (6b) and the denominator D H,V 3 is given by: D V,H 3 =1R V,H 1 R V,H e ihh1 R V,H R V,H 1 e ihh1 R V,H 3 e ih3h. (7)

7 Analysis of electromagnetic wave propagation in forest ANALYTICAL ELECTRIC FIELDS IN CLOSED FORM To evaluate the integrals, it is convenient to convert the range of the integral from (0 ) to( ). Thus the Bessel functions and their derivatives need to be transformed to the Hankel functions and their derivatives. Since the frequency of interest here is 100 MHz or above, the phase k 0 r in free space is πfr/c π r/ = (π/3)r. As the transmitter-receiver distance r is at least the height of the vegetation layer height, i.e. r 0 m, so that the condition k 0 r 1 is always satisfied. Thus, the Hankel functions and derivatives used may be replaced by their asymptotic expressions. In addition, one over D H,V 3 may be expressed in an infinite series form (binomial expansion). For details of the field contributions, we separate the field into two parts E x and E z due to contributions of horizontal ( x ) and vertical ( z ) dipole moments, respectively, and waves associated with TE and TM modes. After exchanging the integration and the summation operations and using the complex transformation λ = k 3 sin β, the field integrals may be written as E z 3z,TM E 3z,TM <z = ωµ o 8π P k 3 z E x 3z,TM E <x 3z,TM E z,t M E <z,t M E x,t M E <x,t M E x,t E E <x,t E E x 3φ,T M E <x 3φ,T M E x 3φ,T E E <x 3φ,T E = ωµ o 8π P x cos φ = ωµ o 8π P z πρ ej π 4 k 3 πρ = ωµ o 8π P x cos φ = ωµ o 8πρ P x cos φ = ωµ o 8πρ P x sin φ = ωµ o 8π P x sin φ L 3z k 3 πρ ej π 4 3π ej 4 L < 3z K 3z L k 3 πρ ej π 4 L <, (8a) K < 3z, (8b), (8c) K 3π ej 4 πk 3 ρ 3π ej 4 πk 3 ρ k 3 πρ ej π 4 K < G G < K 3φ G 3φ G < 3φ K < 3φ, (8d), (8e), (8f), (8g)

8 144 Li et al. where the parameters in the integral domain are given by L 3z L < = q m sin 5 Φ β 3z Φ <, (9a) L L < = q m sin 3 Φ β cos β Φ <, (9b) K 3z K 3z < = q m sin 3 Φ β cos β Φ <, (9c) K K < = q m sin 1 Φ β cos β ρ Φ <, (9d) ρ K 3φ Φ K 3φ < = q m sin 1 β cos β φ Φ <, (9e) φ G G < = q m sin 1 Φ β 1ρ Φ <, (9f) 1ρ G 3φ Φ G 3φ < = q m sin 1 β 1φ Φ <, (9g) 1φ and q m = dβe jk sin β (1) (R ) m V R1 V e jh1 k k3 sin β Γ 0 R3 V e jhk3 cos β R1 V R V e jh1 k k3 m sin β, (10a) q m = dβe jk sin β (1) (R ) m H R1 H e jh1 k k3 sin β Γ 0 R3 H e jhk3 cos β R1 H R H e jh1 k k3 m sin β. (10b) To evaluate the integrals, the steepest descent method and the branch cut integration technique are used. For more information on the method, References 15 and 16 contain a good account of its meaning and application. 3.1 Direct and Multiple Reflected Waves The integrals for the electromagnetic field components have been evaluated in this paper using the steepest descent method and hence expressed in terms of direct and multiple reflected waves. Following similar procedures as in 17 and 5, the saddle point part of the integrals have been evaluated and their solutions are given as (in the

9 Analysis of electromagnetic wave propagation in forest 145 following, stands for or < ): E z,s 3z,TM = ωµ 0P z k 3 5π ej 8 4π k 3 Im {1R V 1,s(β V 1)R,s(β V 1)e bh1 a 1 R,s(β V )R 1,s(β V )e bh1 a R3,s(β V 3) 1R 1,s(β V 3)R,s(β V 3)e bh1 R3,s(β V 4) R,s(β V 4)R 1,s(β V 4)e bh1 E x,s 3z,TM = ωµ 0 4π P k 3 5π x cos φ ej 8 k 3 {1R1,s(β V 1 )RV,s(β 1 )ebh1 I V m E <x,s 3z,TM = ωµ 0 4π P x cos φ a 3 a 4 }, (11a) a 1 c 1 R,s(β V )RV 1,s(β )ebh1 a c R3,s(β V 3 ) 1R1,s(β V 3 )RV,s(β 3 )ebh1 a 3 c 3 } R3,s(β V 4 ) R,s(β V 4 )RV 1,s(β 4 )ebh1 a 4 c 4,(11b) I V m E z,s,t M = ωµ o 4π P z k 3 5π ej 8 k 3 { 1R1,s(β V 1 < )RV,s(β 1 < )ebh1 a < 1 c< 1 R,s(β V < )RV 1,s(β < )ebh1 a < c< R3,s(β V 3 < ) 1R1,s(β V 3 < )RV,s(β 3 < )ebh1 a < 3 c< 3 } R3,s(β V 4 < ) R,s(β V 4 < )RV 1,s(β 4 < )ebh1 a < 4 c< 4, (11c) I V m k 3 k 3 e 3π 8 {1R V 1,s(β 1 )RV,s(β 1 )ebh1 a 1 e 1

10 146 Li et al. E <z,s,t M = ωµ o 4π P z R,s(β V )RV 1,s(β )ebh1 a e R3,s(β V 3 ) 1R1,s(β V 3 )RV,s(β 3 )ebh1 a 3 e 3 R3,s(β V 4 ) R,s(β V 4 )RV 1,s(β 4 )ebh1 a 4 e 4 k 3 k 3 e 3π 8 Im {1R V 1,s(β V 1 < )RV,s(β 1 < )ebh1 a < 1 e< 1 R,s(β V < )RV 1,s(β < )ebh1 a < e< R3,s(β V 3 < ) 1R1,s(β V 3 < )RV,s(β 3 < )ebh1 a < 3 e< 3 R3,s(β V 4 < ) R,s(β V 4 < )RV 1,s(β 4 < )ebh1 a < 4 e< 4 k 3 5π e 8 k 3 {1R1,s(β V 1)R,s(β V 1)e bh1 E x,s,t M = ωµ o 4π P x cos φ I V m },(11d) }, (11e) a 1f1 R,s(β V )R 1,s(β V )e bh1 a f R3,s(β V 3) 1R 1,s(β V 3)R,s(β V 3)e bh1 a 3f3 } R3,s(β V 4) R,s(β V 4)R 1,s(β V 4)e bh1 a 4f4, (11f) E x,s,t E = ωµ o 4πρ P x cos φ I H m e 9π 8 k3 k 3 {1R1,s(β H 1)R,s(β H 1)e bh1 a 1 R,s(β H )R 1,s(β H )e bh1 a R3,s(β H 3) 1R 1,s(β H 3)R,s(β H 3)e bh1 a 3 } R3,s(β H 4) R,s(β H 4)R 1,s(β H 4)e bh1 a 4, (11g)

11 Analysis of electromagnetic wave propagation in forest 147 E x,s 3φ,T M = ωµ o 4πρ P x sin φ I V m e 9π 8 k3 k 3 {1R1,s(β V 1)R,s(β V 1)e bh1 a 1g1 R,s(β V )R 1,s(β V )e bh1 a g R3,s(β V 3) 1R 1,s(β V 3)R,s(β V 3)e bh1 a 3g3 } R3,s(β V 4) R,s(β V 4)R 1,s(β V 4)e bh1 a 4g4, (11h) E x,s 3φ,T E = ωµ o 4π P k 3 5π x sin φ e 8 k 3 Im {1R H 1,s(β H 1)R,s(β H 1)e bh1 a 1 R,s(β H )R 1,s(β H )e bh1 a R3,s(β H 3) 1R 1,s(β H 3)R,s(β H 3)e bh1 a 3 } R3,s(β H 4) R,s(β H 4)R 1,s(β H 4)e bh1 a 4, (11i) R H,V where the intermediate parameters such as θl, β l l,s, a l, b, c l, d ±, e l, f l, and g l have been given in the Appendix. 3. Lateral Waves The solutions of the integrals in (8b) (8g) can be evaluated using the residue theorem in a way similar to that available in the literature 17, 15 and 16. Correspondingly, the branch cut contributions of h = 0 and h 4 = 0 to the fields have been obtained, respectively. The lateral waves along three different ways namely, the lateral waves at the air-canopy interface, at the canopy-trunk interface, and at the trunk-ground interface have been from the analysis. The mixed modes propagation mechanisms of radio waves in the trunk layer radiated by a dipole antenna located inside the trunk layer have been clearly shown in Fig.. For the cases that are considered here, while the lateral waves along the second and third interfaces are highly attenuated due to the

12 148 Li et al. Figure. Mixed mode propagation of the radio waves radiated by a dipole antenna located inside the forest trunk layer. high conductivity of canopy and ground media, the lateral wave at the air-canopy interface is propagating primarily in a lossless medium Along the air-canopy interface Using the parameters given in the appendix, the branch cut contribution due to branch point h 1, which is equivalent to lateral wave along the air-canopy interface, can be expressed as L c1 3z = jn n 1 K c1 3z = L c1 = jn K c1 = G c1 = j π(1 n 1 )1/ n e j π 4 n 1 e jb1 b 3/ (1) m Γ 1 d m 1 a 1 mc 1d, d 1 (1a) 1 n 1 Γ L,c1 3z, (1b) n 1 Γ 1 π n (1 n 1) 3 4 e j π e jb1 4 n 1 b 3/ (1) m Γ 1 d m 1 a mc 1d 3, d 1 (1c) 1 n 1 Γ L c1 n 1 Γ, (1d) 1 π(1 n 1 )1/ n e j π e jb1 4 (1) m Γ 3 d m n 4 a 3 mc d 5, (1e) 1 d 4 b 3/

13 Analysis of electromagnetic wave propagation in forest 149 K c1 3φ = 1 n 1 K c1, G c1 3φ = n 1G c1, (1f) (1g) and K <c1 3z = jn π(1 n 1 )3/ n e j π 4 n 1 e jb1 b 3/ L <c1 = jn π n (1 n 1) 3 4 e j π 4 n 1 (1) m Γ 4 d m 1 e jb1 b 3/ (1) m Γ 5 d m 1 a 4 mc 1d 6, d 1 (13a) a 5 mc 1d 7, d 1 (13b) and the expressions for L <cl 3z, K<c1, G<c1, K 3φ <c1, and G<c1 3φ are given by (1a), (1d), (1e), (1f), and (1g) respectively, with the positions of z and z interchanged. Similar procedures can be employed to obtain the branch cut contributions due to h and h 4, which accounts for the lateral waves at canopy-trunk interface, and trunk-ground interface. The complete solutions of the lateral waves along the two interfaces are presented below. 3.. Along the canopy-trunk interface Using the parameters given in the appendix, L c 3z = jn πe j π e jb1 4 b 3/ Γ 1 e jmh3h n 1 (1 n )1/4 a n 1 n 1 (1 n ) 1/4 a, (14a) K c 3z = L c 1 n Γ L c 3z n Γ, (14b) 1 = j π(1 n )ej π 4 a 1 (1 n ) 1 4 a n 1 e jb1 b 3/ Γ 1 e jmh3h (1 n )1/ n 1 n, (14c)

14 150 Li et al. K c = 1 n n Γ Γ 1 L c, (14d) G c = j π(1 n ) 1 4 e j π e jb1 4 b 3/ a a 1 n 1 n n 3 n Γ 3 e jmh3h, (14e) and K c 3φ = 1 n K c, (14f) G c 3φ = n G c, (14g) 1 n K 3z <c = jn πe j π 4 n L <c e jb1 b 3/ Γ 4 e jmh3h n 1 (1 n )1/4 a n 1 n 4 (1 n ) 1/4 a 3, (15a) = j π(1 n )ej π e jb1 4 a 4 (1 n ) 1 4 a 3 n 1 b 3/ Γ 5 e jmh3h (1 n )1/ n 1 n, (15b) and the expressions for L <c 3z, K <c, G<c, K 3φ <c, and G<c 3φ are given by (14a), (14d), (14e), (14f), and (14g) respectively, with the positions of z and z interchanged Along the trunk-ground interface Again, using the parameters give in appendix, L c4 3z π = jn 4 (1 n 4 )1/ ej π 4 e jb1 b 3/ (1) m Γ 1 a m 3 a 1 mc 1a, a 3

15 Analysis of electromagnetic wave propagation in forest 151 and K c4 3z = (16a) 1 n 4 n 4 L c4 3z with a 1 = e jh3z, (16b) L c4 = j π(1 n 4) 1 4 e j π 4 K c4 = G c4 e jb1 b 3/ (1) m Γ a m 3 a 1 mc 1a, a 3 (16c) 1 n 4 L c4 with a 1 = e jh3z, (16d) n 4 π = j (1 n 4 ) e j π e jb1 1 4 b 3/ (1) m Γ 3 a m 5 a 1 mc a 4, a 5 (16e) K 3φ c4 K c4 n 4 (16f) G 3φ c4 4G c4 (16g) K <c4 3z = j π(1 n 4 )1/ e j π 4 L <c4 = j π(1 n 4) 1 4 e j π 4 e jb1 (1) m Γ 4 a m 3 a 6 mc 1a 7, a 3 (17a) (1) m Γ 5 a m 3 a 8 mc 1a 9. a 3 (17b) b 3/ e jb1 b 3/ Expressions for L <c4 3z, K <c4, G<c4, K 3φ <c4, and G<c4 3φ used as above are given by (16a), (16d), (16e), (16f), and (16g), respectively, with the positions of z and z interchanged. This concludes the analysis of all the radio waves propagating in such a four-layered medium. 4. RESULTS AND DISCUSSION The forest model described can be employed to estimate or predict the performance of communication systems operating within forested environment. This section looks at the transmission loss when the transmitter and the receiver are both located in the same layer (trunk). The radio loss L from an inclined dipole to a vertical receiving antenna

16 15 Li et al. is given by 1 L(dB) = log 10 f 0 log 10 E o 0 log 10 E (18) where f is the frequency in GHz and E o is the reference field in the absence of the forest given by E o = ωµ o 4π P x P z ρ (z z ). (19) Having obtained all the different components of the electric field, we can find the total overall electric field which is given by E 3, total = E 3z E E 3φ. (0) It should be pointed out that the optimum angle obtained in 5 is not correct. It is simply because there is a constant factor missing in the presentation of the electric field z-component. If we consider the total contribution of all the components as given in (0), the expression of the optimum angle in 5 is also incorrect. For a tropical dense forest, the ground has a relative permittivity ɛ r = 50 and conductivity σ = 0.1 S/m, the canopy medium has relative permittivity ɛ r = 40 and conductivity σ = S/m, and the trunk medium has a relative permittivity ɛ r = 35 and conductivity σ = S/m. The transmitter is located at a height of 10 m and the receiver at a height of 15 m. Heights of trunk and canopy layers are typically 10 and 0 m, respectively. Using these parameters for the forest, transmission loss has been computed for different orientations of the dipole transmitter and various frequencies, and depicted subsequently from Fig. 3 to Fig. 8. The path losses for the vertical, horizontal and 45 inclined dipoles are shown in Fig. 3 as functions of the distance between transmitter and receiver at 500 MHz, along the x-direction. As observed, direct and multiple reflected waves dominate the field at near zone. Lateral waves become, however, the dominant radio wave propagation mechanism when the distance increases. Transmission attenuation saturates at large distances since propagation is dominated by lateral wave along the lossless air medium at the air-canopy interface. Above 4 km at this frequency, electric field is primarily determined from the lateral wave. For our case, overall electric field from a horizontal dipole suffers less

17 Analysis of electromagnetic wave propagation in forest 153 Figure 3. Attenuation of total electric field due to a dipole with different orientations in the trunk layer at 500 MHz and φ =0. Figure 4. Attenuation of total electric field due to a vertical dipole in the trunk layer.

18 154 Li et al. Figure 5. Attenuation of total electric field due to a horizontal dipole in the trunk layer at φ =0. Figure 6. Attenuation of total electric field due to a 45 inclined dipole in the trunk layer at φ =0.

19 Analysis of electromagnetic wave propagation in forest 155 Figure 7. Attenuation of total electric field at 500 MHz and ρ =km. Figure 8. Attenuation of total electric field at 500 MHz and ρ =6km.

20 156 Li et al. attenuation as compared to that from a vertical dipole, about 14 db lesser. It is also observed from Fig. 3 that, for given conditions the radio wave due to a horizontal dipole has least attenuation in the far zone and most transmission loss in the near zone as compared to the other orientations, and certainly the wave due to a vertical dipole has the inverse conclusion. The effect of frequency on the transmission loss experienced by the radio wave propagation in the presence of forest is shown in Figs. 4, 5 and 6. As frequency increases, there is a trend of increasing attenuation for all three orientation of dipoles considered here. In the near zone (from 0 to 4 km without respect to the wavelength), the speed of such a attenuation increase is much higher than that in the far zone (from 4 km to 8 km or above). The effect of varying the position of the receiver in the cylindrical angle, φ on the attenuation experienced by the electric field is shown in Figs. 7 and 8. Apparently, the field is not isotropic with respect to the polar angle φ, but periodic instead. For field dominated by direct and multiple reflected wave, received overall field suffers least attenuation at 90 from horizontal dipole, and at around 70 from dipole with a 45 inclination. For lateral wave dominated field, least attenuation occurs at 180 for both horizontal and inclined dipoles. 5. CONCLUSIONS In this paper, the propagation of electromagnetic waves in a medium with four horizontal layers has been studied using dyadic Green s functions. This method presents some advantages as (i) the dyadic Green s functions outside the source region allow a higher degree of flexibility with respect to the space coordinates and assure a good exponential convergence; (ii) the source may contain an arbitrary current distribution; and (iii) the medium may be isotropic or anisotropic. The exact integral solution of the electric field due to an electric dipole of an arbitrarily inclination has been obtained. The asymptotic expressions of all the electric field components are further obtained by the use of the saddle point technique and the branch cut integrations. Correspondingly, three types of waves, i.e., the direct wave, the (multi-)reflected waves, and lateral waves, have been represented in their closed forms. The transmission losses of these waves have been computed numerically and compared each other. It is found that the only the lateral wave propagating along the upper air-canopy interface dominates the

21 Analysis of electromagnetic wave propagation in forest 157 total field in the far zone while the direct wave together with reflected waves of several hops play an important role in the total field in the quite near zone. The results presented in this paper give an insight into the behavior of the electromagnetic waves due to an inclined dipole located in the trunk layer and into the mechanism of the mixed mode propagation of the waves in such a four-layered forest model. APPENDIX. FORMULAS USED IN THE ANALYSIS Note that there is a repetition of the parameters used in the analysis of the saddle point part and lateral wave contributions of the field. A.1 Direct and Multiple Reflected Waves Ray-path distances are given by: r ± = ρ (mh z ± z ), (1a) r ± = ρ (m 1)H z ± z, (1b) r ± < = ρ (mh z ± z), (1c) r < ± = ρ (m 1)H z ± z. (1d) Angles of incidence, β l ( l =1,, 3, 4 ), and angles of total internal reflection, θ l ( l =1,, 3, 4 ) are listed by: ( θ 1 = mh z z ) sin1 r, (a) ( ) θ = (m 1)H z z sin1, (b) r ( θ 3 = mh z z ) sin1 r, (c) ( ) θ 4 = (m 1)H z z sin1, (d) r ( θ 1 < = mh z ) z sin1 r <, (e) ( ) θ < = (m 1)H z z sin1, (f) r <

22 158 Li et al. ( θ 3 < = mh z ) z sin1 r <, (g) ( ) θ 4 < = (m 1)H z z sin1, (h) r < β 1 = π θ 1, β = π θ, β 3 = π θ 3, β 4 = π θ 4, (i) β 1 < = π θ< 1, β< = π θ<, β< 3 = π θ< 3, β< 4 = π θ< 4. (j) Reflection coefficients at the interfaces ( i = 1,, 3 and 4, and again denotes < or ) are: R1,s(β V i )= k k1 k 3 sin βi k 1 k k 3 sin βi, (3a) k k1 k 3 sin βi k 1 k k 3 sin βi R,s(β V i )= k 3 k k 3 sin βi k k 3 cos βi, (3b) k3 k k 3 sin βi k k 3 cos βi R3,s(β V i )= k 4 k 3 cos βi k 3 k4 k 3 sin βi, (3c) k4 k 3 cos βi k 3 k4 k 3 sin βi k R1,s(β H i )= k3 sin β i k1 k 3 sin βi, (3d) k k3 sin β i k1 k 3 sin βi R,s(β H i )= k 3 cos βi k k 3 sin βi, (3e) k 3 cos βi k k 3 sin βi k R3,s(β H i 4 )= k 3 sin βi k 3 cos βi. (3f) k4 k 3 sin βi k 3 cos βi Substituted parameters (where stands for either < or ) are given as follows: Im H,V = R H,V k 3,s (RH,V,s RH,V 1,s ejh1 k3 )

23 Analysis of electromagnetic wave propagation in forest 159 a 1 = ejk3r ρr, b = j k k 3, R H,V 1,s RH,V,s ejh1 k k 3 m (1) m, (4a) a = ejk3r, ρr a 3 = e jk 3r, ρr a 4 = ejk3r, (4b) ρr (4c) c 1 = d ρ, c = d ρ, c 3 = d ρ, c 4 = d ρ, (4d) e 1 = ρ d (r, )3 e = ρ d (r ) 3, e 3 = ρ d (r, )3 e 4 = ρ d (r ) 3, (4e) f1 = ρ(d ) (r, )3 f = ρ(d ) (r ) 3, f 3 = ρ(d ) (r, )3 f 4 = ρ(d ) (r ) 3, (4f) where g1 = (d ) ρr, g = (d ), g ρr 3 = (d ) ρr, g4 = (d ), (4g) ρr d ± =mh z ± z, (5a) d ± =(m 1)H z ± z, (5b) d < ± =mh z ± z, (5c) d < ± =(m 1)H z ± z. (5d) A. Lateral Wave Reflection coefficients at interfaces are provided by: k R,c1 H = 3 k1 k k 1 k 3 k1 k, (6a) k 1 k R3,c1 H = 4 k1 k 3 k 1 k 4 k1 k3, (6b) k 1 R,c1 V = k 3 k k1 k k 3 k1 k3 k k1, (6c) k k 3 k1 R3,c1 V = k 4 k 3 k1 k 3 k 4 k1 k4 k 3 k1, (6d) 3 k k 4 k1

24 160 Li et al. k R3,c H = 4 k k 3 k k 4 k k3, (7a) k R3,c V = k 4 k 3 k k 3 k 4 k k4 k 3 k, (7b) 3 k k 4 k k R1,c4 H = k4 k 1 k 4 k k4 k1, (8a) k 4 k R,c4 H = 3 k4 k k 4 k 3 k4 k, (8b) k 4 R,c4 V = k k 1 k4 k 1 k k4 k k 1 k4, (8c) 1 k k k4 R,c4 V = k 3 k k4 k k 3 k4 k3 k k4. (8d) k k 3 k4 A..1 Along the air-canopy interface Substituted parameters are given by: a 1 = e jhh1 R,c1e V jh3(zh) 1, a = e jhh1 R,c1e V jh3(zh) 1, a 3 = e jhh1 R,c1e H jh3(zh) 1, a 4 = e jhh1 1 R,c1e V jh3(z H ), a 5 = e jhh1 R,c1e V jh3(z H ) 1, (9a) (9b) (9c) (9d) (9e) b 1 = k 1 ρ k 3 (m 1)H z z 1 n 1, (30a) b = k 3 ρ 1 n 1 k 1(m 1)H z z, (30b) c 1 = e jhh1 R3,c1 V R,c1e V jh3h, c = e jhh1 R3,c1 H R,c1e H jh3h, (31a) (31b)

25 Analysis of electromagnetic wave propagation in forest 161 d 1 = R,c1R V 3,c1 V e jhh1 R3,c1 V R,c1e V jh3h, (3a) d = e jh3(zh) R,c1 V e jhh1 R,c1e V jh3(zh) 1, (3b) d 3 = e jh3(zh) R,c1 V e jhh1 R,c1e V jh3(zh) 1, (3c) d 4 = e jhh1 R3,c1 H R,c1e H jh3h R,c1R H 3,c1, H (3d) d 5 = e jhh1 R,c1e H jh3(zh) 1R,c1 H e jh3(zh), (3e) d 6 = R,c1 V e jh3(z H ) e jhh1 1 R,c1e V jh3(z H ), (3f) d 7 = e jh3(z H ) R,c1 V e jhh1 R,c1e V jh3(z H ) 1, (3g) Γ 1 =1R3,c1e V jh3z, Γ =1R3,c1e V jh3z, Γ 3 =1R3,c1e H jh3z, Γ 4 =1R3,c1e V jh3z, Γ 5 =1R3,c1e V jh3z. (33a) (33b) (33c) (33d) (33e) A.. Along the canopy-trunk interface Substituted parameters are given by: a 1 =1e jh3(zh), a =1e jh3(zh), a 3 =1e jh3(z H ), a 4 =1e jh3(z H ), (34a) (34b) (34c) (34d) b 1 = k ρ k 3 (m 1)H z z 1 n, (35a) b = k 3 ρ 1 n k (m 1)H z z, (35b) Γ 1 =1R3,ce V jh3z, Γ =1R3,ce V jh3z, Γ 3 =1R3,ce H jh3z, Γ 4 =1R3,ce V jh3z, Γ 5 =1R3,ce V jh3z. (36a) (36b) (36c) (36d) (36e)

26 16 Li et al. A..3 Along the trunk-ground interface Substituted parameters are given by: a 1 = e jh3z, a =1e jh3z, a 3 = R,c4 V R1,c4e V jhh1 R1,c4R V,c4e V j(hh1h3h), a 4 =1e jh3z, a 5 = R1,c4R H,c4e H j(hh1h3h) R,c4 H R1,c4e H jhh1, a 6 = e jh3z, a 7 =1e jh3z, a 8 = e jh3z, a 9 =1e jh3z, (37a) (37b) (37c) (37d) (37e) (37f) (37g) (37h) (37i) b 1 = k 4 ρ k 3 (m 1)H z z 1 n 4, (38a) b = k 3 ρ 1 n 4 k 4(m 1)H z z, (38b) c 1 = R,c4 V R1,c4e V jhh1, c = R,c4 H R1,c4e H jhh1, (39a) (39b) Γ 1 =(1R1,c4R V,c4e V jhh1 )e jh3(zh) (R,c4 V R1,c4e V jhh1 ), (40a) Γ =(1R1,c4R V,c4e V jhh1 )e jh3(zh) (R,c4 V R1,c4e V jhh1 ), (40b) Γ 3 =(1R1,c4R H,c4e H jhh1 )e jh3(zh) (R,c4 H R1,c4e H jhh1 ), (40c) Γ 4 =(R,c4 V R1,c4e V jhh1 ) (1 R1,c4R V,c4e V jhh1 )e jh3(z H ), (40d) Γ 5 =(R,c4 V R1,c4e V jhh1 ) (1 R1,c4R V,c4e V jhh1 )e jh3(z H ). (40e)

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