Statistic propagation loss model

Several statistic models for estimating the propagation loss have been proposed. Zhang model has been internationally accepted [Reference Zhang7]. The main propagation loss of a troposcatter channel can be given as

(2)$$L = F + 30{\rm 1g}\;f + 30{\rm 1g}\Theta _0 + 10{\rm 1g}\;d + 20{\rm 1g}(5 + \gamma H) + 4.34\gamma h_0 + L_c,$$

where Θ₀ refers to the least scatter angle (mrad), which can be calculated as Θ₀ = θ _t + θ _r + 1000d/a _e, here, a _e is the median effective earth radius. θ _t, θ _r are the transmitter and receiver horizon angle, respectively. F denotes the meteorological parameter (dB), γ represents the atmospheric structure coefficient (km⁻¹), f denotes the frequency (MHz), d refers to the path length between the receiving and transmitting antennas (km). H denotes the height between the lowest scattering point and the connection between receiving and transmitting antennas (km), which can be expressed as H = 10⁻³Θ₀d/4. h ₀ represents the height between the lowest scattering point and ground (km), which can be calculated as $h_0 = 10^{-6}\Theta _0^2 a_e/8$.

The aperture-to-medium coupling loss of an antenna [Reference Li, Wu, Lin, Zhang and Zhao8] can be generally expressed as

(3)$$L_c = 0.07 \times e^{0.055 \times (G_t + G_r)},$$

where G _t, G _r stand for the plane wave gains of TX/RX antennas, respectively. Both of them can be given as

(4)$$G_{t,r} = 101{\rm g}[4.5 \times (D/\lambda )^2],$$

where D denotes the antenna diameter. Therefore, L _Σ can be finally calculated as

(5)$$L_\Sigma = L-(G_T + G_R).$$

Although Zhang model can effectively estimate the propagation loss, the time resolution and spatial resolution are relatively poor.

Dynamic propagation loss model

The received power of a troposcatter link can be defined as [Reference Li, Chen and Liu5, Reference Dinc and Akan9]

(6)$$P_r = P_t\int_V {\displaystyle{{\lambda ^2G_TG_R\sigma _v\rho} \over {{(4\pi )}^3{(r_1r_2)}^2}}} dV,$$

where λ denotes the wavelength, r ₁ (r ₂) represents the distance between the scatter point to TX(RX), σ _v is the scattering cross-section. ρ denotes the aperture-to-medium coupling factor, which can be expressed as L _c = −101gρ. V refers to the common volume, which can be approximated as

(7)$$V = 1.206\displaystyle{{r_1^2 r_2^2 \delta _t\delta _r\phi _t\phi _r} \over {\sqrt {r_1^2 \phi _t^2 + r_2^2 \phi _r^2 \sin \Theta}}}, $$

where Θ is the scattering angle. δ _t,r denotes the half power beamwidth of transmitting and receiving antennas in the vertical direction. ϕ _t,r represents the half power beamwidth in the horizontal direction. They can be modeled as

(8)$$\delta _{t,r}(\phi _{t,r}) = (70{\rm ^\circ} \sim 75{\rm ^\circ} ) \times \lambda /D.$$

According to Kolmogorov theory, σ _v can be expressed as

(9)$$\sigma _v = 2\pi k^4 = \sin ^2\chi ^\Phi (K_s)$$

where k = 2π/λ represents the wavenumber. χ = π/2 for the horizontally polarized EM wave, χ = π/2 + Θ for the vertically polarized EM wave. Φ(k _s) can be calculated as

(10)$$\Phi (k_s) = 0.0033k_s^{-m} C_n^2, $$

where m = (11/3,5). Other parameters can be given as

(11)$$\left\{ \matrix{k_s = 2k\sin (\Theta /2) \hfill \cr C_n^2 (\Lambda_0,M) = 2.8\Lambda_0^{4/3} M^2 \hfill} \right.,$$

where Λ₀ represents the outer scale of atmospheric turbulence, which can be written as

(12)$$\Lambda _0(h) = \left\{ \matrix{0.4h,\quad 0 \le h \le 25\,{\rm m} \hfill \cr 2\sqrt h, \quad 25 \le h \le 1000\,{\rm m} \hfill \cr 63.24,\quad h \ge 1000\,{\rm m} \hfill} \right..$$

In equation (11), M stands for the vertical gradient of tropospheric refractive index. According to the tropospheric theory, n = 1 + 10⁻⁶N, here, n, N refer to the tropospheric refractive index and refractivity, respectively. N = N _dry + N _wet [Reference Gong, Yan and Wang10], here, N _dry, N _wet refer to the dry and wet terms of refractivity, respectively. Both of them can be expressed as [Reference Grabner, Pechac and Valtr11]

(13)$$\left\{ \matrix{N_{dry} = 77.6P/T \hfill \cr N_{wet} = 373256e/T^2 \hfill} \right.,$$

where P denotes the atmospheric pressure in hPa, e represents the water vapor pressure in hPa, and T is the absolute temperature in K. Consequently, M can be deduced as

(14)$$\displaystyle{{dn} \over {dh}} = \displaystyle{{77.6} \over {{10}^6}}\left[ {\displaystyle{1 \over T}\displaystyle{{dp} \over {dh}}-\left( {\displaystyle{\,p \over {T^2}} + \displaystyle{{9620e} \over {T^3}}} \right)\displaystyle{{dT} \over {dh}} + \displaystyle{{4810} \over {T^2}}\displaystyle{{de} \over {dh}}} \right].$$

As displayed in equation (14), estimating M through this way depends on precisely vertical gradient of meteorological parameters. ITU-R P. 835-5 gives a troposphere model based on latitude areas (low-latitude, mid-latitude, and high-latitude) and seasons (summer and winter) [12]. For example, model of mid-latitude area in summer can be expressed as

(15)$$\left\{ \matrix{T_{sum}(h) = 294.9838-5.2159h-0.07109h^2 \hfill \cr P_{sum}(h) = 1012.8186-111.5569h + 3.864h^2 \hfill \cr \rho_{sum}(h) = 14.3542\exp [-0.4174h-0.0229h^2 + 0.001007h^3] \hfill \cr e(h) = \rho (h)T(h)/216.7 \hfill} \right.,$$

where h denotes the altitude. Parameter M at any h can be estimated according to equation (14) and (15). However, both time resolution and spatial resolution are relatively poor, we cannot get the real-time refractivity through this model. Meteorological parameters of three observations in mid-latitude region are selected to display the change law of tropospheric refractivity. Table 1 displays the concrete information of these observations.

Table 1. Information of three observations

Figures 2(a) and 2(b) depict the refractivity of a whole year and first 3 days of a year, respectively. From Fig. 2, we can get that the refractivity of all observations is different and apparently changes with time. Namely, ITU model cannot describe the tropospheric refractivity in more detail. The refractivity model with better performance is in urgent need.

Fig. 2. Refractivity of three observations. (a) Refractivity of a year, (a) refractivity of 3 days.

Hopfield model [Reference Hopfield13] is widely used in satellite system, it can describe N(h) as

(16)$$N_i(h) = \displaystyle{{N_{i0}} \over {H_i^4}} (H_i-h)^4\quad i = {\rm dry,wet,}$$

where H _dry, H _wet denote the equivalent height for hydrostatic and wet refractivity, respectively. They can be given as

(17)$$\left\{ \matrix{H_{dry} = 40136 + 148.72(T_0-273.16){\rm m} \hfill \cr H_{wet} = 11\,000\,{\rm m} \hfill} \right..$$

The vertical gradient of tropospheric refractive index can be further deduced as

(18)$$\displaystyle{{\partial n(h)} \over {\partial h}} = \displaystyle{{-4} \over {{10}^6}}\left[ {\displaystyle{{N_{dry0}} \over {H_{dry}^4}} {(H_{dry}-h)}^3 + \displaystyle{{N_{wet0}} \over {H_{wet}^4}} {(H_{wet}-h)}^3} \right].$$

According to equation (18), Hopfield model can directly estimate M through meteorological parameters of surface. It does not depend on vertical gradient of meteorological parameters. Ref. Reference Liu, Chen, Li and Liu14 employs several observations locating in different zones to show the performance of Hopfield model and ITU model. The consequences indicate that the former has obvious advantages. Table 2 shows bias of annual mean loss estimated by Zhang method and the time-varying method. Here, 1 stands for equator, 2 continental subtropical, 3 marine subtropical, 4 desert, 5 continental temperate, 6 sea, 7 marine temperate. Analogous consequences of different regions indicate that the dynamic method is logical. Detail information of climate zones is same as Fig. 1. Analogous consequences of two methods indicate that the latter method can also effectively estimate the propagation loss.

Table 2. Bias of propagation loss estimated by two models (dB)

Rain attenuation model

Precipitation will affect the performance of many wireless links. Because the frequency of early warning radar always ranges in microwave bandwidth, rain has relatively obvious effect on the troposcatter link [Reference Abdulrahman, bin abdurahman, kamal bin Abdulrahim and Kesavan15].

Figure 3 shows the troposcatter link under rain weather. Only the EM wave passing through rain cell will suffer from the additional attenuation. Without considering the rain scatter, the attenuation rate can be expressed as

(19)$$\gamma _R = k\cdot R^\gamma \;{\rm dB}/{\rm km,}$$

where R denotes the rainfall rate (mm/h), k, γ are the empirical coefficients, the value of them can be found in ITU-R P.676-11 [16]. Figure 4 displays the γ _R changing with frequency. Rain attenuation becomes more obvious with the frequency increasing. Rain attenuation also depends on the length of wave passing in rain cell, [Reference Dinc and Akan17] introduces a reasonable method.

Fig. 3. Troposcatter communication in rain cell.

Fig. 4. Attenuation rate in rain cell.

As stated above, the received power follows

(20)$$S = P_t-L_\Sigma -L_R,$$

where L _R denotes the rain attenuation. Operating range determines the propagation loss, it can be further deduced through equation (20).