Theoretical background

Site diversity

Satellite systems that operate at 10 GHz and above, the most dominant factor that affects signal degradation is rain. Consequently, the channel model for the slant paths prioritizes the impact of rain-induced fading. The key points for the statistical characterization of such channel models take into consideration the following remarks.

• • The statistical analysis is based on the convective raincells model [Reference Kanellopoulos and Koukoulas16].

• • The unconditional Lognormal distribution is used to approximate the random variables of the rain attenuation $A$ (given in decibels) and the rainfall rate $R$ (given in mm/h). The term unconditional point rainfall rate is determined as comprising both raining and non-raining events.

• • To determine the mean and standard deviation features of the rainfall rate a fitting procedure is performed to the local measured data. If these are not available, the rain rates are determined by using ITU-R Recommendation P.837-7 [23].

• • The rain heights (required for each slant path) are extrapolated leveraging ITU-R Recommendation P.839-4 [24].

• • Regarding the homogeneity of the vertical structure of the rainfall medium, Crane’s hypothesis has been considered [Reference Crane25]. In this case, the rain attenuation over the earth projection of a slant path $A$ is associated with the rain attenuation $A_s$ over the slant path according to $A=A_s \cos{\phi}$, where $\phi$ is the elevation angle.

The spatial correlation coefficient of the specific rain attenuation attributes is expressed as

(1)\begin{equation} \rho_{0}(d) = \begin{cases} \frac{G}{\sqrt{G^2+d^2}}, & d \leq D_r \\ \frac{G}{\sqrt{G^2+D_r^2}}, & d \gt D_r \end{cases} \end{equation}

where $D_r$ stands for the diameter of the raincell and $G$ indicates the spatial factor of the rainfall medium $G\in[0.75,3]$ that specifies the spatial correlation. The ${logarithmic}$ correlation coefficient of the rain attenuation for $d\leq$ 50 km is given by

(2)\begin{equation} {\rho_{n_{ij}}=\frac{1}{S_{a_i}S_{a_j}}\ln{\left( 1+\rho_{ij}\sqrt{(e^{S_{a_i}^2} - 1)(e^{S_{a_j}^2} - 1)}\right)}} \end{equation}

where $i,j = 1,2,3$ ( $i \neq j$). Furthermore, $S^2_{a_i}$, ( $i = 1,2,3$), stands for the variance that depends on the rainfall rate statistics, which is expressed as

(3)\begin{equation} S^2_{a_i} = \ln{\left(1 + \frac{H_{1i}}{L^2_{D_i}} \left[ \exp{(b^2_i S^2_{r_i})} - 1 \right] \right)} \end{equation}

where $b_i$, ( $i=1,2,3$), denotes the polarization and frequency dependent attenuation parameter, and $S_{r_i}$ is a lognormal statistical parameter that is extrapolated from rainfall rate distribution $R$ (see Fig. 3). Furthermore, $H_{1i}$ indicates the effective height of the rainfall structure and $L_{D_i}$ stands for the effective path length for the $i$-th slant path ( $i = 1,2,3$).

In the case of $d \gt 50$ km, then, the Paraboni–Barbalisca model is adopted according to

(4)\begin{equation} {\rho_{n_{ij}}=0.94\exp{\left(-\frac{d}{30}\right)}+0.06\exp{\left(-\left(\frac{d}{500}\right)^2\right)}} \end{equation}

In (2), $\rho_{ij}$ is the correlation coefficient factor that is calculated by

(5)\begin{equation} \rho_{ij} = \frac{H_{2ij} = \int_0^{L_{D_i}}{\int_0^{L_{D_j}}{\rho_0 (z,z') \,dz\,dz'}} } {\sqrt{H_{1i}H_{1j}}} \end{equation}

where $z$ and $z'$ denote the integral’s argument features that are associated with the $i$-th and $j$-th effective path length, and $H_{1i}$ is given by

(6)\begin{equation} \begin{aligned} {H_{1i} = \begin{cases} 2L_{D_i}G\sinh^{-1}{\left(\frac{L_{D_i}}{G}\right)} + 2G^2 \left(1 - \sqrt{\left({\frac{L_{D_i}}{G}}\right)^2 + 1} \right), & L_{D_i} \leq D_r \\ 2L_{D_i}G\sinh^{-1}{\left(\frac{D_r}{G}\right)} + 2G^2 \left(1 - \sqrt{\left({\frac{D_r}{G}}\right)^2 + 1} \right) \\ + \frac{G(L_{D_i}-D_r)^2}{\sqrt{G^2+D_r^2}}, & L_{D_i} \gt D_r \end{cases}} \end{aligned} \end{equation}

The same stands for $H_{1j}$ where $i,j = 1,2,3$ ( $i \neq j$). The features indicated in (1)–(6) and the annotations of the effective lengths $L_{D_i}$, given in kilometers, are available in [Reference Kanellopoulos and Panagopoulos26, Reference Panagopoulos, Livieratos and Kanellopoulos27].

The improvement, achieved by the triple site diversity scenario, is assessed by determining the joint probability $P_{1,2,3}$. The latter parameter is expressed in terms of either the single-site probability at the relevant location, or by using the geometric mean probability $P_0$ of the $P_1$, $P_2$, and $P_3$ according to [Reference Kanellopoulos and Koukoulas16]

(7)\begin{equation} \begin{aligned} P_{1,2,3} &= P[A_{s1} \gt x_s,A_{s2} \gt x_s,A_{s3} \gt x_s] \\ &= P[A_1 \gt x_D,A_2 \gt x_D,A_3 \gt x_D] \\ &= \int_{x_D}^{\infty}{\int_{x_D}^{\infty}{\int_{x_D}^{\infty}{f_{{A_1}{A_2}{A_3}}(x_1,x_2,x_3)\,dx_1\,dx_2\,dx_3 } } } \end{aligned} \end{equation}

where $f_{{A_1}{A_2}{A_3}}(x_1,x_2,x_3)$ is the joint density function of the attenuation features $A_1$, $A_2$ and $A_3$. Given that the random variables $\ln{A_1}$, $\ln{A_2}$, and $\ln{A_3}$ approximate a joint unconditional normal distribution, applying a straightforward statistical analysis, the triple site probability can be determined according to

(8)\begin{equation} \begin{aligned} {P_{1,2,3} = \frac{1}{2} \int_{u_0}^{\infty}{\int_{u_0}^{\infty}{f_{{U_2}{U_3}}(u_2,u_3) \text{erfc}{\left( \frac{u_0 - m_1}{\sqrt{2}\sigma_1} \right)} \,du_2\,du_3 } } } \end{aligned} \end{equation}

where $f_{{U_2}{U_3}}(u_2,u_3)$ is the bivariate normal density distribution that is described by

(9)\begin{equation} {f_{{U_2}{U_3}}(u_2,u_3) = \frac{1}{2\pi\sqrt{1-\rho^2_{n_{23}}}} \exp{\left(-\frac{u_2^2+u_3^2-2\rho_{n_{23}}u_2u_3}{2(1 - \rho_{n_{23}}^2)}\right)} } \end{equation}

and $u_0 = \sqrt{2}\text{erfc}^{-1}{(2P_0)}$. The variable $m_1$ and $\sigma_1$ are expressed as

(10)\begin{equation} m_1 = \frac{(\rho_{n_{12}}-\rho_{n_{13}}\rho_{n_{23}})u_2+(\rho_{n_{13}}-\rho_{n_{12}}\rho_{n_{23}})u_3}{1 - \rho_{n_{23}}^2} \end{equation}

(11)\begin{align}\hspace{27pt} \sigma_1 = \left[ \frac{(1-\rho_{n_{12}}^2-\rho_{n_{23}}^2-\rho_{n_{13}}^2+2\rho_{n_{12}}\rho_{n_{23}}\rho_{n_{13}})}{1 - \rho_{n_{23}}^2} \right]^{\frac{1}{2}} \end{align}

and the parameter $u_i$, is expressed as

(12)\begin{equation} u_i = \frac{\ln{A_i}-\ln{A_{m_i}}}{S_{a_i}},~i = 1,2,3 \end{equation}

where $S^2_{a_i}$, ( $i = 1,2,3$), is given by (3) and $A_{m_i}$ is determined by

(13)\begin{equation} A_{m_i} = a_i R^{b_i}_{m_i} L_{D_i} \exp{\left( \frac{b^2_i S^2_{r_i} - S^2_{a_i}}{2} \right)} \end{equation}

where $a_i$, ( $i=1,2,3$), stands for the polarization and frequency dependent attenuation parameter, and $S_{r_i}$ is a lognormal statistical parameter that determined by the rainfall rate distribution $R$. The parameters $A_{m_i}$ and $S^2_{a_i}$ are the statistical attributes of the induced rain attenuation (given in decibels) in each slant path, thereby facilitating the triple site diversity method. It should be pointed out that the probability given by (8) is grounded on the selection combining method, thus outage is represented by (7).

Orbital diversity

The general configuration of the orbital diversity scenario with three geostationary satellites is shown in Fig. 1(b), where $\varphi_1$, $\varphi_2$, $\varphi_3$ are the elevation angles of each slant path. The objective is to determine the service availability improvement using triple-orbital diversity, which is addressed by calculating the exceedance probability given by (6). In essence, the theoretical model presented above can be reversed and instead of having three ground stations, three different satellites, linked to a single ground station, are exploited to improve the performance. The system moderator switches and selects the satellite with the lower attenuation level. Therefore, the required quality is realized with much smaller available link margin from the corresponding one that we have for a single- or double-orbital diversity layout for the same time availability. The difference with the triple site diversity is grounded on the calculation of the correlation coefficient $\rho_{ij}$. In (5), to calculate $H_{2ij}$, the spatial correlation coefficient in the nominator is expressed as

(14)\begin{equation} \rho_0 (z,z') = \frac{G}{\sqrt{G^2 + d(z,z')^2}} \end{equation}

where $d(z,z') = \sqrt{z^2 + z'^2 - 2zz'\Delta\psi_{ij}}$ is the distance between two points belonging to the projections of the slant paths under consideration, whereas the angle $\Delta\psi_{ij}$, $i,j = 1,2,3$ ( $i \neq j$), is given by

(15)\begin{equation} {\Delta\psi_{ij} = \cos^{-1}{((\cos{\theta_{ij}} - \sin{\varphi_i}\sin{\varphi_j}) / \cos{\varphi_i}\cos{\varphi_j})}} \end{equation}

where $\theta_{ij}$ is the angular separation between two pairs of satellites as shown in Fig. 1(b), and $\varphi$ is the elevation angle of the corresponding ground station antenna pointing to each satellite.

Diversity scenarios under consideration

In the island of Cyprus, the diversity scenario comprises two operating teleport terminals in Makarios and Pera. A third location is taken into account in the west part of the island (Pathos site) to aid the suggested triple site diversity concept. In Greece, the experimental terminals are located on the campus of the National Technical University of Athens (NTUA) and Lavrion Technological and Cultural Park (LTPC), respectively. The third site is selected in the Nemea Teleport, which serves the ground terminals of many providers. The corresponding locations and the direct path distance between them, for Cyprus and Greece, are depicted in Fig. 2(a) and (b), respectively.

Figure 2.

Indicative satellite pictures between the selected sites (taken from Google Earth $^\text{TM}$). (a) In Cyprus. (b) In Greece.

The selected site coordinates for both countries are listed in Table 1. The geographical spacing between individual sites plays a key role in determining how well the diversity system performs. In Cyprus, the chosen locations of Makarios and Pera have a spatial separation of 22.6 km, with the eastern part of the Troodos mountain range intervening. Furthermore, the distances between Makarios-Paphos and Pera-Paphos are 87.5 km and 79.9 km, respectively. In Greece, the two main locations of NTUA and LTCP are separated about 36.4 km, whereas the Ymittos mountain creates a natural barrier between them. Accordingly, the separation distance between NTUA-Nemea and LTCP-Nemea is 103 and 126 km, respectively. Nemea teleport is deliberately located between many mountains to achieve a physical isolation from interference in a relatively dry area with low rain events. The created geographic barriers influence the development of the local weather conditions, which is highly likely to lead in higher diversity gains, as they improve the physical isolation between the teleports. The impact is expected to be even more noticeable, adding Paphos and Nemea ground terminals in Cyprus and Greece, respectively, which are both significantly more isolated compared with the other two. Establishing additional sites entails an increased cost of a diversity-based reception method. In this work, the selection of all sites is based on practical considerations (i.e., existing teleport locations or deployed experimental terminals). Conventionally, the choice of the ground stations relies on both climatic conditions and the geometric constraints/characteristics of potential locations [Reference Sakarellos, Skraparlis, Panagopoulos and Kanellopoulos28]. It should be clarified that all the above comments would make sense if there were real experimental facilities to receive real measured data. In this paper, emphasis is given solely on simulated data from theoretical models that do not take into account the weather front in the assumed correlation dependence. There is an ongoing experimental campaign for site diversity reception measured data to prove the influence of the local geographical peculiarities in Greece and Cyprus.

Table 1.

Geographical coordinates of the examined locations in Cyprus and Greece

In Cyprus, long-term experimental meteorological data were collected between 2016 and 2021, including rainfall intensity values locally for Makarios and Pera sites (taken from the Cypriot Meteo Organization), registered at 10-min intervals. In Greece, 4-year data was collected (2016-2020) at 10-min intervals, in both NTUA and LTCP locations (exploiting tipping buckets at each terminal). The data availabilities were 94.96%, 96.43%, 92.33%, and 99.87% for Makarios, Pera, NTUA, and LTCP locations, respectively. The collected data was of high quality as the recorded values maintain consistency with no discontinuities or surges, thereby acting as a reliable baseline feature to be employed in different simulated diversity scenarios enhancing accuracy. A well-established approach is adopted to transform the 10-minute rainfall measurements into corresponding 1-minute rainfall rates [Reference Emiliani, Luini and Capsoni29], which are then leveraged to obtain the point rainfall rate statistics necessary either for the simulations, or for various statistical/empirical attenuation models. The 1-min rate reveals clearly the dynamics of those raining events. For this purpose, the following conversion formula is utilized

(16)\begin{equation} R_\text{1-min} (p\%) = a \cdot R_\text{10-min} (p\%) ^ b \end{equation}

where $a = 0.829$ and $b = 1.097$ according to [Reference Matricciani30].

Figure 3 presents the related exceedance probability results versus rainfall rate for each selected location for Cyprus and Greece. Given that measured data is not available in Paphos and Nemea locations, these are extrapolated taking into account ITU rain-map database [23]. The attributes $R_{m_i}$ and $S_{r_i}$ required in (2) and (13) are determined from the measured data and the rain-maps. The results show that higher rain intensities are encountered in Greece compared with Cyprus locations. More specifically, NTUA and LTCP exhibit the highest rain rates with comparable plots. Another observation is that Makarios and Pera also deliver comparable rainfall rates. This could be probably credited to the rainfall characteristics, whose statistics are not expected to change drastically at relatively small territories, as shown in Fig. 2. However, discrepancies are observed between the measured rainfall properties and the corresponding calculations from ITU, especially in Greece if one compares NTUA and LTCP with Nemea plots. Therefore, for enhanced accuracy and credibility, the simulations should be based on local measured rainfall data, as also suggested in [Reference Moraitis, Papafragkakis, Panagopoulos, Ioannides, Christoforou, Agrotis and Alexandrou1].

Figure 3.

Exceedance probability versus rainfall rate for the locations under consideration.

In the following procedure, vertical polarization is assumed. Furthermore, the selected frequencies are 20 GHz and 40 GHz at Ka- and Q-bands, respectively. In the case of the triple site diversity the selected elevation angles are 43.4 $^{\circ}$ and 45.24 $^{\circ}$ for Greece and 49.42 $^{\circ}$ and 45.45 $^{\circ}$ for Cyprus at Ka- and Q-bands, respectively, based on the associated satellites. Given the short relative distance between the sites and the geostationary orbit of the satellites, these angles are common at each selected ground terminal at Cyprus and Greece. The Makarios ground terminal in Cyprus and NTUA in Greece are considered and set as reference in both site and orbital diversity scenarios.

The performance of the orbital diversity depends on and is subject to limitations upon the availability of the satellites in orbit, and their angular separation. The greater the separation, the highest achievable diversity gain. The terminals of NTUA and Makarios are both considered to point at KaSAT (9 $^{\circ}$E), ALPHASAT (25 $^{\circ}$E), and HellasSat (36 $^{\circ}$E) satellites. Therefore, the angular separations $\theta_{ij}$, are $\theta_{12}=16^{\circ}$, $\theta_{13}=30^{\circ}$ and $\theta_{23}=11^{\circ}$, respectively. The corresponding elevation angles $\varphi_{1,2,3}$ are 43.4 $^{\circ}$, 46.1 $^{\circ}$, 43.2 $^{\circ}$ for NTUA and 42.3 $^{\circ}$, 49.1 $^{\circ}$ and 49.5 $^{\circ}$ for Makarios sites, respectively. Finally, for the rest of the parameters, it is assumed that $G=1.5$ km, $D_r=30$km and $H_r=4.8$ km.

Performance evaluation

The exceedance probability (outage time) versus fade margin in the range of 1–30 dB is calculated at Ka- and Q-bands. Three cases are taken into account – no diversity, 2-site diversity, and 3-site diversity – and similarly, no diversity, 2-orbit, and 3-orbit diversity. The triple site diversity scenario is depicted in Fig. 4, comparing Cyprus and Greece territories.

Figure 4.

Comparison of site diversity performance between the selected sites in Cyprus and Greece. (a) Ka-band. (b) Q-band.

It is evident that, in both cases, leveraging a 3-site diversity there is a remarkable improvement in exceedance probabilities and outage reduction, compared with 2-site diversity and no diversity. As expected higher frequencies induce higher attenuation that is reflected to the related outcome, as Q-band delivers inferior exceedance probabilities compared with Ka-band, as it clearly appears in the corresponding plots in Fig. 4(b) and (a), respectively. Comparing the results in Greece and Cyprus, the former exhibits markedly worser outages, consistently in the three examined cases (no diversity, 2-site, and 3-site). This is more intense at Ka-band where greater outage discrepancies are encountered. The higher rainfall rates in Greece and the greater separation of the selected sites account for these observations. On the other hand, smaller differences between Greece and Cyprus are encountered at Q-band according to Fig. 4(b). To assess quantitatively the performance of the applied site diversity scenarios the improvement factor between outage probabilities is introduced. The specific metric is defined by

(17)\begin{equation} I = \frac{P_\text{b} - P_\text{a}}{P_\text{a}}\times 100\% \end{equation}

where $P_\text{a}$ and $P_\text{b}$ denote the values between the examined exceedance probabilities for scenario $a$ and $b$, respectively (i.e., the improvement between 2-site and no diversity or between 3-site and 2-site diversity). The improvement factors for 10 and 20 dB fade margin are summarized in Table 2.

Table 2.

Outage improvement comparing double and triple site diversity between Cyprus and Greece for 10 and 20 dB fade margin

The numerical values reflect with consistency the previous remarks and the exceedance probabilities plots shown in Fig. 4. Obviously, the adoption of a 3-site diversity scenario delivers improvement factors higher than 99.9%, on average. The outage improvement between 2-site and 2-site diversity is about 4.5%, on average, for 10 dB fade margin at both frequencies and countries. Furthermore, the average improvements at Q-band are marginally better in Greece ( $ \lt 1\%$) as the separations between the selected sites are greater than those in Cyprus. On the other hand, Greece exhibits worser improvements than Cyprus at Ka-band, if 2-site diversity is applied (about 1% and 4.5% for 10 and 20 dB fade margins, respectively), due to the increased rainfall rates. Finally, the results indicated that 3-site diversity is indispensable at Q-band enhancing drastically the outage performance.

The results of the orbital diversity are presented in Fig. 5, in which 3-orbit diversity delivers the best outcome, improving substantially the system outage. The performance behavior between frequencies and countries is similar to the site diversity scenarios (i.e., deteriorated probabilities at Q-band and higher encountered outages in Greece). It is also apparent that site diversity outperforms orbital diversity, if one compares Figs. 4 and 5, which highlights the significance of the former technique that presents much better outages.

Figure 5.

Comparison of orbital diversity performance between the selected sites in Cyprus and Greece. (a) Ka-band. (b) Q-band.

These observations are also evident in Table 3 where the numerical values of the outage improvement are summarized. Much lower values are obtained (between 57% and 74%) comparing the site and orbital diversity performance. However, in any case, the average improvement between 2-orbit and 3-orbit is about 7%, which is better than that in using 2-site and 3-site diversity (4.5%).

Table 3.

Outage improvement comparing double and triple orbital diversity between Cyprus (Makarios) and Greece (NTUA) for 10 and 20 dB fade margin

Furthermore, slightly better improvements are observed at Ka-band (7.5% on average) compared with Q-band (6.5% on average). The higher encountered rain rates in the NTUA location, compared with Makarios, are reflected to the improvement results, which according to Table 3, the former site presents, on average, about 5.5% deteriorated outages. It is also worth remarking that the discrepancies between NTUA and Makarios are more intense if a 2-orbit scenario is adopted (7% better outages for Makarios on average), instead of 3-orbit (4.6%), which suggests that the 3-orbit diversity compensates better the rain-induced attenuation improving the exceedance probabilities.

Orbital diversity real experiment in Athens

In the following, experimental attenuation results are presented solely at the NTUA location, where measurements are available at Ka-band, exploiting ALPHASAT (25 $^{\circ}$E) and KaSAT (9 $^{\circ}$E) satellites’ beacon signal to evaluate orbital diversity. The measurements are then compared with the theoretical model for a 2-orbit exceedance probability. Figure 6 illustrates the measured diurnal in-excess attenuation versus time for two different days where rain events were encountered. The presented orbital diversity statistics are obtained after harmonizing the data to 1 sec time resolution, including only Ka-band attenuation data concurrent for both satellites. The differences in the recorded attenuation levels are more than evident between the two selected satellites, thus being favorable for adopting orbital diversity. The data have been processed according to the procedure described in [Reference Papafragkakis, Panagopoulos and Ventouras32] and [Reference Papafragkakis, Kourogiorgas and Panagopoulos33]. According to Fig. 6, when intense rain events occur, the attenuation values differentiate between the two satellites, which can be leveraged for promising double orbital diversity gains.

Figure 6.

Example of the potential idealized orbital diversity technique for Ka-band at the NTUA location during two rainy days. (a) July 31st, 2023. (b) August 28th, 2023.

The orbital diversity statistics for satellites at two orbital locations $O_1$ and $O_2$ are derived from the measured excess attenuation timeseries according to

(18)\begin{equation} P_{O_1,O_2} = \left[A_{O_1}(t) \gt a, A_{O_2}(t) \gt a\right] \end{equation}

where $P[\cdot]$ is the joint (bi-variate) probability complementary cumulative distribution function (CCDF) that attenuation values $A_{O_1}$ and $A_{O_2}$ at the time instant $t$ exceed the attenuation threshold $a$. The ideal joint minimum attenuation experienced at time instant $t$ is expressed as

(19)\begin{equation} A_{OD} = \min{\left[ A_{O_1}(t), A_{O_2}(t) \right] } \end{equation}

The exceedance probability versus the measured joint excess attenuation for the double orbital scenario is presented in Fig. 7. The measured joint attenuation is in the range of 4–14.2 dB. It should be worth commenting that there is a slight frequency discrepancy between the two transmitted beacons as ALPHASAT transmits at 19.701 GHz with vertical polarization, while KaSAT at 19.680 GHz having horizontal polarization. Nevertheless, one can still obtain a good indication of the performance gain using orbital diversity as a possible countermeasure method.

Figure 7.

Orbital diversity evaluation at Ka-band comparing measurements and theoretical model.

The dashed line designates the exceedance probability versus excess attenuation, applying the theoretical model for double orbital diversity. The plots reveal that the theoretical model approximates quite well the measured values. For example, at 0.1% exceedance probability, the absolute difference between the measured and the theoretical attenuation is 0.04 dB. Higher divergences are observed for attenuation levels greater than 12 dB, although with lower outage values, as it is apparent in Fig. 7. The presented results validate the accuracy of the theoretical model for double orbital diversity, which can be exploited as a useful mathematical tool for robust outage estimations even when measurements are not available at the areas of interest. A different set of measurements at greater angular separations would probably provide a better understanding of the effectiveness of this technique, as suggested by [Reference Matricciani and Mauri31]. Given also the fact that the KaSAT receiver has a lower measurement dynamic range, limits the extent of this evaluation to exceedance probabilities only higher than 0.01%, while it is expected that at lower probabilities the gain could be significantly higher.