Outage Probability versus Carrier Frequency in GeoSurf Satellite Constellations with Radio-Links Faded by Rain

: For sites located in different climatic regions, we estimated the relationship between the annual average probability distributionof exceeding a ﬁxed rain attenuation, and the carrier frequency in the range 16 to 100 GHz, in the zenith paths of GeoSurf satellite constellations. In these constellations rain attenuation is independent of the altitude and number of satellites. Rain attenuation iss calculated with the Synthetic Storm Technique, a reliable prediction method, by using on-site measured rain-rate time series. A suitably deﬁned outage probability factor shows that the outage probability, for ﬁxed power margin, tends to saturate as frequency increases. In wideband radio-links, such as in spread spectrum design, there is very likely a long-term distortion due to the in-band outage probability. The results are oriented to design systems faded by rain attenuation whose value is also the total power margin available due to a mixture of coding and hardware technology, whose combination is not of concern here.


The GeoSurf Satellite Constellations
The GeoSurf satellite constellations belong to the family of Walker Star Constellations and they emulate, for ground stations located at any latitude, the geostationary orbit with zenith paths: Any transmitter/receiver, wherever located, can be linked to a satellite as if it were at the equator and the satellite at its zenith [1]. The GeoSurf constellations can have most of the advantages of the current GEO (Geostationary), MEO (Medium Earth Orbit), and LEO (Low Earth Orbit) satellite constellations without having most of their drawbacks. Table 1 of ref. [1] compares the advantages and disadvantages of GeoSurf constellations with GEO, MEO and LEO constellations. Ref. [2] compared the tropospheric attenuation of GeoSurf satellite paths with those of GEO, MEO, and LEO satellites.
Compared with ref. [2], in this paper we consider only the zenith paths of the GeoSurf constellation and rain attenuation only, without any further comparison with the GEO, MEO, and LEO radio-links. In fact, our aim is to estimate the relationship between the annual average probability distribution P(A) of exceeding a given rain attenuation A (dB) and the carrier frequency f (GHz)-i.e., the center frequency of bandwidth B w -beyond 16 GHz in the GeoSurf paths (local zenith paths). The satellite radio-link can be modeled as an additive white Gaussian noise channel in which the in-band frequencies can be attenuated differently by rain attenuation when the bandwidth is significantly large, i.e., of the order of at least 1 GHz (wideband modulation).
Our results are oriented to system design. If the constellation radio-links have a fixed signal-to-noise power margin, and this margin is attributed only to rain attenuation A, then P(A) is also the outage probability of the radio-links. Because we wish to show system-oriented results, we limited the power margin/attenuation in the range 3 to 30 dBs and the frequency range between 16 GHz and 100 GHz. In other words, we considered Notice that, in our study, it does not matter how the power margin of a radio-link is obtained: it can be a combination of coding and hardware, which can vary very rapidly as technology develops. What counts is the total value, which is here supposed to compensate for rain attenuation.
For illustrating the general characteristics of the relationship P(A) versus f , we report the results concerning the sites listed in Table 1, located in different climatic regions. The rain attenuation expected in GeoSurf radio-links is independent of the particular GeoSurf design (e.g., altitude and number of satellites), because all paths to/from a satellite of the constellation are always vertical (zenith). A was calculated with the Synthetic Storm Technique (SST) [3]. We considered these specific sites because of the availability of the rain-rate time series R(t) (mm/h), with rain rate averaged in 1 min intervals, continuously recorded on-site for several years, although in zenith paths the SST requires only to know the probability distribution of rain rate P(R) (see Equation (29) of ref. [3]). In other words, calculations and models reported below can be derived for any site where on-site measured-best choice-or estimated P(R) is available.
Since the 1960 s, many experiments and studies have been conducted-and reported in a huge numbers of papers-to establish the impact of rain attenuation on the outage probability of fixed (i.e., GEO) and mobile satellite (i.e., MEO and LEO) communications. Because of the saturation of the lower frequency bands currently used, as hardware and transmitter performance improves, satellite communication systems tend to use higher frequencies and larger bandwidths, including THz frequencies [4], for GEO, MEO, and LEO constellations. Unfortunately, rain attenuation, besides clear-sky fading, is still the most serious physical and unavoidable source of outage; therefore, studies, as the one reported in the present paper, or in ref. [5], are always welcome.
After this introductory section, Section 2 reports and discusses the probability distributions of rain attenuation at the reference sites in zenith paths obtained with the SST, Section 3 defines and reports a suitable outage probability factor at the reference sites, Section 4 models the outage probability factor, Section 5 defines the in-band outage probability factor, and finally Section 6 concludes.

The Synthetic Storm Technique Applied to Zenith Paths
Satellite communication links at centimeter and millimeter wave frequencies are faded by rainfall. For a reliable link-budget design, we need to know, at the very least, the annual probability distribution function, P(A), of rain attenuation A (dB) measured/predicted in the up-or down-links to a satellite. Instead of long and expensive measurements of beacon attenuation, prediction models are used for estimating P(A) from locally, measured or estimated, annual probability distributions, P(R), of 1 min rain rate R. The Synthetic Storm Technique [3] is a powerful and accurate tool, as is now recognized, that can produce all the necessary statistics of rain attenuation, not only P(A), but also fade durations and rate of change of attenuation because it provides reliable rain attenuation time series A(t). From knowing the rain-rate time series R(t) (mm/h, averaged in 1 min intervals), recorded at a site, the SST can generate A(t), at any frequency and polarization, and for any slant path above about 10 • . Because it reproduces reliable A(t), it has been used in the last 30 years for many purposes by several researchers [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].
When the Synthetic Storm Technique is applied to zenith paths, the rain attenuation time series lasts exactly as the rain-rate time series, as discussed in ref. [3], Equations (18) and (29).
The vertical (zenith) structure of precipitation is modeled with two layers of precipitation of different depths, a model taken from ref. [21]. Starting from the ground, there is rain (hydrometeors in the form of raindrops, water temperature of 20 • C, called layer A), followed by a melting layer, i.e., melting hydrometeors at 0 • C, layer B. The vertical rain rate R (mm/h, averaged in 1 min) in layer A is assumed to be uniform and given by that measured at ground, i.e., by the rain gauge, R A = R. By assuming simple physical hypotheses, calculations showed that the vertical precipitation rate in the melting layer, termed "apparent rain rate", is also uniform and given by R B = 3.134 × R A [21]. The height of the precipitation (rain and melting layer, H R ) above sea level to be used in the simulation is the latest estimate of the ITU-R [22]. The layer B depth is 0.40 km, regardless of the latitude [3,21]. Now, according to Equation (29) of ref. [3], rain attenuation A(t) (dB) in the zenith path is given by: In Equation (1), k and α are the constants that give the specific rain attenuation Y = kR α -R measured in mm/h and Y in dB per 1 km-in the two layers from the rain rate R measured at the ground, calculated at 20 • C in layer A and at 0 • C in layer B [3], values given in ref. [23].
Finally, because the relationship between A(t) and R(t) is monotonic, the on-site probability distribution P(R) is transformed directly into the P(A) of the zenith path through the variable transformation given by Equation (1). Figures 1-4 show examples of the P(A) obtained at several frequencies at the sites listed in Table 1. Because our analysis is oriented to find the outage probability when the radio-link power margin is not larger than A dB, P(A) is also the radio-link outage probability due to rain.         In the next section, we show how P(A) changes as the frequency increases from 16 to 100 GHz, for a fixed power margin A, by defining a suitable outage probability factor.

Outage Probability Factor
We modeled the relationship between the outage probability P(A) due to rain attenuation A as follows. For a site, let us consider a fixed value A for all frequencies and start by considering, as reference value, the probability P f o (A) exceeded at a reference frequency f o , 16 GHz in our modelling. The reference P(A), for a given A, is therefore that shown in Figure 1 for each site.
We can relate P(A) exceeded at frequencies f ≥ 16 GHz to the reference probability exceeded at 16 GHz, by writing: Now, as f ≥ f o it follows that the probability factor ρ(A, f ) ≥ 1 because the same attenuation is exceeded at frequency f > f o with a larger probability. Figures 5 and 6 show how ρ(A, f ) changes as a function of frequency, for fixed values of A ranging approximately from 3 to about 30 dB in 0.2 dB steps, for some sites. Each curve shown in Figures 5 and 6 corresponds, of course, to a fixed value A. The red curve shows the average values ρ m ( f ), a parameter independent, of course, of attenuation.
shows the average values ( ), a parameter independent, of course, of attenuation.
Because of the different climatic rainfall regions, each site shows quite different values. However-as the average values (red curves) show-as frequency increases, the curves tend to saturate. The saturation values depend on the site considered, but this occurs approximately at about 100 GHz. Notice, however, that the spread of the curves shown in Figures 5 and 6 is not very large, therefore indicating a small sensitivity to the maximum value (30 dBs) of the rain attenuation considered.  In conclusion, if a radio-link has a fixed power margin , as assumed in Figures 5  and 6, as the carrier frequency increases the outage probability also increases across a large bandwidth centered on the carrier frequency , as next section shows.

Model of the Average Outage Probability Factor
The average outage probability factor ( ) shown in Figures 5 and 6, by setting = 16 GHz, can be modeled (best-fit analysis) with the following mathematical expression: Notice that each addend in the right-hand site of Equation (3) has no particular physical meaning; what counts is the sum of the addends. The constants and are reported in Table 2, for each site. For Spino d'Adda, as example, for = 3 dB, (3) = 0.081% (Figure 1), now at 40 GHz the average probability factor, from (3) and Table 2, turns out to be = 9.70; therefore, the estimated  Because of the different climatic rainfall regions, each site shows quite different values. However-as the average values (red curves) show-as frequency increases, the curves tend to saturate. The saturation values depend on the site considered, but this occurs approximately at about 100 GHz. Notice, however, that the spread of the curves shown in Figures 5 and 6 is not very large, therefore indicating a small sensitivity to the maximum value (30 dBs) of the rain attenuation considered.
In conclusion, if a radio-link has a fixed power margin A, as assumed in Figures 5 and 6, as the carrier frequency increases the outage probability also increases across a large bandwidth centered on the carrier frequency f , as next section shows.

Model of the Average Outage Probability Factor
The average outage probability factor ρ m ( f ) shown in Figures 5 and 6, by setting f o = 16 GHz, can be modeled (best-fit analysis) with the following mathematical expression: Notice that each addend in the right-hand site of Equation (3) has no particular physical meaning; what counts is the sum of the addends. The constants a and b are reported in Table 2, for each site. For Spino d'Adda, as example, for A = 3 dB, P 16 GHz (3) = 0.081% (Figure 1), now at 40 GHz the average probability factor, from (3) and Table 2, turns out to be ρ m = 9.70; therefore, the estimated P 40 GHz (3) = 0.787% (outage time about 70 h in an average year) is to be compared to P 40 GHz (3) = 0.954% (Figure 3). In Table 2, notice that when the climatic rainfall regions are very similar, as is the case for Spino d'Adda and Gera Lario, the curves tend to coincide.
Let us asses how good the best-fit given by Equation (3) is by studying the error it gives in considering only the carrier frequency as independent variable. We defined the error as follows (%): Figure 7 shows the mean error versus frequency. In general, the error is very low, for this kind of measurement, not exceeding ∓30%. Notice, however, that the purpose of the modelling (3) is not to predict the result-obviously all data shown in Figures 5 and 6 are available-but to show the general trend.
Telecom 2022, 3, FOR PEER REVIEW 8 Let us asses how good the best-fit given by Equation (3) is by studying the error it gives in considering only the carrier frequency as independent variable. We defined the error as follows (%): Figure 7 shows the mean error versus frequency. In general, the error is very low, for this kind of measurement, not exceeding ∓30%. Notice, however, that the purpose of the modelling (3) is not to predict the result-obviously all data shown in Figures 5  and 6 are available-but to show the general trend. New satellite radio-links-such as those with spread-spectrum design-may occupy a large radiofrequency bandwidth centered at frequency , in which, for a fixed power margin, it is very likely to find different equivalent outages at the extremes of the bandwidth, which may produce signal distortions. In the next section we discuss this issue. New satellite radio-links-such as those with spread-spectrum design-may occupy a large radiofrequency bandwidth centered at frequency f c , in which, for a fixed power margin, it is very likely to find different equivalent outages at the extremes of the bandwidth, which may produce signal distortions. In the next section we discuss this issue.

In-Band Outage Probability Factor
If large-bandwidth (wideband) radio-links are designed, such those in satellite communications with spread spectrum or similar methods, we should know how much the outage probability changes within the large bandwidth, for a fixed power margin A, to assess possible signal linear distortions.
We can estimate the change expected in the bandwidth by studying the in-band probability factor I P (GHz −1 ) defined as: In Equation (5), dρ m ( f )/d f is the frequency derivative of Equation (3) and ρ m ( f ) is given by Equation (3). To estimate from Equation (5) the value of I p in a bandwidth B w (GHz) centered in f = f c it is sufficient to multiply the right-hand side approximately for B w ; therefore, Equation (4) numerically refers to B w = 1 GHz, hence, in finite terms it gives the ratio I p = ∆ρ m ( f c )/ρ m ( f c ) in the 1-GHz bandwidth around f c . Therefore, at the frequency f = f c + 0.5 GHz the outage is greater because the same rain attenuation is exceeded with higher probability than at the lower frequency f = f c − 0.5 GHz. In other words, to avoid outage in all bandwidth, the power margin must be equal to the rain attenuation measured at f = f c + 0.5 GHz, not at f = f c . Figure 8 shows I P for the indicated sites. It is evident that the distortion due to the different in-band outage probability-because of a fixed power margin in the AWGN channel calculated at f = f c -can be significantly different from site to site, and also at a site from frequency to frequency. For Spino d'Adda, at f = 40 GHz, continuing the example of Section 3, I p = 0.044; therefore, the outage time difference between the extreme frequencies in B w = 1 GHz bandwidth is (0.044) ×70 = 3.1 hours in an average year. To a first approximation, this time difference is proportional to B w .
In general, higher frequencies show lower values because a fixed rain attenuation is exceeded with very near probabilities than at lower frequencies. This is evident for Tampa (see Figures 1-4), which gives the lowest curve in Figure 8. The opposite behavior is found when the fixed rain attenuation at higher frequencies is exceeded with different probability than at lower frequencies, as it occurs for Madrid (see Figures 1-4), which gives the highest values in Figure 8. In other words, the slopes of the probability distribution, as those shown in Figures 1-4-i.e., the probability density function of rain attenuation and, in the zenith paths here considered, also of rain rate-affect this kind of distortion more than other parameters.
(see Figures 1-4), which gives the lowest curve in Figure 8. The opposite behavior is found when the fixed rain attenuation at higher frequencies is exceeded with different probability than at lower frequencies, as it occurs for Madrid (see Figures 1-4), which gives the highest values in Figure 8. In other words, the slopes of the probability distribution, as those shown in Figures 1-4-i.e., the probability density function of rain attenuation and, in the zenith paths here considered, also of rain rate-affect this kind of distortion more than other parameters.

Conclusions
We estimated the relationship between the annual average probability distribution ( ) of exceeding a given rain attenuation (dB) and the carrier frequency (GHz) in the range 16 to 100 GHz, in the zenith paths of GeoSurf satellite constellations and reported results oriented to system design. For illustrating the general characteristics of this

Conclusions
We estimated the relationship between the annual average probability distribution P(A) of exceeding a given rain attenuation A (dB) and the carrier frequency f (GHz) in the range 16 to 100 GHz, in the zenith paths of GeoSurf satellite constellations and reported results oriented to system design. For illustrating the general characteristics of this relationship, we have reported the results concerning sites located in different climatic regions. The rain attenuation expected in GeoSurf radio-links is independent of the particular GeoSurf design (e.g., altitude and number of satellites), and was calculated with the Synthetic Storm Technique.
We modeled the relationship according to Equations (2) and (3) by defining a suitable outage probability factor. The mean error of the outage probability factor, Equation (4), depends on site. The main result was that this factor tends to saturate at about 100 GHz, therefore giving about the same outage probability, for a fixed attenuation/power margin, as frequency increases. For a fixed power margin, in wideband radio-links, there is very likely a linear distortion due to in-band outage probability.
Further work should be conducted to apply these concepts to very large bandwidth radio-links, such as those envisaged with spread-spectrum techniques, by making more detailed predictions within the large bandwidth with the SST, as discussed above, especially with the purpose of establishing the main features of the radiofrequency transfer function.