The number of satellite telecommunication systems that make use of frequencies higher than 10 GHz is constantly growing; at frequencies above 10 GHz, attenuation due to rain can be a limiting factor for system availability; the dual-site diversity technique has proved to be quite useful in counteracting rain attenuation. This technique can be extended by adopting a multiple-site configuration. In this paper the performance of small-scale multiple-site diversity systems is investigated through simulations, carried out by exploiting a large database of radar maps of precipitation. Results have shown significant improvements in terms of diversity advantage and site separation reduction with respect to the dual-site configuration. Moreover, if the stations carry data traffic and they are not working at their full capacity, it is possible to recover a significant amount of the traffic that otherwise would be lost.
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 Since the 1970s there has been a steady increase in the number of satellite telecommunication systems that make use of frequencies higher than 10 GHz. It is well known that at frequencies above 10 GHz, the attenuation due to the presence of rain along the Earth-satellite path can be a limiting factor for system availability. Different solutions have been proposed to overcome outage due to rain attenuation. One is site diversity [Hogg, 1967], which consists of two (or more) stations spaced far enough apart to be jointly capable of reducing overall system attenuation, at the annual probability level of interest, to an acceptable value. This technique is based on the physical evidence that the structure of rain is not uniform in space: In particular, the heaviest rain cells have limited extension; the probability of experiencing high values of attenuation simultaneously on both links is sensibly reduced if the links are properly spaced.
 In its classical (dual-site) configuration, a space diversity system consists of two identical stations, connected through a dedicated ground link; their distance is determined depending on the frequency, on the geometry, and on the desired performance (i.e., outage time) of the system. In the past, space diversity has been considered mainly for high-availability telecommunication systems; the high infrastructural costs made the use of more than two Earth stations inconvenient.
 However, it is accepted opinion that in the near future, there will be a strong increase in digital medium-to-low-availability services, offered to private operators or directly to the end users. In this scenario it is not unrealistic for different service providers operating in the same metropolitan area to come to an agreement for the exchange of connection: Their satellite stations could, for example, be linked together in a metropolitan area network (MAN), realizing a small-scale multiple-site diversity system, usually termed wide-area diversity [Dissanayake and Lin, 2000]. Wide-area diversity is ideally suited for data transmission using low-margin very small aperture terminals (VSATs). In fact, they are not normally operated at full capacity all the time; the spare capacity can then be used to support the diversity operation so that terminals can assist each other in the event of rain fading. In this context, there is a motivation for the investigation of the performance of such systems.
 Few direct experimental investigations of the performance of multiple-site diversity systems have been done in the past; Tang et al.  discuss the results of a three-site experiment carried out in Florida at 19 GHz over a period of 29 months. Witternigg et al.  used 12 GHz radiometers to simulate a quadruple-site diversity satellite link. Goldhirsh et al.  describe a three-site diversity experiment at 20 GHz using the Advanced Communications Technology Satellite (ACTS).
 The main drawback of this “direct” approach is that it is expensive and time consuming. Moreover, the geometry of the experiment is fixed, and the results cannot be easily extended to other configurations.
 When direct experimental measurements are not feasible, an effective alternative is simulation. In this respect, meteorological radars can provide very useful information on the space-time structure of rain over a quite large observation window. In this paper an extensive database of radar maps collected during 1989 has been used to evaluate the performance of small-scale, multiple-site diversity systems.
2. Simulation on the Radar Database
 Between February and September 1989, an experimental campaign was carried out with the S band Doppler meteorological radar located in Spino d'Adda near Milan, Italy. The main characteristics of the radar sensor at that time can be found in Table 1. The campaign led to the acquisition of a large database of rain maps, in the form of pseudo–Constant Altitude Plane Position Indicator (CAPPI) images at 1000 m height. These images were obtained by the composition of three PPIs taken at 3°, 5°, and 7° of elevation; this was done by choosing the radar bin whose altitude was closer to 1000 m. To guarantee a good spatial resolution, only echoes up to 47 km of distance were considered. A total of 25 rain events was recorded, corresponding to 6678 radar images; the radar maps were remapped from the polar to a Cartesian grid for convenience, with a resolution of 500 × 500 m2. Radar reflectivity Z (mm6 m−3) was converted into rain rate R (mm h−1) by applying the standard relation Z = 200 R1.6, which corresponds to the Marshall-Palmer drop size distribution [Marshall and Palmer, 1948]. This relation has proven to work quite well in the Padana Valley [Pawlina, 1984] on a statistical basis.
Table 1. Technical Characteristics of the Spino d'Adda Radar During the 1989 Experimental Campaign
S band (2.8 GHz)
Parabolic, 3.6 m
3 PPI at 3°, 5° and 7°
Acquisition time (one PPI)
 The statistical meaningfulness of the radar database has been verified by comparing the cumulative distribution function (CDF) of the radar-derived rain rate PR(R) with that obtained with a colocated tipping-bucket rain gauge over a period of several years (PG(R)). Since the radar operates only when rain is present, the absolute time reference is not available in the radar database; however, the radar-derived PR(R) can be renormalized to the absolute time frame by “forcing” the probability that a given rain rate R0 is exceeded to correspond to the absolute annual probability (at the same rain rate), i.e., to force PR(R0) = PG(R0). This conceptually corresponds to the introduction in the database of the correct number of “empty” radar images that would have been collected during dry days and that were lost since the radar was not operating. We have chosen (somewhat arbitrarily) R0 = 4 mm h−1, forcing PR(R0) = 0.4%. PG(R) and the renormalized PR(R) are plotted in Figure 1; there is a good agreement between the two P(R)s up to R = 120 mm h−1; higher values of R are underestimated by the rain gauge; this is mainly because of the mechanical movement of the tipping bucket [La Barbera et al., 2002].
 Simulations have been carried out assuming N Earth stations pointing at the same geostationary satellite; their layout will be discussed later. The vertical profile of precipitation is assumed to be constant up to the equivalent rain height hR given by ITU-618 [International Telecommunication Union, 1998]; for our location, hR = 3320 m.
 The system under study is “located” on the radar map. Total rain attenuation Ai is calculated by integrating the specific attenuation, obtained from rain rate R through the standard exponential relation [Olsen et al., 1978].
 The cluster of stations is then rigidly translated on the radar map of a fixed distance; we found 2 km (corresponding to 4 pixels) to be a good trade-off between statistical stability and computing time. Attenuation is evaluated for the new position, and the procedure is repeated for all possible positions of the cluster on the radar map; the cluster is translated both in the horizontal and the vertical direction, thus “scanning” the entire map. To avoid the introduction of biases due to a possible preferred orientation of the rain structures, the cluster of stations is rotated by 90°, and another set of simulations is performed. The sequence is then reiterated for the successive map.
 It is now advisable to discuss the layout of the cluster of stations. When only two stations are considered, there is no ambiguity, being that their distance D is univocally defined; the only free parameter is the angle ϕ formed between the baseline (i.e., the line connecting the two stations) and the ground projection of the Earth-space path (see Figure 2).
 When more than two stations are considered, we must define a criterion to compare different geometries to be able to assess the “advantage” of a given configuration over another. In our opinion, this criterion should also take into account the territorial occupation of the cluster.
 With this in mind, we decided to proceed with two different approaches. First, we analyzed the improvement that we get when we pass from two to three stations. The third station is located on a line perpendicular to the baseline joining the two original stations (see Figure 3a).
 Second, when four or more stations are available, these are located inside a circle of a given diameter D (this defines the territorial occupation of the cluster; see Figure 3b). A preliminary analysis has shown that when the diameter of the circle is equal to or smaller than 40 km and for a reasonable number of stations, the best performance (in terms of relative diversity gain, defined later) is obtained when the stations are equally spaced on the circumference. This can be explained as follows: In principle, in a multiple-site configuration, the stations should be placed so as to maximize the distance among them. However, if the ground projections of the Earth-space paths (from the stations up to the equivalent rain height hR) overlap (or they are very close to each other), a strong correlation is introduced in the time series of the attenuation experienced by the stations. This is likely to happen when the diameter of the circle is equal to or smaller than 40 km: The ground projections relative to stations placed inside the circle can (partially) overlap with those relative to stations on the circumference, thus reducing the effectiveness of the technique. These considerations do not apply to bigger circles.
 Attenuation due to gases (oxygen, water vapor, and liquid water) is taken into account by adding its cumulative value as a constant contribution to the rain attenuation statistics and has been computed following ITU-R recommendation 676-3 [International Telecommunication Union, 1997]. The simulations that will be presented refer to systems operating at 20 or 30 GHz; details of the geometries investigated will be given in section 3, where the results will be discussed.
3. Statistics of Attenuation
Hodge  defined a useful parameter for quantifying system performance relative to long-term probabilities, the absolute diversity gain G(P) = A0(P) − AN(P), where A0(P) is the attenuation (in dB) exceeded on the single radio link at a given probability level P and AN(P) is the attenuation exceeded in the N sites diversity configuration (i.e., it is the minimum attenuation experienced by any link in the cluster) at the same probability level. For the definition of these two quantities, it is always true that A0(P) ≥ AN(P). To compare the performance of two different configurations (say, configurations A and B), we define here the relative diversity gain g(P) as
The two configurations can differ for the number of Earth stations and/or their disposition on the ground; note that this definition of relative diversity gain is different from that given by Hodge .
 We have first evaluated the relative diversity gain of a setup of three stations, operating at 30 GHz, with respect to that obtained with two stations. The distance D between the two “original” stations is 10 km; a third station is introduced, according to the geometry shown in Figure 3a; the benefit that we get from the introduction of the third station depends on its distance d from the baseline.
 Results for baseline angle ϕ = 90° and an elevation of 10° are shown in Figure 4; the relative diversity gain (abscissa) is plotted as a function of probability (ordinate); each curve corresponds to a different geometry. Such a low elevation angle can be experienced by stations operating at high latitudes and corresponds to relatively long paths in the troposphere. The label “d4 vs 10 km,” for example, identifies the curve showing the relative diversity gain of the three-station setup for d = 4 km, when compared to a system made of two stations 10 km apart; the three-station setup performs better than the two-station setup, since its relative diversity gain is about +9%. The curve labeled “d4 vs 14 km” shows the performance of the same three-station setup with respect to that of a system made of two stations 14 km apart. In this case, the three-station setup performs worse than the two-station setup, its relative diversity gain being about −4%; negative values of the relative diversity gain, in fact, indicate that the considered geometry has an absolute diversity gain lower than that of the reference geometry, i.e., GA(P) < GB(P). These results show that in general, the performance of a three-station configuration is only slightly worse than that of a system made of two stations, whose distance is D + d.
Figure 5 shows the same results but for an elevation of 40°. With respect to the previous geometry we obtain a significantly higher relative diversity gain for a given distance d; for example, the “d4 vs 14 km” curve in this case shows a relative diversity gain of about 16%. Moreover, in this case, the relative diversity gain of a three-station configuration is consistently better than that of a system made of two stations, whose distance is D + d. In fact, the ground projections of the Earth-space paths are shorter for an elevation of 40° than for an elevation of 10°, and this leads to a lower correlation between attenuation levels experienced by the different stations, thus enhancing the effectiveness of the diversity technique.
 Results for ϕ = 0° and an elevation of 10° are shown in Figure 6; performance of the diversity technique is much better than that observed with ϕ = 90°; this is due to the fact that when only two stations are available, there is a significant overlap of ground projections of the Earth-space paths; this reduces the effectiveness of the diversity scheme. When a third station is added to the cluster, its ground projection is significantly distant from those of the original couple; the performance of the diversity technique is therefore greatly enhanced. This is confirmed by the fact that the relative diversity gain of a three-station configuration is consistently better than that of a system made of two stations, whose distance is D + d.
 Let us now consider four or more stations, located on a circle. The baseline angle in this case is not so important because with an increasing number of stations, the geometry tends to be inherently symmetrical.
Figure 7 shows the cumulative distribution functions of attenuation at 20 GHz and for an elevation of 30°, as a function of absolute time percentage, when stations are located on a circle 10 km in diameter. There is a clear advantage in using more than one station; the incremental gain, however, decreases for an increasing number of stations and “saturates” to an asymptotical curve (i.e., more than five ground terminals do not seem to be effective). A similar behavior has been found at 30 GHz; being that the frequency is higher, for a given value of time percentage, the system experiences a higher attenuation.
Figure 8 gives the same information as Figure 7 but for stations located on a circle 20 km in diameter. For a given level of probability, the attenuation experienced by the cluster of stations is lower than that experienced by the same number of stations located on a circle 10 km in diameter. This is to be expected, since the performance of the diversity technique improves for increasing distances among the stations. In this case, the saturation effect is less evident; this means that by increasing the diameter of the circle, the number of Earth stations that can be usefully linked together increases.
Figure 9 shows the relative diversity gain for a frequency of 20 GHz and an elevation of 30° as a function of the diameter of the circle when two, three, or four stations are located on the circumference. The four curves correspond to absolute time percentages of 0.01%, 0.03%, 0.1%, and 0.3%, and they are relative to a configuration with two stations. Black dots and open diamonds locate the value of the relative diversity gain corresponding to geometries with three and four antennas, respectively. The diameter is indicated as a number under (three stations) or over (four stations) the vertical bars passing through the groups of dots or diamonds. For example, for a distance of 8 km and a time percentage of 0.03%, we have a relative diversity gain of 55% for two stations, 62% for three stations, and 65% for four stations. We can also obtain a relative diversity gain of 55% for a probability of 0.03% by employing two, three, or four stations located on circles of 8, 6, or 5 km in diameter, respectively. If the relative diversity gain is calculated at 30 GHz for the same geometry, the curves show only a negligible difference with respect to those in Figure 9, so these results are omitted here.
4. System Performance
 The statistics of attenuation, presented section 3, tell only part of the story. To evaluate the performance of the system, it is necessary to make some assumptions on the margin of the links; when the margin is known, the CDFs shown in section 3 allow for the calculation of the availability of the system, i.e., of the probability that at least one station of the cluster is working.
 When several stations are linked together to form a cluster, it is reasonable to assume that when one station is in outage, its traffic is handled by the working ones. The performance of the cluster in terms of total traffic therefore depends on the number of stations that are working in a given moment, performance decreasing when the number of stations in outage increases.
Figure 10 shows the outage probability for a cluster of five stations, operating at 20 GHz, as a function of diameter; each link has an elevation of 30° and a margin of 3 dB. Here the cluster is supposed to be “available” if at least N stations are available, regardless of their position. Each curve in Figure 10 refers to a different value of N. When the diameter is zero, all the curves collapse in one point; this coincides with the availability of a single station. The curve labeled “at least 1 stn” coincides with the standard definition of availability, i.e., the probability that at least one station (out of five) is working. When it is required that at least two (or more) stations must be working, the availability decreases; that is, the outage probability increases. It is interesting to note that in this case, if we require four stations to be working simultaneously, the outage probability does not change significantly for increasing diameters of the circle; that is, there is no advantage in using a multiple-site diversity scheme.
 Obviously, when we require all five stations to be working, availability decreases for increasing diameters. Having dispersed the stations in a large area, the probability of having rain (producing a fade level greater than 3 dB) on at least one of the sites increases.
Figure 11 shows the average fraction of stations (from a cluster of N) that are in outage when at least one of them is in outage as a function of diameter. Stations operate with 3 dB of margin, and each curve refers to a different frequency. We found only a very modest dependence of the curves with N, provided that N > 3; Figure 11 is therefore representative of a very general behavior.
 As expected, the average number of unavailable stations decreases steadily for increasing values of the diameter, since the correlation between the sites decreases when the mutual distance increases. At 20 GHz, for example, we find that 70% of the stations are not available on average when the diameter of the cluster is 2 km; this figure drops to 30% when the diameter is 20 km.
 For a given diameter, there is a strong dependence with frequency; this is due to the fact that the rain rate required to experience an attenuation of 3 dB at 10 GHz is much higher than that required at 40 GHz. As already pointed out in section 3, intense (convective) precipitation events tend to have a limited spatial extension and therefore tend to impact a smaller number of stations.
 When Earth stations are used to carry data traffic (access to the Internet, etc.) the average traffic Tm (i.e., the average data rate) will often be smaller than the peak capacity TP. If this is the case, each station has spare capacity that can be “leased” to the station(s) in outage.
 Let us assume that the cluster is composed by N identical stations, having the same normalized maximum capacity TP = 1 and the same average traffic Tm ≤ 1. When Tm = 1, the stations are operating at their full capacity; not being of interest, this case is omitted here.
 Let us investigate what happens when Tm < 1. If M stations are out of service and no diversity scheme is implemented, the total amount of traffic lost is MTm. If the stations are connected in a cluster and can exchange traffic, the total theoretical capacity of the cluster is reduced from NTP = N to (N − M) TP = (N − M); this is an upper limit, valid if we assume that all the stations work constantly at their maximum capacity (i.e., there is no latency in the switch-over process, etc.).
 Since the traffic to be handled is NTm, there are two possible situations. If NTm ≤ (N − M), there will be no loss in traffic; if NTm > (N − M), the capacity of the active stations will not be sufficient, and the amount of traffic lost will be T+ = NTm − (N − M).
 Let us investigate the normalized traffic not served, +, defined as the traffic lost normalized to the traffic that will be lost if no diversity scheme is implemented, i.e., + = T+/(MTm). This quantity gives an indication of the effectiveness (i.e., the gain) of the multiple-diversity scheme with respect to the “no-diversity” situation.
 Let assume that stations operate at 30 GHz with 3 dB of margin and 30° of elevation. Figure 12 shows + for a cluster of two stations as a function of diameter when at least one of the stations is out of service; each curve refers to a different value of average fractional traffic Tm. Figure 12 tells us that for example, when the average traffic of each station is Tm = 0.9 (i.e., a busy situation), the distance between the stations is 6 km and at least one of the stations is out of service, + is 0.95 (95%). That is, by adopting the diversity scheme, we will be able to recover, on average, 5% of the traffic that will be lost without adopting countermeasures. In this case, increasing the distance between the stations introduces only a minor improvement; the maximum amount of traffic that can be retrieved is in fact 9% for a distance of 20 km.
 This fraction increases significantly for decreasing values of Tm; for Tm = 0.75 we will be able to recover almost 20% of the traffic that will be lost. When Tm < 0.75, the amount of recovered traffic increases significantly for increasing distance between the stations.
 If Tm > (N − 1)/N, the curves of + have a theoretical horizontal asymptote for [(TmN) − (N − 1)]/MTm. This is due to the fact that when a station is in outage, the remaining (N − 1) stations are not able to fully absorb its traffic. When Tm = 0.9, for example, it is easy to verify that the asymptotes for a cluster of two, three, four, or five stations are 88.9%, 77.8%, 66.7%, and 55.5%, respectively. On average, the situation will be worse, since more than one station will be out of service.
 It can be noted that the curves for Tm ≤ 0.5 overlap perfectly; this is due to the fact that even if a single station is able to support all the traffic of the cluster, there are situations where both stations are simultaneously out of service. The probability of this event decreases for increasing distance between the stations, and this is clearly reflected in the steady decrease of the amount of traffic not served for increasing distances.
Figure 13 shows the same results when the cluster is composed of three stations; it can be noted that for a given diameter and a given Tm, the percentage of traffic not served is lower than that found with two stations, and the recovered traffic is almost double; this indicates that the effectiveness of the diversity scheme increases significantly when the number of stations of the cluster increases. Also, in this case, the curves relative to the lowest values of Tm overlap perfectly. Figure 14 shows the results for five stations; in this case, the curves do not overlap for any value of Tm because of the relatively high number of Earth stations, showing once more the effectiveness of a multiple-site diversity scheme.
 The dual-site diversity technique has proven to be quite useful in counteracting rain attenuation; this technique can be extended by adopting a multiple-site configuration. Simulations carried out (using a radar database) for small-scale multiple-site diversity systems have shown significant improvements in terms of diversity advantage and site separation reduction with respect to the corresponding dual-site configuration.
 The principal characteristics of a wide-area diversity system can be summarized as follows.
 1. As it is well known, it is always convenient to pass from one to two Earth terminals, provided that their distance is greater than 2 km; the relative diversity gain g(p) is in fact greater than 20% and can be as high as 80%, depending on the separation of the two stations.
 2. When three stations are used, their performance is almost equal to that of two stations whose distance is (D + d), with D the distance between two of the three stations and d the distance of the third station from the line connecting the first two (i.e., the baseline).
 3. When several stations are located in an area whose maximum dimension is smaller than 40 km, the biggest contribution (to the relative diversity gain) comes from the stations located on the perimeter.
 4. The relative diversity gain increases for an increasing number of stations; however, it tends to saturate to an asymptotical value. From an operational point of view, there will be an “optimum number” of stations in the cluster, a trade-off between gain and system complexity.
 5. The global performance of the cluster depends on the number of stations that, at a given time, are in outage. The average number of stations that are unavailable depends dramatically on the size of the circle and on the frequency of operation for a given link margin and only to a very modest extent on the number of stations that compose the cluster.
 6. When the stations carry data traffic and the average data rate is lower than the capacity of the stations, the traffic from stations in outage can be diverted to active stations; this allows us to recover a significant amount of the traffic that otherwise would be lost. The effectiveness of this scheme increases when the average data rate decreases and the number of Earth stations increases. Also, in this case, there will be an “optimum number” of stations in the cluster, a trade-off between traffic recovered and system complexity.