Rain Attenuation Statistics Useful For The Design Of ... - IEEE Xplore

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 2, MAY 1998

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Rain Attenuation Statistics Useful for the Design of Mobile Satellite Communication Systems Emilio Matricciani and Stefano Moretti

Abstract—We have investigated how rain attenuation statistics, necessary to design fixed satellite systems working at frequencies greater than 10 GHz, are transformed to those applicable to the design of mobile satellite systems working in the same frequency bands and weather conditions in the special case of vehicles driven in zig–zag patterns to simulate city streets. The vehicles’ speed has been modeled as a lognormal random variable, a mathematical model derived from measurements performed in freeways or in city traffic. We have used a large number of rainrate time series collected in Italy (Gera Lario and Fucino) and in Canada (Montreal) to simulate rain-rate spatial fields and radio links at 19.77 GHz along a 30.6 slant path. The simulations have shown that a receiving or transmitting terminal moving in zig–zag patterns may experience, in the long term and for a given attenuation, a smaller outage probability, compared to the fixed terminal. For a given rain attenuation, the ratio between the outage probability of the mobile system and that of the fixed system (probability extrapolation factor ) is estimated to be around 0.2–0.3 in the optimistic case in which the probability of encountering rain is the same for all observers (fixed or in motion)—a less optimistic estimate shows that the probability of encountering rain might be twice as large and that, as a consequence, the values of mentioned must be doubled. Conservative values of can be calculated by using average values of rain storm speed and vehicles and average distances covered in the rain by fixed and mobile terminals. The results are less sensitive to changes in the geometrical or other parameters of the simulations. Index Terms—Rain attenuation, road vehicles velocity, satellite mobile communication.

I. INTRODUCTION

U

SING THE work reported in [1], we investigate how rain attenuation statistics, necessary to design fixed satellite systems working at frequencies greater than 10 GHz, are transformed to those applicable to the design of mobile satellite systems working in the same frequency bands and weather conditions. We have simulated different routes for vehicles driven in the rain at realistic speeds according to measurements performed in freeways or in city traffic. While the results discussed in [1] concern vehicles driven in straight lines, the present work reports results concerning a more general zig–zag pattern in a “city.” We apply the “synthetic storm” technique previously tested as a prediction model of rain attenuation for fixed satellite systems with very good results [2] and use a large number of rain-rate time series collected in Italy (Gera Manuscript received March 6, 1996; revised September 23, 1996. E. Matricciani is with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, 32-20133 Milano, Italy. S. Moretti is with the Telecom Italia, Milan, Italy. Publisher Item Identifier S 0018-9545(98)03288-5.

Lario and Fucino) and in Canada (Montreal) to simulate rainrate spatial fields and radio links at 19.77 GHz in a 30.6 slant path. In the simulations, we have assumed the following fundamental hypotheses. 1) Vehicles (i.e., moving observers) and fixed observers in which are all within a fixed geographical area orographic effects are ignored. 2) Vehicles enter and always leave a rain cell. This assumption allows a direct comparison of first-order statistics (cumulative probability distributions of attenuation) and second-order statistics (concerning fade durations and rates of change) predicted for a moving terminal with those predicted or measured for a fixed terminal. For the latter, in fact, rain attenuation is usually measured for the rain storm duration, i.e., as if the fixed terminal in a static were “moving” at the rain storm velocity rain cell. As a consequence, both fixed and mobile rain attenuation time series start and end at zero decibels and a direct comparison is then possible. within . 3) A rain storm moves at constant velocity 4) The synthetic storm technique can be applied [2]. Moreover, we have assumed that the slant path (with elevation angle ) is toward a satellite in the geostationary orbit (GEO) (e.g., Olympus, 19 W, as seen from Spino d’Adda, direction given by the angle in Fig. 1)—the technique developed below can be applied, however, to other geometrical configurations, for instance, to satellites in medium orbits (MEO’s) or low orbits (LEO’s) and to airplanes—in other words, to slant paths with changing elevation angles. In the next section, we report the method used to simulate rain-rate spatial fields and to calculate rain attenuation. In Section III, we report and model data on vehicular speeds in different traffic conditions—this is useful for the present and future simulations. In Section IV, we show the results of simulations and compare the cumulative probability distributions of long-term rain attenuation of the fixed system with those of the mobile system, obtained for optimistic or for more realistic simulations, and in Section V, we draw some conclusions.

II. RAIN-RATE SPATIAL FIELD AND RAIN ATTENUATION The rain-rate time series available to us have been collected continuously for 5 yr at Gera Lario and Fucino during the SIRIO experiment (the results of which are reported in [3] and [4]) and for 10 yr at Montreal [5], but for the latter site only during the spring–summer semesters (see Table I). All the rain-

0018–9545/98$10.00  1998 IEEE

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M

Fig. 1. City streets grid and velocity vectors ~ vR (rain field) and ~v

(vehicle).

TABLE I GEOGRAPHICAL, RADIOELECTRICAL PARAMETERS, AND RAIN-RATE DATABASE FOR THE SIMULATIONS(a)

rate data have been averaged in the standard 1-min interval. We have used these data to simulate both one-dimension (1-D) and two-dimension (2-D) spatial rain-rate fields. The 1-D simulation yields the long-term cumulative proba(decibels) bility distribution that a given rain attenuation is exceeded for a fixed terminal. Because of the hypotheses underlying the synthetic storm technique, this distribution is valid for a homogeneous and isotropic circular area of radius

40–50 km around the rain-gauge site and for a slant path length less than 25–30 km [2]. The latter values are much larger than the average rainy slant path lengths considered by fixed and moving observers (Table I), regardless of their positions, and the so that stationarity may be ensured to calculate (decibels). The vertical attenuation of the mobile terminal structure of precipitation has been modeled with two effective layers (layers A and B) up to the rain height above sea level

MATRICCIANI AND MORETTI: ATTENUATION STATISTICS FOR COMMUNICATION SYSTEMS

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Fig. 2. Cumulative number of rain storms for which the indicated duration (TR ) has been exceeded in Gera Lario (L), Fucino (F), and Montreal (M).

given by the ITU-R for the site, as in [2]—the superior layer (B) models the melting layer [6]. The 2-D simulation yields the rain-rate spatial field in which we make the mobile terminals move randomly as they are being affected by rain attenuation . To make simulations less involved, we have considered for each rain storm a square rain-rate field of side (linear extension of rain) given by (1) is the rain storm duration and is its speed. where Equation (1) has been also used by Drufuca and Zawadzki, who pioneered the studies on the time-to-distance conversion of rain-rate time series and extensively examined the Montreal data [7]. The spatial rain rate changes only along lines parallel to the radio path to the satellite (angle in Fig. 1) according to the synthetic storm technique and remains constant along the orthogonal direction. The latter artificial condition does not affect since this quantity is calculated along the slant path to the satellite so that it always depends on intervals of the true rain-rate space (time) series measured by the fixed observer. Moreover, very unlikely a vehicle will be driven along this orthogonal direction for a significant distance (and time) to experience a constant . The rain-rate field travels rigidly according to its (random) velocity . Fig. 2 shows the cumulative number of rain storms for which the indicated has been exceeded for the three sites (once value of normalized to the total number of rain storms, the curves become the cumulative probability distributions of ). Notice that the value calculated from (1) is at best an estimate of the length of a section, seen by the fixed terminal, of the true rain-rate field. Within a rain storm, this length changes according to the irregular horizontal shape of the

field and to , and it changes from rain storm to rain storm. However, if we consider a large number of rain-rate time series, the average results seeked (e.g., rain attenuation cumulative probability distributions for an average year) are calculated from (1) may be considered reliable [2], and as describing the linear extension of an effective and more regular horizontal shape of the rain-rate field (e.g., square or obtained seem to be circular). Moreover, the values of realistic, as it is shown in the following. We have not tried to infer the size distribution of precipitation areas as done by Yau and Rogers [8] because their method seems to work well only for rain-rate thresholds greater than a few millimeters/hour, that is, for rain modeled as cells of intense rain rate embedded in more widespread rain, the linear extension of which, on the contrary, is of concern in our simulations. In the simulations, we have assumed that in the area the rain-rate statistical features do not change. In a real case, gets large (i.e., many tens or hundreds of when the true kilometers), this is not true. Notice, however, that even if the gets large, we always compare and simulated statistics relative to the same rain-rate environment, and, in our opinion, this comparison is still significant. In a real large rainy and statistics collected area, we must average both in many sites and routes, respectively, with likely different long-term rain-rate features and then compare the results. We have assumed that this comparison can be simply simulated by using the rain-rate field obtained by the synthetic storm technique. For Gera Lario and Fucino, the value of the speed of all the rain storms has been set to 38.2 km/h (10.6 m/s). This value represents the average speed of rain storms estimated by the meteorological radar located in Spino d’Adda (Po

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River Valley) [9]. This average speed has been used to predict at 11.6 GHz accurate long-term probability distributions of for Spino d’Adda, Fucino, and Gera Lario [2]. For Montreal, we have used both the estimated speed of each single rain storm, available with the rain-rate time series [5], and the longterm average value, i.e., 49.9 km/h (13.9 m/s). We anticipate that for any observer (fixed or in motion), the use of the average speed instead of the single rain storm speeds yields the identical attenuation probability distributions, so that the average speed can be used for any site (more details are in [2] and in Section IV for ). As for the motion of rain for storms, we know that there may be preferred directions. In the area around Spino d’Adda, for instance, the angle of arrival is roughly uniformly distributed in a 90 angular interval around the north direction [9]. In the simulations, we have also considered this issue. , we can estimate according to (1). As For a given a consequence the curves of Fig. 2, once normalized, give the too, e.g., by assuming cumulative distributions of km/h for Gera Lario and Fucino, and km/h for Montreal. For the latter site, the values of the speed estimated for each single rain storm yields a probability distribution not significantly different from that calculated with of the average speed. The linear dimensions of the rainy areas obtained are realistic indeed. For example, as the maximum are about 33, 27.2, and 15.4 h for Gera Lario, values of equal Fucino, and Montreal, respectively, we get values of to 1260.6, 1039.4, and 768.5 km. The larger values are usually given by the winter-long rain events characterized by low rain rates. As for Montreal, these events are absent, and then and are statistically smaller. These extreme values agree with indirect measurements of the large scale (mesoscale) linear extensions of rainy areas, e.g., in Italy [10] or in the continental United States [11]. We anticipate that the useful results reported in Section IV are, however, little sensitive to . the absolute value of For each rain storm, the attenuation of the fixed terminal , is calculated as a convolution integral in a at time , direction parallel to the projection at ground of the slant path , to the satellite, with the linear extension of rain equal to as in [2]. Each rain storm thus yields an attenuation time series for the fixed terminal. The attenuation of the moving terminal is calculated by integrating the specific attenuation (decibels/kilometers) along the radio path to the satellite and according to its motion {see [2, (20)] for the simple line integral involved}. The 1-min rain-rate time series yields rain attenuation time series with the same sampling time. As a consequence, (m/s) 60(s) the space sampling distance is given by m, e.g., m (Gera Lario and Fucino) or m (Montreal), which is high enough to introduce an unacceptable quantization error in calculating and . To reduce this error, we have generated smoothed rain-rate time series with an artificial 7.5-s time resolution (eight samples/min) as described in [2]. In the above m examples, the sampling distances become and m. From these space series, we have then generated the spatial rain-rate field as described above.

III. VEHICULAR SPEEDS To obtain realistic values of , we need to estimate the , possibly probability distribution of the speed of vehicles in different traffic and route conditions. Most of the literature reports data more on vehicular traffic (e.g., number of vehicles running through a check point in a given interval of time, etc.) rather than on vehicular speed. In the following, we have summarized and modeled statistical data on vehicular speed collected in Italy, which are very useful for our simulations. We have considered only three types of traffic and roads: 1) urban routes with low-speed traffic (Model I); 2) urban and suburban routes with high-speed traffic (Model II); 3) freeway routes with very high-speed traffic (Model III).

A. Urban Routes with Low-Speed Traffic (Model I) For urban routes with low-speed traffic, we have used data collected in the Italian cities of Naples (13 different routes), Cosenza (7 different routes), and Milan (5 different routes). For Naples and Cosenza, the total sample size amounts to 9802 data and concerns the average speed in short distance (i.e., speed calculated as the ratio between a short distance and the time taken to cover it). For each distance, the average and standard deviation of the average speed are available [12] (see Table II). The measurements refer to lengths of a few hundred meters, short enough to consider the average speed as an instantaneous speed. The results show a relatively large variation in both average and standard deviation. For Milan, we have had raw data available on nine urban routes, collected in February and March of 1991 for several days, including Saturdays and Sundays. Afterwards, because of scarce data during the night, we analyzed only the data collected from 7–20 h, i.e., in the interval between morning and evening busy hours. Moreover, to compare these data with those collected in Naples and Cosenza, we have only considered distances less than 1330 m—in the end, five different routes. For them, we have calculated averages and standard deviations for a total of 819 data (see Table II). . We need now a model of the probability distribution of Fig. 3 shows the probability density histogram of the Milan sample size. It can be modeled by the lognormal model (2)

characterized by the standard deviation (Np) and the average value (Np) of the natural logarithm of . Now, as all the routes of Table II have similar lengths and traffic, we have inferred that the data from Naples and Cosenza might also follow a lognormal model and that an overall lognormal density function can be derived by averaging all the data. We have thus estimated a unique lognormal model (Model I) by calculating the overall average variance (km/h) and the overall average (km/h) of the average speed. From these data, we have then obtained and of the


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TABLE II AVERAGE VALUE (km/h) AND STANDARD DEVIATION (km/h) OF SPEED v IN NAPLES (NA), COSENZA (CS), AND MILAN (MI) FOR THE INDICATED DISTANCES (m). TOTAL NUMBER OF VEHICLES IS 10 621

M

lognormal model from the following relationships (e.g., [13]):

(3) Since the values obtained are km/h and (km/h) , then the lognormal parameters are equal to , i.e., and . Notice that the km/h (i.e., the value median value exceeded with probability 0.5) is almost equal to the average value 30.878 km/h because the skewness of the density is very low. As a matter of fact, a Gaussian density function would also have fit the data as well. We have preferred the lognormal density because it seems to be a better and more natural model as it does not predict negative speeds and because the speed might depend on many multiplicative and independent causes, which, by the Central Limit Theorem, would yield this density.

In using Model I, we have limited the maximum speed to 72 km/h (exceeded with probability 0.006%). B. Urban and Suburban Routes with High-Speed Traffic (Model II) For urban and suburban routes with high-speed traffic, we have used data made available to us from the City of Cesena (Italy). By means of optical sensors, the true instantaneous speed of vehicles was measured in four urban and suburban highway routes for several days in March 1993, 24 h a day, for a total of 142 354 data—a very large sample size. The data have been originally grouped to count the number of vehicles of speed within certain limits in a given hour (see Table III). Integrating this density and plotting the probability of exceeding a given speed on a chart, where the lognormal integral yields a straight line (e.g., [13]), we have noticed that these data are also well fitted by the lognormal model for km/h (the original linear quantization of the histogram is too

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Fig. 3. Histogram of vehicles speed in Milan.

NUMBER OF VEHICLES

WITH

TABLE III

M (km/h) IN THE INDICATED SPEED CLASSES. CESENA. TOTAL NUMBER OF VEHICLES IS 142 354

SPEED v

large for km/h). We have graphically estimated (Model II) and (56 km/h is the median value, and 70 km/h is the value exceeded with probability 15,87%, i.e., it corresponds to the standard variable in the Gaussian integral). Surprisingly, the standard deviation is very close to that found for Model I, i.e., . For uniformity, in the following we have assumed also for Model II. From (3), we have obtained km/h, which, again, is very close to the median 56 km/h. In using Model II, we have limited the maximum speed to 108 km/h (exceeded with probability 0.2%). C. Freeway Routes with Very High-Speed Traffic (Model III) The features of freeway routes are quite different from those of suburban and, especially, urban routes. On freeways, vehicles are likely to run faster and in longer straight routes with smaller speed variations. Since we do not have detailed data available on this type of traffic, we have supposed that it can be also described by a lognormal probability distribution km/h (a realistic value for (Model III), with

Europe [14]) and , as in the other models, and thus ( km/h). This model can be used in rectilinear motions, as discussed in [1]. Table IV summarizes the constants for the three models. IV. SIMULATIONS Let us consider the traffic in a large city where the streets form a grid of squares of side m arbitrarily headed in the east–west and north–south directions and covering all the area over which the square rain-rate field moves (Fig. 1). The 300-m side length is a good estimate of the order of magnitude of real city block dimensions and is also in agreement with the short distances to which our data on speed refer (Table II). The rain-rate field moves at the average speed (38.2 km/h for Gera Lario and Spino d’Adda and 49.9 km/h for Montreal) at an angle . We have used smoothed time (and corresponding space) series with a 7.5-s sampling time. is calculated every 100 m, a choice that yields a variable sampling time according to the value of . The simpler rectilinear motion of vehicles has been investigated in [1].


AVERAGE

mM

(km/h), MEDIAN

MM

TABLE IV

= exp() (km/h), AND STANDARD DEVIATION

To calculate long-term rain attenuation statistics for mobile terminals, we consider the following mutually exclusive hypotheses. 1) A vehicle moving in a zig–zag pattern in encounters rain with the same probability of the fixed observers or vehicles driven in rectilinear motion. We anticipate that this hypothesis yields optimistic rain attenuations exceeded. 2) A vehicle moving in a zig–zag pattern encounters rain with a larger probability than the other observers. We anticipate that the useful results for this case can be obtained, by simple models, from those of case 1) and that they yield less optimistic rain attenuations.

A. Equal Probability of Encountering Rain Equal probability of encountering rain is a good model for fixed and moving observers in rectilinear motion. In fact, a simple model devised by [8] predicts that the number of rain cells encountered along a straight line is proportional to the distance covered in the area . Since fixed and moving observers driven in rectilinear motion cover the same average distance in , it follows that in the long term both observers “see” the same rain storms and, as a consequence, experience the same probability of encountering rain, an assumption made also in [1]. In this section, we assume that the model can be applied to zig–zag routes as well. Following the hypotheses listed in Section I, at time the vehicle enters the grid under a given rain event (actually, it is the slant path that enters it [1]), which is simultaneously moving at the constant velocity at any point of the perimeter of the square rain-rate field, with the same probability, and moves along each element of the grid in a zig–zag fashion (there is no restriction on the zig–zag pattern, except the grid geometry) with different and statistically independent constant speeds along each side of the grid element. The latter hypothesis makes the simulation easier, but it can be far from reality—we do not know any modeling of the random process on successive values of speed. Once the vehicle leaves the rain (at time ), it cannot enter it again (if it did, it would be considered a different vehicle). For the vehicle, the duration , generally different of of the rain attenuation event is then the duration measured by the fixed terminal . Obviously, also and may be different. The hypotheses assumed for this simulation are (reference m, the probability of case): Model I for speed, going straight is 0.5, the probabilities of turning to the left

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M

(Np)

OF

VEHICLE SPEED FOR THE THREE MODELS

or to the right are both equal to 0.25 (no U-turn is allowed, and these values are arbitrary), the angle of arrival of the storms is uniformly distributed in the interval [0 and 180 ], and a different value is extracted for each rain event and kept fixed for that rain event. Now, we need to know how large must the sample size be to estimate the long-term cumulative probability distribution of reliably. While for the fixed terminal the sample size is the number of rain events, i.e., the number of experimental rain-rate time series, our simulations have shown that the exceeded with asymptotic value of the long-term value of a given probability, e.g., as low as 2 10 , is reached by averaging the conditional probabilities (i.e., during rain) of 15 independent vehicles (Fig. 4); of course, this does not mean that for a given rain storm the 15 attenuation time series are statistically independent: as a matter of fact they are not, as in the real world, where we expect that they are partially correlated when the patterns do not differ very much. In our simulations, this correlation may be larger because of the synthetic storm simulation. In any case, the simulated results must be conservative for this problem, i.e., less optimistic than those obtainable in the real world. Higher attenuations (lower probabilities, i.e., rarer events) for a given error require, of course, more vehicles. Compared to the fixed terminal, this behavior is justified by the more degrees of freedom available to the vehicles for their motion. The number of rain attenuation time series generated is always very large. For Gera Lario, for instance, we have considered time series, i.e., a number given by the number of rain storms multiplied by the number of different for each rain storm. As a consequence, realizations of in the following, we have only used results obtained by using 15 vehicles (reference case). is shown in Fig. 5 for An example of the resulting . Both Montreal and compared to the corresponding and have been referred to conditional distributions of an average year by averaging over and making reference to 5 yr for Gera Lario and Fucino, and to 10/2 yr for Montreal (Fig. 5) since, for the latter locality, the observation period was only 1/2 of the total. For Montreal, we have used both the average speed and the single rain storm speeds, and as , the use of the single rain storm anticipated, for a given speeds yields the same cumulative probability distribution obtained by considering the average speed only (Fig. 5). This is also true for the fixed system probability (not shown for brevity). We think that the mathematical explanation of this behavior is that discussed below.

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Fig. 4. Long-term values of A (decibels) exceeded in an average year with probability 4 1003 (lower three curves) or 2 as a function of the number of statistically independent vehicles for Gera Lario (*), Fucino ( ), and Montreal ( ).

2

M

+

0

2 1004 (upper three curves)

Fig. 5. Long-term PF (A) and PM (A) in an average year in a 30.6 slant path at Montreal. Reference case for pS ; pR ; pL ; and angular range. Fixed terminal: upper continuous curve; Model I with average rain storm speed: “*”; Model I with single rain storm speed: lower continuous curve; Model II: “+”; Angular range [0 , 30 ]: “o.”

From the results, such as those shown in Fig. 5, we notice that, for a given , the ’s obtained seem ’s according to the relationship to be scaled from the

(4)

with a constant probability scaling factor , i.e., independent of attenuation, as for rectilinear motion [1]. To assess this finding in Fig. 6, we have reported the values of obtained ’s and ’s of the three sites. from (4) and from the We notice that, for Gera Lario and Montreal, ranges in the interval 0.25–0.32 for a large attenuation range, practically


+

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0

Fig. 6. Values of for Gera Lario (*), Fucino ( ), and Montreal ( ) as a function of attenuation.

a constant value, while a larger spread is found for Fucino. implies that the conditional Notice also that a constant distributions of and are identical [1], although the unconditional ones are not as they refer to different total times and , respectively, in which or in the long term. With these hypotheses, is given by [1]

where is the number of independent vehicles (e.g., ), is the number of steps of length (300 m) covered by a vehicle in the th realization of moving at speed in the th step, and

(5)

is the total distance covered by a vehicle in the th realization. Whereas for a rectilinear motion there is no reason why the fixed and the mobile terminals should statistically cover different route lengths (hypothesis assumed in [1]), notice that in the present case because of the different number of degrees of freedom of the vehicles. Now, the second multiplicative term that appears in the right-hand side of (9) is an unbiased estimate of the average of [13], hence

Let us estimate these two parameters. As the rain storms move at the average speed, is approximately given by (6) is the number of rain-rate space (time) series and is calculated from (1) for the th rain storm. In (6) and (9) below, we have neglected the time necessary to allow the projection at ground, and along the satellite direction , of the average slant rainy path to enter the rain field (usually much ) [1]. Introducing the average value less than

where

(7)

(10)

(11) after the Appendix. Introducing in (9) the average distance (12)

can be written as (8) and (11), we get As for the vehicles,

is given by (13) Hence, from (5), (8), and (13), we get (9)

(14)

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M

Fig. 7. Cumulative probability distributions of D

for Gera Lario (L), Fucino (F), and Montreal (M).

TABLE V VALUES OF THE PARAMETERS NECESSARY TO CALCULATE THE PROBABILITY EXTRAPOLATION FACTOR . CITY TRAFFIC (MODEL I)

where (15a) (15b) In conclusion, to estimate we need to calculate the average distance (or, equivalently, the average duration of rain storms, , as ) covered by the fixed terminal and the average distance covered by the vehicles (or the derived parameter ). Let us estimate these parameters. The probability distribucan be calculated from the curves shown in Fig. 2. tion of Now, Fig. 7 shows the probability distribution of obtained from the simulations. From these curves, we have calculated and , and then from (15), we have derived and , and finally from (14), the corresponding values of . These results are reported in Table V.

The values shown in Table V are in good agreement with those reported in Fig. 6 for which they seem to represent an upper bound and thus yield larger (conservative) values of for system design. In both cases, the three sites do not show significant differences, even if for Montreal we have . Because of its considered statistically smaller values of definition, seems to be insensitive to the absolute values of and . In conclusion, we could calculate the average values (15) and then from (14) and avoid the cumbersome . calculation of Notice that the values of reported in Table V agree with and the results reported in [1], once the same values of are used, for the: 1) so-called “most favorable” case (rain storms and vehicles moving along straight lines and in opposite directions) or 2) for the case in which the vehicles move in straight lines, the direction of which is uniformly distributed in the 0 –360 range, and the rain storms move also in straight


lines with direction uniformly distributed in 0 –180 range, i.e., – . Case 2) may thus simulate the more general zig–zag patterns just considered. Let us assess the sensitivity of the predictions to several parameters such as: 1) average vehicle speed; 2) grid dimension; 3) angular range of arrival of storms; and 4) probabilities of going straight or turning. If instead of Model I we use Model II, we get about the same distributions of Model I (e.g., Fig. 5 calculated from Model II is only 0.5 dB for Montreal) as smaller throughout the range shown, for a given probability. These results agree with “model scaling” by (14) (assuming , or , does not significantly change). that If we restrict the angular range of storm arrival from [0 and 180 ] to [0 and 30 ] for a fixed probability, we get about the same attenuations at higher probabilities and values 1–2 dB larger around 10 (e.g., Fig. 5 for Montreal). If we change the grid dimension from 300 to 600 m or to 1000 m, we get slightly smaller attenuations as increases—similar values are also found when we change the probabilities from the values of the reference case to and (larger attenuations) or to and (smaller attenuations), with the results not shown for brevity. In conclusion the sensitivity to and speed models is of the order of 1-dB peak-to-peak, while a larger sensitivity at low probabilities is found by restricting the angular range of arrival of the storms. B. Larger Probability of Encountering Rain In the following, we derive an estimate of the probability of encountering rain in zig–zag patterns, compared to that of the fixed or moving observers in straight lines, and show that the results obtained in Section IV-A may be extrapolated to the present case, by assuming, again, that the probability of encountering rain is proportional to the average distance ) covered in by a vehicle moving in a zig–zag ( fashion. , we assume the simplifying hypothesis To estimate that, defined an arbitrary orthogonal reference system in , the vehicle moves so that the coordinates and of its position are increasing and/or decreasing functions. In this covered in by a given vehicle case, the total length must satisfy the inequality (16) and thus

satisfies the inequality (17)

Now, since each term in the right-hand side of (17) cannot exceed the average distance covered by the fixed observer ( ) or by the moving observer in rectilinear motion ( ), and the latter two are equal (see Section IV-A), it follows that (18) and, as a consequence, the probability of encountering rain in the zig–zag pattern cannot exceed twice the value estimated

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for the fixed and moving observers in rectilinear motion. Notice that in deriving this result, we have assumed that the two probabilities add, i.e., that they refer to mutually exclusive (disjoint) random events along and , which may be unrealistic, but it has the advantage of being a conservative hypothesis for system design. In conclusion, the values of of Table V might be doubled. Of course, this means higher values of , for a fixed attenuation , and hence less optimistic predictions. If the coordinates and of the vehicle’s position are not increasing and/or decreasing functions (e.g., vehicles driven in circles, or other loops), then the average distances in , and, thus, the the right-hand side of (18) may both exceed probability of encountering rain might be more than doubled, and equal or even greater than that of the fixed terminal. V. CONCLUSIONS While the results discussed in [1] concern vehicles driven in straight lines (rectilinear motion), in the present paper we have reported results concerning more general zig–zag patterns for vehicles driven in the rain at realistic speeds and according to measurements performed in freeways or in city traffic, the probability of which we have modeled as lognormal. We have used a large number of rain-rate time series collected in Italy (Gera Lario and Fucino) and in Canada (Montreal) to simulate rain-rate spatial fields and radio links at 19.77 GHz in a 30.6 slant path. The simulations have shown that vehicles moving in zig–zag patterns may experience, in the long term and for a given attenuation, smaller outage probabilities compared to the fixed system. The probability extrapolation factor , defined in (4), is estimated to be around 0.2–0.3 in the optimistic case in which the probability of encountering rain is the same for all observers (fixed or in motion), regardless of the site considered. Some conservative values of it, independent of attenuation, can be calculated by using average values of ) and vehicles speed ( ) and average rain storm speed ( distances covered in the rain by fixed and mobile terminals and , respectively. The results are less sensitive to changes in the geometrical or other parameters of the simulations. The values of reported in Table V agree with the results reported in [1] once the same values of and are used for the: 1) so-called “most favorable” case (rain storms and vehicles moving along straight lines and in opposite directions) or 2) for the case in which the vehicles move in straight lines, the direction of which are uniformly distributed in the 0 –360 range and the rain storms move also in straight lines with direction uniformly distributed in the 0 –180 range, i.e., – . Reference [1, Case (b)] may thus simulate the more general zig–zag patterns considered. If the probability of encountering rain is larger (as it may be), then the values of mentioned above increase (doubled in the conservative analysis performed), thus yielding less optimistic predictions. Further work, however, is necessary to assess how much these results are realistic or simply optimistic, e.g., by simulating vehicles driven in spatial rain-rate fields closer to reality, as those obtainable from meteorological radar maps.

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APPENDIX By the substitution

, from (2) we obtain

as the integral of the Gaussian density (with average equal to and variance ) equals one, and (3) has been applied.

[8] M. K. Yau and R. R. Rogers, “An inversion problem on inferring the size distribution of precipitation areas from raingage measurements,” J. Atmos. Sci., vol. 41, pp. 439–447, 1984. [9] M. Binaghi and A. Pawlina Bonati, “Modeling of multiple site statistical dependence function through integration of dynamic features of radar rain patterns,” in URSI Commission F Open Symp., 1992, pp. 2.1.1–2.1.6. [10] F. Barbaliscia, G. Ravaioli, and A. Paraboni, “Characteristics of the spatial statistical dependence of rainfall rate over large areas,” IEEE Antennas Propagat., vol. 40, no. 1, pp. 8–12, 1992. [11] R. J. Kane, Jr., C. R. Chelius, and J. M. Fritsch, “Precipitation characteristics of mesoscale convective weather systems,” J. Clim. Appl. Meteor., vol. 26, pp. 1345–1357, 1987. [12] D. C. Festa and A. Nuzzolo, “Analisi sperimentale delle relazioni velocit´a-flusso per le strade urbane,” Le Strade, no. 1226, pp. 459–466, 1990 (in Italian). [13] C. V. Bury, Statistical Models in Applied Science. New York: Wiley, 1975. [14] E. Morello and M. Marcenaro, “AICC autonomous intelligent cruise control—Valutazione mediante microsimulazione,” in PFT2, Genova, Italy, 1995, pp. 2403–2422 (in Italian).

ACKNOWLEDGMENT The authors wish to thank L. Mussone of the Dipartimento Vie e Trasporti, Politecnico di Milano, for having provided us the speed data of Milan and for helpful discussions on vehicular traffic. They are also indebted to the City of Cesena, in particular, to G. Riva, for having kindly provided us the data collected there. REFERENCES [1] E. Matricciani, “Transformation of rain attenuation statistics from fixed to mobile satellite communication systems,” IEEE Trans. Veh. Technol., vol. 44, no. 2, pp. 565–569, 1995. [2] , “Physical-mathematical model of the dynamics of rain attenuation based on rain rate time series and two layer vertical structure of precipitation,” Radio Sci., vol. 31, pp. 281–295, 1996. [3] “Special issue on SIRIO programme,” Alta Frequenza, vol. 47, 1978. [4] “Special issue on the SIRIO program in the tenth year of satellite life,” Alta Frequenza, vol. 56, 1987. [5] G. Drufuca, “Rain attenuation statistics for frequencies above 10 GHz from raingauge observations,” J. Rech. Atmos., vols. 1–2, pp. 399–411, 1974. [6] E. Matricciani, “Rain attenuation predicted with a two layer rain model,” Europ. Trans. Telecommun. Related Technol., vol. 2, pp. 715–727, 1991. [7] G. Drufuca and I. I. Zawadzki, “Statistics of raingage data,” J. Appl. Meteor., vol. 14, pp. 1419–1429, 1975.

Emilio Matricciani was born in Bussi sul Tirino, Pescara, Italy, in 1952. He received the electronics engineering degree (Laurea) in 1978 from the Politecnico di Milano, Milano, Italy. He joined the Politecnico di Milano in 1978 as a holder of a research scholarship and in 1981 as an Assistant Professor of Electrical Communications. In 1987, he joined the Universita’ di Padova as a Professor of Microwaves. Since 1991, he has been with the Politecnico di Milano as a Professor of Electrical Communications. His research activity is mainly in the field of satellite communications at frequencies above 10 GHz, both for the fixed and mobile systems. Besides the institutional activities (lectures on communication systems and research), he teaches technical writing to students.

Stefano Moretti was born in Sondrio, Italy, in 1969. He received the electronics engineering degree in 1995 from the Politecnico di Milano, Italy. His thesis was on fixed and mobile satellite communication systems affected by rain attenuation. Since 1995, he has been with Telecom Italia, Milan, Italy, as a Project Engineer.