Prediction of fading phenomena in land-satellite communication links



[1] This paper addresses the problem of prediction of probability of successful radio communication of any mobile or stationary subscriber located in areas of service such as complex urban environments characterized by nonline-of-sight propagation conditions, which limit GPS, Low Earth Orbit, and Medium Earth Orbit services in land-satellite communications. It presents a self-consistent physical-statistical approach for predicting fading phenomena usually occurring in land-satellite communication links caused by influence of the terrain features on radio signal propagation from the ground-based to the satellite antenna. This approach combines (1) the statistical description of the buildings array located on the rough terrain and the buildings' overlay profile, based on special probabilistic distributions of built-up terrain parameters, and (2) the theoretical description of propagation phenomena, taking into account multiple scattering, reflection, and diffraction mechanisms. A new technique is proposed for predicting the probability of fading phenomena occurring in land-satellite links using the so-called stochastic multiparametric model. Results of theoretical predictions are compared with those obtained from the “pure statistical” Lutz model and physical-statistical Saunders-Evans model, and then with experimental data obtained for different European cities. Obtained results show that the proposed stochastic approach can be used as a good predictor of fading phenomena in land-satellite communication links for different satellite constellation scenarios and elevations of satellites during their movement surrounding the Earth, with respect to the ground-based antenna for different land environments: rural, mixed residential, suburban, and urban.

1. Introduction

[2] Signal propagation in land-satellite communication (LSC) links for the last decades has become an essential issue, especially when high-rate data services are involved [Loo, 1985; Barts and Stutzman, 1992; Corazza and Vatalaro, 1994; Saunders and Evans, 1996; Fontan et al., 1997a, 1997b; Wu, 1997; Evans, 1998; Loo and Butterworth, 1998; Vatalaro and Mazzenga, 1998; Xie and Fang, 2000; Saunders, 2001; Farserotu and Prasad, 2002; Abdi et al., 2003]. Path conditions may indeed yield harmful impairments, severely degrading the GPS performance and availability, and therefore, limitation in mobile subscriber localization and positioning (e.g., servicing). Moreover, since the Low Earth Orbit (LEO) and Medium Earth Orbit (MEO) satellites are positioned at different elevation angles in most practical constellations, the buildings' overlay profile and the density of their distribution at the terrain, become the main parameters to be predicted for successful communication between the desired users' antennas and the satellite antenna [Lutz et al., 1991; Sterling and Harlelid, 1991; Butt et al., 1992; Wiedeman et al., 1992; Singh, 1993; Smith, 1994; Brunt, 1996; Parks et al., 1996; Karasawa et al., 1997; Tzaras et al., 1998].

[3] Thus, LSC systems enable users of handle wireless phones, portable computers, or mobile phones to communicate with one another from any two points worldwide. Propagation conditions may indeed cause harmful impairments that severely corrupt the land-satellite system availability and performance, so that propagation considerations are very important for successful operation. On the other hand, terrestrial land-mobile systems are often subject to a relatively strong multipath effect and they will be power limited and dependent on the line-of-sight (LOS) and nonline-of-sight (NLOS or clutter) components. Such reliability of mobile systems will largely be determined by propagation-induced fading: large-scale (or slow in the time domain) caused by diffraction from land obstructions, and small-scale (or fast in the time domain) due to multiple scattering from stationary and/or moving obstructions, which finally will cause outages and reduction of communications quality.

[4] Therefore, in order to design wireless LSC links successfully, stationary or mobile, it is very important to predict propagation phenomena occurring in such links. This means that we are obliged to give a satisfactory physical explanation of the main parameters of the channel, such as path loss and fading (slow and fast).

[5] As was mentioned by Saunders and Evans [1996], Wu [1997], Evans [1998], Saunders [2001], and Abdi et al. [2003], in such complex terrestrial environments-with high speed satellite movements, it is very complicated to differentiate slow and fast fading effects, as was usually accomplished for land communication links [Blaunstein et al., 2006; Yarkoni et al., 2007; Blaunstein and Christodoulou, 2007]. Therefore, they should be considered together [Saunders, 2001; Abdi et al., 2003].

[6] In this study, we show how to take into account effects of the built-up terrain profile and multiple diffraction and scattering effects for fading description in the LSC channel. As was previously shown by numerous theoretical frameworks (see the full biography in the work of Saunders [2001]), this procedure fully depends on the radio propagation scenarios that occur within each LSC channels and elevation of the satellite above the horizon with respect to the position and location of the desired subscriber. In section 2, we briefly analyze two main concepts on how to account for fading effects on LSC. The first concept is based on the “pure statistical” model introduced by Lutz et al. [1991] [see also Saunders, 2001, and references therein], which deals with two states within the communication link, good and bad. Good state means that the LOS component is dominant, while bad state occurs when the LOS component is small enough and mostly the NLOS component is present due to shadowing and/or multipath phenomena. Another concept is based on the so-called “physical-statistical model” developed by Saunders and Evans [1996], Evans [1998], and Saunders [2001], which deals with the local statistical distribution of built-up profile described by lognormal or Rayleigh probability distribution function, for predicting the outage probability of fading (e.g., shadowing) occurring in land-satellite link.

[7] To cover the existing models and to use them together in our analysis, we have analyzed the problem following the second concept fully described by Saunders and Evans [1996], Evans [1998], Tzaras et al. [1998], and Saunders [2001], which predicts more strictly fading effects in different LSC links with respect to the pure statistical concept [Lutz et al., 1991]. At the same time, the physical-statistical model which is based on a deterministic distribution of the local built-up geometry (see section 2), cannot strictly predict any situation when a satellite moving around the world, has different elevation angles θi with respect to any subscriber located at the ground surface, as shown in Figure 1. As the result, the radio path between the desired subscriber and the satellite crosses different buildings' overlay profiles because of continuously changing elevation angle of the satellite during its rotation around the Earth with respect to the ground-based subscriber antenna. Furthermore, this prediction mechanism of the outage probability of shadowing in real time needs a huge massive of data regarding each building, geometry of each radio path between any individual subscriber and satellite during its movement, and finally high-speed powerful macrocomputers.

Figure 1.

General geometry of the land-satellite link.

[8] In this situation, it is difficult to use the same probability distribution function, lognormal, Rayleigh, or their combination, as was done in previous statistical or physical statistical models [Loo, 1985; Barts and Stutzman, 1992; Corazza and Vatalaro, 1994; Saunders and Evans, 1996; Fontan et al., 1997a, 1997b; Wu, 1997; Evans, 1998; Loo and Butterworth, 1998; Tzaras et al., 1998; Vatalaro and Mazzenga, 1998; Xie and Fang, 2000; Saunders, 2001], taking into account simultaneous multiray effects.

[9] Therefore, we propose in section 3 the same physical-stochastic approach that has been developed for the land-land communication links in terms of statistical distributions of obstructions above the terrain with changing buildings' overlay profile [Blaunstein et al., 2002; Blaunstein and Andersen, 2002; Blaunstein and Christodoulou, 2007; Hayakawa et al., 2008; Blaunstein et al., 2009]. Here, following Blaunstein et al. [2002, 2006], we introduce a new probability distribution function, which takes into account multiple scattering and diffraction and the corresponding buildings' overlay profile, to predict fading phenomena for different satellite elevation angles and conditions of the land environment, rural, suburban and urban.

2. Existing Models: Statistical and Physical-Statistical

2.1. Lutz Statistical Model

[10] In this model, the simple statistics of LOS and NLOS are modeled by two distinct states, good or bad [Lutz et al., 1991], to be appropriate for describing the propagation situations in urban and suburban areas where there is a large difference between shadowed and nonshadowed statistics and is usually observed in LSC links. The LOS condition is represented by a good state, while the NLOS condition by a bad state. In the good state, the signal is assumed to be Ricean distributed with K factor as a ratio of LOS and NLOS components of the total signal envelope, which depends on the satellite elevation angle and the carrier frequency and Pgood = PRicean. In the bad state, the fading statistics of the signal amplitude is assumed to be the conditional Rayleigh distribution pRayleigh (SS0), where S0 varies slowly with a lognormal distribution PLN(S0), representing the varying effects of shadowing in the NLOS component. The transition probabilities are finally based on Markov chain [Lutz et al., 1991], i.e., probability of transition from good state to good state Pgg; probability of transition from good state to bad state Pgb; probability of transition from bad state to bad state Pbb; probability of transition from bad state to good state Pbg.

[11] The sum of these probabilities for any state must sum up to unity, so that

equation image

In the pure statistical model mentioned above, the input data and computational effort are quite simple, because the model parameters fit with measured data. Due to the lack of physical background, such a model, however, only applies with good results in the environments that are very close to the one they have been inferred from.

2.2. Physical-Statistical Model

[12] This general approach relates any channel simulation to the statistical distribution of physical parameters such as buildings' height, width and spacing, streets' width and elevation and/or azimuth angles of the satellite. This approach is usually referred to as the “physical-statistical” approach [Saunders and Evans, 1996; Evans, 1998; Tzaras et al., 1998; Saunders, 2001]. We consider only one of them, which are fully proved by numerous experiments carried out for LSC links performance and called “a model of shadowing based on two-state channel Lutz model.” The geometry of the situation that was analyzed by Saunders and Evans [1996], Evans [1998], and Saunders [2001] is illustrated in Figure 2. It describes a situation where a mobile is situated on a long straight street with the direct ray from the satellite impinging on the mobile from an arbitrary direction. The street is lined on both sides with buildings whose height varies randomly. The PDFs (probability density functions) that were selected to fit with the data are the lognormal-Rayleigh combined distributions with unknown parameters: hm is the mean value of buildings' height and σb is a standard deviation from the mean, both measured in meters. The PDF for the lognormal distribution is [Saunders and Evans, 1996; Evans, 1998; Saunders, 2001]:

equation image
Figure 2.

Geometry for mobile-satellite communication in built-up areas (modified from Saunders and Evans [1996] and Saunders [2001]).

[13] The PDF for the Rayleigh distribution is presented according to Saunders and Evans [1996] and Saunders [2001] as:

equation image

[14] Here all parameters are mentioned above and denoted in Figure 2. The direct ray is shadowed when the building height hb exceeds a certain threshold height hr relative to the direct ray height hs (see Figure 2).

[15] The complementary cumulative distribution function (CCDF) of shadowing denoted here by CCDF(hT), can be expressed in terms of the PDF of the building height Pb(hb) described by equations (2) or (3):

equation image

[16] The definition of hT can be derived by considering the shadowing occurring exactly when the building face blocks the direct ray. Using a simple geometry yields the following expression for hT (see Figure 2):

equation image

where the definitions of hm, θ and ϕ are given in Figure 2.

[17] Then, in this approach the same Markov chain is used, but parameters Pbad and Pgood are obtained from the actual random distribution of the obstructions above the terrain, notably,

equation image

[18] Here the lognormal CDF is for predicting of pure NLOS shadowing, the Rice CDF describes both the LOS and the multipath component, and the Rayleigh CDF describes the multipath component of the total signal when the LOS component is absent [Saunders and Evans, 1996; Evans, 1998; Saunders, 2001].

3. Multiparametric Stochastic Approach

[19] As another example of physical-statistical models, we present here the same stochastic approach, which was successfully used for the prediction of propagation characteristics in land communication channels, rural forested, suburban and urban, and have been confirmed by numerous experiments carried out in various land-land communication links [Blaunstein et al., 2002, 2006, 2009; Blaunstein and Christodoulou, 2007; Hayakawa et al., 2008].

[20] Let us consider that an array of buildings and any other obstructions (trees, hills, etc.) are randomly distributed on the rough terrain. During movement around the Earth, the satellite antenna “illuminates” different land areas with various distributions of obstructions on the ground surface (in the horizontal plane; see Figure 3) and different buildings' overlay profiles (in the vertical plane; see Figure 4) with changes in elevation angle of the satellite with respect to the ground-based subscriber antenna as shown in Figure 5. Let us suppose that z1 is the height of the ground-based subscriber antenna, z2 is the height of the satellite antenna, and h1 and h2 are the minimum and maximum heights of built-up layer (usually, h2 > h1 > z1). With the satellite rotation around the Earth, its height z2 is changed with respect to h2, so that we need to consider the both antennas elevation with respect to the buildings overlay profile [Blaunstein et al., 2002] as well as the main parameters and functions of built-up terrain [Blaunstein et al., 2006, 2009]. The “first-order” parameters of the built-up terrain are: the buildings density ν per km2, the average buildings' length equation image (or width, depending on the orientation to antenna direction), and the buildings' contours density γ0 per km [Blaunstein et al., 2002, 2006]:

equation image

These parameters allow us to obtain the “second-order” characteristics of the urban channel, such as:

Figure 3.

Horizontal map of multipath phenomenon occurring in the urban scene.

Figure 4.

Local effect of buildings' overlay profile.

Figure 5.

Change of the profile function F(z1, z2) in the vertical plane during the movements of a satellite.

3.1. Probability Distribution of Multiray Components

[21] In order to explain the mechanism of deep signal strength variations usually observed in LSC links, we have used the unified stochastic approach [Blaunstein et al., 2002, 2006, 2009; Blaunstein and Christodoulou, 2007; Hayakawa et al., 2008] taking into account multiple reflection and scattering effects from buildings and other obstructions randomly distributed at the terrain as the ordinary flow described by the Poisson law. Such a model allows us to consider the strength of the total field at the receiver as the additive summation of n-time independently scattered waves with independent strengths. The real situation with multipath phenomena occurring in the urban scene is clearly seen from an illustration in Figure 3.

[22] As was obtained by Blaunstein et al. [2002, 2006], in the zones close to the base station (BS) antenna (d < 1 km) the single scattered waves are dominant, whereas the two- and three-time scattered waves become to prevail far from the BS antenna (d > 2 km). Finally, the field strength of the n-reflected (with n > 3) waves exponentially attenuates with distance. As was shown by Blaunstein and Christodoulou [2007], the analysis of numerous experimental data indicates that for microcell communication channels with ranges less than 2–3 km, only single and two-time scattered waves are enough to be taken into account.

[23] However, for macrocell and megacell environments covered at the ground surface by satellite radio beams, we must additionally consider the three-time reflected and scattered waves. As was shown by Blaunstein et al. [2006] and Blaunstein and Christodoulou [2007], the strengths ri of these waves are distributed according to the Gaussian law with the zero-mean value and dispersion σ12 (for single scattered waves), σ22 (for two-time scattered waves) and σ32 (for three-time scattered waves), which depend strongly on the characteristic features of the terrain. The average number of scattered waves involved also dependence on the distance from the BS antenna d. Following Blaunstein et al. [2006] and Blaunstein and Christodoulou [2007], we can obtain now, The average number of single scattered waves

equation image

The average number of two-time scattered waves

equation image

The average number of three-time scattered waves

equation image

where Kn(γ0d) is the MacDonald's function of n order.

[24] The probability to receive single and three-time (i.e., for n = 1, 2, 3) scattered waves at the mobile station (MS) antenna was computed according to the following formula [Blaunstein et al., 2006; Blaunstein and Christodoulou, 2007]:

equation image

[25] In computations of equation (9) we took into account the effects of independent single (the first term), double (the second term) and triple (the third term) scattering of rays with the random amplitude r, as well as their mutual influences on each other (the fourth term), i.e.,

equation image


equation image

is the probability of direct visibility (e.g., LOS component), P1, P2 and P3 are defined by equation (9) combined with (8a)(8c), respectively.

3.2. Buildings' Overlay Profile

[26] The LSC is very sensitive to the buildings' overlay profile as shown in Figure 4, since during its movement around the Earth depending on the elevation angle ϕ, the buildings' profile will be continuously changed leading to different effects of shadowing in the current communication link (see Figure 5).

[27] Taking into account that real profiles of the urban environment are randomly distributed as was shown by Blaunstein et al. [2002], the Ph(z) which describes the probability that the subscriber antenna is located at the urban terrain, can be presented in the following form:

equation image

where the function H(x) is the Heaviside step function which is equal to 1 if x > 0 and is equal to 0 if x < 0.

[28] Using equation (12), we can now introduce the built-up layer profile, measured in meters, that is described by the following function [Blaunstein et al., 2002]:

equation image

[29] The analysis of the height probability distribution function Ph(z) made by Blaunstein et al. [2002], showed that for n ≫ 1 Ph(z) describes the case when the buildings are higher than h1 (this is a very rare case, since most buildings are at the level of a minimal height h1).

[30] The case when all buildings have heights close to h2 (i.e., most buildings are tall), is given by n ≪ 1. For n close to zero or n approaching infinity, most buildings have approximately the same height h2 or h1, respectively. For n = 1 we have the case of building heights uniformly distributed in the range from h1 to h2.

[31] Then, analyzing the built-up layer profile F(z1, z2), we get [Blaunstein et al., 2002]: For the case when the highest antenna height is above the rooftop level, i.e., z2 > h2 > h1

equation image

For the case when the highest antenna height is below the rooftop level, i.e., z2 > h2

equation image

From equations (14a) and (14b) we can determine the average building height as

equation image

which reduces to

equation image

for the case n = 1 of a uniformly distributed profile investigated by Blaunstein et al. [2002]. Then, the cumulative distribution probability function (CDF) of the event that any subscriber located in the built-up layer is affected by obstructions due to shadowing effect can be presented as [Blaunstein et al., 2002]:

equation image

The built-up profile function presented in equation (17) allows accounting for continuous changes of buildings' overlay profile during the movement of the satellite around the Earth and the corresponding changes of its elevation angle with respect to the position of any subscriber located in areas of service. This CDF will be used below to find the total outage probability for predicting successful communication and service by the satellite antenna of any ground-based subscriber.

4. Fading Effects in Land-Satellite Communication Links

[32] Now we present a unified algorithm, which is created to estimate and examine different models for prediction of fading in LSC links and to investigate the performances of different approaches and the corresponding optimal parameters for the unified stochastic approach that are the best fitted to the experimental measurements for various elevation angles and different types of land environment. In our simulations, we used the corresponding CDFs and measured data presented by Butt et al. [1992] and Parks et al. [1996]. The authors performed several series of channel recording experiments in Europe.

[33] Thus, we took experimental data of LSC link above Stockholm, the topographic map and built-up profiles of which were taken from land-land experiments of Aalborg University (Denmark) [see Blaunstein and Christodoulou, 2007] as well as data of measurements carried out in England for two cities, Westminster and Guildford described by Saunders and Evans [1996], Tzaras et al. [1998], and Saunders [2001]. Following these measurements and using a new cumulative distribution function, a combination of two CDFs described by equation (10) and by equations (14a) or (14b) in equation (17) (depending on the height of the satellite with respect to the maximum height of built-up layer h2), we construct a corresponding numerical code to compare it with corresponding measured data. The corresponding complementary cumulative distribution function CCDF(r) = 1−CDF(r), of signal random strength envelope, which describes the total probability to achieve a successful communication between the ground-based and the satellite antennas, can be finally presented as:

equation image

[34] The CCDF gives us the knowledge of stability of received signal with respect to the noise caused by fading phenomena. To compare the measured data obtained in Stockholm for the satellite elevation angle of 13° with results of computations of the stochastic model fully presented by Blaunstein et al. [2002, 2006, 2009] based on different presentation of built-up profile by (12)(14b) and by (17), respectively, we evaluated this model through the corresponding CCDF.

[35] Results of computations were carried out from the following data on the city of Stockholm. As mentioned there, the built-up environment of the city can be characterized as high, dense, and obstructed with the building contour density parameter γ0 of equation (7) varying from 8 km−1 to 12 km−1 with the mean value equation image0 = 10 km−1. The built-up profile parameter n estimated from equation (15) was changed from 0.65 (the case where the tall buildings exceed small buildings) to 1.48 (where the amount of tall and small buildings are roughly the same). All these data obtained from the topographic map of Stockholm, allow us to estimate the standard deviation of CCDF σ, changing from −4 to 4 dB. The results of comparison between the experimental data (in continuous curve) and those obtained numerically for the range in the estimated parameters σ are shown in Figure 6. It is clear from Figure 6 that the experimental data lie between the two theoretical curves computed for σ changing from −4 to −2 dB, which is close to that parameter estimated by analyzing the topographic map of Stockholm.

Figure 6.

Comparison between the experimental data obtained in satellite observations over Stockholm (continuous line) and those obtained theoretically from the stochastic approach.

[36] Another land-satellite experiment was compared with the Saunders-Evans physical-statistical model [Saunders and Evans, 1996; Evans, 1998; Tzaras et al., 1998; Saunders, 2001], and was carried out in England for two cities, Westminster and Guildford. We do not go deeply into the description of these experiments that are fully described in the papers mentioned above. But we will only mention that a comparison showed that the best fit was when CDF of the combined fast/slow fading was described by the corresponding Rayleigh law with the corresponding mean height of buildings equation image =20.6 and standard deviation σb = 17.6 m (for Westminster) and equation image and σb = 6.4 m (for Guildford). Therefore, in our additional comparison with Saunders-Evans physical-statistical model and with experimental data, we accounted for these estimations as well as for the average density of buildings' contours equation image0 ≈ 10.6 km−1 (for Westminster) and equation image0 ≈ 7.5 km−1(for Guildford). We also estimated the corresponding mean height of buildings for Westminster, equation image with standard deviation σb = 7.5 m and for Guildford, equation image and σb = 4.6 m. These estimations have suggested that the two cities have fully different built-up terrain profiles: the parameter of buildings' overlay profile is n = 2 (the amount of small buildings exceeds that of tall buildings) for Westminster, and n = 10 (most buildings are small) for Guilford.

[37] We should note that these parameters were taken in a form of average values, but not exactly, as was done by Saunders and Evans [1996], Evans [1998], Tzaras et al. [1998], and Saunders [2001] using local parameters presented in Figure 2 for each position of the moving satellite. Despite this fact, our estimations are inside the ranges of estimations obtained by Saunders and Evans [1996], Evans [1998], Tzaras et al. [1998], and Saunders [2001] and we can finally compare our computations of CDF = CCDF−1 according to (12)(17) with those obtained by Saunders, Evans, and coworkers using Rayleigh CDF, as a best fit of measured data. This comparison is shown in Figure 7a for Westminster and Figure 7b for Guildford, respectively, where the corresponding data were plotted by dots. Even with the data on terrain features using average parameters of the built-up terrain, not the local parameters, as was done by Saunders and Evans [1996], Evans [1998], Tzaras et al. [1998], and Saunders [2001] and shown in Figure 2, we obtained a satisfactory agreement between the theoretical prediction based on both Saunders-Evans physical-statistical which was based on Rayleigh CDF, and multiparametric stochastic models based on equations (12)–(17) and experimental data. This means that the designers of satellite-land links do not need every time to have information on the local built-up terrain parameters as shown in Figure 2. It is sufficient to obtain average parameters of the terrain during satellite movements above the corresponding city, town, village, and so on, and we can use for prediction of fading effects the stochastic multiparametric approach.

Figure 7.

(a) Comparison between experimental data (dots) obtained in Westminster, physical statistical model (green line), and data obtained from the stochastic approach (blue line). (b) Comparison between experimental data (dots) obtained in Guildford, physical statistical model (blue line), and data obtained from the stochastic approach (green line).

[38] In order to show now how the proposed stochastic multiparametric algorithm is working, we first vary the satellite elevation angle in a “virtual computer experiment.” Figures 8a8c show the increase of the probability to get less path loss with an increase in the number of satellites or with a decrease in elevation angle.

Figure 8.

(a) Probability of fading versus path loss for elevation angle of 40° [Blaunstein and Christodoulou, 2007]. (b) The same as in Figure 8a, but for elevation angle of 60°. (c) The same as in Figure 8a, but for elevation angle of 80°.

[39] Then, we investigated the same probability versus the maximum accepted path loss for satellite networks: Iridium [Brunt, 1996], GlobStar [Wiedeman et al., 1992; Smith, 1994], and ICO [Singh, 1993]. The results are presented in Figures 9a9c, respectively. Along the horizontal axis, the corresponding path loss for each maximum outage probability of fading is presented. It is clear that the differences in results result from the varying satellites altitudes.

Figure 9.

(a) Probability of fading versus path loss for Iridium network [Blaunstein and Christodoulou, 2007]. (b) Probability of fading versus path loss for GlobalStar network. (c) Probability of fading versus path loss for ICO network.

[40] Finally, we show the effect of type of the land usage on the probability to obtain the maximum acceptable path loss. Figures 10a10c show a difference between the obtained results for different land environments: urban, suburban, and open rural. The results were obtained for the low elevation satellite (ϕ = 40°). As was expected, Figures 10a10c indicate that following the knowledge on the kind of terrain topography from good (open) to bad (urban) cases, the main difference is noticeable in the urban environment (as a worst case with strong fading), due to its special propagation features such as multiple diffraction, scattering and reflection from buildings surrounding the subscriber antenna.

Figure 10.

(a) Probability of fading versus path loss for urban environments [Blaunstein and Christodoulou, 2007]. (b) Probability of fading versus path loss for suburban environments. (c) Probability of fading versus path loss for open rural environment.

5. Summary

[41] From the results of simulations, we can conclude that the pure statistical model is less accurate than the physical-statistical models to predict fading phenomena and link budget in LSC links. In addition, it is shown that in most scenarios usually occurring in the LSC links, the physical-statistical model based on our stochastic multiparametric approach is a stricter predictor of fading phenomena and link budget performance with respect to the existing physical-statistical models. It is based on multiple scattering effects occurring at the horizontal ground plane described by equation (10) and on the real buildings' overlay profile in the vertical plane described by (14a)(17). Therefore, the multiparametric stochastic model is the more realistic fitting to measurements as compared with the pure statistical models [Loo, 1985; Lutz et al., 1991; Loo and Butterworth, 1998] and even existing physical-statistical approaches [Saunders and Evans, 1996; Evans, 1998; Tzaras et al., 1998; Saunders, 2001]. Numerical analysis and comparison with experimental data showed that the difference between the physical-statistical model proposed by Saunders and Evans with their colleagues and our proposed stochastic multiparametric model is small, but the multiparametric model is much simpler to use and to compute than the Saunders and Evans model. The reason for this is the fact that for the physical-statistical model we need the knowledge of the PDF of heights of buildings in each local city area, which is a factor that is very difficult to measure in each tested place. Furthermore, the prediction mechanism of the outage probability of shadowing in real time, proposed by Saunders and Evans needs a great massive of data concerning each building, the geometry of each radio path between the individual subscriber and satellite in the process of its movement above a city, and finally, high-speed powerful macrocomputers.

[42] On the other hand, for our multiparametric model we deal with knowledge of the heights of smaller and higher buildings, the average height of buildings, and the building density for each square kilometer of the city area. By using these simple values, we can easily find the significant parameter n of the terrain profile using equation (15) and then the relief function equations (14a) and (14b). As was shown by Blaunstein et al. [2009] that taking average parameters of the built-up terrain in areas of one square kilometer, we obtained practically the same results as using the ray tracing approach, which gives strict information on the local contour of each building.

[43] Therefore, our multiparametric stochastic approach can be used with a great accuracy for predicting fading phenomena, the maximum acceptable path loss, and finally, for link budget design both for personal and mobile land-to-satellite radio communications. Moreover, this approach is much simpler than other physical statistical models, deals with global, not local, average parameters of the built-up terrain and does not need great massive of local built-up terrain data.