Extension of the Misme and Fimbel Model for the Estimation of the Cumulative Distribution Function of the Differential Rain Attenuation Between Two Converging Terrestrial Links

A distribution-free model is presented for the cumulative distribution function of the differential rain attenuation between two co-channel converging terrestrial links operating at frequencies above 10 GHz. This is accomplished through an extension of the Misme and Fimbel model, which determines the cumulative distribution of the rain attenuation on an isolated link from its parameters and data for the radio climatic region, as well as concepts from probability theory. Next, model predictions and experimental results are compared and effects from variations of the angle between the links and of the path length are studied. Finally, rain effects on the cumulative distribution of the C/I ratio between the desired and interference powers at the same receiver are analyzed, considering the angular discrimination of the receiving antenna.


I. INTRODUCTION
Frequency bands above 10 GHz have been increasingly used in point-to-point, high-bandwidth, wireless access of clients to local-and wide-area networks.The access point generally adopts a star topology, with many wireless terrestrial links converging to a common central station.It is well known that rain is the dominant propagation impairment factor to links operating in these frequency bands and that the design of such star networks should consider the possibility that a link can be severely attenuated by rain while experiencing interference by another link operating at the same frequency but under milder conditions.In other words, the design of the network should consider the effects from differential rain attenuation on link availability.It was observed that some rain unavailability calculations for these networks were conservatively performed, leading to a relatively inefficient use of the frequency spectrum.These calculations only considered rain attenuation in the desired link, assuming clear-sky conditions for the interfering link, making it necessary to increase the angle between them.A more realistic consideration would be that rain cells could simultaneously attenuate both links, allowing a decrease of the angle between the links and an increase in spectral Extension of the Misme and Fimbel Model for the Estimation of the Cumulative Distribution Function of the Differential Rain Attenuation Between Two Converging Terrestrial Links Henrique Grynszpan, Emanoel Costa, well as the difference in path lengths to map the cumulative distribution functions of rain attenuation on the two individual links into that of the differential rain attenuation.
The present work will present a distribution-free model based on fundamental principles and on the measured cumulative distribution function of point rainfall rate (without assuming any specific probability law) to determine the cumulative distribution function of the differential rain attenuation between two converging links operating at the same frequency, greater than 10 GHz.The development extends the Misme and Fimbel model [11], which determines the cumulative distribution function of rain attenuation on an isolated link from its parameters and radio climatic data for the region.Thus, the present model at least initially follows a different approach from those leading to the previous methods.Next, the model numerically determines the joint cumulative distribution function of the attenuations on the two links over a fine grid of two-dimensional joint attenuation points.From this partial result, the cumulative distribution function of the differential rain attenuation between the links is easily determined by a simple summation.The present model generalizes the development by Stola [12], also based on the Misme and Fimbel procedures but limited to the determination of the cumulative distribution function of the rain attenuation simultaneously exceeded on two converging links.Following this Introduction, Section II will briefly review the original Misme and Fimbel model [11] and will discuss its extension to the determination of the cumulative distribution function of the differential rain attenuation between two co-channel converging links.Section III will describe experimental campaigns providing input data for comparison with model predictions and present effective values for climatic parameters.Section IV will compare calculation and measurement results.Additionally, it will present model predictions of the effects of variations of the angle between the links and the path lengths on the cumulative distribution function of the differential rain attenuation.The features of the present model make it possible to provide explanations for the observed changes of the cumulative distribution function due to variation on these parameters.Based on the extended Misme and Fimbel model, Section V will analyze the effects of the differential rain attenuation on the availability of co-channel converging links according to the conservative and more realistic procedures, considering the angular discrimination of the receiving antenna.Finally, conclusions will be presented in Section VI.

II. THE MISME AND FIMBEL MODEL AND ITS EXTENSION
The Misme and Fimbel model [11] for the estimation of the cumulative distribution function of the rain attenuation on an isolated terrestrial link adopts the following simplifying assumptions for the rain field: • intense precipitation occurs in the form of circular cylindrical cells with constant rainfall rate R (mm/h); • the average diameter d(R) (km) of a rain cell can be related to the corresponding rainfall rate R through the expression with the parameter values d o = 2.2 km and β = 0.4, as long as d(R) ≤ d max (the maximum cell diameter, equal to 33 km); • the point rainfall rate is a random variable characterized by its measured cumulative distribution function P r (R) that is independent from the observation point within the region of interest; • at each instant of time, attenuation on a link is caused by a single rain cell; • the specific attenuation γ (dB/km) can be related to the rainfall rate R (mm/h) through the expression γ = k R α , with the parameters k and α being determined as functions of the operating frequency and the polarization according to the most recent version of Recommendation ITU-R P.838 [13].
It should be noted that, to simplify numerical procedures, instead of explicitly considering the small contribution from residual precipitation to rain attenuation according to the Misme and Fimbel model specification [10], these effects have been included by extending the validity of expression (1) to rainfall rates below the original threshold of 10 mm/h.Even though relevant developments based on the Misme and Fimbel model have been proposed with the objectives of simplifying or improving the corresponding algorithms [14], the present work will strictly adhere to all the characteristics of the original model, except for the slight modification described above.
As illustrate din Figure 1, the intersection L o1 (km) between a cell with rainfall rate R and link AB with path length D 1 (km) causes the attenuation A o1 = kR α L o1 (dB).When this rain cell is displaced by all possible manners while keeping the length L o1 of the intersection fixed, the cell center describes the dashed geometric locus V 11 V 21 I 1 V 31 V 41 I 2 also shown in Figure 1.The geometric locus limits the area S(A o1 , R) defined by [11] ( ) ( ) The two terms separated by the addition sign in the right-hand side of expression (2) represent the areas of the rectangle V 11 V 21 V 31 V 41 and the two circular segments limited by the geometric locus associated with link AB, respectively.For each assumed value for A o1 , a minimum rainfall rate R min (mm/h) should be found to guarantee the inequalities Its value should be determined from these inequalities and expression (1), considering the upper limit d max .It should be observed that the condition R ≥ R min keeps each term in the right-hand side of expression (2) real and positive or null.It is also noted that a cell with rainfall rate R and center in the interior of the geometric locus will cause attenuation A > A o1 .
Fig. 1.Converging links AB and AC with respective path lengths D 1 and D 2 making the angle θ at the common terminal, associated geometric loci V 11 V 21 I 1 V 31 V 41 I 2 and V 12 I 1 V 22 V 32 V 42 I 2 corresponding to attenuations A o1 and A o2 , respectively, and to a cylindrical cell of rainfall rate R, as well as the intersection V 12 I 1 V 41 I 2 between the two geometric loci.
Based on the assumptions above, geometrical concepts and additional considerations, as well as on the total probability theorem [15], Misme and Fimbel established the following general expression for the cumulative distribution function of rain attenuation on the isolated link In the original integral (reproduced between the equality signs in the above expression) [11], numerically evaluated for A o1 values of interest, p r (R) is the probability density function of the point rainfall rate, assumed to follow a lognormal law with parameters determined from corresponding measurements.A trivial change of variable leads to the last integral of expression (4), which can be numerically evaluated using the measured cumulative distribution function of the point rainfall rate.
That is, without resorting to particular probability distributions and also avoiding possible errors associated with numerical differentiations involved in the determination of p r (R) from P r (R).with the former at the common terminal A, as well as the associated geometric locus V 12 I 1 V 22 V 32 V 42 I 2 corresponding to the same cell with rainfall rate R and attenuation A o2 .Based on the development summarized in the previous paragraphs, the joint probability that the attenuations in the two links simultaneously exceed the respective values A o1 and A o2 can be represented by In the above expression, S ∩ (A o1 , A o2 , R) is the intersection between the surfaces limited by the two geometric loci, represented by the light gray area in Figure 1, and P min is the value of the rainfall rate cumulative distribution function corresponding to R min = max{R min1 , R min2 }, where R min1 and R min2 are the minimum rainfall rates for the two individual links, respectively.Equation ( 5) generalizes the Misme and Fimbel model to consider two converging links and also extends the formulation presented by Stola [12], restricted to the particular case A o1 = A o2 and analyzed in more detail for D 1 = D 2 .Once the configuration of interest has been specified, the integral in expression ( 5) can be numerically calculated for pairs of attenuation values (A o1 , A o2 ) defined over a fine grid with resolution ε (that is, A o1 and A o2 assume the values mε and nε, where m, n = 0,1, … , N, respectively).
The area S ∩ (A o1 , A o2 , R) is exactly determined by means of an algorithm that combines simple computational geometry procedures [16].It is observed that the geometric loci are convex polygons characterized by straight line or circular arc segments.Therefore, these features will be conserved by their intersection, defined by all: (i) vertices of one geometric locus located in the interior of the other (V 12 and V 41 in Figure 1); and (ii) crossings between segments from different geometric loci (I 1 and I 2 in Figure 1).Next, these points are sorted in the counter-clockwise sense and it is determined whether two consecutive points should be connected by a straight line or a circular arc segment.In the latter case, the position of the center of the corresponding circle is also determined.Finally, the intersecting polygon is subdivided into triangles and circular segments.The area S ∩ (A o1 , A o2 ,R) is exactly equal to the sum of the areas of these basic elements.
In principle, the cumulative distribution function of the differential rain attenuation Pr{a 1 -a 2 >A o } would be determined by the double integration of the joint probability density function ( ) in the region of interest, below the straight line A 1 -A 2 = A o , as indicated by Figure 2.That is, { } ( ) where A max is an upper limit beyond which the contribution from the above integral to the cumulative distribution function of the differential rain attenuation can be neglected.Tests have shown that the adopted value A max = 47 dB is adequate for this purpose.As indicated by expression (6), the integral between square brackets should be evaluated for an arbitrary value of A 2 in the interval (0, A max -A o ).
The result, which is a function of A 2 , should then be integrated in the same interval.
Figure 2 indicates that the region of interest can be approximately subdivided into a large number of narrow rectangles in such a way that Note that the values for each term inside the square brackets in the last line of expression ( 7 Figure 2 also shows that the approximated region of integration represented by the narrow gray rectangle differs from the exact one by: (i) incorrectly considering a small triangle above the line A 1 -A 2 = A o ; and (ii) incorrectly neglecting another small triangle below same line.These triangles are images of each other around their common vertex and have the same area, which is small when compared to that of the gray rectangle.Additionally, the joint probability density function of the attenuations on the two links does not display discontinuities and the contributions of the two triangles are approximately equal.Therefore, the last line of equation ( 7) provides a very good approximation to Pr{a 1 -a 2 >A o }.Additionally, Figure 2 indicates that an upper bound to Pr{a 1 -a 2 >A o } can be obtained by keeping the lower left vertices of the narrow rectangles at the line A 1 -A 2 = A o and that a lower bound can be obtained by keeping the upper left vertices of the narrow rectangles at the same line.Performed test have indicated that the two bounds converge very fast to each other and to the results from expression (7) as soon as the value of the differential attenuation A o exceeds a couple of decibels, for the assumed values for the increment ε (0.01 dB for A 1 or A 2 less than 1 dB, since the joint probability displays fast variations in these regions, and 0.1 dB otherwise).It should be observed that the approach described by expression (7) could be used by any model that determines the cumulative distribution function of the differential rain attenuation from the joint cumulative distribution function of the attenuations on two links.
Finally, it is concluded from equations ( 5) and ( 7) that, for a set of link parameters, the extended Misme and Fimbel model maps the directly-measured cumulative distribution function of the rainfall rate onto that of the differential rain attenuation.Therefore, the ideal comparison between experimental results and model predictions should be based on cumulative distribution functions of the rainfall rate and the differential rain attenuation resulting from the same period of measurements.
The results from such comparisons will be presented in the next section.

III. INPUT DATA
In a collection of papers published during the 1970's, Morita and co-workers [1]- [4] reported on measurements performed on three short 19-GHz links converging at the Musashino Electrical Communication Laboratory (ECL), Japan.More recently [7], [8], similar measurements were performed in the City of São Paulo, Brazil, using the following links operated by Empresa Brasileira de Telecomunicações (Embratel) and converging at the central station of Rua dos Ingleses (RIS).The names of the remote stations, frequencies of operation, path lengths, polarizations (representing horizontal and vertical polarizations by H and V, respectively) and azimuths (at ECL) of the links, as well as their periods of operation, are presented in Table I.Angles between links can be determined from azimuth differences.In addition to received signal levels, the rainfall rate data were also acquired at ECL and RIS.The two data sets were processed, Several authors performed rain cell size measurements using meteorological radars located in temperate regions [17]- [20], and a high-elevation slant path at a tropical site [21].Some of the results were presented in the form of a functional relation between the average cell diameter and the rainfall rate and seem to be in reasonable agreement with the assumed values for parameters β and d o of the model represented by expression (1).However, radar measurements performed in the Amazon region provided probabilities that a cell diameter be exceeded for rainfall rate thresholds in the convective regime that are substantially higher than those originating from temperate climates [22].This may be taken as an indication that the original values are not universal.Moreover, it is easily verified that the combination of the assumption that attenuation on a link is due to a single rain cell with the originallyproposed values for β and d o , is not capable of explaining the measured attenuation values, which can reach 33 dB or more during 0.01 % of the time for the RIS experiment.
Ideally, d o and β should be determined from radar measurements, as described by items 2.2 and 2.3, as well as by Figure 1 of the original reference for the Misme and Fimbel model [11].Note that these two parameters should depend only on the rain climate and should not depend on the parameters of any link, either operational or in the planning stage.Unfortunately, radar measurements of rain cell sizes are not available for all climatic regions of the world.To circumvent the absence of information, we described an alternative approximate method for the estimation of the two parameters, based on where attenuation A p and rainfall rate R p values were obtained from the respective measured cumulative distributions for the same percentages of time p. Specific attenuation values γ = kR α were determined considering the appropriate polarizations and operating frequencies according to Recommendation ITU-R P.838-3 [13].For reference purposes, the same calculations were also performed for links converging at the ECL station.Both Figures also show the results from the prediction model proposed by Garcia et al. [8], which estimates the cumulative distribution function of the differential rain attenuation through

IV. COMPARISON BETWEEN CALCULATIONS AND MEASUREMENTS, AND MODEL PREDICTIONS
In equation ( 9), A diff,12 (p) and a i (p) (i = 1,2) are the differential rain attenuation between links 1 and 2, the rain attenuation in link i (i = 1,2) corresponding to the percentage level p.The angle θ between the links should be expressed in radians, the path lengths D i (i = 1,2) in kilometers, and the frequency in Gigahertz.This model is particularly attractive when the cumulative distribution functions of the rain attenuation on both links are immediately available.
The agreement between calculations and measurements displayed in Figures 4 and 5, depending on the tested case, ranges from good to poor.For example, it is good for both models in the cases of the Shakujii-Shinkawa data of Figure 4 for percentages of time greater than 0.004 %.On the other hand, it is poor in the case of the Bradesco2-Barueri data, particularly for the proposed model and percentages of time between 0.01 % and 0.1 %, and for the model of reference [8] in the cases displayed in the right panels of Figure 5, for percentages of time less than 0.02 %.II for each percentage interval, as well as the respective numbers of samples available for the corresponding calculations.The first two lines consider all the pertinent data in Figures 4 and 5, including the relatively long links converging at the RIS station.On the other hand, the last two lines consider only the shorter links converging at the ECL station.The level of agreement observed in Table II is not uncommon in modeling efforts of rain effects on radio links [24], [25].It is seen that the average values and the standard deviations of the errors resulting from the extended Misme and Fimbel model are generally less than the corresponding figures provided by that of reference [8].However, the parameters of the latter model were estimated with basis on nineteen Brazilian links, and most of the ones not converging at the RIS station are considerably shorter than those characterized in the last four lines of Table I.Indeed, nine of the links are shorter than 2.0 km and twelve of them are shorter than 4.5 km.Additionally, expression (9) is applied here to a data set which differs from the original one.These facts may explain why Garcia at al. [8] reported average values and standard deviations of the errors which are substantially less than the ones displayed in Table II.
It should also be stressed that no attempts have been made at minimizing the observed errors through the application of parameter optimization techniques directly to the data in Figures 4 and 5.
This could have been done and would certainly improve the observed agreement.However, this procedure has not been used, since it would be in contradiction with the essential assumptions and methodology leading to the original Misme and Fimbel model and its present extension, which were meant to be upheld.Additional possible factors explaining the observed differences between

Figure 1
Figure 1 also displays a second link AC with path length D 2 (km) that makes an angle θ (degrees)

Fig. 2 .
Fig. 2.Region of interest for the estimation of { } o 2 1 A a a Pr > − and its approximate subdivision into narrow rectangles.
the cumulative distribution functions of the point rainfall rate and the rain attenuation on links simultaneously measured in the region of interest.To extend the applicability of the Misme and Fimbel model to the links converging at the RIS station, parameter values corresponding to the effective rain cell diameter were estimated by least-squares fitting a straight line (y = βx + log d o ) to data rearranged as indicated in the following equation

Figure 3
presents the (x p , y p ) data sets corresponding to the links.The Figure also shows straight-line segments resulting from the least squares fit to data from each link and the estimated effective values of the parameters (log d o ) and β.As expected, the effective values for the parameter d o are larger than the original one for the relatively long RIS links.The present results are consistent with those from similar calculations performed by Sarkar et al.[23] using data from the Indian subcontinent.On the other hand, the effective d o values for the ECL links are closer to those estimated from radar measurements, in reasonable agreement with the original expression (1), and with the assumption that attenuation on shorter links would be mainly due to individual cells.Due to errors in the present alternative approximation method, the results in Figure3clearly indicate that the effective parameter values d o and β derived from equation (8) depend on the path length, polarization and frequency of the links.However, to be consistent with the assumptions of the original Misme and Fimbel model [11], it will be assumed that d o and β are exclusively climatic parameters.Therefore, the present work estimated the effective values d o = 6.80 km and β = 0.52 for calculations involving the RIS links, and d o = 2.80 km and β = 0.14 for those corresponding to the ECL links, by the average values of the samples from the links.It should be remarked that tests indicated that the extended Misme and Fimbel model is relatively insensitive to variations in the parameter β.

Fig. 3 .
Fig. 3. Determination of effective values for the parameters (log do)and β, displayed in the boxed equations in the same order of the respective curves (data corresponding to ECL links adapted from references [1]-[4]).

Figure 4
Figure 4 displays the calculated and measured cumulative distribution functions of differential rain attenuation (DRA), corresponding to the converging ECL links.In this Figure, calculated results, represented by continuous or dashed lines, should be compared with measurement results, represented by filled squares or triangles.It is assumed that A o > 0 and Figure 4 displays both sets of results for Pr{a 1 -a 2 >A o } and Pr{a 2 -a 1 >A o }. Results for the converging RIS links are displayed in Figure 5, using the same conventions.

Fig. 4 .
Fig.4.Calculated (continuous and dashed curves) and measured (filled symbols) cumulative distribution functions for the differential rain attenuation between the converging ECL links described in Section III (adapted from reference[4]).

Fig. 5 .
Fig.5.Calculated (continuous and dashed curves) and measured (filled symbols) cumulative distribution functions for the differential rain attenuation between the converging RIS links described in Section III.

Figure 5
Figure 5 displays several examples of slow variations of the measured differential rain attenuation as the time percentage decreases below 0.01%.This is due to link design and technical characteristics of the receivers, which reach the reception threshold for attenuations near 37 dB.Therefore, only data from Figures 4 and 5 corresponding to time percentages above 0.005 % were used to determine the average value and the standard deviation of the error {DRA meas (p) -DRA calc (p)}, where p represents the common percentage of time.Two percentage intervals were used in the calculations: (i) [0.005 %, 10.0 %], which covers most of the available data, emphasizing large probabilities and low DRA values; and (ii) [0.005 %, 0.03 %], which emphasizes the exceedance probabilities of interest for potential applications of the proposed model.The average values and the standard deviations of the errors for the two models are presented in TableIIfor each percentage interval, as well as the Next, the extended Misme and Fimbel model was used to determine the cumulative distributions of the differential rain attenuation between vertically-polarized 15-GHz links located in the São Paulo region (that is, assuming d o = 6.80 km and β = 0.52) with path lengths D 1 = D 2 = 10 km, for angles varying between 5 o and 50 o with 5 o increments and between 50 o and 180 o with 10 o increments.The resulting distributions for increasing angles are displayed in Figure6from left to right.Fast variations in the curves as a function of the angle between the links are initially observed, for angles less than approximately 60 o .However, the same variation became extremely slow beyond this threshold.Note that, for any fixed time percentage, the differential rain attenuation initially increases and gradually tends to a constant value as the angle between the links increases[4].

Fig. 6 .
Fig. 6.Cumulative distribution functions of the differential rain attenuation presented from left to right for angles varying between 5 o and 50 o with 5 o increments and between 50 o and 180 o with 10 o increments, assuming vertically-polarized 15-GHz links located in the São Paulo region (d o = 6.80 km and β = 0.52) with fixed path lengths D 1 = D 2 = 10 km.

Fig. 7 .Figure 7 .
Fig. 7. Cumulative distribution functions of the differential rain attenuation presented from right to left for increasing values of the path length D 2 (from 3 km to 15 km in 1-km increments), assuming vertically-polarized 15-GHz links located in the São Paulo region(d o = 6.80 km and β = 0.52) with fixed path length D 1 = 10 km and angle θ = 15 o .

TABLE I .
PARAMETERS AND PERIODS OF OPERATION OF THE LINKS

Central Station Remote Station Frequency Path Length Polarization Azimuth Operation
Journal of Microwaves, Optoelectronics and Electromagnetic Applications, Vol.12, No. 1, June 2013 Brazilian Microwave and Optoelectronics Society-SBMO received 23 Jan 2013; for review 28 Jan 2013; accepted 22 May 2013 Brazilian Society of Electromagnetism-SBMag © 2013 SBMO/SBMag ISSN 2179-1074 200 yielding cumulative distribution functions of one-minute point rainfall rate at the central station, attenuations on isolated links and differential attenuations on pairs of links.
Journal of Microwaves, Optoelectronics and Electromagnetic Applications, Vol. 12, No. 1, June 2013 measured distributions are the non-uniformity of the rainfall rate inside the cell, the effects of multiple cells on differential rain attenuation, and errors due to the estimation of d o and β through the alternative approximate methodology described in Section III.

TABLE II .
AVERAGE AND STANDARD DEVIATION OF ERRORS BETWEEN MEASUREMENT AND PREDICTION RESULTS