A rain attenuation time series synthesizer based dirac lognormal

1396 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 3, MARCH 2013

A Rain Attenuation Time-Series Synthesizer Basedon a Dirac and Lognormal Distribution

Xavier Boulanger, Laurent Féral, Laurent Castanet, Nicolas Jeannin, Guillaume Carrie, and Frederic Lacoste

Abstract—In Recommendation ITU-R P.1853-1, a stochastic ap-proach is proposed to generate long-term rain attenuation timeseries , including rain and no rain periods anywhere in theworld. Nevertheless, its dynamic properties have been validatedso far from experimental rain attenuation time series collected atmid-latitudes only. In the present paper, an effort is conducted toderive analytically the first- and second-order statistical proper-ties of the ITU rain attenuation time-series synthesizer. It is thenshown that the ITU synthesizer does not reproduce the first-orderstatistics (particularly the rain attenuation cumulative distributionfunction CDF), however, given as input parameters. It also pre-vents any rain attenuation correlation function other than expo-nential to be reproduced, which could be penalizing if a world-wide synthesizer that accounts for the local climatology has to bedefined. Therefore, a new rain attenuation time-series synthesizeris proposed. It assumes a mixed Dirac-lognormal modeling of theabsolute rain attenuation CDF and relies on a stochastic genera-tion in the Fourier plane. It is then shown analytically that the newsynthesizer reproduces much better the first-order statistics givenas input parameters and enables any rain attenuation correlationfunction to be reproduced. The ability of each synthesizer to re-produce absolute rain attenuation CDFs given by Recommenda-tion ITU-R P.618 is finally compared on a worldwide basis. It isthen concluded that the new rain attenuation time-series synthe-sizer reproduces the rain attenuation CDF much better, preservesthe rain attenuation dynamics of the current ITU synthesizer forsimulations at mid-latitudes, and, if it proves to be necessary forworldwide applications, is able to reproduce any rain attenuationcorrelation function.

Index Terms—Rain attenuation time-series, satellite communi-cation systems, stochastic processes.

I. INTRODUCTION

T HE conventional frequency bands (C, Ku, i.e., 4–15 GHz)used in mid-latitudes for fixed satellite telecommunica-

tion systems are nearly to be saturated. Nevertheless, fixed satel-

Manuscript received June 26, 2012; revised November 16, 2012; acceptedNovember 19, 2012. Date of publication February 07, 2013; date of current ver-sion February 27, 2013. This study has been partly carried out in the frameworkof the European action COST IC0802.X. Boulanger is with the French Aerospace Lab (ONERA), Département

Electromagnétisme et Radar (DEMR), Toulouse 31055, France, and also withthe French Space Agency (CNES), Toulouse 31055, France (e-mail: [email protected]).L. Féral is with the Laboratoire LAPLACE, Groupe de Recherche en Elec-

tromagnétisme (GRE), Université de Toulouse (Paul Sabatier, Toulouse III),Toulouse 31400, France (e-mail: [email protected]).L. Castanet, N. Jeannin, and G. Carrie are with the French Aerospace Lab

(ONERA), Département Electromagnétisme et Radar (DEMR), Toulouse31055, France (e-mail: [email protected]; [email protected];[email protected]).F. Lacoste is with the French Space Agency (CNES), Toulouse 31055, France

(e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2012.2230237

lite services require higher and higher data rates. As a conse-quence, systems operating at higher frequency bands (such asKa or Q/V, i.e., 20–50 GHz) are envisaged to reach higher per-formances in terms of spectrum availability, equipment size, orbandwidth. Besides, in tropical and equatorial areas, where theterrestrial telecommunication infrastructures are less developedthan in temperate regions, the satellite is an interesting alterna-tive as it allows faster and cheaper service deployment.Nevertheless, satellite telecommunication systems at fre-

quencies greater than 5 GHz suffer from tropospheric attenua-tion. The latter increases with the radio link frequency and canreach some tens of decibels at Ka or Q/V bands [1]. Among thevarious tropospheric components, rain is the major contributorto total tropospheric attenuation. Moreover, depending on thelocal climatology, rain attenuation is expected to be more se-vere in tropical and equatorial areas than in mid-latitudes, bothin terms of occurrence and in attenuation level. To counteracttropospheric impairments, a fixed power margin that dependson the required service availability is commonly introduced inthe link budget. Nevertheless, this extra resource allocation isnot optimal for systems above 20 GHz because the requiredmargin becomes too high for continuous operation, resulting incostly user Earth stations and causing excessive interferences.Therefore, to insure a favorable link budget, the fade mitigationtechniques (FMT) (such as adaptive coding or modulation andfrequency diversity [2], [3]) have to be implemented to coun-teract tropospheric impairments. That is the reason why, forsatellite communication systems in centimetric or millimetricbands, adaptive real-time algorithms are being developed [4].Clearly, tropospheric attenuation time series are necessary fortheir development, their evaluation, and their optimization [4].As the local climatology changes from one location to another,experimental attenuation time series are thus needed all aroundthe world, for various radio wave configurations in terms offrequency, polarization, and elevation angle. Unfortunately,propagation experiments above 20 GHz are not numerous, sothat the synthetic rain attenuation time series have to beconsidered.During the 1980s, a stochastic approach was proposed in

[5] to simulate conditional rain attenuation time series (i.e.,only for rainy periods) as a first-order Markov process. Thisapproach has been used in [6]–[8] and has then been extendedin [9]–[11] to generate long-term rain attenuation time series(i.e., including rain and no rain periods). It is important to notethat [9]–[11] and [12] have shown that the dynamics of theirlong-term synthetic rain attenuation time series compares sat-isfactorily (from a statistical point of view) with experimentalpropagation time series collected at mid-latitudes for frequen-cies between 10 and 50 GHz and elevation angles between 25

0018-926X/$31.00 © 2012 IEEE

BOULANGER et al.: A RAIN ATTENUATION TIME-SERIES SYNTHESIZER 1397

and 40 . Although recently adopted by ITU in Recommenda-tion ITU-R P.1853-1 [13], the stochastic approach proposed in[9], [10], and [11] has two shortcomings.First, in [9]–[11] or [13], it is assumed that the long-term (or

absolute) rain attenuation complementary cumulative distribu-tion function (CCDF) is lognormal. Yet, the anal-ysis of experimental rain attenuation time series clearly showsthat the long-term rain attenuation process is not rigorously log-normal as its CCDF falls to 0 for time percentagesbetween 1% and 20 of an average year depending on thelocal probability of rain. To overcome this limitation, [11] and[13] have introduced a posteriori an empirical offset parameter. Therefore, from a conceptual point of view, a contradiction

appears: the synthetic time series does not followthe lognormal CCDF given as an input param-eter as will be shown in Section II. Besides, from rain rate mea-surements collected with disdrometers at various sites locatedfrom middle to tropical and equatorial latitudes, [14] has shownthat the conditional rain rate CCDF islognormal. Disregarding the spatial correlation of rainy eventsand recalling the power relationship between rain intensityand rain specific attenuation [15], it follows that only the condi-tional CCDF of rain attenuation should beconsidered as a lognormal process in compliance with the firstapproach developed in [5] to synthesize conditional rain atten-uation time series (i.e., in presence of rain only).Second, in [9]–[11] or [13], the long-term rain attenuation

time series are modeled as a first-order Markov process, whichimplies that the rain attenuation correlation function is asymp-totically exponential for small time lags whatever the location(i.e., whatever the local climatology), as will be illustrated inSection II. On the one hand, this assumption has been validatedin [9], [11], and [12] from experimental rain attenuation time se-ries collected at mid-latitudes. On the other hand, its validity inother climatic areas (in tropical or equatorial areas for instancewhere the convective nature of rainy events is much more pro-nounced than in mid-latitudes) has not been demonstrated sofar in the literature and might be viewed skeptically. Therefore,for worldwide applications, a rain attenuation time series syn-thesizer that allows any correlation function to be reproducedwould represent a generalization of the current ITU synthesizerdescribed in [13].To overcome the above limitations, a new rain attenuation

time series synthesizer including rain and no-rain periods is pro-posed. It relies on a mixed Dirac-lognormal modeling of the ab-solute rain attenuation CCDF and on a stochastic generation inthe Fourier plane. Contrary to the current ITU synthesizer, thenew model reproduces very accurately the long term rain atten-uation CCDF given as an input parameter and, forworldwide applications, allows any rain attenuation correlationfunction to be reproduced.The paper is organized as follows. First, the rain attenuation

time series synthesizer [13] adopted by ITU-R Study Group 3 isrecalled in Section II. Its conceptual shortcomings are demon-strated from the analytical derivation of first- and second-orderstatistics. Then, a new rain attenuation time series synthesizeris presented in Section III. Its first- and second-order statisticalproperties are derived and a methodology that allows any cor-relation function to be reproduced is presented. Particularly, as

the dynamics of the current ITU synthesizer has been inten-sively tested and validated at mid-latitudes in [9], [11], and [12]from experimental rain attenuation time series, the capability ofthe new synthesizer to reproduce the rain attenuation correla-tion function of the ITU synthesizer is demonstrated. Lastly, theability of each synthesizer to reproduce absolute rain attenua-tion CCDFs given by Recommendation ITU-R P.618 is com-pared on a worldwide basis in Section IV.

II. RAIN ATTENUATION TIME SERIES SYNTHESIS:RECOMMENDATION ITU-R P.1853-1

A. Principle

As mentioned in Section I, Recommendation ITU-RP.1853-1 [13] relies on a stochastic modeling of rain at-tenuation time series. First, a centred reduced Gaussian process

is generated. Second, is low-pass filtered with a cutofffrequency to define a correlated Gaussian process . Asdetailed in [5], generated that way is a centred, reduced,first-order stationary Markov process of which the correlationfunction is exponential and depends only on the lag timeso that

(1)

where is the correlation time. Note that in [5] andthat 5000 s in Recommendation ITU-R P.1853-1 [13].Third, [13] assumes that the absolute rain attenuation CCDF

given as input parameter is well represented bya lognormal distribution with average and stan-dard deviation or, equivalently in terms of natural logarithm,

and standard deviation . In such condi-tions, and the correlatedGaussian process is turned into a correlated lognormalprocess [dB] through

(2)

From classical statistical results, (2) implies that the first-orderstatistics given as input parameters are

(3a)

(3b)

(3c)

(3d)

where is the CCDF of the process andthe inverse complementary error function.Fourth, to be representative of rain and no-rain periods, an

empirical offset parameter is introduced in [13]

with(4)

where is the probability to have rain attenuation on the link.The long-term rain attenuation time series finally given by[10] is

ifotherwise.

(5)


In compliance with (2) and (4), the rain attenuation time seriessynthesizer driven by (5) requires as input parameters ,and . Ideally, the latter two should be regressed from exper-imental rain attenuation CCDF . In practice, as ex-perimental rain attenuation time series are not available world-wide and as recommended by [10], Recommendation ITU-RP.618-10 [16] is used to derive worldwide for prob-abilities in the range 10 % to 5% while the probability to haverain attenuation is approximated by the probability of raingiven by Recommendation ITU-R P.837 [17] anywhere in theworld.

B. Derivation of First Order Statistics

From (4) and (5), it follows that the statistical averagesand of the process are given by

(6a)

(6b)

where is the centred reducednormal probability density function (PDF) and

in compliance with (2). After somemanipulations, (6) finally leads to

(7a)

(7b)

(7c)

Moreover, the derivation of the CCDF of the ITUprocess driven by (5) is straightforward:

(7d)

Therefore, (7a)–(7d) compare satisfactorily with (3a)–(3d) onlyif , which, from a conceptual point of view, prevents

the introduction of an offset parameter. In particular, comparing(7d) and (3d), the introduction a posteriori of makes the cur-rent ITU synthesizer unable to reproduce the lognormal abso-lute rain attenuation CCDF yet given as input parameter. Thispoint will be quantitatively assessed in Section IV where theabilities of the ITU synthesizer and the new model presented inSection III to reproduce rain attenuation CCDFs given by Rec-ommendation ITU-R P.618 [16] are compared on a worldwidebasis.

C. Derivation of Second-Order Statistics

The dynamics of a process such as (5) or (11) is fullydriven by its correlation function. Obviously, the correlationfunction —(1)—of the underlying Gaussian process

drives the correlation function of the ITU rainattenuation process defined by (5). Our objective here is toanalytically assessfrom (5), (7a), and (7c). In particular, defining ,

, , , it followsfrom (4) and (5) that the covariance functionof the ITU process driven by (5) is given by

(8)

where ,, is the joint PDF of the rain

attenuation process and is the bi-variate normal PDF (see equation (A2) in Appendix A for thedefinition).Considering an elementary approximation for [18],

the covariance (8) is derived analytically in Appendix A fromequations (A4), (A13), and (A15). Consequently, using (7a) and(7c), an analytical dependency between and is finallyobtained as a function of , , and .As an illustration, Fig. 1 shows as a function of for

a typical range of values of varying from 0 to 1.5.48 , 1.69, and 1.59 in compliance withSection IV, where a hypothetical radio link at 50 GHz betweenan Earth station situated at Toulouse 43.60 1.44 and ageostationary satellite located at longitude 0 is considered.


Fig. 1. Dependency between and for the ITU rain attenuation time seriessynthesizer driven by (5). 5.48 , 1.69, and 1.59 incompliance with Section IV.

Fig. 2. Analytical correlation function of the ITU process com-pared with estimated from 100 random yearly realizations of orderived from a numerical computation of (8). 5.48 , 1.69,

1.59 in compliance with Section IV. is given by (1) with5000 s in compliance with Recommendation ITU-R P.1853-1 [13].

On the other hand, Fig. 2 shows the correlation functionderived analytically from given by (1) with5000 s (i.e., s ) in compliance with

[9] or [13]. For comparison, the average correlation functionderived from 100 random yearly realizations of is

plotted on Fig. 2. Moreover, a numerical computation of (8) hasalso been conducted to finally give a numerical evaluation of

. The result is also reported on Fig. 2. The three curvesmatch satisfactorily, confirming the validity of the analyticalframework laid in Appendix A for the analytical derivation of

. Fig. 2 shows that once the rain attenuation processis defined in compliance with (5) with given by

the exponential formulation (1), then the correlation functionof has an exponential asymptotic behavior for. This result is confirmed by first asymptotic results

derived from equations (A13), (A15), (A4), (7a), and (7c) thatare not developed here for the sake of conciseness.

Now, and as mentioned in Section I, more flexibility on theshape of might be required if a worldwide rain attenu-ation time series synthesizer that accounts for the local clima-tology has to be defined. Therefore, the exponential definitionof that is a basic assumption of the current ITU rain at-tenuation synthesizer and that defines the shape of is astrong constraint that would be relaxed to accept any analyticaldefinition, particularly the one that best reproduces the local dy-namics of experimental rain attenuation time series. This flexi-bility is one of the advantages of the new rain attenuation timeseries synthesizer detailed in Section III.In addition, note that the analytical framework developed in

Appendix A can be used to assess the dynamic parameter in(1) from rain attenuation measurements. Indeed, once

, , , and have beenderived from experimental rain attenuation time series, an op-timization routine based on equations (A13), (A15), (A4), (7a),and (7c) can be used to find that minimizes the error be-tween of model [10] and the ex-perimental correlation . Consequently, the analyticalderivation of conducted in Appendix A offers an alter-native to the method of the second-order conditional momentthat has been previously used in [9] or [11] or to the method-ology based on the hitting time statistics developed in [19] toinfer .

III. NEW RAIN ATTENUATION TIME SERIES SYNTHESIZER

A. Definition

To overcome the limitations listed in Section II, a new synthe-sizer is proposed. The latter has to generate rain attenuation timeseries including rain and no-rain periods. It must reproducethe first-order statistics—i.e., average , variance , CCDF

—given as input parameters and, for worldwideapplications, must be able to reproduce any correlation function

. First, in compliance with Section I, it is now assumedthat only the rain attenuation conditional PDF islognormal with mean and standard deviation :

(9)

In such conditions, the absolute rain attenuation CCDF givenas input parameter is now supposed to be well represented by amixed Dirac-lognormal distribution:

(10)

where is still the probability to have rain attenuation onthe link. Errors potentially introduced by the mixed Dirac-lognormal modeling (10) will be quantitatively assessed on aworldwide basis in Section IV.Second, a stationary, centred, reduced, correlated Gaussian

process with normal PDF and arbitrary correlation


function is generated in the Fourier domain in com-pliance with the methodology presented in Section III-D.is then turned into a rain attenuation process according to(11), shown at the bottom of the page, where is given by (4).

B. Derivation of First-Order Statistics

From (11) and recalling equation (A6), it is clear that

and . There-

fore, the new model (11) allows rain and no-rain periods to bereproduced without any additional offset parameter. Moreover,for , it can be verified that therandom variable in(11) is normal, with zero average and unit standard deviation.Therefore, the absolute CCDF of the stochasticprocess defined by (11) is given by

(12a)

Contrary to the current ITU model [13], (12a) shows that thenew rain attenuation time series synthesizer driven by (11) re-produces the (mixed Dirac-lognormal) rain attenuation absoluteCCDF —(10)—given as input parameter. Recallingthe classical statistical results (3), it follows from (11) and (12a)that

(12b)

(12c)

(12d)

The full parameterization of (11) now requires the definition ofthe second-order statistics.

C. Derivation of Second-Order Statistics

Similarly to Section II-C, defining ,, the covariance function of the rain atten-

uation process (11) is now given by

(13)

Fig. 3. Dependency between and for the new rain attenuation time seriessynthesizer driven by (11). 5.48 , 0.96, and 1.10in compliance with Section IV. The relationship between and for Rec-ommendation ITU-R P.1853-1 driven by (5)—see Fig. 1—is also plotted forcomparison (dashed line).

In (13),, now refers to the joint PDF

of the rain attenuation process defined by (11), andis still the bivariate normal PDF [see equa-

tion (A2) in Appendix A for the definition]. Unfortunately, dueto the complexity of , (13) cannot be solved analytically sothat a numerical computation is required. It is then convenientto change the variables and to and according to

and . In such conditions,(13) becomes

(14)

Equation (14) is evaluated numerically. The correlation func-tion of the rain attenuation process (11) follows from (12b)and (12d) as a function of , , , and . As an ex-ample, Fig. 3 shows as a function of for a typical range ofvalues of varying from 0 to 1. 5.48 , 0.96and 1.10 in compliance with Section IV where a hypo-thetical radio link at 50 GHz between an Earth station situatedat Toulouse 43.60 1.44 and a geostationary satellite lo-cated at longitude 0 is considered.Clearly, from the explicit dependency between and

illustrated in Fig. 3, the correlation function of the corre-lated Gaussian process in (11) can be derived once is

if

otherwise(11)


Fig. 4. Correlation function that must be given to the Gaussianprocess in (11) to insure that in (11) has the same correlation func-tion (i.e., the same dynamics) as Recommendation ITU-R P.1853-1driven by (5). 5.48 , 1.69, 1.59, 0.96,and 1.10 in compliance with Section IV. The correlation function

of the Gaussian process of Recommendation ITU-R P.1853-1—(1)with 5000 s (dashed line)—is added for comparison.

known. Particularly, if experimental rain attenuation time se-ries are available, then can be derived from the experimentalvalues , , , and

obtained from measured time series incompliance with (14), (12b), and (12d).On the other hand, as experimental data are not available

worldwide and as the dynamics of the current ITU synthesizerhas been intensively tested and validated at mid-latitudes in pre-vious studies from experimental rain attenuation time series [9],[11], [12], the capability of the new synthesizer to reproduce therain attenuation correlation function of the ITU synthesizer isdemonstrated. To do it, must be defined so that the randomprocess (11) reproduces the correlation function of the ITUrain attenuation time series synthesizer driven by (5). This canbe done very easily from the analytical derivations conducted inSection II.Indeed, as an illustration, consider the Recommendation

ITU-R P.1853-1 input parameters 5.48 , 1.69,1.59 in compliance with Section IV. The rain at-

tenuation correlation function of theITU synthesizer driven by (5) is then given in Fig. 2. Theassociated conditional parameters also given in Section IVare 0.96 and 1.10. According to Fig. 3,the one to one correspondence between and of the newtime series synthesizer driven by (14), (12b), and (12d) is thenused to interpolate at . Finally, the correlationfunction that must be given to the underlying Gaussianprocess in (11) to insure that the rain attenuation process

in (11) has the same correlation function —i.e., thesame dynamics—as Recommendation ITU-R P.1853-1 drivenby (5) is shown in Fig. 4.In accordance with Fig. 4, the logarithm of the correlation

function of in (11) shows a linear dependencywith respect to . Therefore, the correlation functionof in (11) is exponential, i.e., accepts the analytical for-mulation (1), but now with 4340 s. Obviously, otherdepartures from the ITU parameter 5000 s have to be

Fig. 5. Worldwide map of the correlation time that has to be given to thecorrelation function —supposed to be exponential—of in (11) toinsure that in (11) reproduces the dynamics of the current ITU synthesizerdriven by (5). The frequency is 40 GHz and the radio link geometrical configu-rations are defined in Section IV.

expected depending on the local values of the parameters ,, , , and .

This point is highlighted worldwide in Fig. 5 where thathas to be given to in (11) to mimic the dynamics ofthe ITU synthesizer is regressed on a worldwide basis consid-ering the satellite radio links operating at 40 GHz defined inSection IV. It is important to note that, as the dynamics of theITU synthesizer have been validated so far only from experi-mental rain attenuation time series collected at mid-latitudes,the validity of Fig. 5 should be limited to mid-latitudes areas.At this stage, an algorithmic scheme to generate stationary cor-related Gaussian processes with arbitrary correlation function

(the one derived in Fig. 4 for instance) is still requiredto make effectual the new time series synthesizer driven by (11).This point is addressed in Section III-D.

D. Generating of a One-Dimensional Correlated GaussianProcess in the Fourier Domain

Our objective is to generate a one-dimensional stationary realGaussian process with zero mean, variance one, and arbi-trary correlation function . The methodology lies on thealgorithmic approach defined by [20] to simulate bidimensionalGaussian processes with arbitrary spatial covariance function.Here, its adaptation to the one-dimensional case is conductedand is fully demonstrated in Appendix B. For numerical imple-mentation, define , whereare the points (or instants) where has to be specified and

is the length of the random process (or durationso that refers to the sampling rate). is constructed using aFourier series:

(15a)

and

(15b)

where and are the direct and inverse Fourier transforms,respectively. The correlated Gaussian random process


is then constructed in the Fourier domain accordingto the following algorithm:1) Generate uncorrelated random complex numbers

which real and imaginary parts are normal,with 0 mean and variance 1. For and , putthe imaginary part of to 0.

2) Complete the definition of so that for.

3) Define except for and where.

4) From the analytical (if available) or the numerical defini-tion of , compute the Fourier transform accordingto (15b).

5) Define

(16)

6) Derive from (15a).Justifications of the algorithm described above are given inAppendix B. In particular, it is demonstrated that the correlatedGaussian process constructed that way has 0 mean, variance1, and correlation function . It is important to note that anycorrelation function can be introduced: as required, theexponential definition of that was a basic assumptionof the current ITU synthesizer is now relaxed.

IV. CAPABILITY OF THE MODELS TO REPRODUCE ABSOLUTERAIN ATTENUATION CCDFS WORLDWIDE

To quantitatively assess on a worldwide basis the ability ofboth rain attenuation synthesizer to reproduce absolute rain at-tenuation CCDFs , satellite radio links operatingat 20, 30, 40, and 50 GHz with three geostationary satellitesarbitrary located at longitudes 80 , 0 , and 80 are consid-ered. As experimental data are not available worldwide, Recom-mendation ITU-R P.837 [17] and ITU-R P.618-10 [16] are usedto predict worldwide (resolution 1.125 ) the probability of rain

and the absolute rain attenuation CCDF , re-spectively. It is important to note that we recall that the validityrange of the worldwide predictions of given byRecommendation ITU-R P.618-10 is limited to 5% incompliance with [16]. Moreover, as noted by RecommendationITU-R P.1853-1 [13], the probability to have rain attenuationis approximated by .On the one hand, for each frequency and each location, the

input parameters and required by the ITU synthe-sizer are derived worldwide from the lognormal regression ofthe absolute CCDFs given by RecommendationITU-R P.618-10. The regression is conducted in compliancewith the methodology recommended in [13], for probabilitiesbetween 10 % and the minimum value between % and

5%. To make the regression reliable, areas whereis lower than 10 % are disregarded. Moreover, to reduce

the computation time, only links with elevation angle greaterthan 25 are considered. For pixels in the coverage intersectionof two satellites, the parameters corresponding to the highestelevation link are kept.On the other hand, the absolute rain attenuation CCDFs

given by Recommendation ITU-R P.618-10 areregressed by a mixed Dirac-lognormal distribution as required

Fig. 6. Absolute rain attenuation CCDF given by Recommen-dation ITU-R P.618-10, reproduced by the current ITU synthesizer, and repro-duced by the new synthesizer for an hypothetical radio link at 50 GHz betweenan Earth station situated at Toulouse 43.60 1.44 and a geostationarysatellite located at longitude 0 .

by the new rain attenuation synthesizer. Worldwide maps ofthe conditional parameters and are then obtained.Fig. 6 is an example of the absolute rain attenuation CCDF

given by Recommendation ITU-R P.618 for an hy-pothetical radio link at 50 GHz between an Earth station situatedat Toulouse 43.60 1.44 and the geostationary satellitelocated at longitude 0 . On the one hand, the lognormal regres-sion on conducted in compliance with [13] leadsto 1.69 and 1.59 while RecommendationITU-R P.837 gives 5.48% . It follows that theoffset parameter in (5) is equal to 2.35 dB in compliancewith (4). On the other hand, the Dirac-lognormal regression of

leads to 0.96 and 1.10.The rain attenuation CCDFs finally reproduced by the cur-

rent ITU synthesizer (7d) and by the new synthesizer (12a) areplotted for comparison. For time percentages between 10 %and 5% (i.e., the highest probability value given by Recommen-dation ITU-R P.618-10), Fig. 6 underlines the greater ability ofthe new model to reproduce the input CCDF givenby Recommendation ITU-R P.618-10. Beyond 5.48%,both model assumes clear sky conditions and the rain attenua-tion CCDF of both synthesizer falls to 0 as expected.In compliance with Fig. 6, note that the probability to

have rain attenuation suggested by Recommendation ITU-RP.618-10 (i.e., derived from an extrapolation) would differfrom the probability of rain 5.48%. Of course, theprobability to have rain attenuation on a slant path that mayextend to a few kilometers depending on the radio link elevationand on the local rain altitude is expected to be greater than thelocal probability of rain but probably not in the proportionsuggested in Fig. 6. Such a difference is clearly acknowledgedin Recommendation ITU-R P.1853-1 and refers to the difficultyto measure and a fortiori to model the probability to haverain attenuation on an Earth–space satellite link. Discussionson this difficult topic are out the scope of the present paper.To quantitatively assess the greater ability of the new model

to reproduce given by Recommendation ITU-RP.618-10, the rms value of the error recommended in


Fig. 7. Ratio(%) computed worldwide for the radio links at (a) 20, (b) 30, (c) 40, and (d) 50GHz. The location of the three satellites considered (SAT 1, SAT 2, SAT 3) arerecalled.

Recommendation ITU.R P.311 [21] is computed for each modelon a worldwide basis. In particular, Fig. 7(a)–(d) shows the ratio

(in

percent) computed worldwide for the radio links at 20, 30, 40,and 50 GHz, respectively. In compliance with the definition of, note that the new synthesizer is all the better with respect to

the current ITU model as tends to 100%. 0% meansno improvement and 0% means that the new model re-produces less accurately the input rain attenuation CCDF givenby Recommendation ITU-R P.618-10 than the current ITU syn-thesizer. For the satellite link at Toulouse considered in Fig. 6,

50.6%.First, according to Fig. 7(a)–(d), 0% whatever the

frequency and whatever the location. Therefore, the newsynthesizer reproduces better the absolute rain attenuationCCDF given as input parameter than the currentITU synthesizer. The improvement is all the better as thefrequency increases in compliance with the analytical deriva-tions conducted in Section II-B. Indeed, recalling that the rainattenuation increases with frequency, the same behavior appliesto and , and it can be concluded from (4) that theoffset parameter increases with the frequency. Therefore,in accordance with equation (7d), the current ITU synthesizerdeparts all the more from given by Recommen-dation ITU-R P.618-10 as the frequency increases. Lastly,the improvement strongly depends on the location. Indeed,varies from about 30% in mid-latitudes and reaches 100%in tropical or equatorial areas.

V. CONCLUSION

From the analytical derivation of the first- and second-orderstatistical properties of the rain attenuation time series synthe-sizer proposed in Recommendation ITU-R P.1853-1, two short-comings have been demonstrated. First, due to the offset pa-rameter, it has been shown analytically that the ITU synthe-sizer does not reproduce the absolute rain attenuation CCDFyet given as an input parameter. Second, in RecommendationITU-R P.1853-1, it is necessary that the correlation functionused to generate the underlying stationary Gaussian process isexponential. Clearly, this assumption does not allow any rain at-tenuation correlation function to be reproduced. More flexi-bility is required, especially if a worldwide synthesizer able toaccount for the local climatology has to be defined and if betterperformances are expected for any climatic area.Therefore, a new rain attenuation time series synthesizer has

been proposed. It relies on a mixed Dirac-lognormal modelingof the absolute rain attenuation CCDF.It has then been shown analytically that the new synthesizer

allows the 1st order statistics given as input parameters to be re-produced. Second, for worldwide applications, the new synthe-sizer is able to reproduce any rain attenuation correlation func-tion . Particularly, as the dynamics of the current ITU syn-thesizer has been intensively tested and validated in previousstudies from experimental rain attenuation time series collectedat mid-latitudes, the capability of the new synthesizer to repro-duce the rain attenuation correlation function of the ITU syn-thesizer has been demonstrated. To make the approach effec-tual and offer a framework allowing potential future improve-ments, a methodology to simulate one- dimensional Gaussianprocesses with arbitrary correlation function has been pre-sented and thoroughly justified.


Then, the ability of each synthesizer to reproduce absoluterain attenuation CCDFs given by RecommendationITU-R P.618 has been compared on a worldwide basis. The newsynthesizer then shows is greater ability to reproduce

worldwide.Therefore, the new rain attenuation time series synthesizer re-

produces better the rain attenuation CCDF given as input param-eter, preserves the rain attenuation dynamics of the current ITUsynthesizer for simulations at mid-latitudes, and, if it provesto be necessary for worldwide applications, is able to repro-duce any rain attenuation correlation function. Consequently,the new synthesizer might be considered as an improvement anda generalization of the current ITU rain attenuation time seriessynthesizer.Now, the ability of the new synthesizer to reproduce statis-

tics derived from experimental rain attenuation time series col-lected worldwide must be intensively tested. In particular, themixedDirac-lognormal modeling of the absolute CCDFmust becarefully investigated on a worldwide basis from experimentalCCDFs. Moreover, at mid-latitudes, the capability of the newsynthesizer to reproduce any rain attenuation correlation func-tion should be used to confirm (or not) that an exponential isdefinitively the analytical formulation that best restitutes the ex-perimental rain attenuation dynamics as reported by [9]–[11] or[12]. Its flexibility should also be used to investigate seasonaleffects. The same exercise must be conducted from equatorialand tropical experimental propagation data to finally define aworldwide parameterization that optimally reproduces the rainattenuation experimental statistics.

APPENDIX AANALYTICAL DERIVATION OF THE CORRELATION FUNCTION OF

THE ITU RAIN ATTENUATION PROCESS

In compliance with Section II, the analytical derivation ofrequires the com-

putation of the covariance function:

(A1)

where , ,and is the bivariate normal PDF given by

(A2)

Let and define

(A3)

so that

(A4)

In such conditions

(A5)

As , (A5) implies that. Recalling that

(A6)

the integration with respect to in (A5) is straightforward andleads to

(A7)

Now, let

(A8)

After some manipulations, (A7) becomes

(A9)

where

(A10)

and

(A11)


Depending on , two cases have to be considered.If : we use the approximation

(A12)

which holds with an error less than for [18].In (A11), , , ,

, and .In such conditions, (A12) in (A9) and recalling (A6) finally

leads to

(A13)

where erfc is the complementary error function, ,, and

If : we use the approximation

(A14)

which holds with an error less than for [18] andwhere a, , , , have been defined in (A12). Therefore,(A14) in (A9) and (A6) finally lead to

(A15)

where as before but now with and.

Depending on the sign of defined by (A10),can be derived analytically from

(A13) or (A15) with good accuracy. The correlation functionof the ITU rain at-

tenuation process defined by (5) is finally obtained using(7a) and (7c). It is important to note that is a functionof , , , and .

APPENDIX BSTATISTICAL PROPERTIES OF THE GAUSSIAN PROCESS DEFINEDBY (15) AND (16) AND JUSTIFICATIONS OF THE ALGORITHMIC

APPROACH TO GENERATE

1) Link between the correlation function and the Fouriercoefficients driven by (15a) and (15b). Defining

and designating by the complex conju-gate, (15) leads to

(B1)

As is real with zero average and variance one andrecalling that the correlation function of the stationaryprocess is an even function, it follows that

. Defining ’,(B1) becomes

(B2)

Consequently

(B3)

It follows from (B3) that

ifotherwise

(B4)

2) Statistical properties of the random process derivedfrom (16).It follows from (15) that

(B5)

In compliance with (16),since is normal with mean 0 so that has zeroaverage.Moreover

(B6)


With ’, (B6) leads to

(B7)Using (B4), (B7) finally leads to

(B8)

In compliance with (B8), the random process con-structed from (16) complies with the requirements, i.e., isGaussian with correlation function , has 0 average andvariance by defini-tion of the correlation function at lag 0.

3) Details and justifications of the algorithm to constructthe correlated Gaussian field in the Fourier domain.By using expansions (15a) and (15b), we are implicitlyassuming that is periodic since . More-over, as is real, it follows that . In suchconditions, the Fourier coefficients in (15b) must verify

(B9)

Moreover, as is an even function, it follows thatfrom (B4) is real. Therefore,

is real and in (16) implies thatand must be real for and in compliancewith step 1 and 2 of the algorithm given in Section III-DFrom (16) and (B4):

(B10)

which holds whatever only if . As the realand imaginary parts of are uncorrelated with variance 1(except for and where the imaginary partof is 0), it follows that except for and

, where . It follows that exceptfor and where in compliance withstep 3 of the algorithm given in Section III-D.

REFERENCES[1] E. Salonen, S. Karhu, P. Jokela, W. Zhang, S. Uppala, H. Aulamo,

S. Sarkkula, and P. P. Baptista, “Modelling and calculation of atmo-spheric attenuation for low-fade margin satellite communications,”ESA J., vol. 16, no. 3, pp. 299–316, 1992.

[2] L. Castanet, M. Bousquet, M. Filip, P. Gallois, B. Gremont, L. D. Haro,J. Lemorton, A. Paraboni, and M. Schnell, “Impairment mitigation andperformance restoration,” ESA Publications Division, SP-1252, COST255 Final Rep. Ch. 5.3, Mar. 2002.

[3] L. Castanet, A. Bolea-Alamañac, and M. Bousquet, “Interference andfade mitigation techniques for Ka and Q/V band satellite communica-tion systems,” presented at the COST 272-280 Int. Workshop SatelliteCommun. from Fade Mitigation to Service Provision, Noordwijk, TheNetherlands, May 2003.

[4] L. Castanet, D. Mertens, and M. Bousquet, “Simulation of the per-formance of a Ka-band VSAT videoconferencing system with uplinkpower control and data rate reduction to mitigate atmospheric propa-gation effects,” Int. J. Satell. Commun., vol. 20, no. 4, pp. 231–249,Jul./Aug. 2002.

[5] T. Maseng and P. M. Bakken, “A stochastic dynamic model of rainattenuation,” IEEE Trans. Commun., vol. 29, no. 5, pp. 660–669, May1981.

[6] F. J. A. Andrade and L. A. R. da Silva Mello, “Rain attenuation timeseries synthesizer based on the gamma distribution,” IEEE AntennasWireless Propag. Lett., vol. 10, pp. 1381–1384, 2011.

[7] G. Karagiannis, A. D. Panagopoulos, and J. D. Kanellopoulos, “Mul-tidimensional rain attenuation stochastic dynamic modeling: Applica-tion to earth-space diversity systems,” IEEE Trans. Antennas Propag.,vol. 60, no. 11, pp. 5400–5411, Oct. 2012.

[8] M. Cheffena, L. E. Braten, and T. Ekman, “On the space-time varia-tions of rain attenuation,” IEEE Trans. Antennas Propag., vol. 57, no.6, pp. 1771–1782, Jun. 2009.

[9] F. Lacoste, M. Bousquet, L. Castanet, F. Cornet, and J. Lemorton, “Im-provement of the ONERA-CNES rain attenuation time series synthe-sizer and validation of the dynamic characteristics of the generated fadeevents,” Space Commun. J., vol. 20, no. 1–2, 2005.

[10] F. Lacoste, M. Bousquet, L. Castanet, F. Cornet, and J. Lemorton,“Event-based analysis of rain attenuation time series synthesizers forthe Ka-band satellite propagation channel,” presented at the ClimDiffConf., Cleveland, OH, USA, Sep. 2005.

[11] G. Carrie, F. Lacoste, and L. Castanet, “A new ‘event-on-demand’ syn-thesizer of rain attenuation time series at Ku-, Ka- and Q/V bands,” Int.J. Satell. Commun. Netw., vol. 29, no. 1, pp. 47–60, Jan./Feb. 2009.

[12] Influence of the Variability of the Propagation Channel on Mobile,Fixed Multimedia and Optical Satellite Communications, L. Castanet,Ed. et al. Aachen: Shaker Verlag, 2008, SatNEx JA-2310 book, ISBN978-3-8322-6904-3.

[13] Tropospheric Attenuation Time Series Synthesis, ITU-R Recommenda-tion P.1853-1, ITU, Geneva, Switzerland, 2009.

[14] H. Sauvageot, “The probability density function of rain rate and theestimation of rainfall by area integrals,” J. Appl. Meteor., vol. 33, no.11, pp. 1255–1262, 1994.

[15] Specific Attenuation Model for Rain for Use in Prediction Methods,ITU-R Recommendation P.838-3, ITU, Geneva, Switzerland, 2005.

[16] Propagation Data and Prediction Methods Required for the Designof Earth-Space Telecommunication Systems, ITU-R RecommendationP.618-10, ITU, Geneva, Switzerland, 2009.

[17] Characteristics of Precipitation for Propagation Modelling, ITU-RRecommendation P.837-5, ITU, Geneva, Switzerland, 2007.

[18] F. G. Lether, “Elementary approximation for erf(x),” J. Quant. Spec-trosc. Radiat. Transfer, vol. 49, no. 5, pp. 573–577, 1993.

[19] S. A. Kanellopoulos, A. D. Panagopoulos, and J. D. Kanellopoulos,“Calculation of the dynamic input parameter for a stochastic modelsimulating rain attenuation: A novel mathematical approach,” IEEETrans. Antennas Propag., vol. 55, no. 11, pp. 3257–3264, 2007.

[20] T. L. Bell, “A space-time stochastic model of rainfall for satellite re-mote-sensing studies,” J. Geophys. Res., vol. 92, pp. 9631–9643, 1987.

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Xavier Boulanger was born in Montpellier, France,in 1986. He received the Diploma of Engineeringdegree in electronics and digital communicationsfrom Ecole Nationale de l’Aviation Civile (ENAC),Toulouse, France, in 2010.In 2009, he conducted his End of Study Internship

in ONERA, France, on the improvement of the mod-eling of the propagation channel for fixed Satcomsystems over temperate climates. Since 2010, he hasbeen working towards the Ph.D. degree in collabo-ration between the French Space Agency (CNES),

France, and ONERA, France. He was also a contributor in the European actionCOST IC-0802.

Laurent Féral, photograph and biography not available at the time ofpublication.

Laurent Castanet, photograph and biography not available at the time ofpublication.

Nicolas Jeannin, photograph and biography not available at the time ofpublication.

Guillaume Carrie, photograph and biography not available at the time ofpublication.

Frederic Lacoste, photograph and biography not available at the time ofpublication.

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