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 Based
on 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 time
series , including rain and no rain periods anywhere in the
world. Nevertheless, its dynamic properties have been validated
so far from experimental rain attenuation time series collected at
mid-latitudes only. In the present paper, an effort is conducted to
derive analytically the first- and second-order statistical proper-
ties of the ITU rain attenuation time-series synthesizer. It is then
shown that the ITU synthesizer does not reproduce the first-order
statistics (particularly the rain attenuation cumulative distribution
function 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 be
defined. Therefore, a new rain attenuation time-series synthesizer
is proposed. It assumes a mixed Dirac-lognormal modeling of the
absolute rain attenuation CDF and relies on a stochastic genera-
tion in the Fourier plane. It is then shown analytically that the new
synthesizer reproduces much better the first-order statistics given
as input parameters and enables any rain attenuation correlation
function 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 is
then concluded that the new rain attenuation time-series synthe-
sizer reproduces the rain attenuation CDF much better, preserves
the rain attenuation dynamics of the current ITU synthesizer for
simulations at mid-latitudes, and, if it proves to be necessary for
worldwide applications, is able to reproduce any rain attenuation
correlation function.
Index Terms—Rain attenuation time-series, satellite communi-
cation systems, stochastic processes.
I. INTRODUCTION
THE 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; accepted
November 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 framework
of the European action COST IC0802.
X. Boulanger is with the French Aerospace Lab (ONERA), Département
Electromagnétisme et Radar (DEMR), Toulouse 31055, France, and also with
the French Space Agency (CNES), Toulouse 31055, France (e-mail: Xavier.
Boulanger@onera.fr).
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: lferal@laplace.univ-tlse.fr).
L. Castanet, N. Jeannin, and G. Carrie are with the French Aerospace Lab
(ONERA), Département Electromagnétisme et Radar (DEMR), Toulouse
31055, France (e-mail: laurent.castanet@onera.fr; nicolas.jeannin@onera.fr;
Guillaume.Carrie@onera.fr).
F. Lacoste is with the French Space Agency (CNES), Toulouse 31055, France
(e-mail: frederic.lacoste@cnes.fr).
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 as
Ka or Q/V, i.e., 20–50 GHz) are envisaged to reach higher per-
formances in terms of spectrum availability, equipment size, or
bandwidth. Besides, in tropical and equatorial areas, where the
terrestrial telecommunication infrastructures are less developed
than 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 can
reach some tens of decibels at Ka or Q/V bands [1]. Among the
various tropospheric components, rain is the major contributor
to total tropospheric attenuation. Moreover, depending on the
local climatology, rain attenuation is expected to be more se-
vere in tropical and equatorial areas than in mid-latitudes, both
in terms of occurrence and in attenuation level. To counteract
tropospheric impairments, a fixed power margin that depends
on the required service availability is commonly introduced in
the link budget. Nevertheless, this extra resource allocation is
not optimal for systems above 20 GHz because the required
margin becomes too high for continuous operation, resulting in
costly user Earth stations and causing excessive interferences.
Therefore, to insure a favorable link budget, the fade mitigation
techniques (FMT) (such as adaptive coding or modulation and
frequency diversity [2], [3]) have to be implemented to coun-
teract tropospheric impairments. That is the reason why, for
satellite communication systems in centimetric or millimetric
bands, adaptive real-time algorithms are being developed [4].
Clearly, tropospheric attenuation time series are necessary for
their 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 around
the world, for various radio wave configurations in terms of
frequency, polarization, and elevation angle. Unfortunately,
propagation experiments above 20 GHz are not numerous, so
that the synthetic rain attenuation time series have to be
considered.
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. This
approach has been used in [6]–[8] and has then been extended
in [9]–[11] to generate long-term rain attenuation time series
(i.e., including rain and no rain periods). It is important to note
that [9]–[11] and [12] have shown that the dynamics of their
long-term synthetic rain attenuation time series compares sat-
isfactorily (from a statistical point of view) with experimental
propagation 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 shows
that the long-term rain attenuation process is not rigorously log-
normal as its CCDF falls to 0 for time percentages
between 1% and 20 of an average year depending on the
local 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 follow
the 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 located
from middle to tropical and equatorial latitudes, [14] has shown
that the conditional rain rate CCDF is
lognormal. Disregarding the spatial correlation of rainy events
and recalling the power relationship between rain intensity
and rain specific attenuation [15], it follows that only the condi-
tional CCDF of rain attenuation should be
considered as a lognormal process in compliance with the first
approach 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, which
implies 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 in
Section II. On the one hand, this assumption has been validated
in [9], [11], and [12] from experimental rain attenuation time se-
ries collected at mid-latitudes. On the other hand, its validity in
other climatic areas (in tropical or equatorial areas for instance
where the convective nature of rainy events is much more pro-
nounced than in mid-latitudes) has not been demonstrated so
far 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 reproduced
would represent a generalization of the current ITU synthesizer
described 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 in
the Fourier plane. Contrary to the current ITU synthesizer, the
new model reproduces very accurately the long term rain atten-
uation CCDF given as an input parameter and, for
worldwide applications, allows any rain attenuation correlation
function to be reproduced.
The paper is organized as follows. First, the rain attenuation
time series synthesizer [13] adopted by ITU-R Study Group 3 is
recalled in Section II. Its conceptual shortcomings are demon-
strated from the analytical derivation of first- and second-order
statistics. Then, a new rain attenuation time series synthesizer
is presented in Section III. Its first- and second-order statistical
properties 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 of
the new synthesizer to reproduce the rain attenuation correla-
tion function of the ITU synthesizer is demonstrated. Lastly, the
ability 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-R
P.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 cutoff
frequency to define a correlated Gaussian process . As
detailed in [5], generated that way is a centred, reduced,
first-order stationary Markov process of which the correlation
function is exponential and depends only on the lag time
so that
(1)
where is the correlation time. Note that in [5] and
that 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 by
a lognormal distribution with average and stan-
dard deviation or, equivalently in terms of natural logarithm,
and standard deviation . In such condi-
tions, and the correlated
Gaussian process is turned into a correlated lognormal
process [dB] through
(2)
From classical statistical results, (2) implies that the first-order
statistics given as input parameters are
(3a)
(3b)
(3c)
(3d)
where is the CCDF of the process and
the 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
if
otherwise.
(5)

In compliance with (2) and (4), the rain attenuation time series
synthesizer 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-R
P.618-10 [16] is used to derive worldwide for prob-
abilities in the range 10 % to 5% while the probability to have
rain attenuation is approximated by the probability of rain
given by Recommendation ITU-R P.837 [17] anywhere in the
world.
B. Derivation of First Order Statistics
From (4) and (5), it follows that the statistical averages
and of the process are given by
(6a)
(6b)
where is the centred reduced
normal probability density function (PDF) and
in compliance with (2). After some
manipulations, (6) finally leads to
(7a)
(7b)
(7c)
Moreover, the derivation of the CCDF of the ITU
process driven by (5) is straightforward:
(7d)
Therefore, (7a)–(7d) compare satisfactorily with (3a)–(3d) only
if , 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. This
point will be quantitatively assessed in Section IV where the
abilities of the ITU synthesizer and the new model presented in
Section III to reproduce rain attenuation CCDFs given by Rec-
ommendation ITU-R P.618 [16] are compared on a worldwide
basis.
C. Derivation of Second-Order Statistics
The dynamics of a process such as (5) or (11) is fully
driven by its correlation function. Obviously, the correlation
function —(1)—of the underlying Gaussian process
drives the correlation function of the ITU rain
attenuation process defined by (5). Our objective here is to
analytically assess
from (5), (7a), and (7c). In particular, defining ,
, , , it follows
from (4) and (5) that the covariance function
of 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 the
definition).
Considering an elementary approximation for [18],
the covariance (8) is derived analytically in Appendix A from
equations (A4), (A13), and (A15). Consequently, using (7a) and
(7c), an analytical dependency between and is finally
obtained 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 with
Section IV, where a hypothetical radio link at 50 GHz between
an Earth station situated at Toulouse 43.60 1.44 and a
geostationary satellite located at longitude 0 is considered.

Fig. 1. Dependency between and for the ITU rain attenuation time series
synthesizer driven by (5). 5.48 , 1.69, and 1.59 in
compliance with Section IV.
Fig. 2. Analytical correlation function of the ITU process com-
pared with estimated from 100 random yearly realizations of or
derived from a numerical computation of (8). 5.48 , 1.69,
1.59 in compliance with Section IV. is given by (1) with
5000 s in compliance with Recommendation ITU-R P.1853-1 [13].
On the other hand, Fig. 2 shows the correlation function
derived analytically from given by (1) with
5000 s (i.e., s ) in compliance with
[9] or [13]. For comparison, the average correlation function
derived from 100 random yearly realizations of is
plotted on Fig. 2. Moreover, a numerical computation of (8) has
also been conducted to finally give a numerical evaluation of
. The result is also reported on Fig. 2. The three curves
match satisfactorily, confirming the validity of the analytical
framework laid in Appendix A for the analytical derivation of
. Fig. 2 shows that once the rain attenuation process
is defined in compliance with (5) with given by
the exponential formulation (1), then the correlation function
of has an exponential asymptotic behavior for
. This result is confirmed by first asymptotic results
derived from equations (A13), (A15), (A4), (7a), and (7c) that
are not developed here for the sake of conciseness.
Now, and as mentioned in Section I, more flexibility on the
shape 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 definition
of that is a basic assumption of the current ITU rain at-
tenuation synthesizer and that defines the shape of is a
strong constraint that would be relaxed to accept any analytical
definition, 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 time
series 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 been
derived 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 analytical
derivation of conducted in Appendix A offers an alter-
native to the method of the second-order conditional moment
that has been previously used in [9] or [11] or to the method-
ology based on the hitting time statistics developed in [19] to
infer .
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 time
series including rain and no-rain periods. It must reproduce
the first-order statistics—i.e., average , variance , CCDF
—given as input parameters and, for worldwide
applications, must be able to reproduce any correlation function
. First, in compliance with Section I, it is now assumed
that only the rain attenuation conditional PDF is
lognormal with mean and standard deviation :
(9)
In such conditions, the absolute rain attenuation CCDF given
as input parameter is now supposed to be well represented by a
mixed Dirac-lognormal distribution:
(10)
where is still the probability to have rain attenuation on
the link. Errors potentially introduced by the mixed Dirac-
lognormal modeling (10) will be quantitatively assessed on a
worldwide 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 be
reproduced without any additional offset parameter. Moreover,
for , it can be verified that the
random variable in
(11) is normal, with zero average and unit standard deviation.
Therefore, the absolute CCDF of the stochastic
process defined by (11) is given by
(12a)
Contrary to the current ITU model [13], (12a) shows that the
new rain attenuation time series synthesizer driven by (11) re-
produces the (mixed Dirac-lognormal) rain attenuation absolute
CCDF —(10)—given as input parameter. Recalling
the classical statistical results (3), it follows from (11) and (12a)
that
(12b)
(12c)
(12d)
The full parameterization of (11) now requires the definition of
the 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 series
synthesizer driven by (11). 5.48 , 0.96, and 1.10
in 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 for
comparison (dashed line).
In (13),
, now refers to the joint PDF
of the rain attenuation process defined by (11), and
is still the bivariate normal PDF [see equa-
tion (A2) in Appendix A for the definition]. Unfortunately, due
to the complexity of , (13) cannot be solved analytically so
that a numerical computation is required. It is then convenient
to 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 of
values of varying from 0 to 1. 5.48 , 0.96
and 1.10 in compliance with Section IV where a hypo-
thetical radio link at 50 GHz between an Earth station situated
at 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 Gaussian
process in (11) to insure that in (11) has the same correlation func-
tion (i.e., the same dynamics) as Recommendation ITU-R P.1853-1
driven 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 experimental
values , , , and
obtained from measured time series in
compliance with (14), (12b), and (12d).
On the other hand, as experimental data are not available
worldwide and as the dynamics of the current ITU synthesizer
has 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 the
rain attenuation correlation function of the ITU synthesizer is
demonstrated. To do it, must be defined so that the random
process (11) reproduces the correlation function of the ITU
rain attenuation time series synthesizer driven by (5). This can
be done very easily from the analytical derivations conducted in
Section 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 the
ITU synthesizer driven by (5) is then given in Fig. 2. The
associated conditional parameters also given in Section IV
are 0.96 and 1.10. According to Fig. 3,
the one to one correspondence between and of the new
time series synthesizer driven by (14), (12b), and (12d) is then
used to interpolate at . Finally, the correlation
function that must be given to the underlying Gaussian
process in (11) to insure that the rain attenuation process
in (11) has the same correlation function —i.e., the
same dynamics—as Recommendation ITU-R P.1853-1 driven
by (5) is shown in Fig. 4.
In accordance with Fig. 4, the logarithm of the correlation
function of in (11) shows a linear dependency
with respect to . Therefore, the correlation function
of in (11) is exponential, i.e., accepts the analytical for-
mulation (1), but now with 4340 s. Obviously, other
departures from the ITU parameter 5000 s have to be
Fig. 5. Worldwide map of the correlation time that has to be given to the
correlation function —supposed to be exponential—of in (11) to
insure that in (11) reproduces the dynamics of the current ITU synthesizer
driven 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 that
has to be given to in (11) to mimic the dynamics of
the ITU synthesizer is regressed on a worldwide basis consid-
ering the satellite radio links operating at 40 GHz defined in
Section IV. It is important to note that, as the dynamics of the
ITU 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 required
to 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 Gaussian
Process in the Fourier Domain
Our objective is to generate a one-dimensional stationary real
Gaussian process with zero mean, variance one, and arbi-
trary correlation function . The methodology lies on the
algorithmic approach defined by [20] to simulate bidimensional
Gaussian processes with arbitrary spatial covariance function.
Here, its adaptation to the one-dimensional case is conducted
and is fully demonstrated in Appendix B. For numerical imple-
mentation, define , where
are the points (or instants) where has to be specified and
is the length of the random process (or duration
so that refers to the sampling rate). is constructed using a
Fourier 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 according
to the following algorithm:
1) Generate uncorrelated random complex numbers
which real and imaginary parts are normal,
with 0 mean and variance 1. For and , put
the 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 according
to (15b).
5) Define
(16)
6) Derive from (15a).
Justifications of the algorithm described above are given in
Appendix B. In particular, it is demonstrated that the correlated
Gaussian process constructed that way has 0 mean, variance
1, and correlation function . It is important to note that any
correlation function can be introduced: as required, the
exponential definition of that was a basic assumption
of the current ITU synthesizer is now relaxed.
IV. CAPABILITY OF THE MODELS TO REPRODUCE ABSOLUTE
RAIN ATTENUATION CCDFS WORLDWIDE
To quantitatively assess on a worldwide basis the ability of
both rain attenuation synthesizer to reproduce absolute rain at-
tenuation CCDFs , satellite radio links operating
at 20, 30, 40, and 50 GHz with three geostationary satellites
arbitrary 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 used
to 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 validity
range of the worldwide predictions of given by
Recommendation ITU-R P.618-10 is limited to 5% in
compliance with [16]. Moreover, as noted by Recommendation
ITU-R P.1853-1 [13], the probability to have rain attenuation
is 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 of
the absolute CCDFs given by Recommendation
ITU-R P.618-10. The regression is conducted in compliance
with the methodology recommended in [13], for probabilities
between 10 % and the minimum value between % and
5%. To make the regression reliable, areas where
is lower than 10 % are disregarded. Moreover, to reduce
the computation time, only links with elevation angle greater
than 25 are considered. For pixels in the coverage intersection
of two satellites, the parameters corresponding to the highest
elevation link are kept.
On the other hand, the absolute rain attenuation CCDFs
given by Recommendation ITU-R P.618-10 are
regressed 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 between
an Earth station situated at Toulouse 43.60 1.44 and a geostationary
satellite located at longitude 0 .
by the new rain attenuation synthesizer. Worldwide maps of
the 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 situated
at Toulouse 43.60 1.44 and the geostationary satellite
located at longitude 0 . On the one hand, the lognormal regres-
sion on conducted in compliance with [13] leads
to 1.69 and 1.59 while Recommendation
ITU-R P.837 gives 5.48% . It follows that the
offset parameter in (5) is equal to 2.35 dB in compliance
with (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) are
plotted 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 of
the new model to reproduce the input CCDF given
by 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-R
P.618-10 (i.e., derived from an extrapolation) would differ
from the probability of rain 5.48%. Of course, the
probability to have rain attenuation on a slant path that may
extend to a few kilometers depending on the radio link elevation
and on the local rain altitude is expected to be greater than the
local probability of rain but probably not in the proportion
suggested in Fig. 6. Such a difference is clearly acknowledged
in Recommendation ITU-R P.1853-1 and refers to the difficulty
to measure and a fortiori to model the probability to have
rain attenuation on an Earth–space satellite link. Discussions
on 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-R
P.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) 50
GHz. The location of the three satellites considered (SAT 1, SAT 2, SAT 3) are
recalled.
Recommendation ITU.R P.311 [21] is computed for each model
on 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% means
no improvement and 0% means that the new model re-
produces less accurately the input rain attenuation CCDF given
by 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 new
synthesizer reproduces better the absolute rain attenuation
CCDF given as input parameter than the current
ITU synthesizer. The improvement is all the better as the
frequency increases in compliance with the analytical deriva-
tions conducted in Section II-B. Indeed, recalling that the rain
attenuation increases with frequency, the same behavior applies
to and , and it can be concluded from (4) that the
offset parameter increases with the frequency. Therefore,
in accordance with equation (7d), the current ITU synthesizer
departs 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-order
statistical 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 CCDF
yet given as an input parameter. Second, in Recommendation
ITU-R P.1853-1, it is necessary that the correlation function
used to generate the underlying stationary Gaussian process is
exponential. 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 to
account for the local climatology has to be defined and if better
performances are expected for any climatic area.
Therefore, a new rain attenuation time series synthesizer has
been proposed. It relies on a mixed Dirac-lognormal modeling
of 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 previous
studies from experimental rain attenuation time series collected
at 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 Gaussian
processes with arbitrary correlation function has been pre-
sented and thoroughly justified.

Then, the ability of each synthesizer to reproduce absolute
rain attenuation CCDFs given by Recommendation
ITU-R P.618 has been compared on a worldwide basis. The new
synthesizer 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 ITU
synthesizer for simulations at mid-latitudes, and, if it proves
to 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 and
a generalization of the current ITU rain attenuation time series
synthesizer.
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, the
mixed Dirac-lognormal modeling of the absolute CCDF must be
carefully investigated on a worldwide basis from experimental
CCDFs. Moreover, at mid-latitudes, the capability of the new
synthesizer to reproduce any rain attenuation correlation func-
tion should be used to confirm (or not) that an exponential is
definitively 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 seasonal
effects. The same exercise must be conducted from equatorial
and tropical experimental propagation data to finally define a
worldwide parameterization that optimally reproduces the rain
attenuation experimental statistics.
APPENDIX A
ANALYTICAL DERIVATION OF THE CORRELATION FUNCTION OF
THE ITU RAIN ATTENUATION PROCESS
In compliance with Section II, the analytical derivation of
requires 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 and
leads 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] and
where 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 function
of the ITU rain at-
tenuation process defined by (5) is finally obtained using
(7a) and (7c). It is important to note that is a function
of , , , and .
APPENDIX B
STATISTICAL PROPERTIES OF THE GAUSSIAN PROCESS DEFINED
BY (15) AND (16) AND JUSTIFICATIONS OF THE ALGORITHMIC
APPROACH TO GENERATE
1) Link between the correlation function and the Fourier
coefficients 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 and
recalling that the correlation function of the stationary
process is an even function, it follows that
. Defining ’,
(B1) becomes
(B2)
Consequently
(B3)
It follows from (B3) that
if
otherwise
(B4)
2) Statistical properties of the random process derived
from (16).
It follows from (15) that
(B5)
In compliance with (16),
since is normal with mean 0 so that has zero
average.
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., is
Gaussian with correlation function , has 0 average and
variance by defini-
tion of the correlation function at lag 0.
3) Details and justifications of the algorithm to construct
the correlated Gaussian field in the Fourier domain.
By using expansions (15a) and (15b), we are implicitly
assuming that is periodic since . More-
over, as is real, it follows that . In such
conditions, the Fourier coefficients in (15b) must verify
(B9)
Moreover, as is an even function, it follows that
from (B4) is real. Therefore,
is real and in (16) implies that
and must be real for and in compliance
with step 1 and 2 of the algorithm given in Section III-D
From (16) and (B4):
(B10)
which holds whatever only if . As the real
and imaginary parts of are uncorrelated with variance 1
(except for and where the imaginary part
of is 0), it follows that except for and
, where . It follows that except
for and where in compliance with
step 3 of the algorithm given in Section III-D.
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Xavier Boulanger was born in Montpellier, France,
in 1986. He received the Diploma of Engineering
degree in electronics and digital communications
from 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 Satcom
systems over temperate climates. Since 2010, he has
been 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 action
COST IC-0802.
Laurent Féral, photograph and biography not available at the time of
publication.
Laurent Castanet, photograph and biography not available at the time of
publication.
Nicolas Jeannin, photograph and biography not available at the time of
publication.
Guillaume Carrie, photograph and biography not available at the time of
publication.
Frederic Lacoste, photograph and biography not available at the time of
publication.

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