Radiocommunication Study Groups
INTERNATIONAL TELECOMMUNICATION UNION
Source: Document 3M/183, Annex 10
15 May 2015
English only
Working Party 3M
FASCICLE
THE PREDICTION METHOD OF THE PROBABILITY OF RAIN ATTENUATION
PROPOSED IN RECOMMENDATION ITU-R P.618
1
Introduction
Recommendation ITU-R P.837-6 predicts the rainfall rate exceeded for a given probability of the
average year and at a given location. In particular, the percentage probability of rain in an average
year, P0, is defined in step 4 from worldwide ERA40 maps. The objective of this fascicle is to
provide information on the basic principles of the method to predict the probability of rain
attenuation on Earth-space links specified in Rec. ITU-R P.618.
2
Modelling of the probability of rain attenuation on Earth-space link
Knowing N equally distributed points on the ground projection of the path of length L and under the
rain height Hr as illustrated in Figure 1, the probability to have non-zero rain attenuation on the
path, P(AR>0), is equal to the complementary probability to not have rain on each point of the path:
P AR  0  lim (1  PR( x0 )  0, Rx1   0,, Rx N   0)
N 
with:
x1  x0 
L cos( )
N
and
PRxi   0  P0 .
(1)
-2-
FIGURE 1
Geometry of the link
2.1
Expression of the probability of rain attenuation from the spatial correlation of
rain rate fields
The joint probability PR( x0 )  0, Rx1   0,, RxN   0 can be expressed from the correlation
between the random variables R(x1), R(x2),…, R(xN). [Jeannin et al., 2011] showed that the rainfall
rate field R(x) can be modelled from locally stationary Gaussian random fields G(x) whose
correlation cG(x) is:
cG ( x)  0.59 exp(  x / 31)  0.41exp(  x / 800)
The rainfall rate field is then obtained by:
(2)
R(x)=f(G(x))
where f is a function that depends on the local cumulative distribution function of rainfall rate.
Consequently,
PR( x0 )  0, Rx1   0,, Rx N   0  PG( x0 )   , Gx1    ,, Gx N    
(3)
with: α = Q−1 (P0 )
(4)
where: Q(x) =
2
∞ −t
e 2 dt
∫
√2π x
1
(5)
So: PR( x0 )  0, Rx1   0,, RxN   0  PG( x0 )   , Gx1    ,, GxN    
(6)
and therefore:
P AR  0  lim (1 
N 
2.2
 




  PG1 , G2 ,GN dG1dG2 dGN )
(7)
Resolution of the integral giving the probability of rain attenuation
The analytical solution of integral (7) is not known and the numerical computation could be
inaccurate and even time consuming, particularly for high values of the correlation. Instead, it is
proposed to use the Markov hypothesis on the rain/non-rain states. Therefore, a 2-states Markov
chain Ui can be defined as illustrated in Figure 2 and so:
PR( x0 )  0, Rx1   0,, R x N   0  P(U i  0) PU 1  0 U i  0 PU N  0 U N 1  0
 (1  p0 )1  
N
(8)
-3-
where  is the transition probability from the state of no-rain attenuation to the state of rain
attenuation and  is the transition probability from the state of rain attenuation to the state of norain attenuation.
FIGURE 2
2-states Markov chain (state 0: no rain, state 1: rain)
P(Ui=1) = P0 which imposes the following constraint:
1  p0   p0 1     p0
and so:
    0 
 1  p0 
 p
then: PR( x0 )  0, R x1   0,, Rx N   0  (1  p0 )1  
N



p
 (1  p0 )1  0  
 1  p0 
(9)
(10)
N
(11)
In order to relate the Gaussian correlation of the rain field and the transition probabilities of the
Markov chain, the covariance of both processes will be assumed to be equal.
On the one hand, the covariance cB of the random process G(xi)> α is:
cB  ,    Gx0    Gxi    
 g 2  g 22  2cG xi  x0 g1 g 2 
 dg1dg 2
exp   1
2

2
2 1  cG xi  x0 
  2 1  cG  xi  x0 



1
 
where
c𝐵 (α, ρ) =

x2 −2ρxy+y2
∞ ∞ −
∫ ∫ e 2(1−ρ2)
2π√1−ρ2 α α
1

(12)
dxdy
is the bivariate normal integral
(13)
On the other hand, the covariance of the random process U is given by U 0U i . So the product
U0U1 is not equal to zero only if U0 and U1 are simultaneously equal to 1, which occurs with a
probability of:
U 0U i  PU 0  1 PU i  1U 0  1
(14)
The probability PU i  1U 0  1 can be obtained thanks to the transition matrix of the Markov
chain:
-4-
1  
T  
 
 

1   
(15)
PU i  1U 0  1
(16)
and so:
0 1T i  PUi  0U 0  1
as:
1
T 
 
then:
PU i  1U 0  1 
and,


1
U 0U i  p  p0 1  p0 1 
 
 1  P0 
i
    1    i

 
 
  

 



 
 

 
(17)
1    i 
(18)
 
i
2
0
(19)
Finally, the equality of the covariances of the processes Ui in (18) and G(xi)>α in (11) leads to:
i



1
L cos 

p  p0 1  p0 1 
   c B   , i
N 

 1  P0 
2
0
and so:
1/ N
  

  c   , i L cos   p 2  
0
  B 
N 
 
  1  
 1  p0 
p0 1  p0 
 
 
 
 
 
 
(20)
(21)
Eventually from (11), the probability to have rain attenuation on the link becomes:
N


 
p0
P AR  0   lim 1  (1  p0 )1 
  
N 

 1  p0  



1

  
 lim 1  1  p0 exp  N log 1  p0 1  exp log c B   
N 
N

  



 c  , L cos   p02 

 1  1  p0    B
p0 1  p0 


3
(22)
p0
Conclusion
Figure 3 illustrates the probability to have rain attenuation on the link in function of the probability
of rain and the ground projection of the link.
-5-
FIGURE 3
Probability to have rain attenuation on the link in function of the probability of rain and
the ground projection of the link
This contribution has presented a model of the probability to have rain attenuation of an Earth-space
link. This type of prediction method can be useful to improve the rain attenuation prediction method
given in Recommendation ITU-R P.618-11 and can be used to feed time series synthesisers such as
the one given in Recommendation ITU-R P.1853-1.
4
References
[Jeannin et al., 2012] Jeannin N., Féral L., Sauvageot H., Castanet L., F. Lacoste : "A large scale
space-time stochastic simulation tool of rain attenuation for the design and optimization of adaptive
satellite communication systems operating between 10 and 50 GHz", International Journal of
Antennas and Propagation, Volume 2012 (2012), Article ID 749829, 16 pages,
doi:10.1155/2012/749829.