An analytical and numerical comparison between two methods for the prediction of distributions of attenuations due to rain on terrestrial paths

Radio Science, Volume 18, Number 6, Pages 1301-1311, November-December 1983

An analytical and numerical comparison between two methods for the prediction of distributions of attenuations

due to rain on terrestrial paths

Emanoe 1 Costa

Centro de Estudos em Telecomunicag•es da Pontiffcia Universidade Cat61ica do Rio de Janeiro

Manoel G. S. Dias

Departamento de Engenharia El•trica da Universidade Federal do Par•

(Received April 13, 1983; accepted June 13, 1983.)

It is shown that the integral proposed by Misme and Fimbel for the prediction of the distribution of attenuations due to rain on a terrestrial path from the knowledge of the rain rate distri- bution in its neighborhood can, under some simplifying assump- tions, be analytically transformed. Although the final expression is relatively involved, it is shown that its most important term agrees with the expression proposed by Assis and Einloft to per- form the same prediction. This indicates that the method proposed by the latter authors is an approximate solution for the Misme and Fimbel integral. Rain attenuation values are also calculated by each prediction method for the same percentage of time, as- suming a large number of combinations of input parameters, in or- der to display the relative difference between the results yielded by them. The obtained results show that the simplifica- tion introduced by Assis and Einloft on the Misme-Fimbel method has no bearing on the agreement of its prediction with those ob- tained by other techniques.

INTRODUCTION [1982] that such a method, besides pro- viding good agreement with experimental

A strong effort has been made worldwide results obtained in different regions, for several years toward the understanding should be easy to apply and should not be of the effects of rainfall in the propa- gation of radio waves with frequencies a- bove 10 GHz. One of the goals of this ef- fort is the establishment of a suitable

method for the prediction of distributions of attenuations due to rain on both ter-

restrial and slant paths. It has been recommended by the CCIR

too critically dependent on the technique used for obtaining rainfall intensity da- ta. It is also desirable that the method

have a physical significance. Several of the proposed methods for the

prediction of rain attenuation distri- butions on terrestrial paths have been compared with experimental data and among themselves, in an attempt to characterize

Copyright 1983 by the American Geophysical those which best fulfill the conditions Union. recommended by the CCIR. One of the most

extensiv. e comparative tests was that in- Paper number 3S1029. volving the methods proposed by Misme and 0048-6604/83/003S-1029508.00 Fimbel [1975], Lin [1975], and Morita and

1301

1302 COSTA AND DIAS: ATTENUATION ON TERRESTRIAL PATHS

Higuti [1976], carried out over Europe [Fedi, 1979]. From the results obtained, the Misme-Fimbel method has been shown to

give good agreement with the experimental data. This method has also been recently extended to predict the attenuation dis- tribution in earth-space links [Misme and Waldteufel, 1980; Waldteufel and Misme, 1981]. However, it has been pointed out [CCIR, 1982] that the Misme-Fimbel method is sensitive to variations in the standard

deviation of the lognormal model chosen to represent the rainfall rate distribution, which appears to be critically dependent on the integration time used to collect rain intensity data [Damosso et al., 1980]. It will also be seen that the

[1979]. Results obtained in a tropical re- gion are also presented. The results clearly indicate that the profound simpli- fication introduced by Assis and Einloft on the Misme-Fimbel method has little in-

fluence on the agreement of its pre- dictions with either experimental data or those obtained by the other technique.

REVIEW OF THE TWO PREDICTION

METHODS

Although both methods will be briefly reviewed here, only those aspects which are essential to the understanding of fu- ture discussions will be presented. Fur- ther information on the methods can be

Misme-Fimbel method requires an extensive easily obtained in the papers referenced set of calculations in order to be applied in this section. to a specific radio link.

A very simple method for the prediction of distributions of attenuations due to

rain on terrestrial paths has been pro- posed by Assis and Einloft [1977]. Al- though these authors recognized that their method was based upon and was developed as a simplification of that by Misme and Fimbel, they made no attempt either to

The Misme-Fimbel method

The Misme-Fimbel method [Misme and Fimbel, 1975; Misme, 1978] makes use of geometric and probabilistic considera- tions, together with the following model for the precipitation system'

show how the Misme-Fimbel integral could Spatial structure of precipitation. Pre- be simplified to yield their expression or cipitation is supposed to consist of cy- to characterize the errors involved in lindrical cells of uniform rainfall rate R this approximation.

It will be show in this paper that the Misme-Fimbel integral can, under mild sim- plifying assumptions, be analytically transformed. Although the final expression is relatively involved, it is observed that its leading term agrees exactly with the expression proposed by Assis and Einloft [1977] to perform the same pre- diction. That is, the expression proposed by these authors is, as they have sug- gested, an approximate solution for the Misme-Fimbe 1 integral.

In order to display the relative differ- ence between the results yielded by each

(in millimeters per hour) and diameters given by the law

D(R) = 2.2(100/R) ø'4 km (1)

imbedded in a more extensive area

exhibiting a residual rainfall rate Rre s defined by the expression [Misme, 1978]

R = 1011-exp(0.0105R) ] mm/h res

which is valid for an area having a maxi- mum linear dimension of 33 km.

Relationship between specific attenua- tion and rainfall rate. The relationship

prediction method, rain attenuation values between specific attenuation ¾ (in deci- have been calculated for the same percent- bels per kilometer) and rainfall rate R age of time by each of them, assuming a (in millimeters per hour) is represented large number of combinations of input pa- by the power law ¾=k R s where the parame- rameters (frequencies, path lengths, ters k and • depend only on the frequency polarizations, and rain rate distri- and the polarization of the link [Misme butions). The assumed ranges for the input and Benoit-Guyot, 1974; Olsen et al., parameters cover those studied by Fedi 1978; CCIR, 1982].

COSTA AND DIAS' ATTENUATION ON TERRESTRIAL PATHS 1303

Point rainfall rate cumulative distri-

bution. Although not essential, it is fur- ther assumed that the (1-min average) con- ditional point rainfall rate cumulative distribution (assuming the occurrence of rain) follows a lognormal law. That is, the associated unconditional probability density function for the point rainfall rate R is

P [ In ] • p(R) = o exp{-

2•oR 2o 2 (3)

where Po is the probability of occurrence of rain (percentage of time rainfall rate exceeds 0.2 mm/h), and R and o are the pa- Fig. 1. Shape of the area S(R,A o) and rameters of the lognormal distribution. geometry for the evaluation of the attenu- These parameters are usually estimated ation distributions P(A>A ) and P'(A>A o) O from experimental distributions.

From the previous assumptions, the Misme-Fimbel method predicts that the per- The Assis-Einloft method centage of time a given attenuation A o (in decibels) in exceeded in a link of known distance d (in kilometers), frequency (in gigahertz), and polarization is given by

P(A>A ) = S(R,A o) dR o

R . is exceeded during [(d/D)P'] percent of mln

the time, where P' is the percentage of

where S(R,A o) is the hatched area in Fig- time the point rainfall rate R is exceed- ure 1, defined by ed. The Assis-Einloft method thus assumes,

3 D2 2 1/2 for simplicity, that rain cells always ob- S(R,A o) = [d-• Lo(R)][ (R)-Lo(R)] struct a radio link along their diameters. A comparison between (6) and (8) shows

1 D • + • (R) arc cos [ Lø(R) that the value of R with produces the at- D(R) ] (5) tenuation A in the Assis-Einloft method o R . is equal to min' defined in (7) There- .

and Lo(R) is the section of the radio link fore, the expression proposed by Assis and obstructed by a cell of diameter D(R) in Einloft [1977] for the prediction of the such a way as to cause an attenuation A o. percentage of time a given attenuation A o That is, is exceeded in a link of known distance d,

frequency f, and polarization can be re- L o (R) [ (Ao/k) d' R • • = - Rres] + [ - Rres] writen in the form

(6)

P(A>A ) = [d/D(Rmin)]P(R>R . ) (9) o m•n where d' = min (d, 33 km). Since Lo(R) .< D(R), the lower limit

Rmi n is obtained from the solution of the equation

D(Rmi n) = L o(Rmin)

The Assis-Einloft method states that the

attenuation A o produced by a cell of di- ameter D(R) in a link of distance d, ap- proximately given by

(4) A = k ReD(R) + k R e [d' - D(R)] (8) o res

where F(K>F'.min ) is the percentage of time rainfall rate Rmi n defined in (8), is ex- ceeded. Assis and Einloft thus introduced

an extreme simplification of the method proposed by Misme and Fimbel [1975]. (7)

1304 COSTA AND DIAS' ATTENUATION ON TERRESTRIAL PATHS

APPROXIMATE EXPRESSION FOR

THE MISME-FIMBEL INTEGRAL •2 0.2

f(o,n;z) = z e

It is easy to see from Figure 1 that the

area S(R,A ) which appears in the Misme- nn=(2n+3)(•-0 4) z= R . /• Fimbel inteøgral is approximately equal to ' m•n the sum of the areas of a rectangle of sides 2x=[D 2 (R)-L 2 (R)]l/2 and y=[d-2Lo(R)] a =1 a =(2n-1)a b =2 with the areas oføtwo semicircles of di- o n n-1 o ameter D To obtain a lower bound (L B ) b =(2n+2)b ß ' ' n n-1 to the area S(R,A o), substract from the approximate area defined above 4 times Here erfc(x) is the complementary error the area of the rectangle ABCD shown in function [Gradshteyn and Ryzhik, 1965] and Figure 1. Similarly, an upper bound (U.B.) the other parameters and functions have can be obtained by subtracting from the been previously defined. Similarly, the same approximate area 4 times the area of maximum absolute error z in the approxi- the triangle ABC. Therefore, a better ap- mation P(A>Ao)= P'(A>Ao)Pis obtained by proximation for S(R,A o) is substituting expression (11) for S(R,Ao)

S(R,Ao)=S'(R A )= 1 (L B + U B ) •D 2 3 D _ x) = 2xy + 4 2 Lo(• (10)

and the absolute error occurring in this approximation is smaller than

erfc (lnz + no) (13)

into (4) and performing the integration a- long the lines sketched in the appendix. The final result is

(14) co a

Po [ ½ f{O,nn;T } Zp = 2--• n=0 n

L From the solution for (7) under the as-

1 • (•_ sumpt ion that Rre s 0 it is seen that both =¾ - = - x) = , Rmi n and d/D(Rmin) increase with A o. It should be further noted that P(R>Rmi n) and

Substituting expression (10) for S(R,A ) (Po/2)f(o,n;z) with values of n close to o

into (4), assuming further that Rres = 0, zero, are of the same order of magnitude and following the steps outlined in the appendix, the expression for P(A>A_) can

. O

be approximately written •n the form

P(A>A )= P'(A>A ) = d o o )

ma.n

2P

ß p(R>Rmin) _ o d • D(Rmin)

ß { • f(o,0;z)-f(o,-0.4;z) oo a

+ f[o, (n n n=0 n

2P

+ { f(o 0;z)-2f(o a-0 4;z)

5 co a + • I ½ f(O,nn;'C>}

n=0 n (12)

where

and that f(o,n;z) decreases as n in- creases. On the other hand, it is observed that the multiplicative factor d/D(Rmi n) (which increases with A o) is absent from the right-hand side of (14). This ex- pression also shows that the lowest-order function in the infinite series is

f(O,no;•) , where •o = 3(•-0.4). For fre- quencies in the range of interest (10 GHz .< f ,< 40 GHz), no is greater than 1.6, while the lowest-order terms in (12) are of order zero in n. Therefore, these arguments indicate that the absolute error zp is small and that P' (A>Ao) , given by ('12), is a good approximation for the Misme-Fimbe! integral P(A>Ao). This indi- cation has been confirmed by the numerical test to be presented in the next section.

Expression (12) is the sum of three terms. The first one is the Assis-Einloft

expression, (9), and the other two are written in the form of a factor multi-

COSTA AND DIAS: ATTENUATION ON TERRESTRIAL PATHS 1305

plying a summation of functions (inside the power margin needed to operate a link curly brackets). The alternating signs in within a previously specified reliability. the second term guarantee that its overall The two methods which are discussed in contribution to P'(A>A o) is smaller than this paper were numerically compared by that from the lowest-order function inside Assis and Einloft [1977] previously. They the curly brackets. Since, as has previ- ously been mentioned, this function (mul- tiplieU by Po/2) is of the order of P(R>Rmin) , the contribution from the Assis-Einloft term to (12) is larger than that from the second term. The contri-

bution from the third term is even

smaller, not only by the same arguments, but mainly because of the absence of the

tabulated, for several percentages of time, the corresponding measured and predicted (by each method) attenuations relative to four terrestrial lznks operat- ing in the 11- to 15-GHz frequency band in England and in France. Their tables clear- ly showed that the two methods provide es- sentially equivalent results, in the sense that the rms errors between the measured

multiplicative factor d/D(Rmin) (which in- attenuations and those predicted by each creases with tAo). Therefore, expression method for the same percentage of time are (p) proposed by Assis and Einloft [1977] approximately equal. Unfortunately, these is the most important term in the expan- data correspond to very limited ranges of sion for the Misme-Fimbel integral frequencies, distances, and rainfall rate presented in (12), whenever the effects of cumulative distributions. the residual precipitation can be In order to broaden the data base upon neglected. which the two methods are compared and to

The basic results in this section remain quantitatively estimate the errors in- valid even when the effects of Rre s are volved in the approximations presented in considered. To show this, the previous the previous section, a computer program derivation was repeated, approximating (2) has been developed to test (9), (12), by a two-section piecewise linear function (14), and their generalizations for the which saturates for intense precipitation. case i•n which R is considered, under

s

Because of space limitations, however, the any set of inpu•eparameters. This program generalized expressions will not be presented.

Although the arguments presented so far indicated the existence of a good agree- ment between the two methods, it also is of fundamental importance that the errors in the previous approximations be ade- quately estimated. This will be done in the next section through a numerical com- parison between the results provided by

has been run for all combinations of input

data in the form (fi,dj,Pg), where the frequency fi has assumed the values 11.5, 14, 18, 22, or 36.5 GHz, the path length dj has assumed the values 10, 15, 20, 25, or 30 km, and the parameters of the point rainfall rate cumulative distribution Pg have assumed the 10 different sets of values characterized in the last three

columns shown in Table 1. The point rain- the two methods and by direct measurement, fall rate cumulative distributions have assuming wide ranges of values for their (common) input parameters and considering the effects of R .

res

NUMERICAL COMPARISON BETWEEN

THE TWO METHODS

Instead of comparing percentages of time, this section will compare attenua-

been taken from the paper by Fedi [1979] and are representative of the precipita- tion systems which have been characterized in European countries. In all cases, the assumed values for the parameters k and • ha•e been those recommended by the CCIR [1982] for a horizontally polarized link.

The results of the numerical comparison have shown that the errors in the approxi-

tions predicted by each method or obtained mation of (4) by the Assis-Einloft expres- by direct measurement, for the same per- sion always decrease with the path length, centage of time. This approach is more whether the effects of Rre s are considered useful in engineering calculations, where or not. It has also been observed that one is usually interested in estimating when Rre s is neglected, the errors are

1306 COSTA AND DIAS' ATTENUATION ON TERRESTRIAL PATHS

TABLE 1. Parameters of the Links Under Test

Region Site

Radio Link Lognormal

d, km f, GHz Polarization P % • mm/h o o • •

1 a Mendlesham, 7.4 10.7 b United Kingdom 7.4 19.4 c 7.4 36.0

2 a Darmstadt, 20.0 12.4 b West Germany 20.0 15.0 c 20.0 29.0

d 20.0 39.0

3 Leidschendam, 12.4 35.5 Netherlands

4 Kjeller, Norway 21.3 18.0 5 Rome, Italy 25.0 11.0 6 a Fucino, Italy 9.5 11.0

b 9.5 17.8

7 a Stockholm, 15.0 17.9 b Sweden 15.0 36.0

8 Graz, Austria 15.1 11.3 9 Dij on, France 53.0 13.0 10 a Paris, France 12.0 13.0

b 58.0 11.7

3.5 0.226 1.583

3.0 0.423 1.461

3.0 0.353 1.575

2.93 0.399 1.438

3.12 1.157 1.583

4.8 1.55 1.223

3.5 0.321 1.563

3.17 0.505 1.466 5.0 0.316 1.691 5.0 0.200 1.768

Data reproduced from Fedi [1979]. H is horizontal. V is vertical.

larger at the extremes of the frequency the same percentage of time in each curve band under study (11.5 GHz and 36.5 GHz). is typically equal to 10% or less. Thus, The behavior of the errors with the fre- the agreement between the results yielded quency becomes more complex when Rre s is by the methods can be considered to be considered. However, it can be said that good, in the ranges of distances, frequen- they are always less than those obtained cies, and parameters (Po, •, o) previously in the absence of the residual precipita- defined. tion. It should be observed that in the test

Figure 2 shows the values predicted by just described, no attempt has been made both methods for the cumulative distribu- to compare experimental data with the tions of the attenuation due to rain in a results predicted by both methods. This is 10-km radio link located in each qf the 10 explained by the fact that, in general, regions characterized in Table 1 and oper- the input data (fi, dj, Pl) do not corre- ating at the frequencies of 11.5 GHz and spond to actually existing radio links. 36.5 GHz, in the more realistic situation However, a direct comparison between ex- in which the effects of Rre s are consid- perimental data and attenuation predicted ered. According to what has been said by each method has also been performed, above, the values presented in this figure based on the results of the European correspond to the largest possible differ- Cooperation and Coordination 25/4 project ences between the results of the two reported by Fedi [1979]. Eighteen links, models for each rain regime, over the path whose parameters (d, f, polarization, Po, length and frequency ranges under study. •o, and o) are characterized in each line It is clearly seen from Figure 2 that the of Table 1, have been selected for study, maximum relative difference between the and the results of the numerical compari- attenuations predicted by each method for son are summed up in Figure 3.

COSTA AND DIAS' ATTENUATION ON TERRESTRIAL PATHS 1307

i0-$ i0-•- i0-1 •0 0 10"3 i0-•- i0-1 I0 0

10-4 10-3 10-2 I0-1 10-4 10-3 10-2 I0-1

10 -3 I0 -'• I0- I 60 ....... ''1 ........ I ........

o $o

........ I ..... .......

I0 0 10-3 10-2 I0-1 I0 0 i ! ! , , i,,• ! w ! , •,,• , , ! ,,,,,,

.

.

- T

.

".'.•...

io-I io-4 io-• io-• o-I

10-3 10-2 I0-1 I0 0 10'3 10-2 I0-1 i ! i ! 1117

I0-1 10-4 10-3 i0-•:

-

.

.

-

.

ß

i i i i ill

I0-1

PERCENTAGE OF TIME ATTENUATION IS EXCEEDED PERCENTAGE OF TIME ATTENUATION I$ EXCEEDED

Fig. 2. Cumulative distributions of attenuations due to rain, calculated by the Misme-Fimbel (solid and dot-dashed curves) and the Assis-Einloft (dashed and dot-dash-dashed curves) methods, assuming d = 10 km, f = 11.5 GHz (dot-dashed and dot-dash-dashed curves), f = 36.5 GHz (solid and dashed curves) and the 10 point rainfall rate cumulative distributions presented in Table 1. The effects of the residual precipitation have been considered. Lower (upper) horizontal scale should be associated with the 11.5-GHz (36.5-GHz) curves.

The two methods have also been tested in to link, situated in the city of Rio de a Brazilian rainy tropical climate Janeiro (22.54øS, 41.15øW) and operating [Einloft and Silva Mello, 1982]. The ex- at 10.9 GHz with a horizontal polariza- perimental data have been collected for tion. The values of the parameters of the three years in the 8.6-km Sumar•-Livramen- lognormal approximation point rainfall

1308 COSTA AND DIAS: ATTENUATION ON TERRESTRIAL PATHS

• 4o•- -I 4o- "-12-. .'• - L•.• ,• -•. • . ß .-•-.. • • - .

10-4 I0-• . 10-2 I0-1 10-4 I0-• I0-• I0-1

• 40 •b 40 •, • ß ._ •._ %x• j

q '. ' '

k , • IO IO

io-5 io-2 io-I io-4 io-5 io-2 io-I

5o

A

m 40

z o $o

z

• •o

4

.

I I I i I I i I . I . . ,

0-4 10-;3 10-2

50

40

,so

Io

io-I io-4

x• IO1• \ 9

i . i i I i i i ! I

io-$ io-2

PERCENTAGE OF TIME ATTENUATION IS EXCEEDED PERCENTAGE OF TIME ATTENUATION IS EXCEEDED

Fig. 3. Cumulative distributions of attenuation due to rain, measured (circles), and calculated by the Misme-Fimbel (solid curves) and the Assis- Einloft (dashed curves) methods, assuming the parameters presented in Table 1. The effects of the residual precipitation have been considered. Experi- mental data reproduced from Fedi [1979].

O-I

rate cumulative distribution simulta- FINAL DISCUSSION

neously measured in the neighborhood of the link are Po=5%, •=3.356 mm/h, and The expressions and arguments presented o=1.164. The agreement between measured in the third section support the claim and calculated results, shown in Figure 4, that the Assis-Einloft equation for the is excellent. prediction of the cumulative distribution

Again, it is clearly seen from the last of attenuations due to rain on a radio figures that the methods yield essentially link from the knowledge of its length, equivalent results (now over a wider data frequency, and polarization, as well as base), in the sense that the rms errors the rainfall rate cumulative distribution between the measured attenuations and in its •neighborhood, is a simplified and those predicted by each method for the approximate solution of the Misme-Fimbel same percentage of time are approximately integral. equal. Results of a numerical comparison be-

COSTA AND DIAS' ATTENUATION ON TERRESTRIAL PATHS 1309

2O

18

16

14

RAINFALL RATE' ._

'1" MEASURED FITTED

--

ATTENUATION: ß MEASURED ._

CALCULATED (MF) x CALCULATED (AE) ._

IO -$ 2. 5 I0 -2 2. 5 I0 -I 2.

240

200 z:

160 '•

120 •

o

5 I0 ̧

PERCENTAGE OF TIME ATTENUATION IS EXCEEDED

Fig. 4. Cumulative distributions of rainfall rate and attenuation due to rain in the Sumar•-Livramento link for the period July 1977 to July 1980. Assis-Einloft attenuation values (crosses) have been calculated directly from the measured rainfall rate distribution (pluses). Misme-Fimbel attenu- tion values (solid curve) have been calculated from the lognormal curve (dashed curve) fitted to the measured rainfall rate. Experimental data re- produced from Einloft and Silva Mello [1982].

tween the two methods point out that the remembered that point rainfall rate cumu- errors occurring in this approximation are lative distributions, except in some re- equal to 10% or less over a wide range of gions of the world, are not currently de- input parameters (10 km<d<30 km, fined with an uncertainty less than these 11.5 GHz<f<36.5 GHz, and point rainfall errors. This is particularly true for rate cumulative distribution typically small percentages of time. measured in European countries). Further, Another important advantage of the comparison between experimental data from Assis-Einloft method is that actually mea- 19 links and values predicted by the two methods [Assis and Ein!oft, 1977], al- though more limited in the range 'of the input parameters than the previous test, suggests that the methods are equivalent (in the sense that their rms errors with respect to experimental data are approxi- mately equal).

sured rainfall rate cumulative distribu-

tions can be directly used in attenuation predictions, without either further numer- ical processing or any a priori assumption regarding its form. This is observed by a comparison between (4) and (9).

The Assis-Einloft method also is consis-

tent with the path length scaling method The errors occurring either between the for rain attenuation statistics recomm-

two methods or between experimental data mended by the CCIR [1982]. Expressions and the values predicted by the Assis- (1), (6), and (7) show that Rmi n is inde- Einloft method are certainly acceptable pendent of the value of the path length d for common application. It should also be when the effects of Rre s are neglected

1310 COSTA AND DIAS: ATTENUATION ON TERRESTRIAL PATHS

and, in general, slightly dependent on d when these effects are considered. Assum-

ing that the other input parameters remain constant, it is easily derived from (9) (exactly in the first case and approxi- mately in the second) that the probability P2(A>A ) that the attenuation A o is ex- ceededøon a path of length d 2 is related to the probability PI(A>Ao) that the same attenuation A o is exceeded on a path of length d 1 by the expression

d 2 P2 (A>Ao) = •11 P1 (A>Ao)

R-ndR exp {- [In(R/•)]2 } 20

ico 2 -x - •o•x e

A

dx (A2)

where A=[ln(o/•)]/(/•o) and n is an index which depends on • and n. This expression

(15) can be integrated with the help of expres- sion 3. 322.1 of Gradshteyn and Ryzhik

This expression has been suggested by the [1965], and the final result can be writ- CCIR [1982] as a path length scaling meth- ten in terms of (13). od of rain attenuation statistics deter-

mined by the occurrence of relatively small cells of heavy rainfall (with the warning that it should be applied with caution for short paths) and employed in other attenuation prediction methods [Crane, 1980].

Therefore, all the previous arguments clearly indicate that the much simpler

Acknowledgements. This work has been sponsored by TELEBtLtS (Telecomunicac6es Brasileiras S.A.) under the contract PUC- TELEBtLIS 88/81. One of us (M.G.S.D.) was partially supported by CAPES (Coordenac•o do Aperfeicoamento de Pessoal do Ensino Superior) during the course of this study. The comments by the referees have been

Assis-Einloft method for the prediction of greatly appreciated. cumulative distributions of attenuations

due to rain on terrestrial paths should be REFERENCES favored over the complex Misme-Fimbel method, for the range of input parameters Assis, M. S., and C. M. Einloft, A simple tested in this paper. method for estimating rain attenuation

APPEND IX

Since 0.<L o (R).<D (R) within the inte- gration range in (4), the height 2x=[D • (R)-Lo 2 (R)]l/2 of the hatched area in Figure 1 can be expanded as follows'

2x= D[1-(Lo/D) • ]:•/2

n=0 n

n+l

] (A1)

where a n and b n have been previously de- fined.

When Rre s is neglected, the term (Lo/D) ß R o . 4 is proportional to -•. Thus, substi-

tuting (10), (11), and (A1) into (4), it is easily seen that P'(A>A o) and • can be written as a summation of integrals p of the form

distribution, Ann. Telecommun., 32, 478- 480, 1977.

CCIR, Attenuation by hydrometeors, in par- ticular precipitation, and other atmo- spheric particles, Rep. 721-1, Int. Telecommun. Union, Geneva, 1982.

Crane, R. K., Prediction of attenuation by rain, IEEE Trans. Commun., COM-28, 1717- 1733, 1980.

Damosso, E,, G. de Renzis, F. Fedi, and P. Migliorini, A systematic comparison of rain attenuation prediction methods for terrestrial paths, Ann. Telecommun., 35, 463-469, 1980.

Einloft, C. M., and L. A. R. Silva Mello, Rain attenuation distribution on terres-

trial radio links' Model and measure-

ments at 11 GHz, paper presented at Seminar on Radiowave Propagation, Empre- sa Brasil. de Telecom., Rio de Janeiro, 1982.

Fedi, F., Attenuation due to rain on a

COSTA AND DIAS' ATTENUATION ON TERRESTRIAL PATHS 1311

terrestrial path, Alta Freq., 48(4), Morita, K., and I. Higuti, Prediction 167-184, 1979. methods for rain attenuation distribu-

Gradshteyn, I. S., and I. M. Ryzhik, Table tions of micro and millimeter waves, of Integrals, Series and Products, Rev. Electr. Commun. Lab., 24(7-8), 651- Academic, New York, 1965. 668, 1976.

Lin, S. H., A method for calculating rain Olsen, R. L., D. V. Rogers, and D. B. attenuation distributions on microwave

paths, Bell Syst. Tech. J., 54, 1051- 1086, 1975.

Misme, P., Affaiblissements dues • la pluie. R•sultats exp•rimentaux, calculs et provisions, Ann. Telecommun., 33, 341-362, 1980.

Misme, P. and G. Benoit-Guyot, Simplifica- tion du calcul theorique de l'affaiblis- sement par la pluie, Ann. Telecommun., 29, 132-138, 1974.

Misme, P., and J. Fimbel, D•termination th•orique et experimentale de l'affai- blissement par la pluie sur un trajet radio•lectrique, Ann. Telecommun., 30, 149-158, 1975.

Hodge, The aRb relation in the calcula- tion of rain attenuation, IEEE Trans. Antennas Propag., AP-26, 318-329, 1978.

Waldteufel, P., and P. Misme, Ameliora- tions du module pour le calcul des at- t•nuations dues & la pluie sur un trajet terre-space et •tude de sensibilit• aux param•tres, Ann. Telecommun., 36, 421- 438, 1981.

Emanoel Costa, CETUC-PUC/RJ, Rua Marqu•s de S•o Vicente 225, 22453 Rio de Janeiro

Misme, P., and P. Waldteufel, A model for RJ, Brazil. attenuation by precipitation on a micro- Manoel G. S. Dias, Dept ø. Eng. El•trica- wave earth-space link, Radio Sci., 15, UFPa, N6cleo Pioneiro do Guam', 66000 655-665, 1980. -- Belem PA, Brazil.