A preliminary weather model is presented for optical propagation through the atmosphere. It can be used to compute the attenuation loss due to the atmosphere for desired link availability statistics. The quantitative results that can be obtained from this model provide good estimates for the atmospheric link budget necessary for the design of an optical communication system. The result is extended to provide for the computation of joint attenuation probability for n sites with uncorrelated weather patterns

Shaik, K. S.

English

NASA Technical Reports Server

N89-20350 
TDA Progress Report 42-95 July-September 1988 
A Preliminary Weather Model for Optical Communications 
Through the Atmosphere 
K. S. Shaik 
Communications Systems Research Section 
A preliminary weather model is presented for optical propagation through the atmo- 
sphere. It can be used to compute the attenuation loss due to the atmosphere for desired 
link availability statistics. The quantitative results that can be obtained from this model 
provide good estimates for the atmospheric link budget necessary for the design o f  an 
optical communication system. The result is extended to provide for the computation of 
joint attenuation probability for n sites with uncorrelated weather patterns. 
1. Introduction 
During the past few decades, the pioneering work of 
Tatarski [ l ] ,  Fried [2,3], Hufnagel and Stanley [4] ,  and 
Ishimaru [5 ] ,  has set the stage for the development of a con- 
sistent theory to evaluate the performance of optical commu- 
nications and imaging through the atmosphere. However, there 
is a conspicuous lack of weather models that may be used to 
obtain even a first-order estimate of statistics for the avail- 
ability of an optical communications link under ambient 
weather conditions. 
The light energy of a beam dissipates as it travels through 
the atmosphere, due to scattering and absorption. If it can be 
assumed that (1) attenuation loss is independent of radiation 
intensity and (2) the absorbing and scattering events occur 
independently of each other, the atmospheric attenuation can 
be expressed by the Bouguer-Lambert law: 
where Z ( v )  is the observed irradiance at optical frequency v ,  
& ( v )  is the irradiance that would have been observed if the 
beam were propagating in a vacuum at a distance Z from the 
source, and y(v ,  z )  is the total extinction coefficient due to 
scattering and absorption by the atmospheric constituents at 
position z .  The magnitude of the argument of the exponential 
in the above equation is known as the optical depth or optical 
thickness, 7, of the medium, i.e., 
Experiments [6] with artificial fog and smoke and diluted 
solutions of milk show that the Bouguer-Lambert law holds 
well for optical thickness 7 < 12. The optical depth can be 
simply related to the attenuation loss, L ,  in dB by the fol- 
lowing approximation: 
L e 4 . 3 4 7  (3) 
21 2 
https://ntrs.nasa.gov/search.jsp?R=19890010979 2020-03-20T03:52:58+00:00Z
The distribution of gas molecules, clouds, fog, haze, aero- 
sols, and other particulates in the transmission path influence 
the computation of attenuation loss. These phenomena at best 
can be interpreted on a statistical basis. The system designer 
may have to  be content with a probability for which the atten- 
uation of the atmosphere due to  ambient weather is less than 
some critical value L dB. Hence we define the probability, 
wl(L) ,  of the random weather variable, W, for a single site as 
w,(L) = P,(f < L )  (4) 
rapidly with increasing concentration of aerosols, fog, 
haze, and clouds. Optical attenuations L > 1000 dB are 
possible, where the validity of the Bouguer-Lambert 
law is questionable. Table 1 shows typical ranges of 
attenuation values for various types of clouds [8].  
Since it is unlikely that an optical communication 
system will be designed for L > 30 dB, the Bouguer- 
Lambert law serves as a good approximation. As shown 
in Fig. 2 ,  the probability of opaque cloud cover over 
the southwestern U.S. is 20% [ 7 ] .  Let us take a pessi- 
mistic view and assume that attenuation loss due to all 
opaque clouds is 100 dB or higher, and use this assump- 
tion to estimate one of the model parameters. 
(4) It is further assumed that the attenuation of the beam 
has an exponential distribution for L 2 Lo .  The hypo- 
thesis is supported by visual observations, and also by 
the experimental data collected at Goldstone in Cali- 
fornia for atmospheric propagation at 8 to 10 Ghz.’ 
where I is the loss variable. w,(L) is the fraction of weather 
conditions at any given site for which the attenuated direct 
beam irradiance Z(v) 2 I o ( v )  exp[-0.23 L ]  . The weather 
model developed below can be used to  compute this proba- 
bility. The result is also extended to obtain the joint weather 
probability, w,(L), that at least one of the sites has extinction 
loss 1 < L for n independent sites receiving simultaneously. 
With the foregoing assumptions in mind, it is now possible 
to postulate a weather model. The weather cumulative distri- 
bution function (CDF) for a single site can be modeled as 
II. Weather Model 
The following observations are made in order to arrive at 
the preliminarv weather model: 
It is assumed that for some fraction of the time p ,  
where 0 < p < 1,  clear weather conditions hold. This 
number can be determined approximately from exist- 
ing data on cloud cover and visibility for potential 
sites. An analysis of two years of GOES satellite data 
by Wylie and Menzel [7] shows that the probability 
of having clear weather in the southwestern U.S. is 
over 60% (Fig. 1). 
The attenuation under clear weather conditions is due 
to scattering and absorption by air molecules and 
sparse particulate matter in the upper atmosphere..The 
attenuation due to molecular absorption is approxi- 
mately 0.5 dB [ 6 ] .  The attenuation due to molecular 
scattering is of the same order. Given that the atmo- 
sphere is never totally free of aerosols and thin cirrus 
clouds, on an average clear day the attenuation loss 
would be in the range of 1 to 3 dB. For simplicity, 
let us assume that the minimum attenuation loss due 
to the atmosphere is 3 dB. Hence, the model defines 
clear-air atmosphere, which occurs with probability 
p ,  as having an attenuation loss Lo  = 3 dB. Note that 
this attenuation loss refers to  near-zenith propagation 
through the entire atmosphere. For any zenith angle 8 ,  
the path attenuation can be written as Lo  sec 8. How- 
ever, the remainder of the discussion assumes near- 
zenith propagation paths, Le., 8 = 0. 
The probability of  not having clear weather i sq  = 1 - p .  
The attenuation loss of the atmosphere increases 
w l ( L )  = 1 - q  exp [-0.23 b ( L  - L o ) ]  (L > L o )  
where w l ( L )  is defined in Eq. (3) above, q is the probability 
fraction when the weather is not clear, b is a sitedependent 
parameter to model the slope of the CDF curve, and Lo is 
the minimum attenuation loss due to the atmosphere. Note 
that w,(L = L o )  = p .  The value of b may vary with geo- 
graphical location and altitude, and can be inferred from 
observed visibility and extinction loss data for potential 
receiving sites. For the southwestern U.S. region, q = 0.4, 
and from the third item above, the attenuation loss is 100 dB 
when w,(L = 100) = 0.8. Using these values in Eq. (9, we find 
that b = 0.03. This estimate of the parameter b will be used in 
numerical examples later. 
Equation (5) can be recast in a more familiar form for an 
optical communication system designer to give the attenua- 
tion loss in terms of the weather probability wl(L), i.e., 
(LO, for w1 ( L )  < p 
L =  
[ L o  + 2 0.23b In[-’] 1 - w , ( L )  ’ for w1 ( L )  > p 
JPL Internal Document 810-5, Rev. D, 1988. 
21 3 
Figure 3 is a plot of attenuation loss as a function of the 
weather probability for the southwestern region of the U.S., 
for which 4 = 0.4, b = 0.03, and Lo = 3 dB. These estimates, 
needless to say, are quite important for the system designer to 
determine the link budget for loss due to the atmosphere for 
Earth-space optical communication paths. 
It is also possible to extend the result to n sites receiving 
simultaneously. From Eq. (4), it is easy to see that the comple- 
ment of the probability wl(L) is given by 
Pw(l > L )  = wf(L) = 4 exp [-0.23 b ( L  - L o ) ]  
The joint probability of n sites, wi(L), that the attenuation 
loss Zi > L for all i can be written as I 
I where the subscripts 1 , 2 ,  . . . , n label the receiving sites. When 
the weather conditions for all sites are independent and iden- 
tically distributed (IID), we have 
w,c(L) = P ( I l  > L )  P (Iz > L )  . . . PW,(Z, > L )  W1 w2 
= [PW(Z > L ) ] "  
Using Eq. (9), the joint probability that at least one site has 
attenuation 1 < L is found to  be 
= 1 -w,c(L = 1 - [ 4  exp [-0.23 b ( L  - L o ) ]  ] 
For a single site with p = 0.6, Lo = 3 dB, and b = 0.03, the 
probability that the attenuation L < 3 dB is w,(L = 3) = 0.60. 
If three such IID sites are chosen, we have w,(L = 3) = 0.94. 
In other words, if a system is designed to absorb an extinction 
loss of 3 dB, a three-site receiving network will be functional 
94% of the time. Figure 4 plots the fraction of the total period 
under ambient weather conditions when the attenuation is 
< 3 dB as a function of the number of sites. Table 2 gives the 
expected dB loss for a desired link availability percentage for 
up to four joint receiving sites. It is, however, not very clear 
how the independence of weather patterns at various sites can 
be insured. It is known that the scale size of weather patterns 
is on the order of a few hundred kilometers, and this measure 
may be used to find sites with uncorrelated weather. Joint ob- 
servation of weather parameters for the probable sites will be 
necessary to make a more accurate determination. 
111. Conclusion 
The virtue of the weather model presented here lies in its 
simplicity. It may be applied with ease to obtain a first-order 
magnitude of probabilities wn(L) for an optical system that 
must operate in the atmosphere. The computation of these 
probabilities, for example, will provide a good statistical esti- 
mate of the attenuation loss to an optical communication 
system designer for link availability. 
The model does not consider frequency dependence, since 
it has been studied thoroughly and well documented in LOW- 
TRAN computer code developed at the Air Force Geophysics 
Laboratory [9] . The model also disregards seasonal variations, 
w h c h  can be incorporated later when adequate data bases 
have been developed. 
Acknowledgment 
The author wishes to thank James K.  Lesh for his comments and discussions, and also 
Dr. Chien-Chu Chen for his helpful suggestions. 
21 4 
References 
[ l ]  V. I. Tatarskii, WavePropagation in a TurbulentMedium, New York: Dover, 1967. 
[2] D. L. Fried, “Propagation of a Spherical Wave in a Turbulent Medium,”J. Opt. SOC. 
Am., vol. 57,pp.  175-180, 1967. 
[3] D. L. Fried, “Limiting Resolution Looking Down Through the Atmosphere,”J. Opt. 
SOC. Am.,  vol. 56, pp. 1380-1384, 1966. 
[4] R. E. Hufnagel and N. R. Stanley, “Modulation Transfer Function With Image Trans- 
mission Through Turbulent Media,”J. Opt. SOC. Am.,  vol. 54, pp. 52-61, 1964. 
[5] A. Ishimaru, Wave Propagation and Scattering in Random Media, New York: Aca- 
demic Press, 1977. 
[6] V. E. Zuev, Laser Beams in the Atmosphere, New York: Consultants Bureau, 1982. 
[7] D. P. Wylie and W. P. Menzel, “Cloud Cover Statistics Using VAS,” SPIES OE- 
LASE88 Symposium on Innovative Science and Technology, Los Angeles, CA, 
January 10-15,1988. 
held in Australia, U.S. Government Printing Office no. 559-065/20988, 1985. 
[8] The Technical Cooperation Program, Volume V- Laser Communications Workshop 
[9] F. X. Kneizys, et al., “Atmospheric Transmittance/Radiance: Computer Code LOW- 
TRAN6,” Rep. AFGL-TR-83-0187,1983. 
21 5 
Table 1. Typical range of values for cloud attenuation 
(adapted from [ E ] )  
Total 
vertical 
attenuation, 
dB 
Cloud base Cloud 
thickness, height, Cloud 
type km km 
Stratus 
Stratocumulus 
Nimbostratus 
Altostratus/ 
cumulus 
Cumulus 
Cumulonimbus 
Cirriform (ice) 
Fog 
0.1 - 0.7 
0.6 - 1.5 
0.1 - 1.0 
2 - 6  0.2 - 2 
0.2 - 0.8 
0.2 - 0.8 
2 - 3 
0.5 - 1.0 
0.5 - 1.0 
6 - 10 
0 
0.5 - 5 
2 - 12 
1.0 - 2.5 
0 - 0.15 
26 - 350 
3 - 100 
260 - 1300 
9 - 260 
22 - 870 
260 - 5200 
1 - 15 
0 -  13 
Table 2. Attenuation loss as a function of desired link 
availability for n sites receiving jointly 
Percentage Attenuation loss, L ,  dB 
n =  1 n = 2  n = 3  n = 4  weather 
60.0 
62.0 
64.0 
66.0 
68.0 
70.0 
72.0 
74.0 
76.0 
78.0 
80.0 
82.0 
84.0 
86.0 
88.0 
90.0 
92.0 
94.0 
96.0 
98.0 
99.0 
3.00 
10.34 
18.07 
26,25 
34.92 
44.16 
54.03 
64.63 
76.08 
88.53 
102.16 
117.23 
134.09 
153.19 
175.24 
201.32 
233.25 
274.40 
332.41 
431.57 
530.72 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
12.55 
23.58 
36.62 
52.58 
73.16 
102.16 
15 1.74 
201.32 
3.00 
3.00 
3.00 
3.00 
3.00 
3 .OO 
3.00 
3.00 
3.00 
3.00 
3.00 
3 .OO 
3.00 
3 .OO 
3.00 
3.00 
3.00 
6.08 
25.4 1 
58.47 
91.52 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
3.00 
11.83 
36.62 
21 6 
PROBABILITY OF CLEAR SKY, PERCENT 
Fig. 1. Contour diagram obtained from 2 years of GOES satellitedata showing the probability 
of clear sky over the United States [A. 
PROBABILITY OF OPAQUE CLOUD COVER, PERCENT 
Fig. 2. Contour diagram obtained from 2 years of GOES satellite data showing the probability of 
opaque cloud cover over the United States (71. 
217 
FRACTION OF TOTAL PERIOD WHEN ATTENUATION 5 L 
Fig. 3. Model curve for the path attenuation as a function of 
weather probability for a single site. Choice of parameters, Lo = 3 
dB, b = 0.03, and p = 0.6 represents estimates for the southwestern 
us. 
5 
4 
C 
vi 
w 3  
k 
v) 
Y 
LT 
W 
0 
I " 2  
3 z 
1 
C I I I 
.6 0.7 9.8 0 3  
FRACTION OF TOTAL PEHIOD WHEl'i ATTENUATION 5 3 dB 
Fig. 4. Weather probability as a function of the number of joint 
receiving sites. L, = 3 dB, b = 0.03, p = 0.6, and L = 3 dB. 
218 


A preliminary weather model for optical communications through the atmosphere

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