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Chapter 15 Atmospheric Channel Effects
and the Impact on Optical Communications
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Table of Contents
Chapter 15 Atmospheric Channel Effects and the Impact on Optical
CommunicationsFormel-Kapitel 1 Abschnitt 1 ................................................... i
15. Atmospheric Channel Effects and the Impact on Optical
Communication ..................................................................................................... 4
15.1. Introduction .......................................................................................... 4
15.2.
Theoretic Description of Atmospheric Turbulence .............................. 7
15.3.
Numerical Simulation of Atmospheric Turbulence ............................ 23
15.3.1.
Introduction .............................................................................. 23
15.3.2.
Method ...................................................................................... 24
15.3.3.
Sampling ................................................................................... 27
15.4.
Aircraft Boundary Layer Effects ........................................................ 28
15.5.
Link Availability ................................................................................ 36
15.5.1.
Attenuation in Fogs................................................................... 36
15.5.1..
Physical Characteristics of Fog ............................................ 36
15.5.1..
Attenuation Due to Fog ........................................................ 37
15.5.1..
Experimental Results ........................................................... 38
15.5.2.
15.5.2..
Attenuation in Clouds and Downlink Availability.................... 45
Attenuation of Clouds .......................................................... 45
15.5.2.. .................................................................................................... 49
15.5.2..
Cloud Coverage Statistics .................................................... 49
15.5.2..
Link Availability Estimated from Cloud Coverage Statistics50
15.5.2..
Availability of Ground Station Diversity Systems ............... 52
15.5.2..
Conclusion ........................................................................... 54
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15. Atmospheric Channel Effects and the Impact on Optical Communication
M. Knapek, F. Fidler, M. Grabner, V. Kvičera, R. Mackey, F. Moll, F.
Stathopoulos
Abstract: Atmospheric index-of-refraction turbulence has a strong impact on optical communication links due the short wavelengths used in these applications.
This chapter presents in detail the atmospheric effects to be expected in HAP-toground links and in inter-HAP scenarios. Links to ground stations will suffer more
significantly from the atmosphere, as the atmospheric density is much higher than
in inter-HAP scenarios. Standard atmospheric models, theoretic formulas, and
simulation results are provided. The boundary layer around moving aerial vehicles
is investigated and their influence on laser communication discussed.
The influence of clouds and fogs on optical links is investigated and availability statistics are given.
15.1. Introduction
The communication channel refers to the medium used to transport information
from the transmitter to the receiver, which in our case is air (in the Earth's atmosphere) as well as vacuum (in space in the case of HAP-to-satellite communication). Each channel shares characteristics, which allow using a common channel
model on how the channel affects the transmitted signal. The optical communication channel through the atmosphere between HAPs or between HAP and satellite
can be modeled by attenuation of the transmitted signal where the random part is
known as fading. The attenuation term is a simplification of the underlying physical processes and captures the change in signal power over the course of the
transmission.
The Earth's atmosphere extends approximately 700 km above the surface and
consists of several distinct layers [1]. Pronounced density is found within the lowest 20 km [2], still influencing HAP-to-satellite or inter-HAP links. When a laser
beam propagates through a turbulent medium like the atmosphere, one observes
several disturbances [2]:
 the laser light is scattered or absorbed (atmospheric attenuation),
 the beam divergence is larger than in the diffraction-limited case (beam
spread),
 the beam is displaced (beam wander), and
 the phase front is distorted (phase fluctuations).
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These phenomena result in loss of power and (the latter two) in intensity fluctuations at the receiver (fading) and - in the worst case - may lead to a link failure.
Absorption and scattering: Absorption occurs when the optical field transfers
energy to the molecular constituents of the atmosphere. It exhibits a strong dependence on wavelength [2,3]. Atmospheric scattering due to molecular sized particles is called Rayleigh scattering. For objects large compared to the wavelength,
Mie scattering occurs. Rayleigh scattering is dominant for clear sky conditions and
- being proportional to λ-4 - for short wavelengths, while Mie scattering does not
depend on the wavelength that strongly [2, 3]. When transmitting an optical signal
on a vertical path through the atmosphere, some 1 to 2 dB of atmospheric loss
have to be expected for example for clear skies at zenith. Much stronger absorption with more than 10 dB have to be expected at low elevation angles
(<20 degrees).
Different weather conditions can also cause variations of the atmospheric loss
by several orders of magnitude. Usually all weather phenomena (and thus also
cloud coverage) happen inside the troposphere, which extends up to a height of
11 km. In reference [4] the maximum cirrus altitude is given with approximately
19 km. The influence of the atmosphere on an optical link from a HAP at 20 km
height to a satellite is therefore much smaller than it is for a link from ground station to satellite.
Fading: Variations of the received signal intensity due to interferometric effects and beam wander are usually called fading, and are caused by changes in the
characteristics of the propagation path with time or space: Turbulent motion of the
atmosphere in the presence of temperature and pressure gradients causes disturbances in the atmosphere's refractive index in the form of eddies, acting as random
optical lenses which refract the propagating light. One may distinguish between
two main effects causing fluctuations in received intensity (and thus fades):
1. Because of random deflections during propagation through turbulent atmosphere, the beam profile moves randomly off the line-of-sight (LOS) between
transmitter and receiver. The instantaneous center of the beam, i.e. the point of
maximum intensity, is randomly displaced in the receiver plane, which is
commonly called beam wander (cf. Fig. 15.1 (a)). Beam wander is caused
mainly by large-scale turbulence near the transmitter and therefore can typically be neglected for downlink scenarios [2].
2. The effect caused by small random index-of-refraction fluctuations is commonly described as scintillation (cf. Fig. 15.1 (b)). It leads to both the temporal variation in received intensity and the spatial variation within a receiver aperture.
Beam spread loss: Atmospheric turbulence causes beam spread (cf. Fig. 15.1
(c)) beyond the diffraction limited divergence θDL, leading to a larger effective divergence angle, θeff, which causes a degradation of the mean received optical power by a factor (θeff/θDL)2.
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Fig. 15.1 (a) Beam wander: Displacement of the center of the beam due to large-scale turbulence. (b) Scintillation: Intensity profile fluctuations due to interference effects within the
beam. (c) Beam spread: Intensity distribution of a propagating laser beam (solid
lines...beam spread due to diffraction, dashed lines...beam spread due to diffraction and turbulence).
Phase-front distortions: When a laser beam propagates through the atmosphere
its phase-front gets perturbed (cf. Fig. 15.2 (a)), which reduces the coupling efficiency into a single-mode fiber [5, 6]. Turbulent eddies, which are relatively small
compared to the beam diameter lead to noticeable phase-front distortions within
the receiving aperture. This is the case for example in a satellite-to-HAP downlink
(cf. Fig. 15.2 (b)). In the uplink case the turbulent eddies are right in front of the
transmitter and comparatively large relative to the optical beam. This leads to a
large beam spread causing the phase-front disturbance to be negligible within the
small receiving aperture at a satellite (cf. Fig. 15.2(c)).
Fig. 15.2 (a) Coupling geometry: A thin lens focuses the incident field E, which is assumed to
have a constant amplitude but a statistically disturbed phase, onto the bare end of a single- mode
fiber. (b) Downlink: Negligible beam spread, large phase-front disturbance over receiving aperture. (c) Uplink: Negligible phase-front disturbance, large beam spreading.
In order to assess the performance of an optical communication link, it is important to find quantitative expressions for all degradations caused by the atmos-
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phere. While some measured data and mathematical models are available in literature [2, 3, 7] for ground-to-satellite links or for horizontal links near ground, such
information is scarce or even non-existent for optical links from or to HAPs. In the
following sections we are going to discuss each degrading effect in detail. Then
we will present analytical, numerical, and empirical methods for the quantitative
estimation of losses and power fluctuations caused by the atmosphere including
several weather phenomena like clouds or fog.
15.2. Theoretic Description of Atmospheric Turbulence
Various phenomena associated with an optical wave propagating through statistically dependent random inhomogeneities of the atmosphere have been studied for
many years (e.g. [2, 8, 9, 10, 11, 12, 13]). Based on those analytical methods and
formulas, we develop methods to calculate parameters which help to describe the
atmospheric impact on laser beam propagation in communication scenarios from
or to HAPs. Table 15.1 shows a set of important input parameters together with
typical default values for optical communication scenarios.
Table 15.2 gives an overview of the output parameters which can be calculated
using the analytical models we will address within this section.
We assume a Gaussian beam shape together with the atmospheric turbulence
spectrums described by Andrews [2] as basis for the theoretical calculation of,
e.g., the scintillation, the fade statistics, beam wander, or the coupling loss into a
single-mode fiber. The lowest-order transverse electromagnetic Gaussian beam is
a solution of the paraxial Helmholtz equation
 2 E  k 2 E  0,
(15.1)
where E is the transversal field of the wave and k = 2π/λ is the wave number related to the wavelength λ. Under the assumption that the change in field distribution
is negligible with propagation distance z, the field of a Gaussian wave can be described as [2]


w
r2 
kr 2 
 exp  jkz  ( z ) 0 ,
E (r , z )  E0 exp   2  exp   j
2
R
(
z
)
W
( z)
W
(
z
)




(15.2)
where we can identify the amplitude

r2 
A0  E0 exp   2 ,
 W ( z) 
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the phase


 kr 2
 0    j 
 kz   ( z ) ,

  2 R( z )
(15.4)
and a normalization factor w0/W(z) which assures that the total power of the beam
along the propagation path z stays constant. The time factor exp (-jωt) of the field
is usually omitted in wave propagation studies [2]. In equation Error! Reference
source not found., r denotes the radial distance from the center line of the beam
(i.e. from the z-axis), w0 is the radius at which the field amplitude falls to 1/e of
that on the beam axis in the plane z = 0, W(z) is the beam radius in a distance z = L
and can be calculated according to [12]
2
 z 
 ,
W ( z )  w0 1  
 w0 
(15.5)
R(z) is the phase-front curvature
2
 w2 
R( z )  z 1   0  ,
 z 
(15.6)
and Φ(z) is the phase term
 z 
( z )  arctan  2 .
 w0 
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Table 15.1 Default input parameter set for optical communication scenario from/to HAPs
Parameter
HAP altitude
Symbol
hHAP
Default value
17 – 20 km
Satellite altitude
hSAT
35786 km
(LEO)
HAP telescope diameter
DHAP
< 30 cm
Satellite telescope diameter
DSAT
< 30 cm
HAP velocity
vHAP
0 – 11 km/min
Geodetic latitude
geo
-
(GEO),
Zenith angle
ζ
0 - 70°
Normalized pointing error
αn
-
Communication wavelength
λ
850 – 1550 nm
Fading threshold
FT
-
Surge threshold
ST
-
Structure constant at ground
2
1.7·10-14 m-2/3
Wind speed at ground
Cn (0)
vwind
3 m/s
Wind speed at tropopause
vT
30 m/s
Selection uplink/downlink
-
-
Selection
untracked/tip-tiltcorr./tracked beam
-
tracked
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400 km
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Table 15.2 Default output parameter set for optical communication scenario from/to HAPs.
Parameter
Fried parameter
Symbol
r0
Unit
[m]
Beam spread loss
abs
[dB]
Coupling loss into SMF
aco
[dB]
Power coupling efficiency into SMF
η
[-]
2
Scintillation index
σI
[-]
Probability of fade
P(F>FT)
[-]
Probability of surge
P(S>ST)
[-]
Expected number of fades per unit time
‹n(FT)›
[-]
Mean fade time
‹tf›
[s]
Angular pointing error
α
[rad]
Mean-square phase variations
σφ 2
[-]
Effective divergence angle
θeff
[rad]
The structure parameter, Cn2(h), represents the total amount of energy contained
in the stochastic field of the refractive index fluctuations [2]. It is a measure of
turbulence strength, required for the calculation of important fading related parameters like the scintillation index or the Fried parameter, and varies as a function of height h above ground. For the calculation of Cn2(h) the Hufnagel-Valley
model,
2
v   h 
 h 
 h 
 h 
16
2
Cn2 (h)  5.94 103 m 8/ 3 s 2  RMS   5  exp 
  2.7 10 exp 
  Cn (0) exp 

 1000m 
 1500m 
 100m 
 27   10 m 
10
(15.8)
is one of the most used models in the field [1], which requires the structure constant at ground, Cn2(0), as an input parameter. As shown in Fig. 15.3 (b), nearground levels may range from 10-17 m-2/3 (during night and weak turbulence conditions) to 10-13 m-2/3 (during day and strong turbulence conditions).
The rms wind speed, vRMS, is required as an input parameter for the HufnagelValley model. It is calculated by the Bufton wind model [2, 14, 15], which we
write as
vRMS

1

15 103

1/ 2
20103

5103
2


  h  h 2  
T

 vwind  vT exp   


dh




d
  T   


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.
(15.9)
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The quantity h is the height, vwind is the ground wind speed, vT is the wind speed at
the tropopause, hT is the height of the tropopause, and dT its thickness. Fig. 15.3 (a)
shows a typical wind speed profile vs. height, calculated using the Bufton wind
model, which is in good accordance with measured data given in [16, 17]. It reveals the relatively mild wind at HAP altitudes between 17 and 22 km, leading to
reduced turbulence, i.e. a small structure parameter (cf. Fig. 15.3 (b)).
When calculating temporal statistics (like the number of fades per second or the
mean fading time), the mean wind speed transverse to the optical beam is required:
 hh
T
vt (h)  vmov (h)  vwind  vT exp  
  dT



2

.

(15.10)
The height-dependent velocity term vmov(h) is caused by the HAP movement relative to the satellite which for example in the case of a GEO can be calculated as
v mov (h) 
 S  hSAT  rEarth 2  rEarth  h 2 sin 2    (rEarth  h) cos 

,
cos 
(15.11)
where rEarth is the Earth's radius and ωS is a height-independent angular velocity of
the laser beam derived from the HAP moving speed (cf. Fig. 15.4):
S 
v HAP cos 
.
L
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Fig. 15.3 (a) Wind-speed profile vs. height, calculated using the Bufton wind model. (b) Structure parameter vs. height, calculated using the Hufnagel-Valley model. (dashed-dotted...vwind =
0 m/s, Cn2(0) = 10-17 m-2/3, solid...vwind = 3 m/s, Cn2(0) = 1.7·10-14 m-2/3, dashed...vwind =
20 m/s, Cn2(0) = 10-13 m-2/3)
Fig. 15.4 HAP moving speed vHAP and angular velocity of laser beam ωs in a HAP-from/tosatellite (or HAP) communication scenario (L...link length, hHAP...HAP altitude,
hSAT...satellite (or second HAP) altitude, rEarth...Earth's radius, ζ = 90-γ...zenith angle,
vn...HAP speed component normal to LOS).
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The Fried parameter (or atmospheric coherence diameter), r0 in [m], is an important quantity used to describe the influence of atmospheric turbulence on a
propagating beam [10]. Two common physical interpretations are:
1. The Fried parameter corresponds to the diameter of an aperture over which
there is 1 rad of rms phase distortion [13].
2. It equals a diffraction limited aperture with diameter r0 which produces the
same divergence angle as atmospheric turbulence would add to the diffraction
limited divergence angle of a telescope with diameter D, resulting in an effective divergence angle θeff = [(λ/D)2+(λ/r0)2] 1/2 [18].
In accordance with [2, 8] one can calculate the Fried parameter as
hSAT


r0  0.423k 2 sec( )  Cn2 (h) dh


hHAP
3 / 5
(15.13)
,
using the optical wave number k in [rad/m], the zenith angle ζ in [rad], the HAP
altitude hHAP in [m], the satellite altitude hSAT in [m], and the structure parameter
Cn2(h) in [m-2/3] at height h. Fig. 15.5 (a) shows the decrease of the Fried parameter with increasing zenith angle, which corresponds to an increase in phase distortion over a certain aperture, i.e. an increase in turbulence. Very large values of r0,
which are of advantage with respect to fading, can be found at high platform altitudes (cf. Fig. 15.5 (c)). While at ground, e.g. at the optical ground station (OGS)
in Tenerife, the Fried parameter varies between 20 mm and 200 mm for strong and
weak turbulence [18], respectively, the Fried parameter for a HAP to satellite link
is larger than 2.5 m even for strong turbulence. Fig. 15.5 (b) illustrates the wavelength dependence of the Fried parameter, showing the advantage of a 1550 nm
communication wavelength compared to shorter wavelengths at 850 nm or
1064 nm. With increasing wavelength the beam divergence increases, leading to
less phase-front distortions (e.g. the amount of rms phase distortion in the uplink
decreases over a certain aperture) and therefore to an increased Fried parameter, to
less scintillation, fading, and beam spread loss.
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Fig. 15.5 Fried parameter r0 for a satellite-from/to-HAP link vs. (a) zenith angle ζ, (b) wavelength λ, and (c) platform altitude hHAP (calculated according to equation (15.13), dasheddotted...vwind = 0 m/s, Cn2(0) = 10-17 m-2/3, solid... vwind = 3 m/s, Cn2(0) = 1.7·10-14 m-2/3,
dashed... vwind = 20 m/s, Cn2(0) = 10-13 m-2/3)
A quantitative measure for the temporal effect of atmospheric turbulence is the
scintillation index, σI2, i.e. the variance of intensity fluctuations normalized to the
square of the mean intensity,
 I2 
I 2 
 1,
I  2
(15.14)
where ‹I› is the temporal mean intensity of the optical wave at the receiver [2].
The scintillation index is generally used to characterize the strength of turbulence
for an optical link. Such, σI2 ≤ 1 corresponds to weak fluctuations, whereas σI2 > 1
is refereed to as moderate-to-strong fluctuation regime. Simulations using the analytical formulas as given in [2] for vertical paths through the atmosphere show that
- contrary to satellite-ground links – the scintillation parameter is typically smaller
than 0.025 for HAP-to-satellite scenarios.
For horizontal links one distinguishes between the regime of weak and strong
turbulence using the criteria given in [2]:
 R2  1 and  R2 5/ 6  1 , weak fluctuation
 R2  1 and  R2 5/ 6  1 , strong fluctuation
conditions,
conditions,
where the Rytov variance σR2
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 R2  1.23C n2 k 7 / 6 L11 / 6
(15.16)
and the parameter Λ

2L
kW 2
(15.17)
depend on the structure parameter Cn², on the optical wave number k, on the path
length L, and on the diffraction limited 1/e² beam radius at the receiver W.
The probability of fade, P(F > FT ), describes the probability that the loss (or
fading depth), F, of the instantaneous received intensity, I, with respect to the received mean on-axis intensity is larger than a fade threshold FT. The term “onaxis” is defined as the line-of-sight (LOS) between the centers of transmit and receive telescope. The fade threshold parameter FT, in [dB], corresponds to the difference between the received on-axis mean intensity ‹I(0;L)› after the transmission
distance L and a smaller intensity threshold level IT , i.e.
  I 0, L  
.
FT  10 log 
 IT

(15.18)
The probability of fade is deduced from mathematical models for the probability
density function (PDF), p(I), of the randomly fading irradiance signal [2, 19, 20,
21]. Assuming that the intensity fluctuation is an ergodic process, the probability
of fade as a function of the threshold level becomes the cumulative probability of
the intensity, i.e.
PF  FT   PI  I T  
IT
 pI dI .
(15.19)
0
In the weak fluctuation regime typical for HAPs, the time-variant intensity of an
optical wave is often described by a lognormal PDF [22]
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   I  2 2 L2  2  2 
  ln 

I

   I (0, L)   W 2  2  




1
eff
 ,
p I  
exp  


2 I2
I 2 I2






(15.20)
Fig. 15.6 (a) illustrates the lognormal PDF in the uplink for various HAP altitudes.
The received intensity is close to the mean on-axis intensity if the HAP is situated
at hHAP = 20 km. The variance of the PDF increases with decreasing HAP altitude,
i.e. with increasing atmospheric turbulence, leading to larger fading levels. Also
the maximum of the PDF function shifts to a lower value, which means that the
instantaneous received intensity is most likely below the mean on-axis intensity.
As shown in Fig. 15.6 (b), the variance of the PDF first increases and then decreases with increasing pointing error. This reflects that the influence of beam
wander is more severe if the (point) receiver is situated at the slope of the Gaussian beam than at its peak or at its tail. Fig. 15.7 shows the probability for a fading
larger than 1 dB as a function of the HAP altitude in the uplink in different turbulence conditions. Below a height of h = 14 km, an untracked beam clearly leads to
a higher probability of fade than a tip-tilt corrected beam or a tracked beam. We
distinguish between three cases:

When speaking of an untracked beam we fully take into account the effects
of beam wander.
 In the case of a tracked beam, we assume the removal of the root-meansquare (rms) beam wander displacement, i.e. a compensation of the
movement of the instantaneous center of the beam with a tracking time
constant much smaller than the time constant of the atmospheric fluctuations.
 In the case of tip-tilt correction tracking is performed by means of closedloop beam tilt control via a tiltable mirror at the transmitter, which removes the rms "tilt" displacement from the far-field beam. This tip-tilt corrected case corresponds to the removal of the Zernike polynomials of the
2nd (x-tilt) and 3rd (y-tilt) order.
We find that below a height of h = 14 km, an untracked beam clearly leads to a
higher probability of fade than a tip-tilt corrected beam or a tracked beam. The often used standard Rytov model [22] approximates the tip-tilt corrected case very
well. At typical HAP altitudes - between 17 km and 22 km - the effect of beam
wander becomes more and more negligible; the probability of fade for an untracked beam, a tip-tilt corrected beam, and a tracked beam become virtually
equal.
The term surge denotes the event when the currently received intensity rises
above the (temporal) mean of the received intensity [23]. The probability of surge,
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P(S ≥ ST), describes the probability that the surge (or excess), S, of the instantaneous received intensity with respect to the received mean on-axis intensity after a
link distance L is larger than the surge threshold ST . The surge threshold parameter, ST in [dB], giving the difference between the received on-axis mean intensity
‹I(0;L)› at α = 0 and a higher intensity threshold level IT , is defined as
  I 0, L  
.
ST  10 log 
 IT 
(15.21)
Following the approach for the probability of fade one can write the expression for
the probability of surge as
PS  ST   PI  IT  

IT
IT
0
 pI dI  1   pI dI .
(15.22)
Fig. 15.6 Normalized lognormal probability density function of the intensity for an untracked
uplink beam: (a) for varying HAP altitudes hHAP (αn = 0), (b) for different normalized pointing
errors αn = α/θeff (hHAP = 20 km).
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Fig. 15.7 Uplink probability of a fading larger than 1 dB vs. HAP altitude for (a) an untracked
beam, (b) a tracked beam, (c) a tip-tilt corrected beam, and (d) when using the standard Rytov
model for calculation. (dashed-dotted...vwind = 0 m/s, Cn2(0) = 10-17 m-2/3, solid... vwind = 3
m/s, Cn2(0) = 1.7·10-14 m-2/3, dashed... vwind = 20 m/s, Cn2(0) = 10-13 m-2/3)
Temporal statistical parameters are, e.g., the expected number of fades per second
or the mean fade time. The number of fades per second gives the mean number of
crossings per second of the received intensity I below a specific threshold value IT,
i.e. the mean number of fades per unit time with a certain fading depth FT. The
mean fade time in [s] represents the average time at which the signal stays below a
defined threshold FT. It depends on the probability of fade as well as on the expected average number of fades per unit time. When calculating temporal statistics, the turbulent eddies are treated as frozen in space which move across the observation path with an rms wind speed1
5010


1

vt (h) 2 dh
3

 50  10  hHAP hHAP

3
vt , RMS
1/ 2
cos( ).
(15.23)
1 This assumption and the formula are in good accordance with the Taylor frozen turbulence hypothesis as described in reference [2].
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The input parameter vt(h) takes into account the moving speed of the HAP and is
calculated according to equation (15.10).
The expected number of fades per unit time larger than a specified fading
threshold level FT is given by [2]
 0.5 2  2 2 L2W  2  0.23F 2
I
eff
T
 nFT    0 exp 
2

 2 I





(15.24)
in the case of weak turbulence, i.e. when the scintillation index is smaller than 1,
the lognormal PDF is used to describe the statistics of the intensity fluctuations.
The formula requires the quasi-frequency
0 
1
2

BI
(15.25)
 I2
as an input parameter, with BI being the second derivative of the temporal covariance function. The average time at which the fading of the received intensity relative to the mean on-axis intensity is larger than a fading level FT is determined by
t f  
PF  FT 
,
 nFT 
(15.26)
where P(F > FT ) is the probability of fade and ‹n(FT)› is the expected number of
fades per second.
Atmospheric turbulence also causes beam spread (cf. Section 15.1) beyond the
diffraction limited divergence. For the calculation of this, additional, beam spread
loss abs in [dB], we compare the diffraction limited beam radius WDL (determined
by the transmit telescope) of a Gaussian beam [12,24] to the effective beam radius
[2]
Weff
 D
 WDL 1   TX
  r0



5/3



3/ 5
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(15.27)
19
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of the same beam but in the presence of turbulence, leading to
 Weff 
.
abs  20 log 10 
W
 DL 
(15.28)
For downlink paths only high-altitude turbulence – which is weak and relatively
far away from the transmitting source in case of a satellite – has an influence on
beam broadening. It is found that in the downlink – where turbulent eddies are
small compared to the beam diameter – the effective spot size, Weff , is essentially
the same as the diffractive spot size, WDL [22]. Hence, beam spread loss is negligible. In the uplink or in HAP-to-HAP scenarios, the size of turbulent eddies situated just in front of the transmitter is large relative to the beam diameter, leading to
a noticeable beam spread loss. Fig. 15.8 illustrates the variation in beam spread
loss with increasing zenith angle, wavelength, and platform altitude in the case of
a HAP-to-satellite uplink and in different turbulence conditions. As expected, the
beam spread loss is larger at zenith angles ζ > 40° and small wavelengths, because
of a smaller Fried parameter r0 under these conditions. The “plateau” which can be
observed in Fig. 15.8 (c) at altitudes between 4 km and 8 km reflects the increasing wind speed at these altitudes (cf. Fig. 15.3 (a)), leading to shear winds and
therefore to additional turbulence.
Fig. 15.8 Beam spread loss abs for a HAP-to-satellite uplink vs. (a) zenith angle ζ, (b) wavelength λ, and (c) platform altitude hHAP. (dashed-dotted...vwind = 0 m/s, Cn2(0) = 10-17 m-2/3,
solid... vwind = 3 m/s, Cn2(0) = 1.7·10-14 m-2/3, dashed... vwind = 20 m/s, Cn2(0) = 10-13 m2/3)
For horizontal HAP-to-HAP links and in the case of weak turbulence the beam radius at the receiver is broadened additionally by the presence of the atmosphere
and its effective 1/e² radius can be calculated as [2]
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1
z

Cn2 ( z )1  
L 0
L


L
Weff  WDL 1  4.35k 7 / 6 L11/ 6 5 / 6
5/3
(15.29)
dz ,
whereas in the strong turbulence regime it is found to be
12 / 5
Weff  WDL
5/ 3
L


1
z

1   4k 7 / 6 L11/ 6  Cn2 ( z )1   dz 


L0
 L


.
(15.30)
For the calculation of the structure parameter Cn²(z) at every point z along the
transmission path the curvature of the Earth has to be taken into account because it
leads to an increased atmospheric turbulence halfway between the transmitter and
the receiver due to a reduced height above ground.
When a laser beam propagates through the atmosphere also its phase-front gets
perturbed, which reduces the coupling efficiency into a single-mode fiber [5, 6].
The spatial phase-front disturbances after a propagation distance L can be described by the phase structure function [25]
D  d , L   1   2  ,
2
(15.31)
where φ1-φ2 denotes the phase difference at two points on the phase front separated by the distance ρd. According to [1] the structure function for downlink channels can be modeled by
D  d , L   2.914k 
2
hSAT
5/3
d
sec   C n2 h dh,
(15.32)
hHAP
which, in combination with equation (15.12) for the Fried parameter r0, allows to
calculate the mean-square phase variation over an aperture of diameter D [13]
D
   1.0299 
 r0 
2
5/3
(15.33)
.
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In the receiver, the optical input field E - which is collected by the receive telescope - is focused by a thin, diffraction limited lens to the bare end of a singlemode fiber. The coupling efficiency η is defined as the ratio of the power carried
by the fiber mode and the power available in the focal plane. It is possible - and
often more convenient - to calculate the coupling efficiency in the aperture plane
A [6], i.e. just in front of the coupling lens (cf. Section 15.1). Then η is given by


A
E r F * r  exp  jr dA
2
(15.34)
where E is the input field normalized to its overall power, and F* is the conjugate
complex of the field distribution of the fiber mode, back-propagated to the aperture plane A. A phase function Φ(r) covers any deviations from an ideal plane
wavefront of the input field E. The coupling efficiency depends also on the lensto-fiber coupling geometry, which can be taken into account via an additional
coupling design parameter ρ/rS, with ρ being the core radius of the fiber and rS being the Airy radius of the beam in the focal plane. The parameter ρ/rS takes into
account the properties of the incoming beam and of the coupling optics, while the
normalized frequency
V
2
,
 nco2  ncl2


(15.35)
with core and cladding refractive indices nco and ncl, determines the characteristics
(e.g. the numerical aperture) of the optical fiber. Without loss of generality we assume for the calculations to follow the fiber to be operated at single-mode cutoff
(V = 2.405) and the lens-to-fiber geometry to be optimized for maximum coupling
efficiency (ρ/rS = 0.5345). In the case of a perfect phase-front, Φ(r) = const, and
when approximating the fiber's eigenmode by a Gaussian field distribution, a maximum coupling efficiency of ηmax = 81.5% can be achieved. When using the exact
field of the fiber's fundamental mode the maximum coupling efficiency amounts
to ηmax = 78.6%, corresponding to a minimum power coupling loss of 1.05 dB [6].
A comparison between the single-mode fiber coupling efficiency in a satellite-toground communication link and a satellite-to-HAP scenario shows the feasibility
of using fiber coupled receivers onboard HAPs due to the reduced impact of atmospheric turbulence at high altitudes. (15.9) gives the coupling efficiency as a
function of the rms phase-front perturbation, RMSλ, expressed in fractions of the
wavelength λ. For the calculations we assumed a wavelength of 1550 nm and a
standard (telecommunications) single-mode fiber with a core diameter of 10 μm, a
core refractive index of 1.46, and a core/cladding refractive indices difference of
0.3%. As expected, the characteristic shown in Fig. 15.9 is rather flat for small
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perturbations, but drops dramatically to very small values for large disturbances.
For typical satellite-to-HAP communication links (i.e. with hHAP > 17 km and
D < 20 cm) the mean coupling loss is always less than 1.2 dB, while for communication links to a ground station it significantly increases to values larger than
7 dB.
Fig. 15.9 Mean coupling efficiency η for a standard single-mode fiber as a function of the rms
phase-front perturbation RMSλ given in fractions of the wavelength λ.
15.3. Numerical Simulation of Atmospheric Turbulence
15.3.1. Introduction
Numerical simulation is a useful tool for investigating the effects of atmospheric
turbulence on beam propagation, particularly in the strong fluctuation regime
where Rytov theory begins to break down. This occurs for low-elevation Earthsatellite links (<30°), where the increase in the propagation path length through
turbulence causes strong fluctuations in phase. In this regime the fluctuations in
optical path length can no longer be approximated using a geometrical optics approach; diffraction effects must also be included to account for the refractive scattering of light rays by turbulent cells less than the coherence length, r0 or the
Fresnel length, L , whichever is smaller [26]. For many general situations it is
often easier to input a set of known physical parameters, such as the Cn2 profile,
the propagation distance, and perhaps even the inner scale, into a numerical simulation and to examine the outputs, than to try and find an analytical
solution. The

drawback
of numerical simulation is the amount of time and computer memory

that is required to generate an adequately large ensemble of independent realizations to calculate quantities of interest, such as the degree of scintillation, the frequency and duration of signal fades etc. However, while the analytical solutions
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for strong fluctuation regime are still being investigated, numerical simulation is a
useful method for evaluating the effect of atmospheric turbulence on wave propagation, in particular in the design and simulation of adaptive optics correction systems.
15.3.2. Method
To simulate wave propagation through atmospheric turbulence, the Cn2 profile (for
example the HV5/7 profile described in Section 15.2) along the propagation path is
discretized to form N slabs of turbulence with a constant Cni2 value over a propagation distance zi, as shown schematically in Fig. 15.10.
z
N
0
i 1
2
2
 Cn  z  dz   Cni zi
(15.36)
The number of discrete layers to use and the layer separation is chosen to ensure
that the scintillation that develops during propagation between layers remains
within the weak fluctuation regime (i.e. a Rytov value R2<< 1). In weak turbulance scenarios, for example optical links at zenith between ground and satellite,
fewer atmospheric layers are required to adequately model the turbulence and the
discretization of the Cn2 profile can be optimized to produce fixed layer heights
with the correct Cn2 weighting by matching the first 7 moments of the integrated
Cn2 profile as described in [27].
The optical path fluctuations over the atmospheric layer thickness of zi are
condensed into a ‘thin’ phase screen, where the term thin indicates that the effects
of diffraction on propagation through the layer are negligible. The optical path
fluctuations are found by integrating the refractive index fluctuations, n1, over the
layer thickness
z
zi
z
zi

i ( x, y, z ) 
2
(15.37)
n1 ( x, y, z )dz
2
The phase fluctuations within the layer are assumed to have Gaussian statistics
and are therefore completely defined by their second moment. The usual method
of producing a phase screen is to use the amplitude of the 2D Kolmogorov power
spectrum as a filter in the frequency domain to weight white Gaussian noise [28].
The 2D filter, 2D, is found by integrating the 3D Kolmogorov power spectrum
over the direction of propagation to give,
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2D ( f x , f y )  0.023k 7 / 6 ( f x2  f y2 )11/ 6
where k = 2/r0 is the atmospheric coherence length, and

(15.38)
f x and f y are spatial
frequencies on the sampled grid with grid spacing f = 1/D, where D is the width
of the grid in meters. The infinite term at the origin represents an unimportant piston term that can be removed by setting its amplitude to zero.
 the range of frequen
When representing the power spectrum on a discrete
grid
cies is limited by the size of the grid and the size of the grid spacing. To increase
the range of frequencies, one may start with a large grid with dimensions of the
outer scale of turbulence and cut out the central region of the phase map. However, as the outer scale of turbulence may be on the order of 100m and the desired
pixel sampling on the order of several mm, this can lead to extremely large grid
sizes. An artificial way of introducing low frequencies is with the addition of subharmonics, as described in reference [29] or by using an interpolation algorithm
that takes into account the 5/3 scaling of the phase structure function, such as the
random mid-point displacement method described in reference [30].
Fig. 15.10 Schematic of the discretization of the Cn2 profile into thin phase screens, Φi, representing atmospheric layers of thickness Δzi.
A split-step Fourier method is used to simulate the propagation of light through
the atmosphere. The method described here follows that of reference [31]. Each
phase screen is positioned at the mid-point of the layer thickness and the optical
field (x,y,z), is allowed to propagate through free-space between phase screens.
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At the positions of the phase screens the phase perturbation of the screen, (x,y,z)
is added to the phase of the complex field to introduce the wavefront aberration
due to the atmospheric turbulence.
x, y, z'  x, y, z  exp i x, y, z 
(15.39)
Propagation over a distance of z through free-space between phase screens is
simulated by convolving the aberrated field, (x,y,z’), with the Fresnel diffraction
kernel [32]

( x, y, z  z )  ( x, y, z ')  exp ik  x 2  y 2   2  z 

(15.40)
where k = 2/ is the wave-number and * denotes a convolution. In the Fourier
domain this is equivalent to multiplying the Fourier transform of the aberrated
field by the Fresnel transfer function,


H  f x , f y   exp iz  f x 2  f y 2 
(15.41)
where fx and fy are the spatial frequencies.
As the beam propagates it expands due to the refractive scattering and diffraction. To try to capture the beam spread within the grid, the grid width is chosen to
be much larger than that of the transmit aperture. For long propagation paths and
spherical wavefronts it is impossible to sample all scales adequately without using
enormous grid sizes. As discussed by Horwath et al. [33], it is necessary to reduce
the angular extent of the grid as the propagation path increases and resample the
grid following the method of Rubio et al. [34] in order to sample all scintillation
scales adequately.
As a safety measure, to avoid reflections of the field at the grid boundary, an
apodizing mask can be applied to the amplitude of the field at each propagation
step. For example, a circular apodizing mask with a flat profile over a diameter of
80% of the grid and with a Gaussian roll-off at the edges could be used. The
apodizing mask is chosen to be flat over much of the grid to try to avoid attenuating any part of the field that may contribute to the field in the receiver plane.
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15.3.3. Sampling
Fresnel propagator: The Fresnel transfer function has a quadratic dependence on
spatial frequency,


H  f x , f y   exp iz  f x 2  f y 2 
(15.42)
To ensure the transfer function is well sampled at the highest spatial frequency the
change in phase between adjacent pixels must be less than  radians. This gives
a requirement that for a fixed pixel size, x, and a fixed interscreen propagation
distance, z, the number of grid points, N, must be
N

z
x 2
(15.43)
Interscreen distance: The maximum interscreen distance is chosen so that the
scintillation that develops over that distance remains within the weak fluctuation
2
regime (i.e. that the Rytov variance,  R 1 ). To ensure weak scintillation the
maximum Rytov value is set at
Rytov value is
  1.23k C z

2
R
7/6
2
n
11/ 6
 R2  0.1. For a plane wave and constant Cn2, the

(15.44)
and thus the maximum interscreen distance is

z 
k
0.254
7/6
Cn2 
(15.45)
6 /11
The spatial coherence length: To ensure adequate sampling of the phase and
scintillation scales at the receiver, the pixel size must be much less than the atmospheric coherence length, r0. The coherence length for a plane wave propagating over a path of length L is given by
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L
 2.91 2

1
r0  
k  cos    Cn2 dz 
0
 6.88

3/ 5
(15.46)
where k = 2/ and  is the angle from zenith.
15.4. Aircraft Boundary Layer Effects
A challenging issue is to calculate the distortions on the propagated laser beam at
the aircraft boundary layer. The distortions are created from the fluctuation of the
air properties inside the flow, specifically the density, which consequently affects
the refractive index. Via extensive simulations we examine the distortion of the
wavefront with regards to various parameters. The main contributions of this work
are: (a) we derive the relation between air properties – e.g. density and pressure –
and the refractive index, and then quantify the affect of the flow field on the laser
beam, (b) we introduce a framework that can be applied in any flying object independently of the kind of flow, (c) using this framework we are able to describe and
quantify the aberration of the laser beam wavefront.
Assuming that the air properties around the aircraft are known, we introduce a
method in order to be able to calculate the wavefront distortions of the laser beam.
From the Gladstone – Dale equation [35]:
G
n 1

or n  1    G
(15.47)
where G is the Gladstone – Dale constant, we can calculate the refractive index n.
The Gladstone-Dale constant depends only on the beam wavelength, according to
the equation [36], [37]:



0
.
0
1
7
8
5
0
7
6
4
G

2
.
1
9
2
5
3
9
 2 

1
0







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(15.48)
Odd page header
Gladstone - Dale const. ( cm3 / g )
Gladstone - Dale constant over wavelength
0.28
0.26
0.24
0.22
0.2 -1
10
0
10
wavelength [µm]
1
10
Fig. 15.11 The Gladstone-Dale constant vs. wavelength
It is obvious from Fig. 15.11 that for a specific wavelength we have a constant ratio between the density and the distance of refractive index from the unit, for two
different points (A,B) of the medium:
A nA 1

B nB 1
(15.49)
That shows us that if we know the density ratio between two points, and the refractive index in one of them (it can be at the free-stream area), we can calculate
the refractive index at the other point.
The optical path length of the laser beam, noted OPL, is related to the refractive
index by:
y
OPL  x, y, z    n  x, y, z dy
(15.50)
0
The Optical Path Difference, OPD, shows the configuration of the wavefront and
is defined as the minimization of OPL:
O
P
D

O
P
LO
P
L
m
in
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(15.51)
29
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Fig. 15.12 Diagram of the wavefront error calculation method
Since we have the OPD we are able to describe the scintillation, aiming to compensate the distortions. In order to get a better aspect of the topic we present the
results of extensive simulations. A number of simulations are performed [38] for
various flow fields from laminar, steady to turbulent flow as well as for different
shapes of an aircraft, trying to reach closer to the aerodynamic shape of a HAP.
The wavelength of the laser beam is considered at 1550 nm.
pressure coefficient over a sphere
pressure coefficient over a sphere
-4
-3
0.8
-3
-2
0.4
-1
0.2
0
0
-0.2
1
-0.4
0
0
1
-0.5
-0.6
2
2
-0.8
3
4
-4
0.5
-1
y- axis
y- axis
0.6
-2
-1
-1
3
-1.2
-2
0
2
4
x- axis
-2
0
x- axis
2
Fig. 15.13 Pressure coefficient around a
Fig. 15.14 Pressure coefficient around an
sphere, r = 1 m (case A)
eclipse, r/R = 1 m/2 m (case B)
In both cases the flow is laminar, steady and incompressible.
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Fig. 15.15 Density around a sphere, R=1 m, Fig. 15.16 Density around a sphere,
u=37.5 m/s, STP
R=1 m, u=54 m/s, air at 20 km
Compressible flows around a sphere (case C)
Optical Path Difference [m] at 0°
5
2.5
10
2
15
-6
x 10
3
5
3
aperture size (cm)
aperture size (cm)
Optical Path Difference [m] at 30°
-7
x 10
3.5
1.5
20
2.5
10
2
15
1.5
20
1
1
25
25
0.5
0.5
30
5
10
15
20
25
30
30
0
5
10
15
20
25
30
0
aperture size (cm)
aperture size (cm)
θ=30°
θ =0°
Fig. 15.17 OPD for a specific elevation angle, for case A, aperture diameter= 30 cm, air at STP.
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Optical Path Difference for elevation angle 0°
40
x 10
35
3.5
30
3
25
2.5
20
2
15
1.5
10
1
-6
0.5
5
5
10
15
20
25
30
35
40
0
θ=0ο
θ=90 ο
Fig. 15.18 OPD for case B, aperture diameter= 40 cm, STP.
θ= 60ο, φ= 90ο
θ= 30ο, φ= 90ο
Fig. 15.19 OPD for case C, aperture amplitude = 40 cm.
Boresight error and mean tilt [Heading??]
From the wavefront configuration, OPD(x,y), we are able to calculate the
boresight error, a(x,y), through the wavefront phase error, e(x,y) [39]:
e
,y
kO
D
,y
x

x

0 P
(15.52)
where K0 is the wave number, and then:
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 



a
x
,
y

 
x
,
y



e
x
,
y



(
x
,
y
)
2

(
x
,
y
)
(15.53)
Knowing the boresight error, we are able to calculate the mean wavefront tilt
along the two coordinates of the wavefront surface (x,y). The mean tilt is the sum
of the two coordinates:
 
2
2
1m
ax,y  
ai and a a
x a
y
mi1
(15.54),
(15.55)
wavefront tilt over elevation angle
12
wavefront tilt [µrad]
10
8
6
4
2
0
0
20
40
60
elevation angle [°]
80
Fig. 15.20 Wavefront tilt over elevation an-
Fig. 15.21 Mean tilt vs the elevation angle,
gle, case A.
case B.
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mean wavefront tilt over velocity (Mach number)
mean beam angle fluctuation over height
12
12
0
30
45
80
8
10
mean wavefront tilt [µrad]
mean wavefront tilt [µrad]
10
6
4
5
30
45
80
8
6
4
2
2
0
0.1
0.15
0.2
Mach number
0.25
0
0
0.3
10
20
height [km]
30
40
Fig. 15.22 Mean tilt over the Mach number
Fig. 15.23 Mean tilt vs height for several ele-
for different elevation angles, case A.
vation angles, case A.
Zernike polynomials [Heading?]
An analytical measure used to characterize the fluctuation of the wavelength are
the Zernike polynomials. They show us the effect of the corresponding Zernike
polynomial making an accurately describing of the wavefront aberrations. In the
next five Figures (Fig. 15.24 to Fig. 15.25) is presented the variation of the first
six coefficients (each one for the corresponding polynomial) for elevation angles
from 0° to 90°.
-9
x 10
value of each coefficient
value of each coefficient
-8
3
2
1
0
-1
0
1
2
3
4
5
zernike polynomial
6
7
15
x 10
9
10
5
0
-5
0
1
θ = 0°
θ = 30°
Fig. 15.24 The value of the first six Zernike polynomials, case A.
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3
4
5
zernike polynomial
34
6
7
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θ= 60ο, φ= 90ο
θ= 30ο, φ= 90ο
Fig. 15.25 The value of the first six Zernike polynomials for case C.
Root Mean Square OPD and Strehl ratio
Another parameter we can research to define the wavefront fluctuations is the difference-Root Mean Square (RMS) of the wavefront, defined as:


2
1
2
O
P
D

O
P
D

O
P
D
d
x

d
y
R
M
S
S
(15.56)
where S is the surface of the aperture.
Τhe ratio of the observed peak intensity to the theoretical maximum peak intensity of the beam, called Strehl ratio, is another determinant. For the calculation of
Strehl ratio complex mathematic equations are required, thus we use the following
equation, accurate for cases with small errors [40]:
2
SR  exp   K  OPD    exp  f 2 


(15.57)
where SR is the Strehl ratio, K is the wave number, OPD is optical path length and
f is the wavefront phase error.
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Fig. 15.26 OPDRMS vs height, case B.
Fig. 15.27 Strehl ratio vs height, case B.
It is obvious that there is still space for more research on the topic of the aircraft
boundary layer effect on the laser beam propagation. A real challenge is to simulate a more realistic scenario of an aircraft and to calculate the distortions of the
beam wavefront. The research is getting more interesting as it combines fluid- and
air-dynamics with optics. But the most stimulating issue is the implementation of
this research for the improvement of optical telecommunications. Furthermore we
can compare but also combine optical links with RF links for better telecommunication results.
15.5. Link Availability
15.5.1. Attenuation in Fogs
Physical Characteristics of Fog
Fog consists of fine water droplets in the atmospheric layer in contact with the
ground. According to the international meteorological definition, a fog occurs
when the horizontal visibility is reduced below 1000 m and when humidity is
close or equal to 100% [41]. The reduction in the visibility depends on the nature
of the fog, on the volume concentration and on the size distribution of the droplets.
Fogs are classified according to the physical process whereby water vapour is
condensed: radiation fog, advection fog, upslope fog, precipitation fog, valley fog,
steam fog, ice fog, freezing fog, hail fog, evaporation fog and mixing fog. Advection fog and radiation fog are two main types of fog. More detailed information
about different types of fogs can be found in [41], [42]. Meteorologists classify
fog in accordance with the visibility V (m). Classification is given in
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Table 15.3.
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Table 15.3 Classification of Fog
light fog
500 m < V < 1000 m
moderate fog
200 m < V < 500 m
thick fog
50 m < V < 200 m
dense fog
V < 50 m
The dependence of the liquid water content LWC (g/m3) on visibility V (km) can
be calculated in accordance with the power law type empirical formula (1), e.g. as
given in [43]
LWC = 0.0165 * V -1.13636
(15.58)
Typical liquid water contents range from 0.003 to 3 g/m3 depending on location,
height in the atmosphere, and meteorological conditions [44]. The ITU-R recommendation [45] gives the liquid water content of a fog as 0.05 g/m3 for moderate
fog (300 m visibility) and 0.5 g/m3 for dense fog (50 m visibility). The radius of
fog droplets varies between 0.1 μm and 20 μm [46], [47].
Horizontal atmospheric visibility is usually measured using either a transmissometer or a diffusiometer at airports so it is possible to obtain visibility characteristics in areas where the FSO systems are to be deployed. The visibility measured
at airports provides a good estimate for the assessment of fog impairment.
The vertical extent of fog usually varies from a few metres to several hundred
meters. In sporadic cases, the vertical extent can be up to 2 km, mainly for the advection fogs [48]. The more detailed information is very scarce. In [49] is stated
that the vertical extent of fog rarely exceeds 100 m, while fog events up to 140 m
in height are reported in [46].
Attenuation Due to Fog
The interaction of the atmosphere with light of free space optical (FSO) links produces the impairment by frequency selective absorption, scattering and scintillation. Molecular absorption is a selective phenomenon and can be negligible in the
preferred optical transmission windows from 690 nm to 10600 nm. It becomes a
serious problem for wavelengths much longer than 10 μm.
The effect of scattering is dominant loss mechanism for fog. The type of scattering is determined by the physical size of fog particles with respect to the wavelength used. Rayleigh scattering occurs when the atmospheric particles are much
smaller than the wavelength. The effect of Rayleigh scattering on the attenuation
is very small. Mie scattering becomes dominant when fog particles are comparable
with the wavelength. Mainly smaller fog particles suffer from Mie scattering.
Non-selective or geometric scattering occurs when the fog particles are much
larger than the wavelength and therefore there is no dependence of attenuation on
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wavelength. Primarily thick and dense fog, rain droplets, snow, hail, and cloud
droplets geometrically scatter laser wavelengths. There is no great variation of attenuation between 500 nm and 10 μm wavelengths. An extensive search of the literature and some full Mie scattering calculations reveal that 785 nm, 850 nm and
1550 nm wavelengths are all in fact equally attenuated in fog with visibility smaller than 500 m [47].
Fog scattering losses can be approximated by four the most used models: Kruse
model [50], Kim model [47] and Al Naboulsi models for advection and convection fog [51], [52]. These models relate attenuation to visibility.
Experimental Results
For the design of FSO links that should meet required availability performance
criteria, attenuation statistics has to be obtained from either attenuation or visibility data measured at particular locations where the planned FSO links are to be
deployed. The following experiment provides an insight into the influence of different type of hydrometeors besides fog on attenuation and its relation to visibility.
Experimental Setup
A FSO link at 850 nm has been operated on a path length of about 850 m since
August 2004 in Prague, the Czech Republic. Transmitted power is 16 dBm, divergence angle is 9 mrad and optical receiver aperture is 515 cm2. The recording fade
margin is about 17 dB. The other experimental FSO link with the wavelength of
830 nm operates on a 100 m long path in the same location. The transmitted optical power is 30 dBm, the diameter of Fresnel lens is 15 cm. A real optical fade
margin is about 20 dB.
The meteorological system is located near the receiver site and is equipped
with VAISALA sensors for the measurement of temperature, humidity and pressure of air, velocity and direction of wind and tipping-bucket rain-gauge for the
measurement of rainfall intensities. The VAISALA PWD11 device is used for the
measurement of visibility from 2000 m to 50 m.
Attenuation and Visibility due to Individual Types of Hydrometeors
The 3-year experimental research (August 2004 – July 2007) of attenuation due to
hydrometeors at 850 nm measured on an 850 m terrestrial path confirmed that fog
is the most significant impairment factor [53]. The obtained cumulative distributions (CDs) of attenuation due to individual types of hydrometeors, i.e. rain (R),
rain with hails (RH), rain with snow (RS), snow (S), fog (F), and snow on the
shields of lens of the FSO device (SSL) and sun (the experimental link is situated
approximately on an east-west axis and therefore the optical receiver oriented towards the sunset was overloaded sometimes during the summer months) for the
entire 3-year period are drawn in Fig. 15.28.
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20
all
S
R
F
RH
SSL
RS
sun
A (dB)
15
10
5
0
0.00001
0.0001
0.001
0.01
0.1
1
10
percentage of time
Fig. 15.28 Obtained cumulative distributions of attenuation due to individual types of hydrometeors for the entire 3-year period
It can be seen in Fig. 15.28 that the occurrences of fog caused dominant attenuation
events. It can be also seen that both the cumulative distribution of attenuation due
to snow and the cumulative distribution due to rain with snow are also significant.
On the other hand, rain attenuation events, rain with hails attenuation events, the
influence of sun and the influence of snow on the shields of lens of the FSO device can be considered to be negligible.
The obtained yearly CD of attenuation due to all hydrometeors together was
analyzed and the results obtain are given in Fig. 15.29 and Fig. 15.30.
20
all together
F
FR
FS
F+FR+FS
R
RS
S
S on lens
sun
A (dB)
15
10
5
0
0.001
0.01
0.1
1
percentage of time
10
100
Fig. 15.29 Yearly CDs of attenuation due to individual type of hydrometeors and other causes
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2000
all together
F
FR
visibility (m)
1500
FS
F+FR+FS
R
1000
RS
S
500
0
0.001
0.01
0.1
1
10
100
percentage of time
Fig. 15.30 Yearly CDs of visibility due to individual type of hydrometeors
It can be seen the reduced visibility due to the occurrence of dense fog, fog with
rain, and fog with snow caused the significant attenuation events. Attenuation of
fog with rain and fog with snow can be taken as attenuation due to fog without
significant error due to the fact that rain intensities and snow intensities respectively, contained in fog were always less than 3 mm/h. This rain intensity can
cause attenuation up to about 3 dB on this path [54]. Therefore, only insignificant
error can arise when the events of fog with rain and fog with snow are taken as fog
events. It is important in the cases when data obtained at airports are processed.
Dependence of Specific Attenuation of Fog on Visibility
The dependence of specific attenuation on visibility for fog only, obtained on a
terrestrial FSO link at 850 nm on an 850 m long path over one-year period (August 2006 – July 2007) [55], was compared with the calculated dependences in accordance with four different models [47], [50] – [52]. The results obtained are
shown in Fig. 15.31. The dependence of specific attenuation on visibility for fog
alone agrees very well with the calculated dependence in accordance with the Al
Naboulsi model for the convection fog [52]. The differences between results of
both Al Naboulsi models are very small, of about 0.2 dB. All the fog events that
occurred on the experimental path were of the convection types.
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25
Specific attenuation for fog (dB/km)
measured
Kruse model
Kim model
20
Al Naboulsi model - advection fog
Al Naboulsi model - convection (radiation) fog
15
10
5
0
500
1000
1500
Visibility (m)
2000
2500
Fig. 15.31 Comparison of measured dependence of specific attenuation on visibility for fog only
and the modelled dependences
Comparison of Measured Attenuation with Models
The CDs of attenuation due to fog were calculated from the measured CDs of visibility due to fog and compared with the measured CDs of attenuation due to fog
[56] on FSO link operating at 850 nm wavelength on a path of 850 m. CDs of visibility due to fog measured in October 2004, December 2004, and July 2005, i.e.
for the months with long-term and dense fog events, were chosen and four different formulas [50], [47], [51], and [52] were used for calculations. Visibility was
varying from 2000 m to 700 m. The results obtained are shown in Fig. 15.32.
20
Oct + Dec + Jul
Kruse model
A F+FR+FS (dB)
15
Kim model
Al Naboulsi model - advection fog
Al Naboulsi model - convection fog
10
5
0
1
10
percentage of time
Fig. 15.32 Comparison of measured and calculated CDs of attenuation due to fog for 3-month
period
It was found that the Al Naboulsi model [52] for convection fog was the best suited model with the r.m.s. value of 0.190 for the above mentioned 3-month period.
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Proposed Empirical Models Based on Experimental Results
Two empirical models based on the results of experimental research in Prague are
proposed – the power-law model and the inhomogeneous model [57].
In the power-law model the specific attenuation A (dB/km) is calculated using
the following equation:
A  10 loge aV b  (c),
(15.59)
where V (km) is atmospheric visibility. In the special case (for parameters a = 3
and b = -1) it is reduced to the Kruse model for the wavelength of λ = 550 nm. An
additional shift parameter c is used to further improve a model fit (the shifted
power-law model).
The inhomogeneous model is based on the assumption of the linear spatial dependence of visibility V(x) along the propagation path and is defined as the mean
value of specific attenuation along the propagation path:
d
d
0
0
A  (1/ d )  A( x)dx  (1/ d ) 10  log(e)  (3 / V ( x)) dx
V ( x)  V0  (aV0  b) x
(15.60)
with parameters a, b and V0 denoting visibility at one site of the propagation path.
Measured time series from 5 dense foggy days (Nov 19, 2006, Dec 28, 2006,
Oct 28, 2007, Jan 2, 2008 and Jan 14, 2008) were processed together and the obtained dependence of the specific attenuation due to fog on visibility (scatter plot
diagram), the Kruse, Kim, Al Naboulsi models and the proposed models are
shown in Fig. 15.33.
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Fig. 15.33 Measured data and comparison of visibility vs. attenuation models, non-parameterized
models (left) and fitted models (right)
The Kruse, Kim and Al Naboulsi models underestimate attenuation values for medium and lower visibilities. The performance of these models is good for visibilities greater than 1 km. The proposed shifted power-law model is the best one with
respect to RMS and it performs better mainly in the region of medium visibilities.
The resulting parameters of this model are a = 6.861, b = -0.858, c = -10.428.
Availability Performances of Simulated 850nm/58GHz and 850nm/93GHz Hybrid
Systems
The availability performances of the FSO link, the back-up radio frequency (RF)
links and the simulated FSO/RF hybrid system can be assessed from the obtained
CDs of attenuation due to all hydrometeors together. Availability performance of
FSO link is reduced mainly by dense fog events while availability performance of
RF link is impaired primarily by heavy rainfall events. Therefore, the RF part of
the hybrid system will mitigate fog events and the FSO part will mitigate rain
events.
The cumulative distributions of attenuation due to all hydrometeors together for
the FSO link, the 58 GHz RF link and the simulated 850 nm/58 GHz hybrid system are plotted in Fig. 15.34 [58].
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25
FSO
58 GHz
hybrid
A all (dB)
20
15
10
5
0
0.00001
0.0001
0.001
0.01
0.1
1
10
percentage of time
Fig. 15.34 Obtained CDs of attenuation due to all hydrometeors for 850 nm link, 58 GHz link
and simulated 850 nm/58 GHz hybrid system
The cumulative distributions of attenuation due to all hydrometeors together for
the FSO link, the 93 GHz link and the simulated 850 nm/93 GHz hybrid system
are plotted in Fig. 15.35.
30
FSO
93 GHz
hybrid
A all (dB)
25
20
15
10
5
0
0.00001
0.0001
0.001
0.01
0.1
1
10
percentage of time
Fig. 15.35 Obtained CDs of attenuation due to all hydrometeors for 850 nm link, 93 GHz link
and simulated 850 nm/93 GHz hybrid system
Let us consider a hybrid system where both the FSO link and the RF link have the
same fade margin FM = 15 dB. The availability performance (AP) of the simulated FSO/RF hybrid system can be estimated from the CDs shown in Fig. 15.34 and
Fig. 15.35. The results obtained are given in Table 15.4.
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Table 15.4 Availability performances of FSO, RF, and simulated FSO/RF hybrid systems
System
850 nm
58 GHz
93 GHz
850 nm/58 GHz hybrid GHz/850
850 nm/93 GHznm
hybrid GHz/850
nm
AP (%)
99.4600
99.9793
99.9608
99.9948
99.9965
It can be seen clearly, that the significant improvement of availability performance
can be achieved by using FSO/RF hybrid system.
Acknowledgments
This work was supported by the Ministry of Education, Youth and Sports of the
Czech Republic under the project No. OC093 in the frame of COST Action 297.
15.5.2. Attenuation in Clouds and Downlink Availability
Attenuation of Clouds
The dependency of clouds is one of the most problematic issues to be solved in
optical free-space communications. At times of overcast sky, a vertical link is very
likely to be blocked due to the high attenuation over the whole optical spectrum.
Only very thin clouds, which are mainly ice clouds at high altitudes, can be handled with an adequate link margin.
Generally, clouds are split into four families. Depending on their base height,
three families are designated to low, middle and high clouds. The fourth family
consists of clouds with high vertical extent. At middle European latitudes low
clouds range from ground to 2 km, middle clouds from 2km to 6 km, and high
clouds from 6 to 13 km. The third family is mainly ice clouds and bear relatively
low attenuation due to low ice water content and thus less scattering and absorbing
particles. The first, second and fourth family are mainly the clouds of concern.
These have high liquid water content and thus bear higher attenuation.
Table 15.5 lists values for the base height, the thickness, and water content of
the most frequent cloud types. Therefore, Stratus, Stratocumulus, and Cumulus
clouds belong to the family of low clouds. Altostratus clouds are found in the family of middle clouds, Cirrus are high clouds; Nimbostratus clouds have high vertical extent and thus are grouped in the forth family.
Table 15.5 Height and thickness of the most frequent cloud types taken from [59] and water content taken from [60].
Cloud type
Base height
[km]
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[km]
46
Water content
[g/m3]
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Stratus
0.1 – 0.7
0.2 – 0.8
0.29
Stratocumulus
0.6 – 1.5
0.2 – 0.8
0.15
Nimbostratus
0.1 – 1.0
2-3
0.65
Altostratus
2–6
0.2 - 2
0.41
0.5 – 1.0
0.5 – 5
1.00
6 – 10
1.0 – 2.5
0.064
Cumulus
Cirrus
For calculation of the wavelength dependent attenuation of a cloud it is necessary
to describe this cloud with a proper model. This comprises the particle size distribution of the water drops in the cloud and the respective wavelength dependent
complex index of refraction of liquid water or ice water, respectively. The shape
of the particles is assumed to be spherical and for simplicity the cloud is considered to be homogenous and not polluted. With these constraints it is possible to set
up a simple to use cloud model which can be used to determine the attenuation
characteristics with the theory of Mie scattering. In general, this theory is applied
for calculating the scattering of an electro-magnetic wave in the presence of particles with spherical shape. To give an example, Fig. 15.36 depicts the particle size
distribution of a Stratus and Stratocumulus cloud as they are also applied in the
widely spread LOWTRAN (LOW TRANsition code) and FASCODE (Fast Atmospheric Signature CODE) models [60]. The shape of these distributions, i.e. the
particle size of maximum occurrence and width of the curve, together with the absolute water content and complex index of refraction determine the attenuation
spectra.
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Fig. 15.36 Particle size distribution for the Stratus and Stratocumulus cloud type. These distributions are generated with the modified gamma function proposed by Deirmendjian [61] using the
parameterization of LOWTRAN and FASCODE [62].
Using the Mie-tool from the LibRadtran simulation suite [63], attenuation spectra
in [dB/km] of these two cloud models together with Cumulus, Altostratus, and
Nimbostratus were calculated. They are plotted in Fig. 15.37. In general, the attenuation is very high and seems to be far beyond being considered in link budgets. However, there happens to be a local minimum at wavelengths between
10 µm and 12 µm for all considered cloud types due to low absorption and scattering on the water droplets. Eventually, this wavelength interval is worth being considered for further investigations.
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Fig. 15.37 Calculated attenuation per km of five water cloud types. For parameterization the
models proposed in LOWTRAN and FASCODE are used [62].
Since clouds have different vertical extent, the knowledge of attenuation per km is
not satisfying.
Table 15.5 lists a range of vertical thickness for the respective cloud types. Thus,
there is an absolute attenuation range every cloud can come up with. A Cumulus
type, for instance, ranges between 500 m and 1000 m which results in attenuation
from ca. 290 dB up to 580 dB in the NIR. Even at the local minimum in the spectrum between 10 µm and 12 µm the attenuation is still far above 200 dB for the
thin Cumulus (500 m). So, optical communication through this cloud type is quite
hopeless. But in times of occurrence of some particular cloud types, like stratus
and stratocumulus types, attenuation may be able to be coped with. The attenuation dynamic of these cloud types is shown in Fig. 15.38. Both types vary in thickness between 200 m and 800 m. In cases where these occur, the attenuation
amounts between 29 dB and 116 dB for the Stratus at 11.13 µm and between
16 dB and 63 dB for the Stratocumulus at 11.05 µm; thus, these clouds may allow
propagation by use of wavelengths in the middle infrared.
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Fig. 15.38 Dynamics of absolute attenuation of Stratus and Stratocumulus clouds. The dashed lines
stand for the attenuation with vertical thickness of 800 m, the solid lines for 200 m. The local minimum of the Stratus is at 11.13 µm (29 dB), the one of the Stratocumulus at 11.05 µm (16 dB) [62].
Cloud Coverage Statistics
Although, there may be a possible wavelength range with lower attenuation than
the usually applied ones in the near infrared, the optical link is likely to be blocked
in either case. Thus, under the assumption that any cloud occurrence in the line of
sight between HAP and ground station blocks the optical link, it is important to estimate future statistical link availability with measurements of cloud amount. This
kind of measurements is available in different metrics. Worldwide, trained weather watchers observe the cloud situation at certain locations four to twenty-four
times a day and send them, together with a packet of other relevant meteorological
parameters, to the national weather services and the WMO (World Meteorological
Organization). The coverage is measured in Okta. 1 Okta corresponds to 12.5%
cloud amount averaged over the visible hemisphere. This kind of measurement has
a relatively good temporal resolution but a low spatial one for the whole field of
view of the observer is taken into account. The second kind of available measurements are based on satellite earth observation. Here, data derived from LEO and
GEO satellite images are to distinguish. The first usually deliver only one image a
day, due to their limited visibility time. Resolution may be in the range of 1 km at
nadir view, which is achieved by the AVHRR (Advanced Very High resolution
Radiometer) on board of the newer NOAA (National Oceanic and Atmospheric
Administration) satellites. The second produce cloud products with higher temporal resolution, like 96 measurements a day, as obtained with the instruments on
the MSG (Meteosat Second Generation) satellite. But spatial resolution is lower
and amounts around 2.5 km for the MSG. All data, no matter if derived from satellite or ground observations, are projected on the Earth surface. Thus, they contain
information about cloud coverage in zenith view and may be used for estimation
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of availability of optical links between ground stations and corresponding HAPs
hovering above them.
Among the two types of cloud measurements mentioned above, there are several other kinds applying cameras, thermopiles, etc, but neither of them is widely
used and thus not valuable for calculating statistics.
It is important to mention that using of different kinds of data may result in deviating results. For instance, cloud coverage happens to be overestimated by
ground observations, but in some cases also underestimated. However, comparisons show that deviations between statistics based on ground observations and
Earth observation products of cloud amount in the ECC (European Cloud Climatology) project stay within a measurement error of 1 Okta [64], [65].
Link Availability Estimated from Cloud Coverage Statistics
For ground stations in Europe, the ECC data is a good source to identify feasible
locations. Fig. 15.39 (left) shows the mean cloud coverage of the year based on
data from the years 1990/1995/2000/2004/2005. In this figure, one can see the typical north-south decrease of the mean cloud coverage. Hence; the amount of cloud
occurrence in higher latitudes is bigger than in lower ones. For instance, the northernmost city in Sweden, Kiruna, has a mean coverage of 74% and the astronomical site on the Calar Alto, Spain, of 38%. Furthermore, geological features like the
Alps, the Carpathians, and the Mediterranean Sea have strong influence on the
north-south course. Eventually, it comes out that the Mediterranean region is to be
examined first if ground station locations are searched for as it is underlined by the
graph in Fig. 15.39 (right). It highlights the locations with over 60% availability
that covers the Mediterranean region quite well.
Fig. 15.39 Mean cloud coverage over Europe derived from ECC data of the years
1990/1995/2000/2004/2005. The right graph shows a binary illustration with a threshold of 60%. The
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white area depicts an availability of 60% and more. The gray area corresponds to the availability below
40% [62].
Further important information is the course of availability over the year. It may
happen that locations with same year averages of cloud cover bear different variation over the year. So for some time periods a specific location may be feasible, or
even extraordinarily good, but for other periods, link availability is just too low.
The course of availability for the German Aerospace Center (DLR) site in Oberpfaffenhofen (Germany) is plotted in Fig. 15.40. As expected, cloud cover tends to
be less during summertime than in wintertime. The best months give an availability of up to 66%, whereas the worst months drop below 30%. The year mean is
40%, the standard deviation 13%. A good example for a favourable site with high
availability is Aix-en-Provence (France) which is chosen here to demonstrate suitable cloud conditions on the mainland with good infrastructural access. Furthermore, this site may fit well into the diversity network presented below. The mean
availability amounts to 68% and remains above 50% for all months. The standard
deviation is similar to the latter one. Another example is a site on the Calar Alto
(Spain). Besides low cloud coverage, this site offers fine seeing conditions. Astronomical facilities are already located here and thus, infrastructure exists. Again,
standard deviation is similar which indicates similar variation of all three stations
over the year. When using a single ground station for communication links, minimizing this value is preferable.
Fig. 15.40 Course of monthly mean cloud cover over Oberpfaffenhofen, Aix-en-Provence, and
Calar Alto. This data is averaged over five years.
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Fig. 15.40 contains statistics averaged over several years. They show the mean
course of availability over the year. Since conditions change not only over the seasons but also between distinct or subsequent years, a closer look reveals large variability as can be seen in Fig. 15.41. The five years, whose data is used to calculate
the year course of Oberpfaffenhofen (Fig. 15.40), are separated here to illustrate the
dynamics. Variations of up to 50% can be observed. This indicates the fact that
any availability statistics are long term and are only valid for long term use of a
dedicated ground station.
Fig. 15.41 Course of cloud cover over the DLR site Oberpfaffenhofen for five different years.
Availability of Ground Station Diversity Systems
To raise link availability one can set up a system of several ground stations. Using
the ECC data, the availability of a ground station diversity system with sites in
southern Europe can be estimated. One condition is to choose sites with an overall
low cloud cover, which restricts the available locations to the marked region in
Fig. 15.39 (right). For weather cells have finite extent and locations close to each
other may be located in the same cell and thus correlate, sites should be separated
in distance so that this effect is negligible. To obtain an indication for a minimum
distance, correlation of cloud condition around Aix-en-Provence was examined in
[66]. Fig. 15.42 shows an exponential best-fit curve of the correlation coefficient
over distance to Aix-en-Provence. It shows that correlation beyond 900 km can be
considered low and thus negligible.
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Fig. 15.42 Best-fit curve for correlation over distance around Marseille [66].
Eventually, this minimum distance is another important condition for the ground
station network since increasing the number of ground stations is senseless for
they always have similar weather conditions if they are to close. What is more,
practical matters like infrastructural facilities have to be obeyed. Together with the
two sites in France and Spain, the resulting example network then hosts two more
sites in Sicilia (Italy) and Crete (Greece). Fig. 15.43 (left) shows the chosen locations.
Fig. 15.43 Location of the stations in two example scenarios. On the left side is a scenario in
Europe. The stations have a least distance of 1000 km to each other and are assumed to be located in uncorrelated weather cells. On the right is a scenario in Germany. Here, the distance between the stations lies between 300 km and 900 km.
Fig. 15.44 shows the course of the resulting mean availability with increasing
number of stations of two network setups, the one in southern Europe and one in
Germany. Looking at the course of the first one, it comes out that there is not
much difference between availability calculation considering and neglecting correlation of weather cells. Thus, the overall availability of the network can be estimated by the pure knowledge of the locations particular mean cloud coverage. The
network in Germany reveals different behavior because the sites are spaced closer.
Calculation of network availability, with and without treatment of correlation, results in two quite different results. Here, it is illustrated clearly that correlation in-
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fluences availability calculation significantly and must be taken care of in that
case.
Fig. 15.44 Increase of availability of ground station diversity systems in Europe and Germany
(left). The dashed lines show the availability when correlation between weather cells is neglected, the solid lines when correlation is taken care of [ 62].
Conclusion
Cloud attenuation spectra and ground station availability statistics are presented.
In most cases, cloud attenuation will be too high, no matter which wavelength is
used and a vertical link from HAP to ground station will be blocked. Thus, availability statistics are needed to estimate link outage frequency, no matter if a single
ground station or a network of ground stations is applied. An example network of
four ground stations in southern Europe is presented and shows that availabilities
over 98% are obtained in that case. However, one has to keep in mind that availability statistics are long term and high variability may occur.
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