Prediction of Drop Size Distribution Parameters for Optical Wireless
Communications through Moderate Continental Fog
Muhammad Saleem Awan1, Roberto Nebuloni2, Carlo Capsoni3, László Csurgai-Horváth4, Sajid Sheikh Muhammad5,
Erich Leitgeb1, Farukh Nadeem1, Muhammad Saeed Khan1
1Graz University of Technology, Graz, Austria
2IEIIT-Consiglio Nazionale delle Ricerche, Milan, Italy
3Politecnico di Milano, Milan, Italy
4Budapest University of Technology and Economics, Budapest, Hungary
5National University of Computer and Emerging Sciences (FAST-NU), Lahore, Pakistan
Wireless Optical Communication Links (OCL), or Free Space Optics (FSO) links involving optical ground stations (OGS) are highly
influenced by the earth atmosphere due to the interaction of the optical wave with particles of different size and shape. Fog, clouds,
rain and snow cause significant signal attenuation thus limiting the performance of OCL. In this paper, we consider the behavior of
OCL in the troposphere under moderate continental fog conditions, which are important for both ground-ground and ground-space
OCL. The impact of the droplet size distribution (DSD) of fog is investigated, by processing laser attenuation measurements carried
out in Milan (Italy) and Graz (Austria). Significant differences are observed between measured and predicted attenuation when using
standard values for the DSD parameters. Hence, new sets of DSD parameters are proposed to model peak, mean and median values of
measured attenuation for moderate continental fog.. These, in turn, can be useful to make accurate link availability predictions, thus
improving the quality of service (QoS) design for OCL.
Index Terms—Drop size distribution (DSD), Optical communication links (OCL), Optical Ground Station (OGS), Specific
attenuation, Visual range,
nterest in free space optics continues to grow for ground-space and ground-ground applications due to its potential in
providing significantly high data rate communication links. OCLs are generally directed line-of-sight links with recent
developments aiming at providing optical multi-input multi-output (MIMO) functionality for the next generation high speed
optical networks [1, 2]. Nowadays, OCL technology is identified as an attractive alternative to complement existing microwave
 and radiofrequency communication links for the backhaul traffic . This technology has some inherent prime advantages
like data rates exceeding easily 100 Gbit/s using WDM techniques, security aspects, EMC/EMI immunity, frequency regulation
issues, small terminal size and weight, small aperture size and low power consumption . It is anticipated that OCL technology
will acquire an important role in satellite networks besides opening up opportunities in emerging technological advancements
such as hybrid optical/radiofrequency or optical/microwave diversity, MIMO for the next generation of optical WLAN, and
diversified applications ranging from extremely short range (e.g. optical links in a PC), to very long range communications (e.g.
links between OGS and the GEO satellites).
Currently, OCL is finding niche applications both in military as well as commercial services sectors; as OCL provide
flexibility of establishing links on fixed as well as on mobile platforms depending upon the user requirements and the dedicated
application scenarios . Typical applications of OCL are secure and speedy service delivery of high bandwidth access to fiber
networks, temporary network installation for special events and purposes, re-establishing high speed connections in case of
emergency or disaster recovery, interstellar communications, ship-to-ship high data rate communications and communications
between ground and spacecraft or between spacecrafts, including elements of a satellite constellation [2, 3, 4]. Moreover,
metropolitan networks may include optical links between office buildings, longer links between a ground station and a building
or a satellite platform, or, again, an inter-satellite link. Finally, OCL provide last mile connectivity throughout metropolitan
To develop the next generation of optical wireless terminals with reduced size and power consumption, and at the same time
increased data transmission rate, several studies were carried out in the last ten years [5, 6]. The optical terminal design has been
specifically optimized for point-to-point fixed as well as mobile platforms links between spacecrafts located at LEO/GEO orbits,
and between high altitude platforms (HAPs) e.g., stratospheric platforms/aircrafts/UAVs and spacecrafts in GEO orbit. Unlike
the previous space-space scenarios, optical link between an OGS and a GEO satellite entails long-distance propagation through
The major drawbacks of OCL are propagation impairments produced by the interaction of the optical wave with the earth
atmosphere. As a result, special techniques have to be developed to cope with the atmospheric effects [7, 8, 9]. Absorption and
scattering on atmospheric particles as fog, rain, snow and clouds  result in such effects as wave front distortion, beam
wandering and beam spreading, that in turn are responsible for significant signal loss. Moreover, even in the absence of
atmospheric particles, air turbulence introduces propagation impairments resulting in an increased BER . Measurements
showed that optical turbulence decreases exponentially with altitude and that significant turbulent activity can be found up to 25
km altitude. The effects of optical turbulence are stochastic in space and time, producing signal fades on a timescale of several
milliseconds in fixed applications . Fade levels can exceed 20 dB in extreme cases depending on the propagation path and on
the link range . The OCL in the troposphere (i.e. the lower part of the atmosphere up to 10-12 km above ground) are mostly
affected by clouds and fog. Nonetheless, the contribution of hydrometeors is not negligible. Attenuation levels as large as 480
dB/km have been measured in dense maritime fog , while they are definitively lower in moderate continental fog conditions
. Rain can cause attenuations up to 20-30 dB/km at a rain rate of 150 mm/h , whereas specific attenuation through falling
snow can exceed 45 dB/km . Finally, attenuation values due to clouds can be much higher than 50 dB/km .
High attenuation due to clouds can be mitigated by ground station diversity to minimize the probability of cloud coverage .
Additionally, using a longer wavelength (e.g. 10 µm instead of 1.55 µm) leads to lower cloud attenuation, increasing link
availability for thin clouds. Long-distance links are affected mainly by the optical turbulence and there exist no wavelength
window in the optical spectrum that can avoid losses induced by turbulence . However, intensity fluctuations and the
corresponding BER (> 10-9) can be decreased by using a longer wavelength since fluctuations decrease as λ-7/6. The short fades
due to turbulence can be mitigated by multi-site diversity, higher power single mode lasers and different error control techniques.
More generally, atmospheric effects can be mitigated by adaptive optics: for instance, the directivity of the antenna can be
improved by applying the appropriate wave front compensation to the transmitted signal .
The choice of the transmission wavelength depends not only on the characteristics of the atmospheric channel, but also on
factors such as the optical background noise and the technologies developed for lasers, detectors and spectral filters. As a result
of recent technology developments, optical sources and detectors operating at short-wave infrared and long-wave infrared have
become available [19, 20]. Furthermore, ground to space laser communications could benefit from modulation schemes less
sensitive to atmospheric effects (e.g., polarization-based modulation schemes) and non-traditional areas such as
optical/radiofrequency or optical/microwave diversity. The transmission of high intensity ultra-short laser pulses through fog and
clouds has recently been demonstrated . Amplification of optical signals in the atmosphere in a similar way like optical
amplification is performed using optical fibers could also be envisaged.
Optical transmission experiments under different atmospheric conditions provide useful insights on the mechanisms of optical
signal loss and permit to predict the reliability of OCL. In this contribution we discuss the results of laser attenuation
measurements carried out in two different sites. We limit our discussion to moderate continental fog, being fog the most
detrimental propagation impairment for OCL. The paper is organized as follows: Section II reviews some microphysical
properties of fog. A new instrument able to measure both drop size and number density of particles is also proposed. The above
quantities are important for modeling fog attenuation. Section III briefly discusses two classes of methods used to predict optical
attenuation by fog. Finally, Section IV describes our experimental activity: the results of optical measurements carried out in
Milan (Italy) and Graz (Austria) are compared. Last, a method of artificial fog droplet simulation is used to compute attenuation
and a comparison is drawn between predicted and measured attenuation.
II. PHYSICAL BACKGROUND
The observation of fog and clouds and the estimation or measurements of the thickness of fog and cloud layers are necessary
for the design of ground-space OCL. The OGS are located in the troposphere namely, the lower part of the atmosphere, where
weather phenomena like rain, snow, fog and clouds occur. The troposphere extends up to 10 – 12 km above ground. Typically,
very thin fog does not require definite consideration on earth-space OCLs due to the fact that the vertical extent and the density of
such fog are small. However, moderate and dense fog conditions pose a real challenge to the OCL as the density of such fog is
significant enough to completely or partially block the optical link for certain amount of time depending upon its spread and
severity. The corresponding attenuation levels can result in complete link blockage for a long time and are certainly not
acceptable for carrier grade communication technology. In order to predict the transmission loss due to fog, the knowledge of a)
the evolution and dissipation of fog, b) its spatial and temporal distribution, and c) its microphysical properties, are important
issues. Among the microphysical properties of interest, the droplet size distribution (DSD) of fog is crucial as both with the
knowledge of the scattering properties of single droplets permits to estimate fog attenuation. However, the measurement of fog
DSD is not trivial and dedicated data are still scarce in the literature. Moreover, the DSD undergoes spatial and temporal changes
and depends upon the atmospheric conditions (relative humidity, temperature), the environment (e.g. continental or maritime),
and the microclimate.
A. Physical properties of Fog
Fog are composed of very fine droplets of water produced by condensation of water vapors on a pre-existing aerosol
distribution of different nature (dust, smoke, volcanic ash etc). The presence of these droplets act to scatter the light and thus
reduce the visual range near the ground. In meteorology, a fog layer is reported whenever the horizontal visual range is less than
1 km. Fog can extend vertically up to a height of 300-400 m above the ground surface, depending on the height of the layer where
temperature inversion takes place. Optical attenuation is highly correlated with fog intensity, and it is particularly affected by the
number density and the size of fog droplets.
Fig. 1 shows the measured time series of temperature, rainfall intensity, relative air humidity and number of transmission error
seconds (#ES) detected along an optical link. Measurements were performed at Budapest, in October 2002 using a 2.3 km optical
link at 0.785 µm. The test signal was a PCM31 format pseudorandom signal, measured with a Wavetek EDT-135 type E1
analyzer. During the rain event there were no errors on the optical link, as there was no significant reduction in visual range. After
the rain ceased, the temperature decreased while the humidity increased, leading to the formation of fog., which, in turn, produced
transmission errors. This measurement confirms that to estimate the attenuation of free space optical links the drop size, air
humidity and the temperature may provide valuable information.
Fig. 1: From top to bottom: temperature, rain intensity and relative air humidity, against number of error seconds (plotted in green) detected
along an optical communication link through a 2.3 km path.
Different fog types can be classified according to the multiple physical processes acting at the same time to produce the
saturation of air. The relative importance of each process changes from case to case and also with time during a fog event. The
two most common types of fog are a) continental or radiation fog, and b) maritime or advection fog. Different fog types result in
different DSDs. Continental fog in temperate areas mostly occur in winter while maritime fog can occur at any time of the year
during day or night. The typical fog droplet diameter ranges from 0.17 µm to 50 µm [10, 22, 23] regardless of fog type. However,
in the case of moderate continental fog droplets, the mode diameter is about 2-4 µm, while for dense continental fog it is about
10-12 µm [24, 25, 26]. As a comparison, a cloud may contain a good proportion of very small water droplets, but the radii of the
drops that dominate extinction and scattering are in the range of 5 µm to 20 µm, whereas the limiting liquid particle diameter of a
cloud is of the order of 200 µm. Larger drops comprise drizzle or rain.
B. Modified Gamma Distribution model for Aerosols
A number of aerosol models are used to predict particle concentration on the basis of actual meteorological conditions or
according to climatology. It is admitted in meteorology that a given aerosol model cannot be used to predict wave attenuation
under all atmospheric conditions, since the local meteorological conditions dominate the aerosol source function . However,
the size distribution of atmospheric particulates is commonly represented by two analytical functions the lognormal distribution
and the modified gamma distribution (MGDSD). The gamma probability density function (PDF) is actually a general PDF that is
used for nonnegative random variables. r is said to be a gamma random number with parameter α (the shape parameter) and β
(the slope parameter), if the following relationship holds:
Γ α =∫
Let us assume that α=m+1 and β=1/Λ, then (1) becomes:
In meteorology, the DSD of fog can be more conveniently described by a frequency distribution c(r) than the probability
distribution f(r), and MGDSD is the most commonly used for several types of fog and clouds . If c(r) represents the fog
particle concentration, then the number of particles per unit volume and per unit radius increment (r to r+∆r) is:
c(r) = N r exp(-), 0<
Here m,σ , N0, Λ are the four adjustable parameters characterizing the MGDSD. m is called the shape parameter and Λ is the
slope or gradient. These parameters allow the MGDSD to fit a large variety of cases. rmod is the mode radius (µm) of fog particles
that is the radius r at which c(r) is a maximum. Multiplying (2) and (3) by NT (total number of fog particles per unit volume), we
get N0 as:
C. Fog drop size and density measurements
In this section we propose a new measurement method to estimate several microphysical properties of fog. We recommend to
carry out at the same time humidity and temperature measurements in order to facilitate the estimation of OCL attenuation by
refined physical models. The main idea of the fog measurement unit is that the suspended fog particles have a similar size as the
one of dust and smoke particles; therefore a dust sensor could be applicable to detect and measure fog density. The proposed
device is a compact optical dust sensor  operating with a diagonally arranged pulsed infrared emitting diode and a photodiode
to measure the light scattered from a small volume of particles. The principle of the measurement is depicted in Fig. 2. The sensor
is calibrated to measure the density of the floating particles in the air. The sensor can detect particles with the size of 1 µm or
higher, therefore it will be appropriate to detect fog particles. The peak voltage of the output pulse response carries the
information of the fog density. Moreover, the pulse pattern informs about the size of the particles.
Fig.2: Operational principle of an optical sensor envisaged to measure the number concentration and the size of fog droplets.
The device is factory-calibrated to measure the mass density of the suspended particles in the air and gives this value in mg/m3.
The sensor should be operated continuously, driving the infrared emitting element with the recommended, 10 ms pulse cycle. The
fog particles are reflecting the light pulse to the receiver element, and after amplification and shaping the output pulse the signal
is ready for the further processing. The fog density can be determined in two ways. On the one hand the factory calibration of the
sensor device ensures that the peak voltage of the output pulse varies linearly with the particle concentration in the air, and the
concentration/voltage characteristic is also given. In addition, the sensor’s pulse cycle is high enough to test the density by
examining the variation of the consecutive output pulses. In case of moderate fog conditions the peak voltage falls down between
two or more detections, while dense fog may cause constantly high output pulse peaks (Fig. 3).
Fig, 3: Detection of fog density. A sequence of short optical pulses is transmitted through a sampling volume filled with fog droplets. As fog
becomes denser, the shape of the received pulse sequence changes according to fog density.
Moreover, the pulse pattern yields information about the size of the particles. The pulsing of the infrared emitter will be
uniformly performed with the recommended cycle and width. It can be foreseen, that the shape of the output pulse (rising and
falling time, peak voltage) will be the function of the drop size, however the effective values can be determined only after the
analysis of the measurement data, i.e. a calibration process is needed.
The measurement device is now in the design phase; a prototype is already in operation. The system consists of a fog sensor,
air humidity and temperature sensor, and an embedded microcontroller. The final system is intended to be installed at the
cooperating partners as a continuously operating measurement device. By comparing the measured data with other different type
of sensors the calibration can be performed and the device applicability can be tested.
III. OPTICAL ATTENUATION BY FOG
Two broad classes of models can be envisaged to estimate fog attenuation: a) empirical models, based on the conversion of
some easy-to-measure atmospheric parameters, as visual range, into optical attenuation, and b) microphysical models, that require
the knowledge of the structure of fog and of the scattering properties of fog particles [24, 28, 29]. As far as optical transmission
through fog is considered, scattering prevails over absorption, because the refractive index of water has a negligible imaginary
part. Furthermore, as the size of fog droplets is the same order of magnitude as the transmitted optical wavelength, Mie scattering
theory must be used (assuming fog droplets are spherical in shape). In the following discussion it is assumed that fog intensity is
uniform along the entire transmission path.
A. Fog attenuations from visual range
Fog attenuation can be estimated, from visual range for a 2% transmission threshold
over the atmospheric path
measurements, using the following relationship:
λ ≅ β λ
( ) = ().
or, for a 5% transmittance threshold the above equation can be re-written as:
λ ≅ β λ=
where V is the visual range in km, λ is the transmission wavelength in nm, βa(λ) is the aerosol scattering coefficient, γ(λ) is the
total extinction coefficient and q is the size distribution coefficient of scattering reported and discussed in the literature [13, 14].
According to Kruse and Kim models, for different visual range regions, q takes up the following values:
γ( ) ( ) =().
Moderate fog Dense fog
The two models have the same value of q for a visual range greater than 6 km. However, the difference between the two lies in
the shorter visual range ranges as Kim model further distinguishes three regions of visual range when V<6 km. It is important to
mention that these two models were not developed specifically to characterize fog. Nonetheless, they provide a reasonable
estimate of fog attenuation for different visual range situations.
B. Atmospheric Transmittance
For OCL concerning ground-ground or ground-space communications, in order to quantify the attenuations experienced by the
optical beam under different fog conditions, we must take into account the thickness of fog. If Pt and Pr are the transmitted power
and the received power, respectively, the atmospheric transmittance T is given by:
where γ is the atmospheric attenuation coefficient and L is path length, assumed to be uniformly filled with fog particles. When
one applies Mie theory to predict optical attenuation by fog, there are few underlying assumptions that, however, are not believed
to have a large impact on the accuracy of calculations [23, 24]. First of all, Mie theory assumes the scatterers are spherical in
shape. Moreover, the scattered light has the same wavelength as the one of the incident light, the particles are acting
independently. Finally only single scattering takes place while multiple scattering effects are negligible [23, 24]. If individual
particles don’t have a high forward scattering efficiency and their number concentration is low enough, path attenuation can be
calculated through the single scattering theory, which assumes that the energy absorbed or scattered by a particle is lost.
However, if this is not the case, the contribution of scattered light to the amount of energy transmitted through the medium cannot
be neglected. Multiple scattering is governed by the radiative transfer equation, stem from the conservation of energy. At very
dense concentration (the volume occupied by particles should be much larger that 1% of the total volume), the diffusion
approximation describes well the process in terms of a random walk of the photons in the medium. Simulations based on a model
of random walk of photons through the atmosphere  show that in the case of thick fog (visual range less than 250 m), the path
attenuation in the first optical window is a few percent smaller (on a dB scale) than the one predicted by a single scattering
In general, the attenuation per unit length γ can be written as:
where αm and αa are the molecular gas and aerosol absorption coefficients, while βm and βa are the molecular gas and aerosol
scattering coefficients, respectively. When calculating fog attenuation, the contributions introduced by molecular gases scattering
and absorption and by aerosol absorption can be ignored, hence (11) simplifies to:
( ) ( )
γ λ β λ
Qnr c r dr
where r = D/2 is the radius of the fog droplets, and γ(λ) is the specific attenuation measured in 1/km calculated by summing up
the attenuation effect of all individual fog droplets present per unit volume and per unit increment of radius (r). nr is the real part
of the complex refractive index of the aerosol particles and Qd is the normalized Mie scattering cross-section and the factor πr2 is
introduced here for de-normalizing with respect to the area of an equivalent sphere.
In Fig. 4 we plotted several MGDSD profiles for different fog conditions. The corresponding MGDSD parameters of
continental fog only are listed in Table 1. The parameters Nr and LWC in Table 1 show the fog droplets number density and the
liquid water content respectively. Fig. 5 shows the specific attenuation for selected wavelengths at 10°C using (13) for moderate
continental fog. The contribution to the whole attenuation caused by fog droplets of radius between r and r + dr, there are slight
indications of wavelength dependent attenuation for optical links under continental fog conditions i.e., at longer wavelengths the
attenuations are smaller when compared with shorter wavelengths.
Radius (micron)Radius (micron)
Specific attenuation (dB/km)
Specific attenuation (dB/km)Specific attenuation (dB/km)
Specific attenuation (dB/km)
650 nm650 nm
750 nm750 nm
850 nm850 nm
950 nm950 nm
1050 nm1050 nm
Fig. 4: Modified gamma DSD for different fog conditions
Fig. 5: Predicted specific attenuation at 10 ºC for moderate
continental fog conditions.
STANDARD VALUES OF MODIFIED GAMMA DISTRIBUTION PARAMETERS FOR CONTINENTAL FOG CONDITIONS [25, 26]
Fog type m N0
It is interesting to analyze in more detail the effect of temperature variations on the complex refractive index and complex
permittivity of water as these two parameters affect Mie scattering calculations. The complex refractive index and complex
permittivity were calculated using the method provided by P. S. Ray  and are plotted for wavelengths up to 10 µm in Figs. 6
and 7 respectively. The real part of the complex refractive index corresponds to scattering while the imaginary part corresponds
to absorption. There are no appreciable variations in the complex refractive index, indicating negligible attenuation by absorption
effect. This justifies the assumption that for optical transmission through fog, the losses due to absorption can be ignored as
compared to the scattering loss. The refractive index is insensitive both to temperature variations (from -10°C to +10°C) and to
wavelength variations (from 0.550 µm to 10µm), its real part changing from 1.330 to 1.342. We recall that temperature remains
almost stable during a fog event and that fog keeps the temperature of the surrounding air steady since its main constituent is a
liquid water droplet, that has an high specific heat capacity (i.e. the amount of heat per unit mass required to raise the temperature
by one degree Celsius). We also computed the complex permittivity for a temperature ranging from 2ºC to +10ºC (Fig. 7). There
are no significant changes in its real and imaginary parts for wavelengths up to 10 µm. The real part of the complex permittivity is
comprised between 5.3272 and 5.359.
Fig. 6: Complex refractive index of water for three different
Fig. 7: Complex permittivity of water for different temperatures
In the following we discuss the results of optical transmission measurements in Milan and in Graz, tailored to investigate the
impact fog attenuation on OCL.
A. Experimental setup
The transmission measurements at Graz (Austria) were carried out along two colocated optical links of different length (79.8 m
and 650 m respectively). The optical transmitters have two independent LED based light sources. The source of the shorter link
operates at 850 nm emitting a 8 mW average optical power. The beam divergence is 2.4 mrad and the average radiated power
after the lens is about 3.5 mW. The longer link transmits a 950 nm optical beam with a 0.8 mrad divergence. In both cases the
received power is sampled once every second. The measurements in Graz were carried out during the winter months in 2004/05
The 319-m optical link in Milan (Italy) is installed within the campus Leonardo of Politecnico di Milano. The experimental
setup consists of a commercial optical link Terescope 3000 formed by two identical transceivers, which can transmit both data up
to 155 Mbps and a single carrier at 785 nm. The Terescope 3000 system was manufactured by Optical Access (now MRV). The
transmitters on each side of the link are assembled in triangular shape and use three identical and independent semiconductor
laser diodes having nominal output power of 10 mW and the beam divergence is 2.5 mrad. The data are sampled every 1 s.
Visual range is measured by an optical transmissiometer (Model 6100) manufactured by Belfort Instrument while the
meteorological quantities (temperature, relative humidity, solar radiation, rain rate, etc) are measured by a weather station
manufactured by Davis. The visual range meter and the weather station are placed near one of the two terminals of the optical
link. Almost continuously from 2003 to 2006, this setup has been collecting data.
Since measured laser attenuation is expressed below as specific attenuation (i.e. per unit length), it is assumed that the
scattering medium is uniform along the measurement path. As the spatial distribution of fog depends on several factors such as
the atmospheric conditions, the environment, and the microclimate in the measurement site, it is not easy to indicate a distance
over which the assumption holds. In the case of the link in Milan and above all of the longer link in Graz, we cannot rule out the
chance that the medium is not uniform along the entire propagation path, especially during the process of fog formation and
dissipation. Similar studies regarding propagation through rain have shown that non-uniform conditions along the path result in
diminished propagation losses .
The measurement database considered in this work consists of six winter months both for Graz and Milan but in different
years. Figs. 8a and 8b show the profiles of visual range and laser attenuation during a fog event occurred in Milan on 11 and 12
January 2005. The system goes into outage every time the measured attenuation exceeds the allowable dynamic range for
atmospheric losses; the dynamic range of the receiver is 21 dB. The optical attenuation, as estimated from visual range through
the models discussed in Section III (Fig 8b), has a maximum value of 154.13 dB/km, while its median value is 34.55 dB/km,
nearly equal to the median value found for the sequence of measured attenuation (34.98 dB/km). Indeed, if we compare the two
time profiles in Fig. 8b, we see that there is a good agreement between measurements and estimates except in the case of the peak
attenuation in the morning of 12th January, when the estimated values are much higher (this behavior could be ascribed to the
sharpness of visual range decrease that may be a symptom of non-uniform visual range conditions along the path. In fact, during
this short event the two curves have a similar profile, but the measured attenuation curve is slightly delayed in time). Figs. 9a and
9b are relative to an heavy fog event occurred in Graz on 2 February 2006. The peak, mean and median values of specific
attenuation, derived from actual measurements made by the transmissiometer, are 224 dB/km, 130 dB/km and 80.24 dB/km,
respectively. These are the largest attenuation values measured during the experimental campaign. Figs. 8c and 9c show the
variations in specific attenuation observed in Milan and in Graz, as obtained by subtracting the attenuation at the nth sampling
time to the one at the (n+1)th sampling time. When attenuations are comparable in the two cases, similar values of attenuation
variations were detected.
23:00 0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00
V is ibility (k m )
23:00 0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00
Attenuation (dB km-1)
23:00 0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00
A ttenuation Diff. (dB km-1
Fig.8: Time profiles of a) visual range, b) specific attenuation, and c) differences in specific attenuation during a fog event occurred in Milan
on 11 and 12 January 2005. In b) two profiles are shown: the measured laser attenuation (black curve) and the attenuation as estimated from
visual range (gray curve).
Fig. 9: (a) Time series of attenuation during a fog event in Graz and b) corresponding
changes in attenuation.
During the measurement period, from September to February, 20 events were recorded in Milan (see Table 2), during which a)
no precipitation was detected, and b) path attenuation exceeded the value corresponding to 1 km visual range (see (7-a)) for more
than 30 minutes. The above visual range value is conventionally used as the upper limit for fog occurrence, according to the
International Visual range Code. In 7 cases, the attenuation exceeded the dynamic range of the receiver. The value of 60 dB/km
corresponds to a visual range around 250 m, that is roughly the limit for moderate fog, lower values being associated with thick
fog and dense fog. In about half of fog events detected in Milan, the maximum attenuation exceeded 60 dB/km. On the whole,
attenuation was in excess of 60 dB/km during about 0.86% of the measurement time, a non-negligible value at all. Two-year
measurements showed that the above value of attenuation was exceeded during 0.3% of time, confirming that the winter months
are the most challenging for optical propagation in the measurement area.
A similar analysis has been carried out on Graz data. the results of which being reported in Table 3. Differently from Milan,
measurements in Graz were not continuous due to limitations of data storage hence some episodes could not be detected during
the measurement period. In Graz, only two events have maximum attenuation less than 60 dB/km, while all but two have a 99%
attenuation above it.
Table 2: Fog episodes detected in Milan from September 2004 to February 2005. The last five columns show laser attenuation values
measured during fog: Maximum and mean attenuation are shown, along with the 50%, 90% and 99% percentiles of the attenuation
distribution. When fog attenuation exceeded the dynamic range of the laser receiver, any or all of the above values could not be calculated.
Table 3: Fog episodes detected in Graz from September 2005 to February 2006. See previous Table 2 for further explanation.
In Fig. 10 we compare the attenuation values detected in the two sites by plotting the median attenuation against the 99th
percentile of each fog event. It can be noticed that Milan’s moderate fog data exhibit a certain degree of correlation between the
two selected values of the attenuation distribution. On the other side, Graz data are much more scattered: even the events that
exhibit median attenuation values similar as the ones in Milan, have a 99th percentile spreading from 50 to more than 200 dB/km.
Therefore not only fog intensity is higher in Graz than in Milan but also the physical process of fog formation, growth and
dissipation seem to be different. The above differences can be explained as follows. Despite Milan and Graz are located in the
same temperate area, there are two major differences which are expected to affect to some extent measurements: a) the climate
during winter is colder in Graz, where daily temperature minima are often below 0°C, while in Milan temperature rarely falls
below 0°C, and b) Milan is a large city and despite the measurement site is located about 3 km from the centre of the city, the
microclimate is that of a dense urban area. On the other side, Graz is a midsize town, with the optical link being located in a
suburban environment with no tall buildings and many wide open areas around. Therefore it is reasonable to declare that fog
episodes are heavier in Graz than in Milan.
99th percentile of fog attenuation (dB/km)
Median fog attenuation (dB/km)
Figure 10: Median laser attenuation against attenuation at the 99th percentile as detected in Milan and Graz during fog events
C. Artificial fog droplet simulation
By comparing the observed attenuation values with the ones obtainable by the DSD model previously described, one can state
whether the model is suitable to describe fog process in Milan and/or Graz eventually after tuning of some parameters, or whether
a different DSD model is necessary. To this aim we have simulated the fog process (DSD), as outlined in the following.
In simulating artificial fog droplets according to a MGDSD, no reference describes how to simulate random numbers from the
modified gamma frequency distribution. However, several methods are available to simulate random numbers from a probability
distribution such as the transformation method  and the rejection method . The method proposed here is employed to
generate modified gamma-distributed variables by considering a monotone transformation h(x). It can be shown that if x has
then y=h(x) follows the gamma density Г(a). Moreover, if g(x) is close to c-x2/2 for some constant c, then a rejection method from a
normal density can be developed. After generating a diameter D that follows the probability gamma distribution, the fog drop
distribution is derived by using the following relationships: f(D)dD
We can then calculate c(D) by binning D into several classes with bin width ∆D. The probability density for each class is
P( ) f(D)dDdD . D
where C∆ and c(∆) are the count and distribution of drops with diameter D1 ≤ D ≤ D2 respectively. From (15), it is robust that
C∆=C(∆). ∆D and the fog DSD is therefore given by:
The fog DSD is correctly reproduced provided the following holds
(c( ). D
D. Predicted DSD parameters analysis
The predicted peak attenuation value using the standard parameters of the MGDSD for moderate continental fog conditions
and the Mie scattering theory, results in attenuation of about 180 dB/km. However, the measured peak specific attenuation value
for Graz was up to 224 dB/km and for Milan it was around 154 dB/km (deduced from visual range). It means that the standard
parameters values are not applicable to model fog attenuation for any fog environment and any location. Therefore the need arises
to find a new set of MGDSD (average) parameters for each of the two measurement locations, since the DSD can vary from
location to location . Hence, before installing a link at a particular location the knowledge of the local environment
parameters, as the fog DSD, should be welcome in order to realistically estimate optical attenuation at that particular location.
Additionally, the knowledge of the frequency of occurrence of a particular fog type and of its seasonal and diurnal dependence is
In our effort here, we try to find a new set of MGDSD parameters for moderate continental fog, to match the observed peak,
median and mean values of measured specific attenuation in Graz and in Milan. The predicted attenuation is obtained through
Mie theory by keeping the mode size of fog droplets to 2 µm, which is assumed to adequately represent moderate continental fog.
We simulated fog droplets with sizes from 0.17 µm to 50 µm with a step size of 0.1 µm. The new sets of MGDSD parameters are
plotted in Figs. 11 and 12 and are also presented in Table 4. The corresponding specific attenuation values are shown in Figs. 13
and 14. It can be noticed that the contribution from 1-5 µm particles is about 96 % of the overall attenuation, while 5-50 µm
particles are responsible for only 4% of attenuation. It means that smaller droplets play a critical role in attenuating the optical
signal: as their size is comparable with normally used optical wavelengths, simple methods are inadequate and therefore use of
Mie theory is mandatory.
It is important to remember that the DSD can change at a same location with time . Furthermore, there are also temporal
variations during the life cycle of a fog event, as well as spatial variations at any given time instant. Therefore, the identical
visibilities during fog formation and dissipation might correspond to different size distributions. This is the reason why we speak
of average DSD parameters.
PROPOSED MODIFIED GAMMA DSD PARAMETERS
This contribution provides newly estimated values for the modified gamma drop size distribution parameters under moderate
continental fog in Graz (Austria) and Milan (Italy), respectively. Additionally, we propose a fog droplet and density measurement
setup which can be useful in improving future OCL design for fog environments. We also provide a comprehensive comparison
between measured and predicted fog attenuation obtained by using Mie scattering theory. In general, it was observed that the
moderate continental fog conditions are stable at both locations because maximum temperature variations during fog events are
limited within 10˚C.
It has been pointed out, although not discussed in detail in this paper, the importance to test the correlation between attenuation
values for different kinds of fog environments and atmospheric parameters like air temperature, complex refractive index, relative
humidity and wind speed. The detailed knowledge of their implications would be very important in view of channel design, allowing
realistic attenuation estimates for different fog conditions. This in turn would improve the availability of OCL by ensuring desired
quality of service demanded by the broadband communication service providers for harsh environments like fog.
Fig. 11: Estimated parameters for MGDSD for peak, standard, mean and median
values of specific attenuations at Graz
Fig. 12: Estimated parameters for MGDSD for peak, standard, mean and median
values of specific attenuations at Milan
Fig.13: Predicted specific attenuations for Graz using MGDSD and Mie
Fig.14: Predicted specific attenuations for Milan using MGDSD and Mie
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