atmospheric scattering and turbulence modeling for ultraviolet … · 2020. 7. 13. · scattering,...

RESEARCH ARTICLE

Atmospheric scattering and turbulence modelingfor ultraviolet wavelength applications

Dario De Leonardis1 & Saverio Mori1 & Silvia Di Bartolo2 & Frank-Silvio Marzano3

# Springer Nature Switzerland AG 2020

AbstractThe recent proliferation of free-space optics (FSO) technologies and development ofpertaining research have led to exploit the whole optical bandwidth from infrared andvisible up to deep ultraviolet (UV) in communications. Within this context, we decided tofocus on UV FSO communication and remote sensing potentials by presenting a phys-ically based single-scattering channel model, UVatmoScat, aiming to include, withrespect to previous analyses, most atmospheric variables like fog, precipitations, aerosols,and turbulence: indeed, they may affect the performance of UV links in short-rangeoutdoor applications adopting non-line-of-sight (NLOS) configuration, as here consid-ered. This analytical single-scattering model computes the temporal impulse response andpath loss and has been validated through Monte Carlo. The former provides the frequencycharacteristic of UV-NLOS propagation links on varying of atmospheric conditions, withdifferent NLOS geometries and receiver apertures. 3-dB bandwidth numerical resultsshow UV-NLOS systems as more significantly affected by the link geometric featuresthan weather perturbations, demonstrating supportive to the choice of frequency constant-envelope modulations. Nevertheless, meteorological perturbations have to be properlyconsidered to better optimize the transmission power in UV telecommunications, as wellas in ozone or aerosol ground-sensing applications. The proposed UVAtmoScat model isa suitable, self-consistent, and effective tool for this purpose.

Keywords Ultraviolet communications . Single scattering . Atmospheric effects . Monte Carlomodel . Physically based channel model . Path loss . Channel bandwidth

1 Introduction

Free-space optics (FSO) technologies are recently emerging in different areas of applicationsuch as wireless communications and remote sensing (Ghassemloy et al., 2012). In the contextof wireless communications, FSO links represent an important solution for high throughput

https://doi.org/10.1007/s42865-020-00010-9

* Frank-Silvio [email protected]

Extended author information available on the last page of the article

Bulletin of Atmospheric Science and Technology (2020) 1:205–229

Published online: 26 March 2020

http://crossmark.crossref.org/dialog/?doi=10.1007/s42865-020-00010-9&domain=pdf

mailto:[email protected]

transmission by atmosphere. The main advantages of FSO technology with respect to radiofrequency (RF) systems consist in higher transmission capacity, no need for frequencyallocation, free licensing, and easy deployment. On the contrary, optical waves propagatingin outdoor conditions may be absorbed and scattered by hydrometeors and aerosols, whichmay generate an extremely high attenuation in the received power (Kalighi & Uysal, 2014;Drost & Sadler, 2014). To mitigate this impairment, a hybrid approach switching between FSOand RF channels has been proposed in recent literature (Usman et al., 2014), but bad weatherconditions still remain one meaningful reason for the link performance collapse.

From another perspective, progress in semiconductor optical sources and detectors hasmotivated interest, within the broader context of FSO systems, in ultraviolet (UV) communi-cation, particularly in the non-line-of-sight (NLOS) configuration (Xu & Sadler, 2008). UVcommunications undergo abundant atmospheric scattering and attenuation, but provide solar-blind operation inside the deep UV band (wavelength of 200∼280 nm). The advantages of aNLOS communication link result evident: UV transceivers working in optically thick atmo-spheric conditions may exploit the scattering effect to reduce power losses (Ding et al., 2009);moreover, due to high attenuation, a UV signal cannot be easily detected out of extinctionrange. The first relevant studies on UV communication have been conducted since the 1960s,fundamentally for long-range communications based on high-power UV light sources (Koller,1965). Only recently there have been advances in low-cost, small size, and low-power devices.In particular, the diffusion of high bandwidth deep UV wave emitting diodes (LEDs) andavalanche photodiodes (APDs) has led to the studies of low-power short-range UV commu-nication (Shaw et al., 2005). An insightful experimental test comparing performance producedby both NLOS and LOS UV communication devices can be found in (Siegel et al., 2004).

On the other hand, by looking at FSO remote sensing applications, UV technology showsto be a meaningful improvement step for research of solving keys in the field of ozone andaerosol ground-sensing, given that it adopts the bandwidth where both these categories ofparticles interact and a very peculiar link geometry, that has been essentially configured toexploit atmospheric scattering phenomenon and eventually provide their concentration along avertical path. Hence, standing to this observation, by speaking of UV-NLOS communicationlinks in the progress of this work, we include at the same time ozone/aerosol ground-sensingapplications, to which the model that is going to be presented can be considered much suitableas to telecommunications.

Modeling of UV-NLOS links has been successfully approached through a single-scatteringanalytical model supported by Monte Carlo simulations with the aim to analyze its perfor-mance in terms of energy path loss through simplified atmospheric models (Elshimy &Hranilovic, 2011). For short-range UVapplications (about 5∼30 m), single-scattering assump-tion is considered reasonably adequate. Nevertheless, in a realistic UV-NLOS link, it must betaken into account not only the absorption and scattering phenomena due to atmospheric gasesbut also attenuation caused by water particles (like raindrops or fog droplets) and mediumfluctuations due to turbulence. Disclosing some details, the semi-closed-form framework forUV-NLOS temporal impulse response can be improved by considering that not only thephysically based scattering effects due to atmospheric changing conditions can be ingestedbut also the turbulence phenomenon can be contemplated as, for instance, modeled by a log-normal random variable scaling the received signal in amplitude (Liao et al., 2015): all theseenvironmental factors evidently contribute to phase-function anisotropy. The closed-formimpulse response model shown by (Sun & Zhan, 2016), from which one may derive bothpath loss and frequency characteristic, cannot be extended to cases where the receiver has a


finite aperture (Elshimy & Hranilovic, 2011), because it has been validated for a point-to-pointtransmission. On the other hand, the multiple-scattering analytical model, supported by MonteCarlo simulations, which is depicted in (Yuan et al., 2016), is restricted to a three-orderscattering effect and is based on very limited hypotheses, such as isotropic conditions ofscattering, clearly simplifying the final path loss expressions.

In order to tackle these open issues, in this study, a more realistic and detailed UV-NLOSmodel, named as UVAtmoScat, has been presented with the aim at analyzing the expectedbehavior of short-range UVoutdoor communications in various environmental conditions. It iswell-documented in the literature the extreme sensitivity of FSO technologies to atmosphericperturbations (Grabner & Kvicera, 2014), becoming more evident when transmissions movetowards UV shorter wavelengths. UVAtmoScat is a physically based single-scattering modelable to simulate most atmospheric events, such as turbulence, aerosol scattering, and ozoneabsorption as well as fog and liquid/solid precipitation, so that their impact can be quantita-tively considered in terms of received signal. By adopting the UVAtmoScat model, the system3-dB bandwidth can be computed together with its dependence on weather changing condi-tions and NLOS link specifications.

This paper is organized as follows. In Section 2, we discuss the channel characterizationand the main parameters to depict exhaustively atmospheric effects and wave-matterinteraction phenomena. Within Section 2, there are subsections 2.1 focusing on absorptionand scattering events, and 2.2, related to turbulence modeling. Section 3 describes theUVAtmoScat model configuration describing a short-range UV-NLOS link working insingle-scattering conditions. This section is subdivided in 2 parts: subsection 3.1 concernsthe Monte Carlo approach, whereas subsection 3.2 looks at analytical modeling. InSection 4, the numerical results of the UVAtmoScat model are discussed in terms ofUV-NLOS path loss (in subsection 4.1) and channel frequency characteristics (insubsection 4.2) assuming various atmospheric channel conditions. Section 5 concludesthe paper.

2 Channel atmospheric characterization

The optical signal propagating through a free-space link is very sensitive to visibility andweather conditions (such as, for instance, fog, turbulence, and rain). This characteristic reflectsinto the difficulty of properly designing FSO links where all stochastic atmospheric featureshave to be accurately considered. Within this framework, it becomes very important to assessthe physical assumptions standing behind the optimal performance in wireless opticsapplications.

2.1 Atmospheric absorption and scattering phenomena

A photon is a quantum of electromagnetic wave energy. Random discrete media, such asatmosphere, are characterized by randomly distributed finite constituents, either molecules orcomplex particles, with a peculiar size distribution. The interaction between photons andatmosphere can yield two different extinction events: (i) absorption; (ii) scattering. Bothphenomena are described by the Beer-Lambert law (Mori & Marzano, 2015) through theabsorption and scattering coefficients, depending on wavelength ⋋: ka(⋋) and ks(⋋), respec-tively. The total extinction coefficient is given by

Bulletin of Atmospheric Science and Technology (2020) 1:205–229 207

ke ⋋ð Þ ¼ ka ⋋ð Þ þ ks ⋋ð Þ ð1Þall measured in [m−1] and representing the inverse of average distance between two successive interaction eventsof absorption or scattering. For an overall characterization, we can define the single-scattering albedo of randomdiscrete medium as

ω0 ⋋ð Þ ¼ ks ⋋ð Þke ⋋ð Þ ð2Þ

Another important optical parameter is the phase-function asymmetry factor g(⋋), whosevalues are between − 1 and 1, indicating the direction of prevailing scattering due to particleensembles (g(⋋) = 1 means forward, g(⋋) = 0 isotropic, g(⋋) = − 1 backward scattering).

By focusing on UV wavelengths, the primary atmospheric constituent involved withabsorption in the solar-blind UV band is ozone (Feng et al., 2008). The ozone absorptioncoefficients are given by:

ka ⋋ð Þ ¼ c ⋋ð ÞPO3 ð3Þwhere c(⋋) is the spectral absorption (see (Inn & Tanaka, 1959) for details); PO3 is the partialpressure corresponding to ozone concentration which has a nominal sea level value between20 and 30 ppb.

On the other hand, scattering by air molecules can be efficiently described through theRayleigh theory formula:

ks Rð Þ ⋋ð Þ ¼24π3 n ⋋ð Þ2−1

h i26þ 3ρn

⋋4N n ⋋ð Þ2 þ 2h i2

6−7ρnð4Þ

where N is number density, in [cm−3], and n(⋋) is the refractive index of air. (6 + 3ρn) / (6 −7ρn) is the depolarization term, also called as King factor. For a standard air (whose pressure is760 mmHg and temperature 288.15 K), N = 2.54743 × 1019 cm−3 and ρn = 0.035 (Feng et al.,2008), whereas air refractive index has the form:

n ⋋ð Þ−1½ � � 108 ¼ 5791817

238:0185−⋋−2 þ167909

57:362−⋋−2 ð5Þ

with ⋋ measured in micrometers.But not only Rayleigh molecular scattering, rather aerosol scattering phenomena can

evidently affect and degrade atmospheric visibility. Aerosol scattering is usually treated byMie theory due to the size of aerosol diameters larger than UV wavelengths. The Miescattering coefficient is parametrically expressed by:

ks Mð Þ ⋋ð Þ ¼ 3:912

Va−ks Rð Þ 550ð Þ

� �⋋550

� �0:585jVa1=3

ð6Þ

where Va is aerosol visibility (in km), ⋋ is measured in nanometers, and ks(R)(550) is theRayleigh scattering coefficient at 550 nm.

In a general atmospheric scenario, a complete parameterization of scattering coefficientscannot exclude hydrometeors. In (Mori & Marzano, 2015), the authors have proposed themicrophysically oriented atmospheric particle scattering (MAPS) model to simulate an amplehost of scattering effects due to atmospheric water particles, like raindrops and fog droplets,


randomly distributed along the FSO link. By using a power-law regression analysis, theaverage behavior of extinction and scattering coefficients at UV wavelengths can be expressedwith respect to particle water contents, precipitation rates, and visibility, in case of fog.

By using MAPS parametric model at 260 nm, extinction and scattering coefficient,phase-function asymmetry factor, and albedo are shown in Fig. 1 for rain drops and inFig. 2 for fog particles. The two figs confirm the irrelevance of absorption phenomenafor these kinds of hydrometeors within the solar-blind UV band since albedo is alwaysvery close to 1. The main contribution to absorption is given by atmospheric gases,substantially by ozone. In this respect, it is not unlikely to consider only the ozone gas(and eventually other aerosols) for describing absorption within UV channel model, asjust observed in (Feng et al., 2008).

An atmospheric scenario is very often a complex combination of molecules, aerosols, waterdroplets, and hydrometeors depending on several environmental factors. If UV scatteringphenomena can be considered statistically independent, the overall contribution to scatteringis given by summing the respective coefficients:

ks ⋋ð Þ ¼ ks Rð Þ ⋋ð Þ þ ks Mð Þ ⋋ð Þ þ ks Hð Þ ⋋ð Þ ð7Þ

Fig. 1 Rain drop simulated Mie coefficients. Extinction and scattering coefficients are measured in [dB/km],asymmetry factor, defined in this case as the asymmetry factor g (that will be considered in the next section),multiplied for scattering coefficient ks, measured in km−1, are shown as function of precipitation rate. The modelpower-law regression curves above-shown (Mori & Marzano, 2015) have been computed at the wavelength of260 nm. CW98 and O78 models utilized for comparison with simulated data can be found in (Mori & Marzano,2015) and reference therein


where ks(H)(⋋) is the scattering coefficient produced by MAPS model when a certain weathercondition is set: e.g., rain or fog—it depends on the different characteristics of droplets (interms for instance of water content) and on their spatial distribution (in terms, e.g., ofprecipitation rate or visibility) (Mori & Marzano, 2015). The Appendix summarizes the mainaspects of MAPS parameterization that are here adopted.

In the following text, we will remove for simplicity the wavelength ⋋ from (2), (7), andasymmetry factor expressions. It is worth pointing out that each scattering contribution (frommolecules, aerosols, or hydrometeors) is also associated with a different scattering phasefunction as it will be discussed in the next section.

2.2 Atmospheric turbulence

Turbulence, also known as scintillation, is due to random fluctuation of the atmosphericrefractive index depending on temperature, atmospheric pressure, and wind speed. Accordingto (Liao et al., 2015), the turbulence attenuation in dB for a generic link distance Δ derives byRytov approximation (Dordova & Wilfert, 2009; Al Naboulsi et al., 2005):

αΔ ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23:17C

2nk7

6Δ

11

6

rð8Þ

Fig. 2 The same analysis (still on the wavelength of 260 nm) which has been shown in Fig. 1, but for fog dropletsimulated Mie coefficients expressed as function of visibility. K62, K01, AlN04a, and AlN04r models utilized forcomparison with simulated data can be found in (Mori & Marzano, 2015) and reference therein


where k = 2π/⋋ is wave number and C2n

is the index of refraction structure parameter,

measured in [m−2/3]. The latter can be estimated through semi-empirical formulas movingfrom the knowledge of atmospheric factors like temperature, pressure, and humidity (Cheru-bini & Businger, 2013; d’Auria et al., 1993): it usually ranges from 10−17 m−2/3 for weakturbulence to 10−12 m−2/3 for a strong one (Vitasek et al., 2011). The linear attenuationcorresponding to (8) is:

LαΔ ¼ 10−αΔ10

ð9Þ

Optical turbulence effects could also produce irradiance fluctuations. In the context of short-range communications, it can be easily approached a log-normal distribution to derive theprobability density function of received intensity. With irradiance denoted by I, the log-normalmodel distribution is given by:

PT Ið Þ ¼ 1

I

ffiffiffiffiffiffiffiffiffiffiffiffi2πσ

2l

r exp

In I=I0ð Þ þσ2l2

264

3752

2σ2l

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

ð10Þ

where I0 stands for the mean of accumulated irradiance and the log amplitude optical varianceterm σ2, for a plane wave, is equal to:

σ2l¼ 1:23C

2nk

76Δ

116 ð11Þ

The model adopting (8) and (11) is valid for a point-to-point communication, as stated in(Dordova & Wilfert, 2009; Al Naboulsi et al., 2005), whereas in our case, with a receiverphysical extension (point-to-receiver), a better approximation is provided by (Vitasek et al.,2011). In the latter, turbulence effect computation considers the diameter of receiver opticallens (i.e., effective receiver area): on the basis of mathematical analysis of turbulence inatmosphere presented by Larry and Andrews (Andrews, 2004), in the case of point-to-receiver photon path, by defining the receiver lens diameter as DR, Eq. (11) can be substitutedby:

σ2lDRð Þ≃exp α1 þ α2ð Þ−1 ð12Þ

where

α1 ¼0:49β

2R

1þ 0:18d2Aþ 0:56β

12=5R

� �7=5ð13Þ


and

α2 ¼0:51β

2R

1þ 0:69β12=5R

� �−5=6

1þ 0:9d2Aþ 0:62d

2Aβ12=5R

ð14Þ

with βR ¼ 0:5C2nk7=6Δ11=6—that is the Rytov variance of optical intensity for spherical

wave—and

dA ¼ffiffiffiffiffiffiffiffiffiffiffiffi2πDR

4⋋Δ

rð15Þ

where Δ can be defined as the distance between source-point of radiation and the center ofreceiver area (in our UVAtmoScat model, it corresponds to the center of lens circle).

Turbulence attenuation along the generic distance Δ, expressed in dB, is given by:

αΔ DRð Þ ¼ −10log10 1−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2lDRð Þ

r� �ð16Þ

so that the corresponding (adimensional) attenuation is:

LαΔ DRð Þ ¼ 10−αΔ DRð Þ

10ð17Þ

which is finally used within model implementation.

3 UVAtmoScat model configuration

Let us consider a generic UV communication link in NLOS configuration like the one depictedin Fig. 3. We define the transmitter and receiver elevation angles as θ1 and θ2, respectively,transmission beam width and receiver field of view (FOV) angles as φ1 and φ2. We denote theportion of volume illuminated by transmitter and receiver patterns as the common volume VQand the communication range, connecting transmitter to receiver aperture center, as distance r.

TxRxr

VQ

θs

Fig. 3 NLOS UV communication link. Generic geometric scheme


In UV-NLOS short-range link transmissions, single scattering is the dominant effect and theother contributions can be considered negligible. Indeed, Van de Hulst criterion asserts thatsingle-scattering approximation can be considered valid until the optical thickness, defined asker, remains ≤ 0.1 (Luettgen et al., 1991). Under this condition, it is possible to use theanalytical framework developed in (Elshimy & Hranilovic, 2011) and set up a comparisonbetween Monte Carlo simulations and analytical approach, as it will be briefly described in thenext subsections.

3.1 Monte Carlo single-scattering model

The Monte Carlo (MC) algorithm, used as reference model in this work, starts underhypothesis of an isotropic source, i.e., a source transmitting uniformly into the solid angleΩT = 2π(1 − cos(φ1/2)), where φ1 is the transmission beam-width angle. The beam axis oftransmitter is tilted, with respect to the ground, by a zenith angle (π/2 − θ1), while its azimuthangle is equal to 0 degrees. After denoting initial photon direction with respect to zenith angleas θi and with respect to azimuth angle as ψi, then, the initial direction from beam axis is

Z ¼ θi− π=2−θ1ð Þ ð18ÞFor an isotropic source, initial angles are respectively given by:

cos Z ¼ 1−Θ 1−cos φ1=2ð Þð Þ ð19Þand

ψi ¼ 2πΨ ð20Þwhere Θ and Ψ are two independent random variables uniformly distributed between zero andone.

The energy quantum that is launched can be defined as a photon packet (Elshimy &Hranilovic, 2011). Each photon packet transports an associated weight, initialized to W0 = 1(because the model has been realized to evaluate the temporal impulse response and needstherefore a unitary normalization). This quantity can be also denoted as the arrival probabilityat initial status.

The photon path is traced by using both spatial coordinates and direction cosines. Wesuppose the transmitter (or light source) as located on generic center of coordinates (x0, y0, z0),but, for simplicity, it often coincides with axis origin. Initial direction cosines are insteadspecified by:

μix ¼ cos θið Þ

μiy ¼ sin ψið Þsin θið Þ

μiz ¼ cos ψið Þsin θið Þ

8<: ð21Þ

Once emitted, the photon propagates along the distance:

Δs ¼ −In ζð Þke

ð22Þ

where ζ is a uniform random variable between 0 and 1.


The photon launch can be hence described by coordinates:

x1 ¼ x0 þ μixΔs

y1 ¼ y0 þ μiyΔs

z1 ¼ z0 þ μizΔs

8<: ð23Þ

After that, photon begins to migrate in space. It has two possibilities: (i) reaching one point thatbelongs to common volume VQ; (ii) remaining out of the receiver FOV and consequentlybeing marked by algorithm as lost. If the photon scatters just within receiver FOV (i.e., itspotential incident angle, with respect to a normal to the aperture area, is less or equal to φ2 /2),then, arrival probability is updated toW1 =W0ω0 exp(−keΔs). The point of coordinates (x1, y1,z1) that has been achieved is a scattering center. As proposed in (Liao et al., 2015), wehypothesize to work with a non-coherent source such that each scattering center can beconsidered a secondary independent point-source emitting photons: under these conditions,turbulence theory can be applied to each path. Through Rytov approximation, the turbulenceattenuation in dB evaluated along the first path is given by (8)—the corresponding linearversion is expressed by (9)—where, for this specific case, Δ =Δs. As consequent, the weightrepresenting photon arrival probability is scaled by the linear attenuation LαΔs and a random

value TΔs following the distribution (10), updating to W Tð Þ1 ¼ LαΔsTΔsW1.

The new migration direction is governed by phase function, deploying the angular intensityof scattered light. The phase function is normalized, for definition, in order to produce anunitary integral. By following (Ding et al., 2009), the phase function can be modeled as aweighted sum of molecular (Rayleigh), aerosol (Mie), and hydrometeor (MAPS) scatteringphase functions:

p τð Þ ¼ k Rð Þs

ksp Rð Þ τð Þ þ k Mð Þ

s

ksp Mð Þ τð Þ þ k Hð Þ

s

ksp Hð Þ τð Þ ð24Þ

where τ = cos (θs): as shown by Fig. 3, θs is the generic zenith angle of scattering (between newphoton path and initial direction). The three-phase functions follow a generalized Rayleighmodel, a generalized Henyey-Greenstein function, and a modified version of the latter comingfrom (Fishburne et al., 1976) (also called as Neer-Sandri function), respectively:

p Rð Þ τð Þ ¼ 3 1þ 3γ þ 1−γð Þτ2½ �16π 1þ 2γð Þ

p Mð Þ τð Þ ¼ 1−g2

4π1

1þ g2−2gτð Þ32þ f

0:5 3τ2−1ð Þ1þ g2ð Þ32

" #

p Hð Þ τð Þ ¼ 1−g02

8π1

1þ g02−2g0τ 3

2

þg0 3τ2−1ð Þ 1−g02

� �2

7

� �1þ g02 þ 2

7

g0

52

0BB@

1CCA

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð25Þ

where polarization factor γ, asymmetry factors g and g′, and backscattering energy fraction fare model parameters that are generally implemented to describe atmospheric constituents’properties (Bucholtz, 1995; Zachor, 1978). There is to notice that a different g′ asymmetry termhas been employed for hydrometeor phase function, because, obviously, hydrometeors present


different scattering properties with respect to aerosols: for its calculation, it is possible to seeEq. (53) in the Appendix and (Bohren & Huffman, 1983).

The deviation angle with respect to incoming direction is determined from a randomvariable ν, uniformly distributed between zero and one:

v ¼ 2π∫τ1−1p τð Þdτ ð26Þwith τ1 = cos (θ), where θ is the effective zenith angle of new photon path with respect to theincoming direction. The corresponding azimuth angle ψ is extracted by a random variablewhich is uniformly distributed between 0 and 2π.

Once provided both θ and ψ, the new direction cosines can be easily calculated as follows:

μx ¼ −sinθcosψffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− μi

x

2 þ μixcosθ

qμy ¼

sinθffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− μi

x

2q μiyμ

ixcosψþ μi

z sinψ� �

þ μiycosθ

μz ¼sinθffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− μi

x

2q μizμ

ixcosψ−μ

iy sinψ

� �þ μi

zcosθ

8>>>>>>>><>>>>>>>>:

ð27Þ

Thus, a new displacement vector, whose magnitude is given byΔs ′ = − ln(ζ′)/ke, with r. v. ζ′uniformly distributed between 0 and 1, is produced and a new set of coordinates is conse-quently computed as follows:

x ¼ x1 þ μxΔs0

y ¼ y1 þ μyΔs0

z ¼ z1 þ μzΔs0

8<: ð28Þ

If the scattering direction defined by the displacement vector connecting scattering center (x1,y1, z1) to the last point achieved (x, y, z) crosses the area of receiver aperture, then the photon isdetected, otherwise, it is not registered. The probability that a photon moves from thescattering point without leaving receiver FOV is given by:

PF ¼ 2∫θMθm φMp cos θsð Þð Þsin θsð ÞU φ2=2−θsð Þdθs ð29Þwhere U (x) is the indicator function: i.e., it counts as 1 when x ≥ 0 and 0 for x < 0. Angles θm,θM, and φM are respectively expressed by:

θm ¼0 if a0≤atan−1

a0−aL

� �if a0 > a

(ð30Þ

θM ¼ tan−1aþ a0

L

� �ð31Þ

ϕM ¼π if θs≤ tan−1

a−a0L

� �cos−1

a20 þ L2tan2θs−a2

2a0Ltanθs

� �if θs > tan−1

a−a0L

� �8><>: ð32Þ


where a =DR/2 is the receiver lens radius, a0 is the projection, on the plane containing receiveraperture, of the segment connecting the scattering center point to receiver area center, and L isthe distance between the scattering center point and the same plane. Moreover, we can denotethe distance between the scattering center point and the center of receiver aperture as Δr. It ispossible to comprehend better all the above passages related to the detection zone of an opticalcommunication link simply by observing Figs. 4 and 5 or by considering Section III.B in(Elshimy & Hranilovic, 2011) for further details.

If the photon is detected, the corresponding photon packet weight is further reduced to

Wr ¼ LαΔrTΔrexp −keΔrð ÞPFWTð Þ1 , where, as before, we consider turbulence statistics—

Eq. (10)—for the distance Δ = Δr. Nevertheless, differently from the previous case, theturbulence attenuation in dB evaluated along this path is given by Eq. (16)—the correspondinglinear expression LαΔr is instead given by Eq. (17)—and the random value TΔr is generated bythe variance expressed in Eq. (12) because the last section of a communication link is moreefficiently described by Andrews’ equations rather than Rytov law.

The arrival time of a photon being scattered is random. Once emittedM photon packets forMonte Carlo simulation, only the weights of N photon packets crossing detector are recordedinto the time slots of length Δti, according to their corresponding arrival times, where i = 1, 2,..., S is the index of time slots dividing impulse response interval. To estimate the temporalimpulse response, the weights that have been recorded into sub-intervals [ti −Δti/2, ti +Δti/2]are divided by M Δti. Finally, it is possible to calculate the corresponding path loss, i.e., theproportion of photons lost during the transit from transmitter to receiver, by inverting theintegral of temporal impulse response.

3.2 Analytical single-scattering model

The analytical framework that we have decided to use in this study directly derives from thegeometric architecture depicted in (Elshimy & Hranilovic, 2011). This work expands theclassical single-scattering propagation model (Luettgen et al., 1991; Reilly, 1976; Reilly &Warde, 1979) to take account of a finite size receiver aperture rather than only a point one.Through these extensions, we are able to take into the right consideration also the non-zerosolid angle subtended by receiver surface from the point of scattering: it shows to be veryuseful in finer UV communication conditions, as well as in high-resolution ozone/aerosol

Fig. 4 Detection zone of an optical communication link. Case where a0 ≤ a


remote sensing applications, i.e., when the transmitter beam width and elevation angles aresmall (e.g., on the order of a degree or even less) and the receiver aperture area is relativelylarge (e.g., with a lens radius of 2.5 cm).

By repeating all the main steps shown in Section II inside (Elshimy & Hranilovic, 2011)(with respect to which we just added turbulence factors corresponding to initial and scatteredphoton paths), the impulse response, according to the same environment, propagation hypoth-eses, and conditions made in subsection 3.1, can be easily computed. We only need to signalthat, under statistic conditions described by Eq. (10), sinceμI ¼ −σ2l =2; then E TΔsf g ¼ E TΔrf g ¼ 1. Therefore, these terms will be removed from thefinal average, over log-normal turbulence fluctuations, of temporal impulse response: it will beshown in the reasoning below.

By envisioning, as in Monte Carlo simulation, an isotropic source with a solid angle ΩT =2π(1 − cos(φ1/2)), the probability that a photon is launched from the source along directionvector ΩT, within a sufficiently small angle δΩT is

δΩT

ΩTð33Þ

The photon begins its path and propagates until it is either absorbed or scattered by a mediumparticle. By recalling Fig. 3, let us consider that the photon interacts with an atmosphericparticle at a given point Q belonging to common volume VQ that is distant Δs from the source,within an infinitesimal interval δs; the probability that this event happens is

ke exp −keΔsð Þδs ð34ÞBy defining δAQ as the elemental cross-sectional area seen by δΩT at interaction point Q, wecan assert that δΩT = δAQ/Δs2. Thus, the joint probability of a photon being emitted in a givendirection and interacting with particles for the first time at point Q, after having undergoneturbulence according to attenuation linear factor LαΔs, is:

LαΔske

ΩTΔs2exp −keΔsð ÞδVQ ð35Þ

Fig. 5 Detection zone of an optical communication link. Case where a0 > a


where δVQ = δAQ δs is the elemental part of common volume VQ. An event can be eitherabsorptive or scattering: the probability that a photon will be scattered rather than absorbed atinteraction point Q is defined by single-scattering albedo given by Eq. (2).

If the photon scatters at Q, then we utilize phase function (distribution of difference anglebetween the incident and scattering directions, as defined in subsection 3.1). After denoting asδΩR the elemental solid angle subtended at point Q and looking towards receiver aperture, theprobability that the photon is scattered, along the direction vectorΩR, under this solid angle, is:

p ΩT ;ΩRð Þ4π

δΩR ð36Þ

where p(ΩT,ΩR) is a more generic definition of the phase function (depending on both zenithangle θ and azimuth angle ψ).

The photon impacts in a single-scattering event before achieving the receiver. The proba-bility that a photon scattered at point Q reaches the detection area after undergoing theturbulence described by attenuation linear factor LαΔr is:

LαΔr exp −keΔrð Þ ð37ÞIn conclusion, we obtain the joint elemental probability of a photon being emitted within agiven solid angle δΩT in the direction ΩT, propagating along a distance Δs and undergoingturbulence according to attenuation linear factor LαΔs, interacting for the first time at point Qinside common volume VQ, then scattering for an energy percentage equal to albedo ω0

towards receiver aperture in the direction ΩR within solid angle δΩR and tracing a distanceΔr, along the which a turbulence level corresponding to attenuation linear factor LαΔr issuffered, to be finally detected, i.e.:

ksp ΩT;ΩRð Þ4πΩTΔs2

LαΔsLαΔrexp −ke ΔsþΔrð Þ½ �δΩRδVQ ð38Þ

After some algebraic manipulations (see Section III in (Elshimy & Hranilovic, 2011) for moredetails), the present expression can be transformed from Cartesian to elliptical coordinates byequations (Abramowitz & Stegun, 1972):

x ¼ r2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiξ2−1

1−η2ð Þcos φð Þq

y ¼ r2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiξ2−1

1−η2ð Þsin φð Þq

z ¼ −r2ξη

8>>>><>>>>:

ð39Þ

where ξ ∈ [0,∞], η ∈ [−1, 1], and φ ∈ [0, 2π). Equation (38) becomes G(ξ)δξ, that can bewritten as:

G ξð Þδξ ¼ ksrexp −kerξð Þ2ΩT

Ip ξð Þδξ ð40Þ

where Ip(ξ) is, after definingL(ξ, η) = LαΔs(ξ, η)LαΔs(ξ, η), equal to:


Ip ξð Þ ¼ ∫η2 ξð Þη1 ξð ÞL ξ; ηð Þ ξ þ η

ξ−η∫φ2 ξ;ηð Þφ1 ξ;ηð ÞPF ξ; η;ϕð Þδφ

� �δη ð41Þ

Its integral limits η1(ξ), η2(ξ), φ1(ξ, η), and φ2(ξ, η) cut out a portion of area over the generic ξ-ellipsoid surface. As shown in (Reilly, 1976; Reilly & Warde, 1979), the transmitter point andreceiver area center are respectively located on the foci of the same ellipsoid. Within thecontext of NLOS FSO schemes, it is indubitably preferable to adopt an elliptical system ofcoordinates, because it allows to map all the photon paths scattered from different pointsbelonging to the same ellipsoidal surface into one only time slot (i.e., ellipsoidal surface can beproperly considered iso-temporal).

The time delay corresponding to ellipsoid ξ is t ¼ rξc (where c is lightspeed). Consequently,

the final expression of temporal impulse response is

H tð Þ ¼ crG

ctr

� �ð42Þ

and the path loss (in decibels) can be derived:

PL ¼ −10log10 ∫þ∞0 H tð Þδt

� �ð43Þ

4 Numerical results and sensitivity analysis

The UVAtmoScat physically based channel model, represented by Monte Carlo simulations andanalytical framework, can be summarized by Fig. 6, where we mark as inputs both the parametersdescribing atmospheric variables (ozone concentration expressed in [ppb], atmospheric refractionindex generating Rayleigh scattering, aerosols and fog visibility in [km], precipitation rate by rain in

[mm/h], and turbulence intensity provided by the index of refraction structure parameter C2n,

measured in [m−2/3]) and the ones related to link geometry and specifications (transmitter beamwidth, receiver FOV, and respective tilt angles, central wavelength). The output of UVAtmoScat isthe temporal impulse response (that we denoted for brevity as TIR inside Fig. 6) from which wederive path loss through the inverse of impulse response area, as shown in Eq. (43), and thefrequency characteristic through the fast Fourier transform of impulse response.

To test the physically based single-scattering UVAtmoScat model by Monte Carlo, it hasbeen built one UV-NLOS configuration system with the following basis setup: λ = 260 nm,asymmetry factor for aerosols g = 0.72, backscattering fraction for aerosols f = 0.5, linkdistance r = 10 m, receiver lens radius a = 4.37 cm, transmitter elevation angle θ1 = 35 degrees,beam-width divergence angle ϕ1 = 20 degrees, and FOVangle ϕ2 = 20 degrees. Both path lossand frequency characteristics will be analyzed in the following subsections for differentatmospheric scenarios and link geometries.

4.1 Path loss

We have computed path loss through Monte Carlo and analytical model in two link config-urations (the former—that we can call as Conf. 1—with receiver elevation angle θ2 = 35


degrees and the latter—corresponding to Conf. 2—with θ2 = 70 degrees) and in differentatmospheric conditions, i.e., on varying of extinction coefficients and index of refraction

structure parameter C2ndenoting turbulence intensity (we have set it to 10−15 m−2/3 for medium

turbulence—indicated for simplicity with M.T.—and 10−12 m−2/3 for a strong kind—indicatedwith S.T.).

Figure 7 depicts how the UV communication link path loss (in dB) varies when ozoneconcentration increases. Ozone has a nominal sea level value, as just said in subsection 2.1,between 20 and 30 ppb, but it can grow until 50 ppb and over in polluted environments. FromFig. 7, we see that ozone absorption phenomena are not so severe when its concentration isbetween 20 and 50 ppb and do not affect the performance of UV-NLOS communication link.We are considering, in this case, a scenario where all the other atmospheric variables are intheir nominal condition, i.e., Rayleigh scattering coefficient is obtained by standard atmo-sphere parameters described in subsection 2.1, aerosol (or Mie) scattering is derived by Eq. (6)computed in the case of clear air or high aerosol visibility (Va = 20 km), whereas fog and rainare absent.

Figure 8 shows the behavior of UV-NLOS link when aerosol visibility increases from 2 to20 km. In this case, ozone concentration is at minimal value, 20 ppb, while Rayleigh scatteringis computed, as before, in standard conditions. Fog and rain are still absent.

Figure 9 shows how the UV-NLOS link behaves in advection fog conditions. Of course, asin previous cases, all the other phenomena are set at minimum (ozone concentration at 20 ppb,

Ozone Concentration

AtmosphericRefraction Index

Aerosols Visibility

PHYSICALLY BASED CHANNEL MODEL

Tx Beam-Width

Rx FOV

Tx Tilt-Angle

Rx Tilt-Angle

TIRPath Loss

Freq. Characteristic

Fog Visibility

Rain Prec. Rate

Turbulence Intensity

Fig. 6 UVAtmoScat physically based channel model scheme

10 15 20 25 30 35 40 45 50

Ozone Concentration [ppb]

70

75

80

PL

[dB

]

Simul - M.T.Model - M.T.Simul - S.T.Model - S.T.Simul - M.T.Model - M.T.Simul - S.T.Model - S.T.Conf. 2

Conf. 1

Fig. 7 Comparison between Monte Carlo simulation and analytical model on varying of link configuration and

turbulence intensity (corresponding to the index of refraction structure parameter C2n

measured in [m−2/3]),

representing the trend of path loss (PL) expressed in dB vs the ozone concentration measured in ppb. Blue isrelative to Conf. 1, while red to Conf. 2


Mie scattering computed when Va = 20 km) or standard conditions (as Rayleigh scattering). Weunderline that, as expected by MAPS model providing the scattering curves shown in Fig. 2,the path loss trend is the same, inside deep UV bandwidth, for both advection and radiation fog(we decided to show only the results corresponding to the former in order to avoid redundan-cy). A discrepancy could emerge if different fog models are applied to parameters’computation—such as AlN04a and AlN04r, cited in (Mori & Marzano, 2015) and referencetherein: “AlN04” stands for Al Naboulsi et al. 2004, a paper that has just been cited, amongother similar ones, as a model landmark within (Mori & Marzano, 2015), while “a” and “r”stand for advection and radiation fog, respectively. These models have been reported for acomparison with the corresponding MAPS regression panels, as shown in Fig. 2. Within thiscontext, scattering coefficients and asymmetry factor for fog droplets—denoted in Eq. (25) asg′—are provided by MAPS model, according to the Appendix.

Figures 10 and 11 respectively show the path loss trend with light and heavy rainprecipitation rate (we adopt the same distinction made in (Mori & Marzano, 2015)), whereall other phenomena are kept in their nominal conditions. Also in this scenario, scatteringcoefficients and asymmetry factor are produced by MAPS model, according to what has beendepicted in Fig. 1 and the Appendix.

In all figures, a good correspondence between Monte Carlo simulations and analyticalframework is observed in different atmospheric conditions and when varying of turbulenceintensity and receiver tilt angle (i.e., in the two link systems above called as Conf. 1 and Conf.2). Our physically based characterization can clearly include at the same time all the atmo-spheric variables discussed inside this work, always by paying attention to Van de Hulst law(Luettgen et al., 1991) and hypothesizing them as independent in order to perform extinctionand scattering overall sum.

4.2 Frequency characteristics and bandwidth

The frequency characteristics corresponding to temporal impulse response H (t), expressed byEq. (42), can be obtained by computing its fast Fourier transform. We consider Conf. 1 and

2 4 6 8 10 12 14 16 18 20

Aerosols Visibility [km]

65

70

75

PL

[dB

]

Simul - M.T.Model - M.T.Simul - S.T.Model - S.T.Simul - M.T.Model - M.T.Simul - S.T.Model - S.T.

Conf. 1

Conf. 2

Fig. 8 As Fig. 7 vs aerosol visibility measured in km

0.5 1 1.5 2 2.5

Adv. Fog Visibility [km]

60

65

70

75

PL

[dB

]

Simul - M.T.Model - M.T.Simul - S.T.Model - S.T.Simul - M.T.Model - M.T.Simul - S.T.Model - S.T.

Conf. 1

Conf. 2

Fig. 9 As Fig. 7 vs advection fog visibility measured in km


Conf. 2 as defined in subsection 4.1 and keep the same notation of medium (M.) and strong

(S.) turbulence (T.) for values of the index of refraction structure parameter C2n

set, respec-

tively, to 10−15 and 10−12 m−2/3. The frequency characteristic is normalized with respect to themaximum value of the best communication performance in each atmospheric scenario. Wedenote it as H (f) and we will consider its absolute value expressed in dB.

Figure 12 depicts the frequency characteristics of the UV-NLOS link in Conf. 1 immersedin fog, computed through the analytical model that has been developed in subsection 3.2,without reporting Monte Carlo simulation markers, just tested in the subsection above. Werefer to the generic term fog because there is no sensitive difference between the advection andradiation fog characterized by MAPS (see (Mori & Marzano, 2015) for details). All the otheratmospheric variables are kept in their nominal magnitude. The frequency characteristic shapedoes not change significantly, once fixed the atmospheric attenuation affecting the link andcorresponding to different combinations of visibility and turbulence intensity. This can beclearly upheld by the computation of 3-dB bandwidth f−3dB, which is ≈ 36 MHz for all theattenuation values set by external phenomena. Nevertheless, a gap of about 6 dB can beobserved between the frequency characteristics computed in the best and worst cases of linkattenuation: the former corresponds to the scenario with medium turbulence and visibility V =0.6 km and the latter to the one with strong turbulence and visibility V = 2.4 km.

An analogous discussion can be argued for Fig. 13. In this case, we are considering thesame environment affected by fog as before, but in Conf. 2. f−3dB ≈ 100 MHz for all theattenuation cases: indeed, the common volume of the present link is closer to the receiveraperture area with respect to Conf. 1. The gap between the best and worst performances interms of link attenuation (corresponding to the same cases shown by Fig. 12) is still equal to6 dB.

Figures 14 and 15 depict the frequency characteristics of the UV-NLOS link operating inrainy environment, when all the other phenomena are kept silent. The former shows thefrequency characteristics of a link that is set to Conf. 1 and confirms f−3dB ≈ 36 MHz for allattenuation values; the latter is instead related to Conf. 2 and shows that f−3dB still ≈ 100 MHz.

10-1 100 101

Light Rain Prec. Rate [mm/h]

66687072747678

PL

[dB

]

Simul - M.T.Model - M.T.Simul - S.T.Model - S.T.Simul - M.T.Model - M.T.Simul. - S.T.Model - S.T.

Conf. 1

Conf. 2

Fig. 10 As Fig. 7 vs light rain precipitation rate measured in mm/h

10 20 30 40 50 60 70 80

Heavy Rain Prec. Rate [mm/h]

60

65

70

PL

[dB

]

Simul - M.T.Model - M.T.Simul - S.T.Model - S.T.Simul - M.T.Model - M.T.Simul - S.T.Model - S.T.Conf. 2

Conf. 1

Fig. 11 As Fig. 7 vs heavy rain precipitation rate measured in mm/h


In these two figs, we simply refer to standard precipitation rate values without distinguishingbetween light and heavy rain typologies, differently from the previous subsection on path lossanalysis. In both figs, the best scenario corresponds to the case with medium turbulence andprecipitation rate R = 47.57 mm/h and the worst to the one with strong turbulence and R =0.56 mm/h; the gap between the best and the worst performances is equal to about 10 dB.

These results support the conclusion that the frequency response shape and its related 3-dBbandwidth, that varies from 36 to 100 MHz in our analysis, depends mainly on the geometriccharacteristics of NLOS links rather than atmospheric conditions. The latter, on the other hand,strongly affect UV-NLOS path loss. Consequently, we can deduce that frequency constant-envelope modulations, i.e., the ones working on invariant shape or envelope characteristicswithin frequency domain, may exploit just this potential of UV-NLOS links and be clearly heldup in the short-range communication context.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f [GHz]

-40

-35

-30

-25

-20

-15

-10

-5

0

|H(f

)| [d

BH

z]

V = 0.6 km - M.T.V = 1.2 km - M.T.V = 1.8 km - M.T.V = 2.4 km - M.T.V = 0.6 km - S.T.V = 1.2 km - S.T.V = 1.8 km - S.T.V = 2.4 km - S.T.

Fig. 12 Frequency characteristic evaluated through analytical framework for an UV-NLOS communicationscenario affected by fog, in Conf. 1. All the curves have been normalized with respect to the peak value of thefrequency characteristic corresponding to attenuation produced by medium turbulence and visibility V = 0.6 km

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

f [GHz]

-40

-35

-30

-25

-20

-15

-10

-5

0

|H(f

)| [d

BH

z]

V = 0.6 km - M.T.V = 1.2 km - M.T.V = 1.8 km - M.T.V = 2.4 km - M.T.V = 0.6 km - S.T.V = 1.2 km - S.T.V = 1.8 km - S.T.V = 2.4 km - S.T.

Fig. 13 As Fig. 12, in Conf. 2


5 Summary and outlook

By summarizing, we have initially studied in terms of path loss the behavior of a short-rangeUV-NLOS link immersed in outdoor conditions, when respectively ozone, aerosols, advection(and radiation) fog, and two intensities of rain precipitation rates are considered: overall, fivescenarios have been shown, where Monte Carlo and analytical framework are compared todemonstrate the good matching between our model and numerical simulations. Then, we havecomputed the FFT transform of temporal impulse response in order to provide a frequencydomain analysis: we showed that the frequency response shape remains always the samedespite the atmospheric changing conditions and this behavior proves its suitability withrespect to constant-envelope frequency modulations.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f [GHz]

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

|H(f

)| [d

BH

z]

R = 0.56 mm/h - M.T.R = 5.62 mm/h - M.T.R = 21.81 mm/h - M.T.R = 47.57 mm/h - M.T.R = 0.56 mm/h - S.T.R = 5.62 mm/h - S.T.R = 21.81 mm/h - S.T.R = 47.57 mm/h - S.T.

Fig. 14 Frequency characteristic evaluated through analytical framework for a UV-NLOS communicationscenario affected by rain, in Conf. 1. All the curves have been normalized with respect to the peak value ofthe frequency characteristic corresponding to attenuation produced by medium turbulence and precipitation rateR = 47.57 mm/h

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

f [GHz]

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

|H(f

)| [d

BH

z]

R = 0.56 mm/h - M.T.R = 5.62 mm/h - M.T.R = 21.81 mm/h - M.T.R = 47.57 mm/h - M.T.R = 0.56 mm/h - S.T.R = 5.62 mm/h - S.T.R = 21.81 mm/h - S.T.R = 47.57 mm/h - S.T.

Fig. 15 As Fig. 14, in Conf. 2


Our interest on UV-NLOS systems moves from the fact that they have been concerned inrecent years as one meaningful solution not only for several short-range communicationenvironments, such as safety links, wireless sensor networks, disaster recovery, and temporarynetwork installations, but also for meteorological and remote sensing applications: for in-stance, we recall the above-mentioned ozone/aerosols ground-sensing (Xu & Sadler, 2008).The goal of this paper has been to investigate UV-NLOS technique in more realistic environ-mental scenarios by using a model-based approach named UVAtmoScat: the latter allows tobetter evaluate UV-NLOS potentials and update the range of its applications. The proposedUVAtmoScat model has been defined as physically based since it incorporates all thoseatmospheric variables that could affect UV-NLOS communication or link performance.Indeed, by MAPS parameterization modeling (Mori & Marzano, 2015), we might also includegraupel and snow effects to further generalize the UVAtmoScat applicability, although they areless frequent than rain and fog. UVAtmoScat model may be effectively used for furtheranalyses and intercomparisons, under the assumption of single-scattering regime. In order tomake adequately use of our model, there are some key uncertainty contributions or limitationsthat need to be monitored: (i) the here-presented framework is a single-scattering model forshort-range applications (5∼30 m); (ii) we have analyzed the behavior of UV-NLOS link underdifferent atmospheric variables when considered separated, but they essentially need a unifiedtreatment; (iii) remote sensing studies looking at ozone/aerosol concentration detecting andtracing are concerned by the model but they are still at the beginning step in state of art; (iv) ameaningful contribution to model validation is clearly represented by experimental tests, stillmissing in our analysis. Therefore, the next steps of this work should consist in: (i) a modelingextension to multiple-scattering conditions, that is able to cover also medium- and large-rangecommunications; (ii) the feasibility investigation, within the field of remote sensing analysis,of UV technologies providing ozone/aerosols concentration trend on vertical path through theelaboration of ground data; (iii) a modeling approach aiming to reduce gradually all thepotential uncertainties due to approximations and base assumptions that have been made inthis work, such as the above-hypothesized un-correlation between all the different atmosphericvariables like ozone and other aerosols: these physical mechanisms cannot be covered yet inthe today state of art by the here-presented framework. Moreover, experimental tests applyingUV devices in outdoor conditions, like the one described in (Siegel et al., 2004), may supportand empirically validate our numerical analysis.

Acknowledgments The authors would like to thank the Superior Institute of the Information and Communi-cation Technologies (ISCTI), Rome, Italy, for the continuous support of this work.

Appendix. Microphysically oriented atmospheric particle scatteringmodel parametrization

By standing at scattering coefficient parametrization of (Mori & Marzano, 2015) and referencetherein, the UVAtmoScat hydrometeor size distribution follows a law called as gamma particlesize distribution (PSD), that is a general model for water particles:

Np rvð Þ ¼ Nervre

� �μe

e−Λervreð Þ ð44Þ


where rv [mm] is the volume-equivalent spherical radius, re [mm] is the effective radius, and Ne

[mm−1 m−3] is the effective particle number concentration defined as

Ne ¼ 103Wp3Λ4þμe

e

4πρp

1

r4e

1

Γ μe þ 4ð Þ ð45Þ

where Wp [g m−3] is the particle mass concentration and ρp[g cm−3] its specific density, whileΛe—adimensional—is the so-called slope parameter, which is related to the “shape” parameterμe as follows:

Λe ¼ Γ μe þ 4ð ÞΓ μe þ 3ð Þ ð46Þ

By defining the particle mass [kg] as

mp rvð Þ ¼ 4

3πρpr

3v ð47Þ

and the Np moment of order n as

mn ¼ ∫rMrm rnvNp rvð Þdrv ð48Þ

where rm [mm] and rM [mm] are the minimum and maximum particle radius, respectively; weobtain the particle mass concentration (or water content)

wp ¼ 4

3πρpm3 ð49Þ

and the effective radius

re ¼ m3

m2ð50Þ

The corresponding particle fall rate Rp [kg h−1 m−2], defined as the particle mass crossing ahorizontal cross-section of unit area over a given time interval is

Rp ¼ ∫rMrm avrbvv mp rvð ÞNp rvð Þdrv ¼ 4

3πavρpm3 þ bv ð51Þ

where av and bv are empirical coefficients taking into account the height-dependent air density.The fall rate can be also expressed by an equivalent height per unit time, as in the case of rainprecipitations: R = Rp/ρp.

Ne and Λe, that are respectively computed by Eqs. (45) and (46), can be derived byanalyzing a set of 1000 simulated PSDs for each particle class (e.g., fog, rain droplets),produced by the uniform random generation of μe ∈ [μe, m, μe, M], re ∈ [re, m, re, M],

Table 1 Regression coefficients of case λ = 260 nm to estimate optical parameters k Hð Þe ; k Hð Þ

s andg0 km−1� �from

visibility range V [km]

Class C1,e C2,e C1,s C2,s C1,g C2,g

Adv. fog 2.995732 − 1.000000 2.995712 − 1.000000 2.575366 − 0.999940Rad. fog 2.995732 − 1.000000 2.995725 − 1.000001 2.557428 − 0.991718


and Wp ∈ [Wp, m,Wp, M], where all the respective minimum and maximum values depend onparticle class and are provided by (Mori & Marzano, 2015).

The hydrometeor extinction coefficient in the UVAtmoScat channel, once set the transmis-sion wavelength, is given by:

k Hð Þe ¼ k Hð Þ

a þ k Hð Þs ¼ ∫rMrm σe rvð ÞNp rvð Þdrv ð52Þ

where σe(rv) = σa(rv) + σs(rv) is the extinction cross-section [m2], composed by the sum ofrelated absorption and scattering cross-sections, that follow the Mie modeling laws for smallparticles exposed in (Bohren & Huffman, 1983). The related hydrometeor asymmetry factor g′is provided by the following expression:

g0 ΩTð Þ ¼ g0 ¼ ∫4π ΩT⋅ΩRð Þp ΩT;ΩRð ÞdΩR ð53Þwhere p is the phase function and ΩT and ΩR are the incident and scattering direction unitvectors.

In case of fog particles, we define the visibility V as the range to which the correspondingtransmittance t, i.e., the ratio between the received intensity and the incident one, is computedaccording to Beer-Lambert law at a given threshold value and becomes t0:

V ¼ −ln t0ð Þk Hð Þe

ð54Þ

where t0 is here considered equal to 0.05: indeed, visibility is defined as the distance to anobject where the image contrast drops to 5% threshold value with respect to the proximity tothe same; in other cases, a 2% threshold value is considered (Mori & Marzano, 2015).

According to power-law regression method provided by MAPS model and considering thecoefficients shown in Tables 1 and 2, we can express the main optical parameters as follows:

k Hð Þe ¼ C1;eX C2;e

p

k Hð Þs ¼ C1;sX C2;s

p

g0k Hð Þs ¼ C1;gX C2;g

p

ð55Þ

where Xp depends on the kind of information acquired about hydrometeors:in our case, Xp = V (visibility) for advection and radiation fog (Table 1) and = R (fall rate)

for light and heavy rain precipitations (Table 2).

References

Abramowitz M and Stegun I, (1972) “Handbook of mathematical functions with formulas, graphs and mathe-matical tables”, Abramowitz and Stegun

Table 2 Regression coefficients of case λ = 260 nm to estimate optical parameters k Hð Þe ; k Hð Þ

s andg0 km−1� �from

prec. rate R [mm/h]

Class C1,e C2,e C1,s C2,s C1,g C2,g

Light rain 0.262696 0.616674 0.262657 0.616646 0.227235 0.616434Heavy rain 0.215844 0.768803 0.215816 0.768766 0.186612 0.768782


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Affiliations

Dario De Leonardis1 & Saverio Mori1 & Silvia Di Bartolo2& Frank-Silvio Marzano3

Dario De [email protected]

Saverio [email protected]

Silvia Di [email protected]

1 Dipartimento di Ingegneria dell’Informazione, Elettronica e Telecomunicazioni (DIET), Sapienza Universitàdi Roma, Rome, Italy

2 Istituto Superiore delle Comunicazioni e delle Tecnologie dell’Informazione (ISCTI), Rome, Italy3 DIET Sapienza Università di Roma and Centro di eccellenza CETEMPS, Rome, Italy


atmospheric scattering and turbulence modeling for ultraviolet … · 2020. 7. 13. · scattering,...

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