Tropospheric scintillation and attenuation on satellite-to-Earth links at Ka and Q band: modeling, validation and experimental applications

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satellite-to-Earth links at Ka and Q band:

modeling, validation and experimental applications

Doctor of Philosophy in Information and Communication Technologies Curriculum: Applied Electromagnetic – XXXII Cycle

Candidate

Augusto Maria Marziani ID number 1152524

Thesis Advisor

Prof. Frank S. Marzano

Co-Advisor

Prof. Nazzareno Pierdicca

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in front of a Board of Examiners composed by: Prof. Guglielmo D’Inzeo (chairman)

Prof. Andrea Bartolini Prof. Andrea Detti Prof. Armando Rocha Prof. Luca Facheris Prof. Paolo Baccarelli

Tropospheric scintillation and attenuation on satellite-to-Earth links at Ka and Q band: modeling, validation and experimental applications

Ph.D. thesis. Sapienza – University of Rome Version: 18 February 2020

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during these years “Gutta cavat lapidem”

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Abstract

Link budget is a crucial step during the design of every communication system. For this reason it is fundamental to identify and estimate the effects of the atmosphere on the electromagnetic signal along the path from the source to the sink. Tropo-sphere represent the bigger source of attenuation and scintillation for signals in the microwave and upper frequency spectrum. During last years we have participated in the European Space Agency “AlphaSat Aldo Paraboni” experimental campaigns to acquire up to date propagation data at two frequencies of interest for future communication systems. We realized two high performance low-noise receiver located in Rome, one at Ka and one at Q band (19.701 and 39.402 GHz) to detect the two signal beacons sent from the AlphaSat geostationary satellite to a wide area over Europe. Collected data from Rome receiving station have been analysed to measure excess attenuation and scintillation along the path. Such statistics collected in a database together with data from other experimenter will be in the near future a useful instrument, giving professionals updated data for their custom application design.

Classical link budget techniques rely on climatological atmospheric statistics based on different time-scales, usually data collected for several years. In the background of the European Space Agency “STEAM” project, we proposed the use of high resolution 3D weather forecast models (up to 166 m pixel resolution) for the calculation of excess attenuation and tropospheric scintillation for satellite to earth link. As a result, the estimation of these electromagnetic parameters to use in link budgets could be given no more as a statistical analysis of past events as in the case of Internation Telecommunication Union recommendation but as time-series forecast specific for the selected receiving station and along the slant path of the transmitted signal. Case studies for the use of this technique have been deeply analysed and results compared with data from the AlphaSat measurement campaign for the Rome and Spino d’Adda receiving station, confirming the validity even in different geographical regions.

In everyday situations, propagation models based on statistics are often replaced by the use of easier to apply parametric models. Those have the advantage of the simplicity and the need of less input parameter to be applied. In particular, for what concerning the tropospheric scintillation, the Hufnagel-Valley refractive index structure constant (C2

n) parametric model is actually the most used, due to

the simplicity and the relative accuracy. We here propose a new Cn2 _polynomial

parametric model (CPP) based just on the altitude z and a function C2

n0(to, RH0)

that allow to calculate the ground refractive index structure constant just using the ground temperature (T0) and the relative humidity (RH0). In this work CPP and

Hufnagel-Valley models are applied to different location around the globe to prove their accuracy. The obtained model could be also used in the future to realize a simulator able to generate random C2

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Ringraziamenti

Considerando che lo svolgimento del mio dottorato, nonostante i vari corsi seguiti all’estero ed il periodo di ricerca trascorso in Portogallo, ha avuto luogo prevalente-mente in Italia, ho deciso di scrivere parte dei ringraziamenti in italiano.

Colgo quindi l’occasione di ringraziare in questa pagina le persone che mi hanno seguito in questi anni, che hanno curato direttamente ed indirettamente la mia crescita professionale e personale.

In primis, sento di dover ringraziare il Prof. Marzano, che in questi anni mi ha seguito e mi ha permesso di partecipare a diversi progetti molto interessanti. In questi hanni mi ha inoltre sempre assecondato nelle mie proposte, aiutandomi a correggere a volte il tiro ed anche a perfezionare un po’ le mie capacità diplomatiche.

Collega inseparabile degli ultimi anni, è Fernando. In questi anni passati presso l’Istituto Superiore delle Comunicazioni mi ha trasmesso una buona parte delle sue conoscenze, sia a livello professionale che personale. Innumerevoli le ore passate insieme ad effettuare misure, realizzare circuiti, controllare le stazioni riceventi e cercare di finire il radiometro a 90 GHz. Prima o poi ce la faremo!

Non posso mancare di ringraziare l’ingegner Pierri, per avermi dato la possibilità di frequentare i laboratori dell’Istituto Superiore delle Comunicazioni e Tecnologie dell’Informazione (ISCTI). Grazie a questa opportunità ho potuto acquisire capac-ità tecniche che altrimenti sarebbe stato difficile apprendere. Il suo supporto alla ricerca e il suo interesse, soprattutto nell’ambito del progetto AlphaSat, sono stati indispensabili.

Vorrei ringraziare ora tutte le persone che, da dietro le quinte, mi hanno aiutato ad affrontare questi anni con il giusto spirito ed i giusti mezzi. Un forte grazie alla mia famiglia, sempre presente durante questi lunghi anni passati fuori casa, anche quando la distanza sembra creare barriere. Un grazie anche agli amici e ai colleghi che mi hanno ascoltato, consigliato, sopportato, tirato su il morale ed accompagnato anche nelle mie attività extracurriculari. A volte una semplice uscita puo’ fare la differenza.

I would like now to say thanks to my Portuguese colleagues. I would like to thanks Susana and Armando who welcomed me at DETI with great attention. A special thanks to Armando who shared several meals with me and also helped me to better understand traditions and ways of Portugal.

Hope to see you again soon. Grazie a tutti!

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List of Figures

1.1 Electromagnetic spectrum. . . 1

1.2 Radio Frequency atmospheric attenuation . . . 2

1.3 Propagation experiment for satellite to earth link. . . 3

2.1 U.S. Standard Atmosphere definition . . . 6

2.2 Example of water drop formation, from nucleation to cloud droplet . 9 2.3 Block diagram of principal precipitation processes[4] . . . 9

2.4 Cloud classification. . . 10

2.5 Energy exchange and distribution in turbulence vortices[10] . . . 13

2.6 Relation between z and height of the turbulence layer . . . 18

2.7 Water vapour and oxygen specific attenuation . . . 19

2.8 Oxygen specific attenuation 60 GHz for several height above sea level: 0 km, 5 km,15 km and 20 km (from ITU-R REC-P676-12) . . . 21

2.9 Satellite communication system segments . . . 26

2.10 Satellite orbits (from Enciclopædia Britannica) . . . 27

2.11 Example of ground station architecture . . . 28

3.1 AlphaSat receiving station in Rome: roof installation . . . 30

3.2 Architecture of the AlphaSat Q-band receiver in Rome . . . 32

3.3 Architecture of the AlphaSat Ka-band receiver in Rome . . . 33

3.4 Test bench for the measurement of the Low Noise Amplifier (LNA) noise figure . . . 34

3.5 Test bench for the measurement of the down converter frequency response . . . 35

3.6 Case study of the 14th _{and 15}th _{of August 2018: AlphaSat beacons} attenuation, visibility and rain rate . . . 36

3.7 Example of Power Spectral Density (PSD) slopes . . . 37

3.8 Power spectral density (PSD) of the Ka-band beacon attenuation . . 38

3.9 Power spectral density (PSD) of the Q-band beacon attenuation . . 38

3.10 Components of the Ka-band attenuation . . . 39

3.11 Components of the Q-band attenuation . . . 39

3.12 Rain rate (red) and scintillation log amplitude (blue) for the Q-band (top) and Ka-band (bottom) beacons . . . 39

3.13 Received power and excess attenuation detrended data. Rome receiv-ing station, Q-band, year 2018 (data-set reduced just for the graphical representation). . . 40

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3.14 Histograms of excess attenuation and scintillation log amplitude of the Alphasat Q-band Rome receiving station. Year 2018. . . 42 3.15 Complementary Cumulative Distribution Function (CCDF) of the

an-nual attenuation and scintillation log amplitude for the Q-band Rome receiving station. Comparison with International Telecommunication Union (ITU) recommendations. . . 43 3.16 Meteorological statistics obtained from the disdrometer. Rome, one

year data analysis. . . 44 4.1 A schematic view of the turbulence spectrum in the horizontal plane

as a function of the horizontal wavenumber magnitude k . . . 49 4.2 WRF model setup used in this work . . . 50 4.3 Block diagram of the NAPS . . . 53 4.4 Horizontal 3D grid of the meteorological forecast parameters: (left)

original data, (right) reduced and re-gridded data . . . 54 4.5 Relation between hydrometeor concentration and specific attenuation

for a satellite link at 39.4 GHz and elevation angle of 41 degrees. . . 56 4.6 Extrapolation of NAPS vertical profile . . . 57 4.7 Extrapolation of IFS-LES (left), GFS-LES (center) and RAOB (right)

atmospheric pressure vertical profiles for Rome, November 14-16, 2017 58 4.8 Extrapolation of IFS-LES (left), GFS-LES (center) and RAOB (right)

temperature vertical profiles for Rome, November 14-16, 2017 . . . . 58 4.9 Extrapolation of IFS-LES (left), GFS-LES (center) and RAOB (right)

RH vertical profiles for Rome, November 14-16, 2017 . . . 59 4.10 Alphasat receiving station in Rome . . . 61 4.11 Alphasat receiving station in Spino d’Adda . . . 62 4.12 Attenuation time series comparison in Rome, 14-15 November 2017 . 66 4.13 Attenuation scatterplots in Rome, 14-15 November 2017 . . . 67 4.14 Attenuation probability comparison in Rome, 14-15 November 2017 68 4.15 Scintillation Probability comparison: models vs Alphasat in Rome,

14-15 November 2017 . . . 69 4.16 Time-series comparison: models vs Alphasat in Rome, 14-15 November

2017 . . . 70 4.17 Time-series comparison: models vs Alphasat in Spino d’Adda, 16-17

June 2017 . . . 73 4.18 Attenuation histogram comparison: models vs Alphasat in Spino

d’Adda, 16-17 June 2017 . . . 74 4.19 Attenuation scatterplot: models vs Alphasat in Spino d’Adda, 16-17

June 2017 . . . 75 4.20 Scintillation histogram comparison: models vs Alphasat in Spino

d’Adda, 16-17 June 2017 . . . 77 4.21 Scintillation time series comparison: models vs Alphasat in Spino

d’Adda, 16-17 June 2017 . . . 78 4.22 Cn2 _{comparison: models vs RAOB (red) in Spino d’Adda, 16-17 June}

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5.1 Mean (blue line) and histantaneous vertical profiles of C2

n for the

Milano Linate RAOB stations during the period 2015-2018 . . . 84 5.2 Comparison between vertical profiles of C2

n: mean value of 4 years

from RAOB of Spino d’Adda (black line), Hufnagel-Valley model (red line) and proposed model (blue line) . . . 86 5.3 Vertical profiles of the C2

n error . . . 87

5.4 Histogram of the C2

n estimation error: Hufnagel-Valley and proposed

model comparison . . . 88 5.5 Scatterplots of the error and correlation of the proposed model and

Hufnagel-Valley model in relation to the RAOB measurements . . . 89 5.6 Comparison of the C2

n0(t) measured from RAOB and C

2

n0(t) obtained

by the mdoel. The x axis represent the sample number (two per day), with a time scale range of 4 years (2015-2018) . . . 91 5.7 Scatterplot of the C2

n0 vs the one obtained by the polynomial function

of T and RH . . . 92 5.8 C2

n vertical profile series over 4 years, comparison of all models . . . 92

5.9 C2

n estimation error for vertical profiles using the Cn20 polynomial

function as input . . . 93 5.10 Scatter plot of C2

nestimation error: Huffnagel-Valley vs CPP proposed

models . . . 93 5.11 Milano Linate vertical profile of C2

n over 4 years: RAOB profiles,

Hufnagel-Valley mean value +/- std, proposed PCC model mean value +/- std . . . 94 5.12 Block diagram of the C2

n additive component generator in the

fre-quency domain . . . 95 5.13 Block diagram of the C2

n additive component generator . . . 95

5.14 Vertical profiles of C2

n obtained using the CPP model together with

the syntetic generator. Yellow line represent the mean value obtained from all C2

n profiles obtained using the CPP model. Ground data

from Milano Linate RAOB station, year 2015. . . 96 5.15 Standard deviation of amplitude scintillation calculated from C2

nusing

the CPP model together with the syntetic generator. Comparison with Spino d’Adda AlphaSat receiving station. Year 2015. . . 96 5.16 Map of the RAOB stations selected for the proposed CPP model

verification . . . 98 5.17 C2

n from Milano Linate RAOB: vertical profiles and mean value for

years 2015, 2016, 2017, 2018 . . . 99 5.18 Comparison between vertical profiles of C2n: mean value from RAOB

of Milano Linate (black line), Hufnagel-Valley model (red line) and proposed model (blue line). Years 2015, 2016, 2017, 2018 . . . 99 5.19 Vertical profiles of estimation error of C2

n from Milano Linate RAOB

for years 2015, 2016, 2017, 2018 . . . 100 5.20 Histogram of the error of the hestimation of the ¯C_n2 otained from

Milano Linate RAOB station. Years 2015, 2016, 2017, 2018 . . . 100 5.21 Scatter plot of estimation error of the ¯C_n2 obtained from Milano Linate

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5.22 Scatter plot of the correlation between ¯C_n2

(RAOB) obtained from

Mi-lano Linate RAOB and ¯C_n2 obtained using the Hufnagel-Valley and

the proposed CPP models. Years 2015, 2016, 2017, 2018 . . . 101 5.23 Vertical profiles of the C2

n, all profile from RAOB, Hufnagel-Valley

and CPP model. Years 2015, 2016, 2017, 2018 . . . 102 5.24 Histogram of C2

nestimation error, Huffnagel-Valley and CPP model.

Years 2015, 2016, 2017, 2018 . . . 102 5.25 Scatterplot of the CPP model C2

n estimation error vs Hufnagel-Valley

estimation error. Years 2015, 2016, 2017, 2018 . . . 103 5.26 Representation of all C2

n profiles from RAOB, mean CPP Cn2

+/-standard deviation, mean Hufnagel-Valley C2

n +/- standard deviation.

Years 2015, 2016, 2017, 2018 . . . 103 5.27 C2

n from Pratica di Mare RAOB: vertical profiles and mean value for

years 2015, 2016, 2017, 2018 . . . 105 5.28 Comparison between vertical profiles of C2n: mean value from RAOB

of Pratica di Mare (black line), Hufnagel-Valley model (red line) and proposed model (blue line). Years 2015, 2016, 2017, 2018 . . . 105 5.29 Vertical profiles of estimation error of C2

nfrom Pratica di Mare RAOB

for years 2015, 2016, 2017, 2018 . . . 106 5.30 Histogram of the error of the hestimation of the ¯C_n2 otained from

Pratica di Mare RAOB station. Years 2015, 2016, 2017, 2018 . . . . 106 5.31 Scatter plot of estimation error of the ¯C2

n obtained from Pratica di

Mare RAOB. Years 2015, 2016, 2017, 2018 . . . 107 5.32 Scatter plot of the correlation between ¯C_n2

Prat-ica di Mare RAOB and ¯C_n2 obtained using the Hufnagel-Valley and

the proposed CPP models. Years 2015, 2016, 2017, 2018 . . . 107 5.33 Vertical profiles of the C2

and CPP model. Years 2015, 2016, 2017, 2018 . . . 108 5.34 Histogram of C2

nestimation error, Huffnagel-Valley and CPP model.

Years 2015, 2016, 2017, 2018 . . . 108 5.35 Scatterplot of the CPP model C2

n estimation error vs Hufnagel-Valley

estimation error. Years 2015, 2016, 2017, 2018 . . . 109 5.36 Representation of all C2

n +/- standard deviation.

Years 2015, 2016, 2017, 2018 . . . 109 5.37 C2

n from Bjornoya RAOB: vertical profiles and mean value for year 2018111

5.38 Comparison between vertical profiles of C2n: mean value from RAOB of Bjornoya (black line), Hufnagel-Valley model (red line) and pro-posed model (blue line). Year 2018 . . . 111 5.39 Vertical profiles of estimation error of C2

n from Bjornoya RAOB for

year 2018 . . . 112 5.40 Histogram of the error of the hestimation of the ¯C2

n otained from

Bjornoya RAOB station. Years 2018 . . . 112 5.41 Scatter plot of estimation error of the ¯C_n2 obtained from Bjornoya

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5.42 Scatter plot of the correlation between ¯C_n2

Bjornoya RAOB and ¯C_n2 obtained using the Hufnagel-Valley and

the proposed CPP models. Year 2018 . . . 113 5.43 Vertical profiles of the C2

and CPP model. Year 2018 . . . 114 5.44 Histogram of C2

n estimation error, Huffnagel-Valley and CPP model.

Year 2018 . . . 114 5.45 Scatterplot of the CPP model C2

nestimation error vs Hufnagel-Valley

estimation error. Year 2018 . . . 115 5.46 Representation of all C2

n+/- standard deviation.

Year 2018 . . . 115 5.47 C2

n from Abu Dhabi RAOB: vertical profiles and mean value for year

2018 . . . 117 5.48 Comparison between vertical profiles of C2n: mean value from RAOB

of Abu Dhabi (black line), Hufnagel-Valley model (red line) and proposed model (blue line). Year 2018 . . . 117 5.49 Vertical profiles of estimation error of C2

n from Abu Dhabi RAOB for

year 2018 . . . 118 5.50 Histogram of the error of the hestimation of the ¯C_n2 otained from Abu

Dhabi RAOB station. Years 2018 . . . 118 5.51 Scatter plot of estimation error of the ¯C_n2 obtained from Abu Dhabi

RAOB. Year 2018 . . . 119 5.52 Scatter plot of the correlation between ¯C_n2

(RAOB) obtained from Abu

Dhabi RAOB and ¯C2

n obtained using the Hufnagel-Valley and the

proposed CPP models. Year 2018 . . . 119 5.53 Vertical profiles of the C2

and CPP model. Year 2018 . . . 120 5.54 Histogram of C2

n estimation error, Huffnagel-Valley and CPP model.

Year 2018 . . . 120 5.55 Scatterplot of the CPP model C2

nestimation error vs Hufnagel-Valley

estimation error. Year 2018 . . . 121 5.56 Representation of all C2

n+/- standard deviation.

Year 2018 . . . 121 5.57 Time series of standard deviation of tropospheric amplitude

scintilla-tion for the Milano Linate locascintilla-tion. Comparison with models, RAOB and AlphaSat measure data. Year 2015, clear air condition . . . 123 5.58 Histogram of the standard deviation of amplitude scintillation. Model

and measurements. Year 2015, clear air condition . . . 124 5.59 Scatterplot of the scintillation obtained from CPP and

Hufnagel-Valley models vs scintillatin obtained from RAOB and measured at the AlphaSat Spino d’Adda receiving station. Year 2015, clear air condition . . . 125

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5.60 Histogram of the estimation error of scintillation from Hufnagle-Valley and CPP models, against the scintillation obtained from RAOB. Year 2015, clear air data. . . 125 5.61 Histogram of the estimation error of scintillation from

Hufnagle-Valley and CPP models, against the scintillation measured at the Spino d’Adda AlphaSat receiving station. Year 2015, clear air data. 126

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Acronyms

3DVAR 3 Dimensional VARiational

DA technique

4DVAR 4 Dimensional VARiational

DA technique

AMOS Air Force Maui Optical

Station

ASI Italian Space Agency

BC Boundary Conditions

CCDF Complementary Cumulative

Distribution Function

CF Correction Factor

CNES Centre National d’Études

Spaciales

CNR Carrier to noise ratio CPP Cn2 polynomial parametric

model

CV5 Control Variable option 5

CW Continuos Wave

DA Data Assimilation

DC Direct Current

DL downlink

ECMWF European Center for

Medium-Range Weather Forecasts

EIRP Efficient Isotropic Radiated

Power

EM Electro Magnetic EOC Edge Of Coverage

EO Earth Observation

ESA European Space Agency GEO Geostationary Earth Orbit GFS Global Forecast System GNSS Global Navigation Satellite

System

HIWE High Impact Weather Events HPBW Half Power Beam Width IC Initial Conditions IFS Integrated Forecasting

System

IF Intemediate Frequency

ISCTI Instituto Superiore delle

Comunicazioni e Tecnologie dell’Informazione

ITU International

Telecommunication Union

InSAR Interferometric Synthetic

Aperture Radar

LAM Limited Area Model LEO Low Earth Orbit LES Large Eddy Scale LNA Low Noise Amplifier

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LST Land Surface Temperature

MEKaP Medium Earth orbit

Ka-band Propagation experiment

MEO Medium Earth Orbit MM5 Fifth-Generation Penn

State/NCAR Mesoscale Model

NAPS Numerical Atmospheric

Propagation Simulator

NCAR National Center for

Atmospheric Research

NCEP National Centers for

Environmental Prediction

NMC National Meteorological

Center

NWP Numerical Weather

Prediction

OoS Out of Service

PBL Planetary Boundary Layer PCB Printed Circuit Board PCI Peripheral Component

Interconnect

PLL Phase Locked Loop PSD Power Spectral Density RAOB RAwindsonde OBservation

RF Radio Frequency

RRTMG Rapid Radiative Transfer

Model For Global

RUC Rapid Update Cycle SAR Synthetic Aperture Radars SBR Satellite Beacon Receiver

SLC-D Submarine Laser

Communications-Day

SM Soil Moisture

SNR Signal to Noise Ratio SST Sea Surface Temperature

STEAM SaTellite Earth observation

Atmospheric Modeling

TDP Technology Demonstration

Payloads

TTC Telemetry and Tele

Command

UHF Ultra High Frequencies

UL uplink

UTC Coordinated Universal Time

WD Wind Direction

WPF Weather Prediction Forecast WPM Weather Prediction Models WRF Weather Research and

Forecasting

WSM6 WRF single-moment 6-class

WS Wind Speed

Z Zulu

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Chapter 1

Introduction

1.1 Context, background and state of the art . . . . 1 1.2 Objective and chapter summary . . . . 4

This chapter gives an overview of the effects of propagation on electromagnetic signals and most common used techniques for estimating excess attenuation and tropospheric scintillation on satellite to earth communication links. The objectives and organization of the work will be also described.

1.1 Context, background and state of the art

The always increasing demand for higher data-rate from the end user is pushing the scientific community to investigate even more the higher part of the microwave spectrum. Millimeter and sub-millimeter waves frequencies represent, in fact, a great resource both in the remote sensing and telecommunication area. Their ability to provide high bandwidth and consequently high bit-rates make them the most attractive candidate for the race to the 5th generation of mobile communication network systems. Figure 1.1 represent the available spectrum. From the picture

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it is clear that the spectrum above the 6 GHz is still quite free to be used. It is important to remind, however, that the use of higher frequency is affected both by higher attenuation due to free space loss (and consequently shorter possible distance achievable) both by peaks in the atmospheric absorption (see figure 1.2)

Figure 1.2. Radio Frequency atmospheric attenuation

Since attenuation and scintillation are crucial parameters, worldwide scientific community is carrying on several investigations on effects of the atmosphere on propagation at frequencies above 6 GHz. This will lead to a better approach to link budgets and communication systems design. Big effort is made both from a theoretical point of view both experimentally collecting data to validate and possibly improve actual propagation models (figure 1.3 shows the mechanism of the experimental activity). Since these wavelengths are quite sensitive to atmosphere hydrometeors and gases, the most relevant measurement campaigns are based on satellite to earth communication links. Among them we can cite the Olympus and Italsat experimentations. The most recent of these campaigns, instead, is the AlphaSat “Aldo Paraboni" propagation experiment. It is a measurement campaign headed by European Space Agency (ESA) in collaboration with several European universities. It consist in the acquisition of two Continuos Wave (CW) beacon signals centered respectively at 19.7 GHz and 39.4 GHz. Signals are sent from the Technology Demonstration Payload 5 of the AlphaSat geostationary satellite over a wide area over the Mediterranean sea [1]. The received power level is continuously monitored at high sample rate, to allow the calculation of both excess attenuation and scintillation due to the presence of the atmosphere (gases, hydrometeors and atmospheric turbulence).

Since the physics behind atmospheric attenuation and amplitude scintillation phenomena is quite complex and requires a vertical characterization of the atmo-sphere along the signal path, simpler solution are adopted. The estimation of these parameters is usually made using tables and models principally based on past years statistical data, like the ones provided by the ITU in their recommendation. These have a quite good accuracy but can only give a statistical approach without any

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interpretation of the event over time or possibility of the use of boundary condition like ground meteorological parameters for the specific site. Focusing on the amplitude scintillation, another solution is to first estimate the value from a parametric model of the refractive index structure constant (C2

n) and then calculate the amplitude

scintillation. This time we have a method that allow to have some kind of input to the system or to discern for example between day and night condition; however parametric models, like ITU recommendations, do not try to guess the exact instan-taneous profile of the C2

n but they are more likely to give a value that can represents

a mean trend of the selected parameter. In literature there are several models for the calculation of the C2

n vertical profile, some of them selected by the simplicity

and recurrence in literature are the Submarine Laser Communications-Day (SLC-D) model, the Brookner model and the most commonly used Hufnagel-Valley model, actually considered the reference parametric model for the C2

n vertical profile.

Figure 1.3. Propagation experiment for satellite to earth link.

The approach of using microphysically-oriented radiopropagation models does not allow to have an estimation of what the value of attenuation or scintillation will be but will remain the most accurate way to calculate both attenuation and scintillation and will be the reference technique for data comparison. As already introduced, for the calculation of the C2

n using the microphysical model, we need the meteorological

characterization of the atmosphere along the signal path. Temperature, atmospheric pressure, relative humidity and wind gradients are used for the characterization of the turbulence through the calculation of the refractive index structure constant. The most effective model is considered to be the Cherubini-Businger model, based on a elaboration of the Tatarski model using the potential temperature theta.

Also the calculation of the attenuation using physics, require the vertical profile of meteorological parameters but, in this case, we are talking of the concentration of all the hydrometeors. Using these, together with regression coefficient it is possible to calculate the extinction coefficient (db/km) and consequently the total attenuation along the path. To get a vertical characterization of the atmosphere, the only way is through the use of radiosonde that record meteorological data during their ascent together with the use of meteorological polarimetric radar.

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1.2 Objective and chapter summary

Core of this work is the study of different solutions for the characterization of attenuation and amplitude scintillation along the signal path of a satellite to earth link. Two different original solutions will be investigated, the first one is the use of high-resolution 3D Weather Forecast Models for the prediction of electromagnetic parameters. A Numerical Atmospheric Propagation Simulator has been realized to use standard netCDF data as input to predict future time series of scintillation and excess attenuation. An additional solution is proposed for the estimation of the refractive index structure constant vertical profile. The proposed parametric model is dependent only on the altitude z and on temperature and relative humidity values at the ground (t0, RH0). All simulation results have been compared with

data acquired by one or more AlphaSat “Aldo Paraboni”receving stations.

In order to give a detailed description of the obtained achievement, the work has been organized as follow. In chapter 2 we give a general overview of the principal aspects of radio-propagation and satellite systems. First the Earth atmosphere and its phenomena are described, the principal notions of the microwave propagation through the Earth atmosphere are given focusing the attention on the radiative transfer theory and scintillation. After that the ITU recommendation for attenuation and scintillation will be exposed. In the end the components of a satellite to earth link will be illustrated.

In chapter 3, we expose the activity related with the AlphaSat experimentation. First there will be an overview of the Rome receiving station, realized by our team. After that we will take a look at some case studies and at the next measurement campaign that is going to start during the second half of the year 2020.

Chapter 4 will be about the use of Weather Prediction Models (WPM) to predict electromagnetic parameters. Used WPM will be depicted highlighting the differences between standard and high resolution models. After that we will switch to the description of the NAPS architecture: used propagation models and potential of the simulator. At the end case studies with results of the simulator and comparison with AlphaSat measured data will be showed.

The chapter 5 will be dedicated to the implemented parametric model of refractive index structure constant. The model will be described, explaining all its working features. It will be used for several case studies comparing results with vertical profile of Cn2 obtained from different radiosonde launch location and the Hufnagel-Valley

parametric model. Results of computed scintillation will be compared with AlphaSat measured scintillation.

In chapter 6 we will point out the main results of the entire work:

• Advantage of managing a whole experimental chain, from receiving hardware to the final data analysis.

• Advantages of the use of NAPS for electromagnetic propagation parameters and its accuracy

• Proposed parametric model of Cn2 vertical profile improvement comparing to

the Hufnagel-Valley model.

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Chapter 2

Satellite communications

2.1 Earth atmosphere . . . . 5

2.1.1 Structure of the atmosphere . . . 5

2.1.2 Clouds and hydrometeors . . . 8

2.2 Microwave propagation . . . . 11

2.2.1 Tropospheric amplitude scintillation . . . 11 2.2.2 Attenuation in the troposphere . . . 19

2.2.3 ITU-R recommendations . . . 25

2.3 Satellite systems . . . . 25

2.3.1 Space segment . . . 26 2.3.2 Ground segment . . . 28

This chapter gives an high level description of the Earth atmosphere in all its components and its effects on a microwave-propagating signal.

Particular attention will be given to the amplitude scintillation and tropospheric attenuation both from a microphysical point of view and from a statistical approach introducing the most recent ITU recommendations.

After that Satellite systems and receiving ground station will be described.

2.1 Earth atmosphere

2.1.1 Structure of the atmosphere

Atmosphere is characterized by several parameters that can be described by physical law along the vertical profile. Pressure and density follow an almost exponential law, decreasing with the increasing of the elevation over the sea level. Also the water vapor density as a similar attitude, even if the variation with height is more irregular and vary a lot on date/time basis, being influenced on time of day (night-time or day-time), season, geographic location, and atmospheric events; however, atmospheric temperature trend along the vertical profile show a periodic attitude, which allow the classification of several atmosphere vertical stratified layer depending on the thermal behaviour [2]. Figure 2.1 shows the atmospheric model first introduced by the “U.S. Committee on Extension to the Standard Atmosphere” in 1958. It is considered the reference model for the static atmosphere defining atmospheric layers and their properties. Usually the border between two layers is identified in a change

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Figure 2.1. U.S. Standard Atmosphere definition

of the sign of the temperature gradient dT along the vertical profile z. The standard layer classification define several region where the boundary is characterized by a change of the temperature gradient from a defined value to zero (-pause regions) to change again at the beginning of the next layer. From the bottom to the top we have [2]:

• Troposphere: it represent the lower layer, the closest to the Earth surface. The temperature gradient is dT/dz ≈ −6.5 KKm−1_{. Usually its extension is from}

0 to 10 km. It is the layer of the atmosphere more involved in meteorological phenomena like rain, snow and cloud presence. Higher layer will be less and less involved in such atmospheric activities. The tropopause define its upper limit and is easily detect by a sharp change of the temperature gradient to zero. Its extension largely depends on latitude and season. A mean value can

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be considered from 10 km to 18 km.

• Stratosphere: it extend from the tropopause to the stratopause where, after two different dT/dz, another zero gradient of the temperature over the elevation stop the absolute temperature to an approximate value of 270.5 °K. Its upper boundary is defined to be at 47 km circa.

• Mesosphere: it’s the third layer of the atmosphere and goes from the stratopause to the mesopause. This layer stop at approximately 90 km where the mesopause begin and the temperature reach its minimum of about 173°K. In this layer the atmospheric composition and molecular weight of the air is almost constant. • Thermosphere: it’s the last layer of the atmosphere. No upper limit for this

region is defined. The air density is very small and the effects of dissociation show with changes in the atmosphere composition.

The use of this stratified model of the atmosphere allows the use of the following models for the vertical profiles of meteorological variables[2]. Each of them can be expressed as a function of z (height above sea level [ km]) and the value of the variable at the ground surface.

The temperature profile for an height up to 32 km can be defined with the function T (z) as following: T(z) =        T(0) + gTz, for 0 ≤ z ≤ 11 km T(11), for 11 km ≤ z ≤ 20 km T(11) + (z − 20)z, for 20 km ≤ z ≤ 32 km

T(0) is the reference atmospheric temperature at sea level (288.15 °K), z is the height

over sea level expressed in km and gT is the temperature gradient of the troposphere dT /dz= −6.5 °K km−1.

For what concerning the density, we have two different model, one for the dry air density profile ρa(z) and another one for the water vapour density ρv(z), both

expressed in kg/ m−3_.

ρa(z) = ρa(0)e(−z/H1) and ρv(z) = ρv(0)e(−z/H2)

where ρa(0) and ρv(0) are respectively the dry air density and water vapour density

at the sea level: 1.225 kg/ m−3 _{and 7.72 kg/ m}−3_{. H}

1 and H2 are the density scale

height for the dry air and water vapour densities, H1 = 9.5 km and H2 = 2 km.

It is important to remember that the water vapour content in the atmosphere is function several atmospheric parameters, the dependence on temperature, on which this model is based is just the strongest one. The reference value just listed is the one proposed by the U.S Standard Atmosphere for middle latitudes.

Also atmospheric pressure vertical profile (expressed in mbar), as air density, can be represented by an exponential function of z. In particular we have:

P(z) = P (0)e(−z/H3)

The atmospheric pressure at sea level P (0) is 1013.25 mbar and the pressure scale height H3 is 7.7 km.

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Table 2.1. Clear air atmosphere normal composition at sea level[2]

Gas Symbol Content Molecular weight

(% by volume) Nitrogen N2 78.084 28.0134 Oxygen O2 20.947 31.9988 Argon Ar 0.934 39.948 Carbon dioxide CO2 0.0314 44.00995 Neon N e 0.001818 20.183 Helium He 0.000524 4.0026 Krypton Kr 0.000114 83.80 Xenon Xe 0.0000087 131.30 Hydrogen H2 0.00005 2.01594 Methane CH4 0.0002 16.04303 Nitrogen oxide N2O 0.00005 44.0128

Ozone O3 Summer: 0 to 0.000007_{Winter: 0 to 0.000002} 47.9982_47.9982

Sulfur dioxide SO2 0 to 0.0001 64.0628

Nitrogen dioxide N O2 0 to 0.000002 46.0055

Ammonia N H3 0 to trace 17.03061

Carbon monoxide CO 0 to trace 28.01055

Iodine I2 0 to 0.000001 253.8088

Atmospheric composition, instead of water vapor variations, can be considered almost constant up to an height of 90 km above sea level. The main constituent of the atmosphere can be divided into dry mixture of the air like gases (see table 2.1) and components that contain water particles [3]. These last one can be divided into

H2O in form of vapour, Hydrosols, aerosols, liquid or iced hydrometeors (rain, snow,

graupel etc) and non-precipitating ice (e.g. convective clouds). Atmospheric clear air condition is characterized by the absence of water particles in the atmosphere. 2.1.2 Clouds and hydrometeors

Clouds comes from condensation of water molecules. The process usually starts with the ascension of an air particle. During the path, the temperature drops while the relative humidity rises due to the expansion of the air. When relative humidity reach the saturation (100%), cloud droplets start to form. This phenomenon evolves with other condensing molecule of water not forming new cloud droplets but attaching to other already existing particles. This is due to the fact that water spend less energy to condense on other particles rather than creating new cloud droplets on its own. This phenomenon is called nucleation and the particles on which water condenses are called cloud condensation nuclei. The possibility of a particle to act as condensation nucleus depends mostly on the size of the particle (but also on the geometry and composition), the bigger the particle, the easier the process. The formation of new cloud droplets in absence of particle to aggregate to, requires a lot of energy and supersaturation condition of several hundred percent of relative humidity. Anyway, usually, the troposphere presents enough particles that act as condensation nuclei.

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Water drops generate when the dimension of particle cluster obtained by nu-cleation is big enough to let other extraneous cloud particles (e.g. aerosols and atmospheric dust) aggregate. So by the condensation molecule of water continue to aggregate till the dimension of the drop increase enough to start the vertical descent. During the first part of the descent, the velocity of the drop is quite low, so the growing process continue by water vapour condensation of the drop. With the increasing speed of the falling drop, the process of coalescence starts and the falling drop absorb smaller drops during its way to the ground. The process of the generation of drops is visible in figure 2.2.

Figure 2.2. Example of water drop formation, from nucleation to cloud droplet Another mechanism is the formation of hail. This comes from the concentration of cloud particles around ice crystals that consequently start to drop. The whole process for each kind of hydrometeor is described in figure 2.3.

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There are several different classification used for clouds. One of the most used is the one described in the International Cloud Atlas. Ten different groups of clouds, called genera, are identified and divided in subgroup named species. Inside each specie we can find different varieties according to the macroscopic geometry and transparency of the cloud. The ten genera are: cirrus, cirrocumulus, cirrostratus, altocumulus, altostratus, nimbostratus, cumulus, cumulonimbus, stratocumulus and stratus[4][5]. The nomenclature comes from Latin words, the word cumulus identify convective clouds that exhibit vertical distribution; stratus is for stratiform clouds with a stronger horizontal distribution; cirrus depicts fibrous clouds and finally nimbus for rainy ones. Figure 2.4 shows the cloud classification while in table 2.2 are listed the main properties for each genera.

Clouds usually occupy the lower part of the atmosphere, the troposphere that can be divided by height into three different overlapped layer. For temperate areas we have: high (5 − 13 km),medium (2 − 17 km), low (up to 2 km)[4].

Figure 2.4. Cloud classification.

Clouds, as already told, are generated by the effect of nucleation due to the change of saturation and temperature along the vertical profile of atmosphere. The different kind of air particle movement through the atmosphere leads to the the formation of different kind of clouds [4]. The three main processes are the following: • Convective clouds are produced by the movement of hot air, in unstable environment, upward in the atmosphere. An example of convective cloud is the cumulonimbus.

• Stratiform clouds are generated by the upward movement of stable air forced by an external agent. Example of stratiform cloud is the nimbostratus. • Orographic clouds are generated by the upward movement of air forced by the

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Table 2.2. Properties of cloud genera, according to the International Coud Atlas [4]

Genera Symbol Formation Liquid Ice Rain rate

area bcontent content [mm/h]

Cumulus Cu Vertical structure_{(0-6 km)} Yes No <30

Cumulonimbus Cb Vertical structure_{(0-15 km)} Yes Yes 10÷100

Stratus St Low Yes No <5

Stratocumulus Sc Low Yes Yes/no <5

Nimbostratus Ns Medium Yes yes <20

Altostratus As Medium Yes Yes <2

Altocumulus Ac Medium Yes Yes <2

Cirrus Ci High No Yes 0

Cirrostratus Cs High No Yes 0

Cirrocumulus Cc High No Yes 0

2.2 Microwave propagation

Almost all radio communication involves propagation through the troposphere for at least part of the signal path. Radio waves traveling through the lowest part of the atmosphere are subject to refraction, scattering and other phenomena[6]. These effects become stronger with the increasing of the signal frequency since dimensions of particles in the atmosphere become comparable with the wavelength of the signal. In the microwave band, that spread from 30 cm to 1 mm wavelength, effect of the atmosphere can be really strong. Part of the transmitted energy is absorbed by gases (O2) and water vapour[6]. The amount of attenuation of the signal depends

on several parameter, first of all the frequency, then on meteorological ones, like temperature, pressure and water vapour concentration. Strong attenuation can be experienced by hydro-meteor absorption, especially in case of severe weather events; however unwanted signal oscillation can appear also in clear air condition, when no rain or other hydrometeor cross the signal path. Atmospheric turbulence, in fact, can affect the transmitted signal with amplitude aleatory oscillation that can reach several dB. This effect is called amplitude scintillation[7].

In this section the theory of amplitude scintillation and atmospheric hydrometeor and gas attenuation will be deeply illustrated for the case of a satellite to earth link in the higher part of the microwave frequency band.

2.2.1 Tropospheric amplitude scintillation

Refractive index structure constant

Scintillation is an atmospheric phenomenon that cause fast fluctuation of ampli-tude and phase of received signals. It is mostly experienced in satellite to earth communication link, because the signal cross the whole atmosphere that can be modeled like stratified. Scintillation is principally caused by the irregularities of the refractive index of the atmosphere along the signal path. It can be caused both by the ionosphere both by the troposphere. Last ones are negligible for signal with

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frequency above 10 GHz[6]. Tropospheric scintillation instead, especially at lower elevation angle, can even cause limitations in the link availability.

In clear air condition, refractive index n is dependent on the relative dielectric constant of the medium r by the relation n =

√

r [8]. Since r is not known, it is

possible to calculate n through the following equation[9]:

n= 1 +A T Pa+ 4810Pe T 10−6 _(2.1) A= 77.6[K/mbar] (2.2) Pe= PaH 0.62198 + H (2.3)

Where Pa is the atmospheric pressure, Pe the water vapour pressure, T the absolute

temperature and H the humidity. Equation 2.1 explicitly show the dependency of the refractive index from the most common atmospheric parameters. It is also possible to calculate the refractive index by use of the refractivity N using: N = (n − 1)106_.

The refractivity can be calculated by the use of atmospheric parameters with the equation[8]: N = 77.6 T Pa+ 4810Pe T (2.4) The refractive index in case of turbulence is to be considered as an aleatory variable that can be expressed like combination of its mean value hni, that can be considered constant, and the aleatory variation n1[9]:

n= hni(1 + n1) (2.5)

The fluctuation are represented by a stochastic function of position and time. Imposing the condition of homogeneous turbulence introduced by Tatarskii, we can write the covariance function Bn[9]:

Bn(δ) = hn(r)n(r + δ)i (2.6)

In equation 2.6, r is the position vector. Covariance function Bnis the Fourier inverse

transform of the spatial spectrum Φn(k) where k is the wave number associated to

the turbulence dimensions[10]. Usually tropospheric turbulence is described using the Kolmogorov spectrum.

Turbulence is made up of three-dimensional vortices of different dimensions. The biggest ones can be generated by wind shear and convective motions of air mass. These vortices are unstable and during the motion turn themselves into ever smaller ones till the dimension is small enough that the energetic exchange between the vortices is dissipated under the form of heat. Figure 2.5 show the exchange of energy between vortices in a turbulent environment (2.5a) and the energy distribution as function of k (2.5b).

The energy distribution is divided into three different region[10]:

• Input range: vortices are bigger than the outer scale L0. It happens for K > K0 = 2π/L0. In this condition turbulence is anisotropic and non

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(a)Energy exchange (b)Energy distribution Figure 2.5. Energy exchange and distribution in turbulence vortices[10]

• Dissipation range: turbulent energy is dissipated into heat exchange. It happens for K < Km = 2π/l0. Inner scale l0 dimension is in the order of

millimeters and can be obtained using the equation: l0 = 7.4(ν3/)1/4 where ν

is the air viscosity and the energy dissipation ratio.

• Inertial range: is the portion of spectrum, between L0 and l0. It is dependent

neither on the input energy nor on the energy dissipation. Since just the the inertial energy transfer is involved, this part of the spectrum is considered isotropic

With the use of dimensional analysis, Kolmogorov, showed that the structure function of wind velocity in the region of inertial range, satisfy the 2/3 power law[10]

Dv(−→r ) =D[v(−→r1 + −→r ) − v(−→r1)]2 E = C2 v−→r 2/3 , l0 << r << L0 (2.7)

where h·i is the average, v(−→r1) is the vector of velocity and −→r is the position vector.

The statistical description of the fluctuation of n due to the turbulence is quite similar to the Dv. For homogeneous and isotropic turbulence, Tatarskii defined the

refractive-index structure function Dn(r)[11]: Dn(r) = D [n(r1+ r) − n(r1)]2 E = C2 nr2/3, l0 << r << L0 (2.8)

that obey to the 2/3 power law. C2

n is the refractive index structure constant and is

measured in m−2/3_{. The values of C}2

n changes with the altitude giving a vertical

profile of the level of turbulence of the atmosphere. C2

n is dependent to the variance

of the aleatory part of the refractive index through the:

C_n2 = 1.91hn2₁iL−2/3₀ (2.9)

Tatarskii, through Weiner-Khintchin theory, found a relation between the refractive index structure function Dn(r) and the power spectrum density function Sn(k):

Dn(r) = 8π Z ∞ 0 dkSn(k)k2 1 −sin(kr) kr (2.10)

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Sn(k) = 1 4π2_k2 Z ∞ 0 sin(kr) kr d dr r2 d drDn(r) (2.11) Putting equation 2.8 into equation 2.11 and solving the derivatives leads to the so called Kolmogorov turbulent power spectrum[12]:

Sn(k, z) = 0.033C_n2(z)k−11/3, 2π/L0<< k <<2π/l0 (2.12)

This model is only valid in the inertial range region, but can be extended to every wave number k imposing l0 = 0 and L0= ∞; however this model suffer of overestimation

of the power spectrum in the dissipation range region. For this reason Tatarskii introduced:

Sn(k) = 0.0033Cn2k11/3e

−k2

km (2.13)

where the correction factor −11/3 reduce the overestimation problem and km is

dependent on the inner scale following the law: km = 5.92/l0.

Amplitude scintillation

We are now going to see the the analytical derivation of amplitude scintillation of a plane wave through a turbulent media. The “layer distribution model” that consider the turbulence localised in just a portion of the link will be used. Generally it is not considered a strong limit as for our study it depicts perfectly the situation of a satellite to earth link. In fact in this case the turbulence is limited to the troposphere, that is localised in a small portion of the path near the receiver antenna. Another important assumption that we made is the one of weak scattering. This approximation was first introduced by Rytov in optical propagation and applied by Obukhov for microwave propagation of electromagnetic wave in random media. It is usually referred as Rytov approximation[13] and allows to express the traveling electromagnetic field as a product of an unperturbed field and the exponential of a surrogate function.

The Rytov condition, confirmed also by Tatarskii and Pisareva is expressed by the equation[13]:

hχ2i <1 (2.14)

where the unit of measure is expressed in Np2_.

We consider now that the turbulent medium is linear and described by a constant permeability that is only space dependent. Under these assumptions we can write the Maxwell equation for electric and magnetic fields as[14]:

∇ × E(r) = −jωµH(r) (2.15)

∇ × H(r) = jωE(r) (2.16)

∇D(r) = ∇(E) = ρ = 0 (2.17)

Eand H are respectively the electric and magnetic fields, ω = 2πf, 0 = 8.854187817×

10−_{12 F m}−_{1 is the dielectric constant of the vacuum and µ}

0= 4π × 10−7Hm−1 is

the permeability of the vacuum. We can now apply the operator ∇× to equation 2.15 to obtain:

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Since: ∇ × ∇ × E(r) = −∇2_E_{(r) + ∇(∇ · (E(r))} _(2.19) and ∇ ·[_r(r)E(r)] = 0 (2.20) equation 2.18 becomes: ∇2_E_{(r) + ω}2_µ 00r(r)E(r) − ∇ _∇ r r · E(r) = 0 (2.21) Rewriting it in terms of n: ∇2E(r) + k2n2(r)E(r) = 2∇ _∇n n · E(r) (2.22) The right side of equation 2.22 represents the polarization effect, that, in clear air condition, can be neglected. We obtain by this way the scalar wave equation:

∇2E(r) + k2₀n2(r)E(r) = 0 (2.23)

The value k0 = 2π_λ₀ is the wave number in the vacuum. We can use the first order

approximation of the refractive index n2 _{= (1 + n}

1)2≈1 + 2nn1 to write the scalar

wave function under the form: ∇2_E_{(r) + k}2

0[1 + 2n1(r, t)]E(r) = 0 (2.24)

We can now use the Rytov approximation to solve the equation of the electric field[15]. It consist in expressing E(r) like a product of the electric field E0(r) measured in

absence of turbulence with an exponential correction factor eΨ(r)_{to take into account}

refractive index irregularities[16]:

E(r) = E(r, t) = E0(r)eΨ(r,t) (2.25)

The correction function Ψ(r, t) can be expanded in series: Ψ(r, t) =

∞ X i=1

Ψi(r, t) (2.26)

Under the already considered assumption of weak scattering, it is possible to consider just the first term of the series. We obtain:

E(r, t) ≈ E0(r)eΨ1(r,t) (2.27)

In absence of fluctuation we have:

n1(r) = 0 (2.28)

E0(r) = eΨ0(r) (2.29)

we need to find a relation between the refractive index fluctuation and the function Ψ(r) that represent the field fluctuation. We can use equation 2.29 together with equation 2.23:

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using again equation 2.23 we can write:

∇2_{Ψ(r) + ∇Ψ(r) · ∇Ψ(r) + k}2

0hni2 = 0 (2.31)

Previous equation should be satisfied in any condition, even in absence of signal fluctuation:

∇2Ψ(r) + ∇Ψ(r) · ∇Ψ(r) + k2₀ = 0 (2.32) We need now to find a solution to this equation. The idea is to find a relation between the source of the signal fluctuation, that is the variability of n, and the function that represent the field fluctuation, Ψ(r). We can write:

∇2[E0(r)Ψ(r)] = [∇2E0(r)]Ψ(r) + 2∇E0(r) · Ψ(r) + E0(r)∇2Ψ(r) = −k₀E0(r)Ψ(r) + 2E0(r)∇Ψ(r) · ∇Ψ(r) + E0(r)∇2Ψ(r) (2.33) We obtain: ∇2Ψ(r) = ∇ 2_[E 0(r)Ψ(r)] E0(r) + k 2 0Ψ(r) (2.34)

Combining the previous two equations we obtain what we where looking for: ∇2[E0Ψ(r)] = k02E0(r)Ψ(r) = −2k02n1(r) (2.35)

This equation is similar to the radiation equation in presence of sources, but in this case the unknown function is the product E0(r)Ψ(r) and the value −2k02n1(r) is

known. The turbulent atmosphere, from a physical point of view, behave like a source o re-irradiation. For this reason the solution to equation 2.35 will be similar to the one that connect the potential vector to the imposed currents:

Ψ1(R, t) = −2k20 Z

d3rG(R, r)n1(r, t) E0(r)

E0(R) (2.36)

Where G is the Green function in free space, R is the receiver location and r is the eddy position from the transmitter (that is considered as the origin of the system). Equation 2.36 represent the formal solution to the problem. Let’s now try to find a easier way to represent it. We can start giving new expressions for the field (dependence from r has been removed for visual clarity):

E = E0eΨ= Aejs (2.37)

Ψ = χ + js = ln(A) + js (2.38)

In the last equation both amplitude and phase components are clearly visible. The value χ is the log-amplitude fluctuation of the field, while s is the phase fluctuation. From now on we will take into account only the log-amplitude fluctuation, since in this work we will not analyse problem related to the phase scintillation.

Using equations 2.36 and 2.38 we can write χ as following:

χ(r) = Z V n1(r0)h(r − r0)d3r0 (2.39) with: h(r − r0) = 1 E0(r)2k 2 0<{E0(r0)G(r − r0)} (2.40)

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Functions χ(r) and n1(r0_{) are aleatory while h(r−r}0_{) is deterministic. It’s a weighting}

function that can be interpreted like an impulsive response in the time domain. Defining r = r2− r1 and r0= r20 − r01 we write:

Bg(r0) = hg(r₁0)g(r0₂)i (2.41) Bf(r0) = hf(r01)f(r

0

2)i (2.42)

Ch(r − r0) = hh(r1− r10)h(r2− r10)i (2.43)

where f(r) is the convolution of g(r) and h(r):

Bf(r0) = Bg(r0) ⊗ Ch(r − r0) (2.44)

We can now introduce the Fourier transform of Bf where k represent the spatial

frequency:

Φf(n) = 1

(2π)3 Z

Bf(r)eik·rd3k (2.45)

Applying now the previous definitions, together with the condition of spatial invari-ance we have[12]: Bχ(r) = hχ(r1)χ(r1+ r)i = Z Φ(k)H2_(k)eik·r_d3_k _(2.46) Bn(r) = hn1(r01)n1(r01+ r0)i (2.47) with: Φχ(k) = Φn(k)H2(k) (2.48)

This could be interpreted like if Φχ(k) is the response of a signal Φn(k) passing

by a filter of impulsive response H2_{(k). The solution is however obtained under}

the condition of three-dimensional spatial invariance. This constraint is not met, because the signal is propagating in the z direction. For this reason a correction factor is needed to adjust the previous result. We define:

Fn(z, kt) = Z

Φ(kt)e−ikz·zdkt (2.49)

and its Fourier transform:

Φn(kt) = _2π1 Z

Fn(z, kt)eikz·zdz (2.50)

where ktand kz are the transverse and longitudinal components of the propagation

vector k. In literature is shown that:

Bχ(rt) = 2π

Z L

0 Z

Φn(kt)H2(L − z)e−ikt·rtd2kt with: rt= xx0+ yy0 (2.51)

Is possible to find again the function Φn(k) as already seen for example using the

Kolmogorov spectrum. From that equation is possible to calculate the variance of χ for λL << l0 that is equal to[10]:

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In case of plane wave, C2

n depends just on the direction of propagation z. In the

transverse plane, eddies are mostly of the order of dimension of the radius of the first Fresnel zone of the link and so are considered constant for the frequencies of our study.

It is possible to extend the Tatarskii theory to the microwave frequencies in the millimeter- and centimeter- wavelengths frequency bands. Since turbulence in satellite to earth communication is most probably located in a subtle layer at a certain altitude, the model will be made of a succession of layers. Considering a subtle layer in between two point z1 and z2 along the path, the variance of scintillation will be[9]:

σ2_χ= 42.25k7/6C_n2(z₂11/6− z₁11/6) (2.53) The value of z is dependent on the height of the turbulent layer by the relation

z= H/sinϑ. It is represented in figure 2.6

Figure 2.6. Relation between z and height of the turbulence layer Equation 2.53 show that the scintillation variance depends on the C2

n, on the

height of the turbulence layer and also on the elevation angle of the antenna. The lower the angle, the bigger the scintillation effect. Also the antenna dimension is part of the equation. For non-ideal and small antenna, an averaging factor needs to be considered. The bigger the antenna the lower the effect of the turbulence on the transmitted field following the equation [7]:

G= 1 − 1.4 Ra

2p

λHEf f (2.54)

where Ra is the antenna diameter and HEf f is the effective height. We can find the

average value simply multiplying the scintillation variance with G. The standard deviation can be found with:

σχ= q

σ2

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2.2.2 Attenuation in the troposphere

Radio belongs to a family of electromagnetic radiation that includes infrared, visible light, ultraviolet, X-rays and the even shorter-wavelength gamma and cosmic rays. Radio has the longest wavelength and thus the lowest frequency of the group. The wavelength of radio frequency, especially in the microwave and upper region, is comparable with atmospheric particles so that various kind of attenuation take place. Atmospheric attenuation, is the driving parameter for the loss computation in a data-transfer link. In order, gases, clouds and precipitation attenuation will be analysed.

Effects due to gases

The main effect of gases at the frequencies of interest is the absorption, in particular in the microwave spectrum water vapour and oxygen are the two stronger contributor. As visible in figure 2.7 there are various peak for each gas. Water vapour has one

Figure 2.7. Water vapour and oxygen specific attenuation

absorption peak at 22.2 GHz and another one at around 183.3 GHz while oxygen presents a peak at 60 GHz and one at 118 GHz. Off course these peaks are the ones of interest for the considered band of frequency. The phenomenon is present both at lower frequencies, with smaller effects, both at higher frequencies, with stronger effects. The attenuation parameter is called absorption coefficient and is measured in dB/km.

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For frequencies below 100 GHz the absorption coefficient of the water vapour, depending on frequency, kH2O(f) can be written as[2]:

kH2O(f) = k(f, 22) + kr(f) (2.56)

where k(f, 22) is the absorption coefficient at 22.3 GHz and kr(f) is the term that

represents the effects of all the attenuation peaks at higher frequencies. We can also write the relation:

k(f, 22) = 2f2ρv(300 T ) 5/2_e−644/t γ1 (494.4 − f2)2+ 4f2_γ2 1 (2.57) and: γ1 = 2.85 _P 1013 300 T 0.626 1 + 0.018ρvT P (2.58) where f is the frequency expressed in GHz and γ1 is the line-width parameter. P is

the pressure expressed in mbar, T the atmospheric temperature in K and ρv the

water vapor density in g m−3_{. The absorption coefficient of the residual part k} r(f),

always expressed in dB is:

kr(f) = 2.4 · 10−6f2ρv

300

T 3/2

γ1 (2.59)

Putting together the two effects, we get:

kH2O(f) = 2f 2_ρ v 300 T 3/2 γ1 " 300 T e−644T 1 (494.4 − f2)2+ 4f2_γ2 1 + 1.2 · 10−6 # (2.60) As already stated, the previous equation is valid for frequencies below 100 GHz, but going up in frequency near that value is better to introduce a correction factor ∆k(f). This is needed to take into account the higher frequency peaks effect, especially the one of the 183.3 GHz. This coefficient, empirically formulated by Gaut and Reifenstein is equal to:

∆k(f) = 4.69 · 10−6_ρ v 300 T 2.1 _P 1000 f2 (2.61)

and just needs to be added to the non-corrected value just obtained for the kH2O to

have the corrected formulation of the specific attenuation[2]:

k0_H₂_O(f) = kH2O(f) + ∆k(f) (2.62)

The second principal contribution to atmospheric gas absorption is molecular oxygen present in the atmosphere. The magnetic moment of the molecular oxygen interact with the transmitted field generating a group of attenuation peaks with centered at approximately 60 GHz. The effect is visible in figure 2.8.

A solution for the specific attenuation due to the oxygen absorption phenomenon, considering an oxygen concentration of 0.21 is[2]:

kO2(f) = 1.61 · 10 −2_f2 P 1013 300 T 2 F0 (2.63)