INVESTIGATION INTO THE EFFECT OF REFRACTIVITY ON PROPAGATION AT UHF AND VHF FREQUENCIES MODELLED AND EXPERIMENTAL ANALYSIS

Transcription

1 INVESTIGATION INTO THE EFFECT OF REFRACTIVITY ON PROPAGATION AT UHF AND VHF FREQUENCIES MODELLED AND EXPERIMENTAL ANALYSIS Thesis submitted for the degree of Doctor of Philosophy at the University of Leicester by Imtiaz Alam Department of Engineering University of Leicester April 2015

2 Abstract The goal of this research was to use weather parameters to compute refractivity variation of the atmosphere that can be used to predict refractivity distribution in the first kilometre of the atmosphere over the English Channel for UHF and VHF propagation and to understand the influence of meteorology on propagation. Different refractivity profiles are constructed based on meteorological data taken from the UK Meteorological Office in order to investigate their effects on wave propagation. The hourly experimental path loss between the transmitter and receiver obtained from the experimental setup comprising of two UHF and two VHF communication links are investigated for a period of one year. The experimental setup comprises of long and short trans-horizon paths (140 km and 50 km) having a transmitter located on Jersey and receivers on Portland and Alderney. In order to investigate the characteristics of refractivity and its impact on wave propagation, a propagation model is developed in MATLAB using parabolic equation method. The model is used to get an hourly modelled path loss corresponding to the experimental path loss for four communication links of UHF Portland, UHF Alderney, VHF Portland and VHF Alderney. The correlation between the modelled path loss and experimental path loss is presented for refractivity distribution recommended by the ITU and for that of a standard atmosphere. The simulated and experimental results showing the influence of evaporation duct upon path loss for frequencies of 240 and 2015 MHz is also included. The results obtained by using the standard atmospheric refractivity profile were a better fit to the experimental observations than the ITU recommended values for some of the investigated links. It is also inferred that evaporation ducts exist up to a height of 10 m for the short path at both frequencies and up to a height of 30 m for the long path at VHF but not at UHF. ii

3 Table of Contents ABSTRACT... II LIST OF FIGURES... VI LIST OF TABLES... XII ACKNOWLEDGEMENTS... XIII GLOSSARY... XIV 1 INTRODUCTION LITERATURE REVIEW Background Diffraction Scattering Refraction Refraction and refractive index Refractivity and modified refractivity Standard refraction Sub refraction Super refraction Ducting Measuring refractivity iii

4 2.3.1 Direct sensing method Remote sensing method Propagation models and techniques Geometrical optics Coupled mode Parabolic equation method MODEL IMPLEMENTATION, VALIDATION AND EXPERIMENTAL SETUP Description of PEM Maxwell s equations Paraxial propagation domain Solution methodology Boundary conditions Model validation Evaporation duct Bi-linear and tri-linear M profiles AREPS Path loss coverage diagrams at operational parameters Experimental setup Conversion factor Meteorological data CHARACTERISATION OF MODEL PARAMETERS General model parameters Effect of EDH on path loss iv

5 4.3 Effect of antenna height on path loss Effect of half power beam width on path loss Effect of beam angle with horizon Effect of change in frequency Band Band Effect of change in range ANALYSIS FOR EVAPORATION DUCT M profiles Annual and monthly analysis for evaporation duct UHF Portland UHF Alderney VHF Portland VHF Alderney CONCLUSIONS Future recommendations REFERENCES v

6 List of figures Figure 2-1 Tropospheric scattering mechanism for transhorizon propagation Figure 2-2 Ray traveling from denser medium to rarer medium while changing its direction Figure 2-3 Classification of different types of refraction using N gradient and M gradient values [from Barclay, 2003] Figure 2-4 Ray propagation under different refractive conditions (after [Gunashekar et al., 2006]) Figure 2-5 M profiles representing different types of ducts (a) Evaporation duct (b) Surface duct (c) Surface based duct (d) Elevated duct [from Patterson, 2003] Figure 2-6 Path loss versus evaporation duct height for the North Sea experiment taken from Hitney and Vieth, [1990] Figure 2-7 Characteristic parameters recommended by ITU for (a) Surface duct (b) Surface based duct and (c) Elevated duct [from ITU-R P , 2012] Figure 3-1 M profiles with evaporation duct heights equal to 0 m, 10 m, 20 m and 30 m Figure 3-2 Path loss coverage diagrams of modelled results for 3 GHz source in the environment of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH vi

7 Figure 3-3 Path loss coverage diagrams of Levy results for 3 GHz source in the environment of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH [from Levy, 2000] (Page 79) Figure 3-4 M profiles representing (a) Bi-linear variations with inversion point at 100 m and (b) Tri-linear variations with inversion points at 100 and 150 m Figure 3-5 Path loss coverage diagrams of modelled results for a bi-linear M profile at (a) 3 GHz and (b) 10 GHz source Figure 3-6 Path loss coverage diagrams of Levy results for a bi-linear M profile at (a) 3 GHz and (b) 10 GHz source [from Levy, 2000] (Page 86) Figure 3-7 Path loss coverage diagrams for a tri-linear M profile obtained using (a) Model (b) AREPS Figure 3-8 Path loss coverage diagrams for a simple M profile obtained using (a) Model (b) AREPS Figure 3-9 Path loss coverage diagrams of modelled results for 2015 MHz source in the environment of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH Figure 3-10 Path loss coverage diagrams of modelled results for 240 MHz source in the environment of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH Figure 3-11 Map of the sites used for the experimental setup over the English Channel vii

8 Figure 4-1 Modelled path loss against different evaporation duct heights for profile 1 of Figure 5-1 for all the communication links with tidal data (excluded) (solid lines) and tidal data (included) (dotted lines) Figure 4-2 Modelled path loss against (a) Transmitting antenna heights and (b) Receiving antenna heights for all the communication links. The evaporation duct height is 0 m Figure 4-3 Characteristic curves of modelled path loss against (a) Half power beam width ranging from 1 to 35 degrees and (b) Half power beam width ranging from 1 to 10 degrees Figure 4-4 Characteristic curves of modelled path loss against beam angle with horizon Figure 4-5 Modelled path loss against frequency range of 1 Hz to 40 GHz for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines) Figure 4-6 Modelled path loss against frequency range of 1 Hz to 1 MHz for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines) Figure 4-7 Modelled path loss against frequency range of 4 MHz to 9 GHz for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines) viii

9 Figure 4-8 Modelled path loss against frequency range of 1 MHz to 3 GHz for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines) Figure 4-9 Modelled path loss against frequency range of 240 MHz to 2015 MHz with a step size of 4 MHz for all the links and with and without tidal data Figure 4-10 Modelled path loss against frequency for a bandwidth of 200 MHz around the centre frequency of (a) UHF (2015 MHz) (b) VHF (240 MHz) Figure 4-11 Modelled path loss against range (distance between transmitting and receiving antennas) for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines). UHF Portland overlaps on UHF Alderney and similarly VHF Portland overlaps on VHF Alderney Figure 5-1 Median of hybrid M profile for July 2009 to June 2010 with different evaporation duct heights (profile 1) Figure 5-2 Median of hybrid M profile for July 2009 to June 2010 with different evaporation duct heights (profile 2) Figure 5-3 Monthly correlation coefficients for different M profiles for UHF Portland in case of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. The bar and line represents the correlation coefficient and p value, respectively Figure 5-4 Modelled path loss and experimental path loss for July for UHF Portland when profile 1 is used as input M profile. The evaporation duct height is 0 m and the correlation coefficient is EPL is shown in red and MPL in black ix

10 Figure 5-5 Modelled path loss and experimental path loss for April for UHF Portland when profile 1 is used as input M profile. The evaporation duct height is 30 m and the correlation coefficient is Figure 5-6 Monthly correlation coefficients for different M profiles for UHF Alderney in case of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. The bar and line represents the correlation coefficient and p value, respectively Figure 5-7 Modelled path loss and experimental path loss for February for UHF Alderney when profile 4 is used as input M profile. The evaporation duct height is 10 m and the correlation coefficient is Figure 5-8 Modelled path loss and experimental path loss for a portion (a highly correlated section) of November for UHF Alderney when profile 4 is used as input M profile. The evaporation duct height is 10 m Figure 5-9 Modelled path loss and experimental path loss for July for UHF Alderney when profile 3 is used as input M profile. The evaporation duct height is 10 m. The correlation coefficient is 0.4. EPL is shown in red and MPL in black Figure 5-10 Modelled path loss and experimental path loss for August for UHF Alderney when profile 3 is used as input M profile. The evaporation duct height is 10 m. The correlation coefficient is EPL is shown in red and MPL in black Figure 5-11 Monthly correlation coefficients for different M profiles for VHF Portland in case of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. The bar and line represents the correlation coefficient and p value, respectively x

11 Figure 5-12 Modelled path loss and experimental path loss for May for VHF Portland when profile 2 is used as input M profile. The evaporation duct height is 30 m and the correlation coefficient is Figure 5-13 Modelled path loss and experimental path loss for May for VHF Portland when profile 4 is used as input M profile. The evaporation duct height is 30 m and the correlation coefficient is EPL is shown in red colour and MPL in black Figure 5-14 Monthly correlation coefficients for different M profiles for VHF Alderney in case of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. The bar and line represents the correlation coefficient and p value, respectively Figure 5-15 Modelled path loss and experimental path loss for July for VHF Alderney when profile 4 is used as input M profile. The evaporation duct height is 10 m. EPL is shown in red colour and MPL in black xi

12 List of tables Table 3-1 Geographical coordinates of the sites along with the heights of the installed antennas Table 3-2 Conversion factor used for each communication link [after Mufti, 2011] Table 3-3 Measured values for different parameters in the experimental setup [after Mufti, 2011] Table 3-4 Weather stations along with their geographical coordinates and distance to the nearest site Table 4-1 General values used in the characterization of the model parameters Table 4-2 Modelled path loss at centre frequencies of UHF and VHF for each communication link with and without tidal data. The path loss values are in db Table 5-1 Number of hours of observations in each month for each communication link Table 5-2 Annual correlation coefficients for all the communication links and M profiles for (a) EDH 0 m (b) EDH 10 m (c) EDH 20 m and (d) EDH 30 m xii

13 Acknowledgements First of all I would like to say thanks to God for all the strengths and capabilities gifted by Him and making me capable of doing this research. I would like to express my deepest gratitude to my supervisor, Dr. David Siddle, for his invaluable support and guidance throughout this research. His continued support and guidance in all forms led me to the right way. I would also like to extend my appreciations to the group members especially Prof. Michael Warrington and Dr. Alan Stocker for their advice and support during my research. I am also grateful to the technical staff from the department of Engineering, University of Leicester, for their help and assistance. I would like to express special thanks to my wife for her sacrifices in every way during the course of my research. I am grateful to my family and colleagues for their unconditional supports and technical help at all stages of this research. I am in indebted to Bahria University Islamabad for providing me the financial support for a period of three years during the course of my research. I gratefully acknowledge the British Atmospheric Data Centre (BADC) and the UK Meteorological Office for providing access to meteorological data. Finally this work is dedicated to my dear parents for their prayers, love and encouragements during the past years. xiii

14 Glossary AMSL APM AREPS BAWH BBC CF CLV Above Mean Sea Level Advanced propagation model Advanced Refractive Environmental Prediction System Beam angle with horizon British Broadcasting Corporation Conversion Factor Channel Light Vessel (A lightship owned and maintained by the UK Meteorological office fixed in the English Channel.) EDH EHF EPL FDM FE GPS HOI HPBW ITU ITU-R LIDAR M MATLAB Evaporation Duct Height Extremely high frequency Experimental Path Loss Finite Difference Method Flat Earth Global Positioning System Height of interest Half power beam width International Telecommunication Union International Telecommunication Union Radiocommunication Sector Light Detection and Ranging Modified refractivity Matrix Laboratory, a programing language suite developed by Mathworks MPL Modelled Path Loss xiv

15 MU n N NASA NOAA NU PEM RADAR RFC RO SPAWAR SPE SSFT TPEM UHF VHF XO M Units Refractive index Refractivity National Aeronautics and Space Administration National Oceanic and Atmospheric Administration N Units Parabolic Equation Method Radio Detection and Ranging Refractivity from Clutter Ray Optics Space and Naval Warfare Systems Centre Standard Parabolic Equation Split Step Fourier Transform Terrain Parabolic Equation Method Ultra High Frequency (300 MHz 3 GHz) Very High Frequency (30 MHz 300 MHz) Extended Optics xv

16 1 Introduction Radio communication links are significantly affected by highly variable propagation conditions of the atmosphere. The goal of this research was to use weather parameters to compute refractivity variation of the atmosphere that can be used to predict distribution of refractivity responsible for these conditions in the first kilometre of the atmosphere over the English Channel for over-sea propagation at two UHF/VHF frequencies in the radio spectrum. Assessing these variable conditions and providing a better prediction of refractivity will potentially help the designers of communication, navigation and radar systems to improve performance. Refractivity predictions are very useful in many applications of wireless communication, navigation and surveillance systems. Such predictions are important in order to cope with the problems encountered where anomalous propagation and unpredicted path loss affect the performance of these systems. The influence of these unpredicted propagation effects are at times so severe that a complete communication breakdown occurs between transmitter and receiver or a radar misses its target completely. It is mandatory for a propagation engineer to take into account the deviation of the propagating wave due to the changes in the distribution of refractivity. Chapter 2 reviews the background review of the propagation of electromagnetic waves at radio frequencies. It gives the detailed description of propagation mechanisms such as diffraction, scattering and refraction. Then the detailed description of the refractive index, refractivity and modified refractivity is given with classification of different 1

17 types of refraction. The phenomenon of ducting where a propagating wave trapped in the form of a duct is classified into four different types with emphasis on the evaporation duct over the sea. Then different methods and techniques for measuring refractivity using refractometer, radar, GPS and lidar etc. are covered. The limitations connected to each of these methods are summarised especially the practical aspect of implementation. Finally a brief introduction is provided to the methodology used in this research for the simulation of radiowave propagation and why the method of parabolic equation is chosen. Chapter 3 starts with the introduction of historically used different propagation models and techniques where the method of geometrical optics and coupled mode are described in brief. It then covers the detailed theory for the core method of parabolic equation. The theoretical and mathematical description of Maxwell s equations, which are the governing equations for the macroscopic electromagnetic phenomena, and their reduction to the paraxial propagation domain used in this research are covered. The simplistic form of the solution for propagation in vacuum including boundary conditions and absorbing effects is demonstrated. Results from the implemented model are then compared to the work performed by other researchers in this area and other available software/computer programs for a number of cases. The results obtained from the simulation were correlated with experimental observations. The final part of chapter 3 describes the experimental setup used to get measurements of path loss for four over-sea communication links in the UHF/VHF frequency band. These links were named as UHF Portland, UHF Alderney, VHF Portland and VHF Alderney. 2

18 Chapter 4 describes the characterization of the parameters used in the implemented model. It covers the effect on path loss by changing some of the basic parameters like evaporation duct height, antenna height, frequency etc. These parameters are changed in a suitable fashion for a range of arbitrary values depending on getting sensible output results in terms of path loss. The characteristic curves are presented for most of parameters used in the model. The experimental and modelled values of path loss are compared in chapter 5. M profiles representing the change in refractivity with altitude in M units per km based on measurements of weather conditions are used as input to the model. The annual and monthly correlation statistics between measurement and model are presented for four evaporation duct heights (0 30 m). Finally, the conclusions of the work are presented in chapter 6. 3

19 2 Literature review Knowing the field distribution of refractivity as accurately as possible is important for making predictions of wave propagation. This distribution is complex as it depends on the prevailing atmospheric conditions at different geographical regions of the world and at different times, seasons and years. It depends on atmospheric pressure, air temperature related to insolation and water vapour concentration related to air temperature, sea temperature and wind speed. Examples of these atmospheric conditions include (1) varying concentration of water vapour at different altitude, (2) different situations produced by heating and cooling of water contents in the air/sea, etc. These varying conditions give rise to changes in refractivity. Wave propagation is very sensitive to such variations in the structure of refractivity. Tracking the changes in refractivity and their resulting impact on propagation is the goal of this research. 2.1 Background Electromagnetic waves are widely used in a range of applications and therefore understanding how they propagate is important. Maxwell s equations [e.g. Jordan and Balmain, 1950], provide a solid foundation for modelling wave propagation. When an electromagnetic wave travels from one point to another through the atmosphere, it is termed a propagating wave and the phenomenon is called propagation. Propagating waves are classified mainly by frequency band; e.g. the UHF (Ultra High Frequency) band extends from 300 to 3000 MHz, while the VHF (Very High Frequency) band lies between 30 and 300 MHz. During the process of propagation the energy contained in the radio wave spread out in the atmosphere from 4

20 the point the wave starts radiating from the aperture of the transmitting antenna. As the composition of the atmospheric structure is complex it is difficult to predict the characteristics of the propagating wave without extensive simulation. The complexity of electromagnetic wave propagation is generally simplified by consideration of a ray. A ray is a line drawn normal to the wavefronts or equiphase surfaces carrying the energy in a wave. When a ray travels through vacuum (an idealized medium where there are no atmospheric particles and whose properties are isotropic, homogeneous and loss free), it travels in a straight line without any absorption or reflection of energy by/from nearby objects. The ray travels at a constant speed as the refractive index of the medium is not changing from one point to another. Free space transmission loss is given by the standard relation of Equation 2-1 [e.g. Matthews, 1965; ITU-R P.525-2, 1994]. Where: ( ) ( ) Equation 2-1 is the free space loss in decibels (db); d is the distance in km; and f is the frequency in MHz. The derivation of Equation 2-1 is given by Freeman, [2007]. When air is the medium of propagation, then a propagating wave confronts a variety of atmospheric particles (e.g. water droplets) and different atmospheric features (e.g. evaporation duct). The effect of this confrontation compels the propagating wave to travel in a curved path instead of straight line. Fortunately, this curving behaviour of the ray path is advantageous as the surface of the Earth is also curved and when the ray has the same curvature as that of the Earth then it can travel for longer distance and ideally can cover the whole globe. However in reality the distance covered by the wave 5

21 is limited by the atmospheric particles and atmospheric features in a variety of ways depending upon the operating frequency of the transmitted signal. The atmosphere is divided principally by gravity in different layers. These layers vary with time and seasons in a variety of ways and each layer has different characteristics that affect radiowave propagation. The lowest layer or region of the atmosphere up to an altitude of 8 km at the poles and 18 km at the equator is called the troposphere. This region contains 80% of the mass of the total atmosphere and 99% of the total water vapour. All meteorological mechanisms, due to clouds, rain, heating or cooling of water bodies etc. occur in this region [Barclay, 2003]. The large scale structure of the troposphere varies much more rapidly vertically than horizontally. In the troposphere under normal atmospheric conditions the air pressure decreases exponentially with height. The temperature decreases linearly with height by about 10 C per km so that layers of cool air are above layers of warm air. At times, however, layers of warm air are formed above layers of cool air and this situation is termed a temperature inversion. Temperature inversions cause ducts of cool air to be sandwiched between the surface of the Earth and a layer of warm air, or between two layers of warm air. Changes in water vapour partial pressure with height depend on the process of condensation. In the absence of condensation, the water vapour partial pressure decreases exponentially with height. However as water vapour partial pressure decreases with temperature and hence with height water condenses to form water droplets or clouds above a certain height. Condensation occurs as air can hold a limited amount of water vapour at a given temperature. Once condensation occurs, the 6

22 temperature falls more slowly with height than before due to the latent heat released by the condensation process [Barclay, 2003]. The net effect of these variations in the atmosphere on the propagating wave is to bend it with a curvature that depends mainly on the structural state of the troposphere (see Section 2.1.3). Since the atmospheric structure is very complex in nature, estimating the wave propagation to a good accuracy is very difficult. Some of the different propagation mechanisms that occur with different atmospheric conditions are described in the following sections Diffraction Diffraction is the tendency of a wave to bend around an obstacle it strikes. Diffraction results in a change of direction of part of the propagating wave from the normal line-ofsight path and makes it possible to receive energy around the edges of an obstacle or at some distances below the highest point of an obstruction. Diffraction was described by Huygen-Fresnel principle [e.g. Born and Wolf, 1999] and the principle of superposition of waves, when the wave encounters an obstacle in its first Fresnel zones [e.g. Clarke and Brown, 1980] (maximum signal power exists in the first Fresnel zone). The loss associated with this mechanism is termed a diffraction loss. This loss can be avoided by providing the Fresnel clearance for line-of-sight communicating antennas which is obtained generally by providing 60% of the first Fresnel zone as unobstructed [Barclay, 2003]. For nonline-of-sight, different models (e.g. ITU-R P , [2012]) are used to calculate the diffraction loss. Diffraction depends on the geometry of the obstructing object as well as on the amplitude, phase and polarization of the wavefront at the point of diffraction [Matthews, 1965]. 7

23 In relation to the long range propagation discussed in this thesis diffraction occurs when a wave travels beyond the visible horizon of the Earth and tends to follow the Earth s curvature. Although the energy in a diffracted radio wave is small compared to the direct signal, it can still be detected by the receiver and may be the dominant propagation mechanisms at times. However diffraction is complicated due to the changes at the interface between the sea/ground and air. Since this research is limited to the propagation over the sea for transhorizon paths, the only object that leads to diffraction is the surface of the sea. Fortunately sea surface is a relatively good reflector of radio waves compared to the ground. So the effect of the changes at the interface between the sea and air are less severe than those between land and air. Also the effect due to the variations in the tides, as well as the effect of evaporation just above the sea surface is taken into account in this research. The effect of tidal variations is included by increasing the heights of the transmitting and receiving antennas simultaneously by the corresponding rise or fall in the level of the tides from datum taken from the available tidal data set. Similarly the heights of the antennas were reduced when the corresponding tidal level in the data set moves down Scattering When a propagating radio wave travels through the troposphere, it collides with air particle (or group of particles). These particles form a cluster which is generally termed a scatter medium and make the energy to produce power fluctuations at the receiver. Scattering from a single object is possible as well but the effect becomes more pronounced when a scatter medium makes the wave to be reflected directly to the Earth s surface as shown in Figure 2-1. The total amount of scattered power at the receiver is the sum of the energy received from each of the particle in the scatter medium. 8

24 The height and size of the scatter medium is determined by the angle of radiation of the transmitting antenna. Several different theories exist to determine scatter power at the receiver as well as the size and location of the scatter medium. However, accurate determination of these values is the challenge for the design engineer to make use of scattering as a dominant propagation mechanism [Booker and Gordon, 1950; Villars and Weisskopf, 1954; 1955]. Since the creation of scatter medium happens in the troposphere due to the large amount of mixing of particles in this region, a more general term used in the radio community for scattering is tropo-scatter which is a short form of tropospheric scatter. The main reason of occurrence of tropo-scatter is the turbulent flow of air, rain drops, fog, snow etc. within the scatter medium which causes the variations in atmospheric temperature, pressure and humidity and hence refractive index [Shen and Vilar, 1995]. Figure 2-1 Tropospheric scattering mechanism for transhorizon propagation Refraction Refraction is the core subject described throughout in this research where the troposphere is studied as a refractive medium characterized by refractive index for 9

25 UHF/VHF propagation. The atmospheric particles are in random motion all the time which causes variations in this refractive medium. The effects of these changes on the propagating wave are covered under this type of propagation mechanism. Variations in temperature, pressure and water contents at different points in the atmosphere are the main cause for the occurrence of refraction. Since it is not convenient to find the temperature of a single particle, the pressure exerted by that particle on nearby particles or the fluctuations in density/distribution of water vapour hence these parameters combined to make them simple for the better prediction of radiowave propagation. For this purpose refractive index is used as described in details in Section Refraction and refractive index Refractive index is a value used to measure the extent of refraction for a medium. By definition, the refractive index, n, of a material is a factor by which the phase velocity of an electromagnetic wave is slowed down relative to vacuum. It was for the first time introduced by a Dutch scientist Willebrord Snell ( ) in his famous Snell s law of refraction presented in 1621 which describes the relationship between the angles made by the ray of incidence and that of refraction. Its value is at the Earth s surface [Barclay, 2003]. Snell s law of refraction provides the basic theory for describing the path followed by electromagnetic wave when it travels from source to destination. It states that when a ray travels from one medium to another medium having different refractive indexes, then it bends either towards or away from the normal to the interface of the two media. This is illustrated in the diagram as shown in Figure 2-2 and mathematically described by the relation in Equation

26 ( ) ( ( )) Equation 2-2 Where: ( ) is refractive index at altitude z and ( ) is the angle made by the ray at that altitude. In Figure 2-2, the atmospheric stratification is characterized by different refractive indexes at different points of the atmosphere. As a result, a wave travels with different velocities in the different layers (e.g. the upper part of the horizontally propagating wavefront may travel faster than its lower part making it to bend downwards) [Barclay, 2003; David and Voge, 1969]. Figure 2-2 Ray traveling from denser medium to rarer medium while changing its direction Refractivity and modified refractivity The refractivity, denoted by N, is widely used instead of refractive index. Refractivity is a scaled up value of refractive index and is related to it by Equation 2-3. Where: ( ) Equation 2-3 is refractivity and is refractive index of the medium. 11

27 Although N is dimensionless, N-Units (or NU) are often used when expressing values of N. Generally, N is computed from Debye equation [Debye, 1929] which is given by Equation 2-4. Equation 2-4 Where: and are the air pressure and water vapour partial pressure in mbar, respectively; and is the temperature in Kelvin [Bean and Dutton, 1966]. Equation 2-4 shows that variation in N is a function of variations in P, T and e. Typical values for P, T and e are 1000 mbar, 293 K and 15 mbar, respectively. Among these three atmospheric variables, water vapour has the greatest effect on refraction followed by temperature while pressure has the least especially in the lower part of the troposphere. Ali et al. [2012] analysed the statistical variation of refractivity due to local meteorological conditions for a period of 10 years from 2000 to 2009 by comparing T and e for a constant value of P and found that a large change of about 360 NU in refractivity can occur under some extreme conditions of high T and e values in the summer. The refractivity of the atmosphere varies with temporal and spatial variations in, and. Spatially it varies in both horizontal and vertical directions. However, the vertical variation is of more direct interest to the researchers as the atmospheric layered structure is more pronounced vertically and horizontal homogeneity can be considered for a limited range in radiowave propagation. The vertical variation in refractivity is measured by N profile which is a representation of refractivity gradient with respect to altitude. 12

28 Sometimes modified refractivity (M), measured in M-Units (MU), is used where the Earth s curvature is taken into account and which is related to N by Equation 2-5. Where: Equation 2-5 is the altitude in km; and are refractivity in NU and modified refractivity in MU, respectively. The main advantage of working with M is that it transforms a spherical propagation problem into a planar one and compensates for the curvature of the Earth s surface making it a flat surface which is comparatively easy to deal with [Pekeris, 1946; Kerr, 1951]. In this research, M is preferably used instead of N. However in some places N is mentioned as well due to common practice. Refraction is characterized by different types of M profiles and is generally classified into different types (or also known as refractive conditions ) as described in the following sections Standard refraction Standard refraction prevails under standard atmospheric conditions, where it is assumed that a linear decrease in temperature with altitude occurs at a rate of 6.5 C per km, air is a perfect and dry gas obeying both Charles Law and Boyle s Law and gravity is constant at all altitudes. It basically gives a hypothetical vertical distribution of atmospheric air temperature, pressure and density as defined by NOAA (National Oceanic and Atmospheric Administration) and NASA (National Aeronautics and Space Administration) for reference purposes [NOAA and NASA, 1976]. Such a standard atmosphere can only be used in theoretical calculations; however, seasonal and geographical standard reference atmosphere can be obtained from ITU-R P.835-5, [2012]. 13

29 Standard refraction occurs where the value of M increases with altitude at a rate of approximately 117 MU per km. For this profile, the value of N decreases with altitude at a rate of 40 NU per km with altitude (see Figure 2-3). The behaviour of the propagating wave following standard refraction is shown in Figure 2-4. A well-known relationship for the representation of standard refraction is given by Bean and Dutton, [1966] as shown in Equation 2-6. Where: ( ) Equation 2-6 in km. is the refractivity at sea level in NU; is the scale height in km; and is the altitude The scale height refers to the height at which N reaches to of. Average mid latitude values for surface refractivity and scale height are 315 NU and 7.35 km, respectively. Bean and Dutton, [1966] used exponentially decreasing values of refractivity. However the temperature in a hypothetical standard profile defined by NASA is linearly decreasing with altitude and the linearly decreasing values of refractivity provides a reasonable approximation to the exponentially decreasing values in the first kilometre of the troposphere. Hence linearly decreasing values are considered in this research except in the case of evaporation ducting (see Section 2.2.5). Global and seasonal values of surface refractivity can be found in ITU-R P , [2012]. 14

30 Figure 2-3 Classification of different types of refraction using N gradient and M gradient values [from Barclay, 2003]. Figure 2-4 Ray propagation under different refractive conditions (after [Gunashekar et al., 2006]). Apart from the standard refraction, non-standard refractive conditions of propagation occur as well. Propagation under these conditions is termed an anomalous propagation. Examples of anomalous propagation are sub refraction (see Section 2.2.3), super refraction (see Section 2.2.4) and ducting (see Section 2.2.5) as shown in 15

31 Figure 2-4 where the ray is bending with different curvatures under different refractive conditions. Mathematically, the radius of curvature (C) for the ray is given by Equation 2-7 [Freeman, 2007]. ( ) Equation 2-7 Where: is the change in refractive index; is the change in altitude, and is the angle made by ray with the Earth s surface. Since is very small for the work done in this research hence the value of ( ) is close to 1 and for a small value of, the radius of curvature of the ray can be approximated to a function of (i.e. N gradient) only Sub refraction Sub refraction is the type of anomalous propagation where the ray bends away from the Earth s surface as shown in Figure 2-4. It occurs when the N gradient value is greater than 0 or when the M gradient value is greater than 157 as shown in Figure 2-3. In other words this happens when the temperature and humidity distribution creates an increasing value of N with height. Sub refraction can result in a considerable attenuation in the signal strength as the direct ray path between transmitter and receiver is obscured by the bulge of the Earth s curvature. Gunashekar et al. [2007] analysed 8340 refraction events in a period of two years ( ) at 2 GHz source for a 50 km sea path over the English Channel and found that sub refraction occurred 439 times (5%). It enhances multipath effect at the receiver as the upwards propagating wave may be reflected back to the Earth s surface by the propagation mechanism of scattering [Patterson et al., 1994]. 16

32 2.2.4 Super refraction As stated earlier, under standard atmospheric conditions, temperature decreases with altitude at a lapse rate of 6.5 C per km. This phenomenon reverses when temperature inversion occurs. Both this and when water vapour contents decrease rapidly with altitude results in decrease in N gradient from the standard rate of -40 NU per km. The wave propagating under such conditions bends towards the Earth s surface at a rate slightly more than the standard rate of refraction as shown in Figure 2-4. This type of anomalous propagation is known as super refraction. It occurs when the N gradient value is between -79 and -157 or when the M gradient value is between 0 and 78 as shown in Figure Ducting The process in which a propagating wave in the open atmosphere is guided or ducted due to some atmospheric conditions is termed as ducting or trapping. Ducting occurs when the N gradient value is smaller than -157 or when the M gradient value is smaller than 0 as shown in Figure 2-3. In other words, if the radius of curvature of a ray is less than the radius of the Earth, then it reaches a maximum height and is trapped between that height and the Earth's surface with a certain thickness. The behaviour of the propagating wave following ducting or trapping is shown in Figure 2-4. Ducting is further classified into the following types and described schematically as shown in Figure

33 (a) (b) (c) (d) Figure 2-5 M profiles representing different types of ducts (a) Evaporation duct (b) Surface duct (c) Surface based duct (d) Elevated duct [from Patterson, 2003] Evaporation duct The existence of an evaporation duct over the sea was first suggested by Katzin et al. [1947] based on measurements of 3 and 9 cm waves (10 GHz and 3.3 GHz). Since then, different researchers have studied and modelled it using different approaches [Hitney et al., 1978, 1985; Hitney, 2002; Richter and Hitney, 1988; Anderson, 1989; Kerans et al., 2002]. It received detailed study in 1970s because of the degradation in line-of-sight digital communication systems due to multipath propagation [Shen and Vilar, 1995]. It occurs due to the change in refractivity caused by saturation of the air with water vapour just above the sea up to a height of few tens of metres. Zhao et al. [2013] 18

34 observed that the probability of occurrence of evaporation duct over the ocean is more than 75%. The probability of occurrence of evaporation duct depends on the wind speed as well as on the altitude from the sea surface. If there is no wind then the duct dies out very quickly [Kerans et al., 2002]. The main factor for characterizing the M profile representing evaporation duct is the evaporation duct height (EDH) which is a function of pressure, temperature, altitude, Jeske s roughness length and surface modified refractivity. The evaporation duct height is defined as that value of height for which M reaches to its minimum value. The relationship that relates these parameters can be derived from rearranging Equation 2-4, Equation 2-5 and Equation 2-6.The world average duct height is 13 m and the tropical climates are more favourable than temperate for the formation of evaporation duct. Under neutral thermal conditions (one where sea temperature is equal to air temperature) the simplified M profile representing evaporation duct at a certain value of EDH is given in Equation 2-8 and plotted in Figure 2-5 (a) [Hitney and Vieth, 1990]. The derivation of Equation 2-8 can be found in Jeske, [1973]. ( ( )) Equation 2-8 Where: is the value of M at the Earth s surface; is the Jeske s roughness length equal, over the sea, to m; and is the evaporation duct height in metres. Roughness length takes the roughness of the sea surface into account where a surface is considered as rough if the variations of the surface are such as to cause variations in the path length of more than an eighth of a wavelength. At the lower frequencies the 19

35 roughness length is small compared to that at higher frequencies for both land and sea surfaces [Kerr, 1951; Matthews, 1965]. The effect of evaporation duct is frequency dependent with signals at higher frequencies more strongly trapped by evaporation ducts than those at lower frequencies [Paulus, 1990]. The effect of evaporation duct also depends on the angle of incidence of the propagating wave to the horizontal direction [Thompson and Haack, 2011]. Once trapped then a signal can propagate substantially further beyond the horizon and, sometimes, signal enhancement occurs where the propagating wave incurs less path loss than the free space loss. Its detailed annual frequency distribution is given by Hitney and Vieth, [1990] for a period of 15 years covering a range of evaporation duct heights from 0 to 40 m with different M profiles. The sites used for these long term studies were the Aegean Sea and the North Sea for a 35.2 km path at different frequencies between 0.6 and 18 GHz, where a decrease in path loss was reported with the increase in evaporation duct height. For instance a 54 db reduction in path loss was found for 23 m increase in evaporation duct height over the North sea at 2.3 GHz as shown in Figure

36 Figure 2-6 Path loss versus evaporation duct height for the North Sea experiment taken from Hitney and Vieth, [1990] Surface ducts Surface ducts are formed by meteorological conditions on the surface of the Earth (i.e. from surface trapping layer) and occur less often than evaporation ducts. A surface duct is simply represented by a bi-linear M profile as shown in Figure 2-5 (b). In the lower portion M decreases linearly with increasing altitude whereas in the upper portion it increases linearly. The shift from decreasing M to increasing M occurs at a certain height which represents the upper boundary of the duct. The height from the surface to this height is generally known as the thickness of the duct. The difference between the value of refractivity at this height and the value at the surface of the Earth is termed the strength of the duct. The thickness (m) and strength (MU) are two important parameters used to characterize surface ducts. Although the wave is propagating in the duct, some of the energy of the trapped signal leaks into the upper atmosphere through the upper boundary of the duct. 21

37 Surface based ducts Some meteorological conditions result in the formation of ducts from elevated trapping layer which are termed a surface based ducts as shown in Figure 2-5 (c). These are actually surface-based elevated ducts represented by a tri-linear M profile and also known as elevated-surface duct. A tri-linear M profile consists of three linear portions. In the first portion M increases linearly with increasing altitude. Then a sudden inversion occurs in the second portion where the value of M decreases with increasing altitude for certain values of height. In the last portion at some height, M starts increasing again. Surface based ducts occur when the air aloft is exceptionally warm and dry compared with the air at the Earth s surface [Patterson, 2003]. Surface based ducts occur in the Aegean Sea about 11% of the time annually, and in the North Sea area about 1.7% of the time. [Hitney and Vieth, 1990]. As this type of duct occurs rarely compared to the surface duct, they are normally considered in the same group as recommended by ITU- R P , [2012]. The meteorological conditions responsible for surface based duct and elevated duct are the same Elevated ducts These occur within hot, dry and high pressurised areas, where meteorological conditions cause a large scale air compression producing heat. As a result a layer of warm dry air overlays a layer of cool moist air often called marine atmospheric boundary layer [Vickers et al., 2001; Skyllingstad et al., 2005] where the ocean and atmosphere exchange large amount of heat, moisture and momentum primarily via turbulent transport [Fairall et al., 1996]. Under such situations the base of the duct occurs above the Earth s surface and hence named as elevated duct. It is shown in Figure 2-5 (d). 22

38 In case of the elevated duct both boundaries of the duct are in the troposphere while in case of the surface based duct (or surface duct) one of the boundaries of the duct is the surface of the Earth. In general, all types of ducts are characterized by some important parameters. For evaporation duct two parameters duct strength and duct base height are defined where the duct base height is more commonly known as the EDH. Both surface based duct and surface duct are characterized by two parameters, namely, strength (S s, MU) and thickness (S t, m) of the duct. An elevated duct is characterized by four parameters, namely, strength (E s, MU), thickness (E t, m), the base height (E b, m) and the height within the duct of maximum M (E m, m). These characteristic parameters except EDH for evaporation duct are shown in Figure 2-7 [ITU-R P , 2012]. Figure 2-7 Characteristic parameters recommended by ITU for (a) Surface duct (b) Surface based duct and (c) Elevated duct [from ITU-R P , 2012]. 23

39 2.3 Measuring refractivity In the last few decades a considerable amount of research has been made in this area using different techniques and algorithms. Among these the most widely used method is the parabolic equation method which is utilized in this research and is found to be the best one for its accuracy, time sampling, vertical and horizontal resolution in space, speed and stability [Grabner and Kvicera, 2011; Dockery, 1988; Valtr and Pechac, 2005; Vasudevan et al., 2007]. Gunashekar et al. [2008] compared PEM (see Section 3.1) predicted results with ITU-R P.526, [2003] and experimentally measured results for three sea propagation paths in the English Channel Islands at 2 GHz to investigate sub-refractive atmospheric conditions. Grabner and Kvicera, [2011] compared PEM predicted results with measured results from 2007 to 2011 in the Czech Republic at 10.7 GHz frequency to investigated atmospheric refractivity related effects. Stagliano et al. [2009] used PEM to remove contaminated data from the radar data caused by the presence of anomalous propagation. Various methods are used to measure refractivity that can be classified into direct sensing methods and remote sensing methods as described in the following sections Direct sensing method In the direct sensing method refractivity is either measured directly by a device called refractometer or derived from the atmospheric conditions measured through radiosondes, rocket sondes, buoys. The standard Abbe refractometer was designed by Ernst Abbe in 1869 which has been upgraded in terms of improvement in features like speed and accuracy to the modern refractometer, however, the basic principle is still the same [Rheims et al., 1997]. 24

40 Measuring refractivity using a direct method is ideal as far as accuracy is required, however, on one hand the refractometer is delicate and hard to calibrate [Levy et al., 1991] and on the other hand getting refractivity values at a high resolution spatially is inconvenient due to limitations on the physical placement of the device. However it is still in use at a small scale especially for the validation of the results obtained by other means. In another form of direct sensing method refractivity is deduced from observations of weather variables obtained through radiosondes or by some other means like in situ measurements where the readings of temperature, pressure and water vapour partial pressure are recorded and refractivity is calculated using Equation 2-4 [Bye and Howell, 1989; Bean and Dutton, 1966]. A radiosonde consists of air sensors that measures different atmospheric parameters; e.g. temperature, pressure, relative humidity, wind speed and direction etc. and send them to a static receiver at the ground. Radiosondes are deployed by either dropping from aircraft or lifting from ground using a small size balloon filled up with hydrogen or helium. Mainly they are used for the weather forecast and for this purpose radiosondes are launched from around 800 sites worldwide, usually twice a day. Unfortunately, radiosondes have many drawbacks, including limited time sampling and limited vertical resolution [Gibson et al., 1997]. Direct methods are reliable but the installation of the conventional devices described above with high spatial resolution is expensive and are not practical for real-time operational use [Douvenot et al., 2010]. The performance of these conventional devices used in the direct sensing methods is severely restricted by different conditions of atmospheric turbulence [Vasudevan et al., 2007]. 25

41 In this research, the refractivity measurements obtained by the direct sensing method are taken from ground based weather stations. More details about these weather stations can be found in Section These weather stations record the weather parameters like temperature, pressure, humidity and wind speed continuously throughout the globe using different instruments. For example thermometer is used for measuring air and sea surface temperature, barometer for measuring atmospheric pressure, hygrometer for measuring humidity and anemometer for measuring wind speed. Automatic measurements are recorded at least once an hour throughout the year. The values obtained from such ground based measurements can be used to obtain information about the structure of refractive index. The instruments used in the weather stations have limitations in terms of installing them at various locations, at different heights, and in recordings of the required data with high resolutions. For example a weather ship to be stationed in the ocean as a platform for surface and upper air meteorological measurements would record weather parameters at the point of its placement only Remote sensing method The direct sensing methods tend to provide estimation of refractivity versus height at a single location while remote sensing method can measure refractivity remotely at different ranges and azimuths. In this method, refractivity is inferred using different techniques like radar, GPS or lidar as described briefly in the following sections RADAR This type of remote sensing technique is more commonly termed a RFC (Refractivity from Clutter) which is a short form of refractivity from RADAR (Radio Detection and Ranging) clutter return or backscatter (reflection of the radar transmitted signal from 26

42 the ground or sea surface). Radar clutter is the term used for unwanted echoes from objects other than the desired target. Ground clutter, weather clutter and sea clutter are the common examples. Factors affecting the sea clutter depend not only on the sea surface layer and surface based atmospheric ducting, particularly evaporation ducting but also on the wavelength, grazing angle and polarization of the transmitted beam [Yardim et al., 2008]. The relationship between radar sea clutter and refractivity is non-linear and ill posed and it is very difficult to get an analytical solution and hence different theories and optimization procedures are in use to get an approximate solution. For example, Krolik et al. [1999] used Markov model [Seymore et al., 1999] for microwave radar clutter returns from the sea surface to perform RFC estimation. Specifically, the parabolic approximation for numerical solution of the wave equation was used to formulate the problem within a non-linear recursive Bayesian state estimation framework and the solution was obtained using sequential importance sampling method. A synoptic characterization of the atmospheric ducts over the spatial extent of the radar was described to overcome the necessity of additional data required in the conventional method of refractivity estimation. However this approach is prone to instability due to the contamination of true data domain which is due to the horizontal variability of the sea clutter radar cross section [Rogers et al., 1999; Vasudevan and Krolik, 2001; Yardim et al., 2006; Vasudevan et al., 2007]. Gerstoft et al. [2003a; 2003b] demonstrated the ability to infer refractivity parameters from radar sea clutter with emphasis on using multiple elevation beams and different grazing incidence techniques. An 11-parameter environmental model was mapped onto an electromagnetic propagation model, which was then mapped into a space of radar clutter observations. Two common refractivity structures; evaporation ducts and surface 27

43 based ducts were investigated. A Genetic Algorithm [Douvenot et al., 2008] was used for optimization of the objective function to get a better match between the observed (determined by Helicopter soundings) and calculated refractivity measurements. Gerstoft et al. then described the advantage of using the remote sensing technique of refractivity estimation which makes it able to sense range varying refractivity at a temporal sampling rate that can track changes in the atmospheric conditions. Fabry et al. [1997] used pair-wise differencing technique which exploits the fact that the phase of a radar return from a target is a function of the path integral of refractive index. Fabry et al. devised a method of estimating the refractive index distributions from the height-gain profile of the received signal strength, which is generated by a surface based duct or an elevated duct. This technique was performed using maximum a posteriori approach where only the linear variations in refractivity parameters can be obtained. Hao et al. [2005] optimized a standard least square algorithm [Douvenot et al., 2008] to modified least square algorithm for refractive index field estimation which is effective in recovering the refractive index field from radar echoes collected by multiple radars from ground targets. The solution to the problem of phase unwrapping, which occurs when a system of multiple radars are used based on the pair wise differencing technique, was provided. In contrast to the conventional in situ measurements, remote sensing measurements require fewer sensors and hence are comparatively less expensive. It provides detailed picture of the dynamics and structure of atmospheric turbulence. Remote sensing predictions are useful in many applications of weather prediction systems; i.e. sensing temperature, humidity, rain forecasts for agricultural purposes in addition to radio 28

44 communication systems etc. Such predictions are important in order to cope with the problems encountered where anomalous propagation and path loss of electromagnetic waves are affecting the performance of the wireless communication systems [Vasudevan et al., 2007; Yardim et al., 2009] GPS Another method of measuring refractivity is that of radio occultation with GPS (Global Positioning System) which is normally combined with radiosonde measurements to improve accuracy. GPS radio occultation is in use since last few decades for sounding the lower part of the atmosphere [Hajj et al., 2002; Ao et al., 2003; Sokolovskiy and Vilar, 2001]. An excellent review of this method is given by Kursinski et al. [1997] and Bevis et al. [1992]. In this method the GPS data is processed which provides a high resolution refractivity data; e.g. one or more satellites operating in LEO (Low Earth Orbit) can yield approximately 500 profiles per LEO orbit daily [Shrestha, 2003]. However, several issues including processing large mass of GPS data and improper calibration of phase delays measured from GPS in order to isolate from the atmospheric delays put a number of limitations on this method [Johny et al., 2009] LIDAR LIDAR (Light Detection and Ranging) is an optical remote sensing technology that measures properties of scattered light to find range and/or other information of a distant target. The word LIDAR comes from combining the words light and radar. The primary difference between radar and lidar is the latter uses much shorter wavelengths of the electromagnetic spectrum, typically in the ultraviolet, visible, or near infrared range. 29

45 The former at VHF and higher frequencies is used extensively for estimation of refractivity. 2.4 Propagation models and techniques Many propagation models and techniques are developed and implemented to solve different types of electromagnetic wave problems. They can be classified into two groups, namely as full wave or low frequency techniques and approximate solution or high frequency techniques. Basic high frequency techniques are Physical Optics (PO) [Mach, 1953], Geometrical Optics (GO) [Born and Wolf, 1999], Physical Theory of Diffraction (PTD) [Ufimtsev, 2007], Geometrical Theory of Diffraction (GTD) [Clarke and Brown, 1980] and Uniform Theory of Diffraction (UTD) [Kouyoumjian and Pathak, 1974]. Low frequency methods have been developed for the numerical discretization and approximation of equations deduced from Maxwell s equations. The Method of Moments (MoM) [Harrington, 1990; 1993], Finite Difference Time Domain (FDTD) method [Yee, 1966; Taflove, 1988] and Finite Element Method (FEM) [Jin, 1993] are the most commonly used low frequency methods. These methods require high computer resources at higher frequencies. A few of these propagation models developed so far are briefly described in the following sections Geometrical optics Geometrical optics involves propagation of light waves, which gives the best analogy to electromagnetic wave propagation and is the simplest conceptual model. Light rays bear same properties of bending as electromagnetic waves when they travel from one medium having some refractive index to another medium with different refractive index. These rays are also reflected and absorbed by surfaces it strike. In fact, they do 30

46 have the diffraction, refraction and scattering properties just like electromagnetic waves. In this type of propagation modelling, energy of the wave is propagated in rays which are trajectories orthogonal to the wavefronts of a plane wave electromagnetic field. These rays are traced from transmitter to receiver for which the path is described by differential equations derived from repeated application of Snell s Law. Hence this type of model is commonly known as Ray Tracing propagation model. There are several limitations to geometrical optics few of which are outlined below. A single ray cannot be split into two There is no amplitude information in a single ray It is a frequency independent technique Difficult to classify rays into families for field calculation Coupled mode This method is also known as waveguide model. It was developed by Baumgartner et al. [1983] and later on improved and extended by Baumgartner, [1983] and Shellman, [1986]. In this method the solution to the propagating wave is found by finding its roots through a Shellman-Morfitt complex root-finding routine. Different propagation modes in a multilayer tropospheric waveguide environment are represented by various eigenvalues. The eigenvalue, for which the model function obtained through the Laplace s expansion is zero, is used to calculate the field strength of the propagating wave [Isaakidis and Xenos, 2004]. Searching for the roots of the model function requires an efficient method and huge computation resources especially when higher frequencies and complicated M profiles 31

47 are involved. This is because the algorithm is very mathematical expensive for finding the poles and zeros of each of the modal function that propagates. This algorithm is incorporated in a multilayer tropospheric propagation program developed by NOSC (Naval Ocean Systems Centre) termed a MLAYER which is used to locate the modes that propagate in the tropospheric ducts [Budden, 1957; Yeoh, 1990; Lee and Han, 1993]. On the contrary in case of parabolic equation method where direct solution of the wave can be easily solved numerically through marching algorithm and neglecting the backward wave propagation is a reasonable approximation especially when treating electromagnetic wave propagation over the sea. The detailed description of PEM with mathematical formulation is given in the following section. 2.5 Parabolic equation method The parabolic equation method (PEM) has been used for many years to model radiowave propagation in the troposphere especially over the sea. This section provides background and a brief introduction to PEM and the detailed theory and procedure for its implementation is given in Chapter 3. PEM was originally proposed by Mikhail Aleksandrovich Leontovich in 1944 [Leontovich, 1944] for long range radiowave propagation and later on in 1946, Leontovich together with Fock [Leontovich and Fock, 1946] provided PEM solution to electromagnetic waves problems. In 1977, Frederick Tappert, [Tappert, 1977] decomposed an elliptical wave equation into two equations through the choice of an arbitrary constant reference wave number, one of which resulted in the development of the standard parabolic equation (also called the narrow angle parabolic equation) [Lee and Pierce, 1995]. This technique gained popularity quite quickly and a number of 32

48 researchers started using it by developing different solution methodologies [Craig, 1988; Kuttler and Dockery, 1991; Craig and Levy, 1991; Ryan, 1991; Barrios 1994; Dockery and Kuttler, 1996]. PEM provides a reliable wave solution for the field in which real refractivity profiles are considered unlike the initially used rays-based solution techniques and mode theory-based solutions techniques for the prediction of electromagnetic propagation. In contrast to PEM, ray-based and mode theory-based solution techniques like geometrical optics, physical optics, normal mode analysis, coupled mode analysis and hybrid methods, resulted in an inappropriate solution [Dockery, 1988]. The basic theoretical development of parabolic equation starts with the reduction of the well-known 3 dimensional Maxwell s equations, representing the existence of a full electromagnetic wave, to 2 dimensional time harmonic Helmholtz equations in range ( ) and altitude ( ). This reduction is performed under the considered paraxial propagation domain (see Section 3.1.2) in which the energy of the propagating wave travels in the form of a cone having vertex at the transmitting antenna and making a small grazing angle (term used for angle made by the wave with the horizontal direction of wave propagation). The horizontal and vertical polarized components of the field are propagating independently inside the cone with a time dependence of ; where is the angular frequency of the propagating wave and is the time [Levy, 2000]. Generally two methods (1) finite difference method (FDM) [Smith, 1978] and (2) split step Fourier transform (SSFT) are used to get the numerical solution of the reduced parabolic wave equation. An excellent comparative description of the two methods is provided by Apaydin et al. [2010]. The first method requires huge computational 33

49 resources because of getting the solution of large system of simultaneous equations in large number of unknowns and the specification of radiation boundary conditions [Israeli and Orszag, 1981] on a closed domain [Villarreal and Scales, 1997]. FDM solves the wave equation explicitly in the time domain only without dropping the carrier frequency and hence a great amount of computing time and storage is needed. In this research SSFT algorithm is chosen as it uses larger range steps, which makes it more efficient computationally. SSFT is a powerful technique pioneered by Hardin and Tappert, [1973] which works on the principle of marching the solution forward in short steps until converged solution is obtained. In other words, the solution to the problem of interest is obtained by splitting the solution in a series of phase screens (steps) orthogonal to the direction of propagation of the field. First the initial field is propagated and then a phase screen modulated by refractive index variations is applied to it. The resulting field is then forward propagated through the medium to the next phase screen, and so on. A more detailed technical mathematical derivation for this technique with its application to tropospheric propagation problems as well as its implementation by using different solution methods can be found in Levy, [2000]. 34

50 3 Model implementation, validation and experimental setup The parabolic equation method was introduced in Section 2.4. This chapter covers the detailed description about its implementation. Then it covers the validation process of the implemented model of parabolic equation method. The implementation is validated by comparing the results generated by PEM with the results obtained from work done by other researchers in this area. Modelled results are validated by comparing the results with another computer program AREPS (Advanced Refractive Environmental Prediction System) as well. Finally the experimental setup used in this research is described. 3.1 Description of PEM In this section the standard parabolic equation is derived from Maxwell s equations. Maxwell s equations provide a mathematical representation of electromagnetic waves. The complete implementation of a solution of interest is complex and requires more computational time if the variation in refractive index with respect to height is considered. The complexity involved in the solution of equations is reduced by doing approximation which depends on the domain of the problem and the subsidiary conditions required for the numerical solution of the problem. Standard parabolic equation is such a reduced mathematical representation of electromagnetic waves and the PEM is the most widely used approximation method in radiowave propagation that utilizes the standard parabolic equation to get solution of the electromagnetic waves. 35

51 3.1.1 Maxwell s equations Maxwell s equations, introduced by James Clark Maxwell, [e.g. Born and Wolf, 1999; Jackson, 1975] characterize an electromagnetic wave at any point in the medium in which it travels. In other words the electromagnetic wave exists if and only if it satisfies the following set of Maxwell s equations presented in both differential and integral forms. Equation 3-1 Equation 3-2 Equation 3-3 Equation 3-4 Where: is the electric field intensity in V/m; is the magnetic field intensity in A/m; is the electric flux density in C/m 2 ; is the magnetic flux density in Wb/m 2 ; is the volume current density in A/m 2 ; is the free volume charge density in C/m 3. Maxwell s equations can be written in the form of scalar general wave equations which is a set of six independent scalar equations for both electric and magnetic fields having components as and in rectangular coordinate system. There are many possible functions that satisfy these scalar general wave equations depending on the nature of the source that is creating the wave. 36

52 In order to reduce the complexity of the wave equation and its solution, different approaches can be used depending on the problem of interest. One such approach is to consider a uniform plane wave (a wave that has same magnitude and direction in a plane containing it at any time) which reduces the scalar general wave equations from a set of six to a set of four equations. This reduction is possible as the magnitude and direction of the wave in a plane containing it is uniform and hence the variation of with respect to and and that of with respect to and is zero. Such a uniform plane wave propagating in the direction reduces these Maxwell s equations to four second order differential equations called Helmholtz s equations. Its advantage is that the wave is now a function of only two variables, direction of propagation ( ) and time ( ). The set of these reduced Helmholtz equations are given in Equation 3-5 to Equation 3-8. Equation 3-5 Equation 3-6 Equation 3-7 Equation 3-8 Where:,, and are the transverse field components; is direction of wave propagation; is time; and are electric permittivity and magnetic permeability of the medium in which the wave travels, respectively. 37

53 Equation 3-5 to Equation 3-8 are true under the following considerations. The medium of propagation of electromagnetic waves is homogeneous having electric permittivity ϵ (also called dielectric constant), magnetic permeability μ and conductivity σ. A medium is said to be homogeneous if it has the same properties at all points in the medium. The medium of propagation is considered to be isotropic and linear. A medium is said to be isotropic if its dielectric constant and magnetic permeability is independent of direction and is said to be linear if is parallel to and is parallel to. Far field approximation is considered which means the field radiated from a transmitting source in the form of spherical wave is approximately a plane wave far away from the transmitter. The set of relations given in Equation 3-5 to Equation 3-8 are similar to one another so their solutions will also be similar to one another and getting a solution for any equation is enough to get the solutions of the remaining equations in the set Paraxial propagation domain In a paraxial propagation domain, energy in the wave propagates from transmitter to receiver making a small grazing angle with the horizontal direction. Conventionally, is taken along the horizontal axis representing range and is taken along the vertical axis representing height of the propagation domain while a time dependence of is used where is the frequency of the source which excites the field. The horizontal and vertical polarized field components of the propagating wave are considered independently. Under these conditions, Equation 3-5 can be written in general as 38

54 ( ) Equation 3-9 Where: is the wave number; ( ) is the index of refraction and is the field component equal to either for horizontal polarized antenna or for vertical polarized antenna. In this research all the antennas used are vertically polarized. Now suppose that ( ) given by the relation in Equation 3-10 is the solution of Equation 3-9. Where: ( ) ( ) Equation 3-10 ( ) is a slowly varying reduced function of with respect to which is a rapidly varying function. Putting Equation 3-10 into Equation 3-9 will result in Equation ( ) Equation 3-11 Equation 3-11 is an elliptical equation and getting its unique solution is very difficult and computationally expensive to achieve until some appropriate boundary conditions are defined. Defining the lower boundary condition at the surface of the Earth is easy and given by the Leontovich or impedance boundary condition [Rokhlin, 1990]. The upper boundary condition is very difficult to define even with the stratified nature of the atmosphere. So Equation 3-11 is approximated by equating the first term ( ) to zero to get the relation given in Equation

55 ( ) Equation 3-12 Equation 3-12 is called Standard Parabolic Equation (SPE) and the method used for getting its solution is termed the Parabolic Equation Method (PEM). PEM is a full wave approach of getting solution of SPE. SPE is a narrow angle approximation method [Zhao, 2013] of the full wave equation as the effect of the approximated first term in Equation 3-11 would be significant for large propagation angles. For example an increase of one order of magnitude in the propagation angle (i.e. from an angle of 10 to an angle of 100 ) would increase the error by four orders of magnitude in the approximation (i.e. increase in error from value of 10-7 to 10-3 ) [Levy, 2000]. The approximation that ( ) is a slowly varying function of governs the condition to be fulfilled in this method as given by Equation Equation 3-13 This means that ( ) changes by a negligible amount in a distance of. The advantage of SPE (Equation 3-12) over elliptical equation (Equation 3-11) is that it is an initial value problem that can be solved easily compared to the boundary value problem of Equation Solution methodology In order to describe the solution methodology, a simple case of propagation in vacuum is considered first. In case of propagation in vacuum SPE (Equation 3-12) is reduced to the form given by Equation

56 The aim is to solve this equation numerically to get a value of the field strength for all possible pairs of height ( ) and range ( ). The numerical solution of Equation 3-14 is obtained through the use of split step Fourier transforms method described in the following steps. Step 1: Fourier transform of Equation 3-14 is taken which is given by ( ) ( ) Equation 3-15 Step 2: Equation 3-15 is solved in the frequency domain whose solution is given by ( ) ( ) Equation 3-16 Step 3: The inverse Fourier transform of Equation 3-16 will result in the solution of SPE in the time domain which can be written in the two forms given by Equation ( ) (( ) ( ( ))) Equation 3-17 Where: ( ) is the initial field containing narrow angle spectral components only; and are the Fourier and inverse Fourier transform operators; is the wave number in time domain and is wave number in vertical direction in frequency domain. The term given as ( ) in Equation 3-17 is known as propagator for vacuum which is modulated with refractive index for the case of non-vacuum. Equation 3-17 provides mathematical description of PEM which is applied repetitively to get the solution of the wave at every range step. Hence it is referred to as split step Fourier transform (SSFT) where the solution is obtained by splitting the solution into a series of phase screens (steps) orthogonal to the direction of propagation ( ). First the initial 41

57 field is propagated through vacuum for a short step and in the next step a phase screen modulated by refractive index variation is applied to it. The resulting field is then forward propagated through the medium to the next phase screen, and so on. A suitable filter must be used with the Fourier transform to avoid the problem of aliasing [Levy, 2000]. The initial field ( ) can be any function (numerical or analytical) of depending on the demand of a specific problem. This initial field which may represent the main lobe of an antenna can be used to obtain the far field of the propagating wave. Example of a Gaussian shaped initial field is given in Equation ( ) ( ) Equation 3-18 Where: is the height of the antenna above mean sea level; is the maximum intensity of the field at height and is the coefficient which determines the beam width of the antenna Boundary conditions To get the solution of a propagating wave, the use of split step Fourier transforms method described in Section is not complete without imposing boundary conditions. For a horizontally propagating wave, two boundary conditions, at the top and bottom of the propagation domain, need to be fulfilled. Ideally, the upper boundary condition is at an infinite height and the lower boundary condition imposed is for a perfectly conducting and smooth surface of the Earth. These upper boundary condition and lower boundary conditions are known as Sommerfeld outgoing radiation 42

58 condition [Sommerfeld, 1949] and Dirichlet boundary condition [Beggs et al., 1992], respectively. The Dirichlet boundary condition states that the initial reduced field ( ) at is zero for all values of. Since water is a good reflector compared to ground and the sea surface is comparatively smoother than ground, so the Dirichlet boundary condition can be applied with reasonable accuracy to over-sea propagation. The Sommerfeld outgoing radiation condition states that the value of the propagating wave at infinity must be zero. To apply this condition at infinity complicates the problem in terms of limitations on computation. However in this research this condition is applied at a specific height termed a Height of Interest (HOI) for every range step to truncate the infinite problem into a real finite propagation domain. For this purpose an absorbing layer is added at HOI in the computational region. The purpose of this additional absorbing layer is to block the reflection of propagating wave from the top boundary into the considered propagation domain of interest. In the implemented model a simple filter is used as an absorbing layer that slowly absorbs the energy in every range step in the upper half of the computation domain. The simple filter used at the HOI is based on the principle of Hanning window satisfying a value of 0 at the top boundary and a value of 1 at the HOI. The slow absorption of energy fulfils the requirement of the boundary condition to be applied ideally at infinity and also resolves the matching problem between the two regions at the interface. The results are valid only in the lower half of the computation domain up to the HOI. The height of interest can be varied at the cost of computational speed and just like other model parameters an optimum value for it is used in this research by obtaining a converged solution to the problem of interest using trial and error method. 43

59 3.2 Model validation The basic concepts and mathematical equations during the development process of the model have been extracted from the work done by Levy, [2000]. As a first step, in the validation process of the model, results generated by the model termed a Modelled results are compared to the results provided by Levy termed a Levy results. Since Levy does not mention some of the required parameters (e.g. horizontal and vertical resolution) used in the Levy results, a trial and error method was used for finding these parameters to get the best match of the output. However, there are still some differences in the detailed output results. In the following modelled results and Levy results are compared with one another for different cases to validate the implemented model Evaporation duct Different values of evaporation duct heights are used in Equation 2-8 with the surface refractivity, equal to 330 MU (as used by Levy, [2000]) to obtain various M profiles as shown in Figure

60 Figure 3-1 M profiles with evaporation duct heights equal to 0 m, 10 m, 20 m and 30 m. For all the M profiles modelled results are obtained at all ranges between 0 and 100 km at an antenna height of 25 m (AMSL). The modelled results for EDH value of 0, 10, 20 and 30 m at 3 GHz frequency is shown in Figure 3-2. For validation of the implemented model the corresponding Levy results are also shown in Figure 3-3. After comparing the corresponding results, it is observed that the modelled results are not only similar to the Levy results, but also provide more detailed description of the path loss with high resolution. Similar tests were also made for a frequency of 10 GHz (results not shown here) which provided identical output results. 45

61 (a) (b) (c) Figure 3-2 Path loss coverage diagrams of modelled results for 3 GHz source in the environment of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. (d) 46

62 (a) (b) (c) Figure 3-3 Path loss coverage diagrams of Levy results for 3 GHz source in the environment of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH [from Levy, 2000] (Page 79). (d) Bi-linear and tri-linear M profiles In this section modelled results for bi-linear and tri-linear M profiles are compared with Levy results up to the height of 200 m for two different frequencies of 3 GHz and 10 GHz. These M profiles are shown in Figure

63 (a) (b) Figure 3-4 M profiles representing (a) Bi-linear variations with inversion point at 100 m and (b) Tri-linear variations with inversion points at 100 and 150 m. The bi-linear M profile is defined in two linear portions by using a constant M gradient at a rate of -118 MU per km from the Earth s surface up to a height of 100 m and 118 MU per km above this height as used by Levy, [2000]. The duct strength (S s ) and duct thickness (S t ) are 10 MU and 100 m, respectively. The decreasing M profile between heights of 0 and 100 m is sufficiently high to trap the energy at both 3 and 10 GHz. The path loss coverage diagram obtained from the model for bi-linear M profile described above is shown in Figure 3-5 for both sources. It can be clearly seen that the modelled results and Levy results are closely similar to one another as shown in Figure 3-5 and Figure 3-6 for 3 GHz and 10 GHz source, respectively. The tri-linear M profile is defined in three linear portions according to the values used by Levy, [2000]. These values are increasing at a standard rate of 118 MU per km from its value of 330 MU at the surface of the Earth up to 100 m. From 100 to 150 m, the profile decreases at a rate of MU per km and then starts increasing at a standard rate of 118 MU per km above height of 150 m. The duct strength (E s ), thickness (E t ), 48

64 height within the duct of maximum M (E m ) and base height (E b ) are 10 MU, 135 m, 100 m and 15 m, respectively. The path loss coverage diagrams for the described tri-linear M profile were obtained (results not shown here) for both 3 GHz and 10 GHz sources and the modelled results and Levy results were found closely similar to one another. These results show that the implementation of PEM used in this thesis produces similar results to that used by Levy. Figure 3-5 Path loss coverage diagrams of modelled results for a bi-linear M profile at (a) 3 GHz and (b) 10 GHz source. 49

65 Figure 3-6 Path loss coverage diagrams of Levy results for a bi-linear M profile at (a) 3 GHz and (b) 10 GHz source [from Levy, 2000] (Page 86) AREPS Advanced Refractive Environmental Prediction System (AREPS) [Patterson, 2003; 2007] is a computer model developed by the US government, Space and Naval Warfare Systems Centre (SPAWAR) which provides a graphical user interface program for Advanced Propagation Model (APM). APM is a hybrid combination of different standalone propagation models like Flat Earth (FE), Ray Optics (RO), extended Optics (XO) and Terrain Parabolic Equation Method (TPEM) [Barrios, 2003]. AREPS provides assessments for electromagnetic wave in the range of LF (low frequency) to 50

66 EHF (Extremely high frequency) especially for determination of radar coverage in real world propagation conditions. The implemented model used in this research is validated with AREPS by comparing modelled results and AREPS results. For example, Figure 3-7 shows modelled results and AREPS results for similar parameters at 2 GHz frequency located at an antenna height of 25 m (AMSL). Both results are similar to one another. Similarly Figure 3-8 shows the coverage diagram for a simple arbitrary M profile obtained using implemented model and AREPS. These results are similar to one another as well. It is concluded from all these comparisons between modelled and Levy results that the results produced by the implemented model of parabolic equation method are better and more qualitative. The model is further used to investigate results for the experiment conducted by the group for a period of one year in for four communication links. (a) (b) Figure 3-7 Path loss coverage diagrams for a tri-linear M profile obtained using (a) Model (b) AREPS. 51

67 (a) (b) Figure 3-8 Path loss coverage diagrams for a simple M profile obtained using (a) Model (b) AREPS. 3.3 Path loss coverage diagrams at operational parameters The results validated in Section are at different parameters (e.g. frequency, path length, antenna height etc.) than used in this work. For instance a different frequency of 3 and 10 GHz is used than the operational frequencies (240 MHz and 2015 MHz) used in the experimental setup. The detailed discussion and analysis on these parameters will be provided later on. However the path loss coverage diagrams at the operational parameters of this work (see Table 4-1) are shown in Figure 3-9 and Figure 3-10 for UHF and VHF frequencies, respectively for a range of evaporation duct heights. The propagation medium is taken as that of a path of 140 km. The comparison is not possible at this scale due to the use of different parameters and propagation domains; however it provides a summary of the path loss at different frequencies and evaporation duct heights. 52

68 (a) (b) (c) Figure 3-9 Path loss coverage diagrams of modelled results for 2015 MHz source in the environment of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. (d) 53

69 (a) (b) (c) Figure 3-10 Path loss coverage diagrams of modelled results for 240 MHz source in the environment of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. (d) 3.4 Experimental setup The map of the sites where the network of the experimental links was setup is shown in Figure Table 3-1 gives full name of the sites, their short name used throughout in this research, their geographical coordinates along with the heights of the installed antennas on each site. 54

70 Figure 3-11 Map of the sites used for the experimental setup over the English Channel. Name of site Short name Latitude Longitude Jersey St John s Quarry Portland Bill Lighthouse Alderney (Isl De Raz) UHF antenna height (m) VHF antenna height (m) Jersey Portland Alderney Table 3-1 Geographical coordinates of the sites along with the heights of the installed antennas. A transmitting site is located at Jersey which consists of a UHF antenna at a height of 16.5 m (AMSL) and VHF antenna at a height of 17.5 m (AMSL). The transmitted signal is received by two receiving sites located at Portland (140 km path length) and 55

71 Alderney (50 km). Each of the receiving sites has a pair of antennas for receiving UHF and VHF signal separately. The receiver antennas both at Portland and Alderney are at heights of 12 and 13.4 m (AMSL) for UHF and VHF, respectively, with vertical polarisation. It can be seen in the map that the path between Jersey and Alderney and that of Jersey and Portland are almost collinear. This type of configuration provided effectively four different fixed point-to-point communication links for the experiment. These communication links are used with the following names in this research at a frequency value of 2015 (UHF) and 240 MHz (VHF). 1. UHF Portland 2. UHF Alderney 3. VHF Portland 4. VHF Alderney For each communication link a set of 6000 values of the received signal strength per hour (i.e. 25 values in 2 seconds, 4 times per minute) were recorded by each receiver at Portland and Alderney during the experimental campaign carried out by the group in The median, 10 th and 90 th percentile for each set of 6000 recorded received signal strengths is calculated and are termed an hourly data. This is done because the meteorological data is available in an hourly format. From the median set of hourly data EPL is calculated for each communication link according to the relation given in Equation Where: Equation

72 is the recorded received signal strength in dbm; and is the conversion factor in dbm from signal strength (dbm) to path loss (db). The value of CF for each communication link is given in Table 3-2 that accounts for all the gains and losses at the transmitters, receivers, amplifiers, feeders etc. The experimental half power beam width of the antennas is 17ᵒ. Further description of how CF is calculated is given in Section Communication Link Conversion factor (dbm) UHF Portland 89.7 UHF Alderney 68.2 VHF Portland 87.5 VHF Alderney 63.5 Table 3-2 Conversion factor used for each communication link [after Mufti, 2011] Conversion factor Conversion factor is a specifically calculated value obtained by using different parameters for each communication link which converts the recorded received signal strength into path loss (see Table 3-2). The formula for getting this value is given in Equation The detailed link budget and calculation of the individual parameters used in Equation 3-20 can be found in Mufti, [2011] (Pages 87 91). The values for all these parameters are tabulated in Table 3-3 to get a corresponding CF value for each communication link. ( ) ( ) ( ) ( ) ( ) ( ) ( ) Equation

73 Where: ( ) is the power of transmitting source in dbm; ( ) and ( ) are the power gains of the amplifiers at the transmitter and receiver systems in db, respectively; ( ) and ( ) are the feeder losses at the transmitter and receiver sites in db, respectively; ( ) and ( ) are the gains of the transmitter and receiver antennas in db, respectively. Link/ Parameter UHF Portland UHF Alderney VHF Portland VHF Alderney ( ) (dbm) ( ) (db) ( ) (db) ( ) (db) ( ) (db) ( ) (db) ( ) (db) Table 3-3 Measured values for different parameters in the experimental setup [after Mufti, 2011] Meteorological data The meteorological data used in this research were taken from four weather stations in Portland, Jersey airport, Guernsey airport and Channel Light Vessel as shown on the map of the sites used for the experimental setup (see Figure 3-11). The geographical coordinates of these weather stations, their heights above mean sea level and distance between each weather station and the nearest radio site is given in Table 3-4. All the weather stations were in the vicinity of the English Channel and were close to the sites of the experimental network. 58

74 Weather station Latitude Longitude Height AMSL (m) Distance to the nearest site (km) Portland Jersey Airport Guernsey Airport Channel Light Vessel (CLV) Table 3-4 Weather stations along with their geographical coordinates and distance to the nearest site. 59

75 4 Characterisation of model parameters This chapter is about the characterisation of different model parameters like frequency, antenna heights etc. In all the cases the characteristic curves for the modelled path loss are obtained by varying that parameter for all the communication links. 4.1 General model parameters The general model parameters as given in Table 4-1 are used in the characterisation of the model parameters unless otherwise stated. Name/symbol of parameter Frequency (f) Altitude (z) Height resolution (zres) Range (x) Range resolution (xres) Evaporation duct height (EDH) Half power beam width (HPBW) Beam angle with horizon (BAWH) Normalization constant (A) Description of parameter Frequency of the transmitting source Maximum Altitude of the propagation domain Vertical resolution of the propagation domain Horizontal distance between transmitting and receiving antennas Horizontal resolution of the propagation domain Height of minimum M value Beam width of the source Angle made by the transmitted beam A constant value used for normalization of the transmitted beam Value/range of values (unit) 2015,240 (MHz) 1 km 1 m 140,50 km 500 m 0 30 m 1 degree 0 degree 1 Radius (R e ) Radius of the Earth km Table 4-1 General values used in the characterization of the model parameters. 60

76 4.2 Effect of EDH on path loss The characteristic curves of modelled path loss for all the communication links against different evaporation duct heights are shown in Figure 4-1. The evaporation duct height is increased from 0 to 40 m in a step size of 1 m. The M profile for each value of evaporation duct height is constructed and used as input to the model to get corresponding modelled path loss. The M profile (i.e. profile 1 of the observed annual median given in Figure 5-1) is the same in all the cases except the changing evaporation duct height. Two cases are categorized for all the communication links which are (1) Tidal data variations excluded and (2) Tidal data variations included. The antenna heights and other parameters are used as given in the table of general parameters (Table 4-1). The result shows that there is no difference between the characteristic curves whether tidal behaviour is included or not. However the behaviour of each communication link is different from the others. There is a little variation at UHF beyond evaporation duct height of 10 m and 20 m for the long and short path, respectively. The effect of evaporation duct height is greater on the VHF where it is extended up to a height of 40 m in a gradual way. For all the characteristic curves the effect of change in EDH on the path loss varies with distance. 61

77 Figure 4-1 Modelled path loss against different evaporation duct heights for profile 1 of Figure 5-1 for all the communication links with tidal data (excluded) (solid lines) and tidal data (included) (dotted lines). 4.3 Effect of antenna height on path loss The transmitting antenna heights are changed from 1 m to 40 m in step size of 1 m keeping the receiving antenna at their normal heights for all the communication links. Then the same is done with the receiving antenna heights keeping the transmitting antennas at their normal heights. The simulations were performed for the two categories of (1) Tidal data variations excluded and (2) Tidal data variations included. The results of the modelled path loss against different transmitting antenna heights and receiving antenna heights are shown in Figure 4-2. In the Figure 4-2, the results represented by the solid lines for all the communication links are for the standard profile without including the tidal variations while those showing with the dotted lines are for the same 62

78 standard profile with including the tidal variations in the sea. The value used for the evaporation duct height for all the characteristic curves is 0 m. (a) (b) Figure 4-2 Modelled path loss against (a) Transmitting antenna heights and (b) Receiving antenna heights for all the communication links. The evaporation duct height is 0 m. The effect of changing antenna height is not following any uniform pattern for all the communication links due to different propagation mechanisms acting in various ways at each antenna height. It can be seen that the UHF links are behaving in a different way compared to the low frequency links. The UHF Portland shows lots of variations in the path loss due to the multi path effects at different antenna heights (both transmitter and receiver). The difference between path loss with tides included and path loss with tides excluded for UHF Portland (Receiving antenna only) is greater too because of severe tidal variations at the receiving site of Portland. This effect is not found for VHF Portland as the presence of tides is less affective at VHF. In all the other links, when the antenna height is increasing the difference between the path loss with tides and without tides is decreasing. For the initial change in both antenna heights up to 5 m, the tidal variations are severe in all the cases which is due to the fact that antenna height is too small and the 63

79 transmitting signal may be blocked by the high tide level at that location. There is strong variation in the path loss for the long path at VHF at m antenna heights for both transmitter and receiver. 4.4 Effect of half power beam width on path loss The half power beam width is changed from 1 to 35 degrees in step size of 1 degree and the path loss for each value is shown in Figure 4-3 (a). In order to see the detailed effect on path loss with varying half power beam width of first 10 degrees, Figure 4-3 (b) is plotted as well. The two results show that the effect with changing half power beam width is varying in a similar fashion for the UHF frequency. For UHF a smaller beam width is desirable otherwise the path loss would increase enormously as can be confirmed in the diagram where the path loss increases to 692 and 702 db at 34 degree half power beam width for UHF Portland and UHF Alderney, respectively. The UHF links are giving the same curves irrespective of the path length and similarly the VHF links are giving the same curves irrespectively of the different paths for Portland and Alderney. (a) (b) Figure 4-3 Characteristic curves of modelled path loss against (a) Half power beam width ranging from 1 to 35 degrees and (b) Half power beam width ranging from 1 to 10 degrees. 64

80 4.5 Effect of beam angle with horizon The beam angle with horizon is changed from 0 to 40 degrees in step size of 1 degree and its effect on path loss is shown in Figure 4-4. In this case the path length changed the nature of the curves and hence the path loss behaviour is different for the long path than the short path. Since the model is based on a narrow angle approximation method (see Section 3.1.2) which gives better output when the beam angle with horizon is smaller (up to 10 degrees), hence the curves are replicating itself after a value of 10 degrees for all the communication links. In the first few degrees the effect of the tidal variations is more on the UHF frequency and smaller effect on the VHF frequency. Figure 4-4 Characteristic curves of modelled path loss against beam angle with horizon. 4.6 Effect of change in frequency The frequency of the transmitting source affects the path loss in various ways so this characteristic parameter is analysed from different perspective to see these effects. A 65

81 larger spectrum of 1 Hz to 40 GHz is shown in Figure 4-5 for all the communication links to get an overall idea of the variations in the modelled path loss with varying frequency for two cases (1) tidal data included and (2) tidal data excluded. It is to note that the legend of UHF Portland or UHF Alderney etc. used in the figure does not mean that the fixed frequency of 2015 MHz for UHF or 240 MHz for VHF is used for these links in the frequency analysis but instead frequency is a variable parameter. These legends are distinguishing other general parameters (like antenna height, path length etc.) associated with each link. Figure 4-5 shows that the path loss decreases in the initial band (1 Hz to 1 MHz) with increasing frequency and it increases in the final band (10 GHz to 40 GHz) with the increasing frequency. Mostly, the curves representing UHF Portland and VHF Portland (long path) and similarly UHF Alderney and VHF Alderney (short path) are close to each other except in two bands (from 316 Hz to 1 MHz termed as band 1 and from 1 MHz to 40 GHz termed as band 2 ). However the different UHF and VHF antenna heights for each path created a bit of difference in the path loss represented by all the curves. Both of these bands are further investigated in greater details in the following sections. 66

82 Figure 4-5 Modelled path loss against frequency range of 1 Hz to 40 GHz for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines) Band 1 Modelled path loss for Band 1 of Figure 4-5 is shown in Figure 4-6 to see it in greater detail. Although the two operational frequencies are not in this range, however it is giving an idea of how the frequency affects the modelled path loss at lower frequencies. It can be seen that the path loss decreases with the increase in frequency for all the cases initially and then there are fluctuations after frequency of 300 Hz. The long path and short path are affecting the path loss in the same fashion but with different path loss values which is higher for the long path as expected. 67

83 Figure 4-6 Modelled path loss against frequency range of 1 Hz to 1 MHz for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines) Band 2 Band 2 contains the two operational frequencies used in this research. As the bandwidth of this band is too large so it is investigated in different parts. However before doing so a closer look at the band 2 of Figure 4-5 is shown in Figure

84 Figure 4-7 Modelled path loss against frequency range of 4 MHz to 9 GHz for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines). In the first part a bandwidth from 1 MHz to 3 GHz in step size of 500 khz is shown in Figure

85 Figure 4-8 Modelled path loss against frequency range of 1 MHz to 3 GHz for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines). In the second part of band 2, a portion of the band from VHF to UHF with a step size of 4 MHz is considered for which the results are shown in Figure

86 Figure 4-9 Modelled path loss against frequency range of 240 MHz to 2015 MHz with a step size of 4 MHz for all the links and with and without tidal data. In the third part, the two operational frequencies of 240 MHz and 2015 MHz are taken as centre frequency. Path loss is shown in Figure 4-10 for both UHF and VHF with a step size of 500 khz and a band width of 200 MHz around each centre frequency (100 MHz on each side of the centre frequency). 71

87 (a) (b) Figure 4-10 Modelled path loss against frequency for a bandwidth of 200 MHz around the centre frequency of (a) UHF (2015 MHz) (b) VHF (240 MHz). Link /Tidal data and centre frequency Path loss at 2015 MHz with tidal data excluded Path loss at 2015 MHz with tidal data included Path loss at 240 MHz with tidal data excluded Path loss at 240 MHz with tidal data included UHF Portland (db) UHF Alderney (db) VHF Portland (db) VHF Alderney (db) Table 4-2 Modelled path loss at centre frequencies of UHF and VHF for each communication link with and without tidal data. The path loss values are in db. The absolute values of the modelled path loss (with tidal data included and excluded) for all the four communication links at centre frequencies of UHF and VHF are given in Table 4-2. In this table the effect of tidal data on modelled path loss is smaller at VHF frequency compared to its effect at UHF frequency. For example for UHF Portland and 72

88 UHF Alderney, this difference in path loss due to the inclusion of tidal data is small for the short path (3 db) compared to the long path where the difference is about 12 db. 4.7 Effect of change in range Range is the horizontal distance between transmitting and receiving radio stations. The effect of range on the path loss is shown in Figure 4-11 by varying it from 40 km to 150 km. As expected, the modelled path loss for UHF Portland and UHF Alderney are identical and overlaps each other when the range is set to be the same for each. This is because they are set to have the same general parameters (e.g. frequency, antenna height etc.) except range. Similarly, the output of VHF Portland is identical with that of VHF Alderney when the range is set to be the same for each. However, this is not the case when the tidal data is included. This is due to the fact that the tidal data is not the same for UHF Portland and UHF Alderney pair and similarly for VHF Portland and VHF Alderney pair. The terms UHF Portland, UHF Alderney, VHF Portland and VHF Alderney do not mean that they represent the long path or short path at UHF/VHF as the path length is a variable quantity. They are used to keep consistency with other parts of the thesis. In the Figure 4-11, for the long path (Portland), the path loss is decreasing with increasing range for VHF with smaller tidal variations and highly fluctuating for UHF frequency with larger tidal variations. For the short path (Alderney), the path loss is increasing with increasing range for UHF and highly fluctuating for VHF frequency. In between the long and short path, around the range of 100 km, the tidal effect for VHF is stronger and not for UHF. 73

89 Figure 4-11 Modelled path loss against range (distance between transmitting and receiving antennas) for all the communication links without including the tidal variations (represented by solid lines) and with including the tidal variations (represented by dotted lines). UHF Portland overlaps on UHF Alderney and similarly VHF Portland overlaps on VHF Alderney. 74

90 5 Analysis for evaporation duct This chapter is about the comparative analysis between modelled path loss (MPL) obtained from the implemented model and experimental path loss (EPL) taken from the hourly data set of the experiment performed by the research group. The annual and monthly correlation coefficients are discussed and analysed for the impact of evaporation duct on propagation for all the considered communication links. 5.1 M profiles Different types of M profiles are constructed to see their effect on propagation as described below. Profile 1 is constructed from the measured meteorological data for each hour up to a height of 102 m. Above this height standard atmospheric gradient of 118 MU per km is used to construct the remaining portion of the profile. For this purpose the hourly meteorological refractivity data, available at the heights of weather stations are used. Although the weather stations are at different geographical locations, however their range difference is insignificant and can be ignored. This is due to the approximation of horizontally independent refractivity as a reasonable case usually applied in PEM [Grabner and Kvicera, 2011]. The range difference is not effective when the variations in refractivity are on a small scale (where the distance between transmitting and receiving sites is less than 100 m). When the variations in refractivity are on a large scale the troposphere is stratified in horizontal layers due to the effect of gravity [Barclay, 2003]. Linear interpolation is performed to get the values of refractivity for heights lying between the heights of weather stations. 75

91 Profile 2 is constructed from the available meteorological data up to a height of 102 m and above this height ITU recommended values are used as described by ITU-R P , [2012] using the monthly mean change in refractivity for February, May, August and November as 40, 50, 50 and 45 NU per km, respectively. For the remaining period a linear interpolation is made between the two available consecutive monthly values. These obtained values for March, April, June, July, September, October, December and January are given as 43.33, 46.66, 50.00, 50.00, 48.33, 46.66, and NU per km, respectively. The ITU-R P , [2012] recommended value for surface refractivity equal to 315 NU is used. Profile 3 is constructed from the actual meteorological data up to 102 m height and standard atmospheric gradient above 102 m in the same manner as profile 1, however the tidal variations included with this profile make it different from profile 1. From the available tidal data (4 values per day) obtained from British Broadcasting Corporation (BBC), linear interpolation is made to get hourly tidal variations for each site for the period of the experiment. The datum for the available tidal data for Jersey, Portland and Alderney is calculated as 6.12 m, 1.09 m and 3.57 m, respectively. The heights of transmitting and receiving antennas for each communication link in each model run were adjusted accordingly to include the effect of high and low tides at the time of occurrence. Profile 4 is the tidal version of profile 2 where compensation for the tidal variations is made by including the available tidal data in the same way as done in the case of profile 3. The four profiles described above are further adapted to include the effect of an evaporation duct up to the altitude of the second highest weather station (i.e. Portland at 76

92 52 m). The evaporation duct heights used are 0, 10, 20 and 30 m. The M profile representing evaporation duct height of 0 m means that no effect of the evaporation duct is included and the meteorological values are considered only. This is done as a reference case. So for each profile the linear variation in refractivity between the two consecutively available meteorological values at a height of 1 and 51 m are adapted by exponentially varying refractivity values using Equation 2-8. For example profile 1 with EDH 10 m means that the profile is a combination of exponential data between 0 m and 10 m, meteorological data from 11 to 102 m and standard data above 102 m. A few such profiles are shown in Figure 5-1 and Figure 5-2 for profile 1 and profile 2, respectively. Since the meteorological values vary for each simulation run, a median value of the whole data set for each weather station is used in the sample hybrid plots. Figure 5-1 Median of hybrid M profile for July 2009 to June 2010 with different evaporation duct heights (profile 1). 77

93 Figure 5-2 Median of hybrid M profile for July 2009 to June 2010 with different evaporation duct heights (profile 2). 5.2 Annual and monthly analysis for evaporation duct The minimum and maximum evaporation duct height used in this research is 0 m and 30 m, respectively. The increase from 0 to 30 m in step size of 10 m is performed to see the effect of the change in the refractivity profile on the propagation at UHF and VHF as described in the following sections. The number of simulation runs for each frequency, link and month is based on how many hourly experimental values are available; i.e. a simulation corresponding to the measured weather data for each hour is performed. Table 5-1 presents the number of simulation runs performed in each month according to the available experimental path loss values. The observed path loss values are correlated with those produced by the simulation. Clearly the amount of data available will affect the correlation. 78

94 Month/Link UHF Portland UHF Alderney VHF Portland VHF Alderney July August September October November December January February March April May June Total Table 5-1 Number of hours of observations in each month for each communication link. The correlation analysis is started by looking at the data for the complete period of the experiment. Although it was not really expected to have some useful output from the analysis of data for a whole year, it was worth looking in case it did. In the annual analysis, all the obtained results of modelled path loss for the whole year of investigation are correlated with the experimental path loss for all the communication 79

95 links and all the M profiles. This is repeated for each of the four selected evaporation duct heights, which resulted in a total of 64 correlation coefficients as given in Table 5-2. The value of correlation coefficient is quantified as insignificant if it lies in the range of -1 to 0.2, significant if it lies in the range of 0.2 to 0.6, and highly correlated if it lies in the range of 0.6 to 1. Since the annual correlation coefficients are insignificant for most of the cases and it is not possible to get any useful information for prediction of refractivity on a large scale hence further investigation on monthly basis is required. A confidence level represented by the p values along with the correlation coefficient values are also presented in the monthly analysis. Link Profile 1 Profile 2 Profile 3 Profile 4 EDH (m) 0 m 10 m 20 m 30 m 0 m 10 m 20 m 30 m 0 m 10 m 20 m 30 m 0 m 10 m 20 m 30 m UHF Portland UHF Alderney VHF Portland VHF Alderney Table 5-2 Annual correlation coefficients for all the communication links and M profiles for (a) EDH 0 m (b) EDH 10 m (c) EDH 20 m and (d) EDH 30 m UHF Portland Monthly correlation coefficients for the four M profiles and evaporation duct heights in case of UHF Portland are presented in Figure 5-3. The p value is plotted with each bar 80

96 of correlation coefficient which shows the confidence level of that value of correlation coefficient. The p value of more than 0.1 represents less than 10% confidence in the output and hence such correlation coefficients are not counted as effective in this analysis. (a) (b) (c) (d) Figure 5-3 Monthly correlation coefficients for different M profiles for UHF Portland in case of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. The bar and line represents the correlation coefficient and p value, respectively. For UHF Portland, there are two months (i.e. July and August) where the correlation coefficient is significant for EDH 0 m. Profile 1 is giving the best correlation among all the profiles. It has some significant correlation in the month of April as well. In April, 81

97 profile 1 is giving significant correlation coefficient for EDH 0 m and insignificant correlation coefficient for EDH 30 m. The annual correlation coefficient value for profile 1 is also the highest (0.2 for EDH 0 m) among the other profiles as shown in Table 5-2. Since they have some of the highest correlation coefficients, July and April are further analysed below. Figure 5-4 Modelled path loss and experimental path loss for July for UHF Portland when profile 1 is used as input M profile. The evaporation duct height is 0 m and the correlation coefficient is EPL is shown in red and MPL in black. 82

98 To get greater details, the modelled and experimental path losses are plotted for July as given in Figure 5-4. This figure shows that there are some fluctuations in the experimental path loss (e.g. around 17 th and 29 th July) which are followed by the fluctuation in the modelled path loss. However, there are some fluctuations (e.g. around 9 th, 10 th and 15 th July) which are not followed by the modelled path loss. These fluctuations in the experimental path loss may not be due to the variations in refractivity. Figure 5-5 Modelled path loss and experimental path loss for April for UHF Portland when profile 1 is used as input M profile. The evaporation duct height is 30 m and the correlation coefficient is

99 Similarly, the results for April for EDH equal to 30 m are shown in Figure 5-5 when profile 1 is used as input M profile to the model. This figure shows that the modelled path loss is almost at a constant level where the experimental path loss is changing. It is due to the very small change in the obtained values of refractivity from the weather stations which effectively yielded an insignificant correlation between the modelled and experimental path losses UHF Alderney Monthly correlation coefficients for the communication link of UHF Alderney for all the M profiles and all the evaporation duct heights are shown in Figure 5-6. There are some bars with high p values showing that those results are outside the 10% confidence level and should not be counted in this analysis. Overall most of the significant as well as highly correlated values are for 0 and 10 m evaporation duct heights. This is evident from the annual correlation coefficient values as shown in the Table 5-2, where the values for 0 m EDH are 0.29, 0.24, 0.32 and 0.25 for profile 1 to 4, respectively. However, for the high evaporation duct heights, the correlation coefficients are comparatively insignificant. 84

100 (a) (b) (c) (d) Figure 5-6 Monthly correlation coefficients for different M profiles for UHF Alderney in case of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. The bar and line represents the correlation coefficient and p value, respectively. There is a very interesting high peak in February where the correlation coefficient is 0.82 for profile 4 and EDH 10 m, for which the detailed modelled and experimental path losses are plotted in Figure 5-7. In this figure the modelled and experimental values are in very good comparison with sample size of 69. Nearly 70% modelled path loss values of the total sample size are seen to be following the experimental path loss while the remaining 30% MPL values are not successfully modelled to follow the EPL values. 85

101 Figure 5-7 Modelled path loss and experimental path loss for February for UHF Alderney when profile 4 is used as input M profile. The evaporation duct height is 10 m and the correlation coefficient is In January and November, the correlation coefficient is 0.47 and 0.45, respectively for the same profile 4 and EDH 10 m. However the sample size of January is 34 and that of November is 175. As there are missing data in November at different intervals so a portion of the modelled and experimental path losses where the correlation between them is the highest is taken. This highly correlated portion of 8 th November is shown in Figure 5-8 with the time of occurrence is given as well. In this figure 15 events were modelled in total for one day, out of which 10 MPL values were successfully modelled with a percentage of 66%, and the remaining MPL values are not following the EPL successfully. 86

102 Figure 5-8 Modelled path loss and experimental path loss for a portion (a highly correlated section) of November for UHF Alderney when profile 4 is used as input M profile. The evaporation duct height is 10 m. It is concluded from this information that profile 4 is giving the best fit between the experimental and modelled path losses for UHF Alderney and evaporation duct height of 10 m is the dominant one. However this cannot be confirmed by the annual correlation coefficients values as shown in Table 5-2 where the value is 0.12 and is insignificant compared to other profiles and evaporation duct heights. Giving less weight to January and February due to their smaller sample size, profile 3 can be taken as the best fit between the experimental and modelled path losses with EDH 10 m in case of UHF Alderney. The month of July and August produces the correlation coefficient of about 0.4 each and for the rest of the period it produces more than 0.2 in most of the cases except December (where sample space is 157). Also the annual correlation coefficient for profile 3 with EDH 10 m is the highest (equal to 0.34) among all the profiles and all the evaporation duct heights. Modelled path loss and experimental path loss for July and August with profile 3 and EDH 10 m are plotted to see the variations in greater details as shown in Figure 5-9 and Figure 5-10, respectively. 87

103 In the Figure 5-9, it can be seen that the fluctuations in the experimental path loss are nearly the same as those of modelled path loss and also the instantaneous values of experimental path loss and modelled path loss are close to each other. The best follow up of the experimental path loss by the modelled path loss can be seen in the 4 th and 5 th panel of the Figure 5-9. Figure 5-9 Modelled path loss and experimental path loss for July for UHF Alderney when profile 3 is used as input M profile. The evaporation duct height is 10 m. The correlation coefficient is 0.4. EPL is shown in red and MPL in black. 88

104 Figure 5-10 gives the absolute modelled and experimental path losses for the month of August when the monthly correlation coefficient value is Although this is not a highly significant correlation, the figure shows very few portions (e.g. 23 rd to 24 th August) where the modelled path loss is not successfully following the EPL. Mostly, a better fit can be seen between the modelled and experimental path losses. Figure 5-10 Modelled path loss and experimental path loss for August for UHF Alderney when profile 3 is used as input M profile. The evaporation duct height is 10 m. The correlation coefficient is EPL is shown in red and MPL in black. 89

105 5.2.3 VHF Portland Figure 5-11 shows monthly correlation coefficients with p values for all the M profiles and evaporation duct heights for VHF Portland. For profile 1, profile 3 and profile 4 the p values are high and hence the correlation coefficients are outside the confidence level of 10%. This is true for all the evaporation duct heights. Even for profile 2 mostly insignificant correlation coefficients exist; e.g. for EDH 10 m all the monthly correlation coefficients are insignificant except in November and April. (a) (b) (c) (d) Figure 5-11 Monthly correlation coefficients for different M profiles for VHF Portland in case of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. The bar and line represents the correlation coefficient and p value, respectively. 90

106 The very small values of annual correlation coefficients (see Table 5-2), for profile 1, profile 3 and profile 4 at all the evaporation duct heights, confirm that only profile 2 is found to be the best profile for VHF Portland. Profile 2 has significant correlation coefficient of 0.49 and 0.38 for EDH 30 and 20 m, respectively. The monthly correlation coefficients for profile 2 at EDH 30 m is shown in the Figure 5-11 (d) where most of the correlation coefficients are above the level of 0.3 with high peak of 0.38 in May. To see the detailed modelled and experimental path loss values of May, these results are plotted as shown in Figure Although there are some changes in the EPL, there are very little or rather no changes in the corresponding value of MPL throughout the month. 91

107 Figure 5-12 Modelled path loss and experimental path loss for May for VHF Portland when profile 2 is used as input M profile. The evaporation duct height is 30 m and the correlation coefficient is The behaviour of smooth MPL as shown in Figure 5-12 is analysed for the whole year (results not shown here), and it is found to be the same throughout the year. Therefore, it is concluded that the effect of variations in refractivity is not severe for the long path at VHF. In order to validate this statement, some more results are shown in Figure 5-13 for the same period with profile 4 (the one with tidal data included), where it can be seen that the tidal patterns are changing the MPL alone when compared to the results 92

108 shown in Figure 5-12 and therefore refractivity variations are not affecting the long path at VHF. Figure 5-13 Modelled path loss and experimental path loss for May for VHF Portland when profile 4 is used as input M profile. The evaporation duct height is 30 m and the correlation coefficient is EPL is shown in red colour and MPL in black VHF Alderney Monthly correlation coefficients with p values for all the four M profiles and evaporation duct heights in case of VHF Alderney are presented in Figure In this figure, profile 1 and profile 4 are giving some significant and highly significant 93

109 correlated MPL and EPL values. The annual correlation coefficients are also the highest for these two profiles as shown in the Table 5-2 (e.g and 0.25 for profile 4 and profile 1, respectively in case of EDH 10 m). (a) (b) (c) (d) Figure 5-14 Monthly correlation coefficients for different M profiles for VHF Alderney in case of (a) 0 m EDH (b) 10 m EDH (c) 20 m EDH and (d) 30 m EDH. The bar and line represents the correlation coefficient and p value, respectively. For VHF Alderney in case of EDH 10 m, there are two months (i.e. January and February) where the correlation coefficient is highly significant (0.61) and significant (0.57), respectively for profile 4. The sample size for these two months consists of only 33 and 69 values each. However, there are some other significant correlation 94

110 coefficients as well like 0.59 in October, 0.55 in September and 0.51 in July where these months have greater sample size (see Table 5-2). Therefore, profile 4 with EDH 10 m is found to give the best fit between the modelled and experimental path losses. From the best profile 4 with EDH 10 m for VHF Alderney, September is selected to show the details of the modelled path loss and experimental path loss as shown in Figure It can be seen in this diagram that most of the fluctuations in the experimental path loss are modelled successfully. For example in the first panel of the figure, the MPL are giving the best fit to the nearly sinusoidal fluctuations in the EPL. Also, around 7 th to 9 th and 18 th to 21 st September, the fluctuations in EPL are random, however, successfully modelled. 95

111 Figure 5-15 Modelled path loss and experimental path loss for July for VHF Alderney when profile 4 is used as input M profile. The evaporation duct height is 10 m. EPL is shown in red colour and MPL in black. 96