radio wave propagation in the marine boundary layer

Alexander Kukushkin

Radio Wave Propagationin the Marine Boundary Layer

Alexander Kukushkin

Radio Wave Propagation

in the Marine Boundary Layer

Alexander Kukushkin

Radio Wave Propagationin the Marine Boundary Layer

Editors

Alexander KukushkinGordon, NSW 2072, Australiae-mail: [email protected]

Cover PictureThe image on the cover is from NASA’s 2.84 GHzSpace Range Radar (SPANDAR) at Wallops Island,Virginia. The image corresponds to a ducting eventon April 2, 1998. Image courtesy of Space and NavalWarfare Systems Center, San Diego, CA.

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V

This book is about the parabolic approximation to a diffraction problem over a seasurface. While the parabolic equation method in radio wave propagation over theearths surface was introduced by V.A. Fok almost fifty years ago, its popularity hasgrown recently due to the development of advanced computational methods basedon the parabolic approximation. Numerous computational techniques have beenevolved and used for analysis of radio- and acoustic wave propagation in either deter-ministic or random media.This book is concerned with the analytical solution to a problem of wave propaga-

tion over the sea surface in the atmospheric boundary layer. Two basic mathematicalmethods have been used, depending on the ease of obtaining a closed analytical so-lution:

1. Expansion of the quantum-mechanical amplitude of the transition into acomplete and orthogonal set of eigen functions of the continuous spectrum.

2. The Feynman path integral.

It is not intended to provide a full step by step mathematical background to theabove methods but, rather, is dedicated to the application and analysis of the physi-cal mechanisms associated with the combined effect of scattering, diffraction andrefraction. The mathematical foundations for the above methods can be found innumerous monographs and handbooks dedicated to quantum mechanics and math-ematical theory.The book is arranged as follows: Chapter 1 presents the basic assumptions used

to describe the propagation media, i.e. the atmospheric boundary layer. It provides asimplified description of the turbulent structure of the refractive index in the atmo-spheric boundary layer and summarises the model of the troposphere to be used inthe analysis of the wave propagation. It introduces some foundation for the compo-sition of the refractive index as two components: a deterministic layered structureand a relatively small-scale random component of turbulent refractive index. A basicclassification of the propagation mechanisms, such as refraction, ducting, diffrac-tion and scattering is briefly introduced according to the presence and value of thenegative gradients of refractivity in the troposphere.Chapter 2 commences with an overview of the mathematical methods developed

for analysis of the problem of wave propagation and scattering in a stratified medi-

Preface

Preface

um with random fluctuations of the refractive index. It also positions the methodintroduced in this book as an extension of the well-known analogy between thequantum-mechanical problem of the quasi-stationary states of the Schr8dingerequation and the problem of radio wave propagation in the earths troposphere. Theadvantage of using this approach is that the Green function to the parabolic equa-tion is expanded over the complete set of orthogonal eigen functions of the continu-ous spectrum. This representation is equivalent to a Feynman path integral which isused in Chapter 3 to investigate the higher order moments of the wave field over thesurface with impedance boundary conditions.Some new physical mechanisms associated with scattering are analysed and

explained in Chapter 3.Chapter 4 introduces a perturbation theory for normal waves in a stratified tropo-

sphere. The problem here is that the common perturbation theory does not work forequations with a potential unlimited at infinity. Such potentials appear in the prob-lem of an electron in a magnetic field or in radiowave propagation over the earthssurface in the parabolic equation approximation. A modified perturbation theory isapplied to the analysis of the spectrum of normal waves (propagation constants) forthe boundary problem with a somewhat arbitrary profile of the refractive index. Theanalytical solution and numerical results are discussed for two practically importantmodels of refractive index in the near-surface domain: the bilinear approximationand the logarithmic profile. Also in Chapter 4, we present a closed analytical solu-tion for a second moment of the wave field (coherence function) in the presence ofan evaporation duct filled with random inhomogeneities of refractive index. Themechanism of interaction between discrete and continuous modes due to scatteringof the random irregularrities in the refractive index is analysed in detail.Chapter 5 deals with the elevated tropospheric duct. We start from a normal

mode structure for the trilinear profile of the refractive index and analyse the wavefield in geometric optic approximation thus introducing rays and modes. The specif-ic case of the presence of two waveguides, elevated duct and evaporation duct, simul-taneously, is analysed in detail by means of presenting the mechanism of exchangeof the wave field energy in a two-channel system.In Chapter 5 we also introduce the mechanism of excitation of the normal waves

in an elevated duct by means of single scattering on turbulent irregularities ofrefractive index. This case may represent significant practical interest in the case ofground–air communication for two reasons: first, the elevated ducts are oftendetached from the surface and the near-surface antenna is ineffective in excitationof the trapped modes, and secondly, strong anisotropic irregularities are often pres-ent in the upper boundary of the elevated tropospheric duct due to the physics of itscreation and, therefore, can produce a significant scattering effect of the incomingwaves from a surface-based antenna.Finally, in Chapter 6 we analyse some non-conventional mechanisms of the over-

horizon propagation. First, the effect of a stochastic waveguide created by anisotrop-ic irregularities in the refractive index. This mechanism is analysed in terms of theperturbation theory presented in Chapter 3. The second mechanism is a single scat-tering of diffracted field in the earths troposphere. This mechanism is rather com-

VI

Preface

plementary to a conventional single-scattering theory; it cannot explain the observedlevels of the signal but, contrary to conventional theory, reveals a correct behaviourwith regard to frequency.The Appendix provides a brief theory of the Airy functions and some asymptotic

representations.The analytical solutions and results considered in this book are chiefly applicable

to radio propagation in the UHF/SHF band, i.e., from 300 MHz to 20 GHz whererefraction and scattering play a major role in anomalous propagation phenomena,such as a waveguide mechanism in tropospheric ducts.I want to thank my former colleagues I. Fuks and V. Freilikher in cooperation

with whom most of the theoretical studies have been performed. It was my privilegeto work with my colleagues in such a productive and encouraging environment.I wish to thank V. Sinitsin who introduced me to research activities in this area andsupported me at the start of my career.Most of all, I amgratefully obliged tomy lovely wife, Galina, formaking it possible.

Alexander KukushkinSydney, Australia, March 2004

VII

IX

Preface V

1 Atmospheric Boundary Layer and Basics of the Propagation Mechanisms 1

1.1 Standard Model of the Troposphere 5

1.2 Non-standard Mechanisms of Propagation 9

1.2.1 Evaporation Duct 9

1.2.2 Elevated M-inversion 11

1.3 Random Component of Dielectric Permittivity 12

1.3.1 Locally Uniform Fluctuations 13

References 17

2 Parabolic Approximation to the Wave Equation 19

2.1 Analytical Methods in the Problems of Wave Propagation in aStratified and Random Medium 19

2.2 Parabolic Approximation to a Wave Equation in a Stratified TroposphereFilled with Turbulent Fluctuations of the Refractive Index 22

2.3 Green Function for a Parabolic Equation in a Stratified Medium 27

2.4 Feynman Path Integrals in the Problems of Wave Propagation inRandom Media 33

2.5 Numerical Methods of Parabolic Equations 38

2.6 Basics of Focks Theory 45

2.7 Focks Theory of the Evaporation Duct 49

References 55

3 Wave Field Fluctuations in Random Media over a Boundary Interface 57

3.1 Reflection Formulas for the Wave Field in a Random Medium over anIdeally Reflective Boundary 58

3.1.1 Ideally Reflective Flat Surface 58

3.1.2 Spherical Surface 61

3.2 Fluctuations of the Waves in a Random Non-uniform Medium above aPlane with Impedance Boundary Conditions 66

Contents

X

3.3 Comments on Calculation of the LOS Field in the General Situation 73

References 74

4 UHF Propagation in an Evaporation Duct 75

4.1 Some Results of Propagation Measurements and Comparison withTheory 77

4.2 Perturbation Theory for the Spectrum of Normal Waves in a StratifiedTroposphere 83

4.2.1 Problem Formulation 84

4.2.2 Linear Distortion 87

4.2.3 Smooth Distortion 89

4.2.4 Height Function 90

4.2.5 Linear-Logarithmic Profile at Heights Close to the Sea Surface 91

4.3 Spectrum of Normal Waves in an Evaporation Duct 92

4.4 Coherence Function in a Random and Non-uniform Atmosphere 99

4.4.1 Approximate Extraction of the Eigenwave of the Discrete Spectrum in thePresence of an Evaporation Duct 99

4.4.2 Equations for the Coherence Function 102

4.5 Excitation of Waves in a Continuous Spectrum in a StatisticallyInhomogeneous Evaporation Duct 108

4.6 Evaporation Duct with Two Trapped Modes 115

References 119

5 Impact of Elevated M-inversions on the UHF/EHF Field Propagation beyondthe Horizon 121

5.1 Modal Representation of the Wave Field for the Case of ElevatedM-inversion 122

5.2 Hybrid Representation 132

5.2.1 Secondary Excitation of the Evaporation Duct by the Waves Reflectedfrom an Elevated Refractive Layer 141

5.3 Comparison of Experiment with the Deterministic Theory of theElevated Duct Propagation 144

5.4 Excitation of the Elevated Duct due to Scattering on the Fluctuations inthe Refractive Index 147

References 151

6 Scattering Mechanism of Over-horizon UHF Propagation 153

6.1 Basic Equations 154

6.2 Perturbation Theory: Calculation of Field Moments 159

6.3 Scattering of a Diffracted Field on the Turbulent Fluctuations in theRefractive Index 164

References 172

Contents

XI

Appendix: Airy Functions 173

A.1 Definitions 173

A.2 Asymptotic Formulas for Large Arguments 176

A.3 Integrals Containing Airy Functions in Problems of Diffraction andScattering of UHF Waves 177

References 189

Index 191

Contents

1

The troposphere is the lowest region of the atmosphere, about 6 km high at thepoles and about 18 km high at the equator. In this book we study radio wave prop-agation along the ocean and can reasonably assume that all processes of propagationoccur in a lower region of the earth’s troposphere. That lower region and the atmo-spheric conditions are of most importance for the subject under study here.

From the perspective of radio communications/propagation we limit our objectiveto an investigation of the impact of the atmospheric structure on the characteristicsof the radio signal propagating through the atmospheric turbulence. All knownmethods of solutions to a similar problem are based on the separation of the space–time scales of the variations in both the refractive index n and electromagnetic field~EE, ~HH in two domains, described in terms of deterministic and stochastic methods.Intuition suggests that the spectrum of turbulent variations in the refractive index nwill have the energy of its fluctuations confined to a limited space–time domain or,at least, have a clear minimum and, desirably, a gap spread over a significant intervalin the time–space domain. It is apparent that the horizontal scales of variations inrefractive index larger than the length of the radio propagation path have noimmediate effect on the characteristics of the radio signal and rather affect its longterm variations over the permanent path. This comes down to an upper boundary ofthe spatial variations of refractive index in a horizontal plane of about 100 km. Thevertical irregularities are of most importance since they are responsible for therefraction and scattering of the radio waves in troposphere. However, there are somenatural limitations on the region of the troposphere which might be of interest in itsimpact on radio propagation. The troposphere is naturally divided into two regions:the lower part of the troposphere, commonly called an atmospheric boundary layer,and the area above, called clear atmosphere.

The electric properties of the troposphere can be characterised by the dielectric per-mittivity e or the refractive index n ¼

ffiffiffie

p. The numerical value of the non-dimensional

parameter n is pretty close to unity, however even a relatively small deviation of therefractive index from unity may have significant impact on radio wave propagation.Therefore, common practice is to use another definition of the refractive indexN = (n–1) . 106 instead of n, measurable in so-called N-units. The refractive indexN, also called the refractivity, has the following relationships with atmospheric pres-sure p, temperatureTand humidity, the mass-fraction of the water vapor, q, in the air:

1

Atmospheric Boundary Layer and Basics of the PropagationMechanisms

1 Atmospheric Boundary Layer and Basics of the Propagation Mechanisms

N ¼ AN �pT

1 þ BNqT

� �(1.1)

where AN = 77.6 N-units . K hPa–1, BN = 7733 K–1. The components p, T and q arerandom functions of the coordinates and time. The stochastic behavior of themeteorological parameters p, T, q and, therefore, the refractive index N is caused byatmospheric turbulence.

There are several reasons to separate the region of the first 1–2 km of atmosphereover the earth’s surface, called the atmospheric boundary layer, ABL. The upperboundary of the ABL is seen as the height at which the atmospheric wind changesdirection due to a combined effect of the friction and Carioles force. Among thosereasons are:

2

a)

c)

900

0 1 2 3 4 5

1000

800

700

q, g/kg

p, hPa

650

700

750

800

850

900

950

1000

1050281 286 291 296

Temperature, K

–1 1 3 5 7 9

Speed, m/sb)

p, hPa

Figure 1.1 Meteorological parameters in the atmospheric boundary layer as function of height(atmospheric pressure, p): Humidity (a), temperature (b) and wind speed (c). All parametersexperience sharp variations at the upper boundary of the atmospheric boundary layer.

. The interaction of the earth’s surface and atmosphere is especially pro-nounced in this region.

. The meteorological parameters such as temperature, humidity and windspeed experience daily variations in this region due to apparent cyclic varia-tions in the sun’s radiation due to the earth’s rotation.

. The ABL can be regarded as an area constantly filled with atmospheric turbu-lence. This is quite opposite to the atmospheric layer above the ABL, the so-called region of clear atmosphere, where turbulence is present only in iso-lated spots.

. The border between the clear atmosphere and the ABL is clearly pronouncedwith sharp variations in all meteorological parameters, as illustrated inFigure 1.1.

The spectrum of turbulent fluctuations in the atmospheric boundary layer isextremely wide: the linear scales of the variations range from a few millimetres tothe size of the earth’s equator, the time scales from tens of milliseconds to one year.Studies of the energy spectrum of the turbulent fluctuations of the meteorologicalparameters (temperature, humidity, pressure and wind speed) [1] have shown thatthe energy spectrum reveals three distinct regions: large scale quasi-two-dimen-sional fluctuations in a range of frequencies from 10–6–10–4 Hz, the meso-meteoro-logical minimum with low intensity of the fluctuations in the range 10–3–10–4 Hzand a small scale three-dimensional fluctuation region with frequencies above10–3 Hz.

Figure 1.2 shows the energy spectrum of fluctuations of the horizontal compo-nent of the wind speed in the atmosphere, taken from Ref.[1], where the ordinatecorresponds to the product of the spectrum density S xð Þ and the cyclic frequency xof the variations in one of the meteorological components, and the abscissa corre-

3

ω S(ω), m2/s2

10–110–210–310–410–510–6

0

5

10

15

20

25

30

10–210–11 101102103

ω , rad/s

hours

Figure 1.2 Energy spectrum of the fluctuations in the windspeed in the atmospheric boundary layer.


sponds to the frequency x ¼ 2p=T, T being a period of variations. As observed inFigure 1.2, there are two major extremes of the function x � SðxÞ: the high fre-quency maximum corresponds to a linear scale of turbulence of the order of tensand hundreds of meters, the low frequency maximum has a time scale of 5–10 dayswhich is caused by synoptic variations (cyclones and anti-cyclones), the respectivehorizontal scale is thousands of kilometres and the vertical scale is of the order of10 km. There is also an extended minimum in the spectrum x � SðxÞ that corre-sponds to the fluctuations with respective horizontal scales from 1 to 500 km and iscalled the meso-pause. The region of low frequency variation is called the macro-range while the region to the left of the meso-pause (high frequency variations) iscalled the micro-range.

The nature of the atmospheric turbulence is different in these two regions: in themacro-range the synoptic processes can be regarded as two-dimensional variations,while in the micro-range, with scales up to a hundred meters, the turbulence isthree-dimensional and locally uniform. The mezo-pause is a transition region wherea combined mechanism is observed. It is important to notice that by describingsmall-scale three-dimensional fluctuations in the micro-range region one can useTaylor’s hypothesis of “frozen turbulence” which allows a transformation from time-to space-fluctuation scales by means of L ¼ 2pv=f , where L is the spatial scale of theirregularities, f is the frequency of time variations in the refractive index, and v isthe mean speed of the incident flow.

The basic conclusion that follows from the above observations is that, to someextent, the refractive index N and the dielectric permittivity e can be presented as asum of a slow varying component e0 ~rrð Þ regarded as a quasi-deterministic functionof the coordinates~rr ¼ x; y; zf g and the random component deð~rrÞ. As observed fromFigure 1.2, the quasi-deterministic component e0 ~rrð Þ still varies in the horizontalplane and the energy of variations in the meso-pause minimum is not negligible.However, these variations have less impact on radio wave propagation than eithervariations of de ~rrð Þ in the micro-range or over-the-height variations in the “determi-nistic” component which may be responsible for a ducting in the troposphere.

Mathematically, the problem of radio wave propagation in a randomly inhomoge-neous medium comes down to solving a stochastic wave equation with dielectricpermittivity e ~rr; tð Þ which is a random function of coordinates and time. In manycases the process of propagation of a monochromatic wave in the troposphere canbe considered in a quasi-steady state approximation, i.e. “frozen” in time. It is thenconvenient to represent the dielectric permittivity in the form e0 ~rrð Þ ” e0ðzÞ þ de ~rrð Þ,where e0ðzÞ ¼ e0 ~rrð Þh i. The angular brackets denote averaging over the ensemble ofthe realisations of e ~rrð Þ. In fact, the mean characteristic of the tropospheric dielectricpermittivity e ~rrð Þh i is commonly understood as a large-scale structure homogeneousin the horizontal plane and practically non-varying over the time over which the sig-nal measurements have been performed and then, as a result of mathematical idea-lisation, e ~rrð Þh i ¼ e0ðzÞ. In radio-meteorology mean characteristics of the meteo-parameters are usually understood to be the values obtained by averaging over a30 min interval [2], i.e. averaging is performed over a frequency interval the lowerlimit of which is positioned within the limits of the meso-meteorological minimum.

4

1.1 Standard Model of the Troposphere

The average characteristic obtained in this way is usually a function of the height zabove the surface (sea, ground) and varies slowly with the horizontal coordinatesand time. Assuming ergodicity and Taylor’s hypothesis, such averaging over a timeinterval corresponds to the averaging over the ensemble of the realisations of e ~rr; tð Þ.

As a compromise in analytical studies of radio wave propagation, a commonapproach is to neglect the residual variations in e ~rrð Þh i over the horizontal coordi-nates, i.e. to regard the e0 ~rrð Þ ” e0ðzÞ. This assumption results in the introduction ofthe traditional model of a stratified atmosphere, in which the average values ofrefractive index N vary over the vertical coordinate z, the height above the ground.This traditional model provides some basis for a classification of the radio wavepropagation mechanisms, in particular, a separation of the propagation into twoclasses: standard and non-standard. The following Sections 1.2 and 1.3 provide abrief analysis of the standard and non-standard models, while Section 1.4 deals witha statistical model for a random component of the refractive index.

1.1Standard Model of the Troposphere

The standard mechanism of radio wave propagation is classified under the conditionwhere the average vertical gradients of the refractive index cN ¼ dN=dz are close tothe value c

stN ¼ –39 N-units km–1. Such conditions of refractivity constitute a model

of standard linear atmosphere defined as

N ¼ 289 � 39 � z (1.2)

and are applicable for heights less than 2 km.Let us consider this model in detail involving a geometrical optic presentation for

wave propagation.Let us define “ray” as a normal to a wavefront propagating through the medium

with varying refractivity n(z). As is known, the ray bends in such a medium and thebending is defined by Snell’s law. Introducing the horizontally stratified medium interms of the set of thin layers with value of refractivity ni, i = 0, 1,..., such as illustrat-ed in Figure 1.3, Snell’s law can be written as

ni cosui¼ const (1.3)

where ui is a sliding angle. Introducing the differentials of the ray direction dz, dSin the ith layer and differentiating both sides of Eq. (1.3) with respect to S:

cosuidnidS

� ni sinuidui

dS¼ 0. (1.4)

Substituting dS ¼ �dz=sinui , we obtain

dui

dS¼ �

cosuiðdn=dzÞni

. (1.5)

5


The radius of curvature at any point, Ri ¼ dS=dui , and using Eq. (1.5) it resultsthat

Ri ¼ni

cosui

1�dn=dzð Þ. (1.6)

For the standard atmosphere with dn/dz = 39 . 10–6 km–1, the radius of curvatureis given by

Ri ¼ 25; 000ni

cosui

. (1.7)

If the launch angle ui is close to the horizontal, the rationi

cosui

»1 and a propaga-

tion path can be described as a circle of radius R = 25,000 km, Figure 1.4(a).By comparison, the radius of the earth’s curvature is a = 6370 km, and1/a = 157 . 10–6 km–1. When the curvatures of both the propagation path and theearth are reduced by 39 . 10–6, as in Figure 1.4(b), the propagation path has an effec-tive curvature of zero (which is a straight line) and that hypothetical earth has aneffective curvature of (157 – 39) . 10–6 km–1 = 118 . 10–6 km–1. The equivalentradius of the sphere ae can therefore be defined as

6

d φ

φi

dS

dφ

dz ni

ni–1

Ri

Figure 1.3 Illustration of Snell’slaw.

a = 6370 km

R = 25000 km

a) b)

ae = 8500 km

Figure 1.4 Introduction of the “effective” radius of the earth:a) Ray refraction in a “normal” atmosphere with “true” earthradius a = 6370 km. b) Effective ray refraction in case when thedifference in curvatures of both ray and earth surface in (a) iscompensated by introduction of the modified earth radiusae = 8500 km.

1.1 Standard Model of the Troposphere

1ae

¼ 1a� 1R¼ 1a� cN�� 10

�6 ¼ 1118

� �· 10

6km= 8500 km, (1.8)

approximately, ae ¼ 4=3a.The modified refractive index can be defined as follows:

nm ðzÞ ¼ nðzÞ þ 1=a. (1.9)

It is apparent that the second term in Eq. (1.9) is a compensation for the earth’scurvature. The modified refractivityM(z) is then given by

MðzÞ ¼ ðnm ðzÞ � 1Þ·106 ¼ NðzÞ þ 0:157z. (1.10)

As observed from Eq. (1.10) for the case of standard refraction, whencN ¼ dN=dz ¼ �39 N-units km–1, cM ¼ dM=dz ¼ 118 N-units km–1. As explainedin Section 2.1, modified refractivity comes logically from the parabolic approxima-tion to the wave equations and effectively results in the introduction of the flat earthand additional curvature of the “rays” associated with radio waves propagating atlow-angles along the earth’s surface, as shown schematically in Figure 1.5.

This traditional linear model of the troposphere provides some means of classifi-cation of the propagation mechanisms based on the value of the gradient of therefractivity in the lower troposphere, both conventional cN ¼ dN=dz and modifiedrefractivity cM ¼ dM=dz. A very broad classification is concerned with the separationof the propagation into two classes: standard and non-standard. Figure 1.6 repre-sents schematically the ray traces for three basic sub-classes of standard mecha-nisms of propagation in the troposphere:

. Standard refraction, when the vertical gradient of refractivity is very close tocs

N¼ �39 km–1, c

s

M= 118 km–1.

. Sub-refraction, when cN

> –39 N-units km–1, cM

> 118 N-units km–1.. Super-refraction, when c

N< –39 N-units km–1, c

M< 118 N-units km–1.

As seen from Figure 1.6 the sub-refraction and super-refraction are associatedwith bending the waves outwards and inwards to the earth’s surface respectively.

The standard propagation conditions can be characterised by the presence of tworegions: the so-called line-of-sight (LOS) region, where the radio signal in the recei-ver has contributions from the direct wave and the wave reflected from the earth’s

7

Figure 1.5 Impact from modified refractivity on the ray trajectories from a point source.


surface; and the shadow region where the radio signal arrives as a result of diffrac-tion over the earth surface. These two regions are separated at the range of horizonwhich can be defined as

ffiffiffiffiffiffiffi2ae

p�ffiffiffiz

p, where the parameter z is the height of the obser-

vation point above the earth’s surface. It should be noted that for a standard refrac-tion

ffiffiffiffiffiffiffi2ae

p= 4.12 . 103 m

1=2compared with the value for optics of 3.83 . 103 m

1=2.

For frequencies above 20 GHz the standard mechanisms include attenuation of theradio waves due to absorption by atmospheric gases (water vapor and oxygen).

Non-standard mechanisms of radio wave propagation are related to some anoma-lies in the vertical distribution of the refractive index, namely the gradients cMbecome negative. These anomalies are in turn caused by some extremes in the verti-cal profiles of either the humidity or temperature.

Under such conditions the waves can be trapped inside a localised region asso-ciated with such negative gradients of modified refractivity, this effect is called “duct-ing” and the radio wave propagation mechanism is somewhat similar to propagationin a waveguide.

8

ae = 8500 km

Standard refraction

ae = 8500 km

Sub-refraction

ae = 8500 km

Super refractionFigure 1.6 Classification of the propagationmechanisms.

1.2 Non-standard Mechanisms of Propagation

1.2Non-standard Mechanisms of Propagation

The term M-inversion is used to characterize the area of negative gradients of theM-profile, where dM/dz < 0. When M-inversions exist over some height interval, aradio-waveguide can be formed. The upper boundary of the waveguide is the heightH at which dM/dz = 0, while the lower boundary zmin is given by M(zmin ) = M(H),see Figure 1.9 later. The waveguide is elevated if zmin > 0. If zmin = 0 or the equationM(zmin ) = M(H) has no solution, then the waveguide is classified as a surface wave-guide, Figure 1.7.

Most often M-inversions are formed in the lower and upper regions of the atmo-spheric boundary layer because of the local interaction between the turbulent airmass and either the surface or the free atmosphere above the ABL [3].

1.2.1Evaporation Duct

One of the most important types of radio waveguide is the evaporation duct abovethe water surface [3] which has a nearly permanent presence in the subtropicalregions of the Pacific Ocean. It is formed as a result of the air in contact with the seasurface becoming saturated by water vapor at the surface temperature. The watervapor content decreases logarithmically with height and the resulting M-profile hasthe form [3]:

MðzÞ ¼ N zrð Þ �Ha lnzzr

� �þ az (1.11)

where zr is the roughness parameter of the sea surface and a ¼ 106=a = 0.157 m–1

and a = 6370 km, the earth’s radius.The M-profile (1.11) is can be constructed from the data of a standard hydro-

meteorological measurement. It describes quite well the height dependence of the

9

0

5

10

15

20

25

30

298 300 302 304

M(z)

he

igh

t, m

∆ M

H

Figure 1.7 Evaporation duct, M-profile.


M-profile under stable and neutral conditions of stratification in turbulence. Thestudies performed in Ref. [3] show that the thickness of the duct (inversion heightH) is a function of the stability parameter of the turbulence at the water level layerand is functionally related to the inversion depth DM = M(zr) – M(H), Figure 1.7.

It should be noted that no analytical solution to the wave equation exists for theM-profile in form (1.11). While a numerical solution can be adopted with somemodifications to Eq. (1.11), in some applications it might be preferable to replaceEq. (1.11) by a profile for which the analytical solution to a wave equation is known.In particular, a common approach is to use a piecewise linear M-profile given by

MðzÞ ¼ MðHÞþaðz�HÞ; z>HMðHÞþDMðH�zÞ=ðH�zrÞ; z £H

�(1.12)

Comparison of the results obtained with both the piecewise linear approximationand the continuous M-profile (1.12) is presented in Section 4.1. It has shown a satis-factory agreement of the results of calculations for the lower order modes. In fact,the discrepancies observed for the high order modes in terms of their propagationconstants and therefore the height functions are not really important because thosemodes are significant in the transition area between the shadow and LOS regions,where, in turn, the mode presentation is hard to utilise because of the very slow con-vergence of the modal series. On the other hand, it was observed that the errorsintroduced by replacement of the continuous profile by a piecewise approximationfor the lower modes do not exceed the errors produced by uncertainties in determin-ing the values of the refractive index from data of standard hydro-meteorologicalmeasurements. The overall conclusion therefore is that the approximation (1.12) isa valid practical approach to analytically evaluate the propagation mechanism in theevaporation duct. In computer based prediction systems, where input radio-meteor-ological information can be collected in nearly real-time with substantial precision,the hydro-dynamical model of the evaporation duct Eq. (1.11) can be easily imple-mented. The common problem of calculating the propagation constants of thetrapped (or significant) modes can be reduced to pre-calculated look-up tables andinterpolation.

Figure 1.8 shows a schematic ray bending in the evaporation duct, which allowsover-the-horizon radio wave propagation.

10

Distance

Tra

jec

tory

Figure 1.8 Ray trajectories in evaporation duct.

1.2 Non-standard Mechanisms of Propagation

1.2.2Elevated M-inversion

The most frequent cause of abnormally high signal level far beyond the horizon(200–500 km) is associated with the presence of elevated M-inversions. Those areformed in the subtropical region above the ocean as a result of powerful motiondownwards in the center of an anticyclone and are usually called depression layers.The elevated M-inversions, also called elevated waveguides or inversion layers, arecharacterized by lower height of inversion Hi, the thickness of the layer with nega-tive M-profile (M-inversion) DH = Hu – Hi and the depth of inversion, the so-calledM-deficit DMi = M(Hi) – M(Hu), as shown in Figure 1.9. Figure 1.10 shows sche-matic ray trajectories in an elevated duct when the source is placed inside its bound-aries.

11

M(z)M(z)

zz

Hu

Hi

Hb

Hu

Hi

∆Mi

∆Mi

a) b)

Figure 1.9 Elevated M-inversion: a) Elevated tropospheric ductin the height interval Hu > z > Hb. b) Surface based troposphericduct in the height interval Hu > z > 0.

Distance

Heig

ht

Modified Refractivity, M

Height

Hk

Hi

Hb

∆M

Figure 1.10 Ray traces in the elevated duct.


The observations indicate that characteristic values for the depression layers areas follows: Hi » 200 � 1500 m, DH » 100 � 300 m, and DMi » 10 � 30 units. Thedepression layers are normally observed as stable atmospheric formations existingover a wide area of the ocean surface. The height of inversion usually coincides withthe height of the atmospheric boundary layer. The elevated M-inversion normallyseparates a convective boundary layer from the free atmosphere above, and in thedirect vicinity of M-inversion turbulence alternates with laminar flow. Due to a pene-tration of convective elements from the lower atmospheric boundary layer the localtemperature and gradients of the wind speed increase and as a result Kelvin–Helm-holtz waves develop. When these waves collapse they generate an intense turbulentstructure.

Characteristic scales of the turbulent inhomogeneities in the depression layer canbe estimated as follows: horizontal scale Lk~500–1000 m, vertical scale Lz~100 m.

There is another important type of elevated M-inversion which is often formed incoastal regions due to the formation of a surface duct as a result of heating of thesea surface at sunrise and elevation of the surface M-inversion at night time. Theelevated M-inversion is then unstable due to a developed convection.

1.3Random Component of Dielectric Permittivity

A comprehensive theory of random fields and methods of their study are providedin Refs. [2] and [4]. Below we summarize the basic assumptions to be used furtherin this book to describe the random field of dielectric permittivity eð~rr; tÞ.

When the random field deð~rr; tÞ is stationary and uniform in time and spacerespectively, a two-point statistical characteristic of this field can be described via aspace–time correlation function

Bð~��; sÞ ¼ deð~rr þ~��; tþ sÞdeð~rr; tÞh i. (1.13)

It is assumed that < de > ¼ 0, and, as a consequence of uniformity and stationar-ity, deð~rr; tÞ is invariant to an offset in the space and time domain, and its correlationfunction (1.13) depends only on the difference in space ~�� and time s between twopoints of observation.

The space–time correlation function Bð~��; sÞ is related to the spectrum densityG x; kð Þ of the field deð~rr; tÞ via a Fourier transform:

Bð~��; sÞ ¼R¥�¥dxRd~kkG x;~kkð Þ � ej ~kk~��xsð Þ

. (1.14)

The function G x; kð Þ is also called a x~kk density, where ~kk is the wave vector offluctuation in de. The reverse transform is given by

Ge x;~kkð Þ ¼ 1

2pð Þ3

Rd

3~��R¥�¥dsB ~��; sð Þe�j ~kk~��xsð Þ

. (1.15)

12

1.3 Random Component of Dielectric Permittivity

Both the spatial spectrum Ue and the frequency (energy) SðxÞ spectrum of thefluctuations de can be expressed via x~kk density:

Ue ~kkð Þ ¼R¥�¥dxG x;~kkð Þ ¼

R¥�¥d

3~��B ~��; 0ð Þ � e�j~kk~��, (1.16)

Se xð Þ ¼R¥�¥d~kkG x;~kkð Þ ¼

R¥�¥dsB 0; sð Þ � ejxs. (1.17)

Consider the case when all time variations in deð~rr; tÞ are caused by transfer of thespatial inhomogeneties d~eeð~rrÞ with constant velocity~vv, then deð~rr; tÞ = d~eeð~rr �~vvtÞ. If amotionless uniform random field d~eeð~rrÞ is described by a spatial correlation functionb~eeð~��Þ ¼ d~eeð~rr þ~��Þd~eeð~rrÞh i then

Bðs;~��Þ ¼ b~ee ~��~vvsð Þ. (1.18)

Substituting Eq. (1.18) into Eq. (1.15) we can find a relationship between x~kk thedensity of the “frozen” field deð~rr; tÞ and the spatial spectrum density U~ee of the stillfield d~eeð~rrÞ:

G x;~kkð Þ ¼ U~ee ~kkð Þd x� k! v!Þ

. (1.19)

Using Eqs. (1.16) and (1.19) we find that both the spatial spectrum of the frozenfield deð~rr; tÞ and that of the motionless field d~eeð~rrÞ are the same

Ue ~kkð Þ ¼ U~ee ~kkð Þ. (1.20)

This is an important relationship since all further studies in this book are basedon the hypothesis of the “frozen” turbulent fluctuations of dielectric permittivitydeð~rr; tÞand all calculations are performed with a motionless field of d~eeð~rrÞ instead ofdeð~rr; tÞ and the symbol ~ can be omitted for deð~rrÞ.

1.3.1Locally Uniform Fluctuations

The random field of dielectric permittivity d~eeð~rrÞ can be regarded as uniform only asa very rough approximation. For example, the statistical characteristics of d~eeð~rrÞ inthe atmospheric boundary layer depend on the height above the ground and, there-fore, another approach, based on the introduction of the structure function, shouldrather be used to describe the turbulent fluctuation of the d~eeð~rrÞ.

The structure function is an average of the square of the module of the incrementof the fluctuations d~eeð~rrÞ and is defined as

D ~rr1 ;~rr2ð Þ ¼ deð~rr1Þ � deð~rr2Þj j2D E

¼ deð~rr1Þ � deð~rr2Þð Þ2D E

. (1.21)

13


The module sign is removed in Eq. (1.21), given the assumption that d~eeð~rrÞ has areal value. The hypothesis of the local uniformity is based on the assumption thatthe increment of the fluctuations of the random field de at two points~rr1 and~rr2 iscaused chiefly by inhomogeneities of de with a sizel less than the distance betweentwo points of observations: l £ ~rr1 �~rr2j j. While the inhomeogeneities of the largescale may exist, they produce nearly the same perturbation in the field de in points~rr1 , ~rr2 and, therefore, a negligible difference in the perturbation in de. When thishypothesis holds, the field de is called “locally uniform” or, using an analogy with arandom process with stationary first increment, we can call de a random field withuniform increment. As a consequence, the structure function D ~rr1 ;~rr2ð Þ dependsonly on the spatial increment ~�� ¼~rr1 �~rr2 :

D ~rr1 ;~rr2ð Þ ¼ D ~��ð Þ. (1.22)

The important advantage of using a structure function is that it makes sense evenwhen the correlation function does not exist, Be 0ð Þ ! ¥. However, if it does existthen the following relationship is valid:

De ~��ð Þ ¼ 2 Be 0ð Þ � Be ~��ð Þ½ �. (1.23)

When Be ¥ð Þ ! 0, De ¥ð Þ ¼ 2Be 0ð Þ and

Be ~��ð Þ ¼ 12De ¥ð Þ �De ~��ð Þ½ �. (1.24)

The structure function De is related to a spatial spectrum Ue via the relationships

De ~��ð Þ ¼ 2Rd~kkUe ~kkð Þ 1 � cos~kk~��ð Þ, (1.25)

Ue ~kkð Þ ¼ 1

16p3k2

R¥�¥d

3� �~kkrDe ~��ð Þ sin ~kk~��ð Þ. (1.26)

When the random field is statistically isotropic, the increment of the fluctuationsin de depends only on the distance between two points ~rr1 and ~rr2 , thenDe ~��ð Þ ¼ De �ð Þ and Ue ~kkð Þ ¼ Ue kð Þ, where � ¼ ~rr1 �~rr2j j, k ¼ ~kkj j are the modules of~�� and ~kk respectively. For locally uniform and isotropic fluctuations of de Eqs. (1.25)and (1.26) can be transformed into the following ones:

De �ð Þ ¼ 8pR¥0

1 � sin k�k�

� �Ue kð Þk2

dk, (1.27)

Ue kð Þ ¼ 1

4p2k3

R¥0

sin k�� k� � cosk�ð Þ dDe �ð Þd�

d�. (1.28)

14

1.3 Random Component of Dielectric Permittivity

The Eqs. (1.19) and (1.20) are valid for a “frozen” and locally uniform random fielddeð~rr; tÞ. The equation for the time–space correlation function (1.18) can be substi-tuted by a similar equation for a space–time structure function

Deð~��; sÞ ¼ De ~��~vvsð Þ. (1.29)

The turbulent fluctuations of dielectric permittivity de ~rrð Þ in a free atmospherecan be modelled as a locally uniform random field. The structure function De �ð Þ offluctuations de obeys a law of 2/3 in the Kolmogorov–Obukhov model [4] for largeenough �:

De �ð Þ » C2e �

2=3; � >> l0 (1.30)

where C2e is the structure constant, a measure of fluctuation intensity and l0 is the

internal scale of the turbulence (the inhomeogeneties with the scale less than l0dissipate rapidly). For small � the structure function follows the square law:

De �ð Þ » C2e �

2; � << l0 . (1.31)

As observed from Eq. (1.30) the structure function De �ð Þ is unlimited by a scaleof fluctuations and allows the existence of turbulent bursts of any scale, which iscontradictory to the applicability of the Kolmogorov–Obukhov model which is itselflimited to an inertial interval of the locally isotropic turbulence.

In order to limit the external scales of the fluctuations to an inertial interval of theturbulence and to make the model consistent one can introduce a maximum scaleL0 of the turbulent fluctuations and therefore a “saturation” feature in the behaviorof the structure function:

De �ð Þ ¼ C2e L

2=30 ; � >> L0 . (1.32)

In fact, a saturation means a limited value of Deð¥Þ and, therefore, a limited valueof variance r

2e ¼ Beð0Þ of the fluctuations de. In this case those values are related as

follows: C2e L

2=30 ¼ 2r

2e .

A combined structure function for locally uniform and isotropic fluctuation indecan be written as follows [4]:

De �ð Þ ¼C

2e l

�4=30 �

2; � << l0

C2e �

2=3; l0 << � << L0

C2e L

2=30 ; � >> L0

8>><>>: (1.33)

Within an inertial interval of locally isotropic turbulence the spatial spectrumUe ~kkð Þ is described by the Kolmogorov–Obukhov model

Ue ~kkð Þ ¼ 0:033C2e k

�11=3exp �k2

k2m

!. (1.34)

15


Here ~kk is the wave vector of fluctuation in de, k ¼ ~kkj j, C2e is the structure con-

stant, a measure of the fluctuation intensity, km ¼ 5:92=l0 and l0 is the internal scaleof the turbulence.

The applicability of Kolmogorov’s model (1.34) for the atmospheric boundarylayer over the ocean surface has been studied in Ref. [5]. The basic observations areas follows:

. A one-dimensional spectrum of the turbulent fluctuations in the horizontalplane follows Kolmogorov’s model up to scales: Lk~10–100 m. The spectruminterval over which the postulates of a locally isotropic turbulence are validdepends on the stability of the atmosphere and decreases greatly when thestability intensifies. In the case of very stable stratification associated with thepresence of M-inversions the turbulent structure reveals significant anisotro-py of the large-scale fluctuations.

. At heights above the surface comparable with the heights of the swells, thestate of the sea surface reveals a significant impact on a form of the spectrumof turbulent fluctuations. This manifest itself in the form of flares in spectraldensity at the frequencies corresponding to the energy-bearing frequencies ofthe sea waves. The amplitude of the wave perturbations decreases with heightand has no practical effect on the form of the spectrum at heights of 15–20 m above the surface;

. The intensity of the high-frequency turbulent fluctuations depends parame-trically on the height above the surface. In a low-frequency domain (the buoy-ancy and energy intervals) one must include additional parameters in themodel of spectrum: the thickness of the atmospheric boundary layer andsome parameters characterizing the roughness of the sea surface.

It can be stated that despite numerous studies, a physically justifiable model for athree-dimensional spectrum of fluctuations in the index of refraction within the at-mospheric boundary layer does not yet exist. The anisotropy of the spatial fluctua-tions in the refractive index is commonly introduced via the anisotropy parametera ¼ Lz=Lk in the following model of the spectrum:

Ue ~kkð Þ ¼0:063 r2

eLzL2k

1þ k2kL

2k þk2

zL2z

� �11=6 (1.35)

where the r2e is a variance of the fluctuations. It can be noted that in the case of

turbulence Lz ; Lk ! ¥ , r2e ! ¥. The generalisation of Kolmogorov’s model to an-

isotropic turbulence can be given as follows:

Ue ~kkð Þ ¼ 0:033C2e?a

k2? þa2k2

z

� �11=6 ¼ 0:033C2

eza5=3

k2? þa2k2

z

� �11=6 (1.36)

where Ce?and Cez are structure constants in the horizontal plane and over heightrespectively, as observed they are not independent and are related via

16

References

Ce? ¼ a1=3Cez . (1.37)

In practical measurements the time of averaging is always finite and so varianceof the fluctuations r

2e and scales L? ; Lz is observed. In fact model (1.35) can be trans

formed into model (1.36) with the substitution r2e ¼ 1:9C

2e?L

2=3? , assuming that the

external scale of the fluctuations in dielectric permittivity L? is confined within themicro-range region of the spectrum of turbulence.

The observations [6] suggest that the value of the anisotropy parameter a itself isa function of the external scale of the turbulent fluctuations and can be estimated tobe of the order of a ~ 0.1 – 0.2 for Lz~ 1 m at the heights close to the water level. Atthe upper boundary of ABL the parameter a may vary from a ~ 0.8 for Lk~15 m toa ~ 0.005 for Lk~ 10,000 m.

17

References

1 Jensen, N.O., Lenshow, D.H. An observa-tional investigation on penetrative convection.J.Atmos.Sci., 1978, 35(10), 1924–1933.

2 Monin, A.S., Yaglom, A.M. Statistical FluidMechanics, Vol.1, MIT Press, Cambridge, MA,1971, p. 769.

3 Gossard, E.E. Clear weather meteorologicaleffects on propagations at frequencies above1 GHz, Radio Sci., 1981,16(5), 589–608.

4 Tatarskii, V.I. The Effects of Turbulent Atmo-sphere on Wave Propagation, IPST, Jerusalem,1971.

5 Gavrilov, A.S., Ponomareva, S.M. TurbulenceStructure in the Ground Level Layer of the Atmo-sphere. Collected Data, Meteorology Series,No.1, Research Institute for MeteorologicalInformation, Obninsk (in Russian), 1984.

6 Kukushkin, A.V., Freilikher, V.D. and Fuks,I.M. Over-the-horizon propagation of UHFradio waves above the sea , Radiophys. Quan-tum Electron. (translated from Russian), Con-sultant Bureau, New York , RPQEAC 30 (7),1987, 597–620.

19

2.1Analytical Methods in the Problems of Wave Propagation in a Stratified and RandomMedium

The analogy between the non-stationary problem of quantum mechanics and theparabolic approximation in a problem wave propagation was likely first explored byFock [1]. It is well known that the methods of classic wave theory were utilised inquantum mechanics in the early stages of its development. At present, the situationis rather the reverse, and the mathematical methods of quantum mechanics havebecome widely adopted in the latest developments of wave theory.

The asymptotic solution to the problem of the diffraction of a plane monochro-matic wave over a sphere of large radius, compared with the wavelength of radiation,was obtained by Fock in 1945 [2]. In obtaining this solution, he developed an asymp-totic theory of diffraction on the basis of Leontovich’s boundary conditions [3] andthe concept of the “local” field [2]. The approach utilised several large parametersinvolved in problem formulation, such as gj j >> 1, m ¼ ðka=2Þ1=3 >> 1,g ¼ eg þ j4pr=x, eg , r, the dielectric permittivity and conductivity of the earth,respectively, x is the cyclic frequency of the radiated wave, k = 2p/k is a wavenum-ber, k is a wavelength, a is the radius of the earth’s sphere. The same results wereobtained by Fock later in Ref. [4] by means of the parabolic approximation to thewave equation which is based on significantly different parameters, with m >> 1,and the characteristic scales of the wave oscillations in two different directions:along the earth’s surface and normal to it. With regards to radio wave propagation ata frequency above 1 GHz the important development was a solution of the parabolicequation for the stratified troposphere, where the refractive index depends solely onthe height above the ground surface [5]. The analysis of the wave field at low alti-tudes z, which are small compared with the radius of the earth’s curvature a, is con-venient to perform by means of the introduction of the modified refractivity:nm(z) = n(z) + z/a. The solution obtained in Ref. [5] is suitable for the very generalcase of the dependence n(z) and impedance boundary conditions at the earth’s sur-face. The solution in Ref. [5] is represented by a contour integral which in the shad-ow region is calculated by a sum over residues in the poles of the integrand, i.e. aseries of normal waves [6]. In the line-of-sight (LOS) region the contour integral is

2

Parabolic Approximation to the Wave Equation

2 Parabolic Approximation to the Wave Equation

commonly calculated using stationary phase methods similar to a geometric opticapproximation [1, 7]. Further study of the wave propagation in a stratified tropo-sphere was dedicated to the development of the effective methods of the field repre-sentation for various height profiles of the refractivity and combinations of the rela-tive location of the transmitting and receiving antennae [8–16]. An alternative analyt-ical method to the solution of wave propagation in stratified medium, the method of“invariant submergence”, was developed by Klatskin [17]. Using that method, aboundary problem, which is, in fact, a problem of diffraction, can be transformedinto a problem of evolution. A significant advantage of this method is its inherentcapability to study the wave propagation through random media, and the method isespecially effective when applied to a randomly stratified medium.

Together with a ducting mechanism, the wave scattering on random inhomogene-ities of the refractive index plays a significant role in the study of radio wave propa-gation in the atmospheric boundary layer at frequencies above 1 GHz. The straight-forward application of either the contour integral or normal mode series to a prob-lem of wave propagation in a random troposphere will face significant difficulties.The major problem will be related to the divergence of the matrix elements of thenormalwave transformation due to a scattering on random fluctuations in the refractiveindex. The deployment of the method of “invariant submergence” also does not lead toan analytical solution to the problem though numerical calculation is possible.

In this book we use the representation of the Green function in the form of anexpansion over the eigen functions of a continuous spectrum which allows one toavoid the above mentioned difficulties of divergence of matrix elements and then toseparate the matrix coefficients with a “nearly discrete” and continuum spectrumwith normalised height gain functions. This approach led to some advances in sol-ving several problems, in particular, the problem of wave propagation in a tropo-spheric duct filled with random irregularities of the refractive index in addition to aregular supper-refractive gradient of the refractivity. The disadvantage of the methodis that the separation of the discrete spectrum of eigen functions is performed in the“unperturbed”, i.e. regular, part of the SchrJdinger operator and, therefore, the spec-trum of eigen functions is left unchanged by the random component of the potentialterm. The study of the spectrum of SchrJdinger’s equation with random potential isa complex problem by itself and most advances so far have been achieved in theone-dimensional problem [18].

The theory of wave scattering in a random medium has been developed duringthe last decades [19–23]. The most significant progress has been achieved in the so-lution to the problems of wave propagation in either a random but unbound medi-um or a uniform medium with a random boundary interface. In the solution ofthese problems a statistical approach is commonly used. This approach is targetedat obtaining the statistical characteristics of the scattered field: probability functionor statistical moments of the field. Among them the first two moments: averagefield (coherent component) and either coherence function or field intensity, are ofsignificant interest in many practical applications. There are three known methodsof obtaining the closed equations for the field moments: Feynman diagrams [19, 24,25], local perturbation [26–28] and the Markov process approximation [29–31].

20

2.1 Analytical Methods in the Problems of Wave Propagation in a Stratified and Random Medium

The study of a coherent component can be done using the modified perturbationtheory [32, 33], a similar result is obtained from the diagram methods [25]. Theadvances in the solution of the higher order moments of the field have beenobtained after application of the parabolic approximation to the wave equation [21,22]. Neglecting back scattering in the parabolic approximation allows one to employthe concept of the Markov process [29–31] and all the following analogies from thestatistical theory of Markov processes.

The path integral in the problem of wave propagation in a random medium wasintroduced in Refs. [34–36]. The advantage of the functional approach, as noted inRef. [22], is that the formal solution to the problem is written via quadrature. Themoment of the field of any order can, in principle, be obtained by averaging therespective dynamic solution which is described by a multiple functional integral inthe space of the virtual trajectories. The solutions to the first two moments of thefield obtained from both the method of closed equations for the moments and themethod of the path integral provide exactly the same result, as shown in Refs. [22,36] under conditions of the Markov process approximation. The study of the fieldmoments of order higher than two is easier with the use of the path integral, sincethis approach does not formally require a solution of the equations for the fieldmoments. In fact, this advantage became apparent in the calculation of themoments of the intensity (signal strength), which are not described by a closedequation.

The current limitations of the path integral method could be rather associatedwith the still limited mathematical methods of analytical calculation of the trajectoryintegrals, despite the serious studies in that area [37, 38].

Finally, we comment on studies concerning scattering on a rough sea surface.The basic conclusion is that, until now, there is no adequate theory that takes intoaccount the combined effect of refraction, diffraction and scattering on randominhomegeneities of the refractive index in the volume of the tropospheric layer aswell as scattering on a rough sea surface. The impact from a rough surface cannotbe regarded as negligible in the general case, at least from a theoretical point ofview. Known attempts are limited to the deterministic model of the sea surface or asemi-empirical approach in treating the scattering on the sea surface in the Kirch-hoff approximation [39, 40] that is applicable to the coherent component of the scat-tered field. While the theory of wave scattering on a random surface is an estab-lished science by itself, see for instance Refs. [19, 41], the studies known to theauthor were concerned with scattering theory on a random surface while treatingthe propagation media in a very simplistic way, basically, as deterministic and uni-form. In applying the scattering theory to a ducting mechanism, the approach devel-oped in Refs. [19, 41] should eventually be modified in order to take into account themultiple effects of refraction, diffraction and scattering. As discussed in Chapter 1,the sea waves could be responsible for the modulation of the turbulence spectrumin the near-surface layer, where the major action of radio propagation takes place.On the other hand, all previous advances in theory have been driven by some unre-solved problems in experiments on the radio wave propagation phenomenon. So far,the currently available radio coverage prediction system [39] is reported to be in rea-

21


sonably good agreement with the results of existing models, at least for measuredsignal strength.

2.2Parabolic Approximation to a Wave Equation in a Stratified Troposphere Filledwith Turbulent Fluctuations of the Refractive Index

Let us consider the vertical electric dipole in a spherical coordinate system ðr;J;uÞwith the spherical axis coming through the dipole. Assume that the dielectric per-meability e is a function of the radius only, e.g. e ¼ eðrÞ. As shown in Ref. [1], theclassical solution to the electromagnetic field can be derived by introduction ofDebye’s potentials U and V. The field components are given by the following equa-tions:

Er ¼ 1rD

�U;

EJ ¼ � 1er

@2

@r@JerUð Þ þ j k

sinJ@V@u

;

Eu ¼ � 1er sinJ

@2

@r@uerUð Þ � jk @V

@J;

(2.4)

Hr ¼ � 1rD

�V ;

HJ ¼ j kesinJ

@U@u

þ 1r@

2ðrVÞ@r@J

;

Hu ¼ �jke @U@J

þ 1r sinJ

@2ðrVÞ@r@u

:

(2.5)

We assume the field to be a harmonic function of time – exp �jxtð Þ, x is a cyclicfrequency of the electromagnetic radiation, k ¼ x=c is a wave number, and c is thespeed of light. The notation D

�represents a Laplace operator on a sphere

D� ¼ 1

sin J@

@Jsin J

@

@J

� �þ 1

sin2J@

2

@u2 . (2.6)

The Maxwell equations

rot~HH ¼ jke~EE ;

rot~EE ¼ �jk~HH(2.7)

will be satisfied if potentials U and V obey the following equations:

1r@

@r1e@ erUð Þ

@r

� �þ 1

r2D

�U þ k

2eU ¼ 0, (2.8)

1r@

2 rVð Þ@r2

þ 1

r2D

�V þ k

2eV ¼ 0. (2.9)

22

2.2 Parabolic Approximation to a Wave Equation in a Stratified Troposphere Filled with Turbulent…

The approximate Leontovitch’s boundary conditions for U and V have the form

@ erUð Þ@r

¼ �jk effiffiffig

p rU, (2.10)

@ rVð Þ@r

¼ �jk ffiffiffig

prV (2.11)

with r = a, where a is the earth’s radius.Let us assume that dielectric permittivity eð~rrÞ is stratified on average, i.e.

eð~rrÞh i ¼ e0ðrÞ. This means that eð~rrÞ is a random function of the space coordinates~rr ¼ r;J;uf g: e ~rrð Þ ¼ e0 rð Þ þ de ~rrð Þ, where de ~rrð Þ is a random component present ineach realisation of e ~rrð Þ, and de ~rrð Þh i ¼ 0. Assume also that de=ej j << 1. The angularbrackets here and below mean averaging over the statistical ensemble of the fluctua-tions de.

Let us assume that, the relationships between field components ~EE, ~HH and poten-tials U, V are still given by Eqs. (2.4) and (2.5). Strictly speaking, the Maxwell equa-tions (2.7) cannot be satisfied for an arbitrary function deð r*Þ, and one can seekapproximation to Eq. (2.7) where the divergent terms are small.

Introduce vector~xx ¼ aJ;ursinJf g. The divergent terms in Eq. (2.7) will be smallwhen

@e@xi

��<< k (2.12)

where xi is any of component of the vector ~xx. When the inequality (2.12) holds, thepotentials U and V are governed by Eqs. (1.8), (1.9)where e ¼ eo rð Þ þ de ~rrð Þ.

Let us examine the vertically polarised field for which V = 0, U „ 0. For high fre-quencies when ka >> 1, we can select a “dedicated” direction of wave propagationx ¼ aJ aligned with the direction along the arc of the earth’s radius. With ka >> 1,the radial component of the electric field Er is related to the potential U via:Er»� ka

2U.

Following Ref. [1], introduce a new function

U1 ¼ erU (2.13)

which is governed by the equation

@

@r1e@U1

@r

� �þ 1

er2

@2U1

@J2 � @U1

@J2e@e@J

� cosJsinJ

� �þ

U11

e2@e@J

� �2

� 1e@

2e

@J2 �cosJe sin J

@e@J

!266664

377775

þ 1

er2sin J@

2U1

@j2 � 2e@U1

@j@e@u

�U1

e2@

2e

@j2 �2e

@e@j

� �2 !" #

þ k2U1 ¼ 0 (2.14)

and obeys the boundary condition

23


@ U1

� �@r

¼ �jk U1ffiffiffiffiffiffiffiffiffigþ1

p , with r = a. (2.15)

Let us isolate the slow varying complex amplitudeW1 in the wave field by

U1 ¼W1ejkaJ

(2.16)

and introduce coordinate y ¼ rsinJð Þj. The amplitude W1 is then given by equa-tion

@2W1

@r2 þ 2jk

r@W1

@Jþ @

2W1

@y2 þ k

2e� k

2 a2

r2

� �W1 ¼

@e@r

@W1

@r� 1r2

@2W1

@J2 þ j krW1

2e@e@J

� cosJsinJ

� �þ

W1

r2

1e2

@e@J

� �2

� 1e@

2e

@J2 �cosJe sinJ

@e@J

� �þ

1r2

2e@W1

@y@e@y

�W1

e@

2e

@y2 �

2e

@e@y

� �2 !" #

(2.17)

at the right-hand side of which we have a correction term. Using qualitative argu-ments, it can be shown that in the case of a stratified medium e ¼ eðrÞ the terms onthe left have an order of magnitude of k2W1/m

2 while on the right the terms contain-ing derivatives are of the order of k2W1/m

4. The right-hand side terms containingsinJ in the denominator will also be small if the following inequality holds

x >>am

. (2.18)

The problem now is to evaluate the terms in the right-hand side of Eq. (2.17)for the case of random deð~rrÞ caused by turbulence. Let us introduce coordinatez = r – a. The characteristic scales L of variations in W1 over x, y, z can be estimatedas follows:

Lx~am; Ly~

ffiffiffiffiffikx

p; Lz~

mk. (2.19)

respectively. Having compared the terms in the left- and right-hand sides ofEq. (2.17) we obtain the inequalities

@e@x

� �2

<<m2

a2 ;@e@y

� �2

<<1kx

;@e@z

� �2

<<k2

m2 (2.20)

which, being satisfied, allow one to treat the terms in the right-hand side ofEq. (2.17) as negligible. Therefore, with fulfilment of inequalities (2.20) the right-hand side of Eq. (2.17) can be replaced with zero.

The inequalities (2.20) put limitations chiefly on the intensity of small-scale fluc-tuations deð~rrÞ. We can assume that the fluctuations in dielectric permittivity deð~rrÞ

24

2.2 Parabolic Approximation to a Wave Equation in a Stratified Troposphere Filled with Turbulent…

are locally uniform and isotropic. Let us introduce a spatial spectrum Ue of fluctua-tions deð~rrÞ by the equation

Ue ~kkð Þ ¼ 1

16p3

Rd3~�� grad de ~rr þ~��ð Þ � de ~rrð Þ½ �2

D E~kk sin ~kk~��ð Þk

(2.21)

where~kk ¼ kx ; ky ; kz� �

is the wave vector of the fluctuations in the dielectric permit-tivity, k ¼ ~kkj j.

We can evaluate the gradients in de ~rrð Þ by

@e@xn

� �2� �~Rd3~kk k2

nUe ~kkð Þ (2.22)

where xn is any of the coordinates x, y, z. For the spatial spectrum of fluctuationsde ~rrð Þ we can use the formula related to the equilibrium interval of the locally-isotro-pic turbulence [42]:

Ue ~kkð Þ ¼ 0:033C2e k

�11=3exp � k2

k2m

!(2.1)

where cz is a structure constant, km = 5.92/l0, l0 is an internal scale of turbulence.Substituting Eq. (2.1) into Eq. (2.20) we obtain

0:033C2e k

4=3m <<

kx;k2

m2 ;m2

a2 . (2.24)

Assuming reasonably short distances and high frequencies of radiation, e.g.m >>1 and x £ 103 km, we can see that the major impact comes from variations ofthe z-coordinate, and the inequality (2.24) thus reduces to

0:033C2ep

a2

� �2=3 kl0

!4=3

<< 1. (2.25)

Here we took into account that k=m ¼ k2=3

a=2ð Þ1=3 . Estimating the parametersC

2e ¼ 10

�14cm

�2=3, a = 8500 km, we observe that the coefficient with k=l0ð Þ4=3 has

an order of 10–9.Now we can conclude that with fulfilment of inequalities (2.18), (2.25) and

z/a << 1, the complex amplitudeW1ð~rrÞ obeys the approximate equation

j2k@W1

@xþ @

2W1

@y2� @

2W1

@z2þ k

2em ðzÞ � 1 þ de ~rrð Þð ÞW1 ¼ 0 (2.26)

and boundary condition

@W1

@z¼ �jk W1ffiffiffiffiffiffiffiffiffi

gþ1p with z = 0. (2.27)

The potential U is given by

U ~rrð Þ ¼ ejkaJ

erW1ð~rrÞ. (2.28)

25


The next step is to figure out the correct representation of the field amplitude W1

at small distances from the source in order to obtain an initial condition. In particu-lar, at the distance where the curvature of the earth can be neglected as well as theray’s refraction we have to obtain a reflection formula in the parabolic approxima-tion [1]. Introducing the coordinates of the source: x = 0, y = y0, and z = z0, andassuming that g >> 1 the reflection formula will be satisfied when

W1ðx;~��Þ ¼ 2eðx;~��Þx

a exp jkx

~��~��0ð Þ2� �

(2.29)

where �* ¼ y; zf g, �*0 ¼ y0 ; z0f g.

Taking into account the relationship

limjx!x

0k

2p x�x0� � exp j

k2 x�x0� � ~��~��

0� �2

( )¼ d ~��~��

0� �

where d ~��ð Þ is a Dirac’s delta-function defined by

d ~��ð Þ ¼ 1

4p2

Rd2~kk exp j~kk~��ð Þ, (2.30)

we make a transition in Eq. (2.29) with x ! 0 and obtain an expression for singular-ity at the source

limjx!0W1ðx;~��Þ ¼ 4pje 0;~��0ð Þ a

kd ~��~��0ð Þ. (2.31)

Equation (2.29) shall take place in a line-of-sight region, i.e. with x <<ffiffiffiffiffiffiffiffiffiffi2az0

p:

Taking into account that the inequality (2.18) is to hold as well, we obtain the limita-tions at the height of the source z0 under which the expression (2.31) is valid:

z0 >> a/m2 . (2.32)

It should be noted that in the case of very small heights of the source, anotherapproach can be used and the solution to Eq. (2.26) should be tailored with the solu-tion of Veil-Van-der-Paul [1] at small distances x.

In conclusion, one should note that there is some ambiguity with the definitionof the complex amplitude via Debye’s potential U. Indeed, equations similar to Eq.(2.26) can be obtained for the other functions W2 and W3 defined by the followingexpressions:

W2 ¼ erffiffiffiffiffiffiffiffiffiffiffiffiffisinJ

pe�jkaJ

U;

W3 ¼ er sinJe�jkaJU:(2.33)

The area of applicability of the parabolic approximation for W3 is defined by thesame set of inequalities (2.18), (2.24) while for W3 the inequality (2.18) can bereplaced by a less stringent one:

x >> m/k. (2.34)

26

2.3 Green Function for a Parabolic Equation in a Stratified Medium

Let us summarize the major steps undertaken in this section:

1. The components of the electromagnetic field ~EE and ~HH can be represented viaDebye’s potentials U and V. Even in the presence of a random turbulent com-ponent of the refractive index the de-polarization effects can be neglectedand, therefore, the field components are governed by independent equationsfor U and V.

2. The wave equations for both potentials U and V can be approximated by para-bolic equations for a slow varying complex amplitude. In this approximationall radiated waves are travelling only in one direction, away from the source.The Leontovich boundary conditions are used along with the parabolic equa-tion. The conditions in the source are also modified in a parabolic approxima-tion.

3. During the transformation from the Helmoltz equation to a parabolic equa-tion we actually replaced the earth sphere with a cylinder of the same radiusand then flattened the cylinder by means of compensating the earth’s curva-ture by “ray’s curvature” in the opposite direction. This is a well-known “flatearth” approximation with modified refractivity and locally Cartesian coordi-nates.

2.3Green Function for a Parabolic Equation in a Stratified Medium

Let deð~rrÞ ¼ 0 in Eq. (2.26) and take into account that for the frequencies above1 GHz the boundary condition (2.27) is reduced to Wð~rrÞ ¼ 0 at z ¼ 0. By definition,a Green function Gð r*;~rr 0 Þ is governed by the equation

2jk@G@x

þ D?Gþ k2em ðzÞ � 1½ �G ¼ �4pdð~rr �~rr

0 Þ, (2.35)

where D? ¼ @2

@y2þ @

2

@x2 ,

obeys the boundary condition

Gð r*;~rr 0 Þ ¼ 0 with z ¼ 0,

and is a continuous and limited function for all~rr „~rr 0 .Introducing a Fourier transform over the transverse coordinates x, y :

Gð~rr;~rr 0 Þ ¼ 1

4p2

R¥�¥

dpR¥�¥

dqG_

p; q; z; z0

� �exp jpðx � x

0 Þ þ jqðy� y0 Þ

� �and taking into account the definition of the d-function, we obtain

d2G_

dz2þ k

2 ðem ðzÞ � 1Þ � 2kp� q2

h iG_

¼ �4pdðz� z0 Þ. (2.36)

27


We will seek a solution to Eq. (2.36) as a composition over functions WE ðzÞ whichare governed by the equation

d2W

dz2þ k

2 ðem ðzÞ � 1Þ½ �W ¼ EW (2.37)

and satisfy the boundary conditions

WE ðzÞ ¼ 0; z ¼ 0; WE ðzÞ ¼ 0; z! ¥. (2.38)

Equation (2.37) is similar to the one-dimensional SchrJdinger equation for a par-ticle in a field with potential energy which is proportional to the term em ðzÞ � 1ð Þ.As known from Refs. [43, 44], in the case of unlimited potential (in Eq. (2.37)em ðzÞ � 1ð Þ ! 2 z=a with z! ¥ )), the motion of the particle in a stationary state is

infinite to z! þ¥ and the spectrum of the eigen values E is purely continuous.Therefore, the function G

_ðp; q; z; z0 Þ can be thought of as a composition of the

eigen functions of the continuous spectrum which obey Eq. (2.37) and the boundaryconditions (2.38):

G_ðp; q; z; z0 Þ ¼

R¥�¥

dEBðp; q;E; z0 ÞWE ðzÞ. (2.39)

The eigen functions of a continuous spectrum satisfy the orthogonality condi-tions

R¥0

WE ðzÞW�E1ðzÞdE ¼ dðE � E1Þ (2.40)

and “completeness”

R¥�¥

WE ðzÞW�E ðz

0 Þ ¼ dðz� z0 Þ. (2.41)

The conditions (2.40) and (2.41) are known from quantum mechanics [44] wherethey are proven for potentials limited at infinite z. Using methods similar to thosein Refs. [43, 44] one can prove Eqs. (2.40) and (2.41) for a potential with a linearincrement at infinity. The sign � means a complex conjugate, here and below.

Substituting Eq. (2.39) into Eq. (2.36) and taking into account Eqs. (2.40) and(2.41) we obtain

Bðp; q;E; z0 Þ ¼ � 4pW�E ðz

0 ÞE�2kp�q2

, (2.42)

and then performing the integration in Eq. (2.39) we obtain

Gð~rr;~rr 0 Þ ¼ 2pkðx � x

0 Þ

!1=2

exp jp

4þ j

k2y�y0� �2x�x0

( ) R¥�¥

dEWE ðzÞW�E ðz

0 ÞejE2kðx�x0 Þ

.(2.43)

In all practical problems related to radio wave propagation and scattering in thetroposphere, the non-uniformities of the refractive index are localised in the bound-

28


ary layer of the atmosphere and em ðzÞ � 1 can be approximated by a linear function2 z=a with z! ¥. Thus Eq. (2.36) has two solutions: v

þE ðzÞ and v

�E ðzÞ behaving

asymptotically as Airy functions w2E

l2 � lz

� �and w2ðE=l

2 � lzÞ, where l = k/m.

Following Ref. [1], we define the function w1(t) via a contour integral

w1ðtÞ ¼1ffiffiffip

pRC

dn exp � n3

3þ tn

!(2.44)

where contour C comes from infinity along the ray arg n ¼ �2p=3 to zero and thenfrom zero to infinity n ! þ¥ along the real axis n. The function w2ðtÞ is defined bya complex conjugate of Eq. (2.44).

The functions vþE ðzÞ and v

�E ðzÞ describe eigen waves coming to and from infinity.

The regular (with z = 0) solution for WE ðzÞis given by the composition

WE ðzÞ ¼ AðEÞ v�E ðzÞ � SðEÞ vþE ðzÞ

h i(2.45)

where AðEÞ ¼ 1=2ffiffiffiffiffiffipl

pis a coefficient defined from normalisation to the delta-func-

tion Eqs. (2.40) and (2.41). The term S(E) (normally called the S-matrix in quantummechanics) is determined by a kind of non-uniformity em ðzÞ � 1 and the boundarycondition at z = 0. In the case of an ideal boundary condition at z = 0, Eq. (2.38), thelaw of conservation for a number of particles holds true and SðEÞj j ¼ 1. Therefore,for real E, SðEÞ ¼ exp ð�2jdðEÞÞ, where dðEÞ is the real phase of scattering.

The S-matrix S(E) has poles En in the upper half-space of E and residues in Encompletely define the field in the shadow region. The value of WE ðzÞ in the poleE = En provides a normalised height function of normal wave with number n:

WEnðzÞ ¼ vEn ðzÞ ” vnðzÞ. (2.46)

The normal wave vnðzÞ with complex value of the propagation constantEn(Im En > 0) is unlimited at infinite z. This unlimited growth is a consequenceof the exponential decay of the field over the x-coordinate, since the field at z! ¥is created by radiation propagating from points distant to �¥ over the x-coordinate.

Let us calculate integral (2.43) for normal refraction when em ðzÞ ¼ 1 þ 2z=a atz > 0 and the eigenfunctions WE ðzÞof the continuous spectrum (2.45) are definedby

WE ðzÞ ¼1

2ffiffiffiffiffiffipl

p w2E

l2 � lz

� ��w2

E

l2

� �

w1

E

l2

� �w1E

l2 � lz

� �2664

3775. (2.47)

One can expect that, in a line-of-sight region, expression (2.43) shall represent a“reflection formula” [1] in accordance with geometric optics, i.e. represent the fieldas a sum of the direct wave and the wave reflected from the earth’s surface.

Consider ray equations corresponding to the waves radiated either in an upwarddirection from the source (z > z0) or downward from the source (z > z0), and definethe relation between the grazing angle of the ray and stationary value of E. We

29


define the hit-angle J to be a sliding angle of the ray relative to the surface z > z0 atthe point of the source location~rr0 ¼ 0; y0 ; z0f g.

The wave propagating in the direction up from the source can be expressed asfollows

Vþ~ e jxE=2kw1

E

l2 � lz

� �w2

E

l2 � lz0

� �. (2.48)

Using an asymptotic expression for the Airy function we obtain the trajectoryequation by defining the extremum of the phase in Eq. (2.48)

zþðxÞ ¼ x2l3

4k2þ xk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil

3z0 � E

qþ z0 . (2.49)

A similar equation can be obtained for the ray in the direction down from thesource

z�ðxÞ ¼ x2l3

4k2� xk


3z0 � E

qþ z0 . (2.50)

Differentiating Eqs. (2.49) and (2.50) we can find the equation for angle J and itsrelation to the stationary value of E:

J ¼ � a tan1k


3z0 � E

q� �, (2.51)

E ¼ l3z0 � k

2tan

2J. (2.52)

As observed from Eq. (2.51) the values of E satisfying the inequality E £ l3z0

determine the real hit-angles J, those, in turn, represent uniform plane waves withangle J=m between the normal to their wavefront and tangential to the radiusr = a + z0 at the point of source location. Trajectory (2.50) has a minimum in

xmin ¼ 2kl3


3z0 � E

q, and zminðxminÞ ¼

E

l3 . From that it follows that the values

E < 0 determine the waves reflected from the earth’s surface. The sector

�a tan� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kz0=m3

q �< J < a tan

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikz0=m

3q �

determines the “space” rays (directwave) related to a stationary value of E from the interval

0 < E < l3z0.

The ray trajectories described by Eqs. (2.49) and (2.50) are shown in Figure 2.1.Therefore, in the line-of-sight region, i.e. x <

ffiffiffiffiffi2a

p ffiffiffiz

pþ ffiffiffiffiffi

z0p� �

we can select twoareas of integration in Eq. (2.43):

I1 ¼R0�¥

dEWE ðzÞW�E ðz0Þe

jE2kx, (2.53)

I2 ¼R¥0

dEWE ðzÞW�E ðz0Þe

jE2kx. (2.54)

30


Let us substitute Eqs. (2.47)–(2.49) into Eq. (2.53) and assume z > z0 and

kz0/m >> 1. (2.55)

The integration in Eq. (2.53) can be performed using the method of stationaryphase similar to the approach in Ref. [1]. The result is given by

I1 ¼ hðffiffiffiffiffi2a

p ffiffiffiz

p� ffiffiffiffiffi

z0p� �

� xÞI11 þ I12 , (2.56)

I11 ¼ 2lA2ffiffiffiffiffiffiffiffi2kpx

rexp �jp

4þ j k

2xz� z0ð Þ2þj kx

2azþ z0ð Þ � j 1

12mxa

� �3� �;

I12 ¼ffiffiffiffika

r ffiffiffiffiffiffiffi�n

p

4ffiffiffip

p 1ffiffiffiffiffiffiffizz04p ·

exp j km

� �3=2z3=2 þ z

3=20

� �þ j kx

2az þ z0ð Þ � j 1

12mna

� �3� �(2.57)

where n ¼ x �ffiffiffiffiffi2a

p ffiffiffiz

pþ ffiffiffiffiffi

z0p� �

.Consider integral (2.54) and make use of the asymptotic expression for Airy func-

tions w1;2 ðE=l2Þ for large positive arguments:

w1E

l2

� �» w2

E

l2

� �»

E

l2

� �1=2

exp2

3l3 E3=2

� �. (2.58)

Substituting Eq. (2.57) into Eq. (2.47) we obtain S(E) = 1. Let us introduce theAiry function vðtÞ by the relationship

2jtðtÞ ¼ w1ðtÞ � w2ðtÞ (2.59)

and the variable s ¼ E=l2�lz0 . The integral (2.54) can then be transformed into

31

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25 30 35

Distance, km

He

igh

t, m

minz

minx

Figure 2.1 Ray trajectories in the case of normal refraction.


I2»14pejx2k

l3z0 R¥

�lz0

ejxs2k

l2

t sð Þt s� l z� z0ð Þð Þds. (2.60)

Without significant error the low limit in the integral (2.60) can be extended to�¥. Substituting into Eq. (2.60) the integral representation for t tð Þ

t tð Þ ¼ 12ffiffiffip

pR¥�¥

exp jn3

3þ jnt

!dn (2.61)

and performing the integration, we obtain

I2»ffiffiffiffiffiffiffiffik

2px

rexp � j p

4þ j

k2x

z� z0ð Þ2� j kx2a

zþ z0ð Þ � jmxa

� �3� �. (2.62)

Equation (2.62) describes the direct wave in the range of distancesffiffiffiffiffi2a

p ffiffiffiz


z0p� �

< x <ffiffiffiffiffi2a

p ffiffiffiz

pþ ffiffiffiffiffi

z0p� �

and transforms into Eq. (2.55) withx ¼

ffiffiffiffiffi2a

p ffiffiffiz


z0p� �

. Therefore in a whole line-of-sight region the Green function(2.43) can be represented by superposition of the direct wave and the wave reflectedfrom the earth’s surface.

Let us consider the Green function’s representation (2.43) in a shadow region, i.e.x >

ffiffiffiffiffi2a

p ffiffiffiz

pþ ffiffiffiffiffi

z0p� �

. In this case the contour of integration over dE can be closedin the upper half-plane and the integral is represented by a residue sum over thepoles of S(E). In the particular case of normal refraction and ideal boundary condi-tions, the poles are given by

w1E

l2

� �¼ 0 (2.63)

and En ¼ k2

m2 snejp=3

, s1 ¼ 2:338; s2 ¼ 4:02; :::

As a result, we obtain the Green function representation in terms of the normalwave series

G ~rr;~rr0

� �¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p

k x � x0

� �sexp jp

4þ j k

2

y� y0

� �2

x � x0

8><>:

9>=>;·

X¥n¼1

exp jEn2k

x � x0

� �� w1E

l2 � lz

� �

w01

E

l2

� � w1E

l2 � lz0

� �

w01

E

l2

� � :

(2.64)

In conclusion, the Green function has been built as an expansion over the set ofcomplete and orthogonal eigenfunctions of the continuous spectrum. In the case ofnormal refraction, such a representation produces final formulas for the field of thepoint source that are similar to those introduced by Fock [1] on the basis of the clas-sical approach of the contour integral.

It is worthwhile to emphasise the purpose of the above presentation for a Greenfunction. In a problem of multiple scattering the whole path is involved in the scat-tering process and one needs to have a representation for the field (or Green func-tion) which is equally applicable in the line-of-sight and shadow regions. Also, in the

32

2.4 Feynman Path Integrals in the Problems of Wave Propagation in Random Media

process of obtaining the closed set of equations for statistical moments of the field,the Green function has to possess some properties of orthogonality and complete-ness which can be achieved by using the above expansion over the set of eigenfunc-tions of the continuous spectrum.

2.4Feynman Path Integrals in the Problems of Wave Propagation in Random Media

In this section we introduce an alternative representation for a Green functionbased on Feynman path integrals. This approach is most suitable for the analysis ofthe wave propagation and scattering in a line-of-sight region. This section providessome basics for study in an unbounded medium, while later in Section 3 we intro-duce a path integral representation in the presence of a boundary.

The basic Feynman postulate [45] states that the probability amplitude for thetransition of a particle-wave from point~rr0 to point~rr can be represented by contribu-tions made by individual trajectories over which the particle-wave can propagate be-tween points ~rr0 and ~rr. The contribution of each trajectory is proportional toexp jS=�hgf , where S is a classical action and �h is Planck’s constant. As is well known,the quantum-mechanical probability amplitude of transition is equivalent to theGreen function G ~rr;~rr0ð Þ of the Helmholtz equation written in a parabolic approxima-tion, after the time has been replaced by the x-coordinate along the selected direc-tion of propagation and m=�h has been replaced by the wavenumber k = 2p/k, wherem is a particle’s mass, k is the wave number of the admitted wave, and k is the wave-length. Turning to the trajectory continuum, we can write

G ~rr;~rr0ð Þ ¼RD~�� xð Þ exp jkS ~�� xð Þ½ �f g, (2.65)

where ~rr ¼ x;~��f g, ~rr0 ¼ 0;~��0f g are the coordinates of the observation point andsource, respectively, ~�� ¼ y; zf g, ~�� ¼ y0 ; z0f g are the vectors in the plane orthogonalto the direction of propagation x, and D~�� xð Þ is the differential in the space of contin-uous trajectories. The action S is given by

S ~�� xð Þ½ � ¼Rx0

dx0L0 x

0;~�� x

0� ��

þ 12de x

0;~�� x

0� �� + ,

, (2.66)

where L0 x;~�� xð Þð Þ ¼ 12

d�dx

� �2

þU0 x;~�� xð Þð Þ is an unperturbed (when de ¼ 0) Lagran-

gian in the small-angle approximation, U0 x;~�� xð Þð Þ is an unperturbed potential,U0 x;~�� xð Þð Þ ¼ 1=2 e x;~�� xð Þð Þ � 1h i, the angle brackets :::h i here and later denote aver-aging over the ensemble of dielectric permittivity, e x;~��ð Þ, de x;~��ð Þ is a random com-ponent of e x;~��ð Þ, deh i ¼ 0.

Consider calculation of the first two moments of the Green function (2.65) in thecase of an unbounded stationary medium filled with random inhomogeneities ofdielectric permeability eðx;~��Þ. In continuous notation the average Green function

33


G ~rr;~rr0ð Þh i and coherence function C x;~��1 ;~��2 ;~��00 ;~��

000

� �¼ G ~rr1 ;~rr

00

� �G

�~rr2 ;~rr

000

� �D Ewe obtain:

G ~rr;~rr0ð Þh i ¼RD~�� xð Þ exp jkS0 ~�� xð Þ½ �f g exp j

k2

Rx0

dx0de x

0;~�� x

0� �� !* +

, (2.67)

C x;~��1 ;~��2 ;~��00 ;~��

000

� �¼ G ~rr1 ;~rr

00

� �G

�~rr2 ;~rr

000

� �D E¼

ZD~��ðxÞ

ZD~RRðxÞ exp jk

Zx0

dx0 d~RRdx

0d~��

dx0

8><>:

9>=>;·

exp j k2

Zx0

dx0de x

0;~RR x

0� �

þ~�� x

0� �2

0@

1A� de x

0;~RR x

0� �

�~�� x

0� �2

0@

1A

24

35

8><>:

9>=>;

* +:

(2.68)

In Eq. (2.68) we have introduced the sum ~RRðxÞ ¼ 1=2 ~��1 xð Þ þ~��2 xð Þð Þ and the dif-ference ~��ðxÞ ¼ ~��1ðxÞ �~��2ðxÞ of the trajectories with the following boundary condi-tions

~RRðx0 ¼ 0Þ ¼ 12

~��00 þ~��

000

� �~��ðx0 ¼ 0Þ ¼ ~��

00 �~��

000 ,

~RRðx0 ¼ xÞ ¼ 12

~��1 þ~��2ð Þ~��ðx0 ¼ xÞ ¼ ~��1 �~��2 .

We can assume that the fluctuations in de are statistically uniformde x1 ;~��1ð Þde x2 ;~��2ð Þh i ¼ Be x1 � x2 ;~��1 �~��2ð Þ and have different correlation scales:

Lx in direction x and L? in the plane (y,z). If x >> L the integralRx0

dx0de x

0;~�� x

0� ��

represents a Gaussian random value in accordance with the central-limit theorem.Hence

exp jk2

Rx0

dx0de x

0;~�� x

0� �� !* +

= exp �cs� �

, (2.69)

exp j k2

Zx0

dx0de x

0;~RR x

0� �

þ~�� x

0� �2

0@

1A� de x

0;~RR x

0� �

�~�� x

0� �2

0@

1A

24

35

8><>:

9>=>;

* +¼

exp �Dsð Þ;(2.70)

where

cS ¼ k2

8

Rx0

dx0 Rx0

dx00

de x0;~�� x

0� ��

de x00;~�� x

00� �� D E

(2.71)

34


is the variance of the phase fluctuations along the trajectory ~��ðxÞ and

DS ¼ k2

8

Rx0

dx0 Rx0

dx00

de x0 ;~RR x0� �

þ~�� x0ð Þ

2

� ��de x0 ;~RR x0

� ��~�� x

0ð Þ2

� �+ ,�

de x00 ;~RR x00� �

þ~�� x00ð Þ

2

� ��de x00 ;~RR x00

� ��~�� x

00ð Þ2

� �+ ,* +

(2.72)

is a structure function of the phase difference along the trajectories ~RR xð Þ þ~�� xð Þ2

and~RR xð Þ �~�� xð Þ

2.

Introduce a two-dimensional spatial spectrum of the fluctuations de in a plane(y, z):

Fe x0 � x

00;~kk?

� �¼ 1

4p2

Rd2~��Be x

0 � x00;~��

� �e�j~kk~��

. (2.73)

Then for cs and Ds we obtain

cS ¼ k2

4

Rx=20

dgR2g

�2g

dnRd2~kk?Fe n;~kk?ð Þ exp j~kkn

d~��dg

� �, (2.74)

DS~RR;~��h i

¼ k2

4

Rx=20

dgR2g

�2g

dnRd2~kk?Fe n;~kk?ð Þ exp j~kkn

d~RRdg

!·

cos ~kknd~��dg

� �� cos ~kk~��ð Þ

+ ,. (2.75)

To analyse Eqs. (2.74) and (2.75) let us introduce the parameter hR ¼ maxd~RRdx

" #,

the characteristic angle of the ray’s trajectory along the x-axis. The value of hR is con-tributed to by the three characteristic parameters: hd , the gradient of the ray trajec-tory due to either the position of the correspondent or regular refraction (if any), incase of normal refraction hd is given by Eq. (2.51); hF ¼ 1=

ffiffiffiffiffikx

p, the angular size of

the Fresnel zone; and hs , the characteristic angle of scattering on the fluctuationsde, hs~ 1=kL? . Therefore hR ¼ max hd ; hF ; hs

� �.

Let us also introduce a parameter of anisotropy a in the fluctuations of de:a ¼ L?=Lx The effective width of function Fe n;~kk?ð Þ over n does not exceed a corre-lation scale Lx . When inequality

hR << a (2.76)

holds, we can assume that ~kk?nd~��=dg;~kk?nd~RR=dg << 1 in a significant region over nin Eqs. (2.74) and (2.75). As a result, we obtain the following approximations:

cSðxÞ ¼pk2x

4

Rd2k?Ue 0; k?ð Þ; (2.77)

and

35


DS~RR;~��h i

”DS ~��½ � ¼ pk2

2

Rx0

dx0 Rd2k?Ue 0; k?ð Þ 1 � cos k? �

*ðx0 Þ� �h i

(2.78)

where

2pUe 0;~kk?ð Þ ¼R¥�¥

dnFeðn;~kk?Þ (2.79)

and Ue ~kkð Þ is a three-dimensional spatial spectrum of fluctuations in de. We canreasonably assume that hR~ hs , and the inequality (2.76) takes the form

kL2?

Lx>> 1 (2.80)

which means that the longitudinal correlation scale has to be small compared withthe distance kL

2? where the diffraction on the irregularities of the scale L? become

significant. The expressions (2.69) and (2.78) provide the fundamental solution tothe equation of the coherence function C in the Markov approximation:

@C@x

� jk@

2C

@~RR@~�� pk2

4H ~��ð ÞC ¼ 0, (2.81)

H ~��ð Þ ¼ 2Rd2k?Ue 0; k?ð Þ 1 � cos k? �

*ðx0 Þ� �h i

and DS ~��½ � ¼ pk2=4Rx0

dx0H ~�� x

0� ��

. “Local” conditions of the Markov approximation

are boundedby inequalities (2.67) and (2.77), for an average fieldwe also need smallnessof the attenuation of the average field over the distance of Lx : cSðLx Þ << 1. “Non-local“ conditions were obtained in Ref. [21] and lead to the inequality

Rc >> k, (2.82)

where Rc is a coherence radius of the scattered field in a plane (y, z), which dependson the distance x, Rc ”Rc ðxÞ. The parameter Rc can be found from

Ds(Rc) = 1. (2.83)

To illustrate the path integral approach, we obtain a known solution to Eq. (2.81)for the unbounded medium filled with statistically uniform fluctuations of deðx;~��Þ:Using Eqs. (2.68), (2.80) and (2.78) we obtain

C x;~��1 ;~��2 ;~��00 ;~��

000

� �¼RD~��ðxÞ

RD~RRðxÞ · exp jk

Rx0

dx0 d~RRdx0

d~��dx0

�DS ~�� x0

� �h i( ).

(2.84)

The integral over trajectories R(x¢) represents a continuous Fourier transform ofthe delta-function dðd2~��=dx2Þ [45]. A Lagrangian in the exponent can be expandedinto a functional series up to the second order terms in the vicinity of the trajectory

36


~��0 x0

� �¼ ~��x

0=x þ~��0 1 � x

0=x

� �. Integrating then over ~�� xð Þ, we obtain the well-

known solution for an unbounded medium:

C x;~��1 ;~��2 ;~��00 ;~��

000

� �¼ k2

4p2x2 expjk

~RR�~RR0ð Þ ~��~��0ð Þx

�

pk2

4

Rx0

dx0H ~��x0

xþ~��0 1� x

0

x

� �� 8>>><>>>:

9>>>=>>>;: (2.85)

The relationship between the average of the path integrals (2.67), (2.68) and theaverage over the Fermat paths was analysed in Refs. [34,36]. The asymptotic evalua-tion of the path integral representation for the Green function (2.67) may be per-formed using the method of stationary phase. The asymptotic method consists offinding the extreme path ~��

�ðxÞ which renders the minimum value of the phaseS ~��ðxÞ½ � given by Eq. (2.66). It can be noted that in obtaining the solution for thecoherence function in an unbounded medium (2.85) we actually did not face anydifficulties with integration over R(x) due to an isotropy of the medium,e ~rrð Þh i ¼ const: However, this is not the case for more complex media. The varia-

tional problem for determining ~�� ðxÞ gives rise to the Euler equation for the Fermat

paths

d2~��

dx2 þrdeðx;~��ðxÞÞ ¼ 0, (2.86)

where r ¼ @

@y;@

@z

� �.

The rigorous mathematical solution to the problem encountered difficulties in ageneral case of the multi-scale inhomogeneities of de [34] and, therefore, the basicphysics has to be involved for a qualitative analysis.

As mentioned above, there are two characteristic transverse scales of the phasefluctuations in S ~��½ �. The first is the Fresnel zone size resulting from the first term inEq. (2.67). The second term is due to a random component of the dielectric permit-tivity de and is of the order of the coherence scale Rc introduced by Eq. (2.83).

We assume a small value of the mean-square variation of the fluctuations in aphase difference of the fields approaching any points separated in space by a dis-tance of the order of the Fresnel zone size. Hence, this assumption is rendered bythe inequality

dS1 ¼ k2p4

Rx0

dx0H

ffiffiffiffiffix0

k

r !<< 1. (2.87)

As known [21], the meaning of this inequality is that the Fresnel zone size is thecharacteristic region in the plane x

0 ¼ constant, from which the rays arrive at thereceiving point in phase, even in the presence of the de fluctuations. It also meansthat the integration in Eq. (2.67) can be fulfilled along a single canal-ray tubebounded in lateral cross-section by the Fresnel volume. We assume the extreme tra-jectory ~��

�ðxÞ to deviate slightly from the unperturbed path ~��0ðxÞ, defined by theunperturbed Lagrangian L0 ~�� xð Þ½ �, therefore, we should require the fluctuations inthe arrival angle of the wave defined by ~��

�ðxÞ to be small compared with angular

37


size of the Fresnel zone ~ kxð Þ�1=2. In a functional representation this requirement

is equivalent to

dS2 ¼ k2p8

Rx0

dx0d~��

2x0

� �r2H ~�� x

0� ��

£ k2p8x

2 Rd2~kk?Ue 0;~kk?ð Þ << 1. (2.88)

In Eq. (2.71) we evaluated the deviation d~�� ¼ ~�� ~��0 to be of the order of the

Fresnel zone size.Having satisfied inequalities (2.87) and (2.88), the integration in Eq. (2.67) can be

performed along the unperturbed ray trajectories ~��0ðxÞ which are the Fermat pathsfor a free wave particle propagation (in the particular case of an unbounded medi-um):

~��0ðx0 Þ ¼ ~��~��0ð Þ x

0

x. (2.89)

Such a straight-line approximation of trajectories in a path integral is similar tothe extended Huygens–Fresnel principle introduced in Ref. [46], and used in Ref.[47]. The applicability of extended Huygens–Fresnel principle was analysed in [35],where shown that this approximation besides yielding the exact solution for the firsttwo moments, provides a qualitatively correct solution for the high-order momentsof the wave field.

2.5Numerical Methods of Parabolic Equations

Among many numerical methods used in the problems of applied electromagneticsand wave propagation one may distinguish two methods most widely used in aVHF/UHF propagation in the atmospheric boundary layer, namely: split-step-Fouri-er and split-step PadP methods. Both methods are based on the parabolic approxi-mation to a wave equation and differ in the method of obtaining the approximationof the exponential operator. Initially, the split-step parabolic approximation wasintroduced in a problem of applied acoustics [48] where the wave propagation can bedescribed by a scalar wave equation of elliptical type

@2u

@x2 þ@

2u

@z2 þ k2n

2u ¼ 0 (2.90)

where u is the spectral amplitude of the wave component with frequency x, k ¼ x=cis the wavenumber, c is the group velocity of the propagation and n is the refractiveindex of the medium and may vary with the coordinates, we assume, for now, a con-stant value of n, n ” 1. Equation (2.90) is written for a two-dimensional case of thepropagation since this case is the one most widely considered in applications of thesplit-step approximation.

38

2.5 Numerical Methods of Parabolic Equations

Equation (2.90) can be factorised as follows

@

@xþ j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ @

2

@z2

s24

35 @

@x� j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ @

2

@z2

s24

35u ¼ 0, (2.91)

representing the forward and backward propagating waves respectively. We may alsoremove fast phase variations with distance by introducing the slow-varying ampli-tude ~uu, u ¼ ~uu exp ðjkxÞ. The truncated and factorised equation for amplitude ~uu takesthe form

@

@xþ jk 1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ 1

k2@

2

@z2

s0@

1A

24

35 @

@xþ jk 1 �


k2@

2

@z2

s0@

1A

24

35~uu ¼ 0. (2.92)

By retaining the only forward propagating wave we obtain the forward parabolicequation

@

@xþ jk 1 �


k2@

2

@z2

s0@

1A

24

35~uu ¼ 0 (2.93)

which has a formal solution

~uuðx þ DxÞ ¼ exp jkDx


k2@

2

@z2

s� 1

24

35

8<:

9=;~uuðxÞ (2.94)

where Dx is a range increment. As apparent from Eq. (2.94) the forward propagatingwave at range x +Dx can be obtained from values of the wave amplitude at the pre-vious distance x by applying the exponential operator in Eq. (2.94). In order to actu-ally calculate the wave field, the exponential operator in Eq. (2.94) should be approxi-mated in a form suitable for computations. Let us introduce a notationZ ¼ 1=k

2@

2=@z

2and, depending on the type of approximation, we obtain three

known types of the split-step approximation to parabolic equations:

(a) The standard or narrow angle approximation, obtained by expanding thesquare root in the exponent into a Taylor series and retaining the linear term:

ffiffiffiffiffiffiffiffiffiffiffiffi1 þ Z

p¼ 1 þ Z

2. (2.95)

This approximation is used in a split-step Fourier method to be discussedlater, and is normally valid for narrow angles of propagation relative to axis x,not exceeding 15Q–20Q.

(b) The Claerbout approximation in the form

ffiffiffiffiffiffiffiffiffiffiffiffi1 þ Z

p»

1þ 3

4Z

1þ Z

4

(2.96)

39


which is reported to be valid for wider angles up to 30–40Q [49, 50]. Andfinally:

(c) The split-step PadP approximation, when exponential is expanded into series

exp jkDxffiffiffiffiffiffiffiffiffiffiffiffi1 þ Z

p� 1

4 5» 1 þ

PMm¼1

amZ1þ bmZ

(2.97)

where the coefficients am, bm are determined numerically in the complexplane using the approach developed in Ref. [49]. The split-step PadP approx-imation is reported [50, 51] to be valid for angles of propagation relative tothe x-axis of up to 90Q.

All the above methods have been realized in a very powerful computational tech-nique well suited for parallel computing and widely used in numerous applications.With regard to the problem of radio wave propagation in the earth’s troposphere themost established computational approach is based on the split-step Fourier methodequivalent to a standard narrow angle approximation of the square root in the expo-nential operator. Whilst the wide angle approximations, such as Claerbout and PadPapproximations have been used in the problems of scattering on objects submergedin a free-space [50, 52], the author does not know of any systematic derivation of thePadP approximation in the case of radio wave propagation in a stratified troposphereover terrain or the sea boundary surface. We may notice that while the fundamentalelliptic equation of type (2.90) can be obtained for potentials (Hertz functions) underfree-space propagation conditions, the equations for the Debye’s potentials take adifferent form in the case of a stratified troposphere [1]. In the presence of a randomcomponent of the refractive index the main equation for the slow varying envelopeis given by Eq. (2.26).

Consider Eq. (2.26) and retain the term1

r2@

2W1

@h2 on the right-hand side of the

equation. While this is not very consistent with the arguments used in Section 2.1,we retain this term in order to get in line with the approach of other authors toobtaining the parabolic equation. We may notice that to strictly follow the procedurewe have to leave the other terms in the right-hand side of Eq. (2.26) in order to beconsistent in the accuracy of the approximation. Introducing the same coordinatesx ¼ ah, z ¼ r � a and y ¼ u a sinJ we obtain

@2W1

@x2 þ j2k@W1

@xþ D?W1 þ k

2em ðzÞ � 1ð Þ þ de ~rrð Þ½ �W1 ¼ 0. (2.98)

Then we may introduce the operator

L ¼ 1

k2D? þ k

2em ðzÞ � 1ð Þ þ de ~rrð Þ½ �

h i(2.99)

and factorize Eq. (2.98) in a form similar to Eq. (2.90),

@

@xþ jk 1 þ

ffiffiffiffiffiffiffiffiffiffiffi1 þ L

p� �+ , @

@xþ jk 1 �


p� �+ ,W1 ¼ 0:

40


Then, following the above described procedure, we may retain the forward propa-gating wave obeying the truncated equation

@W1

@x¼ jk


p� 1

� �W1 (2.100)

with formal solution at the marching step in the form

W1ðx þ DxÞ ¼ W1ðxÞexp jkDxffiffiffiffiffiffiffiffiffiffiffi1 þ L

p� 1

� �4 5. (2.101)

The exponential can then be expanded into a series similar to Eq. (2.97) and thenext range step solution is given by

W1 x þ Dx; y; zð Þ ¼ W1ðxÞ þPMm¼1

amL1þbmL

W1ðx; y; zÞ. (2.102)

Following the approach in Ref. [50], we may introduce the auxiliary function

fm ðx þ Dx; y; zÞ ¼ amL1þbmL

W1 x; y; zð Þ (2.103)

which can be found by multiplying both parts of Eq. (2.103) by 1 þ bmL and solvinga second-order differential equation for fm. Substituting fm into Eq. (2.102) we have

W1 x þ Dx; y; zð Þ ¼ W1ðxÞ þPMm¼1

fm ðx þ Dx; y; zÞ. (2.103)

It is important to notice that all the functions fm can be solved independently andin parallel at each step along the distance x. The marching equation (2.103), as wellas the forward propagating wave equation (2.93), shall be appended by both bound-ary conditions (at z = 0 and ~rrj j ! ¥) and initial conditions at x = 0. A comprehen-sive treatment of the PadP series can be found in Ref. [51].

A somewhat modified approach to the implementation of the PadP method is tosubstitute a PadP series with a PadP product

PMm¼1

amL1þbmL

¼QMm¼1

1þkmL1þlmL

. (2.104)

This approach has been developed in Refs. [48, 53] and was shown to be well suit-ed for finite-difference computational schemes.

Now, we concentrate on the split-step Fourier method. Consider Eq. (2.26) andassume the presence of deterministic and stratified refractivity in the medium, therandom component de is set to 0. Assume first that the propagation takes place inan unbounded medium, i.e., we have only requirements on the appropriate decay ofthe field at infinity. The next assumption normally employed in the split-step meth-od applied to radio wave propagation in the troposphere is to assume the medium tobe stratified over the z-coordinate, i.e. a stratified troposphere, thus removing thedependence on the y-coordinate and truncating Eq. (2.26) to a two-dimensional (x, z)parabolic equation.

41


Equation (2.26) in this case can be presented in the form

@W@x

¼ jk2

em ðzÞ � 1ð Þ þ 12k

@2

@z2

" #W . (2.105)

In a vertically stratified medium the formal solution to Eq. (2.105) is then givenby

Wðx þ DxÞ ¼ exp jDxk2

em ðzÞ � 1ð Þ þ 12k

@2

@z2

" #( )WðxÞ (2.106)

where Dx is a range increment. Equation (2.106) forms the basis for a split-stepFourier implementation.

We follow the basics of the split-step Fourier implementation reported in Ref.[54]. Let us define a Fourier transform of the envelopeW(x, z) as follows:

~WWðx; pÞ ¼ =Wðx; zÞ½ � ¼R¥�¥

dzWðx; zÞexp �jpzð Þ. (2.107)

Apparently p is a vertical component of the wave vector of the incident fieldW(x, z).

The fundamental assumption in a split-step solution to Eq. (2.106) is that aFourier transform of the envelope W(x, z) is performed while treating the termk2

em ðzÞ � 1ð Þ as a constant, i.e., no dependence on the z-coordinate. With that

assumption, we obtain a marching solution

~WWðx þ Dx; pÞ ¼ ~WWðx; pÞexp � k2em ðzÞ � 1ð Þ � p

2� �Dx

2jk

+ ,. (2.108)

The inverse Fourier transform is given by

Wðx þ Dx; zÞ ¼ exp jk2

em ðzÞ � 1ð ÞDx+ ,

=�1 ~WW x; pð Þexp �jp2 Dx2k

� �� . (2.109)

In the above equation the termk2

em ðzÞ � 1ð Þ is no longer a constant, its variations

actually results in variation of the angular spectrum p of the marching fieldW(x + Dx, z) with the distance.

The boundary condition in the case of ideal reflection (a perfectly conductingboundary surface) can be realized by adding the mirror image of the field W(x, z) inthe upper half-space (z > 0) into the lower half space (z < 0) with an appropriatephase to reproduce the odd or even image of the field W relative to z = 0. The oddcomposition satisfies a condition W(x, z = 0) = 0 for a vertically polarized field,while the even composition results in the boundary condition @W=@zjz¼0 for hori-zontal polarization. In these cases of perfectly conducting boundary the Fouriertransform reduces to a one-sided sine or cosine transform, respectively

~WWoðx; pÞ ¼ �2j=Woðx; zÞ½ � ¼ �2jR¥0

dzWoðx; zÞsin pzð Þ,

42


~WWeðx; pÞ ¼ 2=Weðx; zÞ½ � ¼ 2R¥0

dzWeðx; zÞcos pzð Þ (2.110)

where the subscripts e and o indicate that the field is even or odd, which, in turn,corresponds to horizontal or vertical polarization, respectively.

The impedance boundary condition (2.27) was also treated in Ref. [54] where itwas shown that such a boundary condition can be realized using a mixed Fouriertransform

~WWqðx; pÞ ¼R¥0

dzWðx; zÞ q sin ðpzÞ � p cos ðpzÞ½ �

where q ¼ jmffiffiffiffiffiffiffiffiffigþ1

p (see Section 2.2). The marching solution for W(x, z) is then givenby [54]:

Wðx þ Dx; zÞ ¼ exp jk2

em ðzÞ � 1ð ÞDx+ ,

·

exp jq2 Dx

2k� qz

� �KðxÞ þ 2

p·

R¥0

dpq sinðpzÞ�p cosðpzÞ

q2þp2expð�j p

2

2kDxÞ ·

R¥0

dz0Wðx; z0 Þ q sinðpz0 Þ � p cosðpz0 Þ

� �

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

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

where the term K(x) is defined as [54]:

KðxÞ ¼ 2qR¥0

dzWðx; zÞexp �qzð Þ; ReðqÞ > 0, (2.112)

KðxÞ ¼ 0; ReðqÞ < 0.

It is apparent that the implementation of the split-step Fourier algorithm employ-ing the formulas (2.111) and (2.112) will be more complicated than the rather sim-ple implementation for a perfectly conducting interface using formula (2.110). Asshown in Ref. [54] and many other publications (we may refer to a practical imple-mentation of the numerical methods in radio coverage prediction systems [39, 55,56]), the value of the implementation of the impedance boundary conditionsbecomes pronounced at vertical polarization at smaller distances and at lower VHFfrequencies for long-range propagation, where the impact of the surface wave pro-duced by the distributed images of the source provides a substantial contribution tothe received field strength.

The important development in an implementation of the split step Fourier trans-form method is the capability to treat an irregular terrain profile, as reported in nu-merous publications (see, e.g. Refs. [55, 57–59]. The approach was initially devel-oped in underwater acoustics [60] and then implemented in the studies of radiopropagation in the troposphere [55, 57].

43


Following Ref. [55], let us consider the field of horizontal polarization over an ir-regular terrain with the terrain profile given by the function of the distance T(x).(We may note that with qj j >> 1 the boundary condition is Wðx; z ¼ 0Þ ¼ 0for both polarizations.) The range dependent boundary condition is then given byW(x, z = T(x)) = 0. The approach is then to map the irregular terrain profile to asmooth or more flat one with consequent modifications of the wave equation. Themapping is made by introducing a change of variables. Let the new height and rangevariables be represented by

v ¼ x; y ¼ z� TðxÞ. (2.113)

In the new coordinate system the slow varying envelope W(x, z) can be written inthe form

Wðx; zÞ ¼ wðv; yÞexp jhðv; yð Þ (2.114)

where hðv; yÞ is a phase correction term due to the irregular terrain. Substituting thenew variables into Eq. (2.26) and following the procedure described in Ref. [55] weend up with the following equation for a new envelope

@2w

@y2 þ j2k@w@v

þ k2

em ðy þ TðvÞÞ � 1ð Þ þ 2yT00vð Þ

h iw ¼ 0 (2.115)

and

h v; yð Þ ¼ �kyT 0vð Þ � k

3=2 Rv0

T0að Þ

h i2da. (2.116)

As observed, the implementation of the split-step Fourier transform can be donein a way similar to the case of a smooth terrain, the difference from Eq. (2.26) is thepresence of the second derivative of the boundary interface, 2yT

00 ðvÞ, in the expo-nential operator. Nonetheless, given that the split-step Fourier transform algorithmtreats this term as a constant, the addition of the second derivative does not make adifference in the implementation though ensuring the important correction of thewave front due to scattering on the irregular terrain. Specific issues in computer-based implementation of the split-step Fourier algorithm are discussed in Refs.[55, 56].

We may notice apparent similarities between the split-step Fourier method andthe discrete implementation of the path integral method [36, 45]. In the case of prop-agation over the boundary interface, considered in Section 3, implementation of theboundary conditions into the path integral approach will result in the appearance ofthe mirrored source, similar to the above formulas (2.110) and (2.111). In fact,numerical calculation of the discrete version of the path integral is practically rea-lised by applying the fast-fourier transform algorithm to the exponential propagatorat given step along the range x.

Finally, we can state that the above numerical methods constitute a powerfulframework for quantitative analysis of radiowave propagation through the tropo-

44

2.6 Basics of Fock’s Theory

spheric boundary layer in the very general case of the refractivity condition. In par-ticular, the methods described above can incorporate either non-uniformity of therefractivity profiles in the horizontal plane as well as in an irregular terrain. The lim-itation of the above methods, as in fact of all numerical methods, is that while theyare quite helpful in a quantitative estimation of the propagation phenomena whenthe physical mechanism is clearly understood they lack the capability to provide aframework for qualitative analysis, especially in the case of combined effects ofrefraction and scattering on random irregularities of refractive index or a randomlyrough sea surface.

2.6Basics of Fock’s Theory

In this section we briefly reproduce the basic results obtained by Fock in Chapter 12of Ref. [1] for the vertical dipole in the case of normal refraction, in order to have areference model for the further studies described in this book.

Following Fock [1], we introduce a non-dimensional distance and height n; h,respectively:

n ¼ mxa

; h ¼ kzm

(2.117)

where x ¼ ah and z = r – a , the same physical coordinates as defined in Section2.2. The relation between the Debye’s potential U and the slowly varying attenuationfactor (envelope) V is given by

U ¼ exp ðjkahÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2hsin h

p Vðn; h1 ; h2 ; qÞ (2.118)

where h1and h2 are the non-dimensional heights of the receiver and transmitter,respectively, above the earth’s surface, n is the distance between them along theearth’s surface. A normalised impedance q is defined by

q ¼ jmffiffiffiffiffiffiffiffiffigþ1

p . (2.119)

The attenuation factor V is given by a contour integral in a plane of complex para-meter t:

Vðn; h1 ; h2 ; qÞ ¼ exp �jp4

� � ffiffiffinp

r RC

dt exp jntð ÞF t; h1 ; h2 ; qð Þ. (2.120)

The contour C comes from infinity in the second quadrant of the t-plane passingaround and below all the poles of the integrand and then goes to infinity in the firstquadrant of the t-plane. In the case of h2 > h1, the function F is given by

F t; h1 ; h2 ; qð Þ ¼ j2w1ðt� h2Þ w2ðt� h1Þ �

w02ðtÞ�qw2ðtÞ

w01ðtÞ�qw1ðtÞ

w1ðt� h1Þ" #

. (2.121)

45


In the case h1 > h2 we need to swap the h1 and h2 in the expression for F above.As shown in Ref. [1], three characteristic regions can be selected in the space ofn; h1 ; h2 or, over the distance n, given fixed values of h1, h2. The first region is aline-of-sight region with n <

ffiffiffiffiffih1

pþ

ffiffiffiffiffih2

p, where the field is composed by superposi-

tion of the direct wave and the wave reflected from the earth’s surface. The secondregion is a shadow region at n >

ffiffiffiffiffih1

pþ

ffiffiffiffiffih2

p, where the electromagnetic field prop-

agates only due to a diffraction mechanism and, finally, the third region is a transi-tion region with n»

ffiffiffiffiffih1

pþ

ffiffiffiffiffih2

p, which separates the line-of-sight region from the

shadow region. These regions are shown schematically in Figure 2.2. The transitionregion, also called a “shade cone”, is filled with dots in the figure. Fock obtained anasymptotical expansion of the integral (2.120) for all three regions [1]. We reproducehis results for both the line-of-sight and the shadow regions.

In the shadow region the function F can be written in the following form:

F t; h1 ; h2 ; qð Þ ¼ w1ðt� h2Þf ðh1 ;tÞ

w01ðtÞ�qw1ðtÞ

(2.122)

where

f ðh1 ; tÞ ¼ w01ðtÞ � qw1ðtÞ

h ivðt� h1Þ � v

0 ðtÞ � qvðtÞh i

w1ðt� h1Þ. (2.123)

With h1 = 0 the function f and its derivative have the values

f ð0; tÞ ¼ 0, df =dh1 ¼ �q. (2.124)

The above equations can be obtained using the representation the Airy functionw1 via the Airy functions u and v , see Appendix 1. From Eq. (2.124) one can findthat if t ¼ tn is a root of the equation

w01ðtÞ � qw1ðtÞ ¼ 0, t ¼ t1 ; t2 ; ::: (2.125)

the value of the function f is given by

f ðh1 ; tsÞ ¼ fsðh1Þ ¼w1 ts�h1

� �w1 tsð Þ (2.126)

46

1h2

h

Line of sight region

Shadow region

21ξ = hh +

Figure 2.2 Geometry of wave propagation.

2.6 Basics of Fock’s Theory

and can be regarded as a height gain function of the mode with number s. In twolimiting cases q = 0 and q ¼ ¥ the roots ts can be estimated from asymptotic expres-sions valid for large s:

ts;q¼0@ 3p2

s� 34

� �h i2=3exp jp

3

� �;

ts;q¼¥@ 3p2

s� 14

� �h i2=3exp jp

3

� �;

(2.127)

and the first roots are as follows: t1;q¼0=1.01879ejp=3

; t1;q¼¥=2.33811ejp=3

;

The integral (2.120) can then be calculated as a sum of the residues in the polesof the integrand given by Eq. (2.125):

Vðn; h1 ; h2 ; qÞ ¼ exp jp

4

� �2ffiffiffiffiffiffipn

p P¥s¼1

e jnt

1�tsq2

w1 ts�h1

� �w0 tsð Þ

w1 ts�h2

� �w0 tsð Þ . (2.127)

Equation (2.127) represents the series of the normal modes of the diffractedwaves converging rapidly in a shadow region.

Consider the field in the line-of-sight interference region, n <ffiffiffiffiffih1

pþ

ffiffiffiffiffih2

p. In

that region we have to obtain the reflection formula corresponding to the reflectionfrom the spherical surface. The integral (2.120) can be represented by the sum oftwo terms

Vðn; h1 ; h2 ; qÞ ¼ Vd � Vr ,

Vd ¼12

exp jp

4

� � ffiffiffinp

r RC

dt exp jntð Þw1 t� h2ð Þw2ðt� h1Þ, (2.128)

Vr ¼12

exp jp4

� � ffiffiffinp

r RC

dt exp jntð Þw02ðtÞ�qw2ðtÞ

w01ðtÞ�qw1ðtÞ

w1 t� h1ð Þw1ðt� h2Þ. (2.129)

Assuming that the major contribution to integrals (2.128) and (2.129) comesfrom the interval of large and negative t we may use the asymptotic formulas (A.21),(A.22) for the Airy functions w1 and w2. Then for direct wave Vd we obtain

Vd ¼12exp j

p

4

� � ffiffiffinp

r RC

dtexp jjðtÞð Þ

h1�t� �

h2�t� �4 51=4 . (2.130)

The phase of the integrand j tð Þ is given by

j tð Þ ¼ ntþ 23h2 � tð Þ3=2� 2

3h1 � tð Þ3=2 .

The stationary value of ts is determined from j0tsð Þ ¼ 0. The stationary value of

the phase we denote as j ” j tsð Þ and it is given by

j ¼ h1�h2

� �24n

þ 12n h1 þ h2ð Þ � n3

12. (2.131)

47


The value of j has a simple geometric meaning of the difference between thephase of the direct wave along the distance R between the correspondents and thedistance x ¼ ah along the earth’s surface, Figure 2.3:

j ¼ k R� xð Þ. (2.132)

Application of the stationary phase method to integral (2.130) leads to the follow-ing expression for the direct wave

Vd ¼ exp ðjjÞ. (2.133)

Consider the term Vr . Using a similar asymptotic expression for the Airy functionin the integrand we obtain

Vd ¼12exp j

p

4

� � ffiffiffinp

r RC

dtexp jwðtÞð Þ

h1�t� �

h2�t� �4 51=4 q� j

ffiffiffiffiffiffi�t

p

qþ jffiffiffiffiffiffi�t

p(2.134)

and

w tð Þ ¼ ntþ 23h2 � tð Þ3=2þ 2

3h1 � tð Þ3=2� 4

3�tð Þ3=2 . (2.135)

The root of the equation w0tð Þ ¼ 0 we define as t = –p2, where p > 0. The station-

ary point in terms of p will be the root of the equationffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih1 þ p

2q

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2 þ p

2q

¼ 2pþ x. (2.136)

The stationary value of the phase can then be defined in terms of the parameter p:

w ¼ �3p2nþ 2p h1 þ h2 � n

2� �

þ n h1 þ h2ð Þ � n3

3. (2.137)

The application of the stationary phase method to Eq. (2.134) then results in thefollowing expression for reflected wave:

48

γ

1r

2r

aθx =

R

1z2

z

Figure 2.3 Ray traces in the line-of-sight region.

2.7 Fock’s Theory of the Evaporation Duct

Vr ¼q�jpqþjp

ffiffiffiffiA

pexp ðjwÞ (2.138)

with

A ¼ px

3pxþ x2�h1�h2

. (2.139)

The formula (2.138) has the following geometric meaning. The parameter p isp ¼ m cos cð Þ, where c is an incidence angle as shown in Figure 2.3. The factorq� jp=qþ jp is a Fresnel coefficient of reflection taken with the opposite sign. Theparameter

ffiffiffiffiA

pis a correction term accounting for divergence of the beam after

reflection multiplied by a facto R/r1, where r1 is the distance along the ray from thesource to the reflection point, Figure 2.3. The phase w can be written in a form sim-ilar to Eq. (2.132):

w ¼ k r1 þ r2 � xð Þ. (2.140)

Combining both terms in V we obtain the reflection formula

V ¼ exp jjð Þ � q�jpqþjp

ffiffiffiffiA

pexp ðjwÞ. (2.141)

The actual stationary value of p is given by

p ¼ 12x

h1 þ h2 �12n2 þ 4�

2sin

2 a3

� �� (2.142)

where

�2 ¼ 1

3n2 þ 2h1 þ 2h2

� �; sin að Þ ¼ n h1�h2

� ��2 ; �p

2< a <

p2. (2.143)

The reflection formula (2.141) is valid for large enough p, practically it gives agood approximation for p > 2.

As will be discussed further, experimental results at frequencies above 1 GHzsuggest little difference in the propagation of the vertically and horizontallypolarised fields. As may be seen from the definition and values of m and g, para-meter q is actually large for sea water at frequencies above 1 GHz, which suggeststhat, at least in the case of long range propagation, i.e. at distances beyond the opti-cal horizon, the impedance boundary conditions can be approximated by conditionsfor a vertically polarised field with q! ¥, practically suitable for both polarizations.

2.7Fock’s Theory of the Evaporation Duct

In the case of the super-refraction associated with the evaporation duct the M-profilehas one minimum at some height Zs above the sea surface, called the height of the

49


evaporation duct. Considering the case where q ¼ ¥, the attenuation factorVðn; h1 ; h2 ;¥Þ is given by Eq. (2.120) with the integrand F t; h1 ; h2 ;¥ð Þ given by

F t; h1 ; h2ð Þ ¼ j2f1ðt; h2Þ f2ðt; h1Þ �

f2ð0;tÞf1ð0;tÞ

f1ðt; h1Þ+ ,

. (2.144)

and the height-gain functions f1, f2 are the solutions to the equation

d2 f

dh2 þUðhÞf ¼ tf . (2.145)

The function U(h) is assumed to be an analytical function of its argument. Weconcentrate further on h and t where the equation UðhÞ � t ¼ 0 has two roots:h ¼ b1 and h ¼ b2 . For real t in the interval U Hsð Þ < t £ U 0ð Þð Þ, the above roots arealso real.

Consider an asymptotical integration of Eq. (2.145) for height-gain functionswhich is valid for all values h and t, including t ¼ UðHsÞ, i.e. the point of the mini-mum of the U-profile. The asymptotical solution in this case should be built uponan “etalon” equation, which behaves similarly to Eq. (2.145), i.e. has a single mini-mum, and, on the other hand, has a known analytical solution. The “etalon” equa-tion can be expressed in the form of an equation for functions of a parabolic cylin-der

d2g

dy2 þ 14y2 þ m

+ ,g ¼ 0. (2.146)

The relation between Eqs. (2.146) and (2.145) is established by a transformationof h into y in a form which ensures that two conditions are met: (a) the parameterUðhÞ � t in Eq. (2.145) has the same roots as 1=4 y

2 þ m, and (b) with large valuesof the above parameters, either Eq. (2.145) or Eq. (2.146) results in the same asymp-totic expression for their respective solutions f and g. Such conditions are satisfiedwith the following substitution

Rhb1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU hð Þ � t

pdh ¼ 1

2

Ry�2 j

ffiffim

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 þ 4m

qdy (2.147)

with parameter v bounded by following equation

Rb2b1


pdh ¼ 1

2

R2 j ffiffimp

�2 jffiffim

p


qdy. (2.148)

The value integral in Eq. (2.146) results in the relationship between roots b1, b2,and v

jpm ¼Rb2b1


pdh . (2.149)

Expanding the right-hand side of Eq. (2.149) into a Taylor series in the vicinity ofthe minimum of UðhÞ we obtain v as a holomorphic function of t

50


m ¼ U Hsð Þ�tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2U00 Hsð Þ

q þ ::: (2.150)

Since U00Hsð Þ > 0, we can read formula (2.149) in the following way: v > 0 when

t < U Hsð Þ and v < 0 when t > UðHsÞ; with UðHsÞ < t < U 0ð Þ we then have

m ¼ � 1p

Rb2b1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit�U hð Þ

pdh . (2.151)

Then, introducing the notations for the phase terms

S ¼Rh0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUðhÞ � t

pdh, S0 ¼ 1

2

Rb10


pdhþ 1

2

Rb20


pdh, (2.152)

the substitution (2.147), which in fact defines the relation between h and y, can beexpressed in the form

SðhÞ � S0 ¼ 12

Ry�2 j

ffiffim

p


qdy ¼ 1

4yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2 þ 4m

qþ m ln y þ


q� �� m

2ln 4mð Þ.

(2.153)

The respective solutions of Eqs. (2.145) and (2.146) are related as follows

f ¼ffiffiffiffiffidhdy

rg (2.154)

where function g, the solution to Eq. (2.146), can be expressed via a function of aparabolic cylinder Dn(z), given by the series

DnðzÞ ¼2�n

2�1

Cð�nÞ e�z

2

4 P¥m¼0

Cm�n

2

� �C mþ1ð Þ 2

m2 �zð Þm . (2.155)

Function Dn(z) will obey Eq. (2.146) when n ¼ jm� 1=2 and z ¼ ye� jp=4

.The appropriate solutions to Eq. (2.145) are then given by

f1ðh; tÞ ¼ c1 mð Þffiffiffiffiffiffiffiffiffi2 dhdy

rg1 yð Þ;

f2ðh; tÞ ¼ c2 mð Þffiffiffiffiffiffiffiffiffi2 dhdy

rg2 yð Þ

(2.156)

and

g1 yð Þ ¼ Djm�1=2 ye�j p=4

� �;

g2 yð Þ ¼ D�jm�1=2 yejp=4

� �;

(2.157)

51


c1 mð Þ ¼ exp �pm4

þ jp8þ j 1

2m� 1

2m ln mð Þ � S0

� �+ ,;

c2 mð Þ ¼ c1 mð Þ�¼ exp �pm4

� jp8� j 1

2m� 1

2m ln mð Þ � S0

� �+ ,:

(2.158)

The coefficients c1ðmÞ and c2ðmÞ are defined by Eq. (2.158) with the aim being toobtain a correct asymptotic expression for f1 h; tð Þ, f2 h; tð Þ with large values of theargument h and holomorphic functions (2.156) in the vicinity of v ¼ 0.

With large positive y, that corresponds to large heights h, well above Hs , theheight gain functions take the following asymptotic forms:

f1 h; tð Þ ¼ e j p=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU hð Þ�t4

p exp j S� 2S0ð Þð Þ;

f2 h; tð Þ ¼ e�j p=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU hð Þ�t4

p exp �j S� 2S0ð Þð Þ: (2.159)

Inside the evaporation duct and, strictly speaking, far below the inversion heightHs when y is large and negative, the asymptotic representations for the height-gainfunctions take the form:

f1 h; tð Þ ¼ v1 mð Þ ejp=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

U hð Þ � t4p exp j S� 2S0ð Þð Þ þ e

�pm e�jp=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU hð Þ � t4

p exp �jSð Þ;

f2 h; tð Þ ¼ v2 mð Þ e�jp=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU hð Þ � t4

p exp �j S� 2S0ð Þð Þ þ e�pm e

jp=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU hð Þ � t4

p exp jSð Þ(2.160)

where

v1 mð Þ ¼ffiffiffiffiffiffi2p

p

C12� jm

� � exp �pm2

þ j m� m ln mð Þð Þh i

;

v2 mð Þ ¼ffiffiffiffiffiffi2p

p

C12þ jm

� � exp �pm2

� j m� m ln mð Þð Þh i

:

(2.161)

The asymptotic representation of the integrand (2.144) is then given by

Fðt; h1 ; h2Þ ¼j2

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU h1ð Þ � t4

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU h2ð Þ � t4

p ·

ejS h1ð Þ �Ke

�jS hð Þ+ ,

e�jS h2ð Þ � e

jS h2ð Þ+ ,

1 �K

(2.162)

with Hs > h1 > h2 , where

K ¼ je�pmþ2jS0

v1mð Þ . (2.163)

52


The poles in the integral (2.120) with the integrand in the form (2.162) will begiven by the roots of the characteristic equation

1 �K ¼ 0. (2.164)

In many practical cases the evaporation duct parameters are such that the valuesof parameter y are of the order of 1, or even less, in Eqs. (2.162) and (2.164). There-fore, some doubts can be raised as to the validity of the asymptotic formulas (2.163)and (2.164) in this case. Fock [1] calculated the propagation constants given by Eq.(2.164) and the exact equation

g1 0; tð Þ ¼ 0, (2.165)

expressed via functions of a parabolic cylinder. A sample of the calculations of prop-agation constants for the hyperbolic profile

UðhÞ ¼ h�Hsð Þ2

hþhl, (2.166)

where hl is a parameter , is presented in Table 2.1

Table 2.1 Comparison of the calculated propagation constants using theasymptotic Eq. (2.164) and the exact formula (2.165).

Hs + hl Asymptotic t1 Exact t1

23.11 –0.085 + j0.466 –0.107 + j0.44325.24 –0.148 + j0.262 –0.158 + j0.26948.07 –0.104 + j0.224 –0.113 + j0.227

As observed, the agreement between the asymptotic formulas and the exact solu-tion is satisfactory, especially with regard to the imaginary part of the propagationconstants, Im tnð Þ. The relation between the attenuation rate of the nth mode, cn ,and Im tnð Þ is given by

cn ¼ 0:434k

m2 Im tnð Þ, dB/km. (2.167)

Consider Eq. (2.164) for trapped modes which have a small imaginary componentof the propagation constant tnand are in the interval: UðHsÞ £ Re tn < Uð0Þ.The respective values of parameter v are negative. For v < 0 we defineln mð Þ ¼ ln �mð Þ þ jp and from Eq. (2.152) we obtain

S0 ¼Rb10


pdh� j

p

2m ”S1 � j

p

2m. (2.168)

53


Equation (2.164) then takes the form

je2jS1 ¼ v mð Þ @ 1 . (2.169)

For large negative m, v mð Þ ! 1, and we obtain

S1”Rb10


pdh ¼ m � 1

4

� �p; m ¼ 1; 2; ::: (2.170)

For either large positive m or complex m with positive Re m, it follows thatRe t < U Hsð Þ. In this case the phase S1 can be defined as

S1 ¼ S0 þ jp2m (2.171)

and Eq. (2.164) will again be truncated to Eq. (2.170).

A basic conclusion followed from the study of Eq. (2.164) and its truncated form(2.170) (performed by Fock in Ref. [1]) is that the attenuation rate of the field in theevaporation duct is affected not only by the height Zs and the M-deficit of the evapo-ration duct but also by a curvature of the M-profile at the level of inversion heightZs . The attenuation rate cmof the mth mode from Eq. (2.170) can be estimated asfollows:

cm ¼ 2p m � 14

� �10

�3 ZsM00 Zsð Þ

bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Mð0Þ�M Zsð Þð Þ

p H ¼ 2pkm

10�6Z

2s M

00Zsð ÞH (2.172)

where

H ¼ kmk

Im tm ; (2.173)

km ¼ b 103Zsm�1=4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Mð0Þ �M Zsð Þð Þ

pis a critical wavelength of the mth mode trapped in the evaporation duct formed by agiven M-profile, k is the wavelength of the radiated field. The parameter b is intro-duced by

b ¼R10

qðzÞdz; qð0Þ ¼ 4; qð1Þ ¼ 0; (2.174)

qzZs

� �¼ 4

MðzÞ�MðZsÞMð0Þ�MðZsÞ

. (2.175)

For most M-profiles the value of b is close to 1.

54

References 55

References

1 Fock, V.A.ElectromagneticDiffraction and Propa-gation Problems, PergamonPress,Oxford, 1965.

2 Fock, V.A. Distribution of currents excited by aplanewave on the surface of conductor, J. Exp.Theor. Phys., (Nauka), 1945, 15 (12), 693–702.

3 Leontovitch, M.A. Method of solution to theproblems of em wave propagation over theboundary surface, Izv. Acad. Nauk, Ser. Fiz.,1944, 8 (1), 16–22.

4 Leontovitch, M.A., Fock, V.A. Solution to aproblem of em wave propagation over theEarth’s surface using the method of parabolicequation, J. Exp.Theor. Phys., (Nauka), 1946,16 (7), 557–573.

5 Fock, V.A. Theory of wave propagation in anon-uniform atmosphere for elevated source,Izv. Acad. Nauk, Ser. Fiz., 1960, 14 (1), 70–94.

6 Krasnushkin, A.E. The expansion over normalwaves in a spherically-stratifiedmedium,Dokl.Acad.Nauk SSSR, 1969, 185 (6), 1262–1265.

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2 Parabolic Approximation to the Wave Equation56

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57

In this chapter we study wave propagation over the earth’s surface in a line-of-sightregion in the presence of a random component of refractive index. In the absence offluctuation this problem may be regarded as having been solved many decades agoand numerous publications are available. Nonetheless, it may be noted than even ina classical formulation of the problem, i.e. just a point source of vertical or horizon-tal polarization above a smooth terrain in the absence of super-refraction, the actualsolution is not quite simple and can be described in terms of reflection formulasonly in a “true” line-of-sight region, at distances not too close to the horizon. Fock [1]demonstrated that the reflection formulas in a form of superposition of direct andreflected waves are valid at distances of the order of a=m before the horizonffiffiffiffiffi

2ap ffiffiffi

zp

þ ffiffiffiffiffiz0

p� �, i.e. outside the “shade cone”. In the case of super-refraction, single

reflection formulas are also valid in the range of distances before the horizon,which, in turn, is also modified in the presence of refraction. In particular, in thecase of the evaporation duct, there are two separate horizons for the direct andreflected waves [1]. In practical applications [2, 3], signal strength calculations in aline-of-sight region are performed by means of ray theory that can be applied to avery general profile of refractivity.

The problem becomes even more complicated in the presence of random fluctua-tions in the refractive index. The number of publications on this topic is rather lim-ited and all studies known to the author are concerned with a plane boundary inter-face [4–6]. The applicability of the results discussed in Ref. [4] is limited to “weak”fluctuations of the scattered field. Strong fluctuations in the scattered field are con-sidered in Ref. [5] using the method of “local perturbations”. In Ref. [6] the authorsuse a perturbation theory for auxiliary functions, treating direct and reflected wavesseparately. That approach resulted in obtaining an expression for fluctuations inamplitude and phase applicable in the region of the interference minima.

In this chapter we use path integrals to obtain the second- and fourth ordermoments for the wave field in a random medium above the plane and sphericalboundary interface.

3

Wave Field Fluctuations in Random Media over a BoundaryInterface

3 Wave Field Fluctuations in Random Media over a Boundary Interface

3.1Reflection Formulas for the Wave Field in a Random Medium over an Ideally ReflectiveBoundary

3.1.1Ideally Reflective Flat Surface

Let us examine the field in a randomly non-uniform medium above a plane inter-face. In this case, the Green function G ~rr;~rr0ð Þ can be presented as a superposition ofthe Green function for a point source G

þ~rr;~rr0ð Þ situated at the point~rr0 ¼ 0; y0 ; z0f g

and that for the mirror-reflected point source G�~rr;~rr0ð Þ located at~rr0 ¼ 0; y0 ;�z0f g:

G ~rr;~rr0ð Þ ¼ Gþ~rr;~rr0ð Þ � G

�~rr;~rr0ð Þ . (3.1)

The boundary condition at the surface z = 0

G ~rr;~rr0ð Þjz¼0

¼ 0 . (3.2)

The boundary conditions (3.2) have the following impact on a continual represen-tation of the Green function: the mirror reflection of the point source requires a mir-ror reflection of the medium as well, it results in the introduction of the fluctuationsin dielectric permittivity to be a function of the modulus of the z-coordinate normalto the surface of separation, z = 0, i.e. de ~rr xð Þð Þ ¼ de x; y xð Þ; z xð Þj jð Þ. We will define~��

xð Þ , the trajectories of the waves departing from the sources in the upper and

lower half-space. For G

~rr; r*

0

� �we have

G~rr;~rr0ð Þ ¼

RD~��

xð Þexp jkS0 ~��

xð Þ

h iþ j

k2

Rx0

dx0de x

0; �

x0

� �� ( ), (3.3)

We have introduced the notation ~��xð Þ ¼ y

xð Þ; zðxÞ

n o. The action S0 is given

by

S0 ~�� xð Þ½ ¼Rx0

dx0L0 x

0;~�� x

0� ��

þ 12de x

0;~�� x

0� ��

(3.4)

where L0 x;~�� xð Þð Þ ¼ 12

d�dx

� �2

. As we can see from Eq. (3.3), the expressions for G+

and G– are of identical form but differ in terms of the initial conditions for the tra-jectories:~��

þx ¼ 0ð Þ ¼ y0 ; z0f g; ~��

�x ¼ 0ð Þ ¼ y0 ;�z0f g.

Consider a calculation for the second moment of the Green function (3.1). UsingEq. (3.3), we obtain:

G ~rr;~rr0ð ÞG�~rr;~rr0ð Þ

� �¼ G11h i þ G22h i � G12h i � G21h i (3.5)

where

58

3.1 Reflection Formulas for the Wave Field in a Random Medium over an Ideally Reflective Boundary

G11h i ¼RD~��

þ1 ðxÞ

RD~��

þ2 ðxÞ · exp jk S

þ1ð Þ � S

þ2ð Þ

h i�M11 x;~��

þ1 ðxÞ;~��

þ2 ðxÞ

h in o,

(3.6)

G12h i ¼RD~��

þ1 ðxÞ

RD~��

�2 ðxÞ · exp jk S

þ1ð Þ � S

�2ð Þ

h i�M11 x;~��

þ1 ðxÞ;~��

�2 ðxÞ

h in o,

(3.7)

M11 ¼ pk2

4

Rx0

dx0H ~��

þ1 x

0� �

�~��þ2 x

0� ��

, (3.8)

M12 ¼ pk2

4

Rx0

dx0H ~��

þ1 x

0� �

�~��2 x

0� ��

. (3.9)

Here the subscripts 1 and 2 correspond to the first and second sources, respec-tively; the symbolic notations S

ð1Þ and Sð2Þ have the meaning of the actions

from the direct source (1) and mirror-reflected source (2). The remaining terms inEq. (3.5) can be written in a way similar to Eqs. (3.6) and (3.7).

Thus, the coherence function for the field of the point source above the ideallyreflective surface is represented by the superposition of correlators between thedirect and reflected sources. In the functional space of the trajectorieszðxÞ ¼ 1=2 z1ðxÞ þ z2 xð Þð Þ and yðxÞ ¼ z1ðxÞ � z2ðxÞ we can isolate two regions inthe presence of a reflective surface.

X1ðxÞ � z2ðxÞ > y2ðxÞ

4

( ), X2ðxÞ � z

2ðxÞ < y2ðxÞ4

( ). (3.10)

In the region X1ðxÞ each term in Eq. (3.5) obeys Eq. (2.35) and the integrals inEq. (3.5) are similar to those in Eq. (2.68). In the region X2ðxÞ instead of H ~�� xð Þð Þwe need to use HðyðxÞ; 2zðxÞÞ. In both regions, X1ðxÞ and X2ðxÞ, the criteria forapplicability of the Markov approximation (2.76) and (2.80) remain the same as foran unbounded medium.

Subsequent calculations are performed here for partially saturated fluctuations[7], engendered by atmospheric turbulence. While the mean-square value of thephase fluctuations is large the following two conditions have to be satisfied.

First we assume the smallness of the mean-square value of the fluctuations in thephase difference at the base to be of the order of the Fresnel zone size:

pk2

4

Rx0

dx0H

ffiffiffiffiffix0

k

r !¼ 0:73C

2e k

7=6x11=6

<< 1. (3.11)

The inequality (3.11) means that even in the presence of phase fluctuations theFresnel zone remains a characteristic region in the plane x

0 ¼ const from which therays arrive in phase. As a consequence, no stochastic multipath occurs and integra-tion in Eq. (3.5) is carried along a single ray-tube limited by a Fresnel zone volume.

59


The second condition is to require a small fluctuation of the arrival angle com-pared with the angular size of the Fresnel zone. In path integral representation thisrequirement is equivalent to

pk2

8

Rx0

dx0~��

2x0

� � d2H ~��ð Þd~�� 2 £ pk2

8x2 R

d2~kk?Ue 0;~kk?ð Þ~kk2

? ¼ 0:037C2e kx

2l�1=30 << 1

(3.12)

where l0 is an internal scale of turbulence. Therefore, when inequalities (3.11) and(3.12) hold, the integration in Eq. (3.5) can be performed along non-perturbed trajec-tories. For turbulence in the atmosphere and radio frequencies above 10 GHz, theinequalities (3.11) and (3.12) hold at distances x £ 300 km.

Non-perturbed trajectories ~�� xð Þ represent solutions to Euler equations and aregiven by

~��

x0

� �¼ ~�� 0 þ ~��~�� 0ð Þ x

0

x. (3.13)

Introducing the sum- and difference-coordinates of the corresponding points weobtain

M11 ¼ pk2x4

R10

dnH ~��nþ~�� 0 1� nð Þð Þ , (3.14)

M12 ¼ pk2x4

Rn10

dnH y0 þ n 2z� y0� �

; ynþ y0 1� nð Þ� �

þ

R1n1

dnH 2z0 1� nð Þ þ ny; ynþ y0 1� nð Þð Þ

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

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

, (3.15)

M22 ¼ pk2x4

Rn20

dnH y0 1� nð Þ þ ny; ynþ y0 1� nð Þ� �

þ

Rn1n2

dnH 2ðzþz0Þn�2z0n;ynþy0 1�nð Þ� �

þ

R1n1

dnH ynþy0ð1�nÞ;ynþy0 1�nð Þ

� �

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

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

, (3.16)

where y; yf g ¼ ~�� 1 �~�� 2 , y0 ; y0� �

¼ ~��00 �~��

000 , n1;2 ¼

z0�y0

2

z� y

2þ z0 �

y0

2

are the dis-

tances, expressed in units of xalong the surface from the source to the point ofreflection. The equation for M21 is similar to Eq. (3.15) while n1 is replaced with n2 .

A similar, but rather simple, equation follows for the intensity of the wave fieldfrom the point source Jðx; z; z0Þ ¼ 4p

2=k

2G 1ð Þj j2

D E, when~�� ¼ ~�� 0 ¼ 0:

J x; z; z0ð Þ ¼ 2

x2 1� exp �M12 x; z; z0ð Þð Þcos DS x; z; z0ð Þð Þ½ (3.17)

60


where DSðx; z; z0Þ ¼ 2kzz0x

,

M12 ¼ M21 ¼ pk2x4

R10

dnH dpn� �

. (3.18)

The M12 is a mean-square fluctuation in a phase difference between the directand the reflected waves, dp ¼ zz0=ðzþ z0Þ is the maximum possible separation inthe vertical plane between the trajectories of the direct wave and the wave reflectedfrom the surface, in fact dp is the height at which the direct ray passes the point ofmirror-reflection for the reflected wave. In the case of turbulence fluctuations in dewe obtain

M12ðx; z; z0Þ ¼ 0:869C2e k

2xd

5=3p . (3.19)

The expression (3.19) coincides with the structural function of the phase for thebase dp [8] with accuracy to the numerical coefficient.

As observed from Eq. (3.17) the interference-like structure of the field is kept untilthe direct and reflected waves are correlated in phase, M12 << 1. It can be noted thatthe mean square of the phase fluctuations in either direct or reflected waves maynot necessarily be small, cS >> 1. As the correlation between the direct and thereflected waves diminishes, the first term in Eq. (3.17) predominates, yielding a fieldintensity that is twice as large as that in a free space.

3.1.2Spherical Surface

The second moment for the field over a spherical surface is determined by Eqs. (3.5)to (3.9) where the Lagrangian is

L0 ~�� xð Þð Þ ¼ 12

dydx

� �2

þ 12

dzdx

� �2

þ zðxÞj ja

, (3.20)

a is the curvature radius of the spherical surface, z is the height above the surface.In fact, the sphere is replaced by a cylinder with an infinitely long generatrix parallelto the y-axis. The applicability of such an approximation is examined in Ref. [1]. Thedifference between the given problem and that examined in the previous section isfound in the presence of the potential term zj j=a in the Lagrangian (3.20). This termtakes into account the spherical boundary surface in a parabolic approximation. Thepresence of this term leads to the introduction of two segments on the trajectoryz�ðx0 Þof the reflected wave (departing from the imaginary mirror-reflected source),

and these two segments are separated by the reflection point x0.

3.1.2.1 Trajectory EquationsFollowing Fock [1], let us introduce the dimensionless coordinates

~xx ¼ mxa

, ~xx0 ¼ mxa

, y ¼ kzm

, y0 ¼ kzm

(3.21)

61


and consider the ray in the direction upwards from the source. The phase is givenby

Sþ ¼ ~xxtþ 2

3y� tð Þ3=2� 2

3y0 � tð Þ3=2 . (3.22)

The derivative over t gives the equation for the stationary value of t

S0 þ ¼ ~xx � ffiffiffiffiffiffiffiffiffiffi

y� tp þ ffiffiffiffiffiffiffiffiffiffiffiffi

y0 � tp

(3.23)

that leads to the solution

ffiffiffiffiffiffiffiffiffiffiffiffiy0 � t

p ¼ y�y0�~xx2

2x. (3.24)

The equation for the trajectory yþð~xx0 Þ becomes

yþð~xx0 Þ ¼ y0 þ ~xx

0 2 þ ~xx0

~xxy� y0 � ~xx

2� �

(3.25)

or, in physical coordinates,

zþðx0 Þ ¼ z0 þ

x02

2aþ x0

xz� z0 �

x2

2a

!. (3.26)

For the ray directed downwards from the source we can separate two regionsalong ~xx

0: before and after the reflection point ~xx0 . For ~xx

0< ~xx0 we obtain

S� ¼ ~xx

0tþ 2

3y0 � tð Þ3=2� 2

3yð~xx0 Þ � t� �3=2

(3.27)

and the stationary value of the phase is given by

S0 � ¼ ~xx

0 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiy ~xx

0� �

� t

r� ffiffiffiffiffiffiffiffiffiffiffiffi

y0 � tp ¼ 0 . (3.28)

This leads to an equation for the ray trajectory in the region ~xx0< ~xx0

y ~xx0

� �¼ y0 þ ~xx

0 2 � 2~xx0 ffiffiffiffiffiffiffiffiffiffiffiffi

y0 � tp

. (3.29)

Assume now that t = t0, the solution to Eq. (3.27) for fixed y, y0 and ~xx. In that caseyð~xx0 ¼ ~xx0Þ ¼ 0, where ~xx0 ” ~xx0ðt ¼ t0Þ, and

y ~xx0

� �¼ y0 þ ~xx

0 2 � x0

x0

x20 þ y0

� �(3.30)

becomes the equation for the ray trajectory in the region x0< x0 , when ~xx0 is used as

a stationary parameter instead of t0.The equation for ~xx

0 ‡ ~xx0 can be obtained in a similar way using the boundary con-dition yð~xx0 ¼ ~xxÞ ¼ y:

y ~xx0

� �¼ ~xx

0 � ~xx0

� �2þ ~xx0�~xx0

~xx�~xx0

y� ~xx � ~xx0ð Þ2� �

. (3.31)

62


In physical coordinates, Eqs. (3.30) and (3.31) take the form

z x0

� �¼ z0 þ

x02

2a� x0

x0

x20

2aþ z0

!, (3.32)

z�

x0> x0

� �¼ x0�x0

� �22a

þ x0�x0

x�x0

z� x�x0

� �22a

!. (3.33)

Now we need to find a reflection point x0 from the equations for the stationaryphase (3.28).

The general case solution is obtained by Fock [1] and is provided in Chapter 2. Letus consider here approximate formulas for small dy ¼ y� y0 , dyj j << 1. Introducet ¼ t0 þ dt and expand Eq. (3.28) into series over dy and dt. Leaving only first orderterms we obtain

S� 0 ¼ x � 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy0 � t0

pþ 2

ffiffiffiffiffiffiffiffi�t0

pþ

dt 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy0 � t0

p � 1ffiffiffiffiffiffiffiffi�t0p

� �� dy 1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy0 � t0

p ¼ 0:(3.34)

Equating the terms of the same order we obtain

ffiffiffiffiffiffiffiffi�t0p ¼ 4y0�x

2

4x¼ y0

x� x

4, (3.35)

dt ¼ dy2

ffiffiffiffiffiffiffiffi�t0

pffiffiffiffiffiffi�t

p�

ffiffiffiffiffiffiffiffiffiffiy0�t

p� � ¼ �dy4y0�x

2

4x2 . (3.36)

The stationary value t0 corresponds to the case of equal heights of the transmit-ting and receiving antennas above the surface, dy ¼ 0, in this case the reflectionpoint x0 is apparently x0 ¼ x=2. The correction term to t0 is given by Eq. (3.36) andsubstitution of dt into Eq. (3.28) will lead to the correction term dx to the distance x0

to reflection point:

dx ¼ �dyx

4y0þx2 . (3.37)

Similar approximations can be obtained from the general solution (Chapter 2) whenthe inequality dy x=�

3<< 1 holds, which in turn is equivalent to dy << 1 since x

and �3are terms of the same order of magnitude. Finally, we have

~xx0 ¼ ~xx2� ðy� y0Þ

~xx

4y0þ~xx2 (3.38)

and

x0 ¼ x2� ðz� z0Þ

x

4z0þx2

2m

. (3.39)

63


3.1.2.2 Moments of the FieldUnder the above conditions of Eqs. (3.11) and (3.12) we can use the equations forunperturbed trajectories of direct and reflected waves (3.26), (3.32), (3.33) in the cal-culation of the field’s moments.

As a less cumbersome case of calculation of the second moment, let us considerthe intensity of the field. In the line-of-sight region we obtain an equation similar toEq. (3.17)

J x; z; z0ð Þ ¼ 2

x2 1� exp �Msph12 x; z; z0ð Þ

� �cos DS

sphx; z; z0ð Þ

� �h i(3.40)

where

Msph12 ¼ pk2x

4

R10

dnH dsphn� �

, (3.41)

DSsph ¼ k � x3

32aþ z�z0� �2

8xþ zþz0� �

x4a

� z2

2 x�x0

� �� z20

2x0

" #. (3.42)

The parameter dsph is given by

dsphðxÞ ¼ z0 �x20

2a¼ z� x�x0

� �22a

. (3.43)

As observed from Eq. (3.43) and the geometry of the problem, parameterdsphðxÞ ! 0 when the distance x between corresponding points approaches the hori-zon x ! xm ¼

ffiffiffiffiffi2a

p ffiffiffiz

pþ ffiffiffiffiffi

z0p� �

. The phase difference DSsph ! 0 and the field

intensity Jðx; z; z0Þ ! 0 when x ! xm , in accordance with the laws of geometricoptics. The numerical value of Eq. (3.41) is given by Eq. (3.19) where the parameterdp is replaced by dsph (x).

Let us examine the fluctuations in the field intensity J2

D Eabove either plane or

spherical boundaries with ideal reflection. Using representation (3.1) we obtain

J2

D E¼ G

211

D Eþ G

222

D Eþ 2 G11G22h i þ G

212

D Eþ G

221

D Eþ 2 G12G21h i�

2 G11G12h i � 2 G12G22h i � 2 G21G11h i � 2 G21G22h i:(3.44)

As an example, the fourth-order correlator for a direct wave has the form

G211

D E¼RD~RR

0 ðxÞRD~RR

00 ðxÞRD~��

0 ðxÞRD~��

00 ðxÞ�

exp ikRx0

dx0 d~RR0

dx0d~�� 0

dx0þ d~RR00

dx0d~�� 00

dx0

" #( )�

exp �pk2

4

Rx0

dx0H ~��

0x0

� �� þH ~��

00x0

� �� h i( )�

exp

�pk2

2

Rx0

dx0Rd2~kk?Ue 0;~kk?

� �exp j~kk?

~RR0 x0� �

�~RR00 x0� ��

�

cos ~kk?�*0 ðx

0 Þ � �*00

ðx0 Þ2

!�cos ~kk?

�*0 ðx

0 Þþ �*00 ðx

0 Þ2

!" #8>>>><>>>>:

9>>>>=>>>>;

(3.45)

64


where ~��0 ðxÞ ¼ ~��

þ1 xð Þ �~��

þ2 xð Þ, ~��

00 ðxÞ ¼ ~��þ3 xð Þ �~��

þ4 xð Þ, ~RR

0 ðxÞ ¼ 1=2 ~��þ1 ðxÞþ

�~��

þ2 ðxÞÞ, ~RR

00 ðxÞ ¼ 1=2ð~�� þ3 ðxÞ þ~��

þ4 ðxÞÞ. The remaining terms in Eq. (3.44) have a

form similar to Eq.(3.45).Under the same conditions defined by inequalities (3.11) and (3.12) we can obtain

the mean-square value of the fluctuations in intensity of the wave fieldr

2J ¼ hJ2i � Jh i2 :

r2J ¼ 2

x2 1� exp �2M12ðx; z; z0ð Þ½ ·

1� exp �2M12ðx; z; z0ð Þcos 2DS x; z; z0ð Þð Þ½ (3.46)

and for the scintillation factor b2J ¼ r2

J

Jh i2respectively

b2J ¼ 1

21�exp �2M12ðx;z;z0

� �$ %1�exp �2M12ðx;z;z0

� �cos 2DS x;z;z0

� �� $ %1�exp �M12ðx;z;z0

� �cos 2DS x;z;z0

� �� $ %2 (3.47)

Here the phase difference DS and the mean-square of the fluctuations in a phasedifference between the direct and reflected waves M12 are determined by the respec-tive formulas for a plane or spherical interface, either Eqs. (3.17) and (3.18) or (3.41)and (3.42).

Given the values of z and z0 let us introduce the points xmin and xmax at a distancex where the unperturbed field has a maximum and minimum respectively. Thesepoints can be derived from the equations DS xmax ; z; z0ð Þ ¼ p 2nþ 1ð ÞandDSðxmin ; z; z0Þ ¼ 2pn. The value of the scintillation factor b

2J at those points we

define as b2J ðx ¼ xmaxÞ ” b

2Jmax and b

2J ðx ¼ xminÞ ” b

2Jmin respectively. As observed

from Eq. (3.47), if M12 >> 1 and, therefore, the direct and reflected waves are com-pletely uncorrelated, the fluctuations in intensity are uniform in space. In this casethe scintillation factor b

2Jmax ¼ b

2Jmin ¼ 1=2 at both the maximum and minimum of

the unperturbed field. In another limiting case when M12 << 1, the values of thescintillation factor at the minimum and maximum of the unperturbed field aresignificantly different. b

2Jmax ¼ 1=2 M

212 and b

2Jmin ¼ 2; and r

2J ðxminÞ ¼ 8M

212 while

J xminð Þh i2¼ 4M212 . Such behaviour of the field fluctuation is of a general nature.

When the only phase fluctuates significantly, which is the case, the amplitudefluctuations dAof the total field at the minimum xmin are contributed by thefluctuation of the phase difference between the direct and reflected wavesdA~Du12 ¼ dS

þð1Þ � dS�

2ð Þ and dAð Þ2D E

~ Du212

D E; J

2xminð Þ

D E~ Du

412

D E. With

normal distribution of the fluctuations in the phase it follows thatr

2J ðxminÞ ¼ 2 Du

212

D E¼ 2 Jðxminh i2 , and consequently b

2Jmin ¼ 2.

65


3.2Fluctuations of the Waves in a Random Non-uniform Medium above a Planewith Impedance Boundary Conditions

Let us introduce the Cartesian coordinate system (x,y,z) and the attenuation factor uby equation Ez ¼ u � e jkx .

The slow varying field amplitude u is governed by the equation

2jk@u@x

þ D?uþ deð~rrÞu ¼ 0 (3.48)

with the impedance boundary conditions at the surface separating two media

@u@z

þ jqu ¼ 0 (3.49)

and initial condition (at the source) uðx ¼ 0; y; zÞ ¼ ð2p=kÞ dð~��~�� 0Þ, where~rr ¼ fx;~��g; ~�� ¼ fy; zg; ~�� 0 ¼ fy0 ; z0g, D? ¼ @

2=@y

2 þ @2=@z

2, q ¼ k= ffiffiffiffiffi

egp ,

k ¼ 2p=k, k is the wavelength, eg is the effective dielectric permittivity of the lowerhalf-space (z < 0), deð~rrÞ is the random component of the medium’s dielectric per-mittivity in the upper half-space (z > 0), eh i ¼ 0, and the angle brackets denote aver-aging over the ensemble of the deð~rrÞ realisations.

Following the representation introduced by Malyuzhinetz [9], the boundary prob-lem (3.48), (3.49) reduces to a problem with ideal conditions of reflection via intro-duction of the Malyuzhinetz transformation:

u ¼ uþ0 ðx; y; zÞ þ u

�0 ðx; y; zÞ þ uqðx; y; zÞ (3.50)

where

uqðx; y; zÞ ¼ �2jk � ejqzRz

¥ �e jp=4dy � e jkyu�0 ðx; y;�yÞ . (3.51)

Here uþ0 is the field of the incident wave, and u

�0 is the field of the wave generated

by the mirrored source. The uþ0 and u

�0 fields determine the solution of the bound-

ary problem (3.1), (3.2) when eg&& &&! ¥ , i.e., the field of the vertical electrical dipole

in the case of a randomly non-uniform medium above an infinitely conductingplane. The last term in Eq. (3.50) takes into consideration the corrective factor forthe finite value of egwhich we can refer to the field of the impedance source.

Let us examine the average intensity Ið~rrÞ of the scattered field at the point~rr ¼ fx; 0; zg; excited by a vertical electric dipole situated at the point~rr0 ¼ f0; 0; z0g:

Ið~rrÞ ¼ uð~rrÞj j2D E

¼ I0ð~rrÞ þ uqð~rrÞuþ0 ð~rrÞ

�D E

þDuqð~rrÞu

�0 ð~rrÞ

�Eþ c:c:þ uqð~rrÞ

&& &&2D E.

(3.52)

Here, I0ð~rrÞ is the average intensity of the wave field above the ideally reflective

surface, I0ð~rrÞ ¼ uþ0 þ u

�0

&&& &&&2D E: The fields u

0 can be represented in the form of

66

3.2 Fluctuations of the Waves in a Random Non-uniform Medium above a Plane…

Feynman trajectory integrals [10]. A trajectory integration is described in Section 3.1for the boundary problem with ideal reflective conditions, and I0ð~rrÞis given by

I0ð~rrÞ ¼1

x2 1þ exp �M12ðx; z; z0Þ½ cos DS x; z; z0ð Þð Þf g (3.53)

where DS ¼ 2kzz0=x represents the phase difference between the direct andreflected waves. The variance of the fluctuations in phase difference between thedirect and reflected waves M12ðx; z; z0Þ is given by

M12ðx; z; z0Þ ¼ M12ðx; dÞ ¼pk2x4

R10

d �HðndÞ (3.54)

where H(�) is a structure function of the fluctuations de, d ¼ zz0= zþ z0ð Þ is theheight at which the direct beam passes above the surface at the point of mirrorreflection. Expression (3.53) and the subsequent formulas of this section have beenderived using the approximation of smooth perturbations, i.e. on the validity of theinequalities

pk2

4

Rx0

H

ffiffiffiffiffix0

p

k

!dx

0<< 1; (3.55)

pk2x2

8

Rx0

Ue 0; k?ð Þk2?d

2k2? << 1; (3.56)

where Ue kð Þ is a three-dimensional spectrum of fluctuations in the dielectric per-mittivity deð~rrÞ, k ¼ kx ; k?f g is the wave vector of the fluctuations, k? ¼ ky ; kz

� �.

Let us examine the correlation function for the field of the impedance source uqand the direct wave u

þ0 . After averaging, trajectory integration and expanding the

phase of the derived equation in a series over m ¼ z� y, we obtain

uquþ0 �

D E¼ 2jq

x2 exp jDSðx; z; z0Þ �M12ðx; dÞ½ ·

R0C

exp jkm2

2x� jqm 1þ ffiffiffiffiffi

egp tan ðwÞ

� �� kma� km

2J2s x; dð Þ

( )dm

(3.57)

where the contour C emanates from infinity along the ray ej5p=4

, tan ðwÞ = ðzþ z0Þ/x, w is the angle of reflection. The coefficient J2

s ðx; dÞ ¼ C2e xd

�1=3has the meaning

of the angular width of the scattered field for a base equal to d, the parametera ¼ J2

s ðx; dÞ=c determines the ratio of the angular spectrum width to the width ofthe interference lobe c ¼ 1=kd.

Let us assume that fluctuations deð~rrÞ are caused by turbulence and d > l0 where l0is an internal scale of the turbulence, and for the structure function Hð~��Þ we willassume Hð~��Þ ¼ C

2e �

5=3, where Ce is a structure constant for the fluctuations de. For

coefficients a, J2s ðx; dÞ and M12(x,d) we have

67


a ¼ 1:46kxC2e

25=3 n1n2

� �z2=3n5=3

1þz2=30 n5=3

2

� �� 5

8ð2zÞ2=3n8=3

1þð2z0Þ

2=3n8=32

� �þ

ð2z0Þ2=3n5=3

21�2n2

1

� �264

375

(3.58)

J2Sðx; dÞ ¼ 1:46kxC

2e

116

25=3

z�1=3

n8=31 n

22 þ z

�1=30 n

2=32 n

31

� �n1 �

611

� ��

53

ð2Þ2=3ðz�1=3n8=31

n2þz�1=3

0 n2=32

n31Þþð2z0Þ

2=3n8=32

� �þ

524

ð2zÞ�1=3n8=31

þð2z0Þ�1=3n8=3

2ð1þ4

5n2Þ

� �

26666664

37777775(3.59)

M12ðx; dÞ ¼ 0:869C2e k

2xd

5=3(3.60)

where the parameters n1 and n2 correspond to the distance of the mirror reflectionpoint x0 from the source and the receiver respectively:

n1 ¼ x0

x¼ z0zþz0

and n2 ¼ x�x0

x¼ fzþz0

.

When d < l0 the structure function can be approximated by H �ð Þ �C2e �

2l�1=30 and

the coefficient a ¼ 0, J2s ðx; dÞ ” r

2s ðxÞ, where r

2s ðxÞ is a variance of the fluctuations

in the angle of incidence in the vertical plane.Let us calculate the integral (3.57) for large numerical distances, i.e., when

qxj j >> 1. Integrating by parts, we obtain

uquþ0

D E¼ 1x2 exp �M12 x; dð Þ þ jDSðx; z; z0Þf g�

� 2

1þjgð Þ 1þ ffiffiffiffiffieg

p tan ðwÞ� � þ 2jg 1þjbð Þ

kx 1þjgð Þ3 1þ ffiffiffiffiffieg

p tan ðwÞ� �3 þ

6e2gkxð Þ2

1

1þjgð Þ5 1þ ffiffiffiffiffieg

p tan ðwÞ� �5

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

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

(3.61)

Parameter b ¼ 2kxJ2s determines the ratio of the width of the angular spectrum

of the scattered field J2s to the angular size of the Fresnel zone h

2F ¼ 1=kx. In the

light of the validity of inequality (3.56) we have b << 1. The parameter

g ¼ a

1ffiffiffiffiffieg

p þ tan ðwÞ !

68


determines the ratio of the angular spectrum width of the scattered field to the prod-uct of the angular width of the interference lobe c and the effective grazing angle ofthe reflected wave 1= ffiffiffiffiffi

egp þ tan ðwÞ.

The correlation function of the impedance-source field uq with reflected field u�0

is calculated similarly to Eqs. (3.57)–(3.61). Assume that the following inequalitieshold true:

max kl0f >> ffiffiffiffiffieg

p&&& &&&; kl0 tan ðwÞ >> 1g. (3.62)

In this case the trajectories determining the principal contribution to the trajectoryintegrals, uq and u

�0 actually coincide, and for the structure function H(�) we can

use the quadratic approximation Hð�Þ ¼ C2e �

2l�1=30 . Then uqu

��0

� �can be written as

follows

uqu��0

� �¼ 1x2

� 2

1þ ffiffiffiffiffiegp tan ðwÞ� � þ 2jeg 1þ jgsð Þ

kx 1þ ffiffiffiffiffieg


6e2gkxð Þ2

1þ jgsð Þ2

1þ ffiffiffiffiffieg

p tan ðwÞ� �5

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

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

(3.63)

where g2s ¼ r

2s xð Þ=h2

F . The parameter r2s ðxÞ ¼ 0:82C

2e l

�1=30 x is a variance of the fluc-

tuations in the arrival angle of the incident wave.Let us examine the average intensity of the impedance-source field uq

&& &&2D E.

Using an approach similar to the calculation of Eq. (3.57) we carry out averagingand continuous integration under the condition of validity of the inequalities (3.55)and (3.56). Expanding the phase of the expression under the integral sign into a se-ries over the distance from the upper limit zand retaining the quadratic terms, weobtain

uq&& &&2D E

¼ 4 qj j2

x2

R¥ ejp=4

0

duR2u

�2u

dv expjkm2

2x�jm ku

x�ReðqÞ�k tan ðwÞ

� �þ

2m ImðqÞ�km2r2s

8>><>>:

9>>=>>;dm. (3.64)

Let us assume here and below that Im eg ¼ 4pr=x¼ 0, r is the conductivity ofthe lower medium (z < 0Þ and x ¼ 2pf is a circular frequency of radiation. Thiscondition does not limit the generality of the results since the solution in quadra-tures for the average intensity (3.52) has already been determined by Eqs. (3.53),(3.57) and (3.64). In the meanwhile such an approximation significantly simplifiesthe asymptotic expressions for the total field. In the case of radio frequencies above10 GHz and with the sea surface assumed to be a boundary between the two media,the main contribution to eg comes from displacement currents and Im eg << Re eg .For large distances qxj j >> 1 we obtain

69


uq&& &&2D E

¼ 1

x2

(� 4



8egr2s


p tan ðwÞ� �2 �

1þ 1

1þ ffiffiffiffiffiegp tan ðwÞ� �2

0B@

1CA� 20e2g

kxð Þ2

). (3.65)

Collecting all components of Eq. (3.52) we obtain

Iðx; z; z0Þ ¼ 1x2

1þ R0

&& &&2þ2 exp �M12ðx;dÞ$ %

Re RsejDS

h iþ2 exp �M12ðx;dÞ

$ %�

Re ejDS 2jeg 1þ jbð Þ

kx 1 þ jgð Þ3 1 þ ffiffiffiffiffieg

ptan ðwÞ

� �3 þ6e

2g

kxð Þ21

1 þ jgð Þ5 1 þ ffiffiffiffiffieg

ptan ðwÞ

� �5

264

375

8><>:

9>=>;

þ 12e2g

kxð Þ2 1þ ffiffiffiffiffieg

ptan ðwÞ

� �5þ

8r2s eg


ptan ðwÞ

� �2 1� 11þ ffiffiffiffiffiegp tan ðwÞ þ 1


ptan ðwÞ

� �2

0B@

1CA�20e

2g

kxð Þ2

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

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

(3.66)

Here R0 ¼ ffiffiffiffiffiegp tan ðwÞ � 1= ffiffiffiffiffiegp tan ðwÞ þ 1 is a Fresnel coefficient of reflectionin the parabolic approximation for the wave field polarized in a plane of incidence.The coefficient

Rs ¼ffiffiffiffiffieg

p tan ðwÞ�1þjg ffiffiffiffiffieg

p tan ðwÞþ1� �

ffiffiffiffiffieg

p tan ðwÞþ1� ��

1þjgð Þ(3.67)

also has the meaning of the reflection coefficient with provision for wave scatteringin the media above the surface. When the angle of incidence w becomes equal to aBrewster angle w ¼ a tan ð1= ffiffiffiffiffi

egp Þ, R0 ¼ 0, and the coefficient Rs is defined by the

ratio of the angular width of the scattered field to the width of the interference lobe.

Rs ¼jg

1þjgð Þ» jJ2s ðx;dÞ2c

ffiffiffiffiffiegp . (3.68)

Let us consider some limiting cases. One of the limiting cases is when de ¼ 0.We assume in Eq. (3.66) that rs ; g; b ¼ 0. In the area of applicability of reflection

formulas derived by

kx tan ðwÞ » kd >> ffiffiffiffiffiegp (3.69)

70


we can drop the terms of the order of eg =kx and eg =kx� �2

. Then

I x; z; z0ð Þ ¼ 1

x2 1þ R0j j2þ2R0cos DSð Þh i

. (3.70)

For sliding angles with ffiffiffiffiffiegp tan ðwÞ << 1 and

kd << ffiffiffiffiffieg

p (3.71)

the field intensity (3.66) can be expressed in terms of the familiar attenuation func-tion [9] wðxÞ ¼ 2jeg =ðkxÞ :

Iðx; z; z0Þ ¼ IðxÞ ¼ 1

x2 wðxÞj j2 . (3.72)

In this limiting case, R0 » � 1, DS << eg =ðkxÞ << 1.

Let us examine Eq. (3.66) in the presence of fluctuations of the refractive index,deð~rrÞ „ 0: The variance of the phase difference between the direct and reflectedwaves can be written in the form of a ratio of the angular width of the scattered fieldJ2s ðx; dÞ at the base d to the angular width of the interference lobe c:

M12ðx; dÞ ¼ J2s ðx; dÞ=c

2. (3.73)

Let us define three characteristic distances on the propagation path: xs ; xg and xcfrom the following relationships:

gðxg ; dÞ ¼ 1, M12ðxc ; dÞ ¼ 1, r2s ðxsÞ ¼ 1. (3.74)

Then

xs ¼ k�1=2

C�1e l

1=80 , xg ¼ kd

2=3C

2effiffiffiffiffieg

ph i�1, xc ¼ k

2d5=3

C2e

h i�1. (3.75)

One finds that xc is the distance where the variance of the phase difference at thebase d reaches the order of unity, or the angular spectrum of the scattered fieldbecomes wider than the interference lobe; xs is the distance where the variance ofthe fluctuation of the angle of incidence r

2s becomes equal to the square of the angu-

lar size of the Fresnel zone h2F ; and xg is the distance from which the angular spec-

trum of the scattered field becomes wider than the product of the width of the inter-ference lobe c and the effective grazing angle of the reflected wave tan ðwÞ= ffiffiffiffiffi

egp .

Because of the validity of inequality (3.56), the range of distances under considera-tion is such that x < xs . To analyse the value of the parameters we will define theratios

xsxg

¼ k1=2

d2=3

Ceffiffiffiffiffiegp l

1=60 , (3.76)

xsxc

¼ k3=2

d5=3

Ceffiffiffiffiffieg

pl1=60 , (3.77)

from which it follows that in the range of radio frequencies when k » 1 cm�1

andunder conditions of the earth’s atmosphere C

2e=10–14 cm–2/3, eg

&& && » 102, the ratio

71


xs=xg << 1. Given the fact that we have used the method of smooth perturbations(inequalities (3.8) and (3.56)), we have always g << 1, at the same time xc=xs < 1,when d > 103 cm.

In the area of sliding angle propagation when inequality (3.71) is satisfied, theparameter M12ðx; dÞ < 1 for all x < xs .

Let us assume that xc > xs , in this case M12ðx; dÞ < 1, g << min1; ffiffiffiffiffiegp tan ðwÞn o

, and Rs»R0 . In the interference region (inequality (3.69)) the fieldintensity can be approximated as follows:

I x; z; z0ð Þ ¼ 1

x2 1þ R0j j2þ2R0 cos DSð Þ exp �M12ðx; dÞ½ h i

. (3.78)

In the case where ffiffiffiffiffiegp tan ðwÞ << 1, when inequality (3.71) holds, Rs »R0 » � 1.In this case there are two small parameters M12ðx; dÞ and eg =kx. Let us introduceyet another characteristic distance xM , where M12ðxM ; dÞ ¼ eg =kxM :

x2M ¼ eg

k3C2e d

5=3 . (3.79)

When inequality (3.62) holds, xM < xs . Assuming x < xM the field intensity canbe approximated by a composition of two terms: one is an attenuation factorwðxÞand the other is a weighted variance of the angle fluctuation of the scatteredfield:

I x; z; z0ð Þ ¼ 1

x2 w xð Þj j2þ8egr2s ðxÞ

h i. (3.80)

In the range of distances xs < x < xM , the main contribution to the sliding anglepropagation mechanism is provided by scattering on the non-uniformity of therefractive index, de ~rrð Þ, and the field intensity in that region is given by

I x; z; z0ð Þ ¼ 1

x2 kdð Þ2J2s ðx; dÞ þ 8egr

2s ðxÞ

h i. (3.81)

In the regime of non-coherent composition of the direct and reflected waves wehave M12ðx; dÞ >> 1, and xc < xs . In this range the wave parameter kd

2=x must be

sufficiently large kd2=x >> 1. The interference structure of the wave field in this re-

gion is entirely distorted by the large fluctuation of the phase difference between thedirect and reflected waves and the intensity of the field is composed of two terms:

I x; z; z0ð Þ ¼ 1

x2 wðxÞj j2þ8r2s ðxÞ

h i. (3.82)

In conclusion, one can note that the limitation caused by the quadratic approxi-mation of the structure function H �ð Þ (inequality (3.63)) is not a fundamental oneand serves only to simplify the derivation of the asymptotic expansion for themoments uqu

��0

� �and uq

&& &&2D E. The analytical solution for the coherence function

and the moments of higher order can be obtained similarly to the solution for inten-sity provided in this section, however, the final expressions are exceedingly cumber-some.

72

3.3 Comments on Calculation of the LOS Field in the General Situation

3.3Comments on Calculation of the LOS Field in the General Situation

From the above study it is apparent that in the general case of the stratified tropo-sphere filled with random fluctuations in the refractive index, the line-of-sight fieldcan be calculated in a similar way, applying the ray theory. This approach can, inprinciple, be applied to calculation of the field in the tropospheric duct at distancesof up to a few hops. In the general case of refractivity, the intensity of the line-of-sight field can be presented in the form

Jðx; z; z0Þ ¼PNn¼1

PNm¼1

AnA�mexp jk Sn � Smð Þ �Mnm½ (3.83)

where An, Sn are the amplitude and phase of the nth ray, Mnm is a structure functionof the phase difference of the phases along the ray’s trajectories:

Mnm ¼ 0:73C2e k

2 Rx0

ds rnðsÞ � rm ðsÞj j5=3 (3.84)

and z, z0 are the heights of the receiving and transmitting antennas respectively, x isthe distance between them. The phase Sn along the nth ray is given by:

Sn ¼ jk2

Rx0

dsd2rnds2

þ em rn sð Þð Þ" #

(3.85)

and the trajectories rn(s) are given by a solution to the Euler equation

d2rnds2

¼ demdrn

(3.86)

with boundary conditions

rnð0Þ ¼ z0 , rnðxÞ ¼ z . (3.87)

Amplitude An accounts for the divergence (or convergence) of the rays

An ¼ AnðxÞ ¼Dn 0ð ÞDnðxÞ

�1=2(3.88)

where Dn(s) is a cross-section of the ray tube at a distance s along the nth ray. For thewaves reflected from the sea surface, the amplitude An is modulated by a reflectioncoefficient

An ¼ AnðxÞ � R hnð Þ (3.89)

where hn is the angle of incidence of the nth wave in the point of reflection. In mostcases, Eqs. (3.83)–(3.87) can be solved by applying computer-based algorithms.

Apparently, the above equations are not applicable with small grazing angles inthe case of impedance boundary conditions and should be modified before beingapplied in the vicinity of caustics.

73

3 Wave Field Fluctuations in Random Media over a Boundary Interface74

References

1 Fock, V.A. Electromagnetic Diffraction andPropagation Problems, Pergamon Press,Oxford, 1965.

2 Hitney, H.V., Richter, J.H., Pappert, R.A.,Anderson, K.D. and Baumgartner, G.B. Tro-pospheric radio propagation assessment,Proc. IEEE, 1985, 73 (2), 265–283.

3 Belobrova, M.V., Ivanov, V.K., Kukushkin,A.V., Levin, M.B. and Fastovsky, J.A. Predic-tion system on UHF radio propagation condi-tions over the sea, Institute of Radio Astron-omy, Ukrainian Acad. Sci., Preprint No 31,1989, 39 pp.

4 Bass, F.G., Braude, S.Ya., Kaner, E.A. andMen, A.V. Fluctuations of em waves in a tro-posphere in the presence of the boundaryinterface, 1961, Usp. Sov. Phys. Sci., 1961, 73(1), 89–119.

5 Puzenko, A.A., Chaevsky, E.V. Function ofmutual coherence in a problem of small graz-ing angle wave propagation in random medi-

um over boundary interface, Radiophys.Quantum Electron., 1976, 19 (2), 228–239.

6 Kostenko, N.L., Puzenko, A.A. and Chaevsky,E.V. Correlation of the amplitude and phaseof the scattered field in case of propagationthrough turbulent medium over boundaryinterface, Preprint of the Institute of Radio-physics and Electronics, Ukrainian Academyof Science, No 153, 1980, 36 pp.

7 Dashen, R. Path Integrals for waves in ran-dom media, J. Math. Phys., 1979, 20 (5),894–918.

8 Tatarskii, V.I. The Effects of the Turbulent Atmo-sphere on Wave Propagation, IPST, Jerusalem,1971.

9 Feinberg, E.L. Radio Wave Propagation alongthe Earth’s Surface, Nauka, Moscow, 1961.

10 Feynman, R.P. and Hibbs, A.R. QuantumMechanics and Path Integrals, McGraw-Hill,New York, 1965.

75

The most important characteristic of the troposphere in terms of radio wave propa-gation is a refractivity profile averaged over the coordinates tangential to the sea sur-face, i.e., an M-profile in a “stratified” troposphere:

MðzÞ ¼ 106 eðzÞ�1

2þ za

� �¼ 10

6 eðzÞ�12

� �þ 0:157z

where z is the height above the sea surface, the gradient 0.157 N-units m–1 takesinto account the curvature of the earth in the case of normal refraction. According toradio-meteorological and refractometer world-wide data [1–3], the gradient of themodified refractivity gm ¼ dM=dz is less than the critical gradient gc = –0.157 N-units m–1 for more than 50% of the time of observation over the sea surface. Inthose conditions the surface M-inversion forms an evaporation duct, which signifi-cantly affects the process of radio wave propagation by trapping the waves radiatedin a selective frequency band.

The simplest characteristic of the ducting properties of the surface based M-inver-sion is the “critical” wavelength kc defined as [4]:

kc ¼4310

�3bZs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 M 0ð Þ �M Zsð Þð Þ

p, m (4.1)

and

b ¼R10

ffiffiffiffiffiffiffiffiffiqðyÞ

pdy.

Parameter y is introduced in Ref. [4] as y ¼ z=Zs thus providing a meaningful def-inition for

qzZs

� �¼ 4

MðzÞ�MðZsÞMð0Þ�MðZsÞ

with q(0) = 4, q(1) = 0.Frequently observed surface M-inversions of height up to 15 m are capable of

ducting radiowaves in a cm band with wavelength k £ kc ~ 5 cm. With increasingwavelength the impact of the evaporation duct is weakened, though the attenuationof the waves with wavelength k > kc may still be significantly less than in the case

4

UHF Propagation in an Evaporation Duct

4 UHF Propagation in an Evaporation Duct

of normal refraction. Fock [4] showed that the attenuation of the field in the shadowregion follows the exponential law exp ð�cxÞ, where x is the distance from the hori-zon, and the magnitude of the attenuation exponent c depends on the curvature ofthe M-profile at the minimum point at z ¼ Zs [4]:

c � 2p10�3 ZsM

00 ðz¼ZsÞbffiffiffiffiffiffiffiffiffiffiffi2DM

p H (4.2)

where H is a coefficient determined by the imaginary part of the propagation con-stant tn.

The above estimates are qualitative in nature. To calculate the field we need tosolve the boundary problem (2.26), (2.27).

In a stratified troposphere, calculation of the electromagnetic field beyond the ho-rizon is truncated to the determination of the complex propagation constants En andthe height-gain functions vnðzÞ of the normal waves. The total field is then com-posed of a superposition of the normal waves, which converge in a shadow region.The analytical solution to the problem is known only for a few etalon problems witha limited number of selected M-profiles [4, 5]. In the general case the modal formal-ism will require a numerical solution of the characteristic equation for propagationconstants and, in many cases, numerical integration of the differential equations forthe height-gain functions [4–7].

This chapter is arranged as follows: First in Section 4.1, we discuss some resultsof the propagation measurements and comparison with the prediction based onexisting propagation models in order to highlight the current status of the theoryand the remaining problems in modelling the propagation phenomena. Then weintroduce the perturbation theory for the normal waves and propagation constantsin Section 4.2. The perturbation theory provides a means of analytic study of thespectrum of the propagation constants and the height-gain functions with smallvariations of the M-profile from the etalon profile for which the solution is known.

Section 4.3 is dedicated to the determination of the spectrum of normal waves inan evaporation duct. The solution is limited to the case of a stratified tropospherethat is commonly used in existing radio coverage prediction systems.

In Section 4.4 we will study the impact of random fluctuations in refractive indexon propagation inside an evaporation duct. The results suggest that the impact maybe significant in the frequency range 10 GHz and above.

Section 4.5 deals with the height gain structure of the field inside and outside theevaporation duct for the case of scattering on the turbulent fluctuations in the refrac-tive index. It is shown that scattering on random inhomegeneities of the refractiveindex leads to a smoother height dependence of the field beyond the horizon com-pared with duct theory. The results seem to be closer to observations.

76

4.1 Some Results of Propagation Measurements and Comparison with Theory

4.1Some Results of Propagation Measurements and Comparison with Theory

Systematic studies of radiowave propagation over the sea surface started in the1940s, driven by the development of radar, and are still underway due both to theimportance of the problem and advances in technology and applications. Severalcomprehensive programs of radio-meteorological measurements performed in the1940s and 1950s [8–10] still provide a benchmark reference for further studies dueto the exceptionally high quality of the obtained results. Among relatively recentstudies we may reference [2], see also several references to reports of the NavalResearch Laboratory in Ref. [2].

Here we briefly review the major results obtained in Ref. [10]. The propagationmeasurements were performed at two frequencies 3 GHz and 10 GHz over a seapath extending to several hundred kilometres and were supported by shipboardmeteorological measurements of the temperature, water vapor and air pressure. Therestored M-profile was then averaged over several measurements along the propaga-tion path and that averaged profile was approximated by a linear-exponential M-pro-file (Peceris’s model [11]) in order to perform a theoretical calculation of the propa-gation loss and to estimate the attenuation rate of the signal beyond the horizon. Asample of the comparison of the experimental attenuation rates with Peceris’s ductmodel is presented in Figure 4.1 for 3 and 10 GHz measurements.

As observed from Figure 4.1, the measured attenuation rates of the 3 GHz signaltend persistently to be less than predicted from the linear-exponential model of anevaporation duct. In contrast, the measured attenuation rate significantly exceededthe theoretical estimates. The comparison of the absolute values of the signalstrength data with the Peceris model was also performed in Ref. [10]. The conclu-

77

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5

Calculated attenuation, dB/km

Measu

red

att

en

uati

on

, d

B/k

m

3 GHz data

10 GHz data

Figure 4.1 Comparison of the measured and modelled attenuation rates at 3 and 10 GHz [10].


sion drawn in 1949 was that there was a significant quantitative discrepancy be-tween the theory and the measurements.

A second revision of the reference data from Ref. [10] was performed in the 80swith close involvement of the author. A new advanced computer-based model of anevaporation duct was implemented to calculate the propagation constants andheight gain functions. For comparison of the theory and measurements it wasdecided to limit to the distances to less than 150 km. The analysis of the measureddata led to the conclusion that the evaporation ducts restored at the time of measure-ment may explain the behavior of the signal in that sub-range, while at larger dis-tances the observed signal levels might be explained by either single-scattering the-ory in the case of low level signals or the presence of the elevated M-inversion. Thelast was difficult to analyse since there were no adequate radiosound measurementsperformed at the time of the radio measurements.

Figure 4.2 shows the result of a comparison of the same measured data as forFigure 4.1 at 3 GHz with a computer-based calculation for bilinear approximation ofthe averaged M-profile. As observed, the measured attenuation rates are still lessthan those predicted from the evaporation duct model. Major discrepancies are ob-served for lower duct heights, in conditions of unstable stratification. One of themeasured samples is shown in Figure 4.3, where the restored M-profile reveals anevaporation duct with parameters: Zs = 6 m, and DM = 2 N-units. Such discrepancytends to be persistent in other measurements and cannot just be explained by una-voidable errors in measurements of the meteorological data, see for instance Ref.[12]. We may also refer to the Ref. [5] where similar observations were reported. Inprinciple, the bilinear model tends to underestimate the attenuation rate comparedwith the more realistic linear-logarithmic model of the evaporation duct. This leads

78

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5


Measu

red

att

en

uati

on

, d

B/k

m

Figure 4.2 Comparison of the measured and modelled attenua-tion rates at a frequency of 3 GHz. The evaporation duct is mod-elled by a bilinear M-profile.


to the conclusion that the other models of propagation may provide a somewhatintermediate situation when the evaporation duct is not sufficiently strong to sup-port the trapping mechanism by itself, however, negative gradients of the refractivityprovide significant enhancement of the propagation beyond the horizon. One of theunusual but possible mechanisms is studied in Chapter 6.

It should be also noted that, in the presence of a strong evaporation duct withheights exceeding 10 m, the waveguide mechanism is more pronounced at a fre-quency of 3 GHz, and comparison of the evaporation duct model with measure-ments is rather satisfactory. A typical example is shown in Figure 4.4 for an evapora-

79

-60

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

20 25 30 35 40 45 50 55 60

Distance, kmS

ign

al

str

en

gth

rela

tive t

o f

ree s

pace,

dB

Figure 4.3 Comparison of the measured and predicted signallevels at 3 GHz in the presence of an evaporation duct withheight 6 m and M-deficit of 2 N-units.

-15

-10

-5

0

5

0 20 40 60 80 100 120 140 160

Distance, km

Re

ce

ive

d f

ield

str

en

gth

re

lati

ve

to f

ree

sp

ac

e,

dB

Theory

Measurements

Figure 4.4 Comparison of the measured and predicted signallevels at 3 GHz in the presence of an evaporation duct withheight 14 m and M-deficit of 7 N-units.


tion duct with height Zs = 14 m, and DM = 7 N-units. The heights of the transmit-ting and receiving antennas are 18 m and 6 m, respectively.

Figure 4.5 shows the comparison of the revised attenuation rates for frequency10 GHz. As observed, the spread of the data points relative to the bisector is widerthan in Figure 4.2 for the 3 GHz data. In most cases the measured attenuation ratesare higher than predicted and sometimes the actual propagation mechanism can be

80

0

0.5

1

1.5

0 0.5 1 1.5


Measu

red

att

en

uati

on

, d

B/k

m

Figure 4.5 Comparison of the measured and modelledattenuation rates at frequency 10 GHz. The evaporationduct is modelled by a bilinear M-profile.

-60

-50

-40

-30

-20

-10

0

10

10 20 30 40 50 60 70 80 90

Distance, km

Sig

na

l s

tre

ng

th r

ela

tiv

e t

o f

ree

sp

ac

e, d

B

(2)

(1)

Figure 4.6 Comparison of the measured (1) and predicted(2) signal levels at 10 GHz in the presence of an evaporationduct with height 9 m and M-deficit of 3 N-units.


questioned, for example in Figure 4.6. In general, the evaporation ducts with aheight of over 10 m are supposed to provide a waveguide mechanism at 10 GHzwith attenuation rate of several hundredth dB km–1 (chiefly due to a finite impe-dance of the sea surface). On the other hand, the attenuation rates measured duringthe experiment were never less than 0.2 dB km–1 at 10 GHz, even in the presenceof a very strong and stable with distance evaporation duct.

Figure 4.7 shows the results obtained when the height of the receiving antennaswas changed during the experiment. The evaporation duct structure was stable overthe distance and the assumption of the uniformity of the duct in a horizontal planewas quite reasonable during the experiment. The evaporation duct with parametersZs= 14 m, and DM = 7 N-units forms two trapped modes at frequency 10 GHz.The interference between these modes is responsible for the ripples in the theoreti-cal curve for the received signal at a height of 12 m. At a height of 6 m the interfer-ence is not pronounced since the second mode has a minimum at this height. Areceiver placed above the evaporation duct at a height of 18 m is supposed to receivethe em field by penetration through a potential barrier, or, in other words, by a“leak” of the trapped modes. Theoretically predicted signal levels at 18 and 6 mshould then differ by 20 dB, as shown in Figure 4.7. On the other hand, the mea-sured data do not reveal any significant difference with the height of the receivingantenna. According to the assessment in Ref. [2] this situation is not unique and isobserved in other experiments. One possible explanation of the above phenomena isprovided in Section 4.5.

Referring to recent studies described in Refs. [13–20] we may conclude that signif-icant development in modelling the evaporation duct structure in the lower part ofthe marine boundary layer resulted in the availability of computer-based tools capa-

81

-30

-20

-10

0

10

20

10 20 30 40 50 60 70 80 90 100 110

Distance, km

Sig

nal

str

en

gth

rela

tive t

o f

ree s

pace,

dB

Measured data at 6 m



Theory 6 m

Theory 12m

Theory 18 m

Figure 4.7 Comparison of measured and predicted data at10 GHz in the presence of an evaporation duct with height 14 mand M-deficit of 7 N-units. The transmitting antenna is mountedat a height of 6 m.


ble of reasonably good prediction of the shipboard radar coverage in nearly real timeover a wide range of frequency bands from VHF to EHF.

References [16, 17] provide comprehensive analysis of the propagation experi-ments performed in the Mediterranean and North Sea areas. The statistical assess-ment of the evaporation duct contribution reported in Ref. [16] also demonstratesunexpectedly high signal levels beyond the horizon in the case of weak ducting con-ditions at frequencies 0.6 GHz and 3 GHz, where the evaporation ducts withheights less than 10 m are too weak to ensure a single trapped mode. Nonetheless,the received signal tends to be higher than predicted, as shown in Figure 4.8 fromRef. [16]. The same statistics are reported in Ref. [17]. It was suggested in Ref. [16]that the observed high level signals might be caused by propagation mechanismsother than an evaporation duct. One of the conclusions emphasised in Ref. [17], andearlier in Ref. [2], is that the height dependence of the received field is not as signifi-cant as may follow from the evaporation duct theory and, basically, high altitudeantennas are preferable. In terms of the optimal frequency the results of the mea-surements suggest the 10–20 GHz band is an optimal band where the evaporationduct makes a strong impact without being counteracted by gaseous absorption athigher frequencies.

A clear demonstration of the advances in radar coverage prediction and range detec-tion enhancement due to an evaporation duct is published in Refs. [2, 12, 14] for18 GHz radar measurements. Similar results are reported in Ref. [13] for propagationexperiments in a range from 3 to 94 GHz. The predicted values of the path loss aredepicted as a curve in Figure 4.9. The ripples for the higher duct’s height are caused bythe presence of several trapped modes and corresponding restructuring of the heightgain function of the total trapped field in the waveguide. While overall agreement be-tween prediction andmeasurements is very good the prediction tends to underestimatethe path loss for lower duct heights. It might be noted that the scattering on the roughsurface is modelled in a rather conservative way using the Kirchhoff approximation [2,21], which is applicable to a coherent field component. Nonetheless, an additional

82

140

150

160

170

180

190

200

210

220

0 4 8 12 16 20 24

Evaporation duct height, m

Path

Lo

ss ,

dB

2.3 GHz

0.6 GHz

6.8 GHz

Figure 4.8 Path-loss versus evaporation duct height for the North Sea experiment [16].

4.2 Perturbation Theory for the Spectrum of Normal Waves in a Stratified Troposphere

attenuation factor, like scattering at random fluctuations in the refractive index, mightbe taken into account for possible improvement in the prediction model. It may also bepointed out that the spread of the data points in Figure 4.9 tends to be more close andsymmetrical relative to the prediction curve, which may again suggest the importanceof the full treatment of the scattering mechanism. We recall these arguments later inSection 4.3where the scatteringmechanism is described.

Another experiment in the millimetre wave range was reported in Ref. [15] for94 GHz propagation. The results obtained demonstrated the validity of the evapora-tion duct mechanism in this frequency band and gave good agreement between themeasurements and the theory of the evaporation duct, with some underestimationof predicted path loss with an average value of error of 10 dB. It might be noted thatwhile the prediction model has made use of the Kirchhoff theory of the scattering ata rough sea surface, the model did not take into account scattering on turbulent fluc-tuations in the refractive index.

4.2Perturbation Theory for the Spectrum of Normal Waves in a Stratified Troposphere

The presence of stratified inhomogeneities of the refractive index results in a consid-erable change in the field structure in the region of geometric shadow. A completesolution to the problem of wave propagation in a stratified troposphere, in whichthe refractive index depends only on the height h above the earth’s surface,was obtained by Fock [4]. He showed that for large values of the parameterm ¼ ðka=2Þ1=3 >> 1, here k is the wave number and a is the earth’s radius, the prob-lem can be simplified by transition from a spherically stratified to a plane-stratifiedmedium. Such a simplification is provided through the use of the parabolic approx-imation and introduction of the modified permittivity

em ðhÞ ¼ eþ 2ha.

83

Figure 4.9 Observed path losses for evaporation duct conditions at 17.7 GHz, from Ref. [14].


Even in this simplified formulation an analytical solution can be obtained onlyfor a few standard problems with a limited set of permittivity profiles em ðhÞ. In thegeneral case the analysis of the propagation conditions comes down to a numericalsolution for an eigenvalue problem with complex propagation constants. In severalcases the function em ðhÞ differs little from the “etalon” function, as for instance inthe case of weak refraction. Therefore, some need arises to construct a perturbationtheory for “open” system for which the spectrum of eigenvalues (propagation con-stants” is complex and eigenfunctions (height factors for the wave field vnðhÞ) growexponentially with height, vnðhÞ ! ¥; h ! ¥. A similar problem arises in quantummechanics with the study of the decay of quasi-stationary states. Perturbation theoryfor that case was developed by Zeldovitch [22], although the essential assumptionwas made on the finiteness of the potential (the analogue of em ðhÞ) at h ! ¥. Sucha condition is not satisfied in the problem of wave propagation in a stratified tropo-sphere:

em ðhÞ ! 2h=a as h ! ¥.

Here we describe a generalisation of the perturbation theory [22] to the case ofpotentials unlimited at h ! ¥.

4.2.1Problem Formulation

The field attenuation factor V for a point source in the spherically stratified mediumcan be represented by superposition of normal waves

Vðx; y; y0Þ ¼ 2ffiffiffiffiffiffipx

pejp=4 P¥

n¼1

ejxtn vðy;tnÞvðy0 ;tnÞv

ð0;tnÞ@

@tnvð0;tnÞ

(4.3)

where we introduce dimensionless coordinates x ¼ mD=a; y ¼ kh=m, k ¼k0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieðh ! ¥Þ

p, k0 is the wave number in a vacuum, and D is the distance along the

earth’s surface from the source of radiation. The transmitter (point source) is situ-ated at height h0 and the receiver at height h, respectively. The height functionsvðy; tnÞ are governed by the equation

d2

dy2vðy; tnÞ þ UðyÞ � tn½ �vðy; tnÞ ¼ 0 (4.4)

and satisfy the following boundary conditions

vð0; tnÞ ¼ 0,ddy

arg vðy; tnÞð Þf gy!¥

> 0 . (4.5)

Equations (4.4) and (4.5) determine a discrete spectrum of propagation constants tn.We represent the modified refractive index UðyÞ ¼ m

2ðeðyÞ � 1Þ þ y, as shown in Fig-ure 4.10, in the form UðyÞ ¼ U0ðyÞ þ dUðyÞ, where U0(y) is the unperturbed index ofrefraction for which the solution to the boundary value problem Eqs. (4.4) and (4.5) isknown. We shall treat dUðyÞ as the perturbations to the etalon profile U0(y).

84


Following the approach in Ref. [22], we introduce the logarithmic derivative

znðyÞ ¼dvðy;tnÞ

dy� v�1ðy; tnÞ (4.6)

through which the height function is expressed as

vðy; tnÞ ¼ limr!0vðr; tnÞ � expRyznðy

0 Þdy0 !

. (4.7)

For zn(y) we can obtain the equation

z0nðyÞ þ z

2nðyÞ þU0ðyÞ þ dUðyÞ � tn ¼ 0 . (4.8)

We seek zn(y) and tn in the form

znðyÞ ¼ z0nðyÞ þ dznðyÞ; tn ¼ t

0n þ dtn (4.9)

where z0nðyÞ and t

0n correspond to the height function of the unperturbed Eq. (4.4)

and dzn~ dtn~ dU: Substituting Eq. (4.9) into Eq. (4.8), we obtain the equation forcorrection to the height function

dz0n ¼ �2z

0nðyÞdznðyÞ þ dtn � dUðyÞ. (4.10)

The solution to Eq. (4.10) has the form

dznðyÞ ¼ v�20 ðy; t0nÞ

Ry0

v20ðy

0; t

0nÞ dtn � dUðy0 Þh i

dy0. (4.11)

It is reasonable to assume that perturbations dUðyÞ to a standard profile can beneglected at sufficiently large heights, then the solution to Eq. (4.4) as y ! ¥ hasthe same form as that of the standard problem v0ðy; tnÞ:

85

H)1( 3µ+

H

y

)(yU

Figure 4.10 Refractivity profile with distortion.


dznðyÞjy!¥ » d v�10 ðy; t0n þ dtnÞ ddy v0 y; t

0n þ dtn

� �� ¼ dtn

v2o ðy; t

0nÞ

·

voðy; t0nÞ

@2voðy; t

0nÞ

@y@tn� @voðy; t

0nÞ

@y

@voðy; t0nÞ

@tn

" #:

(4.12)

Substituting Eq. (4.12) into Eq. (4.11), we obtain the equation for correction to thepropagation constant

dtn ¼ N�1limjy!¥

Ry0

dUðy0 Þv2nðy0 Þdy0 (4.13)

where vnðyÞ ” v0ðy; t0nÞ.

Equation (4.13) differs from the usual equation of perturbation theory in twoaspects: first, there is a v

2nðyÞ under the integration instead of the square of the abso-

lute value vnðyÞ 2 and second, instead of the norm, which does not exist in the pres-

ent case, there is the finite term N equal to

N ¼ limjy!¥

Ry0

v2nðy

0 Þ � vnðy0 Þ @

2vnðy0 Þ

@ _yy@tnþ @vnðyÞ

@y0@vnðyÞ@tn

" #dy

0( )

¼ @vn@y

@vn@tn

� �y¼ 0

(4.14)

In the case of the troposphere the variation of the modified refractive indexbecomes linear at large enough heights and Eq. (4.14) takes the form:

N ¼ limjy!¥

Ry0

v2nðy

0 Þ � w21 ðt

0n � y

0 Þh i

dy0

( )� w

01ðt

0nÞ

h i2,

where w1ðt0n � yÞ is an Airy function [4] which is a solution to the standard problem

(4.4) and (4.5) with U0ðyÞ ¼ y.In the case of normal refraction- U0ðyÞ ¼ y; vnðyÞ ¼ wlðt

0n � yÞ for all 0 < y < ¥.

The parameter a shall be understood as equivalent to the earth’s radius. The propa-gation constants in this case are determined from the solution to a transcendentequation w1ðt

0nÞ ¼ 0 and for large numbers n they have an asymptotic form

t0n ¼ sne

jp

3 , sn ¼ 3=2 n� 1=4ð Þp½ �2=3 .Equation (4.14) is then reduced to

N ¼ limjy!¥

Ry0

w21 ðt

0n � y

0 Þdy0 þ ðt0n � yÞw21 ðt

0n � yÞ � w

01ðt

0n � yÞ

h i2( ). (4.15)

Calculating the integral appearing in Eq. (4.15),Rw21 ð�xÞdx ¼ xw

21 ð�xÞ þ w

01ð�xÞ

h i2(4.16)

and using the boundary condition (4.5), we obtain N ¼ � w01ðt

0nÞ

h i2and

dtn ¼ � w01ðt

0nÞ

h i�2R¥0

dUðyÞw21 ðt

0n � yÞdy . (4.17)

86


Equation (4.17) allows one to find explicitly the correction terms dtn to the com-plex propagation constants t

0n of the normal wave for arbitrary distortion dUðyÞ of

the linear profile of the refractive index. Of course the limitation is that the correc-tion terms prove to be small, i.e. dtnj j << t

0n � t

0n�1

. It should be noted that in thecase of large perturbation dUðyÞ a new branch of the spectrum of the normal modescould arise, that case is outside the scope of the problem considered here.

4.2.2Linear Distortion

Let us consider a linear distortion

dUðyÞ ¼ ð1þl3ÞðH�yÞ; y £ H0; y > H

((4.18)

where 1þ l3is the gradient of the modified refractive index in the height range

0 < y £ H as shown in Figure 4.10. The equation for the spectrum tn of the normalmodes was derived by Fock [4], its solution cannot be obtained analytically andrequires application of numerical methods. Application of perturbation theory inform (4.17) allows investigation of the tn-spectrum in a considerably simple way.

Let the height of distortion be small, assuming that H << sn and sn >> 1. Thenthe phase of the argument of the Airy function lies within the sector p=3; 2p=3ð Þand the asymptotic form of w1ðt

0n � yÞ is as follows:

w1ðt0n � yÞ~ðt0n � yÞ�1=4

e23ðt0n�yÞ3=2

þ je� 23ðt0n�yÞ3=2

( ). (4.19)

Substitute Eqs. (4.18) and (4.19) into Eq. (4.17) and expand the exponentials inEq. (4.19) in a series in powers of y=t

0n . Retaining the linear terms, which is valid

when inequality H2<<

ffiffiffiffiffisn

pholds, we obtain

dtn ¼ jð1þ l3Þ t

0n

� ��1=2w01ðt

0nÞ

h i�2H

2 þ 4t0n

� ��12� e

�2Hffiffiffit0n

pþ e

2Hffiffiffit0n

p� �� .

(4.20)

When the stronger inequality H2sn << 1 applies we can obtain from Eq. (4.20)

dtn ¼ �ð1þ l3Þ H4

12þH5 t0n

90

( ). (4.21)

As observed from Eq. (4.21), increase in the height H or depth ð1þ l3ÞH of inver-

sion in the refractive index leads to a decrease in the real and imaginary componentsof the propagation constant tn, that corresponds to an increase in the phase velocityof the normal wave and a decrease in its attenuation.

The applicability of the perturbation theory is limited by a small variation to thepropagation constant

dtnj j << t0n � t

0n�1

» pffiffiffiffiffisn

p . (4.22)

87


Inequality (4.22) in conjunction with H2sn << 1 yields the limits of applicability

of Eq. (4.21):

H << min s�1=2n ;

12

ð1þl3Þ

� �1=4

s�1=8n

" #(4.23)

and, for Eq. (4.20)

H <<1ffiffiffiffiffisn

p ln16s3=2n

1þl3

!. (4.24)

Let us consider another limiting case, when the height of distortion is largeH >> sn . We represent the integral in Eq. (4.17) as the sum of integrals over the arcof radius H in the sector of angles 0;p=3ð Þ and over the ray arg y ¼ p=3:

RH0

ðH � yÞw21 ðt

0n � yÞdy ¼ I1 þ I2 ,

I1 ¼RHej p=3

0

ðH � yÞw21 ðt

0n � yÞdy ,

I2 ¼ jHRp=30

ð1� ejjÞw2

1 ðt0n �He

jjÞdj .

In the integral I1 we make a change in the variable y ¼ sejp=3

and use the propertyof the Airy function w1ðxe

jp=3Þ ¼ 2ejp=6

vð�xÞ :

I1 ¼ 4ej2p=3 RH

0

ðH � sejp=3Þv2ðs� snÞdy . (4.25)

The upper limit of integration in Eq. (4.25) can be extended to infinity since theAiry function vðs� snÞ decreases exponentially for s > sn. Then, using Eq. (4.16) weobtain

I1 » 4ej2p=3

HR¥0

v2ðs� snÞdy ¼ �4e

j2p=3H

ffiffiffiffiffisn

p. (4.26)

To estimate integral I2 we use the asymptotic behavior of the function w1ðt0n � yÞ

for large negative arguments

I2 ¼ �H2 Rp=30

ð1� ejjÞðHe jj � t

0nÞ�1=2

ejWðjÞ

dj (4.27)

where WðjÞ ¼ 4=3H3=2

ej 3u=2 � 2

ffiffiffiffiffiH

pej u=2

t0n . The main contribution to the integral

(4.27) comes from the region of small j where the only linear terms can be retainedin the expansion of the integrand. Then we obtain

I2 @1

4H3=2 exp j43H

3=2 � 2 jffiffiffiffiffiH

pt0n þ j

p2

� �. (4.28)

Substituting Eqs. (4.28) and (4.26) into Eq. (4.17) we obtain

88


dtn ¼ ð1þ l3ÞH 1� j

16snH5=2 exp j

43H

3=2 � 2 jffiffiffiffiffiH

pt0n

� �" #. (4.29)

As observed from Eqs. (4.29) and (4.21), the correction factor dtn grows exponen-tially with number n since the imaginary part of the propagation constant isIm tn~n

3=2, and hence the inequality (4.22) ceases to be satisfied, starting with cer-

tain n . It can be shown, however, that this result is a consequence of the “non-physi-cal” nature of the broken linear profile. For smooth functions U(y) that actuallydescribe realistic behavior of the refraction index, the dtn -dependence on a wave-number n has a fundamentally different character (see, for instance, Eq. (4.40)below).

In fact, performing integration by parts in Eq. (4.17), we can write

dtn ¼ �N�1 R¥0

ddUðyÞdy

FnðyÞdy (4.30)

where FnðyÞ ¼Ry0

w21 ðt

0n � y

0 Þdy0 behaves as 1=2t0n exp ð2t0n ffiffiffi

yp Þ for large numbers n:ffiffiffiffiffi

snp

y >> 1. If dUðyÞ is a function continuous along y altogether with its derivativesand which decreases sufficiently at infinity, then integration in Eq. (4.30) will resultin values of dtn which decrease for large n.

In the case where dUðyÞ is abruptly reduced to zero at y = H (a discontinuity ofthe first kind), a singularity of a delta-function type appears in the integrand inEq. (4.30) at this point, and at large tn it make the main contribution to the integral:

dtn~�N�1Fnðy ¼ HÞ~ exp 2H

ffiffiffiffit0n

q� �1

2 t0n� �3=2 . (4.31)

This result can be generalized to the case of discontinuity of the kth derivative:

dtn~ exp 2Hffiffiffiffit0n

q� �1

2 t0n� �3=2 1

2k t0n� �k=2 . (4.32)

4.2.3Smooth Distortion

In the atmosphere the averaged dielectric permittivity eðhÞ is a sufficiently smoothfunction that normally decreases with height. Let us consider a perturbation dUðyÞdescribed by a continuous function, which permits analytical expansion into the re-gion of complex y. To calculate the correction term it is convenient to use, instead of

the function dUðyÞ, its Laplace transformation: FðsÞ ¼R¥0

dye�sy

dUðyÞ:

dtn ¼ N�1 Rrþj¥

r�j¥dsFðsÞ

R¥0

esyw21 ðt

0n � yÞdy (4.33)

Let the function dUðyÞ be such that F(s) has no singularities in the right halfplane s or on the imaginary axis. Then the integration contour over ds can be shifted

89


into the left half plane, while the contour in the integral over dy can be deformedinto a ray with argðyÞ ¼ p=3. Substituting the asymptotic form (4.19) into Eq. (4.33)we obtain

R¥0

w21 ðt

0n�yÞesydy ¼ 4e j2p=3

R¥0

v2ðy�snÞ exp ðsyejp=3Þdy þ

limjR!¥

R�3=2 exp sRþ4sn

3

ffiffiffiffiR

p� �h i:

(4.34)

In line with the above assumptions, Re s < 0, so the second term in Eq. (4.32)vanishes. Using an inverse Laplace transformation, we obtain

dtn ¼ 4ej2p=3

N�1 R¥

0

dUðyejp=3Þv2ðy� snÞdy. (4.35)

It should be noted that Eq. (4.35) could be used when the convergence sector ofthe function dUðyÞ is wider than p=3.

Let us consider the application of Eq. (4.35) to the example of perturbation givenby dUðyÞ ¼ U0exp �y=Gð Þ. We set sn >> M1 and H << 1. Substituting the asymp-totic form vðy� snÞ, we obtain

dtn ¼ 2U0ejp=3

N

Ry0

dy exp� ye

jp=3

H

!1þ sin

43

sn � yð Þ3=2� ��

. (4.36)

When inequality H2<<

ffiffiffiffiffisn

pis satisfied Eq. (4.36) can be reduced to

dtn ¼ 2U0e jp=3

Nffiffiffiffiffisn

p H 1� 1

1�4snH2e

� j 2p=3

" #. (4.37)

When H2sn << 1 Eq. (4.37) converges to a simple form

dtn ¼ �U0 2H3 þ 8snH

5ejp=3

h i. (4.38)

In the opposite case,H2sn >> 1, we obtain

dtn @U0H2sn

e�jp=3

. (4.39)

4.2.4Height Function

The correction to the height function v0ðy; t0nÞ for known dtn is determined by the

integral (4.11), which allows one to obtain simple equations for dznðyÞ in variouslimiting cases.

For example, let us consider the case when the antennas are placed sufficientlylow above the earth’s surface, i.e. y ~ y0 and y

2sn << 1, while the perturbation dUðyÞ

is given by the function (4.18). Expanding v0ðy; t0nÞin a series in powers of y, we

obtain

90


vðy; tnÞ » v0ðy; t0nÞexp

dtny2

6� ð1þ l

3Þ y3

36

" #. (4.40)

As observed from Eq. (4.40), the new height function vðy; tnÞ is localized nearthe surface to a greater degree that the unperturbed one v0ðy; t

0nÞ.

4.2.5Linear-Logarithmic Profile at Heights Close to the Sea Surface

Consider another specific kind of perturbation to the refractivity profile, which canbe applied to the estimation of the impact from large gradients of humidity in theimmediate vicinity of the sea surface. Refractometer measurements or measure-ments of the meteorological parameters are difficult to perform at these heightsbecause of the roughness of the sea surface. On the other hand, this layer can becharacterised by extremely rapid changes in refractivity due to the gradients ofhumidity. As follows from hydrodynamic theory of the evaporation duct [5, 23], themodified refractivity profile in the layer close to the sea surface can be modelled by alinear-logarithmic function:

UðyÞ ¼ y� Ys lnyyr

� �(4.41)

where Ys ¼ kZs=m and Zs is the evaporation duct’s height, yr ¼ kzr=m, and zr is theheight above the sea surface below which it is not feasible to make the measure-ments. Normally zr is associated with a sea roughness parameter. Now we considerthe impact from the gradient of humidity in a layer below yr as a perturbation dU toa linear profile:

dU ¼�Ys ln

y

yr

� �; y £ yr

0; y > yr :

8><>: (4.42)

In fact, we may approximate the initial profile (4.41) with a linear segment in theinterval 0 < y < yr as discussed later in Section 4.3, and regard the combined profileas the “true” profile while considering the logarithmic part as the perturbation.

With the perturbation in form (4.42) and if y2r sn << 1 we obtain the correction

term to the propagation constant of the first mode

dtn ¼ � Ysy3r

9þ y5r t

0n

75

!. (4.43)

We may now evaluate the reference height zr below which the exact shape of theprofile does not make a difference to the calculation of the field attenuation beyondthe horizon. The impact perturbation is negligible if the variation in attenuation ofthe first mode due to perturbation (4.42) is small, i.e.

91


kx

2m2 Im tn » 6 � 10�6 k

6z5rm7 x << 1. (4.44)

In the range of distances x ~ 100 km for frequencies of the order of 10 GHz , wemay see that the above inequality is satisfied with zr £ 2 m. Therefore, we may statethat the refractivity measurements can be performed from heights above zr = 2 mand the M-profile close to the surface at heights below zr can be approximated by alinear function or any other function approaching a finite value at z = 0.

4.3Spectrum of Normal Waves in an Evaporation Duct

As shown in Chapter 2, the attenuation function of the wave field produced by apoint source in a shadow region is determined by the sum of residues in the polesof the S-matrix. The pole’s position in a plane of complex t ¼ m

2=k

2E is given by

the roots of the equation

vþ

0; tð Þ ¼ 0. (4.45)

The function vþðh; tÞ is a solution to Eq. (2.36) representing the wave outgoing to

infinity, h ! ¥. The shadow region is defined as the region of the distances n > nh ,where nh is a horizon of the wave reflected once from the earth’s surface.

In the case of linear approximation

UðhÞ ¼ l31ðHs�hÞ; h £Hs

l32ðh�HsÞ; h>Hs :

8><>: (4.46)

Eq. (4.45) can be truncated to the following

1� RsðtÞRg ðtÞ ¼ 0 (4.47)

where

Rs ¼ �l1w1ðx2Þw

01ðx1Þ þ l

2w01ðx2Þw1ðx1Þ

l1w1ðx2Þw

02ðx1Þ þ l

2w01ðx2Þw2ðx1Þ

, (4.48)

Rg ¼ �w1ðx0Þw2ðx0Þ

,

x1 ¼ t

l21

; x2 ¼ t

l22

; x0 ¼t�l3

1Hs

� �l21

.

Here, Rs and Rg are reflection coefficients of the waves from the boundaries h ¼ Hs

and h = 0, respectively. It should be noted that we assume an ideal reflection at theboundary h = 0, since the boundary impedance q is assumed to be infinite, qj j ! ¥.

92

4.3 Spectrum of Normal Waves in an Evaporation Duct

Note that here and throughout the book we use a “shifted” M- and em ðzÞ profileto define a non-dimensional profile U(h). In particular, the “shifted” eem ðzÞ profile isgiven by eem ðzÞ ¼ em ðzÞ �min em ð0Þ; em ðZsÞf g. The non-dimensional U-profile isthen determined as

UðhÞ ¼ m2eem ðyÞ � 1½ � ¼ m

2eðyÞ � 1�min e 0ð Þ; eðHsÞ½ �½ � þ y .

The difference between the conventional U-profile and the shifted one is an addi-tional phase shift in the propagation constant that can be accounted for as a phasecorrection in the definition of the envelopeW1 given by Eq. (2.16).

Consider positive values of t. With Re(t) > 0, in accordance with the general prin-ciple of quantum mechanics, the movement of a particle in a potential (4.36)becomes finite, i.e. the wave is reflected from the potential barrier. With t >> l

21 ; l

21 ,

Rs @� 1þ 2jCðtÞ (4.49)

where

C ¼ �l1w1ðx2Þt

0 ðx1Þ þ l2w01ðx2Þtðx2Þ

l1w01ðx1Þw2ðx2Þ þ l

2w02ðx2Þw1ðx1Þ

(4.50)

and Eq. (4.47) is equivalent to

tðx0Þ � Cw1ðx0Þ ¼ 0. (4.51)

We can use perturbation theory to solve the above equation for small values ofC(t) which correspond to large positive values of the parameter t. Let us assume firstthat C = 0. In this case the roots of the truncated Eq. (4.51) are defined by a series ofpositive real numbers:

t0n ¼ l

31Hs � l

21nn (4.52)

where n1 ¼ 2:338, n2 ¼4.088, n3=5.521; with n >> 1, nn ¼ 32p n� 1

4

� �� 2=3. The

first correction term dtn to the value of t0n takes the form

dtn ¼ �l21C t0n� �

t0 �nnð Þ� �2 » l2

1ffiffiffiffiffinn

p l31þl3

2

16 t0n� �3=2 exp � 4

3l31

t0n

� �3=2" #þ jcn

8><>:

9>=>; (4.53)

and

cn ¼l21

4ffiffiffiffiffinn

p exp � 43

l31þl3

2

l31l32

t0n

� �3=2( ). (4.54)

Parameter cn has the meaning of the square of the module of the coefficient ofpenetration through the potential barrier.

93


Equation (4.51) has a finite number N of roots with Re tn > 0, which determinesthe number of the trapped modes in the evaporation duct. This number can be esti-mated as

N ¼ entier23p

l1Hs

!3=2 þ 14

� �. (4.55)

The contribution of the trapped modes to the total signal strength is given by thesum of the residues of integrand (2.40) in the poles tn ¼ t

0n þ dtn :

Wðx; z; z0Þ ¼ffiffiffinp

rejp=4 PN

n¼1

ej ntn Fnðh; h0Þ (4.56)

The function Fn h; h0ð Þ is given by the following equations:With h; h0 £Hs ,

Fn h; h0ð Þ ¼l1ffiffiffiffiffinn

p tðl1h� nnÞtðl1h0 � nnÞ ; (4.57)

with h > Hs ; h0 £ Hs ,

Fn h; h0ð Þ ¼ffiffiffiffiffil1

l2

sn�1=4n c

1=2n w1

tnl22

� l2 h�Hsð Þ !

tðl1h0 � nnÞ ; (4.58)

and with h; h0 > Hs ;

Fn h; h0ð Þ ¼ cnl21

w1tnl22

� l2 h�Hsð Þ !

w1tnl22

� l2 h0 �Hsð Þ !

. (4.60)

For arbitrary heights and M-deficits of the evaporation duct the propagation con-stants tn can be calculated by using numerical solutions of Eq. (4.47). In relation to aproblem of quantum mechanics such a study was presented in Ref. [23] and asso-ciated references. In a problem of wave propagation through the troposphere a rele-vant study was presented in Ref. [24]. While the case of elevated M-inversion wasconsidered in Ref. [24], it seems important that two branches of the propagation con-stant can be distinguished in that case. One branch corresponds to the “whisperinggallery” modes, i.e. trapped modes, while the second branch corresponds to the dif-fraction modes. Figure 4.11 shows a similar study of the propagation constants tnfor an evaporation duct with a linear M-profile (4.46), where the circles show theposition of the propagation constant in a complex plane t as a function of the ductheight Zs with constant gradients l1 ; l2 . In this case we can also distinguish twobranches of the propagation constants. As observed from Figure 4.2, the diffractionmodes are concentrated near the ray argðtÞ ¼ p=3, while the second branch of theroots of Eq. (4.47) traverses from the left half-plane of t to the right half-plane withincreasing depth of the M-inversion, asymptotically reaching the positions corre-

sponding to the trapped modes tn ¼ l31Hs � l

21nn , where nn ¼ 3p=2 n� 1

4

� �� 2=3.

The waveguide mode with n = 1 is common to both branches. The contribution ofthe diffraction modes becomes negligible with increasing depth of the M-inversion

94


compared with the contribution of the residues in the poles in the left hand-side ofthe t-plane.

We may also obtain the asymptotic solution for the characteristic equation (4.45)in the case of the smooth M-profile using Fock’s theory [4]. In particular we areinterested in the asymptote of Eq. (4.45) for small values of Re tn , i.e., in the vicinityof the minimum of the M-profile where it is most significantly different from theother approximations, especially the bilinear one. Using the WKB approximation forthe height gain functions in Eq. (4.45) for a smooth but arbitrary profile we obtain

jp2� pm

2� jmþ j m ln mþ 2jS1 � C0 þG ¼ j 2pn (4.61)

where

C0 ¼ 0:91894; G ¼ lnC12þ jm

� �; m ¼ tnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2U00Hsð Þ

q ;

S1 ¼ 2

3l31

U0 � tnð Þ3=2� jp m2

.

(4.62)

For a linear-logarithmic M-profile under conditions of neutral stratification theparameters in Eq. (4.62) are defined as follows:U

00ðHsÞ ¼ l

32=Hs and U0 ¼ l

32Hs ln Hs=Hrð Þ, where Hr ¼ kzr=m and zr is the mini-

mum height at which the radio meteorological measurements can be carried out,the height zr is normally associated with sea roughness.

The results of the calculation of the propagation constant t1 of the first mode forboth bilinear and linear-logarithmic M-profiles are shown in Table 4.1. The linear-logarithmic M-profile corresponds to the case of stable stratification when the valueof the M-deficit DM and the duct height Zs are related via the approximate expres-sion: DM ¼ gsZs , where gs ¼ 0:3 N-units m–1.

95

0

1

2

3

4

5

6

-4 -2 0 2 4

Re t

Im t

Diffraction modes

Guided modes

Figure 4.11 Evolution of the propagation constant in the t-plane with changes in the duct height.


Table 4.1 Propagation constants of the first mode for bilinear and logarithmic M-profiles.

Duct height, m Bilinear Model Linear-logarithmic Model Asymptotic

Re t Im t Re t Im t Re t

20 6.992 0 6.994 0 6.99819 6.237 0 6.27 0 6.26818 5.529 0 5,562 0 5.55917 4.838 0 4.871 0 4.86716 4.167 0 4.199 0 4.19515 3.515 0 3.548 0 3.45314 2.887 0 2.919 9 2.91413 2.285 0.0000467 2.316 0.00000154 2.30912 1.717 0.00086 1.741 0.0000098 1.73111 1.194 0.00857 1.199 0.000095 1.18410 0.733 0.0452 0.7 0.0016 0.679 0.335 0.144 0.027 0.032 0.198 0.00002 0.33 -0.04 0.133 -0.0237 –0.25 0.631 –0.043 0.46 –0.596 –0.85 1.04 –0.043 0.5 –0.89

The value of the M-deficit can be determined by approximating the logarithmicM-profile in the interval 0 < z < Zs by a linear M-profile with gradient equal toM

0 ðzÞz¼zr

, i.e., at the sea roughness level. As can be seen from comparison of the

results in Table 4.1, for very strong M-inversion, when the mode is deeply trappedinside the duct, i.e., Re >> tn >> 1, the propagation constants for the linear-logarith-mic and bilinear profiles and asymptotical values are practically equal to each otherand to asymptotical values for the linear profile t

01 ¼ l

31Hs � l

212:338, where

l31= a � 10

�6gs . In the case of weak trapping, when Re tn £ 1, the difference between

the two profiles becomes significant. This is caused by the behavior of the reflectionof trapped waves from a potential barrier in the vicinity of the minimum of U(h). Inparticular, the imaginary part of the propagation constant of the linear-logarithmicprofile is less than one obtained from a linear profile since the thickness of thepotential barrier is greater in the case of the logarithmic profile, Figure 4.12.

The contribution of the first mode in Eq. (4.46) in the case of the linear-logarith-mic profile can be written in a form similar to Eqs. (4.57) to (4.60):

WithHs << l22 t1j j1=4 , then for all h; h0

F1 h; h0ð Þ ¼ 1

4l22

ffiffiffiffiffin1

q exp jS h; t1ð Þ þ jS h0 ; t1ð Þ þ jp3

� �. (4.63)

When h; h0 £ Hs

F1 h; h0ð Þ ¼ � 2DK

sin S h; t1ð Þ½ � sin S h0 ; t1ð Þ½ � (4.64)

where

96


D ¼ U hð Þ � t1ð Þ � U h0ð Þ � t1ð Þ½ ��1=4, K ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiU0�t1

pl31

.

When both antennas are located above the duct, i.e., h; h0 > Hs , then

F1 h; h0ð Þ ¼ T �D2K

exp jS h; t1ð Þ þ jS h0 ; t1ð Þ½ � (4.65)

where

T ¼ 1� Rs t1ð Þj j2 , Rs ¼1

1þe2pmt1. (4.66)

And, finally, when only one antenna is placed above the duct, say h > Hs, we obtain

F1 h; h0ð Þ ¼ jD2K

sin S h0 ; t1ð Þ½ � exp jS h; t1ð Þ½ �. (4.67)

In the case of an arbitrary M-profile and a finite value of the surface impedance,the solution to Eq. (4.45) can be found by using numerical methods. In this case wemay introduce a normalised height gain function f instead of the function v

þðh; tÞ

f ðh; tÞ ¼ vþðh;tÞvþð0;tÞ

. (4.68)

Function f is then governed by the uniform equation

d2 f

dh2þ UðhÞ � t½ � f ¼ 0 (4.69)

and satisfies the following boundary conditions at h = 0

f ð0; tÞ ¼ 1;dfdh

h¼0

¼ q (4.70)

where q ¼ jm=ffiffiffig

p.

97

M∆

sZ

rz

)(zM

z

Figure 4.12 Smooth M-profile.


Note that for horizontal polarization the parameter q should be replaced byq0 ¼ jm

ffiffiffig

p. At the upper boundary of the duct, at some height ho , ho ‡Hs , the func-

tion f should satisfy the condition of the outgoing wave, i.e., d=dh argðf ðh; tÞÞ > 0.While this height can be arbitrarily chosen, in practice the profile U(h) is approxi-mated by a linear one, thus allowing one to represent the outgoing wave via the Airyfunction w1 .

Equation (4.69) for the height-gain function f can be integrated by standardnumerical methods making use of boundary conditions (4.70) and the analyticaltransition of the function and its derivative at the height ho . The application ofboundary conditions at both boundaries h = 0 and h ¼ ho leads to the characteristicequation Q(tn) = 0 for the propagation constants tn.

The attenuation factor W in the shadow region can then be represented as thesum of normal waves equivalent to the residue sum of the integral (3.43) in thepoles of the S-matrix:

Wðn; h; h0 Þ ¼ 2ejp=4 ffiffiffiffiffiffi

pnp P¥

n¼1

ejntn dtn

dqf h; tnð Þf ðh0 ; tnÞ (4.71)

where parameterdtndq

¼ � ddh

@f@tn

� �h¼0

" #�1

is called the excitation coefficient of the

normal wave with number n.

The characteristic equation Q(tn) = 0 in the general case can be realised in com-puter-based calculations by simultaneously solving Eq. (4.69) for height-gain func-tions with a trial value of tn. In the case of the bilinear profile (4.46) the characteristicequation Q(tn) = 0 is exactly the same as Eq. (4.47). In the case of the linear-logarith-mic profile, as shown in Section 4.1, the refractivity profile close to the sea surfacecan be replaced at the heights h < hr by a linear segment with appropriate gradientwithout appreciable errors in the estimation of the propagation constant. In thatcase the resulting profile consists of three segments: the linear profile in the interval0£ h < hr , the logarithmic profile UðhÞ ¼ U hrð Þ � k ln g=hrð Þ, and the linear profileat heights h > ho . In this case the solution to Eq. (4.52) should be tailored in termsof the continuity of the function f and its derivative df =dh with analytical representa-tion of the function f via Airy functions in linear segments.

The numerical algorithm for finding the propagation constants can be implemen-ted in the following way. Assume that we know the etalon solutions t

0n for a specific

profile U0(h). The respective characteristic equation we denote as Qðt0nÞ. Instead ofthe equation Q(tn) = 0 for the propagation constant of the problem we may use

Qðtn ; rÞ ¼ 1� rð ÞQðt0nÞ (4.72)

introducing a new parameter r varying between 0 and 1. The roots of Eq. (4.72)depend on thevalue of r. When r = 0 we have a solution of the etalon problem, i.e.tn ¼ t

0n , with r = 1 the solution to Eq. (4.72) gives “true” propagation constants.

With variation in r from 0 to 1 the roots of Eq. (4.72) traverse along some trajectoriesin the complex plane t from t

0n to tn . The equation for the trajectories is

98

4.4 Coherence Function in a Random and Non-uniform Atmosphere

dtndr

¼ � Qðt0nÞ@

@tnQ tn ;rð Þ

. (4.73)

In practice, the parameter r should be related to one of the inherent parametersof the problem, i.e., this might be the impedance q, the height of the duct Hsor thegradient of the M-profile above or below Hs. In that case the trajectories of the rootstnðrÞ are likely to be restrained by their own valley in the module profile ofQðtn ; rÞj j in the plane of complex t [7]. The algorithm based on the evolution of theheight of the evaporation duct has been implemented in a radio coverage predictionsystem [25, 26] with the close participation of the author. In that system we used abilinear approximation of the linear-logarithmic M-profile for the evaporation ductdescribed in terms of a single parameter, the height of the duct,Hs . The approximat-ing linear profile reveals an almost constant gradient within h < Hs thus allowingone to use one parameter Hs to build the evolution algorithm similar to Eq.(4.55)and create look-up tables for pre-calculated propagation constants tn Hsð Þ.

Various well-developed methods of calculation of the propagation constants canbe found in the literature. The formalism of the duct propagation in a stratified tro-posphere has been developed in a number of classical works started in the 1940s [6,27, 28]. Until now, those publications provide a benchmark reference and a frame-work for the detailed study of duct propagation. The practical algorithms incorporat-ing the mode theory have been implemented in various computer-based programsand prediction systems [29, 30]. Among known models of the M-profile describingthe average refractivity in the atmospheric boundary layer, the linear-logarithmicprofile is accepted as the most adequate M-profile model for the evaporation duct tobe utilised in development of the radar coverage prediction systems.

4.4Coherence Function in a Random and Non-uniform Atmosphere

4.4.1Approximate Extraction of the Eigenwave of the Discrete Spectrum in the Presence ofan Evaporation Duct

Consider Eq. (2.35) in the case when the evaporation duct is present in the heightinterval ð0;HsÞ. Let us seek a solution to the attenuation function W in the expan-sion over the system of eigenfunctions of the continuous spectrum defined in Sec-tion 2.2

Wð~rrÞ ¼R¥�¥

dE � Aðx; y;EÞWE ðzÞ , (4.74)

Aðx; y;EÞ ¼R¥0

dz �Wðx; y; zÞWE ðzÞ . (4.75)

99


The function WE ðzÞ obeys Eq. (3.37), boundary conditions (2.38) and conditions(2.40), (2.41) for orthogonality and completeness respectively. It also has singulari-ties of pole type in an upper half-space of E, asymptotically approaching rayarg E ¼ p=3 with Ej j ! ¥ when em ðzÞ ! 2 z=a with z ! ¥. Because of the pres-ence of M-inversion in a near-surface layer, an evaporation duct, some finite numberof poles En, (n = 1, 2,...N) lie close to the real axis of E. The value of Re En , denotedfurther as ~EEn , belongs to the interval k

2em ðHsÞ � 1ð Þ < ~EEn < k

2em ð0Þ � 1ð Þ. Such

poles correspond to waves trapped in the waveguide channel created by the evapora-tion duct. Without loss of generality we consider here the situation when N = 1, i.e.a single mode waveguide (evaporation duct). It is the most common case for theevaporation duct propagation of radio waves in a range of about 10 GHz.

Let us split the interval of the integration in Eq. (4.49) into two domains and setW = W1 + W2 where

W1ð~rrÞ ¼R¥Em

dE � Aðx; y;EÞWE ðzÞ ,(4.76)

W2ð~rrÞ ¼REm�¥

dE � Aðx; y;EÞWE ðzÞ ,

and Em ¼ k2em Hsð Þ � 1ð Þ. Using Eq. (4.75) we can evaluate the contribution of the

term W1. In the shadow region x >ffiffiffiffiffi2a

p ffiffiffiz

pþ ffiffiffiffiffi

z0p !

, where z, z0 are the heights ofthe transmitting and receiving antennas respectively, the major contribution to theintegral (4.51) comes from the E-interval k

2em ðHsÞ � 1ð Þ < E < k

2em ð0Þ � 1ð Þ and

the height interval z < Hs :

W1¼R¥Em

dE�R¥0

dz0Wðx;y;z0 ÞWE z0 !

WE ðzÞ »

RE0Em

dE�RHs

0

dz0Wðx;y;z0 ÞWE z0 !

WE ðzÞ:(4.77)

Within the interval Em < E < E0 we can distinguish a function u0E ðzÞ, which

depends little on E

WE ðzÞ » CðEÞu0E ðzÞ (4.78)

and normalised by the following equation

RZs0

u0E ðzÞu

0E ðzÞdz ¼ 1 . (4.79)

The meaning of Eqs. (4.78) and (4.79) is that we approximated the functionWE ðzÞ of the continuous spectrum by the eigenfunction of discrete spectrum u

0E ðzÞ

localised inside the evaporation duct in such a way that the residue of Eq. (4.74) inthe pole E ¼ ~EEn would approximately provide the contribution of the trapped wave.

100


Using Eq. (4.78) the termW1 is given by

W1 ¼RE0Em

dE � CðEÞj j2u0E ðzÞ

RHs

0

dz0Wðx; y; z0 Þu0

E z0

� �. (4.80)

Now assuming that em ðzÞ ¼ 2 z=a for z > Hs we obtain

WE ðzÞ ¼1

2ffiffiffiffiffiffipl

p w2E

l2� lðz�HsÞ

� �� SðEÞw1

E

l2� lðz�HsÞ

� �� (4.81)

where l ¼ k=m. The coefficient S(E) is determined by combining the solution (4.81)for a large height with the boundary conditions at the surface z = 0. In the vicinityof the pole En ¼ ~EEn þ jdn the coefficient S(E) has the form [22]:

SðEÞ » BðEÞE�~EEnþjdn

E�~EEn�jdn(4.82)

where B(E) is a slowly varying function of E. Returning to Eq. (4.80) and using Eqs.(4.82), (4.79) and (4.81) we obtain an explicit expression for CðEÞj j2:

CðEÞj j2¼ �2jdS

dESþ S

W

0E ðZsÞ

h i2�EW2

E ðZsÞh i

. (4.83)

Retaining the resonant term 2 jdS

dES in Eq. (4.58) as dn ! 0, we obtain

CðEÞj j2¼ dn

E�~EEn� �2

þd2n. (4.84)

Since the function (4.84) has a sharp maximum at E ¼ ~EEn and

R¥Em

CðEÞj j2dE»R¥�¥

CðEÞj j2dE ¼ p, (4.85)

function (4.84) can be approximated by a delta function

CðEÞj j2» d E � Enð Þ. (4.86)

Then, substituting Eq. (4.86) into Eq. (4.80) we obtain forWð~rrÞ

Wðx; y; zÞ @u0E1ðzÞR¥0

Wðx; y; z0 Þu0E1ðz0 Þdz0 þ

REm�¥

Aðx; y;EÞWE ðzÞ . (4.87)

As observed from Eq. (4.87), the contribution of the region of E > Em into integral(4.74) is represented by expansion over the eigenfunctions of the discrete spectrumu0En

that actually represent the boundary problem (2.37), (2.38) with term 2 z=a inem ðzÞ equal to zero. The function u

0EnðzÞ coincides, up to exponentially small terms,

with the real eigenfunction u1ðzÞ of the discrete spectrum in the height regionz << l

2=d

21 . Therefore, for consistency of the approximation made, we have to

exclude scattering in the troposphere’s layer at the height z >> l2=d

21 . For a single

scattering it results in the limitation the distance

101


x << 2k=d1 . (4.88)

For frequencies of about 10 GHz and evaporation duct heights Hs of the order of10–15 m, the value of d1 usually does not exceed 10

�3k=mð Þ2 , in which case the

inequality (4.63) is satisfied at the distances x << 104km.

4.4.2Equations for the Coherence Function

Lets consider the equations for the coherence function

C x;~��1 ;~��2ð Þ ¼ Cw x;~��1 ;~��2 !a2

(4.89)

and Cw x;~��1 ;~��2ð Þ ¼ W x;~��1ð Þ �Wx;~��2ð Þ

$ %, where ~��1 ¼ y1 ; z1f g, ~��2 ¼ y2 ; z2f g are

the coordinates of the observation points at the distance x from the source. Theclosed equation, similar to Eq. (2.68), for the coherence function can be obtained forCw ðx;~��1 ;~��2Þ:

@Cw

@x� j2k

ðD?1 � D?2ÞCw � pk2

4H ~��1 �~��2ð ÞCw ¼ 0 (4.90)

where

D?i ¼@2

@y2iþ @

2

@z2iþ k

2em ðziÞ � 1ð Þ; i ¼ 1; 2, (4.91)

and H ~��ð Þ is a structure function of the fluctuations in de. The conditions of applic-ability of Eq. (4.90) are similar to those listed in Section 2.3 and are given by a set ofinequalities:

a) ðkHsÞ2>> 1, b) kL?ð Þ2 >> 1,

(4.92)

c)kH2

s

Lx>> 1, d)

kL2?Lx

>> 1.

The inequality (4.92 c) means de-correlation of the consecutive acts of scatteringof the wave: the interval of the longitudinal correlation Lx should be less than thelength of the cycle K in the waveguide, K~Hs=Jc , and Jc~1=ðkHsÞ is a characteristicsliding angle of the trapped wave. The condition (4.92 (d)) means that the Fresnel-zone size

ffiffiffiffiffiffiffiffikLx

pis less than the vertical scale of the inhomogeneities L? , i.e. be-

tween consecutive acts of scattering, the distance between which is of the order ofLx , the wave propagates as in a uniform medium, and we do not take into accountdiffraction at the inhomogeneities of de.

As follows from Eq. (4.87), the coherence function Cw x;~��1 ;~��2ð Þ can be presentedas the superposition of the discrete and continuum eigenfunctions:

102


Cw x;~��1 ;~��2ð Þ ¼ gdðx; y1 ; y2Þudðz1Þud

ðz2Þþ

2ReREm�¥

dE�gcd E;x;y1 ;y2 !

WE ðz1Þudðz2Þþ

REm�¥

dE1 �REm�¥

dE2 �gc E1 ;E2 ;x;y1 ;y2 !

WE1ðz1ÞW

E2ðz2Þ:

(4.93)

In Eq. (4.93) the term gd determines the part of the coherence function carried bytrapped modes, gc is contributed by the waves of the continuum spectrum, and gcd isa term responsible for the combined mechanism of transfer of the coherence func-tion.

Substituting Eq. (4.93) into Eq. (4.90), introducing the variables: Y ¼ ðy1 þ y2Þ=2and y = y1 – y2 and the Fourier transform of gd over variable Y:

gdðx; y1 ; y2Þ ¼R¥�¥

dp � ~ggd x; p; yð Þe�jpY . (4.94)

For the other terms, gc and gcd, the Fourier transforms are introduced in similarway. Let us also introduce the notations

c0 ¼ pk2

2

Rd2~kk?Ue 0;~kk?ð Þ, g ¼ yþ p

xk, D kz ; gð Þ ¼ Ue 0;~kk?ð Þe

jkgy ,

we obtain the system of coupled equations, which describes the energy exchangebetween the trapped waves and the waves of the continuous spectrum due to a scat-tering on the fluctuations of de:

@~ggdðx;p;gÞ@x

þ c0~ggdðx; p; gÞ ¼Rd2~kk?D ~kk? ; gð Þ ·

~ggdðx;p;gÞ V11 kzð Þ

2þ REm�¥

dE1~ggcd

x;E1 ;p;g !

V11 kzð Þ� Vþ1 E1 ;kz ! !þ

REm�¥

dE2~ggcd

x;E2 ;p;g !

V11 kzð Þ�V�

1 E1 ;kz !

þ

REm�¥

dE1

REm�¥

dE2~ggc x;E1 ;E2 ; p; gð ÞVþ1 E1 ; kzð ÞV�

1 E2 ; kzð Þ

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

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

, (4.95)

@~ggcdðx;E;p;gÞ@x

þ c0 �j2k

E1 � Eð Þ� �

~ggcdðx;E; p; gÞ ¼Rd2~kk?D ~kk? ; gð Þ ·

~ggdðx; p; gÞV11 kzð ÞV�1 E; kzð Þþ

REm�¥

dE1~ggcd x;E1 ; p; gð Þ � V11 kzð ÞVE1 ;E; kzð Þþ

REm�¥

dE2~ggcd x;E2 ; p; gð Þ � V�1 E; kzð ÞVþ

1 E1 ; kzð Þþ

REm�¥

dE1

REm�¥

dE2~ggc x;E1 ;E2 ;p;g !

Vþ1 E1 ;kz !

V� E2 ;E;kz !

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

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

, (4.96)

103


@~ggc ðx;E;E0;p;gÞ

@xþ c0 �

j2k

E � E0

� �� ~ggc ðx;E;E

0; p; gÞ ¼

Rd2~kk?D ~kk? ; gð Þ ·

~ggdðx; p; gÞ V�1 E; kzð Þð ÞV�

1 ðE0; kzÞþREm

�¥dE1~ggcd x;E1 ; p; gð Þ � ðV�

1 ðE0; kzÞÞ

V

ðE1 ;E0; kzÞþ

REm�¥

dE2~ggcd x;E2 ; p; gð Þ � V E2 ;E; kzð ÞV�1 ðE0

; kzÞþ

REm�¥

dE1

REm�¥

dE2~ggc x;E1 ;E2 ;p;g !

V E1 ;E;kz !

V E2 ;E0;kz

!

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

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

. (4.97)

The functions

V11 kzð Þ ¼R¥0

dzejkz zu

dðzÞu

dðzÞ , (4.98)

V�1 E; kzð Þ ¼

R¥0

dze�jkz zu

dðzÞWE ðzÞ , (4.99)

V E1 ;E2 ; kzð Þ ¼R¥0

dzejkz zWE1

ðzÞWE2 ðzÞ (4.100)

are the coefficients of scattering on inhomeogeneties de with vertical scaleslz ¼ 2p=kz for waveguide modes (4.98), waveguide modes into the waves of the con-tinuous spectrum (4.74), and continuum into continuum (4.100). The relative valueof the contribution of each of the terms ~ggd , ~ggcd , and ~ggc to the total field at the observa-tion point depends on the form of the initial distribution, Cðx ¼ 0;~��1 ;~��2Þ, and theposition of the receiver relative to the evaporation duct.

It is reasonable to assume, when the points of observations are located inside theevaporation duct, z1 ; z2 < Hs , that the major contribution to the coherence functioncomes from the trapped waves. Therefore, in a first order approximation, we canneglect the contribution from gc and gcd to gd and the total coherence function Cw ,considering the multiple scattering of the trapped waves only. In this case we obtain

@~ggd

@xþ cs �

Rd2~kk � D ~kk; gð Þ V11 kzð Þj j2

" #~ggd ¼ 0 . (4.101)

Substituting the solution to Eq. (4.101) into Eq. (4.93), we can use the orthogonal-ity feature between the waves of the discrete and continuous spectra. Performingthe Fourier transform, inverse to Eq. (4.94), we obtain

Cw x;y;Y ;z1 ;z2 !

¼ k

2pxudðz1Þud

ðz2ÞR¥�¥

dy0R¥�¥

dY 0Cw ðx¼0;y0 ;Y 0 Þ ·

exp jk

xy�y0 !

Y�Y 0 !�Pðx;y0 ;yÞ

h i:

(4.102)

104


The parameter P(x, y¢, y),

Pðx; y0 ; yÞ ¼ pk2

2

Rd2~kk?Ue 0;~kk?ð Þ ·

x � V11 kzð Þj j2Rx0

dx0exp jky y

x0

xþ y

01� x0

x

� �� ( ), (4.103)

after integration over dy0determines the attenuation exponent of the coherence

function with distance x. The first term in Eq. (4.103) describes the attenuation ofthe average field due to energy transfer to the incoherent component. The secondterm determines the incoherent contribution of the energy scattered back to thewaveguide mode in the direction of propagation.

Let us consider the intensity of the field at the point~rr ¼ x; 0; zf g produced by apoint source located at ~rr0 ¼ 0; 0; z0f g. The initial distribution of the field in Eq.(4.77) takes the form:

Cw x ¼ 0; y0;Y

0� �

¼ a2

4k2u2

dz0ð Þd y

0� �

d Y0

� �. (4.104)

Now, we can assume that em ðzÞ is described by a bilinear function with gradientm ¼ dem=dz, for z < Hs. While computation of Eq. (4.96) can be performed with anyregular function em ðzÞ, the bilinear approximation provides an analytical solutionuseful for qualitative analysis of the scattering mechanism. Introducing the parame-ters:

l1 ¼ a mj j=2, Hs ¼ kZs=m, h ¼ kf=m, h0 ¼ kz0=m, and s1 ¼ �l212:338þ l

31Hs we

obtain a solution for the intensity of the field, normalised on the intensity in a freespace:

J ¼ x2

a2 Wðx;~rrÞj j2¼ x

2

a2 Cw x; 0; zð Þ ¼ kx

8pm2s1

t2 s

l21

� l1 Hs � h0ð Þ !

·

t2 s

l21

� l1 Hs � hð Þ !

exp �cdx

� �:

(4.105)

Here, the function udis expressed via the Airy function tðxÞ of the real argument

since the pole E1 is regarded as a real one, Im E1f g ¼ 0. The attenuation exponent inEq. (4.105) is given by

cd¼ pk2

2

Rd2~kk?Ue 0;~kk?ð Þ 1� V11ðkzÞj j½ � (4.106)

where~kk? ¼ ky ; kz& '

.Consider the calculation of c

dwith the spectrum given by Eq. (1.36) for locally

uniform anisotropic fluctuations de and introduce the non-dimensional variablet = z / Z0, where Z0 = m0 / k, the characteristic scale of the variations in the func-tion u

dðzÞ, m0 ¼ k= mj jð Þ1=3 . Performing the integration over ky and introducing the

variable q ¼ kzZ0 , we obtain

105


cd¼ 0:17k

2C2e?

Z0

a

� �5=3

A (4.107)

where A is a constant of the order of unity, the exact value of which is defined by atrue behavior of the height function u

dzð Þ:

A ¼R¥0

dq � q�8=31�

R¥0

dt � ejqtu2

dtð Þ

" #2. (4.108)

In the case of the bilinear model of em ðzÞ, Eq. (4.108) takes the form

A ¼R¥0

dq � q�8=31� 1

s21

R¥0

dt � ejqtt2 t� s1ð Þ

" #2

, (4.109)

and calculation of Eq. (4.107) results in the value A = 1.51. Hence, for cdwe obtain

finally

cd¼ 0:264k

8=9C2e?a

�5=3mj j�5=9

. (4.110)

Equation (4.110) is valid for locally uniform turbulent fluctuation de, even whenthe external scales of the turbulence are infinite: Lz ; L? ! ¥, r2

e ! ¥. As discussedin Chapter 1, real measurements of the fluctuations de are always limited in time,and for the purpose of comparison with experiment another model of spectrum(1.35), with finite values of r

2e and external scales, can be used instead of Eq. (1.36).

The match between models is achieved when r2e ¼ 1:9C

2e?L

2=3? .

The calculation of Eq. (4.106) can be simplified when Lz << Zs . In this case, thesecond term can be neglected and the attenuation factor is entirely determined bythe attenuation of the coherent component of the wave field:

cd» cS ¼ pk2

2

Rd2~kk?Ue 0;~kkð Þ ¼ 0:374r

2e k

2L? . (4.111)

The apparent reason for this is that the scattering on the small-scale fluctuations,with the scattering angle greater than the critical angle of the waveguideJc~1=ðkHsÞ, leads only to a flow of energy from the waveguide.

Until now we have assumed that the fluctuations in de were statistically uniformover the height over the surface, i.e., parameters Ce ; a as well as Lz do not dependon the height z. However, from the theory of atmospheric turbulence [31], it followsthat the external vertical scale of the fluctuations can be regarded as a linear functionof height: Lz = bz, where b is a coefficient with numerical value less than unity. Tosome extent, a quasi-uniform behavior of the fluctuations de can be accounted forby using the values of Ceand a at the height zm, where the scattering is moreintense, i.e. at the point of the maximum of the first normal wave u

d. In the case of

the bilinear approximation, zm ¼ 1:32 m0=k. Thus,

a ¼ bzmLx

¼ 1:32b

k2=3vj j1=3Lx (4.112)

and, instead of Eq. (4.110), we obtain

106


cd¼ 0:166k

2C

2e?L

5=3x b

�5=3. (4.113)

In fact, the attenuation factors, defined by both Eqs. (4.113) and (4.111), will be equalsince the vertical scale of the fluctuations dewill be less than the thickness of the evapo-ration duct, Lz << Zs . From comparison with Eqs. (4.113) and (4.111), the value of bcan be estimated as b = 0.4 which agrees well with the measurement, and, therefore,Eq. (4.86) can be used to estimate the attenuation of the radio wave in an evapora-tion duct. Figure 4.13 [32] shows some results of a comparison of the field strength Jin the evaporation duct relative to one in a free space at the frequency 10 GHz.

107

-60

-50

-40

-30

-20

-10

0

10

20

0 50 100 150 200 250 300 350

Range, km

Sig

na

l s

tre

ng

th,

dB

Figure 4.13 Signal strength in an evaporation duct at 10 GHz:jMeasured signal;* Calculated, de „ 0; – Calculated, de ¼ 0.

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8 1

Measured attenuation, dB/km

Calc

ula

ted

att

en

uati

on

, d

B/k

m

Figure 4.14 Measured attenuation factor vs. that calculated according to Eq. (4.111) [32].


As observed from Figure 4.13, the calculation of the field strength in an evapora-tion duct using only the mean M-profile, when de ¼ 0, provides unrealistically highlevels of the signal beyond the horizon. The curve marked by the squares is cal-culated with Eq. (4.80) using Eq. (4.86). The measurements of the fluctuation deperformed at the time of the radio measurements provided the following values:CN = 0.09 N-units cm

�1=3, L?= 48 m at a distance less than 100 km; and CN =

0.11 N-units cm�1=3

, L?= 52 m at a distance in the range between 100 and 200 km.Here CN ¼ 1=2 10

6Ce .

Figure 4.14 from Ref. [32] shows results of a comparison between the measuredcm and the calculated (according to the theory provided in this section) attenuationfactors c

dat the frequency 10 GHz in the presence of an evaporation duct over the

ocean. The data were collected from 16 tests when radio measurements were per-formed synchronously with refractometer measurements of the fluctuations in thenear-surface layer of the troposphere. The correlation coefficient between the mea-sured and calculated values of the attenuation factor is 0.8, which suggests that thewave scattering at the fluctuations of de provides a significant contribution to themechanism of the radio wave propagation in an evaporation duct.

4.5Excitation of Waves in a Continuous Spectrum in a Statistically InhomogeneousEvaporation Duct

In the absence of fluctuations in refractive index, de, the signal strength in the shad-ow region above the evaporation duct is exponentially small because of the small-ness of the “sub-barrier leakage” of the trapped modes. The contribution of thewaves of the continuous spectrum, initially excited by the transmitter, can beneglected because of the diffraction attenuation in the shadow region. The scatter-ing of the waveguide trapped modes by random non-uniformities of the refractiveindex gives rise to incoherent exchange of energy between trapped modes and exci-tation of waves in the continuous spectrum. In addition, the initially excited wavesof the continuous spectrum may reach beyond the horizon due to a scattering in theupper layers of the troposphere (Booker–Gordon’s single scattering mechanism). Inthis section, we study the contribution of the waves of the continuous spectrum,whose source is a waveguide mode, to the intensity of the field, i.e. the mechanismof the excitation of waves of the continuous spectrum by trapped modes of the evap-oration duct scattered on the imhomogeneities in the refractive index.

Recalling the equations for the coherence function from the previous section, weconcentrate on investigation of the spectral amplitude of the component of thecoherence function related to a continuous spectrum:

~ggc x;E1 ;E2 ; p; yð Þ ¼ 12p

R¥0

dz1R¥0

dz2R¥�¥

dY � C x;Y; y; z1 ; z2ð ÞWE1 ðz1ÞWE2 ðz1Þ exp jpYð Þ

(4.114)

108

4.5 Excitation of Waves in a Continuous Spectrum in a Statistically Inhomogeneous Evaporation Duct

As is well known [31], in the presence of random fluctuations of the refractiveindex, de, the coherent component of the waves of the continuous spectrum attenu-ates with distance due to incoherent multiple scattering. The decrement of attenua-tion in the coherent component is given by

cc ¼pk2

2

Rd2~kk?Ue 0;~kk?ð Þ (4.115)

where Ue 0;~kk?ð Þ is the spectrum density of the fluctuations in de. As observed inSection 2.3, in the case of free-space propagation with random irregularities of de,the attenuation of the coherent component in the field intensity is completely com-pensated by the inflow of the incoherently scattered field from other directions, andthe total intensity of the wave field does not reveal exponential attenuation. Sincethe propagation above the evaporation duct is free-space-like we may use the aboveargument in support of neglecting multiple scattering of the wave of the continuousspectrum in a space above the evaporation duct. However, the effect of multiple scat-tering of the wave of the continuous spectrum can obviously be neglected if wedemand a smallness of the attenuation of the coherent component of the CS spec-trum wave, i.e., cc xa << 1, over the path xa of the wave from the point where scatter-ing of the wave guide mode into the CS wave occurred up to the point of observa-tion. Evaluating the maximum value of the distance xa as xamax ¼

ffiffiffiffiffiffiffiffi2az

p, where z is

the height of the observation point, we obtain the limitation z << 2ac2c

� ��1. In the

case of Kolmogorov’s turbulence cc ¼ 2:66k2C

2nL

5=30 . Assuming C

2n=10–15 cm�2=3 ,

L0 = 104 cm, k = 2 cm–1, we obtain the result that the maximum heights under con-sideration should not exceed the value zmax ~ 103 m. Thus, the approximation inwhich the estimate is obtained below for the waves of the continuous spectrum isapplicable for altitudes of the receiving point z satisfying the inequality z £ zmax.

Let us study the intensity of the field of the waves of the continuous spectrumIc ðx; zÞ ¼ C x;~��;~��ð Þ, ~�� ¼ 0; zf g, setting the y-coordinates of the receiving and trans-mitting antennas to zero. Then, integrating Eq. (4.97) and employing the expression(4.104) for the component of the coherence function associated with waves of thediscrete spectrum, Cd , we obtain

Ic ðx; zÞ ¼ka2

16xudðz0Þ

2exp �cdx

� �

·Rx0

dx0 R¥�¥

dqR

Q<� qj j=2dQBðq;QÞWQþq=2 zð ÞW

Q�q=2ðzÞexp jqx0

2kþ c

dx0

� �(4.116)

where we have introduced q ¼ E1 � E2 ; Q ¼ 1=2ð Þ E1 þ E2ð Þ,

Bðq;QÞ ¼R¥�¥

dkyR¥�¥

dkzUeð0; ky ; kzÞ ·

R¥0

dz1R¥0

dz2WQþq=2ðz1ÞWQ�q=2ðz2Þud

ðz1Þudðz2Þ exp jkz z1 � z2ð Þð Þ:

(4.117)

109


The integral B(q, Q) determines the coupling between the waves of the discreteand continuous spectra due to the scattering on the inhomogeneities of de. Themajor contribution to B(q, Q) comes from the spectral components of de with thewavenumbers kzj j > l ¼ �dU=dzð Þ1=3 with z < Zs and the region of the altitudesz1,2 in the neighbourhood of the turning point zd of the mode of the discrete spec-trum, where U zd

!¼ Ed . We use the WKB approximation for the waves of the con-

tinuous spectrum and the Airy function representation for the waveguide mode inthe vicinity of the turning point:

WE ðzÞ ¼ffiffiffi2p

rUðzÞ � E½ ��1=4

exp �j 3p4

� �sin

Rz0

dz0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU z

0� �

� E

r !, (4.118)

udzð Þ ¼ 1

Nd

t l z� zd ! !

; Nd ¼R¥0

dz udzð Þ

2 .The spatial spectrum of the fluctuations in the refractive index we define in the

form (1.36) for the case of turbulent fluctuations de isotropic in the (x, y)-plane

Ue ~kkð Þ ¼ 0:033C2ka k

2k þ a

2k2z

� ��11=6(4.119)

we substitute into Eqs. (4.117) and (4.118) the integral representation of the Airyfunction

t tð Þ ¼ 12ffiffiffip

pR¥�¥

dnexp jn3

3þ jnt

!

and require the inequalities

kzj j > l and 2ffiffiffiffiffiEd

p mk>> 1 (4.120)

to hold, which is necessary for the entire scheme of the solution obtained in Section4.3. The meaning of inequality (4.120) is that we do not take into account the scatter-ing of the leaky modes, assuming that the waveguide field is composed only oftrapped (non-attenuated) modes. The B(q, Q) can then be truncated to

Bðq;QÞ ¼ C2e a

�5=3·0:089

sin qKd Qð Þ� �

qN2d Ed � Q !4=3 ; Dk

kd

<< 1

0:053cos qKd Qð Þ� �

l3N

2d Ed �Q !5=6 ; Dk

kd>> 1

8>>>><>>>>:

(4.121)

where

Kd ¼ 12

Rzd0

dz0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUðz0 Þ � Q

q, kd ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqd � Q

p, Dk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUð0Þ �Q

p�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEd � Q

p.

(4.122)

Parameter kd determines the Bragg scattering angle hs from the waveguide modeinto the wave of the continuous spectrum with energy Q at the height of the turningpoint z = zd, i.e., 2sin hs=2 ¼ kd=k. The parameter Dk is a difference between themagnitude of the scattering angle at the surface z = 0 and at the altitude of the turn-

110


ing point z = zd. Thus the inequality Dk=kd << 1 requires that the change in thescattering angle owing to refraction in the scattering volume, i.e. in the region of themode’s localization, be much smaller than the scattering angle itself at the altitudezd.

The relationship between the characteristic “energy” Q and the sliding angle h ofthe wave front of the wave of the continuous spectrum relative to the surface z ¼ Zs

is given by the equation Q ¼ �k2=m2tan

2h . The parameter KdðQÞ determines the

distance along x traversed by the wave of the continuous spectrum with “energy” Qas it propagates from z = 0 to z = zd. Thus, B(q, Q) defines the angular distributionof the scattered field, i.e., the scattering function in the “directions” E1, E2.

To calculate the intensity of the field lc(x, z) we employ the WKB approximation(4.118) and (4.121) in Eq. (4.116). We leave aside the constant terms for later andconcentrate on the integration in Eq. (4.116). The integration over the “energy” dif-ference q leads to the appearance of terms that place the limits on the region of thedistances x

0from which the scattered field is collected in the receiving antenna.

WithffiffiffiffiffiffiffiffiffiffiffiUð0Þ


qdp

=ffiffiffiffiffiqd

p<< 1 we obtain

Ic ðx; zÞ ~R0�¥

dQ qd �Q !�4=3

UðzÞ �Qð Þ�1=2Rx0

dx0exp c

dx0

� �

·

12h Kd þK� x0

k

� �� 12h Kd �K� x0

k

� ��

h Kd �x0

k

� �cos 2

Rz0

dz0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU z

0� �

�Q

r !8>>>>><>>>>>:

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

(4.123)

and for

ffiffiffiffiffiffiffiffiffiffiffiUð0Þ

p� ffiffiffiffiffiqdpffiffiffiffiffiqd

p >> 1

Ic ðx; zÞ ~R0�¥

dQ qd �Q !�5=6

UðzÞ �Qð Þ�1=2Rx0

dx0exp c

dx0

� �

·

12d

x0

kþKd �K

� �þ 12d

x0

k�Kd þK

� �þ 12d

x0

k�Kd �K

� �

�dx0

k�Kd

� �cos 2

Rz0


0� �

� Q

r !8>>>>><>>>>>:

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

(4.124)

where

hðxÞ ¼ 1; x > 01=2; x¼ 00; x < 0

8>><>>:

and

K ¼ KðQ ; zÞ ¼ 12

Rz0


0� �

�Q

r.

111


We introduce the dimensionless variables s ¼ �Qm2=k

2, sd ¼ Edm

2=k

2,

h ¼ kz=m, b ¼ m2cs=k and the modified profile UðhÞ ¼ m

2=k

2UðzÞ. We then

require that the inequality

kcdKdð0Þ << 1 (4.125)

holds. Since Kdð0Þ‡Kd Ed !

, where kKdðEdÞis the length of the cycle of a waveguidemode and the contributing values of Q are limited by the inequality Qj j£Ed, the con-dition (4.125) demands the smallness of the attenuation of the waveguide mode atthe distance equal to its cycle. It may be noted that in the opposite case there is nowaveguide mode structure to sustain scattering, the mode will be destroyed duringone cycle.

Finally we obtain the intensity of the wave of the continuous spectrum normal-ized at the intensity of the free-space field

Jc ðx; zÞ ¼ Ic ðx; zÞx2

a2¼ 0:078k

�5=3xC

2eka

�5=3m

11=3 udz0 ! 2N2d

0B@

1CA·

KðhÞexp �cdx

� �(4.126)

and the function KðhÞ, which commands that the height distribution of the scatteredfield is given by two equations obtained from Eqs. (4.123) and (4.124).

WithffiffiffiffiffiffiffiffiffiffiffiUð0Þ


sdp

=ffiffiffiffiffisd

p<< 1,

KðhÞ¼R¥0

ds sdþs

� ��4=3UðhÞþsð Þ�1=2K

dðsÞ ·

exp bKðs;hÞð Þ� cos 2Rh0

dh0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU h0 !

þsq !( )

;

(4.127)

and forffiffiffiffiffiffiffiffiffiffiffiUð0Þ


sdp

=ffiffiffiffiffisd

p>> 1

KðhÞ ¼ p �dU

dh

h i�2=3h¼hd

R¥0

ds sdþs

� ��5=6UðhÞþsð Þ�1=2 ·

exp bKðs;hÞð Þ�cos 2Rh0

dh0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU h0 !

þsq !( )

:

(4.128)

We shall now study the total field in the shadow region as being the result of com-position of the waves of the discrete and continuous spectra excited due to the ran-dom scattering of the waveguide modes.

For the intensity of the waveguide mode the relation below follows:

Jd ¼ x

8pk Nd

4 udðz0Þ

2 udðzÞ

2exp �cdx

� �. (4.129)

112


We define the total intensity as

Jtot ¼ Jd þ Jc ¼x

8pk Nd

2 udðz0Þ

2exp �cdx

� �SðzÞ (4.130)

where

SðzÞ ¼udðzÞ

2N2d

þ 0:139k�2=3

C2e a

�5=3m

11=3KðzÞ. (4.131)

The function S(z) determines the height distribution of the total intensity of thefield. We examine the behavior of the function S(z) in the case of a bilinear depen-dence of U(z). We assume the following values for the parameters of the problem:k = 2 cm–1, a = 8500 km, Zs = 11 m, DeM ¼ eM ð0Þ � eM ðZsÞ ¼ 5.8 Q 106. The modi-fied profile U(h) can be defined as

UðhÞ ¼ l31ðHs�hÞ; h ‡Hs

h�Hs ; h>Hs

((4.132)

where Hs ¼ kZs=m ¼2.42, l31 ¼ m

2DeM=Hs = 2, sd ¼ Uð0Þ � l

21s1 ¼ 1, Uð0Þ =

m2DeM = 4.34, s1 ¼ 2.338.Consider the height dependence of the waveguide mode, i.e., the first term in

S(z). Thus far in the analysis of the waveguide field we have neglected the leakage ofthe trapped waves of the discrete spectrum through the potential barrier. Theattenuation caused by the effect of the leakage is accounted for in the imaginary partof Ed, which can be written as

dd ¼ ImEd ¼l2l2

1

4ffiffiffiffiffis1

p exp � 43s3=2d

1þl31

l31

( ). (4.133)

Sub-barrier leakage has virtually no effect on the height structure of the trappedmode inside the waveguide channel for h < Hs, where

udðhÞ ¼ 1

Nd

tsd

l21

� l Hs � hð Þ !

. (4.134)

Outside the waveguide channel the height structure of udðhÞ corresponds to an

outgoing wave with amplitude proportional to the leakage factor dd:

udðhÞ ¼

l�1=3

1

lNd

s1=41 d

1=2d w1 sd þ j

dd

l2� h�Hsð Þ

� �. (4.135)

Examining the second term in S(z) given by Eq. (4.131), we can express the coeffi-cient in front of K(z) in terms of the non-dimensional attenuation coefficient busing Eq. (4.101) for c

dfrom the previous section:

b ¼ m2cd=k ¼ 0:264k

�2=3C2eka

�5=3m

11=3l�5=91 .

Then for S(z) we obtain

SðzÞ ¼udðzÞ

2N2d

þ 0:492 l5=91 bKðzÞ. (4.136)

113


In principle, in order to determine the coefficient b it is necessary to know themagnitude of the structure constant Ce and the anisotropy parameter of the irregu-larities of the refractive index in the volume of the waveguide channel. In somecases when the radio signal strength is measured, the measured data may bedeployed to estimate the height dependence of the received signal. In particular,when the magnitude of the field attenuation per unit length cx ¼ dI=dx is knownfrom the measured signal, then c

dis related to cx via cx ¼ 4:34c

d, and given know-

ledge of the average M-profile we can construct a theoretical height structure of thefield intensity S(z). According to the data presented in Ref. [10], the cx magnitudefor centimeter waves lies in the range cx = 0.2–0.5 dB km–1. The corresponding lim-its for b are b = 0.207–0.52 for k = 2 cm–1 and a = 8500 km (normal refraction abovethe evaporation duct). Thus under real conditions the coefficient in front of K(z) inEq. (4.111) is of the order of unity and, consequently, the contribution of the wavesof the continuous spectrum to S(z) can be significant.

Figure 4.15 shows the result of the calculation of the resulting height distribution10 log S(h) based on Eq. (4.136) for b = 0.207 and b = 0.52 with U(h) given by Eq.(4.132) and its parameters defined above. The figure also shows the height depen-dence of the first term in Eq. (4.136), i.e., the contribution of the waves of the dis-crete spectrum alone. As observed from the figure, the intensity of the field abovethe turning height hd (in this case hd = 1.67) is a contribution from the waves of thecontinuous spectrum, i.e. second term in Eq. (4.136). Apparently, scattering on ran-

114

0

2

4

6

8

10

-30 -20 -10 0 10

10 log(S), dB

Heig

ht,

m 1

2

3

Figure 4.15 Height structure of the wave field in the presenceof an evaporation duct: f = 10 GHz, Curve 1, no scattering oninhomogeneities of the refractive index, curves 2 and 3, scatter-ing is taken into account.

4.6 Evaporation Duct with Two Trapped Modes

dom fluctuations in the refractive index leads to a significant change in the heightdependence of the signal strength; in the presence of random fluctuations de in therefractive index there is no sharp exponential decay in the average signal strengthoutside the duct and for sufficiently strong fluctuations the de field outside the ductmay increase to the order of magnitude of the field inside, thus revealing a rathersmooth unpronounced height dependence compared to the case of deterministicduct propagation.

At relatively large altitudes, h�Hs >> sd , the function S(h) has the following as-ymptotic form

SðhÞ~ 14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih�Hs � sd

p exp � 43s3=2d

l31 þ 1

l31

þ 2ddl2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih�Hs � sd

q" #

þ1:58l

5=91 b

s1=3d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffih�Hs

p exp bffiffiffiffiffiffiffiffiffiffiffiffiffiffih�Hs

p� �:

(4.137)

Beginning with the altitudes h1 ¼ Hs þ sd þl4

2d2dand h2 ¼ Hs þ

1

4b2, the expo-

nential growth predominates over the cylindrical divergence in either the first or sec-ond terms, respectively. The exponential growth of those terms in Eq. (4.137) is dueto the arrival of the waves from shorter distances x

0< x, at which the field of the

waveguide mode was exponentially large, compared with its value at distance x. Thesecond term makes the main contribution in Eq. (4.137), since the exponential fac-tor b exceeds the sub-barrier leakage factor d by at least an order of magnitude.

As mentioned before, we ignored the scattering of the waves in the space abovethe waveguide. While, in principle, it should be taken into account for the sake ofconsistency, the qualitative character of the total field will not be expected to changedrastically. We may assume however that additional scattering of the waves of thecontinuous spectrum in the space above the evaporation duct will smooth the expo-nential dependence of the scattered field in Eq. (4.137) for large altitudes. At thesame time additional divergence of the waves due to a broadening of the angularspectrum can be compensated by the arrival of the scattered waves from the shadowregion, i.e. from distances x

0< x � a=m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffih�Hs

p.

4.6Evaporation Duct with Two Trapped Modes

Finally, we provide a closed form solution for the field intensity in the case when theevaporation duct can trap two modes. In this case the intensity in the two-modewaveguide is given by

Jðx; z; z0Þ ¼ 4px I1 � S1 þ I2 � S2½ � (4.138)

where

115


I1ðxÞ ¼ A1ek1 x þ A2e

k2x ,

I2ðxÞ ¼ A3ek1 x þ A4e

k2x ,

k1;2 ¼ �C1þC2

2� 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2 � C1ð Þ2�4W12

q,

A1 ¼ � I10 k2þC2

!�W12I20

� �k1�k2

,

A2 ¼ I10 k1þC1

!�W12I20

� �k1�k2

,

A3 ¼ I20 k2þC2

!�W12I10

� �k2�k1

,

A4 ¼ I20 k1þC1

!�W12I10

� �k1�k2

,

C1 ¼ 1:603 � As , C2 ¼ 3:98 � As ,W12 ¼ 0:316 � As , and As ¼ k�2=3

C2e a

�5=3m

11=3.

Coefficients In0 ¼ vn h0ð Þ 2are the coefficients of the excitation of the mode with

number n and function Sn(h) is a height distribution of the total intensity of the nthmode:

Sn hð Þ ¼ vn hð Þ 2þ0:492Cnl

5=31 PnðhÞ. (4.139)

The height gain function vn is given by

vnðhÞ ¼ffiffiffiffiffil1

p

snv

tn�UðhÞl21

!, h < Hs (4.140)

vnðhÞ ¼ d1=2

w1tn�UðhÞ

l22

!, h ‡ Hs

where d is the coefficient of penetration through the potential barrier into the spaceabove the duct:

d ¼l1

4snexp � 4

3

l32þl3

1

l31l32

t3=2n

" #, (4.141)

sn ¼ 32p n� 1

4

� �� 2=3.

The function Pn(h) is given by the integral

116

4.6 Evaporation Duct with Two Trapped Modes

PnðhÞ ¼ p=l2

1

R¥0

ds tn þ sð Þ�5=6UðsÞ þ sð Þ�1=2

eK s;hð Þ � cos 2Qðs; hÞð Þ

h i(4.142)

where

Q s; hð Þ ¼ 2

3l31

U0 þ sð Þ3=2 � UðhÞ þ sð Þ3=2h i

, h < Hs (4.143)

Q s; hð Þ ¼ 23

1

l31

U0 þ sð Þ3=2 �l31þl3

2

l31l32

s3=2 þ UðhÞ þ sð Þ3=2

" #, h ‡ Hs

K s; hð Þ ¼ 1

l31

U0 þ sð Þ1=2� 1

l31

UðhÞ þ sð Þ1=2 , h < Hs (4.144)

K s; hð Þ ¼ 1

l31

U0 þ sð Þ1=2�l31þl3

2

l31l32

s1=2 þ UðhÞ þ sð Þ1=2 , h ‡ Hs .

It should be noted that the above formulas actually represent the incoherent sumof two trapped modes. Under very moderate assumptions for the intensity of theatmospheric turbulent fluctuations (as in Figures 4.16 and 4.17 below) the inter-ference term in the coherence function may be neglected since phase fluctuationsin this case are strong enough to make the above approximation valid.

117

-15

-10

-5

0

5

10

15

20

0 50 100 150 200 250 300

Distance, km

Sig

na

l s

tre

ng

th r

ela

tiv

e t

o f

ree

sp

ac

e,

dB

(1)-z=10 m, uniform duct

(4) z=150 m, non-uniform duct

(3) z=10m, non-uniform duct

(2) z=150 m, uniform duct

Figure 4.16 Range dependence of the signalstrength at 10 GHz in the presence of an eva-poration duct with two trapped modes. Thetransmitter antenna is mounted at 10 m abovethe sea surface. Curves (1) and (2) correspondto the range dependence of the received fieldstrength for the receiving antennas at the

heights inside (1) and outside (2) the duct forthe case of a uniform duct in the absence ofrandom fluctuations in the refractive index.Curves (3) and (4) correspond to the sameantenna elevations for a non-uniform duct withrandom and anisotropic fluctuations of therefractive index: Ce

2 = 10–14 cm–2/3, a = 0.1.


To illustrate the effect of scattering on random fluctuations in a two-mode evapo-ration duct we calculated the signal strength at frequency 10 GHz in the evaporationduct with parameters Zs = 15 m and DM = 5 N-units. Figure 4.16 shows a rangedependence of the signal strength in the absence and presence of the fluctuations inthe refractive index. As observed from the figure, at a height 10 m inside the ductthe composition of the two trapped modes produces an interference pattern, creat-ing fades. The second curve shows the range dependence at the height 150 m wherethe second trapped mode is dominant and therefore no interference pattern is ob-served. The height gain pattern in this case is typical for one that follows from themodel of the evaporation duct, namely, revealing the maximum of the signalstrength inside the duct, Figure 4.17. In the presence of fluctuations we observe sig-nificant attenuation of the field with distance as well as considerable changes in theheight gain distribution, namely, the height structure becomes largely uniform.

In the range of frequencies from 3 GHz to 20 GHz the evaporation duct in mostcases can support only a few trapped modes. The calculation of the field strength fora finite number of modes can be performed in a way similar to that described in thissection. The surface-based and elevated duct created by an advection mechanismand normally associated with strong elevated inversion of temperature may trap ahundred modes and the above approach is inefficient. In that case other methods

118

0

20

40

60

80

100

120

140

160

180

200

-40 -20 0 20

Signal strength relative to free space, dB

heig

ht,

m

(1) Uniform duct

(2) Non-uniform

duct

Figure 4.17 Height-gain dependence of thesignal strength at 10 GHz in the presence ofan evaporation duct with two trapped modesat 100 km from the transmitter. The transmit-ter antenna is mounted at 10 m above the seasurface. Curve (1) shows the received field

versus the height in the absence of randomfluctuations in the refractive index. Curve (2) isa height-gain dependence for a non-uniformduct with random and anisotropic fluctuationsof the refractive index: Ce

2 = 10–14 cm–2/3,a = 0.1.

References

based on the application of diffusion theory for a parabolic equation can beemployed to solve the problem of scattering in a multimode waveguide.

119

References

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3 Gossard, E.E. Clear weather meteorologicaleffects on propagation at frequencies above4 GHz, Radio Sci., 1981, 16 (5), 589–608.

4 Fock, V.A. Electromagnetic Diffraction andPropagation Problems, Pergamon Press,1965.

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7 Bocharov, V.G., Kukushkin, A.V., Sinitsin,V.G. and Fuks, I.M. Radio Propagation in Sur-face Tropospheric Ducts, IRE, Ukrainian Acad.Sci., Preprint No 126, 1969, 44 pp.

8 Kerr, D.E. Transmission along the Californiacoast, in Propagation of Short Radio Waves,Kerr, D.E. Ed., McGraw Hill, New York, 1951,pp. 328–335.

9 Report on Factual Data from the CanterburyProject, Department of Science and IndustrialResearch, Wellington, New Zealand, 1951.

10 Braude, S.Ya., Ostrovsky, I.E., Sanin, F.S. andShamfarov, Ya.L. Propagation of CentimetreWaves over the Sea in the Presence of Super-refractivity, Morskoi Vestnik, Leningrad, 1949,No2, 103 pp.

11 Peceris, C.L. Wave theoretical interpretationof the propagation of the 10 cm waves in low-level ocean ducts, 1947, Proc. IRE, 1947,pp. 453–462.

12 Paulus, R.A. Practical application of an evapo-ration duct model, 1985, Radio Sci., 1985, 20(4), 887–896.

13 Heemskerk, H.J.M. Boekema, R.B. The influ-ence of the evaporation duct on the propaga-tion of electromagnetic waves low above thesea surface at 3–94 GHz, Eighth InternationalConference on Antennas and Propaga-tion, 1993, Vol. 1, pp. 348–351.

14 Anderson, K.D. Radar measurements at16.5 GHz in the oceanic evaporation duct,IEEE Trans. Antennas Propagation, 1989, 37(1), 100–106.

15 Anderson, K.D. 94 GHz propagation in theevaporation duct, IEEE Trans. Antennas Propa-gation, 1990, 38 (5), 746–753.

16 Hitney, H.V., Vieth, R. Statistical assessmentof evaporation duct propagation, IEEE Trans.Antennas Propagation, 1990, 38 (6), 794–799.

17 Hitney, H.V., Hitney, L.R. Frequency diversityeffects of evaporation duct propagation, IEEETrans. Antennas Propagation, 1990, 38 (10),1694–1700.

18 Anderson, K.D. Radar detection of low alti-tude targets in a maritime environment,IEEE Trans. Antennas Propagation, 1995, 43(6), 609–616.

19 Shen, X., Vilar, E. Path loss statistics andmechanisms of transhorizon propagationover a sea path, Electron. Lett., 1996, 32 (3),259–261.

20 Brookner, E., Ferraro, E. and Ouderkirk, G.D.Radar performance during propagation fadesin the mid-Atlantic region, 1998, IEEE Trans.Antennas Propagation, 1998, 46 (7),1056–1064.

21 Beckmann, P., Spizzichino, H. The Scatteringof Electromagnetic Waves from Rough Surfaces,Pergamon and MacMillan, New York, 1963.

22 Baz, A.I., Zeldovich, Y.B., Perelomov, A.M.Scattering, Reactions and Decays in Nonrelati-vistic Quantum Mechanics, 1980, AcademicPress, New York, 1980.

23 Brocks, K., Jeske, H. The meteorological con-ditions on electromagnetic wave propagationabove the sea, in Electromagnetic Distance

4 UHF Propagation in an Evaporation Duct120

Measurements, Pergamon Press, London,1967, pp. 122–136.

24 Ott, R.H. Roots of the modal equation for emwave propagation in a tropospheric ducts,1980, J. Math. Phys., 1980, 21 (5), 1255–1266.

25 Belobrova, M.V., Ivanov, V.K., Kukushkin,A.V., Levin, M.B. and Fastovsky, J.A. Predic-tion system on UHF radio propagation condi-tions over the sea, Institute of Radio Astron-omy, Ukrainian Academy of Science, PreprintNo 31, 1989, 39 pp.

26 Kukushkin, A.V., Fastovsky, J.A. and Levin,M.B. Propagation effect analysis: programsystem, Seventh International Conference OnAntennas and Propagation, Conference Publi-cation No 333, Part 1, 1991, pp. 535–539.

27 Budden, K.C. The Waveguide Mode Theory ofWave Propagation, Pergamon Press, London,1961.

28 Wait, J.R. Electromagnetic Waves in StratifiedMedia, Pergamon Press, New York, 1970.

29 Baumgartner, G.B., Hitney, H.V. and Pappert,R.A. Duct propagation modelling for the inte-grated refractive effects prediction system(IREPS), Proc. IEE, 1983, 130 Part F,630–642.

30 Marcus, S.W. A model to calculate em fieldsin tropospheric duct environment at frequen-cies through SHF, Radio Sci., 1982, 17,895–901.


32 Kukushkin, A.V., Freilikher, V.D. and Fuks,I.M. Over-the-horizon propagation of UHFradio waves above the sea , Radiophys. Quan-tum Electron. (transl. from Russian), Consul-tant Bureau, New York , RPQEAC, 1987, 30(7), 597–620.

121

The elevated refractive layer, the so-called M-inversion, is frequently associated withan anomalousy high level of the received signal at UHF frequencies over the hori-zon. It is apparent that the methods of prediction of either the parameters of theelevated layer or the signal level are needed in many applications, such as radar,surveillance and communications.The methods of the analytical solutions to the problem of wave propagation in the

presence of elevated M-inversion are less well developed than those applied to prop-agation in an evaporation duct. The major reason for this is that such a waveguidehas a multi-mode or multi-ray nature that, in turn, may require a different approachto obtaining the analytical solution, depending on the geometry of the problem(positions of the transmitter and receiver relative to the elevated duct “boundaries”,range and frequency).

For instance, when the transmitter and receiver are based close to the groundwith the distance between them x << 2

ffiffiffiffiffiffiffiffiffiffi2aZi

p, see Figure 5.1, where Zi is the height

of the minimum in the M-profile of the refractivity, the most effective approach is toapply the method of multiple reflections [1, 2], that, in turn leads to the approxima-tion of the geometrical optic with k! ¥. In contrast, at longer distances, the mosteffective method uses normal waves providing the modal representation of the wavefield.

In a sub-tropical region of the world’s oceans both evaporated and elevated ductsmay exist simultaneously thus complicating the situation. In Ref. [3] this situation isanalysed by applying the normal wave method to the M-profile approximated by apiece-wise linear profile. An alternative approach is described in Section 5.2, wherethe contribution of evaporation duct is presented by trapped modes while the reflec-tion from elevated M-inversion is analysed in terms of geometric optics.This chapter is arranged as follows. The modal representation of the wave field

for the case of elevated M-inversion is presented in Section 5.1. Section 5.2 intro-duces the hybrid, ray and modes, representation of the wave field in the problem ofa two-channel system. Some results of the measurements versus prediction are dis-cussed in Section 5.3 and, finally, in Section 5.4, we introduce a method of estimat-ing the excitation of the elevated duct due to scattering of the direct wave on thefluctuations in refractivity in the vicinity of the upper boundary of the atmosphericboundary layer.

5

Impact of Elevated M-inversions on the UHF/EHF FieldPropagation beyond the Horizon

5 Impact of Elevated M-inversions on the UHF/EHF Field Propagation beyond the Horizon

5.1Modal Representation of the Wave Field for the Case of Elevated M-inversion

Consider a piece-wise linear model of the M-profile as shown in Figure 5.1 andintroduce the dimensionless coordinates n ¼ mx=a, h ¼ kz=m. The parameters ofthe M-profile can be expressed in terms of h-coordinates and the scaled potentialUðhÞ ¼ 2m

210

�6MðzÞ:

Hk ¼ kZk=m, Hi ¼ kZi=m, Hs ¼ kZs=m,

Uk ¼ 2m210

�6MðZkÞ, Ui ¼ 2m

210

�6MðZiÞ, U0 ¼ 2m

210

�6DM.

Let us also introduce the gradients of the M-profile in each of the layers betweenZs ; Zi and above Zk respectively:

G2 ¼ dM=dz, with Zs < z < Zi

G4 ¼ dM=dz, with z > Zk .

The respective gradients of the dimensionless profile U(h) are given by

l31 ¼ U0=Hs , l

32 ¼ a � 10�6G2 , l

33 ¼ Ui �Uk

� �= Hk �Hi� �

, l34 ¼ a � 10�6G4 .

122

Zk

Zi

Zs

M(z)

Figure 5.1 Piece-wise linear approximation of the M-profile with two M-inversions.

5.1 Modal Representation of the Wave Field for the Case of Elevated M-inversion

Following the approach described in Chapter 2, we expand the attenuation factorW n; h; h0ð Þ over the set of eigenfunctions of a continuum spectrum. Using theresults of Section 2.3 and introducing the dimensionless “energy” t ¼ m2

E=k2we

obtain

W n; h; h0ð Þ ¼ e� jp

4R¥�¥dtW ðt; hÞW� ðt; h0Þe

jnt(5.1)

where the eigenfunction W t; hð Þ obeys the equation

d2W

dh2þ UðhÞ � t½ W ¼ 0 (5.2)

and the boundary conditions

W t; 0ð Þ ¼ 0 and Wðt;¥Þ ¼ 0 . (5.3)

As known from analogy with a quantum-mechanical problem [4], the solution toEq. (5.2) is given by a superposition of the waves, outgoing (v

þ) to infinity (h! ¥)

and incoming (v�) from infinity:

W t; hð Þ ¼ 12ffiffiffiffiffiffiffiffipl

4

p v�t; hð Þ � SðtÞvþ t; hð Þ

h i. (5.4)

It is also observed that v�t; hð Þ ¼ v

þt; hð Þ

� �, where the sign � indicates a complex

conjugate. The factor1

2ffiffiffiffiffiffiffiffipl

4

p in Eq. (5.4) comes from a delta-function normalisation

of the eigenfunction of the continuum spectrum, Section 2.3. From the boundarycondition (5.3) we have

SðtÞ ¼ v� t;0ð Þvþ t;0ð Þ

. (5.5)

Given the linear approximation to the M-profile in each of the layers:Hs < h £ Hi Hs < h £ Hi , Hi < h £ Hk and h > Hki the solution for v

�can be pre-

sented via superposition of the Airy–Fock function. Taking into account the continu-ity of both the function v

�and its derivative at the boundaries of the layers with con-

stant gradient of U(h), the outgoing wave can be written as follows:

1. With h > Hk

vþðhÞ ¼ w1

t�UðhÞl24

!. (5.6)

2. WithHk ‡ h > Hi

vþðhÞ ¼ A w1

t�UðhÞl23

!þ Rkw2

t�UðhÞl23

!( )(5.7)

where

123


A ¼ w2 xþk

� �w1 x

�k

� �þRkw2 x

�k

� � , (5.8)

Rk ¼ �w02 x

�k

� �þQ

kw2 x

�k

� �w01 x

�k

� �þQ

kw1 x

�k

� � . (5.9)

Here xþk ¼ t�Uk

l24

, x�k ¼ t�Uk

l23

, Rk is a reflection coefficient of the wave incident to

the boundary h ¼ Hk from h < Hk , the parameter Qk is the surface impedance atthe boundary h ¼ Hk and is given by

Qk ¼l4

l3

w01 x

þk

� �w01 x

�k

� � . (5.10)

3. WithHi ‡ h > Hs

vþðhÞ ¼ B w1

t�UðhÞl22

!þ Riw2

t�UðhÞl22

!( )(5.11)

where

B ¼ Aw2 xþi

� �þR

kw1ðx

þi Þ

w1 x�ið ÞþRiw2 x

�ið Þ , (5.12)

Ri ¼ � w01 x

�ið ÞþQiw1 x

�ið Þ

w02 x

�ið ÞþQ

kw2 x

�ið Þ , (5.13)

Qi ¼l3

l2

w01 x

þi

� �þR

kw1 x

þi

� �w02 x

�ið ÞþR

kw2 x

�ið Þ . (5.14)

Here xþi ¼ t�Ui=l

23 , x

�i ¼ t�Ui=l

22 , the other parameters have the same mean-

ing as above.

4. With h £ Hs

vþðhÞ ¼ C w2

t�UðhÞl21

!þ Rsw1

t�UðhÞl21

!( )(5.15)

where

C ¼ Bw1 xþs

� �þRiw2ðx

þs Þ

w2 x�sð ÞþRsw1 x

�sð Þ , (5.16)

Rs ¼ �w02 x

�sð ÞþQsw2 x

�sð Þ

w01 x

�sð ÞþQsw1 x

�sð Þ , (5.17)

Qi ¼ �l2

l1

w01 x

þs

� �þRiw1 x

þs

� �w02 x

�ið ÞþR

kw2 x

�ið Þ . (5.18)

124


Here xþs ¼ s=l

22 , x

�s ¼ s=l

21 , the other parameters have the same meaning as

above.The solution toWðn; h; h0Þ is obtained in principle and given by Eqs. (5.1)–(5.18).

However, the direct calculation of Eq. (5.1) is not practical because of the problem ofconvergence. Therefore, a significant amount of research, see for example Refs. [3,5, 6–10] has been dedicated to finding effective methods of calculating the electro-magnetic field in the presence of elevated M-inversion.

Here we consider a modal representation of the attenuation function Wðn; h; h0Þin a two-channel system when both elevated and surface-based M-inversion are pres-ent. In this approach, the integral (4.1) is calculated as a sum of the residue at thepoles of the S-matrix in the upper half-space of variable t. Taking into account Eqs.(5.5) – (5.18) we obtain the expression for the S-matrix in this particular problem:

S ¼ �RgA�w2

2 x�sð Þ� 1�R�

i T�sð Þ� 1�R�

i Tþs

� �� 1�R�

i R�gð Þ

Aw21 x

�sð Þ� 1�RiT

þs

� �� 1�RiT

�s �ð Þ� 1�RiRgð Þ

(5.19)

where Rg ¼ �w1ðx0Þ=w2ðx0Þ is the reflection coefficient of the ground,x0 ¼ t�U0=l

21 , T

þs ¼ �w2ðx

þs Þ=w1 x

�sð Þ is the reflection coefficient of the wave

incoming from the upper space from the boundary h ¼ Hswith ideal boundary con-ditions on it; T

�s ¼ �w1 x

�sð Þ=w2ðx

þs Þ is a similar coefficient of reflection from the

boundary h ¼ Hs ,but for the wave incoming from the space below the boundaryh ¼ Hs .The resonant terms in the denominator of the S-matrix determine the spectrum

of the normal waves in a two-channel system. Apparently, the propagation constantsof the normal waves are defined by the position of the poles of the S-matrix in thet-plane. The waves trapped in an evaporation duct (surface-based M-inversion) havepropagation constants tn in the interval Re tn 2 ð0;U0Þ. Waves localised in the ele-vated duct between Hs and Hk have propagation constants tn in the intervalRe tn 2 ð0;UiÞ. And, finally, the normal waves localised in a channel formed by theground surface h = 0 and the upper boundary of the elevated layer h ¼ Hk are in theinterval Re tn 2 ðUk ; 0Þ.

Now we consider the resonant terms of the S-matrix in more detail and assumethat Uk > 0. The resonant term 1� RiT

þs determines the spectrum of the normal

waves for the elevated channel with boundaries h ¼ Hs and h ¼ Hk . With U0 „ 0localisation of the modes in the surface-based channel is possible, in principle, andthe spectrum of these modes is defined by the equation 1� RsRg ¼ 0. In the casewhere UkU0 , both series of propagation constants are clearly separated in the wave-number space. More detailed analysis of this case is provided in Section 5.2. Nowwe concentrate on the modes of the elevated channel (duct). The characteristic equa-tion for the spectrum of propagation constants is given by

1� RiTþs ¼ 0 . (5.20)

With s=l22 >> 1 we can estimate T

þs ¼ �1þOðe

� 4

3t3=2n =l2

2 Þ and Eq. (5.20) canthen be reduced to 1þ Ri ¼ 0. Assume that the propagation constants of interestsare such that Re tn �Uk=l

22 >> 1. These normal waves experience complete reflec-

125


tion from the boundary h ¼ Hk, in fact the respective rays turn back long beforereaching the boundary Hk. The reflection coefficient can then be estimated asRk » � 1. Taking into account Eq. (5.12) for Ri we can obtain instead of Eq. (5.20)the following characteristic equation for the waves trapped in an elevated duct

t0 ðx�i Þ þ qitðx

�i Þ ¼ 0 (5.21)

where qi @ l3=l2 t0 ðxþi Þ=tðx

þi Þ, tðxÞ ¼ 1=2Þ j w1 xð Þ � w2 xð Þ½ ð . Equation (5.21) can

be further simplified using the asymptotic formulas for the Airy function t. Theresult is given by

sin yþi þ y

�i þ p

2

� ¼ 0; (5.22)

where

yþi ¼ 2

3l33

Ui � tð Þ3=2 , y�i ¼ 2

3l32 Ui � tð Þ3=2 .

The solution to Eq. (5.22) provides a spectrum of the propagation constants tnofthe elevated duct:

tn ¼ Ui �32

l2l3

� 3l32þl3

3

p n� 12

� �264

3752=3

. (5.23)

As observed, Eq. (5.23) is obtained by neglecting the leaking of the modes intothe space outside the duct. The number N1 of trapped modes is limited by the con-dition tn > Uk , therefore,

N1 ¼ entier23

l32þl3

3

l2l3

� 3 Ui �Uk� �3=2 þ 1

2

264

375. (5.24)

Counting the next terms of the asymptotic for both Tþs and Rk we can obtain

from Eq. (5.20) some correction terms to the propagation constants tn (5.23). Thesecorrection terms provide an estimate for a phase shift and attenuation factor for themodes due to the limited thickness of the potential barrier. Omitting the simple butcumbersome calculations we provide the final result for the imaginary part of thepropagation constants:

cn ¼ Im tn ¼ 14

l32þl3

3

l2l3

� 3 exp � 43

l33þl3

4

l3l4

� 3 tn �Uk� �3=2

8><>:

9>=>;. (5.25)

Let us consider another situation and assume now that Uk < 0. We pay attentionto the modes of the channels formed by the earth’s surface h ¼ 0 and the upperboundary of the M-inversion, h ¼ Hk . As discussed above, the propagation constants

126


of these modes lie in the interval Re tn 2 ðUk ; 0Þ. The spectrum of the propagationconstant is determined by the characteristic equation 1� RsRg ¼ 0. Using asymp-totic expressions for Airy functions we obtain

Rs @RiDe�jd

, (5.26)

Rg @� exp jp

2þ j 4

3l31

U0 � tð Þ3=2" #

where

d ¼ pþ 43

l32þl3

3

l32l33

�tð Þ3=2 , D ¼ 12�RiT

þs,

Tþs @� exp j

p

2� j 4

3l32

�tð Þ3=2" #

.

Let us expand the term D into a series

D ¼P¥m¼0

�1ð Þm ð1� RiTþs Þm . (5.27)

The series (5.27) takes into account the multiple effect of secondary reflection ofthe waves in the channel Hs £ h £Hk due to a leakage of the energy of the normalwaves localised in the channel 0 < h£Hk due to partial reflection from the bound-ary h ¼ Hs. In the first and rough approximation this effect of mutual coupling oftwo channels can be neglected, at least this approximation will be good enough forRe tnj j >> 1. Under this condition we can retain only the first term in series (5.27).As a result we obtain

1� RsRg » 1� RiRg e�jd ¼ 0. (5.28)

The last equation can be further simplified to the form

23

l32þl3

3

l32l33

Ui � tnð Þ3=2þ 2

3l31

U0 � tnð Þ3=2� 23

l32þl3

1

l32l31

�tnð Þ3=2¼ p n� 14

� �. (5.29)

The limiting case when surface M-inversion is absent is accounted for by Eq.(5.29) if we assume that U0 = 0. We can also observe that the number n satisfyingEq. (5.29) starts from n = N1 + N2 + 1, where N2 is the number of modes trapped ina surface-based channel, 0 < h £ Hs .To conclude, we may state that the characteristic equations (5.20) and (5.29) deter-

mine the limited, yet large, (for high frequencies and strong inversions of tempera-ture) set of trapped modes in a two-channel system. While in the general case thetwo modal series in the evaporation and the elevated duct are coupled, for modeslocalised deep in respective channels the mutual coupling can be neglected in thefirst approximation. In this way, the trapped modes of the evaporation duct can beestimated as the modes formed by a surface-based inversion only, Figure 5.2, andthe analysis is similar to that provided in Section 4.1.

127


Let us obtain a residue of the integrand in Eq. (5.1) in the pole of the S-matrix.Utilising the asymptotic expression for Airy functions incorporated into W ðt; hÞ andW

�ðt; h0Þ, the residue with Re tn > Uk is truncated to

Re s W ðt; hÞW� ðt; h0Þ t¼tn

¼ 2j Im tnf gvþ tn ; hð Þvþ tn ; h0ð Þ (5.30)

Next we obtain an asymptotic expression for the height-gain functions vþtn ; hð Þ.

First, define the coefficients (5.8) and (5.12) for Re tn > Uk:

AðtnÞ» jffiffiffiffiffil4

l3

sexp

D

2

� �(5.31)

where

D ¼ 43

l33þl3

4

l33l34

tn �Uk� �3=2

and, for Re tn > 0,

BðtnÞ »Affiffiffiffiffil3

l2

s1

cos ðnpÞ. (5.32)

For 0 > Re tn > Uk ,

BðtnÞ» 2jffiffiffiffiffil3

l2

ssin

3

2l32

Ui � tnð Þ3=2 þ p4

!1� exp j

4

3l32

Ui � tnð Þ3=2 þ jp2

( )" #·

exp �j 2

3l32

Ui � tnð Þ3=2 � jp4

( ): (5.33)

While obtaining Eq. (5.32) we take into consideration that Ri tnð Þ ¼ �1 at thepole. Assume that there is no evaporation duct present, i.e. Hs = 0, U0 = 0, and con-sider a height-gain function v

�tn ; hð Þ with h! 0 for 0 > Re tn > Uk. As follows

from Eq. (5.11),

vþtn ; h ¼ 0ð Þ ¼ 2B cos S1 � dð Þ exp jdð Þ (5.34)

where

128

Hs

Uo Figure 5.2 Isolated surface-based M-inversion.


S1 ¼ 2

3l32

�tnð Þ3=2þp

4, d ¼ 2

3

l32þl3

3

l32l33

Ui � tnð Þ3=2 .

Taking into account that at the pole, as follows from Eq. (5.22) with U0 = 0, thearguments of the cosine in the term S1 � d ¼ �npþ p=2, we obtain the boundarycondition of interest, v

þðh ¼ 0Þ ¼ 0. With U0 „ 0 and Re tn < 0, from Eq. (5.15) itfollows that

vþtn ; 0ð Þ » cos 2

3l31

U0 � tnð Þ3=2 � 23

l31þl3

2

l31l32

�tnð Þ3=2�p

4

" #. (5.35)

The equation for the poles takes the form RsRg ¼ 1 in this case. The relationshipbetween Rs and Rg is given by Eq. (5.26) from which we obtain that the argument ofthe cosine in Eq.(5.35) has the value npþ p=2 at the pole and, thereforevþtn ; 0ð Þ ¼ 0.In the case of positive Re tn > 0; tnj j >> 1, the behavior of the height-gain func-

tion vþtn ; hð Þ with h! 0 is governed by the exponent factor

vþtn ; hð Þ

h!0

» exp � 2

3l32

�tnð Þ3=2 !

! 0

and the boundary condition at h = 0 is satisfied asymptotically.Finally, we can present an explicit expression for the attenuation function

W n; h; h0ð Þover the sumof the normalwaves that can be used for computer calculation:

W n; h; h0ð Þ ¼ e jp=4ffiffiffinp

r14l

4

l32l33

l32þl3

3

PNn¼1ejntn�cn nvn hð Þvn h0ð Þ. (5.36)

The number of trapped modes N is determined by the number of real roots ofEqs. (5.22) and (5.29). The eigenfunction vn hð Þ is given by the following equations:

1. With h > Hk

vn hð Þ ¼ w1tn�UðhÞ

l24

!exp �D

2

� �. (5.37)

2. WithHk > h ‡ Hi

vn hð Þ ¼ 2

ffiffiffiffiffil4

l3

sttn�UðhÞ

l23

!. (5.38)

3. WithHi > h; tn > 0

vn hð Þ ¼ 2jBðtnÞ ttn�UðhÞ

l22

!(5.39)

and for Hi > h ‡ Hs ; tn < 0

vn hð Þ ¼ BðtnÞ w1tn�UðhÞ

l22

!þ RiðtnÞw2

tn�UðhÞl22

!" #. (5.40)

The factor BðtnÞ is defined by Eqs. (5.32) and (5.33).

129

5 Impact of Elevated M-inversions on the UHF/EHF Field Propagation beyond the Horizon130

0

50

100

150

200

250

300

350

-30 -20 -10 0 10 20 30

M(z)

z,

m

Figure 5.3 M-profile from Ref. [10].

(1)

(2)

Figure 5.4 Range dependence of the received field at thewavelength 9.1 cm inside the duct (1) and above the duct (2).


4. WithHs > h; tn < 0

vn hð Þ ¼ ffiffiffiffiffil1

py�1=4n t l1 h� ynð Þ

� �, (5.41)

where yn ¼ l1Hs � tn=l21 .

The representation (5.36) is valid in a “geometric shadow” region, i.e. at a distanceexceeding the maximum length of the cycle of the trapped modes Kmax~2

ffiffiffiffiffiffiffiffiffiffi2aZi

p. In

the opposite case, at distances less than Kmax , the “leak” modes of a higher orderprovide a substantial contribution to the sum of the normal waves. These modes,though attenuating with distance, are not localised in the elevated duct, their ampli-tude grows exponentially with the order of the mode. The sum of leaked modes con-verges slowly as a result of interference of these modes. This leads to a need for pre-cise determination of the relative phases of the modes and, in turn, to a sophisti-cated calculation of the complex propagation constants [5]. An alternative method tocalculate the field at the “line-of-sight” distance x < Kmax is to use the method ofstationary phase applied directly to integral (5.1). In a transition region at distances

131

Figure 5.5 Coverage diagram for 10 GHz source at 30.5 mfor the surface-based duct formed by M-profile in Figure 5.3.The return path loss is 220 dB.


close to Kmax , the most practical approach is to use interpolation over values of theattenuation function obtained in either region. While the asymptotic expression canbe obtained for the transition region it is very cumbersome and practically does notprovide significant advantage over a simple interpolation.To illustrate the modal representation described in this section, we apply the

above formulas to the case of radio wave propagation at wavelength k = 9.1 cm in asurface based tropospheric duct created by M-profile, shown in Figure 5.3. In factthis profile corresponds to the conditions of the experiment reported in Ref. [10].Figure 5.4 shows the range dependence of the received signal strength at two eleva-tions: 100 and 500 m. The calculated signal strength clearly illustrates the multi-mode character of the field inside the duct at the height 100 m and the contributionof a few modes only for the receiving antenna above the duct (at 500 m). Figure 5.5shows the impact on the radar coverage diagram at 10 GHz in case of the above tro-pospheric duct. It is observed that strong ducting mechanism allows for a targetdetection at the distances of several hundred miles.

5.2Hybrid Representation

In this section we use a Fock’s contour integral representation for the attenuationfactor Wðn; h; h0Þ. As shown in Chapter 2, both the contour integral representationand the expansion over the set of eigenfunctions of the continuous spectrum areequivalent in representing the attenuation factor Wðn; h; h0Þ in the absence of ran-dom inhomogeneities of the refractive index. From practical point of view, in a deter-ministic problem the manipulations with a contour integral are a bit less cumber-some than eigenfunctions of the continuous spectrum, since the latter containsterms, such as waves v

�ðt; hÞ coming from infinity that produce a negligible contri-bution to the integral for positive n in the absence of random fluctuations.

In line with the study in Ref. [11], we consider a slightly more general case of theM-profile for the hybrid ray-mode representation, as shown in Figure 5.6.The contour integral for the attenuation factorWðn; h; h0Þcan be written as

Wðn; h; h0Þ ¼ffiffiffinp

re� j p=4 R

CKðh; h0 ; tÞe

jntdt. (5.42)

The function Kðh; h0 ; tÞ satisfies the equation

d2

dh2K þ UðhÞ � t½ K ¼ dðh� h0Þ (5.43)

and the conditions

dKdh

þ qK� �

h¼0¼ 0, (5.44)

ddh

argK

h!¥

> 0,

132

5.2 Hybrid Representation

where q ¼ jmZ and Z is a surface impedance of the interface h = 0. The integrationcontour embraces singularities of the integrand lying in the first quadrant. The pre-cise form of K depends on the shape of the profile U(h) and the relative valuesh; h0 ;Hs and Hi . If both the transmitter and the receiver are within the lower inver-sion layer, i.e. h0 < h < Hs , then K takes the form

Kðh; h0 ; tÞ ¼ � f2ðh0 ;tÞþRg f1 h0 ;t� �# $

� f1ðh;tÞþR�f2 h;tð Þ# $

Wðf1 ;f2Þ 1�RðtÞRg ðtÞ# $ , (5.45)

where f1 and f2 are the independent solutions of uniform equation (5.43) specifiedwithin 0 < h < Hs , and

W ¼ f1df2dh

� f2df1dh

. (5.46)

The values Rg and R are defined as

133

h

U(h)

kH

iH

sH

+

lh

–l

h

+

mh

–m

h

mt

,2lt

,1

Figure 5.6 Smooth M-profile for a two-channel model of refractivity.


Rg ¼ � f02 0;tð Þþqf2 0;tð Þf 01 0;tð Þþqf1 0;tð Þ , (5.47)

R ¼ �f 01 Hs ;tð Þ�f1 Hs ;tð Þj

01 Hs ; tð Þ þ Ri � j

02 Hs ; tð Þ

j1 Hs ; tð Þ þ Ri � j2 Hs ; tð Þ

f 02 Hs ;tð Þ�f2 Hs ;tð Þj01 Hs ; tð Þ þ Ri � j

02 Hs ; tð Þ

j1 Hs ; tð Þ þ Ri � j2 Hs ; tð Þ

, (5.48)

Ri ¼j01Hi ;t� �

�j1Hi ;t� �

�w01 Hi ; tð Þ

w1 Hi ; tð Þ

j02Hi ;t� �

�j2Hi ;t� �

�w01 Hi ; tð Þ

w1 Hi ; tð Þ

. (5.49)

The functions j1 and j2 are independent solutions of the uniform equation(5.43) for Hs < h < Hi, and w1 is the solution for h > Hi which satisfies the condi-tion

ddh

arg w1 > 0 with h! ¥. (5.50)

The derivative sign f0in Eqs. (5.47)–(5.49) means the derivative over the h-vari-

able. Apparently, Rg, R and Ri can be regarded as the reflection coefficients ath ¼ 0; h ¼ Hs and h ¼ Hi , respectively.

Now we can get to the hybrid representation of the received field, as manifestedin the beginning of Section 5.2.

As a first step toward implementing this program we will express the factorð1� RiRg Þ

�1in terms of a partial geometric series,

ð1� RiRg Þ�1 ¼ 1þ

PNn¼1

RiRg� �n þ RiRg

� �Nþ11�RiRg

(5.51)

and substitute Eq. (5.51) into the integral (5.42):


re�jp=4

ZC

dtejntW

�1ðf1 ; f2Þ ·

F1ðh0 ; tÞ f1 h; tð Þ þ Rf2 h; tð Þ½ þPNn¼1

RRg� �n

F1ðh0 ; tÞ f1 h; tð Þ þ Rf2 h; tð Þ½ þ

RRg� �Nþ1

Fðh;h0 ;tÞ1�RRg

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

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

(5.52)

Here

F1ðh0 ; tÞ ¼ f2ðh0 ; tÞ þ Rg f1 h0 ; tð Þ;(5.53)Fðh; h0 ; tÞ ¼ �Kðh; h0 ; tÞWðf1 ; f2Þð1� RRg Þ

and N is an arbitrary integer whose choice is dictated by the length of the propaga-tion path. It represents the maximum number of reflections from the upper “bound-ary” h = Hi to be taken into account.

134


If there is no elevated M-inversion (cf. the dashed profile in Figure 5.6), theattenuation functionWðn; h; h0Þ would take the form


re�jp=4 R

Cdte

jnt F1ðh0 ;tÞ f1 h;tð ÞþRs f2 h;tð Þ# $

Wðf1 ;f2Þ 1�RsRg� � , (5.54)

where Rs denotes the reflection coefficient at the “upper wall” h = Hs of the surfaceduct, i.e.,

Rs ¼ �f 01 Hs ;tð Þ�f1 Hs ;tð Þj

01 Hs ; tð Þ

j1 Hs ; tð Þ

f 02 Hs ;tð Þ�f2 Hs ;tð Þj01 Hs ; tð Þ

j1 Hs ; tð Þ

. (5.55)

Of the two independent functions above Hs , i.e. j1 and j2 , only j1 will remainbecause of the “new” condition

ddh

argj > 0 with h > Hs . (5.56)

To isolate the set of eigenwaves of a solitary surface duct from the solution (5.52),we multiply and divide the first term of Eq. (5.52) by the “resonant denominator” ofEq. (5.54), i.e., ð1� RsRg Þ, and represent R as

R ¼ Rs � RiK1þRsK2

1þRiK2

¼ Rs � Rim (5.57)

with

K1 ¼f 01 Hs ;tð Þ�j

2Hs ;tð Þ�f1 Hs ;tð Þ�j0

2Hs ;tð Þ

f 02 Hs ;tð Þ�j1Hs ;tð Þ�f2 Hs ;tð Þ�j0

1Hs ;tð Þ,

(5.58)

K2 ¼f 02 Hs ;tð Þ�j

2Hs ;tð Þ�f2 Hs ;tð Þ�j0

2Hs ;tð Þ

f 02 Hs ;tð Þ�j1Hs ;tð Þ�f2 Hs ;tð Þ�j0

1Hs ;tð Þ.

Now consider the function g(h, t) defined as the solution of

d2

dh2g þ U1ðhÞ � t½ g ¼ 0 (5.59)

with the conditions

dgdh

þ qg� �

h¼0¼ 0, (5.60)

gðHs ; tÞ ¼ CðtÞj1ðHs ; tÞ (5.61)

where U1(h) corresponds to the dashed M-profile of Figure 5.6. We can obtain that

135


1� RsRg ¼

dgdh

þ qg� �

h¼0

gð0; tÞ f 01 ð0; tÞ þ qf1ð0; tÞh i �

f1 0; tð Þ �f01 Hs ; tð Þ � f1 Hs ; tð Þ

j01Hs ;tð Þ

j1Hs ;tð Þ

f02 Hs ; tð Þ � f2 Hs ; tð Þ

j01Hs ;tð Þ

j1Hs ;tð Þ

f2 0; tð Þ

266664

377775

(5.62)

and hence Eq. (5.60) is equivalent to the characteristic equation

1� RsRg ¼ 0

specifying the modes of the surface duct. Further, we bring the integrand ofEq. (5.52) to the form

F1ðh0 ; tÞ f1 h; tð Þ þ Rs f2 h; tð Þ½ 1� RsRg

� F1ðh0 ; tÞRimf2ðh; tÞ1� RsRg

�Fðh; h0 ; tÞRsRg

1� RsRg

þXNn¼1

Fðh; h0 ; tÞRng R

ns þ

Xnp¼1

np

� ��1ð ÞpRn�ps R

pi mp

24

35þ

RRg� �Nþ1

Fðh; h0 ; tÞ1� RRg

(5.63)

withnp

� �being binominal coefficients, and express the first integral as a sum of

the residues at t ¼ ttn , or the zeros of the equation 1� RsRg ¼ 0 (the bar serves toindicate their distinction from the roots of 1� RRg ¼ 0). Representing1� RsRg� ��1

in the second and third term as another partial geometric series, gen-erally speaking with a number of terms different from Eq. (5.51), we assemble simi-lar terms and arrive at


re� jp=4

W01 þ T2 þ T3 þ T4ð Þ (5.64)

where termW01 represents the sum of the normal modes of the surface duct

W01 ¼ 2pjP¥n¼1

exp jttnn� � dttn

dqgðh;ttnÞgð0;ttnÞ

gðh0 ;ttnÞgð0;ttnÞ

(5.65)

and

T2 ¼ �RCdte

jntW

�1ðf1 ; f2ÞF1ðh0 ; tÞRim � f2 h; tð Þ, (5.66)

T3 ¼PNn¼1

RCdt

ejnt

Wðf1 ;f2ÞRng ·

Fðh; h0 ; tÞPnp¼1

np

� ��1ð ÞpRn�ps R

pi mp � F1ðh0 ; tÞRimR

ns f2ðh; tÞ

" #, (5.67)

136


T4 ¼RCdt

e jnt

Wðf1 ;f2ÞFðh;h0 ;tÞ1�RRg

RRg� �Nþ1� F2ðh;h0 ;tÞ

1�RsRgRsRg� �Nþ1" #

(5.68)

where

F2ðh; h0 ; tÞ ¼ F1ðh0 ; tÞ f1ðh; tÞ þ Rs f2ðh; tÞ½ . (5.69)

The term T2 can be interpreted as the field arriving at the observation point due toa single reflection from the elevated inversion while T3 is a set of waves multiplyreflected between the boundaries h ¼ Hi , h ¼ Hs , and h = 0. T4 is a remainder termwhose smallness will be secured by the proper choice of N for every specific separa-tion from the source. It should be emphasized that in bringing Eq. (5.52) to theform of Eq. (5.64) we made no approximation, and hence Eq. (5.64) is an exact repre-sentation of the field in the structure analysed ( to be more precise, it is exact in thesame sense as the initial formula Eq. (5.42), which itself corresponds to a parabolicequation approximation). The set of normal modes associated with the surface duct,the entire infinite spectrum, has been separated from Eq. (5.42) solely by transform-ing the integrand (5.45). The fields given by T2 and T3 so far cannot be interpretedin the ray optical form, since rays appear only at the stage of asymptotic evaluationof the integrals.

As for the numerical truncation of the modal series, the question seems almosttrivial, since in the absence of a reflecting upper wall the higher-order modes areevanescent and their respective eigenvalues have the ordering Imttnþ1 > Imttn .

As observed from Eqs. (5.66) and (5.67), the integrands T2 and T3 contain no sin-gularities. Depending on the number of significant saddle points, each of these inte-grals splits into several terms representing waves which are radiated either up ordown from the transmitting antenna. Further along the path, they are reflectedfrom the boundaries h = 0 , h = Hs and h = Hi to arrive at an observation point alongsome of the ray trajectories of Figure 5.7. For example, T2 splits into two terms, eachcorresponding to a wave that is reflected once from the elevated layer. The firstleaves the source in an upward direction, reaches a reflection point in the upperlayer, and comes to the observation point along the ray 1, Figure 5.7. Anotheremerges downwards from the source downward, undergoes reflections from theearth’s surface and the elevated layer, and arrives at the observation point along tra-jectory 2, Figure 5.7.The different components of the angular spectrum of waves excited by the trans-

mitter dipole are subject to a kind of spatial Mfiltration’ in the nonuniform tropo-sphere. We can define the critical angle h0 as the angle at which the rayQO

0emerging from the source turns at the height h = Hs, as shown in Figure 5.7.

The rays with departure angles h < h0 are trapped by the surface duct; those withh > h0 leave it freely. The ray with h ¼ h0 þ d, d ! 0 comes back to earth, uponreflection from the elevated layer, at the maximum range attainable to a singlereflection from the elevated layer, we define this range as n1m . Real rays arriving atgreater distances than n1m can do so only through a higher number of reflectionsfrom the “boundaries” h = Hi, h = 0 and h = Hs. Generally, each group of rays of a

137


given reflection multiplicity, l, has its own Mhorizon’, i.e. the limiting distance stillcorresponds to real stationary points in the integrand of T2 (for l = 1) and/or T3(l > 1). It is in the vicinity of the lth horizon that the contribution of l times thereflected wave is the highest. Indeed, the grazing angle with respect to the elevatedlayer assumes the lowest possible value when the observation point is near the hori-zon. As a result, one can distinguish, along the path, characteristic zones of the“first hop” (i.e. near n ¼ n1m where is the dominant contribution to T2 + T3 + T4, forthe N terms in T3, only those with l £ N Mwork’). Moving beyond the lth horizon, anobserver finds himself in an umbral zone where the field of the l + 1 hop has notyet formed while that of the lth hop can no longer propagate except by diffraction.The magnitude of this diffractional field is determined by the small contribution ofcomplex rays corresponding to the complex stationary points of T2 and/or T3, butmainly by the value of the remainder integral T4. For evaluating this latter it seemsconvenient to represent it as

T4 ¼ T41 þ T42 (5.70)

with

T41 ¼R¥�¥dte jntF1ðh0 ;tÞ1�RsRg

f1ðh; tÞ þ f2ðh; tÞRNþ1�RNþ1s

RN�RNs

" #RNþ1g RðRN � RNs Þ, (5.71)

T42 ¼R¥�¥dte jntF1ðh0 ;tÞðRsRg Þ

Nþ1

1�RsRg� �

1�RRg� � f2ðh; tÞ þ f1ðh; tÞ

1Rs

' (RðR� RsÞ. (5.72)

As can be seen, T41 is the contribution of waves N + 1 times reflected from theupper layer, and hence it should be small near the horizon of l £N-reflected waves.

138

0θ

QO

O

Range

sH

iH

kH

iΛ

2

2

1

1

Figure 5.7 Schematic of the ray trajectories in the structureshown in Figure 5.6; Ki is the maximum range attainablethrough a single reflection from the upper layer.


The number of reflections, N, which are to be taken into account at a specific dis-tance from the transmitter can be estimated as follows: Let us assume for simplicitythat the reflection coefficient Ri depends on the grazing angle h as

RiðhÞ ¼ 1; h£ h0RiðhÞ ¼ 0; h > h0

(5.73)

where h is with respect to the lower edge of the layer, ri ¼ aþ Zi . Consider the fieldTN due to reflections from the upper layer, for h ¼ h0 ¼ 0 and n @ 2

ffiffiffiffiffiffiHi

p(i.e., near

the horizon of singly reflected waves), i.e.,

TN ~ exp jntþ j 43N Hi � tð Þ3=2 � j 4

3N �tð Þ3=2

' (. (5.74)

The saddle point of Eq. (5.74) is

tðNÞ ¼ �Hi

N2�1� 2

4N2 . (5.75)

Recalling the relation between the grazing angle and t, i.e.,

cosh ¼ kaþmtkri

(5.76)

and the fact that real saddle points of real rays satisfy the inequality

t £ min h; h0ð Þ (5.77)

we can obtain from Eq. (5.76) an estimate at t! 0 for the grazing angle h of Ntimes the reflected waves

hðNÞmin @

2Hia

� �1=2N2þ12N

. (5.78)

To have Ri hminð Þ„ 0, it is necessary that

2Hia

� �1=2N2þ12N

£ h0 . (5.79)

The order of magnitude of h0 is 10–2, and that of 2Hi=að Þ1=2 @ 5· 10�3 to 10–2,whence the number of terms to be retained in Eq. (5.64) near the horizon l = 1 isN = 1–3.We leave evaluation of the magnitude of term T42 in Eq. (5.72) until the next sec-

tion, noting here that its contribution cannot be described in pure ray optical terms.In fact this term describes a peculiar physical effect of secondary excitation of thesurface (evaporation) duct.

A better illustration of the relative importance of the different terms in Eq. (5.64)can be achieved through numerical analysis. Figure 5.8 represents the calculatedattenuation function for f = 10 GHz in a two-channel structure of the refractivity.The surface M-inversion is characterised by Zs = 9 m, DM ¼ Mð0Þ �MðZsÞ = 2 M-units. The elevated M-inversion lies at the height Zi = 500 m and has been approxi-

139


mated with a discontinuity DMi = 10 M-units of no thickness (i.e., Zk = Zi). Anapproximation like this obviously results in an overestimated value of the reflectioncoefficient Ri(t). The heights of the transmitting and receiving antennas arez0 = 8.5 m and z = 5 m, respectively, apparently both inside the evaporation duct.The normalised impedance q of the earth’s surface is assumed to have a value corre-sponding to that of sea water: Req = 6, Im q = 94 . The calculated imaginary part ofthe first-mode eigenvalue is found to be Im t1 = 0.4. For a given combination of theantenna heights the optical horizon would be at » 20 km. As observed from Figure5.8, there is a region beyond that horizon (extending to about 60 or 70 km) wherethe field level is determined by the lower-order mode of the surface duct alone.Further along the path, the contribution of waves reflected from the elevated layerbecomes substantial. In the zone of the first hop (i.e. 60 km £ x £ 200 km) the max-imum field strength becomes close to the free space level. The remainder term T4reaches a –40 dB level near distance 200 km with N = 1.The mechanism of the radio wave propagation in the above two-channel system

can be summarised as follows. At relatively smaller distances from the transmitterthe dominant contribution to the received signal strength comes from evaporationduct propagation alone. Further along the path, the waves, departing with anglesexceeding the critical angle h0 and reflected by the elevated M-inversion, contributeto the received field at an appropriate distance of the single hop. These waves inter-fere with each other as well as with the waves propagating through the evaporationduct producing the deep fades caused by large phase difference. For example, in Fig-ure 5.8, the deepest fades are observed in the range 60–100 km where the wavepropagating through the evaporation duct and superposition of the waves reflectedfrom the elevated M-inversion are comparable in amplitude. Then, along the dis-tance, the field reflected from the elevated M-inversion takes over and provides themajor contribution to the received field strength.

140

-80

-70

-60

-50

-40

-30

-20

-10

0

0 20 40 60 80 100 120 140 160 180 200 220

Distance, km

Re

ce

ive

d s

ign

al le

ve

l re

lati

ve

to

a f

ree

-

sp

ac

e, d

B

Figure 5.8 Calculated signal strength level for a two-channelsystem at frequency 10 GHz: Zs = 9 m, DM = 2 M-units,Zi = 500 m and DMi = 10 M-units.


5.2.1Secondary Excitation of the Evaporation Duct by the Waves Reflected from an ElevatedRefractive Layer

Referring to the discussion in the previous section and Figure 5.7, we state that therays with departure angles from the source less than critical angle h0 are trapped inthe surface duct, those with h > h0 leave it freely. We also note that rays with h > h0reach the observation point at the distances n < n1max . At distances greater than thehorizon of first hop (n1max , single reflection) there are no single reflected rayslaunched at the angles of departure h > h0 . Hence, between the first and secondhop there is a range of distances where the receiving field is formed by anothermechanism represented by term T42 in Eq. (5.72) and evaluated in this section. .

Consider the range of distances n ‡ n1max concentrated on trapped waves/rays.The trapped waves with departure angles h < h0 turn inside the surface duct, asshown in Figure 5.8. Since the potential barrier is finite the trapped waves can leakto the space above the surface duct as shown schematically in Figure 5.9, theirrespective phases are complex due to the leaking mechanism distributed along thedistance. The field created by these waves in the space above the surface duct can beimagined as a cluster of rays distributed along the distance and sliding along thetangents to circles of radius r = a + h2, where the height h2 is given by the relation2m2 10–6 M(h2) = t for h2 > Zs, where t is a dimensionless Menergy’ corresponding tothe angle h.

In evaluating the maximum magnitude of T42 and therefore the impact from sec-ondary excitation at n ‡ n1max we shall set N = 1. Taking into account that the obser-vation point lies within the surface layer, we will represent the integral (5.72) as asum of residues at the zeros of 1 – Rs Rg = 0.

141

Distance

Height

M(z)

2h'

2h ''

Height

Zs

Figure 5.9 Schematic diagram of the penetration of “leaked”modes through the potential barrier in an evaporation duct.


Using Eq. (5.57) we can isolate the coefficient of reflection from the upper bound-ary of the surface duct Rs in R and present the denominator of Eq. (5.72) in the form

ð1� RRg Þð1� RsRg Þ ¼ 1� RsRg� �2

1� RiRg m

1�RsRg

" #. (5.80)

The term m may be interpreted as a coupling factor between the lower and upperducts. For low attenuating modes (Imtn << 1) it can be estimated as

mðtnÞ » exp � 43lþ1lt3=2n

� �(5.81)

where l ¼ dU=dhj j is a gradient of profile U(h) for 0 < h < Hs. The greater Re tnthe lower is the turning height h1, the lesser the amount of leaking from the surfaceduct and, in accordance with a reciprocity theorem, such waves are harder to gener-ate by a source distributed across the reflective layer but outside the surface duct.To finish evaluation of T42, we write a partial geometric series for

1� RsRg� ��1

1� RRg� ��1

with the help of formula (5.80), namely,

ð1� RsRg Þ�1ð1� RRg Þ

�1 ¼ ð1� RsRg Þ�2·

1þXP�1p¼1

�RiRg m� �p1� RRg� �p þ �RiRg m

� �P1� RRgð Þ 1� RsRg

� �P�124

35 (5.82)

and substitute it in Eq. (5.72).Now we restrain the evaluation by the first term of the expansion (5.82) using the

same arguments of angular filtration as in the previous section. Then we assumethat Hi �Hs >> 1 and a=Hi � 10

�6DM << 1, where DM ¼ Mðz ¼ 0Þ �MðZsÞ is the

M-deficit of the surface duct. Therefore, we assume that the waves reflected fromthe elevated refractive layer can be regarded as a plain waves on their arrival at thelevel of the surface duct, and the horizons of departure of the leaked waves h2 inFigure 5.9 are not to too different from Hs compared with Hi, i.e., max h2ð Þ=Hi << 1.This allows us to use the same magnitude of the reflection coefficientRi tð Þ » Rið0Þ for all rays from the cluster of leaking waves, expanding the ampli-tude and phase of the reflection coefficient into a series over powers of t. Taking intoaccount that the observation point lies within the surface layer, we will represent theintegral (5.72) as a sum of residues at the zeros of 1� RsRg

� �2¼ 0

T42 @ 2pj Rið0Þ exp j 4

3Hi �Hsð Þ3=2þd 0ð Þ

h i) *XSs¼1

exp jntts� �

1þ qð Þm tts� � dtts

dq

� �2

·

gðh;ttsÞgð0;ttsÞ

gðh0 ;ttsÞgð0;ttsÞ

m0 ðttsÞmðttsÞ

þ jnþ 3g�1ð0; ttsÞ

@gð0;ttsÞ@ tts

þ dttsdq

@

@ tts

dttsdq

� ��1" #

þ @

@ tts

gðh;ttsÞgð0;ttsÞ

gðh0 ;ttsÞgð0;ttsÞ

' (8>>>><>>>>:

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

(5.83)

Here S is the number of modes trapped in the surface duct.

142


Equation (5.83) provides an estimate of the secondary excitation of the surfaceduct by a limited number of the modes of the surface duct, “trapped”, from the pointof view of geometrical optics but, in fact, “leaking” from the duct along the distanceof propagation due to the finite thickness of the potential barrier in U(h). Thesewaves, then being reflected from the elevated M-inversion, reach the surface ductfrom above and “leak” back into the wave guide reproducing itself in the surfaceduct. Of course, the symmetry in the secondary excitation of the surface duct relieson the uniformity of the refractive index in the horizontal plane along the distance.Analysing Eq. (5.83) we may note that for the shorter wavelength k of the electro-magnetic radiation, the trapping condition of the surface duct becomes stronger andleaking of the trapped modes is reduced, as is the attenuation of the ducted fieldwith distance. This leads to an even smaller contribution from the secondary excitedfield described by term T42, Eq. (5.83). Increase in the wavelength k leads, on theone hand, to a greater attenuation of the ducted field and stronger leaking of thetrapped modes from the duct and, on the other hand, to better conditions for reflec-tion from the elevated layer and stronger secondary excitation of the surface duct.The major contribution to the secondary guided field is from the modes withRe tnj j << 1. These modes have relatively high attenuation compared with deeplytrapped modes, however they still provide a significant contribution at smaller dis-tances from the source and, being reflected from the elevated layer, at distancesn ‡ n1max: Evaluating this result numerically, we have found that the secondaryguided field given by Eq. (5.83) may reach a maximum level of –50 dB relative to afree space condition with ideal reflection from the elevated layer, Ri

¼ 1.The rather exotic though observable situation is shown in Figure 5.10 where we

have extremely strong elevated M-inversion and surface-based evaporation duct si-multaneously. Here Zs = 18 m, DM = M(0) – M(Zs) = 6 M-units, Zs = 500 m, Zk =

143

-4

-2

0

2

4

6

8

10

12

14

0 50 100 150 200 250 300

Distance, km

Sig

na

l s

tre

ng

th r

ela

tiv

e t

o f

ree

sp

ac

e,

dB

Figure 5.10 The range dependence of the attenuation factor ina two-channel system:h, both antennas located inside the eva-poration duct;~, antennas above the evaporation duct.


600 m, DMs = 60 M-units. The calculation was performed for radio frequencyf = 3 GHz and the following combination of antenna heights: z0 = 16 m, z = 10 mand 50 m, respectivly.

Figure 5.10 shows that the field reflected from elevated M-inversion dominates atdistances of the order of the reflection cycle length Ki despite there being lots oftrapped modes of the elevated duct. At shorter distances the total field is a combina-tion of the modes of the evaporation and elevated ducts with predominance of theevaporation duct. It also might be observed that at distances exceeding the first hopdistance but still shorter than the second hop distance (in the above case withx > 200 km) the secondary excited field of the evaporation duct may provide a sub-stantial contribution revealing a distinct change in the attenuation rate with distanceand somewhat repeating the field at short range.

Figure 5.7 shows the ray trajectories for this situation. As observed, in the vicinityof the first hop the waves reflected from the elevated M-inversion provide a domi-nant contribution to the received field.

5.3Comparison of Experiment with the Deterministic Theory of the Elevated DuctPropagation

Among numerous experiments we select Refs. [10–14] where the measured datahave been compared with theoretical calculation of the propagation in the evapora-tion duct. In Refs. [10, 13], the authors studied the height structure of the receivedsignal strength inside the elevated duct and good agreement between theory andmeasured data was observed for the location of the receiving antenna inside theduct. However, in the vicinity of the channel borders the measured data exceed thelevels predicted from deterministic theory.

In Ref. [10], themeasured data for frequencies 65, 170, 520 and 3300 MHz have beencompared with the theory for the respective frequencies. The elevated M-inversionforms the singlemodewaveguide at the frequency 65 MHz. The characteristic values ofthe M-profile measured in Ref. [10] are as follows: Zi= 180 m, Zk= 302 m,DMi = 40 M-units; the waveguide channel is created in the interval of heights0 < z < Zk, i.e., the duct belongs to the class of surface-based elevated ducts. Figure 5.11fromRef. [10] shows very good agreement of the height dependence of the received sig-nal strength with the theoretical calculations. It might be noted that in this frequencyband the effects of scattering on the fluctuations of the refractivity are relatively small.

Figure 5.12 shows the measured data for 3.3 GHz in the same duct. As observedfrom the figure, at this frequency the elevated duct reveals the multimode structureof the received signal level. Inside the duct the measured signal strength levelsexceed the level of the field due to the condition of “normal” refraction at 50–30 dB.The results of theoretical calculation (solid line) depicted in Figure 5.12 show goodagreement with the measured data, fitting very well between the maximum andminimum signal levels observed in the experiment inside the duct. As seen fromthe same figure, the discrepancy between measured and predicted results becomes

144

5.3 Comparison of Experiment with the Deterministic Theory of the Elevated Duct Propagation

significant when the receiving antenna are located close to boundaries inside theduct or outside the duct. The rapid decay of the calculated signal level at heightsz > 250 m is caused by a destructive interference of many modes. At the same timethe amplitude of each mode may decay at a lesser rate compared with the compositeof all modes. As pointed out in Ref. [10], the phase relationship between modes maybe distorted by non-uniformity of the refractivity structure in a horizontal plane.The predicted path loss at frequency 3 GHz in Ref. [10] shows good agreement

with the measured data for the transmitter and receiver located within the duct.Such good agreement with theory based on a rather crude model of stratified refrac-tivity is likely due to the fact that slow variations in the refractive structure of theevaporation duct may result in “adiabatic” restructuring of the trapped modes andmutual transfer of energy between the resulting mode structure still largely formedby trapped modes. The result of this superposition in the received field strengthwould be close to one in the case of a horizontally uniform duct for the transmitterand receiver located far enough from the duct borders.

145

0

100

200

300

400

500

600

-40 -30 -20 -10 0

Received signal level relative to a free space, dB

Heig

ht

, m

Figure 5.11 Height dependence of the received field level at adistance 111.2 km from the transmitter at a frequency of65 MHz: solid line, measured data;s, calculated using a split-step Fourier method.


Figure 5.13 from Ref. [10] shows the results of another experiment with an ele-vated duct detached from the ground surface. The parameters of the M-profile inthis experiment were as follows: Zk= 800 m, zmin = 600 m, DMi = 20 M-units. Thetransmitting antenna was located at a height 20.7 m while receiving antenna was ata height 914 m above the sea surface, i.e. outside the elevated duct. With such place-ment of the receiving and transmitting antennas the trapped modes of the elevatedduct cannot practically be generated by the radiating field of the transmittingantenna in such a configuration of the antennas in relation to the duct boundariesZk, zmin Figure 5.13 shows the result of the theoretical calculation of the receivedsignal strength at frequency 3 GHz versus distance for a given placement of theantennas. As observed the calculated signal is most likely composed of the highorder “leaked” modes with a rather great attenuation rate along the distance. At thesame time, the measured signal level significantly exceeded that theoretically pre-dicted from the deterministic theory of the stratified refractivity. The measured dataalso shown in Figure 5.13 do not attenuate with distance, behavior that is ratherassociated with trapped modes of the elevated duct.

146

0

100

200

300

400

500

600

-40 -30 -20 -10 0 10 20 30 40

Signal level relative to free space, dB

he

igh

t, m

Figure 5.12 As Figure 5.11 but for frequency 3.3 GHz. Twosolid curves withh show minimum and maximum signal levelsobserved during the measurement interval.

5.4 Excitation of the Elevated Duct due to Scattering on the Fluctuations in the Refractive Index

In order to evaluate another mechanism for long range radio wave propagation,the authors of Ref. [10] plotted in Figure 5.13 the level of the single-scattered field inaccordance with the standard assumptions of the Booker–Gordon theory [15].

In a following study [13] the authors attempted to explain the observed signal lev-els in Figure 5.13 in the above experiment by the presence of an evaporation duct atthe time of mthe easurements. Nonetheless, as discussed in Ref. [13], during theradio experiment there was no adequate measurement of the meteorological datawhich are required to restore the M-profile near the sea surface. The authors usedinstead some routine meteorological data available for that area and restored threepossible M-profiles of the evaporation duct, one of which provides satisfactory agree-ment with the measured data in Figure 5.13.

It might be worthwhile to consider an alternative interpretation of the measureddata in the above experiment, related to excitation of the trapped modes in the ele-vated duct due to scattering on the turbulent fluctuations of the refractive index in-side the elevated duct. The respective theory and estimates are the subject of thenext section.

5.4Excitation of the Elevated Duct due to Scattering on the Fluctuations in the RefractiveIndex

Consider the piece-wise linear mode of the M-profile, introduced at the beginning ofthis chapter, Figure 5.1, and assume Zs = 0, DM = 0. In this section we investigatethe mechanism of excitation of the elevated duct that can be formulated as follows:The characteristic values of the thickness, Zk – Zi, of the elevated M-inversion and

depth, DMi, in most cases are themselves sufficient to ensure a waveguide mecha-

147

120

140

160

180

200

220

240

260

280

0 200 400 600

Distance, km

Re

ce

ive

d s

ign

al

lev

el,

dB

m

Series1

Series2

Series3

Series4

Series5

Figure 5.13 The received field strength vs. distance for elevatedM-inversion at frequency 3.0877 MHz [10]: (1) signal strength ina free-space; (2) measured data; (3) theoretical results usingaveraged deterministic M-profile; (4) the level of the troposcat-tered signal; (5) results of the calculation using formula (5.90).


nism for radio wave propogation in the range of 1–10 GHz of the frequency spec-trum. As discussed in Chapter 1, the elevated duct location is defined in the intervalof heights Zmin £ z £ Zk . The efficiency of the excitation of the guided waves none-theless depends on the position of the transmitting antenna relative to the ductboundaries. If the antenna hight z0 is outside the duct, say z0 < Zmin, it is “impossi-ble”, from the geometric optic point of view, to excite the guided waves. This meansthat the radiated field of the direct wave propagates through the elevated refractivelayer experiencing slight refraction within, Figure 5.14. In this case the trapping ofthe waves inside the elevated duct is only possible due to scattering on the fluctua-tions of the refractive index, Figure 5.14.The distances of interest are far beyond the horizon relative to the scattering vol-

ume, therefore we take into account only a single scattering into trapped modes ofthe elevated duct. For the same reason, the size of the scattering volume in theheights is limited by the interval of localisation of the trapped mode, i.e.,Zmin £ z £ Zk . Let us assume that the width of the antenna pattern ha in azimuth iswide enough, i.e.,

JaD >> Lk (5.84)

where D is the distance between the transmitter and the scattering volume, Lk is anexternal scale of the fluctuations in de in the plane (x, y). In the current estimate ofthe scattering field we use an approximation for a free-space attenuation functionW0 of the direct wave. In this approximation for W0 we do not take into accountrefraction (other than normal refraction) along the path between the transmittingantenna and a point within the scattering volume. In the problem of interest theimpact of refraction is insignificant, since there is no trapping of the incident directwave within the elevated refractive layer. We also will not take into account the effectof the boundary z = 0 on the scattered field. The presence of the interface z = 0

148

+ n

h

− n

h

nt

)(hU

kH

iH

sθ

1D

2D

VX

h

Distance

Figure 5.14 Schematic diagram of excitation of the elevated duct via a scattering mechanism.

5.4 Excitation of the Elevated Duct due to Scattering on the Fluctuations in the Refractive Index

leads to a lobed structure of the incident field. Such a complication, while easy toaccount for, is not worthwhile in the evaluation of the effect on the order of magni-tude, which is the purpose of this investigation. Therefore, we take the artificial caseof a grounded antenna z = for further study in this section.The Green function of the scattered field we define as the superposition (5.36) of

the trapped modes of the elevated duct. The height-gain functions vnðzÞ are thendefined by Eq. (5.38) within the scattering volume.

Another important assumption is that the major contribution to scattering comesfrom the inhomogeneities of de in the vicinity of the upper boundary of the elevatedrefractive layer, i.e., with Zi £ z£Zk . This assumption is founded on observations ofthe sharp increase in intensity of the fluctuation of de in the vicinity of the upperboundary of the elevated layer [17], which is also associated with the upper boundaryof the marine boundary layer. As a result, scattering into the trapped mode withnumber n is chiefly produced by the inhomogeneities located in the vicinity of theupper turning point z

þn of the wave associated with the nth mode, Figure 5.14:

zþn ¼ m

kl33

Ui � tn � l33Hi

� . (5.85)

In fact, the upper limit of integration gþover the height of the scattering volume

for the nth mode is effectively limited by the height of the turning point zþn and,

from below, by the characteristic scale Kz of the oscillations in the height-gain func-tion Kz~m=kl3 . Let us demand the following inequalities to be true

Lz << a=mtn¼ Kx , Lk << XV ¼ D2 � D1 (5.86)

where Lz is the characteristic external scale of the fluctuations in de. The parameterKx in Eq. (5.89) is a characteristic scale of diffraction of the mode field due to thecurvature of the earth, in the 10 GHz range Kx~10 km; XV is the length of the scat-tering volume, Figure 5.14, normally about several tens of km. Under condition

hr << hs (5.87)

where hs ¼ k=Lz is the scattering angle in the vertical plane and hr ¼ffiffiffiffiffiffiffiffiffiffiffiffiZi=2a

pis the

characteristic angle of refraction. The wave front of the incident wave has to beturned to an angle hr in order for that wave to be trapped in the elevated duct. Theinequality (5.87) means that the angular width of the scattering exceeds the charac-teristic angle of refraction. In this case, the inequality

Kz

2Zi

h2rh2s

<< 1 (5.88)

holds and we can neglect the changes in the characteristic angle of refraction hrwithin the scattering volume. Let us estimate the common values involved in prob-lem formulation: the height of the elevated layer Zi is normally about Zi ~102 – 103 m; the Fresnel zone size

ffiffiffiffiffiffikD

p@ k

1=22aZið Þ1=2 is about 100 m in the

10 GHz range; the external size Lkof the inhomogeneities de in the horizontal plane

149


is about Lk » 5 Q 102–103 m [16] at heights of the order of Zi; the vertical scale Lz ismuch smaller and is of the order of tens of meters because of the strong anisotropypresent in vicinity the of the upper border of the atmospheric boundary layer [16].We can also assume that

Kz >> Lz (5.89)

which is most certainly satisfied for lower-order trapped modes, and then the inten-sity of the single-scattered field can be obtained in the form

J ¼ Wj j2D E

¼ 0:07k�2=3

C2e a

�5=3 2mð Þ11=3 g23-nv

� ·

PNn¼1

vn zð Þe�cn nv n22;n

�n21;n

� hþn-nþn

� ��8=3:

(5.90)

Here hþn ¼ kzþn =m, nþn ¼ ðn1;n þ n2;nÞ=2, g23 ¼ l2l3

� �6=ðl32 þ l

33Þ

2, Ce is a struc-

ture constant of the fluctuations of de in the vicinity of the upper border of the mar-ine boundary layer z~Zi. The parameters n1;n ; n2;n determine the non-dimensionalborders of the effective scattering volume. To define them we use the geometry ofthe problem. Let us assume that the half-width of the antenna pattern in the verticalplane (E-plane) (x, z) is equal to he . The ray which departs from the source at anangle he can be related to the stationary value of the non-dimensional energyt ¼ m2

E=k2(see Section 2.2), equal to t1 ¼ �m2

h2e (analog of the formula (2.52)).

For the ray tangential to the earth’s surface, as follows from Section 2.2, the station-ary value is t = t2 = 0.The ray with stationary value t1 reaches the upper h

ðþÞn and lower h

ð1Þn heights of

the turning level for mode of number n and the respective distances

nð�Þ1;n ¼ 1

l32

ffiffiffiffiffiffiffiffiffiffiffiffiffiffitn � t1

p, (5.91)

nðþÞ1;n ¼ 1

l32

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUi � t1

pþ 1

l33

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUi � t1

p� ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tn � t1p� �

,

and the heights hðþÞn , h

ð�Þn are given by

hðþÞn ¼ 1

l33

Ui � tn þ l33Hi

� , (5.92)

hð�Þn ¼ 1

l32

tn .

The near boundary n1;n of the scattering volume in Eq. (5.90) can then be definedas n1;n ¼ ðnð�Þ

1;n þ nðþÞ1;n Þ=2. The far boundary n2;n can be defined in similar way using

the Eq. (5.91) with t1 replaced by t2 = 0.Curve (5) in Figure 5.13 shows the estimate of the scattered field in the elevated

duct using Eq. (5.90) with the value of structure constant C2e ¼ 10

�15cm

�2=3. As

150

References

observed from the figure, the strength of the estimated scattered field is 25 dB lessthan the level of the measured signal. This might be explained by underestimationof the intensity of the fluctuations in refractive index in the vicinity of tZk, whichpotentially can reach a value of two orders of magnitude higher than the valueC2e ¼ 10

�15cm

�2=3used here. Such values of C

2e have been reported in Ref. [17]. Of

course, any attempt to match the theory (5.90) with the measured data [10] would bespeculative, since information about fluctuations in the refractive index was notreported in Ref.[10]. The main purpose of the above comparison is to give a some-what different explanation of the mechanism of the propagation that can hardly beexplained by the deterministic model of a stratified troposphere.

151

References

1 Wait, J.R. Electromagnetic Waves in StratifiedMedia, Oxford, Pergamon Press, Oxford,1962, 372 pp.

2 Bremmer, H. Terrestrial Radio Waves, Elsevier,New York, 1949, 343 pp.

3 Dresp, M.R., Rather A.S. Tropospheric ductpropagation beyond the horizon, Proc. URSICommission, Sect. F. Open Symp., La Baule,France, 1977, Pt.1, pp. 31–36.

4 Baz, A.I., Zeldovich, Y.B., Perelomov, A.M.Scattering, Reactions and Decays in. Nonrelati-vistic Quantum Mechanics, Academic Press,New York, 1980.

5 Ott, R.H. Roots of the modal equation for emwave propagation in a tropospheric duct,J. Math. Phys., 1980, 21 (5), 1256–1266.

6 Migliora, C.G. Felsen, L.B., Cho, S.H. HighFrequency propagation in elevated tropo-spheric duct, IEEE Trans. Antennas Propaga-tion, 1982, 30, 1107–1120.

7 Baumgartner, Jr., G.B., Hitney, H.V., PappertR.A. Duct propagation modelling for the inte-grated refractive effects prediction system(IREPS)., Proc. IEE, 1983,130, Pt.F, 630–642.

8 Felsen, L.B. Hybrid ray-mode fields in inho-mogeneous waveguides and ducts, J. Acoust.Soc. Am., 1981, 69 (2), 352–361.

9 Marcus, S.V. A model to calculate em fields intropospheric duct environments at the fre-quencies through SHF, Radio Sci., 1982, 17,895–901.

10 Pappert, R.A., Goodhart, C.L. Case study ofbeyond-the-horizon propagation in tropo-spheric duct environments, Radio Sci., 1977,12 (1), 75–81.

11 Kukushkin, A.V., Sinitsin, V.G. Rays andmodes in non-uniform troposphere, RadioSci., 1983, 18 (4), 573–581.

12 Guinard, N.W., Ransone, J., Randall, D. et al.Propagation through an elevated duct. Trade-wind 3, IEEE Trans. Antennas Propagation,1964, 12 (4), 479–453.

13 Hitney, H.V., Pappert, R.A., Hattan, C.P.Evaporation duct influences on beyond-the-horizon high altitude signals, Radio Sci.,1978, 13 (4), 669–675.

14 Chang, H.T. The effect of tropospheric layerstructures on long-range VHF radio propaga-tion, IEEE Trans. Antennas Propagation, 1971,19 (6), 751–756.

15 Booker, H., Gordon, W. A theory of radioscattering in the troposphere, 1950, Proc.IRE, 1950, 38 (4), 401–412.

16 Gavrilov, A.S., Ponomareva, S.M. TurbulenceStructure in the Ground Level Layer of the Atmo-sphere, Collected Data, Meteorology Series,1984, No.1, Research Institute for Meteorolo-gical Information, Obninsk, (in Russian).

17 Deardorff, J.W, Willis, G.E. Further resultsfrom a laboratory model of the convectiveboundary layer, Bound. Layer Met., 1985, 35,205–236.

153

As discussed in Chapter 1, the analytical study of wave propagation through a turbu-lent troposphere is complicated due to the high dynamic range of the turbulent fluc-tuations of the refractive index as well as the presence of the boundary surface.In Chapter 4 we studied the impact of the random component of the dielectric

permittivity of the troposphere on UHF propagation in an evaporation duct. It wasshown that wave scattering on random inhomogeneities of the refractive index leadsto a non-coherent redistribution of the energy between the waveguide modes, theloss of coherence results in additional attenuation of the trapped modes. Using thelanguage of quantum mechanics, we have found the perturbations to the eigenva-lues of the discrete spectrum of the localised states. The results obtained in Chapter4 are valid at distances from the source, on the one hand, long enough to filter allleaked modes and, on the other hand, not too long so that the scattering in theupper layers of the troposphere can be neglected in comparison with the field car-ried by the trapped modes.In the absence of tropospheric ducts it is sometimes convenient to solve the trans-

port equations for the coherence function in the continuum spectrum. In Ref. [1]such a solution has been obtained for multiple wave scattering in a random medium(uniform on average) over the flat boundary surface. The results of Ref. [1] demon-strated that anisotropic inhomogeneities of the refractive index may have a signifi-cant impact on the signal level. The multiple scattering in the case of strong aniso-tropy was studied in Ref. [2], where the following was shown:Compared with the case of isotropic fluctuations of de ~rrð Þ, when the attenuation

of the coherent component of the received field is defined by the square of the phasefluctuations, multiple scattering on anisotropic fluctuations with a factor of anisotro-py a ¼ L?=Lk << 1 results in lesser attenuation of the coherent component by a fac-tor of kL?að Þ with the same value of Lk and the scale of the frequency correlationincreases kL?að Þ�2 times. Similar results were obtained in Ref. [2], where theauthors derived integro-differential equations for the field moments which accountfor diffraction on the random inhomogeneties and variations in the trength of scat-tering with small scattering angles. The important result of Ref. [2] is that the Mar-kov approximation is not applicable with extremely large parameters of anisotropy.In such cases the two-scale model [3] may provide a sufficient tool for analysis.

6

Scattering Mechanism of Over-horizon UHF Propagation

6 Scattering Mechanism of Over-horizon UHF Propagation

In this chapter we suggest a technique for calculating the coherent signal compo-nent, with allowance for both diffraction by the earth and wave scattering on ran-dom inhomogeneities of the refractive index. Theoretically, the coherent componentis defined as the complex amplitude averaged over a statistical ensemble of realisa-tion. If the ergodicity and locally frozen hypotheses hold, the ensemble averaging isequivalent to such over an infinitely long interval. For practical purposes, however,the component coherent over a finite interval of time is of interest. According todata provided in Refs. [4–7], in about 60% of experiments performed on trans-hori-zon links up to 300 km in length, the received amplitude distribution is differentfrom Rayleigh law. This difference is ascribed to the effect of the coherent compo-nent. Therefore, it seems necessary to analyse all the factors influencing the coher-ent component, its range and wavelength dependences, etc.The approach can be formulated in the following way. For the electromagnetic

field component, coherent over time T, we derive, using the Markov approximation,a parabolic-type equation allowing for the additional decay of the coherent signaldue to scattering on small-scale fluctuations of the refractive index. Further, we ana-lyse the possibility of replacing in the equation the average (over time T ) dielectricconstant, which is a “slowly varying” random function of all three variables, with arandom function depending on a single coordinate, i.e. height over the interface.Through this procedure, the problem of wave propagation through a three-dimen-sional random medium is reduced to that for a random stratified medium. The fieldcomponent averaged over time T can be represented in this approach as a normalwave (i.e. modal) series with random propagation constants and height gain func-tions for each mode. These are determined with the aid of the perturbation tech-nique formulated in Chapter 3, in which the unperturbed refractive index profile isthat averaged over the ensemble of the realizations, and the random stratification due tolarge-scale anisotropic inhomogeneities is considered as a perturbation. The approachpermits one to obtain closed-form expressions for statistical moments of any order andanalyse the correlation between the signal levels and the turbulent troposphere.It seems noteworthy that at decimetre wavelength the attenuation rate of the

coherent component (T £ 1 min ) is not normally very high, hence at links about200 km long the coherent intensity practically coincides with the whole decimetresignal. The basic observation is that the approach suggested in this chapter is bettersuited to wave propagation effects at frequencies below 1 GHz, where the effect ofthe ducting is also week.

6.1Basic Equations

Consider a vertically polarized field whose attenuation function (2.16) is governedby the parabolic equation

2j@W@x

þ D?W þ k2eM x; y; zð Þ � 1½ W ¼ 0 (6.1)

where eM x; y; zð Þ ¼ e x; y; zð Þ þ 2 z=a.

154

6.1 Basic Equations

As is known, scattering on the fluctuation in the dielectric permittivity bringsabout additionnal attenuation of the coherent component of the field which can bedescribed by the factor e

�cx. In the Markov process approximation, the decrement of

attenuation c is given by [8]

c ¼ pk2

4

Rd2~kkUe 0;~kk?ð Þ , with~kk ¼ kx ; ky

� �.

Substitution of a Kolmogorov turbulence spectrum in the above equation wouldresult in a divergence of the integral at small ~kk. The reason for this is that theensemble average usually implied in theoretical calculations allows for arbitrarilylarge phase distortions due to very large inhomogeneities or over an infinitely longtime interval. In practice, however, the coherence over a finite time interval and thedependence of the coherent amplitude on the time of average is of interest.Let us average Eq. (6.1) over a finite time T, denoting the average values as :::h i

T,

and make use of the locally “frozen”, stationary turbulence hypothesis [9]. Thishypothesis implies that variations of the random field e ~rr; tð Þ with time result solelyfrom the motion of the turbulent flow at a velocity ~tt which can be a random valueitself. Introducing deT ¼ eM ð~rrÞ � eM ð~rrÞh i

Tand transforming Eq. (6.1) to the inte-

gral equation form we obtain

W ~rrð Þh iT¼ W 0; y; zð Þ exp j

k2

Rx0

dx0deT x

0; y; z

� � !* +T

� 12jk

Rx0

dx0exp j

k2

Rxx0dx

00deT x

00; y; z

� �0@

1A D? þ k

2eM ~rrð Þh i

T�1

h ih iW ~rrð Þ

* +T

:

(6.2)

Instead of averaging the functionals depending on deT over time T, one can per-form averaging over the ensemble of e as de does not contain greater time-scalesthan T. On the other hand, the term eM ~rrð Þh i

Tmay be considered invariant over

times t £ T. Assuming statistical independence of the fluctuations in deT and ~tt, wecan perform averaging over ~tt independently of deT . Suppose we wish to first per-form the averaging over deT . Introduce definition WT ” W ~rrð Þh i

Tand then, as has

been shown in Ref. [8], the termWT will obey the equation

2 j@WT

@xþ D?WT þ k

2eM x; y; zð Þh i

T�1

� �� 2jkcT

h iWT ¼ 0 (6.3)

provided that the wave propagation can be regarded as a Markovian process, i.e. theconditions below are met

Lk << kL2z , r

2e k2L2k << 1. (6.4)

Here, Lk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2x þ L

2y

q, Lz are, respectively, the horizontal and the vertical scales of

the inhomogeneities and r2e is the root mean square magnitude of fluctuations in e.

155


The attenuation rate cT averaged over the fluctuations in Eq. (6.3) is given by therelation

c » pk2

4

Rkyj j>kyT

dky

Rk2z>k

2zT

dkzUe 0; ky ; kz

� �(6.5)

and

kyT ¼ tyT� ��1

; kzT ¼RT0

T � sð ÞBtzðsÞds !�1=2

; (6.6)

ty can be estimated as a mean value of the horizontal transfer velocity

tk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2x þ t

2y

q, while the vertical component of the velocity ~tt can be regarded as

solely random with the correlation function Btz . We can assume, for further esti-mates, that kzT » rvzTð Þ�1 . The conditions (6.4) were obtained in Ref. [8] for a spa-tially uniform field of de. The inequalities (6.4) impose some limitations to an aver-aging time T, since under “frozen” turbulence conditions Lk ¼ tkT; Lz ¼ rtzT ,therefore instead of Eq. (6.4) we obtain

k2r2vzTtk

<< 1; k2C2e t8=3k T

8=3<< 1 (6.7)

and

r2e ¼ 1=2 C

2e L2=3k

for “locally uniform” turbulence.The function eM ð~rrÞh i

Tinvolved in Eq. (6.3) is a much slower function of x and y

than eM ~rrð Þ, at least with sufficiently long averaging time. It should be noted thatvertical variations of the averaged dielectric permittivity are much more rapid thanthe horizontal-plane variations [9,10].Establish at which times of averaging the x and y dependence of eM ð~rrÞh i

Tcan be

totally neglected in Eq. (6.3). In other words, for which T the term eM ð~rrÞh iTcan be

replaced by a vertical profile eTiðzÞ related to some fixed (generally, arbitrary) valuesx = xi and y = yi, i.e.

eTiðzÞ ¼ e xi ; yi ; zð Þ� �

T. (6.8)

The set of functions eTiðzÞ can be regarded as an ensemble of realisations ofsome random function eT zð Þ. Averaging over that ensemble will be denoted herein-after by a horizontal bar above the character, e.g. �eeT zð Þ. The substitution of eT zð Þinstead of eM ð~rrÞh i

Tcorresponds to the introduction of a vertically stratified random

medium whose “instantaneous” profile eT zð Þ is the same along the entire propaga-tion path. The slow variations of eM ð~rrÞh i

Tare equivalent to altered realisations of

eT zð Þ. Obviously, the transition of the vertically stratified medium can only be justi-fied if the amount of fluctuations (over time T) in the coherent component that aredue to the difference between eM ð~rrÞh i

Tand eT zð Þ is small. Introducing

156

6.1 Basic Equations

e1 x; y; zð Þ ¼ eM x; y; zð Þh iT�eT zð Þ (6.9)

we can presentWT (x, y, z) as

WT ð~rrÞ ¼ W0T ð~rrÞ exp Wð~rrÞð Þ (6.10)

where W ¼ V þ jS is a complex phase andW0T ~rrð Þ satisfies the equation

2j@W0

T

@xþ D?W

0T þ k

2eT zð Þ � 1ð Þ � 2jkcT

h iW0T ¼ 0 . (6.11)

Let W0T ~rr;~rr0ð Þ denote the propagation factor of the field from a point source at

~rr0 ¼ 0; 0; z0f g. Within the first-order approximation of Rytov’s method, the complexphase of Eq. (6.10) is [8]:

W ¼ � k2

W0T ~rr;~rr0� � Rx

0

dx0 R

d2~��

0G ~rr;~rr

0� �

e1_~rr~rr

� �W0T ~rr

0;~rr0

� �, (6.12)

where

~��0 ¼ x

0; y

0n o

,~rr0 ¼ x

0;~��

0n o

, G ~rr;~rr0

� �¼ � 1

4p x�x0� �W0T ~rr;~rr

0� �

. (6.13)

We will consider the case when the reception point~rr0is shadowed from the trans-

mitter by the terrestrial sphere. The complex phase fluctuations represented by Eq.(6.12) depend essentially on the vertical profile of e1 ~rrð Þ. If e1 ~rrð Þ contains spectralcomponents characterised by small vertical scales lz << m=k, then the mechanismof trans-horizon propagation is a resonance scattering in the higher troposphericlayers, i.e. the troposcatter described by the Booker and Gordon theory [11]. In thiscase the fluctuations in the log-amplitude V are very high.To reduce the standard of fluctuations in the log-amplitude V determined by

Eq. (6.10) it is necessary to suppress the multi-scale random variations of e1 ~rrð Þ bychoosing a sufficiently long averaging time T. The minimum vertical scale sizes ofthe e1 ~rrð Þ inhomogeneities which are related to the average time as lz ¼ rtzT shouldmeet the condition lz >> m=k. The resultant condition on the averaging time is

T >>m

krtz. (6.14)

In contrast to the “standard” troposcatter mechanism [11], the scattering of wavessatisfying the above conditions occurs in the region of the troposphere which is inthe shadow zone with respect to both receiver and transmitter. Hence, the propaga-tion factor W

0T ~rr;~rr0ð Þ and the Green function G ~rr;~rr

0� �

are most conveniently repre-sented as modal series, Chapters 2 and 3. Rough estimates for W

0T ~rr;~rr0ð Þ and

G ~rr;~rr0

� �values can be obtained by retaining just the first terms of these expansions,

i.e.

157


W0T ~rr;~rr

0� �

¼ �2 exp jp4

� �mk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipm x � x

0� �

a

vuut vðzÞvðz0 ÞN

·

exp jq x � x0

2kþ j k2

y� y0

� �2x � x

0 � cT x � x0

� �8><>:

9>=>;:

(6.15)

Estimating the integral in Eq. (6.12) we will first assume eT zð Þ to correspond tothe standard tropospheric refraction, i.e. eT zð Þ ¼ 1þ 2 z=a. Explicit expressions forthe values involved in Eq. (6.15) are

q ¼ k2=m

2� �

s1ejp=3, s1 ¼ 2:338, N ¼ �4m=k

ffiffiffiffiffis1

pe

jp=3and

v zð Þ = w1 s1ejp=3 � kz=m

� �,

where w1 is the Airy function defined in the Appendix. Performing the integrationin Eq. (6.15) and averaging over the ensemble of e1 ~rrð Þ on the assumption of statisti-cally independent fluctuations thereof, we can arrive at the following results.With

T >>

ffiffiffiffiffikx

p

vk(6.16)

the mean-square log-amplitude fluctuations are

V2

D E¼ 4 10�4p3=2C2e1 L

�7=3k x

3(6.17)

where Ce1, is the structure constant of e1 . The structure function has been assumed

to obey the “two-thirds” law, i.e. the spectrum of e1 takes the form (1.34). Thus, pro-vided the following inequalities hold,

a) 4 10�4p3=2C2e1 t�7=3k T

�7=3x3<< 1,

b) k2C2e t8=3k T

8=3<< 1, (6.18)

c) T maxmktz

;

ffiffiffiffiffikx

p

tk;tk

kr2tz

( ),

the term eM x; y; zð Þh iTin Eq. (6.3) can be replaced by eT zð Þ without introducing

considerable error in WT ~rrð Þ. Substituting C2e1 »C

2e ¼ 10

�14cm–2/3, k = 10 cm,

T = 30 s, tk ¼10 m s–1, rtz=1 m s–1, we see that the principal inequality (6.18a)

can hold at x £ 103 km.

158

6.2 Perturbation Theory: Calculation of Field Moments

6.2Perturbation Theory: Calculation of Field Moments

Thus, calculation of the coherent field amplitude averaged over time T has beenreduced to solving Eq. (6.11). If eT ðzÞ is a regular function, Eq. (6.11) can be solvedby known methods, with the solution presentable in the form (6.14). Actually, theproblem consists in determining the propagation constants q and the height-gainfunctions vðzÞ. Depending on the specific form of eT ðzÞ, this can be done eitheranalytically or numerically. However, this is a rare occasion. Generally, eT ðzÞ is arandom function whose only known parameters are the vertical stratification scale

Lz and the root mean square fluctuation strength eT ðzÞ2

D E1=2. We further assume

that averaging over the statistical ensemble is equivalent to averaging over time,

e2T

D E¼ e

2T i.e. the validity of the ergodicity theorem. An analytical solution to

Eq. (6.11) cannot be written, we should aim instead at evaluating the mean level of

WT ~rrð Þ and the root mean square fluctuations rW ¼ffiffiffiffiffiffiffiffiW2T

qthereof from knowledge

of the eT ðzÞ statistics. Averaging over the statistical ensemble we can write

eT ðzÞ ¼ eT ðzÞ þ DeðzÞ, De ¼ 0 (6.19)

assuming specifically

eT ðzÞ ¼ e0ðzÞ ¼ 1þ 2za. (6.20)

To determine the propagation constants and height gain functions, we will makeuse of the perturbation theory described in Section 3.1. Representing q and v as

q ¼ q0 þ dq , (6.21)

vðzÞ ¼ v0ðzÞexpRz0

dz0f z

0� � !

, (6.22)

where q0 and v0 are governed by the unperturbed Eq. (6.11) with eT ðzÞ ¼ e0ðzÞ. Wecan obtain the random correction term dq and fðzÞ in the form

dq ¼ � k2

N2R¥0

DeðzÞv20 zð Þdz, (6.23)

f zð Þ ¼ 1

v20ðzÞRz0

dq � k2De z

0� �h i

v20 z

0� �

dz0. (6.24)

Now we single out of WT(x, y, z; z0) the value W0, i.e. the solution to Eq. (6.11)with eT ðzÞ ¼ e0ðzÞ, to obtain

WT ðx; y; z; z0Þ ¼ W0ðx; y; z; z0Þ exp jdq2k

x þRz0

f z0

� �dz

0 þRz00

f z0

� �dz

0þ( )

: (6.25)

The factorWT (x, y, z; z0) is a random function of a “slow” time t > T. Within eachrealization of the signal, the factor WT is the coherent component of the total signal

159


received during the “short” intervals t £ T. During the “long” time-interval t >> T ,the factor WT undergoes relatively slow random variations owing to changes of rea-lisation of eT . The part ofWT fluctuating at the change of realisations DeðzÞ is givenby the exponent of Eq. (6.25).We can specify the intensity J0 of the coherent component WT (x, y, z; z0) given

e0ðzÞ in the form (6.20):

J0 ¼ W0j j2¼ pmx4as1

w1 s1ejp=3 � kz

m

& '((((((((2 w1 s1e

jp=3 � kz0m

& '((((((((2 ·

exp � mxa

s1ffiffiffi3

p� 2cT x

& ' (6.26)

where 2cT is the extra attenuation rate due to the energy transfer to the incoherentpart of the signal. Generally, the attenuation rate cT given by Eq. (6.5) undergoesrandom variations as the medium realisation changes, because of the non-unifor-mity in the small scale fluctuations of eM ~rrð Þ. In the following study, we shall restrictourself to the case where the eM ~rrð Þ fluctuations of scale sizes lk < tkTand lz < tzTare statistically uniform, i.e. cT ¼ const.As can be observed from Eqs. (6.23) to (6.25), the random valueWT should be dis-

tributed log-normally, in view of the central limiting theorem. Note that similar dis-tributions are actually observed in the experiment. For instance, according to datareported in Ref. [5], the integral distribution of the amplitude measured over 1 to 5min intervals reveals log-normal statistics.To simplify further the derivations, we will assume the receiver and transmitter

heights to be equal, i.e. z = z0, and consider the intensity JT of the coherent (overtime T) signal component averaged over the statistical ensemble, viz.

JT ¼ WTj j2D E

¼ J0exp

� x2

8k2dq � dq

�� 2D Eþ

jx2k

dq � dq�� Rz0

f z0

� �þ f

�z0

� �h idz

0* +

þ

2Rz0

dz0 Rz0

dz00

f z0

� �þ f

�z0

� �h if z

00� �

þ f�

z00

� �h iD E

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

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

(6.27)

Further calculations are straightforward but cumbersome. As a specific example,consider one of the correlators involved in Eq. (6.27):

Rz0

dz0 Rz0

dz00

f z0

� �f z

00� �D E

¼ dqð Þ2D E Rz

0

dz0

v20

z0� � Rz

0

0

dz00v20 z

00� �2

435�

2k4

N

Rz0

dz0

v20

z0� � Rz

0

0

dz1v20 z1ð Þ

Rz0

dz00

v20

z00� � Rz

00

0

dz2v20 z2ð Þ

R¥0

dz3v20 z3ð Þ De z2ð ÞDe z3ð Þh iþ

k4 Rz0

dz0

v20

z0� � Rz

0

dz00

v20

z00� � Rz

0

0

dz1Rz000

dz2v20 z1ð Þv20 z2ð Þ De z1ð ÞDe z2ð Þh i: (6.28)

160


Other terms in the exponent of Eq. (6.27) can be expressed in a similar way.After lengthy but straightforward calculations, somewhat simplified with

z << m=kffiffiffiffiffis1

p, we can arrive at

JT ¼ J0exp

� x2

8k2dq � dq

�� 2D Eþ j

xz2

6kdqð Þ2

D E� dq

�� 2D Eh iþ

jkxz4

90m2s1e

�jp

3 dq�� 2D E

� s1ejp

3 dqð Þ2D E" #

�

kx

m2s1

ffiffiffi3

p

90z4

dqj j2D E

þ z4

18dq þ dq

�� 2D E

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

9>>>>>>>=>>>>>>>;: (6.29)

The corresponding terms dq can be expressed via the Fourier transform of DeðzÞ,viz.

~ee kð Þ ¼ 12p

R¥�¥

dze�jkz

De zð Þ, (6.30)

dq ¼ �k2 R¥�¥

dk ~ee kð ÞV kð Þ (6.31)

where the function V kð Þ is given by

V kð Þ ¼ 1N

R¥0

dz v20 zð Þejkz . (6.32)

It has the meaning of the scattering coefficient from the first mode back to itselfagain, due to inhomogeneities De zð Þ of the scale-size l ¼ 2p=k.According to the inequality (6.14), the major contribution to the random compo-

nent of the vertically stratified De zð Þ is given by sufficiently large inhomogeneitieswith k ¼ kz << k=m. For this case V kð Þ can be represented by an asymptotic expan-sion

V kð Þ ¼ 1þ jmkk

s1ejp=3 þO

mkk

� �2� �. (6.33)

Defining the spectral density of the fluctuations as

U1 kð Þ ¼ffiffiffiffiffiffi2p

pr2DeLz exp � k2L2z

2

!, (6.34)

with Lz ¼ rvzT , we shall substitute Eq. (6.33) into Eq. (6.29) with an account of thespectrum in the form (6.34). Then the average intensity JT becomes

JT ¼ J0 exp3px2

4L2zm2r2Des

21 � x

k5z4r2Des1ffiffiffi3

p

180m2� 43pr

2De kzð Þ4

( ). (6.35)

The range of validity of Eq. (6.35) is dictated, according to the perturbation meth-od employed, by the demand that the second-order correction to WT (x, z; z0) besmall.

161


Analyzing Eqs. (6.27) and (6.29) we find that the corrections to the propagationconstants take the major role in the second-order corrections overall, and these canbe evaluated as

dq2ð Þ»m2

k2dqð Þ2 . (6.36)

Noting that dq ~ k2rDe we will demand that the terms containing the range (dis-

tance) squared in the exponent are small, whence

r3Dek

2x2m2<< 1. (6.37)

Substituting r2De » 10�13 , we find that the requirements of Eq. (6.37) can be met

for x £ 150 km at k = 10 cm (f = 3 GHz) and x £ 700 km at k = 30 cm (f = 1 GHz).Shown in Figure 6.1 are the range dependences of the average intensity JT as cal-

culated from Eq. (6.35) for f = 3 GHz and f = 1 GHz. The parameter valuesassumed for the calculation are z = z0 = 10 m, r

2De » 10�13 , Ls = 20 m and

a = 8500 km. Range dependences J0 are also shown for comparison. As can be seenfrom Figure 6.1 and from Eq. (6.35), the presence of random gradientsdeT =dz ~ rDe=Lz in the refractive index results in a sharp increase in the signalstrength. This can be regarded as a kind of “trapping” or localization of the radiatedfield near the earth’s surface, however, in this case it is of a random nature.Making use of Eqs. (6.25), (6.27) and (6.35), we can evaluate the square of the

standard of the intensity fluctuations, viz.

162

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 20 40 60 80

Distance , km

Re

ce

ive

d s

ign

al

str

en

gth

re

lati

ve

to

fre

e

sp

ac

e,

dB

Series1

Series2

Series3

Series4

Figure 6.1 Range dependence of the averageintensity of the received field, 10log(JT ): (1)Received field strength at frequency 3 GHz inthe absence of fluctuations in the refractivity(r

2De=0); (2) received field strength at fre-

quency 3 GHz, 10log(JT ) in the presence of

fluctuations in the refractivity r2De»10

�13; (3)

received field strength at frequency 1 GHz inthe absence of fluctuations in therefractivity(r

2De=0); (4) received field strength

at frequency 1 GHz, 10log(JT ) in the presenceof fluctuations in the refractivity r

2De»10

�13.


r2J ¼ J

2T � JT

2 ¼ JT2exp

3px2

2L2zm2r2Des

21 � x

k5z4r2Des1ffiffiffi3

p

90m2� 29pr

2De kzð Þ4

" #� 1

( ).

(6.38)

Figure 6.2 provides range dependences of the fluctuation standard r2J ðxÞand the

variation index b2J ðxÞ ¼ r

2J ðxÞ= JT

2ðxÞcalculated for the same parameter values. Thelatter magnitude b

2J characterizes the relative fluctuations in the intensity (i.e. the

depth of “slow” fading). As observed from Eq. (6.38) and from Figure 6.2, the meansquare magnitude of the intensity fluctuations is proportional to the average inten-sity and grows with the range as the first and major term in the exponent (6.38).

We believe that Eqs. (6.35) and (6.38) can provide an explanation for the increase inthe fading depth which is observed experimentally with increase in the mean levelsof the signal [5]. According to the field measurements discussed in Refs. [4, 5], thefading depth increases with the path length up to ~ 200 km, such behavior is also inagreement with the theoretical result provided by Eq. (6.38). The above theory, devel-oped in this section, introduces a two-scale model of fluctuations in the refractiveindex: small-scale fluctuations treated as a Kolmogorov turbulence and large-scalefluctuations De which are treated as a random stratification De zð Þ.Within the limits of this theory, the signal strength should demonstrate strong

correlation with the fluctuation standard r2De of the dielectric permittivity near the

earth’s/ocean’s surface. As r2De increases, the field intensity should grow, even with

small gradients of the average profile of the refractive index, i.e. without ducting orenhanced refraction. This increase in the signal strength is due to multiple scatter-

163

-60

-40

-20

0

20

40

60

80

0 20 40 60 80 100

Distance, km

No

rma

lis

ed

flu

ctu

ati

on

s,

dB

Series1

Series2

Series3

Series4

Figure 6.2 Range dependences of the fluctua-tion standard and the variation index in thepresence of fluctuations in the refractivity withr2De»10

�13: (1) Fluctuation standard r

2J ðxÞ at

frequency 3 GHz; (2) fluctuation standardr2J ðxÞ at frequency 1 GHz; (3) variation index

b2J ðxÞat frequency 3 GHz; (4) variation index

b2J ðxÞat frequency 1 GHz.


ing of the lowest propagation mode on the “layered” inhomogeneities whose verticalscale sizes are commensurate with Kz ¼ m=k, the vertical scale size of the modeoscillations, as shown in Figure 6.3. A one-dimensional analogy (along the z-coordi-nate) to this phenomenon is the stochastic parametric resonance considered in Ref.[12].To analyse cases of ducted radio wave propagation in an atmospheric duct over

the sea’s surface, the common approach is to attempt to compare the measured datawith theoretical predictions for eM ðzÞprofiles recovered from meteorological mea-surements. To a certain degree, controlled by the validity of the hydrodynamic theoryof an evaporation duct [13], such profiles correspond to the ensemble averageddependences eT ðzÞ, with the random stratification DeðzÞ apparently disregarded.Yet, as can be seen from the above theory, the random component of eðzÞ can playthe dominant part in cases where the average profile does not reveal a strong near-surface inversion, like an evaporation duct or for frequencies significantly below10 GHz when an evaporation duct, even if present, is normally insufficient for theducting mechanism at these frequencies.

6.3Scattering of a Diffracted Field on the Turbulent Fluctuations in the Refractive Index

Long-range tropospheric propagation due to re-emission of the energy of electro-magnetic waves by inhomogeneities of the refractive index has been known sincethe 1940s [4]. The simple mechanism of the single scattering was first developed byBooker and Gordon [12] and then updated taking into account the Kolmogorov the-ory of the turbulent spectrum of fluctuations in the refractive index [9]. The theoryof single scattering was proposed to explain the phenomenon of the long-rangepropagation in the absence of super-refractive anomalies in the refractive index pro-files. It takes into account scattering by inhomogeneities located in the regionformed by the intersection of the directional diagrams of the receiving and transmit-ting antennas, as shown in Figure 6.4.According to Ref. [12], the mean intensity Js of the scattered field Is normalised on

the intensity in a free space IFS is expressed as follows:

Js ¼I0IFS

¼ 16 r0V

R2(6.39)

164

m1ϑ

max =∆

=

Figure 6.3 Schematic representation of multiple scattering of a diffracted field.

6.3 Scattering of a Diffracted Field on the Turbulent Fluctuations in the Refractive Index

where V is the effective scattering volume and R is the distance between the receiverand transmitter, and r0 is the effective scattering cross-section.The effective scattering cross-section r0 from a unit volume to a unit solid angle

is given by

r0 ¼pk2

2Ue ~qqð Þ, (6.40)

where~qq is the scattering vector and Ue ~qqð Þ is the spectral density of the fluctuationsin the dielectric permittivity de ~rrð Þ. The scattering vector ~qq has the componentsqx = qy = 0 and qz ¼ 2k sin J=2ð Þ, where x and y are coordinates along the surface ofthe earth and z is along the normal to it, k is the wavenumber and J is the scatteringangle, Figure 6.4.For the inertial interval of the turbulence spectrum s0 is determined by the rela-

tion

r0 ¼ 0:052k1=3

C2e 2 sin

J2

& '�11=3, (6.41)

where Ce is a structure constant, Section 2.The above Booker–Gordon theory provides rather good estimates of the signal

strength of the scattered field as well as describing the range dependence of thereceiving signal strength. However, there are several factors which, in the majorityof observations, are not in agreement with the theory of single scattering, for exam-ple, the dependence on wavelength, elevation angle, and the cumulative distributionof the scattered field [5, 7].In this section we attempt to estimate the intensity of the signal over the horizon

due to scattering of the waves in the volume located in the geometric shadow relativeto both the transmitter and receiver, as shown in Figure 6.5.

165

ϑ

z

x0xT x

R

fsVV =

Figure 6.4 Geometry of single scattering in the troposphere.


Consider the case of normal refraction, in which the modified dielectric constantof the troposphere eM x; y; zð Þ ¼ e0ðzÞ þ de x; y; zð Þ, e0 ¼ 1þ 2 z=a. Here de x; y; zð Þis a random component of the dielectric permittivity, deh i ¼ 0.We shall define the mean intensity Js of the scattered field normalised at an inten-

sity in a free space via the attenuation factorW:

Js ¼Ish iIFS

¼ Wj j2D E

. (6.42)

The attenuation factor can be represented in the formW = W0 + W1, whereW0 isthe solution of the equation

2jk@W0

@xþ @

2W0

@y2þ @

2W0

@z2þ k

2e0 zð Þ � 1½ W0 ¼ 0. (6.43)

andW1 is defined in the Born approximation by the expression

W1¼�k2Rxx0

dx0 R¥�¥

dy0R¥0

dz0G x�x0;y;z;y0 ;z0

� �·

de x0;y0 ;z0

� �W0 x0�x0 ;y

0;z0 ;y0 ;z0

� � (6.44)

where~rr and~rr0 are the vector-coordinates of the receiver and transmitter respectively,~rr ¼ x; y; zf g,~rr0 ¼ x0 ; y0 ; z0f g. Later we will set x0 = 0 and y = y0 = 0.We also restrict the integration volume for discussion of the scattering of a dif-

fracted field in the shadow region by the inequalities:

z0< min

x0�Dx0� �2

2a;

x�x0�Dx� �2

2a

( ), (6.45)

Dx0 < x0< x � Dx.

166

z

x0

zl

λϑ ~

kaax z

κϑ = ∆ ~

Figure 6.5 Geometry of the single scattering of the diffracted field.


Here Dx0 ¼ffiffiffiffiffiffiffiffiffiffi2az0

p, Dx ¼


p. We note that the Booker–Gordon theory takes

account of scattering only in the radiated region bounded by the inequality

z0>

x0�Dx0� �2

2a;

x�x0�Dx� �2

2a

( ). (6.46)

We can define the Green function Gð~rr;~rr 0 Þ in terms of the attenuation factor,

G ~rr;~rr0

� �¼ 1

4p x�x!

� �W0 ~rr;~rr0

� �, (6.47)

which is represented in shadow region in the series of normal waves, Section 2,

W0~rr;~rr 0� �

¼�2e jp=4pm x � x

0� �

a

24

351=2

mkexp j

k y� y0

� �22 x � x

0� �

8<:

9=; ·

P¥n¼1exp jqn

x � x0

2k

0 1vn zð Þvn z0ð Þ

Nn

(6.48)

where qn is a complex propagation constant

qn ¼ k2

m2sne

jp=3, sn ¼ 3

2p n � 1

4

& '2 32=3, Nn ¼ �4m

kffiffiffiffiffisn

pe

jp=3. (6.49)

The height-gain functions vn zð Þ satisfy the equation

d2vn

dz2þ k

2e0 zð Þ � 1ð Þ � qn

h ivn ¼ 0 (6.50)

and boundary conditions

vn z ¼ 0ð Þ ¼ 0, ddzarg vnðzÞ > 0 with z ! ¥. (6.51)

For the case of normal refraction vnðzÞ can be represented in terms of the Airyfunction, vnðzÞ ¼ w1 tn � lzð Þ, where l ¼ k=m and tn ¼ qn=l

2.

We shall substitute Eqs. (6.47) and (6.48) into Eq. (6.44) and take into account inthe double summation over the modes only diagonal terms which do not oscillatewith distance. Integrating in Eq. (6.44) over dy and assuming statistical uniformityof the fluctuations in the refractive index and their isotropy in the x-, y-planes, wecan obtain the equation

W1

(( ((2D En¼ 2 pkð Þ2m

axF z;z0� �

exp �mxasn

ffiffiffi3

p& '·

Rx1 =20

dx0 x0�Dx0� �2þ x�Dx�x0� �2h i R¥

�¥dky

R¥�¥

dkzUe 0;ky ;kz

� �Vn kz ;x

0� �(( (( (6.52)

167


for the intensity of the scattered field in the nth mode W1j j2D E

n, where F z; z0ð Þ =

m2

k2 Nnj j2vn zð Þ(( ((2 vn z0ð Þ

(( ((2 is a factor describing the dependence of the mean intensityof the scattered field on altitude and x1 ¼ x � Dx � Dx0 .

The function Vnðkz ; x0 Þ in Eq. (6.52) has the meaning of a coefficient of re-scatter-

ing from the nth mode into the nth mode by inhomogeneities with the scalelz ¼ 2p=kz and is defined by

Vn kz ; xð Þ ¼ 1Nn

Rx2=2a

0

dzv2nðzÞ exp jkzzð Þ. (6.53)

With kz < k=msn and x >> a=mð Þ ffiffiffiffiffiffiffiffiffiffisn=2

pthe major contribution to integral

(6.53) comes from the altitude’s interval 0 < z < msn=2k. The upper limit in Eq.(6.53) in this case can be replaced by infinity by means of introducing a smooth lim-iting function which compensates the growth of v

2nðzÞ at z ! ¥. Let us introduce

such a function in the form of exponent exp �bzð Þ. After transition to non-dimen-sional variables a ¼ b m=k, f ¼ kz=m and q ¼ kzm=k we can deform the contour ofintegration over f in Vn kz ; xð Þ into a ray with arg f ¼ p=3. With a„ 0 we canneglect the contribution from integration over the arc of infinite radius, and forVn kz ; xð Þ we then obtain

Vn kz ; xð Þ » Vn kzð Þ ¼ 1ffiffiffiffiffisn

pR¥0

dft2f� snð Þe�q

�f, (6.54)

where q� ¼ a� jqð Þe jp=3

, t f� snð Þ is the Airy function. We can further put para-meter a = 0 and expand the exponent into a series over powers of q

�f

Vn kzð Þ ¼ 1ffiffiffiffiffisn

p e�q

�sn P¥

m¼1Pm

q�ð Þm

m!�1ð Þm , (6.55)

where

Pm ¼R¥

�sn

dx xmt2

xð Þ. (6.56)

The recurrent formulas for integrals (6.56) are provided in the Appendix.With qj jsn << 1, we can retain only the first term in series (6.55). Taking into

account that t �snð Þ ¼ 0, we obtain the asymptotic expression

Vn kzð Þ @ 1þ j3

kzmsnk

� �exp j

p3

& 'þO

kzmsnk

� �2� �which determines the contribution of non-resonant scattering by inhomogeneitieshaving vertical scales lz greater than the maximum characteristic scale msn=k ofoscillations in v

2nðzÞ.

When ks2n=m << kz << 2 kx=a, the stationary point of the integrand in Eq. (6.53):

zst ¼mk

sn2þ kzm

2k

� �2� �makes the main contribution to Vn kz ; xð Þ. In this case the

resonant scattering of the nth mode occurs at the altitude zst at which the scale of

168


oscillations of v2nðzÞ is equal to the vertical scale of the inhomogeneities, and we

have for Vn kz ; xð Þ the expression

Vn kz ; xð Þ @ 12

kkzmsn

& '1=2e

jp=12H zsð Þ exp �j

kzmtnk

� j112

kzmk

� �30 1, (6.57)

H zsð Þ ¼Rzs

�¥exp jn

2� �

dn, zs ¼1

4ffiffiffiffiffikz

pmk

� �3=2 2kxa

& '2�k

2z

2 3. (6.58)

When kz > 2kx=a, the upper limit of integration makes the main contribution tothe integral Vn kz ; xð Þ:

Vn kz ; xð Þ @ 12

a2mkxsn

& '1=2e

jp=12exp �j

mxtna

� j23

mxa

& '30 1. (6.59)

This equation corresponds to non-resonant scattering by inhomogeneities withscales lz < pa=kx.Let us substitute into Eq. (6.52) the asymptotes (5.57) and specify the spectrum of

fluctuations in the refractive index Ueð~kkÞ by Eq. (1.35):

Ue ~kkð Þ ¼0:063r2eLzL

2k

1þk2kL2kþk2zL

2z

� �11=6 , (6.60)

where kk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2x þ k

2y

q. This spectrum takes into account the finite external scales

along the vertical Lzð Þ axis and horizontal plane ðLk Þ and the anisotropy of the inho-mogeneities a ¼ Lz=Lk „ 1.Taking then account of the fact that for x >> a=msn (for f~10 GHz,

a=msn ~ 10 km) small-scale inhomogeneties with kz >> k=msn make the main con-tribution to the scattering, and bearing in mind that msn >> 1, we obtain for theintensity of the scattered field Js the following expression

Jsd » W1j j2D E

¼ 0:055p3C2e a

� 5=3

12s31

a2

� �1=3k�1=3

x1

a

� ��8=3·

Fðz;z0Þ exp �mn

as1

ffiffiffi3

p� � (6.61)

where n ¼ffiffiffiffiffi2a

p ffiffiffiz

pþ ffiffiffiffiffi

z0p� �

. We have taken into account the contribution of the firstmode with n = 1, the scattering of the other modes can be estimated similarly, how-ever, the contribution of the modes with higher indices n decays as n–2.The contribution of the large-scale inhomogeneities to the intensity of the scat-

tered field is exponentially small which can be explained as follows: The wave dif-fracted over the sphere’s surface creates the secondary waves sliding along the tan-gent to that surface. These waves then scatter on the imhomogeneities of the refrac-tive index at the scattering angle J~ k=lz . The scattered wave touches the sphericalsurface at the distance from the point of initial detachment Dx ~ aJ ¼ a kz=k, andthereafter diffracts along the earth’s surface arriving at the receiving point. Alongthe path x1 derived along the geodesic curve, the wave attenuation is determined byexp �mx1=a s1

ffiffiffi3

p� �. For the scattered wave the distance along the geodesic x1 is

169


given by x1 = x – Dx, since at the interval Dx the wave propagates under free-spaceconditions, as shown in Figure 6.5. As observed, for kz < k=msn, Dx < am=sn ,hence the larger the inhomogeneities participating in wave scattering, the shorterthe “free-space” interval and, therefore, the larger the attenuation along the geodesicinterval. When x >> am=sn , the contribution of the large-scale inhomogeneties tosingle scattering, with kz < k=msn , can be neglected. With greater kz the scatteringtakes place in the higher layers of the troposphere at larger angles J, thus increasingthe “free-space” interval and decreasing the attenuation of the scattered wave.In contrast to Eqs. (6.39) to (6.41), the dependence of the signal strength on the

altitude z enters not only through the scattering angle J ¼ x1=a but also throughthe height-gain function vn zð Þ. In the range of altitudes z << x

2=2a and distances

x a=mð Þffiffiffiffiffiffiffiffiffiffisn=2

p, the scattering angle J can be assumed to be independent of altitude,

i.e. J ~ x=a; then the dependence of Jsd on z is determined by the factorexp � m=að Þ


psn

ffiffiffi3

p5 6 vn

(( ((2 . Substituting the asymptotes of vnðzÞ into Eq. (6.60)and assuming z0 >> m sn=2k, we obtain

Jsd ~ z2k2=3, with z <<

mkffiffiffiffiffisn

p , (6.62)

Jsd ~ z�1=2

k , with z >>m sn2k,

and with z ¼ msn=2k function Jsd(z, k) has a maximum in which the value of vn zð Þ(( ((

is of the order of one.We note that in the majority of experiments the wavelength dependence of the

scattered field is proportional to the wavelength, Jsd kð Þ ~ km, where 0.7 < m < 1.4 [5].

Let us consider the case of elevated antennas when z; z0 > ms1=2k and comparethe intensity of the scattered diffracted field Jsd with the intensity Js calculated fromformula (6.39) with the spectrum defined by expression (6.60). Extracting from Eq.(6.61) the effective scattering cross-section rd with anisotropy factor accounted for,i.e. rd ¼ 0:052C2e k1=3a�8=3

2 sinJ=2ð Þ�11=3 , we can represent the intensity of thescattered diffraction field in a form similar to Eq. (6.39)

Jsd ¼ 16 rdVd

x2, (6.63)

where Vd is the effective scattering volume, which is defined by the equation

Vd ¼ 0:0087 p3x3a

16ms41kffiffiffiffiffiffiffiffiffiz z0

p . (6.64)

Without even numerical calculation we may conclude that the contribution of thediffracted field will be an order of magnitude less than the contribution of the scat-tering in a “free-space” volume, defined by formula (6.39) according to the Booker–Gordon theory. The value of the theory developed in this section is that it providesthe correct frequency and height dependence of the scattered field, compared withthe “free-space” single scattering theory.

170


The correct estimation of the scattered field should require calculation of the addi-tional terms in the scattered field not accounted for here. We may notice that givenpractical antennas we always have two terms in the incident field : direct wave +reflected from the ground wave( comprising the line-of-sight mechanism) and dif-fracted field. For simplicity, the reflected wave is omitted in the Booker–Gordon sin-gle-scattering theory, and the contribution from the transition (between line-of-sightand shadow region) is also omitted. The intensity of the total scattered field shouldcontain four terms:

Jtotal ¼Itotal

IFS¼ 16r0

Vfs

x2þ 16rd

Vd

x2þ 16rd;0

Vd; fs

x216r0;d

Vfs;d

x2(6.65)

where the first term represents the contribution from the scattering in a free-spacevolume Vfs, as given by Eq. (6.39) ðVfs ”VÞ, the second term is scattering of the dif-fracted field given by Eq. (6.64) and two last terms represent scattering of the line-of-sight waves into the diffracted field in the volume Vfs,d. (Figure 6.6) and the dif-fracted field into the free-space waves in the volume Vd,fs. A combination of thesefour terms may provide observable levels of the scattered field as well as frequencyand height dependence in agreement with experiment.

171

z

x0

ϑd

V

dfsV

, fsdV

,

Figure 6.6 Schematic diagram of the scattering volumes fordiffracted-to-diffracted field (Vd), diffracted to free-space field(Vd,fs) and free-space to diffracted field (Vfs,d).

6 Scattering Mechanism of Over-horizon UHF Propagation172

References

1 Ivanov, V., Kinber, B., Korzenevitch, I. andStepanov, B. Impact of the earth surface onlong-range tropospheric propagation, Radio-tech. Electron., 1980, 25 (10), 2033–2042.

2 Saitchev, A., Slavinsky, M. Equations formoment-functions of waves propagating inrandom media with anisotropic inhomogene-ities, Radiophys. Quant. Electron., 1985, 28 (1),75–83.

3 Kukushkin, A., Fuks, I. and Freilikher, V.Impact from random stratification on a coher-ent component of the field over horizon,Radiophys. Quant. Electron., 1983, 28 (8),1064–1072.

4 Long-range UHF Propagation in the Tropo-sphere, Eds. Vvedensky, B., Kolosov, M., Kali-nin, A. and Shifrin, J., Soviet Radio, Moscow,1965, 416 pp.

5 Shur, A. Signal Characteristics of TroposphericRadiolinks, Swyaz, Moscow, 1972, 105 pp.

6 Shifrin, J. Problems of Statistical Antenna The-ory, Soviet Radio, Moscow, 1970, 384 pp.

7 Kalinin, A., Troitzky, V. and Shur, A. Thestudy of long-range UHF tropospheric propa-gation, in Radiowave Propagation , Eds. Kolo-sov, M., Armand, N., Katzelenbaum, B. andSokolov, A., Nauka, Moscow, 1975, pp. 127–153.

8 Rytov, S.M., Kravtsov, Y.A. and Tatarskii, V.I.Introduction to Statistical Radiophysics: Part 2,Random Fields, Nauka, Moscow, 1978, 464 pp.


10 Feinberg, E. Radiowave Propagation over theEarth’s Surface, Nauka, Moscow, 1961, 547 pp.

11 Booker, H., Gordon, W. A theory of radio scat-tering in the troposphere, 1950, Proc. IRE,1950, 38 (4), 401–412.

12 Klatskin, V.I. Stochastic Equations and Wavesin Random Media, Nauka, Moscow, 1980,336 pp.

13 Rotheram, S. Radiowave propagation in theevaporation duct, 1974, The Marconi Rev.,1974, 37 (192), 18–40.

173

A.1Definitions

Studying the behavior of light in the neighbourhood of caustics in 1838, Sir G. B.Airy introduced the famous integral

vðtÞ ¼ 1ffiffiffip

pR¥0

cosx3

3þ xt

!dx (A.1)

which represents one of the solutions to the differential equation

w00 ðtÞ � twðtÞ ¼ 0, (A.2)

namely, one which decreases at positive infinity more rapidly than any finite powerof t. A second independent solution to Eq. (A.2) can be named u(t) and will bedefined later.Following Fock [1], the solution to Eq. (A.2) can also be defined via the integral:

wðtÞ ¼ 1ffiffiffip

pRC1

dz exp tz � z3

3

!(A.3)

where contour C1 goes by the ray with arg z ¼ �2p=3 from infinity to 0 and then toinfinity along the real and positive z-axis, Figure A.1.Consider the behavior of the function w(t) with real argument t. First, let us find

the sector of convergence of the integral (A.3) in order to perform useful deforma-tions on the integration contour. With zj j ! ¥ we may neglect the first term in theintegrand (A.3) then the convergence sector is determined by the condition

Re z3> 0. (A.4)

We may define z ¼ zj je ja(�p < a < p), then the condition (A.4) results in

cos 3að Þ > 0, (A.5)

Appendix: Airy Functions


which, in turn, determines three sectors of convergence of the integral (A.3) in thez-plane:

1) �p6< a <

p6,

2)p

2< a <

5p6, (A.6)

3) � 5p6

< a < �p2.

The contour C1 in Eq. (A.3) passes through the middle of the first and the thirdsector and therefore coincides with one of the lines of steepest descent of the inte-grand’s phase.We can now deform the contour C1 within the sector of convergence without

changing the value of the integral. Let us turn the lower part of contour C1 to coin-cide with the negative imaginary axis of z. Then integral (A.3) can be written in thefollowing form


pR¥0

dz exp tz � z3

3

!þ 1ffiffiffi

pp

R0� j¥

dz exp tz � z3

3

!. (A.7)

Changing the variable in the second integral (A.7) to y ¼ jz, we obtain


pR¥0

dzfexp tz � z3

3

!þ jffiffiffi

pp

R¥0

dy exp �j ty � y3

3

! !. (A.8)

We may now select the real and imaginary parts in w(t):

wðtÞ ¼ uðtÞ þ jvðtÞ. (A.9)

174

Re z

Im z

2Γ

1Γ

Figure A.1 Sectors of convergence for the Airy integral.

A.1 Definitions

where

uðtÞ ¼ 1ffiffiffip

pR¥0

dz exp tz � z3

3

!þ 1ffiffiffi

pp

R¥0

dz sin tz þ z3

3

!, (A.10)

vðtÞ ¼ 1ffiffiffip

pR¥0

dz cos tz þ z3

3

!¼ 12ffiffiffip

pR¥0

dz exp �j tz þ z3

3

! !. (A.11)

As observed from Eqs. (A.9) to (A.11), functions u(t) and v(t) represent two inde-pendent solutions to Eq. (A.2), and v(t) is indeed an original Airy integral (A.1). Thefunctions u(t) and v(t) are related via the following equation

u0 ðtÞvðtÞ � uðtÞv0 ðtÞ ¼ 1. (A.12)

The set of two independent solutions to Eq. (A.2) can also be found in terms ofanother contour integral of Eq. (A.2). Further we redefine the first integral (A.3) as afunction w1(t) while the second integral we define in the following form

wðtÞ ”w2ðtÞ ¼1ffiffiffip

pRC2

dz exp tz � z3

3

!. (A.13)

The contour C2 is a mirror image of the contour of C1 relative to the real axis of zshown by the dashed line in Figure A.1. With real values of the arguments, func-tions w1(t) and w2(t) are complex conjugate and

w2ðtÞ ¼ uðtÞ � jvðtÞ. (A.14)

For functions w2(t) and w1(t) there exists an equation similar to Eq. (A.10)

w01ðtÞw2ðtÞ � w

02ðtÞw1ðtÞ ¼ �2j (A.15)

and

w1 tej 2p=3

� �¼ e

jp=3w2ðtÞ. (A.16)

While functions u(t) and v(t) are real functions with real values of the argument t,these are also natural transcendent functions valid for complex values of t. The rela-tions (A.9) and (A.14) hold for complex t. Numerous formulas, useful for the treat-ment of Airy functions, have been provided by Fock [1] and some of them are listedbelow for convenience:

w1 tejp=3

� �¼ 2e

jp=6vð�tÞ , (A.17)

w1 tejp

� �¼ uð�tÞ þ jvð�tÞ , (A.18)

w1 tej 4p=3

� �¼ 2e

jp=6vðtÞ , (A.19)

175


w1 tej 5p=3

� �¼ e

jp=3uð�tÞ � jvð�tÞ½ � . (A.20)

The above relations represent the values of the function w1(t) at the raysarg t ¼ np=3 (n = 0, 1, 2, 3, 4, 5) in the complex t-plane via real functions u(t) andv(t) with real values of the argument t.We may also note that the functions u(t) and v(t) are equivalent to Airy functions

Ai(t) and Bi(t) [2]:

AiðtÞ ¼ 1ffiffiffip

p vðtÞ; BiðtÞ ¼ 1ffiffiffip

p uðtÞ.

A.2Asymptotic Formulas for Large Arguments

The asymptotic expressions for the Airy function for large arguments have beenderived by Fock and are provided below.Frequently used formulas for large negative t are:

w1ðtÞ ¼ �tð Þ�1=4 exp j23

�tð Þ3=2þ jp4

� �, (A.21)

w2ðtÞ ¼ �tð Þ�1=4 exp �j23

�tð Þ3=2� jp4

� �. (A.22)

Let us introduce the definition

x ¼ 23

t3=2

and coefficients an, bn to be used in the forthcoming formulas:

a1 ¼572; a2 ¼

5 11ð Þ 71 2 72ð Þ2

; a3 ¼5 11 17ð Þ 7 13ð Þ1 2 3 72ð Þ3

;

an ¼ 5 11::: 6n�1ð Þð Þ 7 13::: 6n�5ð Þð Þ1 2 :::n 72ð Þn ,

b1 ¼772; b2 ¼

7 13ð Þ 51 2 72ð Þ2

; b3 ¼7 13 19ð Þ 5 11ð Þ1 2 3 72ð Þ3

;

bn ¼ 7 13::: 6nþ1ð Þð Þ 5 11::: 6n�7ð Þð Þ1 2 :::n 72ð Þn .

Then a comprehensive asymptotic expression for Airy functions of large positiveargument t can be written as follows:

uðtÞ ¼ t�1=4

ex1þ a1

xþ a2

x2þ :::

, (A.23)

176

A.3 Integrals Containing Airy Functions in Problems of Diffraction and Scattering of UHF Waves

u0 ðtÞ ¼ t

1=4ex1� b1

x� b2

x2� :::

, (A.24)

vðtÞ ¼ 12

t�1=4

e�x

1� a1xþ a2

x2� a3

x3þ :::

, (A.25)

v0 ðtÞ ¼ � 1

2t1=4

e�x

1þ b1x� b2

x2þ b3

x3� :::

. (A.26)

For large negative t we have

uð�tÞ ¼ t�1=4

cos x þ p4

� �1� a2

x2 þ

a4x4 �

a6x6 þ :::

þ

t�1=4

sin x þ p4

� � a1x

� a3x3 þ

a5x5 �

a7

x7 þ :::

;

(A.27)

u0 ð�tÞ ¼ t

1=4sin x þ p

4

� �1þ b2

x2 �

b4x4 þ

b6x6 � :::

þ

t1=4cos x þ p

4

� � b1x

� b3x3 þ

b5x5 �

b7

x7 þ :::

;

(A.28)

vð�tÞ ¼ t�1=4

sin x þ p4

� �1� a2

x2 þ

a4x4 �

a6x6 þ :::

�

t�1=4

cos x þ p4

� � a1x

� a3x3 þ

a5

x5 �

a7

x7 þ :::

;

(A.29)

v0 ð�tÞ ¼ t

1=4cos x þ p

4

� �1þ b2

x2 �

b4x4 þ

b6x6 � :::

þ

t1=4sin x þ p

4

� � b1x

� b3x3 þ

b5x5 �

b7

x7 þ :::

:

(A.30)

A.3Integrals Containing Airy Functions in Problems of Diffraction and Scattering ofUHF Waves

In this section we obtain the asymptotical expansions of the integrals containing theproduct of the Airy–Fock functions [1]:

V ¼R0�¥

wðt � yÞwðt � y0Þ exp ðjntÞdt (A.31)

where w(x) is any one of the solutions to the Airy equation

w0 ðxÞ � xwðxÞ ¼ 0 (A.32)

defined by one of the integrals below with desirable behavior at infinity

177


wðxÞ ” w1ðxÞ ¼1ffiffiffip

pRC

dz exp xz � z3

3

!, (A.33)

w1ðxÞjx!¥ ~ �xð Þ�1=4 exp j23

�xð Þ3=2þ jp4

� �, (A.34)

wðxÞ ” vðxÞ ¼ 12ffiffiffip

pR¥�¥

dz exp jz3

3þ jzx

" #, (A.35)

vðxÞjx!¥ ~ xð Þ� 1=4 exp � 23

xð Þ3=2� �

. (A.36)

In integral (A.33) the contour C elapses from ¥ along the ray exp j 4p=3ð Þ towards0 and then along the real axis of z to þ¥. Figure A.1. Along with the functionsw1(x), v(x) we use function w2(x) = w1

*(x), where the sign * denotes complex conju-gate.Integrals of type (A.31) are often calculated in problems of UHF diffraction and

propagation in the atmospheric boundary layer. This is because in an analyticalapproach the model of the average refractivity can be described in terms of the linearapproximation to a whole height profile of the averaged refractivity or, at least, tosome sections of the profile. For instance, the integral of type (A.31) appears whenone need coefficients of the energy transformation between modes (coupling coeffi-cients) due to scattering on the random fluctuations of the refractive index in themedium. Then the eigenfunctions of the operator

Lz jz

�¼ d2

dz2� k

2e0 � gezð Þ

" #jnðzÞ ¼ EnjnðzÞ, (A.37)

jnðz ¼ 0Þ ¼ 0, jnðz ! ¥Þ ¼ 0 (A.38)

associated with the discrete spectrum of eigenvalues correspond to the waves propa-gating with almost negligible attenuation, i.e. waveguide modes (trapped modes inthe common terminology of tropospheric duct propagation). In the case of linearprofile eðzÞthese functions can be expressed via the Airy function

jnðzÞ ¼ CnvEn�l3ðz�HÞ

l2

!(A.39)

where z is the height above the boundary surface (z = 0), l3 ¼ age=2 the gradient of

the refractive index inside the waveguide, i.e. for 0 < z < H, H the thickness of thewaveguide formed by a negative gradient of the refractive index ge , ge =� de=dz þ 2=að Þ .The coefficient of re-scattering Vm,n between the modes with numbers m and n

due to scattering on a spatial Fourier-component of the fluctuation in the refractiveindex with vertical wavenumber k is then given by

178


Vm;n kð Þ ¼R¥0

dzvEn�l3ðz�HÞ

l2

!v

Em�l3ðz�HÞl2

!exp jkzð Þ. (A.40)

After apparent transformations, the evaluation of integral (A.40) will result in cal-culation of the integral

V1 ¼R¥0

dt v t � nnð Þv t � nmð Þ exp jqtð Þ . (A.41)

Similar arguments can be applied to evaluation of the coupling coefficients of theeigenfunctions of the continuous spectrum WE1

ðzÞ and WE2ðzÞ. These coefficients

will result in integrals of the following kind:

V2 ¼R0�¥

dt w1 t � n1� �

w1 t � n2� �

exp jqtð Þ , (A.42)

V3 ¼R0�¥

dt w1 t � n1� �

v t � n2� �

exp jqtð Þ , (A.43)

V4 ¼R0�¥

dt w1 t � n1� �

w2 t � n2� �

exp jqtð Þ ¼ J1 � 2jJ3 . (A.44)

We can note that

V5 ¼R0�¥

dt w2 t � n1� �

w2 t � n2� �

exp jqtð Þ ¼ J2ð�q; n1 ; n2Þ (A.45)

for real q; n1 and n2 . Therefore, the basic set of integrals of interest is J1, J2 and J3.

A.3.1Integral V1

Making use of the integral representation (A.35), transform integral V1 into a tripleintegral and perform integration over the variable t as well as over one of the newlyintroduced variables of integration. As a result we obtain

V1 ¼exp jp=4

� �4ffiffiffip

pR¥�¥

dsffiffis

psþqð Þ exp j

s3

12� jsm� j

d4s

!(A.46)

where the contour of integration over s encircles the singularities s = 0 and s = q inthe upper half-plane of the variable s; m ¼ n1 þ n2

� �=2, d ¼ n1 � n2 . With mj j >> 1

integral (A.46) can be calculated using the method of stationary phase.Assume m >> 1, d >> 1. In the integral (A.1.1) we observe four stationary points

s1;2;3;4 ¼ �ffiffiffiffiffin1

p�

ffiffiffiffiffin2

p� �. The pair s3;4 ¼ �

ffiffiffiffiffin1

p�

ffiffiffiffiffin2

p� �» � d

2ffiffiffim

p may come closeto the square root singularity in the point s = 0.

Let us evaluate the contribution of the stationary points

s1;2 ¼ �ffiffiffiffiffin1

pþ

ffiffiffiffiffin2

p� �» 2

ffiffiffim

p 1� d2

32m2

. We may notice that

179


V1 ¼exp � j p=4

� �4ffiffiffip

p I1 m; d; qð Þ þ jI�1 m; d;�qð Þ

� �, (A.47)

where

I1 ¼R¥0

dsffiffis

psþqð Þ exp j

s3

12� jsm� j

d4s

!. (A.48)

Withd2ffiffiffim

p << 1

I1ðm; d; qÞ@ I1ðm; 0; qÞ ¼exp j 4=3m3=2

� �ffiffiffiffiffiffiffiffiffi2ffiffiffim

pp R¥�¥

dxxþ2

ffiffiffim

pþqð Þ exp j

s3

2� jsm� j

d4s

!

(A.49)

Taking into account the known representation for the error integral [2], we obtain

I1ðm; d; qÞ@ � jpffiffiffiffiffiffiffiffiffi2ffiffiffim

pp 1þ 2ffiffiffip

p exp jp

4

� � Rm1=4

s1þqð Þ= ffiffi2p

�¥dx exp �jx

2� �8><

>:9>=>; (A.50)

and, respectively, a uniform asymptotic for the contribution to J1 from stationarypoints s1;2 ¼ � 2

ffiffiffim

p:

V1 @ jpffiffiffiffiffiffiffiffiffi2ffiffiffim

pp

exp �j43m3=2 þ j

ffiffiffim

p

2s1;2 þ q

2� �

·

1þ2e jp=4ffiffiffip

pRm

1=4s1;2þqð Þ= ffiffi2p

�¥exp �jx

2� �

dx

264

375�

j exp �j43m3=2� j

ffiffiffim

p

2s1;2 þ q

2� �

·

1þ 2e�jp=4ffiffiffip

pRm

1=4s1;2þqð Þ= ffiffi2p

�¥:ffiffiffi2

p exp jx2

� �dx

264

375

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

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

: (A.51)

Now, let us evaluate the contribution to V1 of the region in the vicinity of the sta-tionary points s3;4»� d=2

ffiffiffim

p. We may notice that with d >> 2

ffiffiffim

pevaluation of the

integral can be performed similarly to the method used in obtaining Eq. (A.51).Therefore, attention will be paid to the case d << 2

ffiffiffim

p, which, being applied to the

problem of scattering on random fluctuations in the refractive index, corresponds tothe situation where the transversal wavenumbers of the mode’s pair are large,though the distance between the modes in a wavenumber space is small. This situa-tion occurs in a multimode tropospheric duct for higher-order modes when the ductis formed, for instance, by inversion of temperature.

180


Taking into account that the major contribution to the integral comes from theregion of small s, s~ 1=m, we may expand exp ðjs3=12Þ into a Taylor series:

V1 @ � e� jp=4

4ffiffiffip

pR¥�¥

dsexp �jsm�j

d2

4s

ffiffis

psþqð Þ 1þ j

s3

12� j

s9

288þ :::

!

¼ ejp=4

4ffiffiffip

p I2 m; d; qð Þ þ 112

@3

@m3I2 m; d; qð Þ þ :::

" #:

(A.52)

We may observe that

I2 ¼ ej mq

I3 ; I3 ¼R¥

�¥dsexp � j mðsþqÞ�j

d2

4s

ffiffis

psþqð Þ . (A.53)

Now, introduce a new function I4 using the relation

I4 ¼@I3@m

jejmq. (A.54)

The function I4 is then determined via the integral

I4 ¼R¥0

dsffiffis

p cos wðs; m; dÞð Þ þ sin wðs; m; dÞð Þ½ � (A.55)

where wðs; m; dÞ ¼ ms þ d=4s. The integral (A.55) is tabulated [3] and given by

I4 ¼ffiffiffiffiffiffiffi2pffiffiffim

pr

cos dffiffiffim

pð Þ.

Then, the reverse operation leads to

I2 ¼ 2ffiffiffip

pe� j 3p=4þjmq Rm

¥dge

j gq cos dffiffiffig

p� �ffiffiffig

p þ C. (A.56)

The value of constant C, which does not depend on parameter q, can be chosenusing the norm of J1 with q ! 0. Then,

C ¼ j2pffiffiffi

qp exp j

d2

4q� j

p

4

!.

After some transformation expression for I2 the following results:

I2 m; d; qð Þ ¼ � 2ffiffiffip

pffiffiffiq

p exp �j 3p4

þ jmq þ j d2

4q

·

erf ejp=4 ffiffiffi

qp

m� d2ffiffiffiq

p � �

þ erf ejp=4 ffiffiffi

qp

mþ d2ffiffiffiq

p � � ! (A.57)

where erf ð:Þ is the error integral [2].It should be noted that the second term in Eq. (A.52) has an order Oð1=m3=2Þ and

may be neglected for the following reason. We seek the estimate of the integral J1 asa sum of Eqs. contributions from the stationary points, four in total. Therefore, the

181


result is a combination of (A.51) and (A.52). Taking into account that the contribu-tion of the stationary points s1;2 ¼ �2

ffiffiffim

pin Eq. (A.51) evaluated with precision up

to the terms of order O 1=ffiffiffim

pð Þ, we should neglect the second term in Eq. (A.52),

thus substituting the expression (A.57) for I2 in Eq. (A.52).Finally with m >> 1, d << 1, the integral V1 m; d; qð Þ is represented by a sum of

contributions of four stationary points:

V1 m; d; qð Þ»V1 js¼ s1þV1 js¼ s21

þV1 js¼ s3þV1 js¼ s4

,

where V1 js¼ s1and V1 js¼ s2

are defined by Eq. (A.51) and V1 js¼s3and V1 js¼s4

by Eqs.(A.52) and (A.57).In the limiting case d ¼ 0 and qm << 1 we obtain

V1 m; d; qð Þ ¼P¥p¼0

2p e jpp=2

2pþ1ð Þ qpm

pþ1=2 þ O1ffiffiffim

p

. (A.58)

With q = 0, m ¼ nn (recall that nn is a propagation constant of the nth mode), theexpression (A.58) equates to the known asymptotic of the norm of the eigenfunc-tions of the discrete spectrum with nn >> 1:

Nn ¼R¥0

dtv2

t � nnð Þ ¼ffiffiffim

p. (A.59)

a) Let us consider the integral J1 with small parameters d; m: m << 1, d << 1.Expanding exp ð�jms � jd

2=4sÞ and s þ qð Þ

�1into a series, we obtain

V1 m; d; qð Þ ¼P¥

m¼0

P¥p¼0

P¥r¼0

Bpmr�1ð Þr e� jppþm=2

p!m!22mqrm

p(A.60)

where

Bpmr ¼ 12ð Þm�pþr3 C

p � m þ r3

� 16

·

exp jp6

p � m � r � 12

� � þ exp j 5p

6p � m � r � 1

2

� � � �:

In the particular case when q ¼ m ¼ d ¼ 0, integral (A.1.15) equates to

V1 ¼C

5

6

� �2�4=3ffiffiffi

pp

31=6(A.61)

which is the same as the known integral [2]:

R¥0

dxv2ðxÞ ¼ �

ffiffiffip

p

31=3C1

3

� �24

352 ¼ C

5

6

� �2� 4=3ffiffiffi

pp

31=6. (A.62)

b) With qm >> 1 we may expand (s + q)–1 over the reverse powers of q thusobtaining the series representation

182



m¼0

P¥p¼0

P¥r¼0

Bpmr�1ð Þr e� j p pþm=2

p!m!22mmpd2m

q rþ1 (A.63)

where

Bpmr ¼ 12ð Þm�pþr3

� 16C

p � m þ r3

� 16

·

exp jp6

p � m � r þ 12

� � þ exp j 5p

6p � m � r þ 1

2

� � � �:

(A.64)

c) When m ~ 1, d << 1, we may expand exp jsmþ j d=4sð Þ over a series of Besselfunctions, thus obtaining


m¼�¥�jð ÞmJm d

ffiffiffim

p

2

2ffiffiffim

p

d

m

Bm qð Þ (A.65)

where Jm(x) is a Bessel function of the first kind and coefficient Bm(q) is given by theintegral

Bm qð Þ ¼Rþ¥

�¥ds

exp js3

12

!

sþqsm� 1

2, (A.66)

which can be represented via Gamma-functions similar to the coefficient in Eqs.(A.61) and (A.64).

A.3.2Integral V2

Let us consider integral (A.42) where n1and n2 have real and positive values, q is aparameter of real value.

V2 ¼R0�¥

dt w1 t � n1� �

w1 t � n2� �

exp jqtð Þ

We use the integral representation for Airy functions (A.33) to transform the inte-gral (A.42). The contour of the integration C1 in (A.33) can be defined as elapsingfrom � j¥ to 0 and then to ¥ along the real axis. After integration over variable t weobtain

V2 ¼1p

RC

RC

dy1dy2

exp �n2y1�n

1y2�y

31 þ y

32

3

!

y1þy

2þrþjq

. (A.67)

We have introduced an additional small parameter r, r > 0, in order to shift thepole from the contour of integration into point � r þ jqð Þ and make a transforma-tion of the contour C1 into contour C

01 as shown in Figure A.2.

Introducing a new variable s ¼ y1 þ y2 and redefining for convenience y ” y1 , wemay find that integration over variable y can then be performed given the conver-gence after changing the order of integration:

183


V2 ¼1ffiffiffip

pRC2

ds

exp � s3

12� s

n1 þ n22

þ ðn1 � n2Þ2

4s

!ffiffis

pðs þ jq þ �Þ . (A.68)

The contour C2 for the newly introduced variable s can be obtained from contourC1 by shifting C1 to a value y2 . When y2 varies along the imaginary axis, the contourC1 is shifted down along that axis to y2

## ##. When y2 ‡ 0, the contour C1 should beshifted right along the real axis, i.e. contour C2 of the integration over s seems todepend on the value of y2 . We may demonstrate that all these variations in contourC2 are equivalent to contour C1.Consider first the case, when the contour C1 is shifted along the imaginary axis s

to the value � y2## ## as shown in Figure A.3. Inside the closed contour in Figure A.3,

integral V2 has no singularities and along the branches C1 and C2of the contour itsmagnitude is negligible. Therefore, according to Cauchy’s theorem, we may performintegration over contour C1 instead of C2 , i.e. these contours are equivalent in termsof integration.Using similar arguments we may show that when the contour C2 is obtained by a

shift of C1 along the real axis, both contours are equivalent. Therefore, whatever val-ue y2 takes along contour C, contour C2 can be deformed into contour C1 which, inturn, no longer depends on value y2 , thus allowing, in principle, the changes in theorder of integration in Eq. (A.68).In order to perform the integration we need to deform the contours C and C2

within their sector of convergence in such a way that will ensure convergence of theintegral over variable y. New contours will have the following shape: contour C goesfrom negative ¥ along the ray e

� j 2p=3to 0 and then along the real axis to ¥; contour

C2 goes from �j¥ to 0 overtaking the pole –jq from the right (in a counter clockwisedirection) and then to ¥ along the ray e

�j p=6.

The internal integral over y is then a Poisson’s integral with apparent solution. Inthe remaining single integral we then make a further transformation of the contour

184

Re z

Im z

1Γ'

σ−

jq−

32πj

e∞

0

−(σ + jq)

Figure A.2 Contour of integration for inte-gral V2.


of integration C2 , namely we can turn the contour C2 through an angle of p=2 in acounterclockwise direction and change the path along the contour in the oppositedirection, then parameter r can be put to 0.Finally, we end up with the following integral

V2 ¼ � ejp4ffiffiffip

pRC3

ds

exp �js3

12þ jsm� j

d2

4s

!ffiffis

ps�qð Þ (A.69)

with contour C3 shown in Figure A.4.Now we can evaluate integral (A.69) in some limiting cases.

a) Consider the case when m >> 1, d << 1.The integrand in Eq. (A.69) along the imaginary axis represents a rapidly attenu-

ating function without any singularities at that section of integration. The majorcontribution to Eq. (A.69) hence follows from the integration along the ray e

�jp=6

(or along the real axis of s, since that section of the contour C3can be further trans-formed to one along the real axis).

185

Re s

Im s

1Γ ′

(σ ) jq+ −

−σ

jq−

jqej

− ∞ 3

2π

0

1C

2C

jqjej

− − ∞ 2

32

ξπ

Figure A.3 Transformation of the integration contour in V2.

Re s

Im s

3Γ

q

0

Figure A.4 Final integration contour in V2.


The asymptotic evaluation of the integral (A.69) is then composed of a contribu-tion from the stationary points: s1 ¼ 2

ffiffiffim

p1� d

2=32m

� �and s3 ¼ d=2

ffiffiffim

p:

V2»V2 js1þV2 js3 (A.70)

where

V2 js1 @� e jp=4ffiffiffip

p I5 m; d; qð Þ (A.71)

and

I5 m; d; qð Þ ¼R¥0

dsexp �j

s3

12þ js m

ffiffis

ps�qð Þ . (A.72)

Using an approach similar to that introduced in Section A.3.1, we obtain theasymptotic for I5 m; d; qð Þ:

I5 m; d; qð Þ @ jp

ffiffiffiffiffiffiffi2ffiffiffim

pr

exp �j43m3=2 � j

ffiffiffim

p

2s1 � qð Þ2

·

1þ e�jp=4ffiffiffip

pRm

1=4s1�qð Þ=2

0

exp ðjs2Þds

264

375: (A.73)

A contribution from the stationary point s3 into V2 will be evaluated when the sta-tionary point s3 is close to a square root singularity, i.e when d << 2

ffiffiffim

p. Expanding

exp �j s3=12

� �into a Taylor’s series in the vicinity of s ¼ 0 we obtain

V2 js3 @� ejp=4

4ffiffiffip

p P¥n¼0

e� jpn=2

12nn!

R¥�¥

dsexp jsmþj

d2

4s

ffiffis

ps�qð Þ s

3n

¼ � ejp=4

4ffiffiffip

p I6 m; d; qð Þ þ 112

@3

@m3I6 m; d; qð Þ þ :::

" # (A.74)

Then using similar arguments as in the derivation of Eq. (A.56) we obtain for I6the following expression

I6 m; d; qð Þ ¼ ej 3p=4 ffiffiffi

pp Rm

�¥dxexp �jxqþ jd

ffiffiffix

pð Þffiffiffi

xp þ C . (A.75)

The value of constant C is determined from the condition of the finite value of V2with q ! 0:

C ¼ �ffiffiffip

h

rexp j

d2

4q� j

p

4

!, (A.76)

and the asymptotic for V2 js3 is finally given by

V2 js3 @ 2ffiffiffiph

rexp jmq þ j

d2

4q

!erf e

jp=4 ffiffiffim

pq þ d

2ffiffiffiq

p

. (A.77)

186


b) For small parameters m; d, i.e., m << 1, d << 1, we may use a series expansionfor V2 similar to the approach in Section A.3.1.

For the case qm >> 1,

V2 ¼ � ejp=4ffiffiffip

p P¥m¼0

P¥p¼0

P¥r¼0

Bpmre� jppþm=2�jp

p!m!22mm pd2m

qrþ1 (A.78)

where

Bpmr ¼ 4 12ð Þpþr�m3

þ 56C

p � m þ r3

þ 16

·

exp jp2

p � m þ r � 32

� �� þ exp jp

6p � m � r þ 1

2

� � � �:

(A.79)

For the case qm << 1,

V2 ¼ � e jp=4ffiffiffip

p P¥m¼0

P¥p¼0

P¥r¼0

Bpmre jppþm=2

p!m!22mm

pd2m

qr

(A.80)

where

Bpmr ¼ 4 12ð Þpþr�m3

� 76C

p � m þ r3

� 16

·

exp jp2

p � m þ r � 52

� �� þ exp �jp

6p � m � r � 1

2

� � � �:

(A.81)

With m ~ 1 and d £ 1 , the asymptotic of V2 can be obtained via a series of Besselfunctions in a way similar to the approach in Section A.3.1.

A.3.3Integral V4

We use the integral representations (A.3), (A.11) of Airy functions w1ð:Þ and vð:Þ tocalculate V4. We may choose a contour of integration for w1ð:Þ in Eq. (A.3) to passalong the negative imaginary axis to 0 and then along the real axis to ¥. The integralrepresentation for vð:Þ can be chosen as follows:

vðxÞ ¼ 12ffiffiffip

pRj¥

�j¥exz� z

3

3 dz. (A.82)

Then we can perform integration over t in V4 and, in a remaining double integral,we can change the variables s; y similar to those used in Section A.2. Using similararguments for contour transformation we truncate the integral V4 to a single inte-gral given by

V4 ¼e

j3p4

2ffiffiffip

pR¥�¥

ds

exp �js3

12þjsmþj

d2

4s

!ffiffis

ps�qð Þ . (A.83)

187


The contour of integration takes into account the overpass of both the square rootsingularities and the pole over the upper half-circumference in a clockwise direction.The asymptotes for V4 can be obtained in a way similar to that used for V2 in SectionA.2, taking into account the contribution of the stationary points s4 ¼ �2

ffiffiffim

pand

s5 ¼ �d=2ffiffiffim

p. We may also observe that V4 ¼ 2V

�1 ðm; d;�qÞ.

A.3.4Integral V5

As mentioned earlier, the integral

V5 ¼R0

�¥dt w2 t � n1

� �w2 t � n2� �

exp jqtð Þ (A.84)

can be expressed via the basic integral V2. Here, we obtain the asymptotic form of V5for small values of the parameter d, d << 1.Let us introduce a new variable s ¼ t � n2 and transform the contour of integra-

tion in Eq. (A.84) to a ray ¥ej 2p=3

. Taking into account known transformations of theAiry functions [1],

w2 sej 2p=3

� �¼ 2e

jp=6vðsÞ,

w2 sej 2p=3 � d

� �¼ 2e

� jp=6vðs� de

� j 2p=3Þ

we obtain

V5 ¼ 4ej p=3� jn2 q R¥

n2

ds v sð Þv s� de�j 2p=3

� �exp �qs

ffiffiffi3

pþ j

� �2

. (A.85)

Assume both d << 1, qj j << 1,

V5 ¼ 4ejp=3�jn2 q P¥

m¼0

P¥n¼0

�1ð Þnexp �jp2nþ7m

2

� �3

0@

1A

n!m!qmd

n·

R¥n2

ds smvðsÞ dn

dsn vðsÞ:

(A.86)

As observed, the task is truncated to calculation of the integrals of the followingtype:

Qm;n ¼Rsm

v sð Þ dn

dsn vðsÞds . (A.87)

In the case when n = 0, m ‡ 3, integrals Qm;0 can be easily calculated using therecurrent formula:

188

References

Qm;0 ¼m

2mþ1sm�1

v0 ðsÞvðsÞ � m�1

2sm�2

v2ðsÞ � 1

msm

v0 ðsÞ

h i2�

12

m � 1ð Þðm � 2ÞQm�3;0

264

375: (A.88)

For m £ 3 we have

Q0;0 ¼ sv2ðsÞ � v

0sð Þ

h i2,

Q2;0 ¼152svðsÞv0 sð Þ � v

2sð Þ � s

2v0sð Þ

h i2þs

3v2sð Þ

h i, (A.89)

Q1;0 ¼ vðsÞv0 sð Þ � s2

2v0sð Þ

h i2þ s2

2v2sð Þ � 3

2Q2;0 .

Integrals Qm;1containing the higher order derivative can also be defined via recur-rent formulas:

Qm;1 ¼sm

2v2sð Þ � m

2Qm�1;1 , for m‡ 1, (A.90)

Q0;1 ¼12

v2sð Þ .

We used a known feature of the Airy function, that is that a derivative of any ordern,n > 1 of an Airy function v sð Þ can be derived via polynomials of s, function v sð Þand its first derivative v

0sð Þ.

This concludes the calculation of the integrals of the Airy function products.

189

References

1 Fock, V.A. Electromagnetic Diffraction andPropagation Problems, Pergamon Press,Oxford, 1965.

2 Abramovitz, M. and Stegun, I. Handbook ofMathematical Functions, NBS, Applied Mathe-matics Series-55, Washington, 1964.

3 Bateman H. and Erdellyi A. Tables of IntegralTransforms, McGraw-Hill, New York, 1954.

191

aAiry equation 177Airy function 29, 30, 31, 46, 47, 48, 86, 87,

88, 98, 105, 110, 123, 126, 127, 158, 167,168, 175, 176, 178, 183, 188, 189

Airy integral 173, 174, 175anisotropy 35, 114, 153, 169anisotropy parameter 17atmosphere 1

standard linear 5 ff.stratified 5

atmospheric boundary layer (ABL) 1 ff., 12,16, 20, 38, 99, 121, 150

attenuation, coherent component 109, 153 ff.attenuation factor 45, 72, 83, 106, 107, 108,

132, 166attenuation function 71, 129attenuation rate 53, 54, 77 ff., 146

bBessel function 183, 187Booker–Gordon theory 157, 164, 165, 167,

170Born approximation 166boundary condition 23 ff., 27, 28, 42, 44, 58,

62, 66, 73, 84, 98, 100, 101, 129, 167ideal 32impedance 43, 49, 73Leontovitch’s 23, 27

Bragg angle 110Brewster angle 70

cCauchy’s theorem 184central limit theorem 34coherence function 34, 59, 102, 104, 108,

109, 117, 153coherence scale 37coherent signal component 154, 159

correlation function 14, 67, 156space-time 12, 15spatial 13

cyclic frequency 3, 19 ff.

dDebye’s potential 22, 26, 40, 45dielectric permittivity 4, 19, 23, 33, 58, 66,

89, 156, 163anisotropic fluctuations 153ensemble of realisations 4isotropic fluctuation 15mean characteristic 4random field 12, 14spatial spectrum of fluctuations 25, 36spectrum of fluctuations 13, 67turbulent fluctuation 13, 15variance of fluctuations 16

eearth

curvature 6“effective” radius 6flat 7

eigen functionscontinuous spectrum 20, 28, 32, 99 ff.,

123, 132discrete spectrum 20, 100 ff.

elevated duct 11, 121, 126, 131, 147, 149, 150error integral 181Euler equation 37, 60, 73evaporation duct 9, 10, 49, 52, 54, 75, 76,

77 ff., 82, 94, 99, 100, 104, 107, 108, 109,114, 118, 121, 139, 140, 143, 147, 153, 164linear-logarithmic 78, 91

fFermat paths 37, 38Feynman diagrams 20

Index

192

Feynman integrals 67Feynman path integrals 33Fourier transform 36, 42, 103, 104, 161Fresnel coefficient 70Fresnel volume 37, 59Fresnel zone 35, 37, 38, 59, 68, 71, 102, 149

gGamma function 183Gaussian, random value 34geometrical optic presentation 5Green function 20, 27, 32, 33, 37, 58, 149,

157, 167

hheight-gain function 47, 50 ff., 76, 78, 82, 97,

98, 128, 129, 149, 154, 159, 167horizon 7, 49, 57, 76, 108humidity 1 ff., 8Huygens–Fresnel principle 38

kKelvin–Helmholtz waves 12Kirchhoff approximation 82Kirchhoff theory 83Kolmogorov–Obukhov model 16

locally uniform 15

lLagrangian 33, 36, 37, 61Laplace operator 22Laplace transformation 89, 90line-of-sight (LOS) region 7, 10, 19, 29, 30,

32, 46, 48, 57, 64, 73, 131local uniformity hypothesis 14

mMalyuzhinetz transformation 66Markov approximation 36, 59, 153, 154Markov process 20, 155Maxwell equations 22meso-meteorological minimum 3meso-pause 4mode

fraction 94leaked 146, 153normal 87trapped 53, 81, 82, 94, 96, 103, 108, 110,

113, 115, 117, 118, 126, 127, 143, 148,150, 153

waveguide 94, 104, 108, 110, 112,113, 153

pPadC product 41parabolic cylinder function 50 ff.path integral 21, 36, 37, 44, 57, 60Peceris duct model 77permittivity, modified 83perturbation theory 76, 84, 86, 87perturbations, smooth 57, 71Poisson’s integral 184pressure, atmospheric 1 ff.profile, refractivity 85propagation, non-standard mechnisms 9

standard mechanism 5 ff.propagation constant 53, 76, 78, 84, 86, 87,

89, 94, 95, 96, 98, 125, 126, 127, 154, 159,167discrete spectrum 84

rRayleigh law 154refraction 84, 149, 164

normal 29, 31, 32, 45, 86, 114, 166, 167standard 7sub – 7super – 7, 57

refractive index 1, 4, 8, 57, 73, 76, 83, 108,117, 118, 132, 153, 154, 162, 164fluctuations 71modified 7, 84, 87spatial spectrum of fluctuations 110, 169

refractivity 162depression layers 11 f.elevated M-inversion 11, 121, 125, 139,

143, 144, 147inversion depth of M-profile 10M-deficit 79, 94, 142M-inversion 9, 16, 75, 94, 100, 127modified 7M-profile 9, 49, 54, 75, 78, 92, 95, 97, 99,

108, 114, 121, 144, 146surface M-inversion 125, 139

Rytov’s method 157

sscattering cross-section 165, 170scattering volume 148, 149, 150, 165, 170SchrEdinger operator 20SchrEdinger’s equation 20scintillation factor 54sea surface, roughness parameter 9shadow region 7, 10, 46, 98, 100, 108, 112,

131, 166S-matrix 29, 92 ff., 128

Index

193

Snell’s law 5spatial spectrum 14 ff.split-step approximation

Claerbout 39Fourier 38, 40, 41, 43, 44marching solution 42, 43PadC 38, 40, 41standard 39

stationary phase method 31structure constant 15 ff., 67, 114, 150, 158structure function 13, 14 ff., 67 ff., 102

space-time 15surface, rough 21

tTaylor series 181Taylor’s hypothesis 4temperature 1 ff., 8, 127terrain, irregular 44troposphere 1, 75, 76, 108, 153

linear model 7stratified 19 ff., 41, 73, 76, 83, 84

tropospheric duct 132, 153turbulence 67 ff.

atmospheric 1 ff., 59, 106inertial interval 15 f.

internal scale 15, 25, 60locally isotropic 15 f., 25

turbulent variations 1anisotropy 16macro-range 4micro-range 4synoptic 4

vVeil-Van-der-Paul solution 26

wwave equation 38

parabolic approximation 19 ff., 38waveguide channel 113wavelength, critical 75waves

continuous spectrum 103 f., 108 ff., 114discrete spectrum 104, 109 ff., 114normal 19, 29 ff., 121, 125trapped 103

wind speed 3spectrum of fluctuations 3

WKB approximation 110, 111

Index

radio wave propagation in the marine boundary layer

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