Parabolic Equation Methods for Electromagnetic Wave Propagation
This book is the first to present the application of parabolic equation methods in electromagnetic wave propagation. These powerful numerical techniques have become the dominant tool for assessing clearair and terrain effects on radiowave propagation and are growing increasingly popular for solving scattering problems. The book gives the mathematical background to parabolic equation modelling and describes simple parabolic equation algorithms before progressing to more advanced topics such as domain truncation, the treatment of impedance boundaries and the implementation of very fast hybrid methods combining raytracing and parabolic equation techniques. The last three chapters are devoted to scattering problems, with application to propagation in urban environments and to radar cross section computation. This book will prove useful to scientists and engineers who require accurate assessment of diffraction and ducting on radio and radar systems. Its selfcontained approach should also make it particularly suitable for graduate students and other researchers interested in radiowave propagation scattering.
Inspec keywords: electromagnetic wave propagation; parabolic equations
Other keywords: parabolic equation method; scalar wave equation; rays; tropospheric radiowave propagation; twodimensional scattering; parabolic equation framework; irregular terrain modelling; electromagnetic wave propagation; oversea propagation; rough sea surface; impedance boundary modelling; domain truncation; parabolic equation algorithm
Subjects: Engineering mathematics and mathematical techniques; Electromagnetic wave propagation
 Book DOI: 10.1049/PBEW045E
 Chapter DOI: 10.1049/PBEW045E
 ISBN: 9780852967645
 eISBN: 9781849193986
 Page count: 352
 Format: PDF

Front Matter
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1 Introduction
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This book is concerned with the application of PE methods to tropospheric propagation problem. Radiowaves in the VHF to millimetric range are affected by atmospheric refraction, and by terrain diffraction and reflection effects. Because of the very large size of the domains of interest relative to the wavelength, it is not possible to compute exact solutions of Maxwell's equations, and approximations must be sought. For many years, solutions were found in terms of geometrical optics and mode theory for refraction problems, and in terms of geometrical theory of diffraction for terrain problems. These have now largely been superseded by PE algorithms, which provide a fast and efficient numerical solution to most longrange radiowave propagation problems. Describe the building blocks which are necessary to construct efficient PE models. For longrange radiowave propagation, PE methods have reached sufficient maturity to make such an organized description possible. Object scattering applications are more recent, and the exposition there is by necessity less systematic. This book is concerned with the twodimensional scalar parabolic wave equation, to which many radiowave problems can be reduced.
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2 Parabolic equation framework
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This chapter discusses the twodimensional electromagnetic problems where the fields are independent of the transverse coordinate y. There are then no depolarization effects, and all the fields can be decomposed into horizontally and vertically polarized components propagating independently.
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3 Parabolic equation algorithms
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In Chapter 2, we have seen how the parabolic equation could be interpreted in terms of Fourier transforms for the case of propagation in vacuum. The situation gets more complex when refractive index variations are present, since they cannot be dealt with directly with Fourier transform techniques. Section 3.2 introduces the powerful splitstep/Fourier technique pioneered by Hardin and Tappert, which replaces the original problem with propagation through a sequence of phase screens. At this stage the discussion is limited to the narrowangle case, starting from the standard parabolic equation.
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4 Tropospheric radiowave propagation
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One of the main applications of parabolic equation techniques is the calculation of radio coverage in the troposphere. Specific approximations need to be made in order to derive an appropriate scalar parabolic equation for tropospheric radiowave propagation from Maxwell's equations. This give a very brief overview of the basic notions of radiometeorology in Section 4.2. Scalar wave equations for horizontally and vertically polarized radiowaves are derived in Section 4.3. The next stage is to choose a coordinate system which simplifies the representation of structures following the Earth's surface. This is the purpose of the Earth flattening transformation, which is derived in Section 4.4. This have now reduced several types of radiowave propagation problems to the twodimensional scalar wave equation. The various frameworks are summarized in Section 4.5. Go back to tropospheric propagation in Section 4.6, where we set up the parabolic equations for tropospheric propagation of horizontally and vertically polarized radiowaves. Finally, we considered normalization of the output PE field in Section 4.7, relating the initial field to the farfield beam pattern.
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5 Rays and modes
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For many years, assessment of refractive effects on tropospheric links was based mainly on geometrical optics and mode theory methods. Although ray optics do not in general give a reliable estimate of field strength, they provide an excellent qualitative picture of propagation conditions, bearing in mind of course that diffraction phenomena are not represented by the geometrical optics approximation. Even now that PE techniques have become the dominant tool for radiowave applications, geometrical optics is still extremely useful. It provides a fast and accurate solution for propagation angles at which refractive effects are not severe, and this makes it invaluable in the construction of efficient hybrid propagation models. In many cases, raytracing also provides crucial information on the behaviour of the wave front. This is used to model angledependent reflection effects. In section 1, we give a brief derivation of the ray equations in the simple case of rangeindependent media with a piecewise linear refractive index profile. In section 2, we outline the basic principles of mode theory for the rangeindependent case and derive the mode theory solution for a piecewise linear refractive index profile. A more complete discussion of the modal series is given in Appendix C.
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6 Oversea propagation
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The simple narrowangle splitstep sine transform PE algorithm described in Chapter 3 is already powerful enough to treat many problems of oversea propagation. The assumption that the sea has infinite impedance is adequate at frequencies above 300 MHz or so regardless of polarization, provided we only consider small grazing angles on the surface and we neglect roughness effects. All the examples considered in this chapter are treated with a sine transform algorithm, neglecting polarizationdependent surface reflection effects. We look at commonly encountered ocean environments: evaporation ducts are considered and surface and elevated ducts given by bilinear and trilinear refractive index profiles are also treated. PE results are compared with raytracing and mode theory. Rangedependent environments are treated and examples involving measurements in situ are given. The chapter ends with a discussion of absorption modelling.
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7 Irregular terrain modelling
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Accurate modelling of radiowave propagation over irregular terrain is crucial for the planning of cellular communications networks and the prediction of radar performance in coastal environments. Many operational models are based on the very efficient simplified Deygout solution for multiple knifeedge diffraction [43, 44]. More sophisticated diffraction models representing the ter rain profile as a succession of knifeedges, wedges, cylinders or flat strips are also available [159, 98, 143, 165]. Another important family of models uses an integral equation formulation [120, 118], which can be greatly simplified by making a paraxial approximation [60]. All of these techniques assume propaga tion in a homogeneous or linear atmosphere. By contrast parabolic equation methods model the combined effects of terrain diffraction and atmospheric refraction [85, 107, 11] while remaining straightforward to implement.
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8 Domain truncation
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For many radiowave propagation problems the bottom boundary of the domain is the physical ground/air interface, often represented by a surface impedance boundary condition, while the top boundary is a computational artifact needed to limit the integration domain in height. The top boundary must represent at a finite distance the Sommerfeld radiation condition it must be perfectly transparent, letting all the energy coming from below the boundary escape to infinity. The earliest implementations of domain truncation in PE modelling were based on absorbing layers, which work well for small propagation angles but become quite expensive for larger angles from the horizontal. Absorbing layers are still the most popular truncating technique in view of their ease of implementation.
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9 Impedance boundary modelling
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Although the perfectly conducting ground model is adequate for many applications, it is not universally applicable. A more accurate model is required for surface wave propagation, for correct modelling of reflection effects at vertical polarization or for the modelling of propagation over rough surfaces. Applications to lineofsight propagation of vertically polarized waves and to surface wave propagation are also discussed.
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10 Propagation over the rough sea surface
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For parabolic equation modelling of forward scatter over a rough surface, we are concerned with the average field, corresponding to specular reflection. Most applications involve a nonhomogeneous propagation medium, a situation which is extremely difficult to treat. Although some progress has been made on rough surface scattering in idealized ducting environments, there is no rigorous solution in general. Moreover in the case of propagation over the rough sea surface, which constitutes the most important application, insufficient information is available on the complex boundary layer phenomena which cause both the refractive effects and the rough surface formation. Further complication is added by the fact that it is usually very small grazing angles which are of interest for longrange radiowave applications.
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11 Hybrid methods
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This chapter presents the integration of parabolic techniques in electromagnetic wave propagation. As PE integration times depend on frequency, propagation angles and domain size, calculations become prohibitively expensive for such large domains. The Radio Physical Optics model is presented in Section 11.2. With hybrid techniques merging vertical PE (VPE), HPE and raytracing algorithms, quite general propagation problems involving terrain and atmospheric refraction can be solved quickly. In Section 11.4, we show how HPE methods can be combined with raytrace techniques to solve high antenna problems very efficiently, using the ideas of chapter 8 to model sources above the vertical PE domain boundary. Finally we apply hybrid techniques to Earthspace propagation problems in Section 11.5.
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12 Twodimensional scattering
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This chapter is concerned with scattering by twodimensional objects. Inegration domains are generally far smaller than for longrange applications, perhaps a few hundred wavelengths rather than thousands or millions, and it is modelling of object boundaries that becomes crucial, rather than that of refractive index variations.
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13 Threedimensional scattering for the scalar wave equation
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In this chapter we consider threedimensional scattering problems which can be modelled with the scalar wave equation. Acoustic scattering by nonelastic objects is included in this category, but of course most electromagnetic scattering problems involve coupling between scalar components of the field through the boundary conditions on the object and need vector solutions. Some important electromagnetic applications can however be treated with the scalar wave equation.
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14 Vector PE
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This chapter discusses a vector version of the parabolic equation method, which is required to treat general threedimensional electromagnetic problems. The vector PE is obtained by coupling component scalar parabolic equations via suitable boundary conditions on the scatterers. This allows accurate treatment of polarization effects within the paraxial constraints.
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Appendix A: Airy functions
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In this appendix the following are discussed: the Airy differential equation; and zeros of the Airy function.
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Appendix B: Farfield expressions
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This appendix discusses the propagation of electromagnetic fields and its farfield formulations.
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Appendix C: Theoretical derivation of mode series
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This appendix presents the theoretical derivation of mode series. In Section C.2, we show that for a linear profile, the modes always have a unique asymptote along the line of argument π/3. Section C.3 gives the function analysis framework for the derivation, looking at the modal problem in terms of operators on a suitable Hilbert space. Section C.4 proves the validity of modal expansions for linear profiles. Finally Section C.5 deals with perturbations of a linear profile.
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Appendix D: Energy conservation
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We consider twodimensional problems, threedimensional problems and unicity of parabolic equation solutions.
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Back Matter
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