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 These have now largely been superseded by PE algorithms, which provide a fast and efficient numerical solution to most long-range radiowave propagation problems. Describe the building blocks which are necessary to construct efficient PE models. For long-range 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 two-dimensional scalar parabolic wave equation, to which many radiowave problems can be reduced.

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 split-step/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 narrow-angle case, starting from the standard parabolic equation.

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, ray-tracing also provides crucial information on the behaviour of the wave front. This is used to model angle-dependent reflection effects. In section 1, we give a brief derivation of the ray equations in the simple case of range-independent media with a piecewise linear refractive index profile. In section 2, we outline the basic principles of mode theory for the range-independent 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.

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 knife-edge diffraction [43, 44]. More sophisticated diffraction models representing the ter rain profile as a succession of knife-edges, 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.

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 line-of-sight propagation of vertically polarized waves and to surface wave propagation are also discussed.

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 ray-tracing 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 ray-trace 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 Earth-space propagation problems in Section 11.5.

In this chapter we consider three-dimensional scattering problems which can be modelled with the scalar wave equation. Acoustic scattering by non-elastic 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.

In this appendix the following are discussed: the Airy differential equation; and zeros of the Airy function.

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.