Parabolic equation methods for electromagnetic wave propagation

Title:

Personal Author:

Levy, Mireille

Series:

IEE electromagnetic waves series ; 45

Publication Information:

United Kingdom : The Institution of Electrical Engineers, 2000

ISBN:

9780852967645

Subject Term:

Differential equations, Parabolic -- Numerical solutions

Tropospheric radio wave propagation -- Mathematical models

Added Corporate Author:

Institution of Electrical Engineers

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010099190	QC973.4.T76 L48 2000	Open Access Book	Book	Searching... Unknown

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 clear-air 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 ray-tracing 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 self-contained approach should also make it particularly suitable for graduate students and other researchers interested in radiowave propagation scattering.

self-contained approach should also make it particularly suitable for graduate students and other researchers interested in radiowave propagation scattering.self-contained approach should also make it particularly suitable for graduate students and other researchers interested in radiowave propagation scattering.self-contained approach should also make it particularly suitable for graduate students and other researchers interested in radiowave propagation scattering.

Author Notes

Mireille Levy holds a PhD in mathematics from the Paris University Pierre et Marie Curie. She is the head of the modelling group at the Radio Communications Research Unit of the Rutherford Appleton Laboratory

Preface	p. xi
1 Introduction	p. 1
2 Parabolic equation framework	p. 4
2.1 Introduction	p. 4
2.2 Basic derivation	p. 5
2.2.1 Paraxial wave equation	p. 5
2.2.2 Cylindrical coordinates	p. 9
2.3 Approximations of the square-root operator	p. 10
2.3.1 The standard parabolic equation	p. 10
2.3.2 Overview of wide-angle schemes	p. 11
2.4 Propagation in vacuum	p. 12
2.4.1 Angular spectrum	p. 12
2.4.2 SPE in vacuum	p. 14
2.4.3 Square-root operator	p. 16
2.5 Analogy with Fresnel-Kirchhoff diffraction	p. 17
3 Parabolic equation algorithms	p. 20
3.1 Introduction	p. 20
3.2 Split-step formulation of the SPE	p. 21
3.2.1 The range-independent case	p. 21
3.2.2 Range dependence	p. 23
3.2.3 Implementation of operators	p. 24
3.3 Split-step sine transform solution	p. 24
3.3.1 Waveguides: discrete sine transform solution	p. 27
3.3.2 Discretization of the infinite problem	p. 29
3.3.3 Sampling and domain truncation	p. 31
3.4 Wide-angle split-step algorithm	p. 32
3.5 Finite-difference methods	p. 35
3.5.1 Narrow-angle code	p. 37
3.5.2 Claerbout approximation	p. 39
3.6 Source modelling	p. 40
4 Tropospheric radiowave propagation	p. 42
4.1 Introduction	p. 42
4.2 Radio refractive index	p. 42
4.2.1 Atmospheric absorption	p. 45
4.3 Wave equations for radiowave propagation	p. 46
4.3.1 Boundary conditions	p. 52
4.4 Earth flattening transformation	p. 53
4.5 Two-dimensional wave equation summary	p. 57
4.6 PE for tropospheric radiowave propagation	p. 58
4.7 Path loss	p. 58
4.8 Aperture field	p. 61
5 Rays and modes	p. 63
5.1 Introduction	p. 63
5.2 Ray-tracing	p. 64
5.2.1 Basic principles	p. 64
5.2.2 Piecewise linear profiles	p. 67
5.3 Mode theory	p. 69
5.3.1 Basic principles	p. 69
5.3.2 Piecewise linear profiles	p. 73
6 Oversea propagation	p. 77
6.1 Introduction	p. 77
6.2 Evaporation ducts	p. 77
6.3 Bilinear and trilinear profiles	p. 83
6.4 Range dependence	p. 85
6.5 Measured refractivity data	p. 88
6.6 Atmospheric absorption	p. 92
7 Irregular terrain modelling	p. 95
7.1 Introduction	p. 95
7.2 Terrain models	p. 96
7.2.1 Staircase terrain modelling	p. 96
7.2.2 Piecewise linear terrain	p. 97
7.2.3 Conformal mapping	p. 100
7.3 Finite-difference implementation	p. 102
7.4 Examples	p. 103
7.4.1 Diffraction by the Earth	p. 103
7.4.2 Multiple knife-edge diffraction	p. 104
7.4.3 Combined effects of terrain and ducting	p. 111
8 Domain truncation	p. 115
8.1 Introduction	p. 115
8.2 Absorbing layers	p. 116
8.3 Perfectly matched layer	p. 118
8.4 Non-local boundary conditions	p. 120
8.5 Diffractive non-local boundary conditions	p. 121
8.5.1 Homogeneous medium	p. 121
8.5.2 Linear medium	p. 122
8.5.3 Implementation	p. 125
8.5.4 Examples	p. 128
8.6 Transmitting NLBCs	p. 131
8.6.1 High antennas in linear media	p. 133
8.6.2 Elevated duct example	p. 134
8.7 Split-step incoming energy PE	p. 136
9 Impedance boundary modelling	p. 141
9.1 Introduction	p. 141
9.2 Boundary conditions at a horizontal interface	p. 142
9.2.1 Leontovich boundary conditions	p. 145
9.2.2 Small angle GIBC	p. 147
9.2.3 Reflection coefficient	p. 148
9.3 Finite-difference implementation	p. 149
9.4 Mixed Fourier transform	p. 150
9.4.1 Continuous mixed Fourier transform	p. 150
9.4.2 Discrete mixed Fourier transform	p. 152
9.5 Examples	p. 158
9.5.1 Line-of-sight propagation	p. 158
9.5.2 Surface wave	p. 158
9.5.3 Recovery effect	p. 160
10 Propagation over the rough sea surface	p. 162
10.1 Introduction	p. 162
10.2 Random rough surfaces	p. 163
10.3 Sea surface spectrum	p. 163
10.4 Surface roughness and specular reflection	p. 164
10.5 HF propagation over the rough sea	p. 165
10.5.1 Small perturbation method	p. 165
10.5.2 Propagation at 10 MHz	p. 166
10.6 Ducting propagation over the rough sea surface	p. 167
10.6.1 Roughness reduction factor	p. 167
10.6.2 Estimation of grazing angles	p. 170
10.6.3 Surface duct propagation	p. 171
11 Hybrid methods	p. 175
11.1 Introduction	p. 175
11.2 Radio Physical Optics	p. 176
11.3 Horizontal PE	p. 179
11.3.1 Homogeneous medium	p. 182
11.3.2 Linear medium	p. 183
11.3.3 Spectral decomposition and numerical performance	p. 184
11.3.4 Examples	p. 186
11.4 High antennas	p. 188
11.5 Earth-space paths	p. 192
12 Two-dimensional scattering	p. 199
12.1 Introduction	p. 199
12.2 Pade schemes	p. 201
12.2.1 Pade-(1,1) scheme	p. 205
12.2.2 Pade-(1,0) scheme	p. 207
12.2.3 Pade-(2,1) scheme	p. 211
12.3 Split-step Pade method	p. 214
12.4 Wide-angle NLBCs	p. 218
12.4.1 Split-step Pade implementation	p. 222
12.5 Modelling the scattered field	p. 224
12.5.1 Reflecting facet model	p. 224
12.5.2 Rotating PE method	p. 225
12.5.3 Boundary conditions	p. 227
12.5.4 Initial field	p. 228
12.5.5 Numerical implementation	p. 228
12.5.6 Near-field/far-field considerations	p. 229
12.6 Examples	p. 230
12.6.1 Cylinders	p. 230
12.6.2 L-shaped object	p. 232
12.6.3 2D aircraft shape	p. 233
13 Three-dimensional scattering for the scalar wave equation	p. 237
13.1 Introduction	p. 237
13.2 Axial symmetry parabolic equation	p. 238
13.2.1 Domain truncation	p. 239
13.3 Application to X-ray optics	p. 242
13.4 General 3D case	p. 248
13.4.1 Implementation of the 3D Pade-(1,0) scheme	p. 253
13.4.2 Double pass method	p. 256
13.5 Rough surface modelling	p. 257
13.6 Building scatter	p. 260
13.6.1 Normal incidence	p. 260
13.6.2 Oblique incidence	p. 261
13.6.3 Millimetre-wave building scatter	p. 261
14 Vector PE	p. 267
14.1 Introduction	p. 267
14.2 Vector PE framework	p. 267
14.3 Implementation aspects	p. 271
14.4 Examples	p. 271
14.4.1 Bistatic scattering geometry	p. 271
14.4.2 Circular cylinder	p. 272
14.4.3 Spheres	p. 273
14.4.4 NASA almond	p. 278
14.4.5 F117 aircraft	p. 279
Appendix A Airy functions	p. 283
A.1 The Airy differential equation	p. 283
A.2 Zeros of the Airy function	p. 284
Appendix B Far-field expressions	p. 287
B.1 Two-dimensional case	p. 287
B.1.1 Far-field formulation	p. 288
B.1.2 Near-field/far-field transformation	p. 289
B.1.3 Bistatic RCS	p. 291
B.2 Three-dimensional case	p. 291
B.2.1 Far-field formulation	p. 292
B.2.2 Near-field/far-field transformation	p. 293
B.2.3 Bistatic RCS	p. 293
Appendix C Theoretical derivation of mode series	p. 294
C.1 Introduction	p. 294
C.2 Modes for linear profiles	p. 295
C.3 Functional analysis framework	p. 297
C.3.1 Complex heights	p. 297
C.3.2 The resolvent of T	p. 298
C.4 Mode expansions for the linear case	p. 300
C.4.1 The self-adjoint case	p. 300
C.4.2 Connecting the solution	p. 303
C.4.3 Finite impedance case	p. 304
C.5 Perturbations of the linear case	p. 310
Appendix D Energy conservation	p. 314
D.1 Two-dimensional problems	p. 314
D.2 Three-dimensional problems	p. 317
D.3 Unicity of PE solutions	p. 318
Bibliography	p. 319
Index	p. 333

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