 | Franceschetti G. Scattering, natural surfaces, and fractals / G.Franceschetti, D.Riccio. - Burlington; San-Diego; London: Elsevier Academic Press, 2007. - xv, 282 p.: ill. - Ref.: p.277-482. - ISBN-10 0-12-265655-5; ISBN-13 978-0-12-265655-2
|
Preface ...................................................... xiii
1 The Scattering Problem ....................................... 1
1.1 Introduction and Chapter Outline ........................ 1
1.2 The Scattering-Problem Definition ....................... 2
1.3 Motivations ............................................. 3
1.4 Surface Models and Electromagnetic Methods .............. 4
1.5 Deterministic versus Stochastic Models for the Natural
Surfaces ................................................ 5
1.5.1 Surface Deterministic Models ..................... 5
1.5.2 Surface Stochastic Models ........................ 6
1.6 Deterministic versus Stochastic Evaluation for the
Scattered Field ......................................... 9
1.6.1 Scattered-Field Deterministic Descriptions ....... 9
1.6.2 Scattered-Field Stochastic Descriptions ......... 10
1.7 Analytic versus Numerical Evaluation of the Scattered
Field .................................................. 12
1.8 Closed-Form Evaluation of the Electromagnetic Field
Scattered from a Natural Surface ....................... 14
1.9 Book Outline ........................................... 18
1.10 References and Further Readings ........................ 19
2 Surface Classical Models .................................... 21
2.1 Introduction and Chapter Outline ....................... 21
2.2 Fundamentals of Stochastic Processes ................... 22
2.2.1 Stochastic Processes: Definition ................ 22
2.2.2 Stochastic Processes: Relevant Averages ......... 23
2.2.3 Stochastic Processes: a Relevant Property ....... 25
2.3 Spectral Characterization of Stochastic Processes ...... 26
2.4 Isotropic Surfaces ..................................... 30
2.5 Classical Models for Natural Surfaces: First-Order
Stochastic Characterization ............................ 33
2.6 Classical Models for Natural Surfaces: Second-Order
Stochastic Characterization ............................ 34
2.7 Physical Counterpart of Natural-Surfaces Classical
Parameters ............................................. 40
2.7.1 Standard Deviation .............................. 41
2.7.2 Correlation Length .............................. 41
2.8 Surface Classical Models Selection for
Electromagnetic Scattering ............................ 44
2.9 References and Further Readings ........................ 44
Appendix 2.A Surface Classical Models ....................... 45
2.A.1 Gaussian Autocorrelation ........................ 46
2.A.2 Exponential Autocorrelation ..................... 47
2.A.3 Intermediate Gaussian-Exponential
Autocorrelation ................................. 48
2.A.4 Power-Law Autocorrelation ....................... 50
2.A.5 Multiscale Gaussian Autocorrelation ............. 51
2.A.6 Multiscale Exponential Autocorrelation .......... 55
2.A.7 Mixed Gaussian-Exponential Autocorrelation ...... 56
3 Surface Fractal Models ...................................... 61
3.1 Introduction and Chapter Outline ....................... 61
3.2 Fundamentals of Fractal Sets ........................... 63
3.2.1 Hausdorff Measure ............................... 64
3.2.2 Fractal Dimension ............................... 65
3.2.3 Scaling Properties .............................. 67
3.3 Mathematical versus Physical Fractal Sets .............. 68
3.4 Deterministic versus Stochastic Fractal Description
of Natural Surfaces .................................... 71
3.5 Fractional Brownian Motion Process ..................... 72
3.5.1 Mathematical fBm Processes ...................... 72
3.5.2 Physical fBm Processes .......................... 91
3.6. Weierstrass-Mandelbrot Function ........................... 94
3.6.1 Mathematical WM Functions ....................... 94
3.6.2 Physical WM Functions ........................... 96
3.7 Connection between fBm and WM Models ................... 98
3.8 A Chosen Reference Fractal Surface for the
Scattering Problem .................................... 100
3.9 Fractal-Surface Models and their Comparison with
Classical Ones ........................................ 101
3.9.1 Classical Parameters for the fBm Process ....... 103
3.9.2 Classical Parameters for the WM Function ....... 105
3.10 References and Further Readings ....................... 106
Appendix 3.A Generalized Functions ......................... 106
Appendix 3.B Space-Frequency and Space-Scale Analysis of
Nonstationary Signals ................................. 107
3.B.1 Introduction ................................... 107
3.B.2 fBm Wigner-Ville Spectrum ...................... 108
3.B.3 fBm and Wavelet Approach ....................... 110
4 Analytic Formulations of Electromagnetic Scattering ........ 115
4.1 Introduction and Chapter Outline ...................... 115
4.2 Maxwell Equations ..................................... 116
4.3 The Integral-Equation Method .......................... 118
4.4 Incident and Scattered-Field Coordinate-Reference
Systems ............................................... 125
4.5 The Kirchhoff Approximation ........................... 128
4.6 Physical-Optics Solution .............................. 132
4.7 Extended-Boundary-Condition Method .................... 134
4.8 Small-Perturbation Method ............................. 138
4.9 References and Further Readings ....................... 140
5 Scattering from Weierstrass-Mandelbrot Surfaces:
Physical-Optics Solution ................................... 143
5.1 Introduction and Chapter Outline ...................... 143
5.2 Analytic Derivation of the Scattered Field ............ 144
5.3 Scattered-Field Structure ............................. 148
5.3.1 Number of Modes Significantly Contributing
to the Scattered Field ......................... 149
5.3.2 Mode Directions of Propagation ................. 153
5.3.3 Modes Amplitude and Phase ...................... 155
5.4 Limits of Validity .................................... 156
5.5 Influence of Fractal and Electromagnetic Parameters
over the Scattered Field .............................. 157
5.5.1 The Role of the Fundamental-Tone Wavenumber .... 159
5.5.2 The Role of the Tone Wave-Number Spacing
Coefficient .................................... 162
5.5.3 The Role of the Number of Tones ................ 164
5.5.4 The Role of the Overall Amplitude-Scaling
Factor ......................................... 166
5.5.5 The Role of the Hurst Exponent ................. 168
5.6 Statistics of the Scattered Field ..................... 169
5.7 References and Further Readings ....................... 170
6 Scattering from Fractional Brownian Surfaces:
Physical-Optics Solution ................................... 171
6.1 Introduction and Chapter Outline ...................... 171
6.2 Scattered Power-Density Evaluation .................... 172
6.2.1 Persistent fBm ................................. 175
6.2.2 Antipersistent fBm ............................. 176
6.3 Scattered Power Density ............................... 179
6.4 Scattered Power Density: Special Cases ................ 180
6.4.1 Scattering in the Specular Direction ........... 181
6.4.2 Brownian Surfaces (H=1/2) ...................... 181
6.4.3 Marginally Fractal Surfaces (H → 1) ............ 181
6.4.4 Quasi-Smooth Surfaces (kT << 1) ................ 182
6.5 Backscattering Coefficient ............................ 182
6.6 Validity Limits ....................................... 184
6.7 Influence of Fractal and Electromagnetic Parameters
over the Scattered Field .............................. 186
6.7.1 The Role of the Spectral Amplitude ............. 188
6.7.2 The Role of the Hurst Exponent ................. 189
6.7.3 The Role of the Electromagnetic Wavelength ..... 191
6.8 References and Further Readings ....................... 191
7 Scattering from Weierstrass-Mandelbrot Profiles: Extended-
Boundary-Condition Method .................................. 193
7.1 Introduction and Chapter Outline ...................... 193
7.2 Profile Model ......................................... 195
7.3 Setup of the Extended-Boundary-Condition Method ....... 196
7.3.1 Incident Field ................................. 196
7.3.2 Integral Equations ............................. 197
7.3.3 Surface-Field Expansions for WM Profiles ....... 201
7.4 Surface-Fields Evaluation ............................. 204
7.5 Fields Expansions ..................................... 205
7.6 EBCM Equations in Matrix Form ......................... 207
7.7 Matrix-Equations Solution ............................. 209
7.8 Matrices Organizations ................................ 210
7.9 Scattering-Modes Superposition, Matrices Truncation,
and Ill-Conditioning .................................. 213
7.10 Influence of Fractal and Electromagnetic Parameters
over the Scattered Field .............................. 215
7.10.1 The Role of the Fundamental-Tone Wavenumber .... 219
7.10.2 The Role of the Tone Wavenumber Spacing
Coefficient .................................... 219
7.10.3 The Role of the Number of Tones ................ 223
7.10.4 The Role of the Overall Amplitude-Scaling
Factor ......................................... 225
7.10.5 The Role of the Hurst Exponent ................. 227
7.11 References and Further Readings ....................... 229
Appendix 7.A Evaluation of the Dirichlet- and Neumann-
Type Integrals ........................................ 230
7.A.1 Evaluation of the Dirichlet-Type Integral ...... 231
7.A.2 Evaluation of the Neumann-Type Integral ........ 234
8 Scattering from Fractional Brownian Surfaces: Small-
Perturbation Method ........................................ 239
8.1 Introduction and Chapter Outline ...................... 239
8.2 Rationale of the SPM Solution ......................... 240
8.3 Extended Boundary Condition Method in the
Transformed Domain .................................... 243
8.4 Set up the Small Perturbation Method .................. 248
8.5 An Appropriate Coordinate System ...................... 251
8.6 Zero-order Solution ................................... 252
8.7 First-order Solution .................................. 256
8.8 Small Perturbation Method Limits of Validity .......... 260
8.9 Influence of Fractal and Electromagnetic Parameters
Over the Scattered Field .............................. 262
8.9.1 The Role of the Spectral Amplitude ............. 263
8.9.2 The Role of the Hurst Exponent ................. 264
8.9.3 The Role of the Electromagnetic Wavelength ..... 266
8.10 References and Further Readings ....................... 267
Appendix A: Mathematical Formulae ............................. 269
Appendix B: Glossary .......................................... 273
Appendix C: References ........................................ 277
Index ......................................................... 283
|
|
