Chapter 1. Statistical Preliminaries ............................ 1
1.1. Introduction ............................................... 1
1.2. Random Variables ........................................... 1
1.3. Transformation of Variables ................................ 4
1.4. Wiener-Khinchin Theorem .................................... 7
1.5. The Karhunen-Loeve Expansion ............................... 8
1.6. Statistical Independence .................................. 10
1.7. Characteristic Functions and Generating Functions ......... 10
1.8. Detection ................................................. 11
References ..................................................... 14
Chapter 2. The Gaussian Process ................................ 15
2.1. Introduction .............................................. 15
2.2. Independent Gaussian Variables ............................ 15
2.3. Correlated Gaussian Variables ............................. 16
2.4. Higher-Order Correlations ................................. 17
2.5. Gaussian Processes ........................................ 18
2.6. Complex Gaussian Processes ................................ 20
2.7. Joint Statistical Properties .............................. 21
2.8. Properties of the Derivative of a Complex Gaussian
Process ................................................... 26
2.9. Joint Phase Derivative Statistics ......................... 29
References ..................................................... 33
Chapter 3. Processes Derived from Gaussian Noise ............... 35
3.1. Introduction .............................................. 35
3.2. Rice Variables ............................................ 35
3.3. Rice Processes ............................................ 38
3.4. Gamma Variables ........................................... 40
3.5. Gamma Processes ........................................... 42
3.6. Statistics of the Derivative of a Gamma Process ........... 44
3.7. Compound Variables ........................................ 45
3.8. Other Commonly Encountered Distributions .................. 47
References ..................................................... 51
Chapter 4. Scattering by a Collection of Discrete Objects:
The Random Walk Model .............................. 53
4.1. Introduction .............................................. 53
4.2. The Incident Wave ......................................... 53
4.3. Moments of the Field Scattered by a Fixed Number of
Small Particles ........................................... 55
4.4. The Probability Distribution of the Scattered Wave ........ 58
4.5. Variations in Step Length: Illumination by a Gaussian
Beam ...................................................... 62
4.6. The Effect of Variations in Step Number ................... 63
4.7. Intensity Correlation ..................................... 66
4.8. Partially Developed Speckle ............................... 73
References ..................................................... 78
Chapter 5. Scattering by Continuous Media: Phase Screen
Models .............................................. 81
5.1. Introduction .............................................. 81
5.2. Scattering Geometries ..................................... 83
5.3. Time-Dependent Scattering ................................. 85
5.4. Scattering into the Fresnel Region ........................ 86
5.5. Fraunhofer Scattering ..................................... 90
5.6. Scattering in Non-Gaussian Regimes ........................ 93
5.7. Surface Scattering ........................................ 94
References ..................................................... 98
Chapter 6. Scattering by Smoothly Varying Phase Screens ....... 101
6.1. Introduction ............................................. 101
6.2. "Smooth" Models for the Phase Correlation Function ....... 102
6.3. Qualitative Features of Scattering by Smoothly Varying
Phase Screens ............................................ 103
6.4. Calculation of the Scintillation Index in the Fresnel
Region ................................................... 105
6.5. Predicted Behavior of the Scintillation Index in the
Fresnel Region ........................................... 109
6.6. Higher-Order Statistics in the Fresnel Region ............ 1ll
6.7. Spatial Coherence Properties in the Fresnel Region ....... 113
6.8. Calculation of the Scintillation Index in the
Fraunhofer Region ........................................ 114
6.9. Predicted Behavior of the Scintillation Index in the
Far Field ................................................ 118
6.10.Higher-Order Statistical Properties in the Far Field ..... 120
6.11.Coherence Properties in the Far Field .................... 121
6.12.Phase Statistics ......................................... 126
References .................................................... 129
Chapter 7. Scattering by Fractal Phase Screens ................ 131
7.1. Introduction ............................................. 131
7.2. Scattering into the Fresnel Region by a Fractal Phase
Screen ................................................... 133
7.3. Scattering into the Fraunhofer Region by a Fractal
Phase Screen ............................................. 139
7.4. Subfractal Phase Screens ................................. 141
7.5. Ray Density Fluctuations beyond a Subfractal Phase
Screen ................................................... 144
7.6. Coherence Properties of the Intensity Beyond a
Subfractal Screen ........................................ 146
7.7. Outer Scale Effects ...................................... 148
7.8. Scattering into the Fraunhofer Region by a Subfractal
Screen ................................................... 151
7.9. Concluding Remarks ....................................... 154
References .................................................... 156
Chapter 8. Other Phase Screen Models .......................... 159
8.1. Introduction ............................................. 159
8.2. Scattering by Smoothly Varying Gamma Distributed
Phase Screens ............................................ 160
8.3. Scattering by Fractal Gamma Distributed Phase Screens .... 162
8.4. Telegraph Wave Phase Screens ............................. 164
8.5. Scattering by Telegraph Wave Phase Screens ............... 167
8.6. Phase Statistics ......................................... 171
8.7. Concluding Remarks ....................................... 172
References .................................................... 173
Chapter 9. Propagation through Extended Inhomogeneous
Media .............................................. 175
9.1. Introduction ............................................. 175
9.2. Single Phase Screen Approximation ........................ 175
9.3. Power-Law Models for the Refractive Index Spectrum ....... 177
9.4. Multiple Phase Screens ................................... 180
9.5. Propagation of Electromagnetic Waves through
Turbulence ............................................... 184
9.6. Intensity Fluctuations ................................... 187
References .................................................... 192
Chapter 10.Multiple Scattering: Fluctuations in Double
Passage and Multipath Scattering Geometries ........ 195
10.1.Introduction ............................................. 195
10.2.Multiple Scattering in Particulates ...................... 196
10.3.Enhanced Backscattering through Continuous Media ......... 198
10.4.A Phase Screen Model for Enhanced Backscattering ......... 199
10.5.Fluctuations in Double Passage Configurations ............ 205
10.6.Multipath Near Surfaces .................................. 208
References .................................................... 210
Chapter 11.Vector Scattering: Polarization Fluctuations ....... 213
11.1.Introduction ............................................. 213
11.2.The Polarization Characteristics of Gaussian Speckle ..... 214
11.3.Non-Gaussian Polarization Effects in Scattering by
Particles ................................................ 217
11.4.Correlation of Stokes Parameters in Particle
Scattering ............................................... 220
11.5.Scattering from Particles Near an Interface .............. 223
11.6.Polarization Fluctuations: Particles Near an Interface ... 225
References .................................................... 227
Chapter 12.K-Distributed Noise ................................ 229
12.1.Introduction ............................................. 229
12.2.Experimental Evidence .................................... 230
12.3.A Population Model for Scatterer Number Fluctuations ..... 233
12.4.Properties of the Scattered Intensity .................... 237
12.5.Related Distributions .................................... 238
12.6.Statistical Mechanics .................................... 240
References .................................................... 242
Chapter 13.Measurement and Detection .......................... 245
13.1.Introduction ............................................. 245
13.2.Temporal Averaging of a Gamma-Lorentzian Process ......... 246
13.3.Approximations for the Effect of Temporal and
Spatial Integration ...................................... 247
13.4.Averaging Signals with More than One Scale ............... 249
13.5.Enhancement of Fluctuations Caused by Filtering .......... 250
13.6.The Effect of Finite Dynamic Range ....................... 251
13.7.The Effect of Finite Measurement Time .................... 254
13.8.Noise in Frequency Demodulation .......................... 258
13.9.Detection ................................................ 259
13.10.Quantum Limited Measurements ............................ 264
References .................................................... 269
Chapter 14.Numerical Techniques ............................... 271
14.1.Introduction ............................................. 271
14.2.The Transformation of Random Numbers ..................... 271
14.3.Gaussian Random Numbers .................................. 272
14.4. The Telegraph Wave ...................................... 273
14.5. The Gaussian Random Process ............................. 274
14.6. Non-Gaussian Processes .................................. 282
14.7. Simulation of Wave Propagation .......................... 287
14.8. Multiple Phase Screens .................................. 291
References .................................................... 294
Index ......................................................... 297
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