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5 5.1 5.2 5.3
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Steepest Descent Method Real Symmetric Positive Definite Matrix General Real Matrix and Complex Matrix References and Additional Readings
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1.1 1.2 1.3 1.4 2 2.1 2.2 2.3
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Scattering of Waves Obliquely Incident on Periodic Rough Surfaces: Integral Equation Approach
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Preliminaries 3-D Green's Function in 3-D Lattices 3-D Green's Function in 2-D Lattices Numerical Results References and Additional Readings
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1 1.1 1.2 1.3 1.4
Perfect Electric Conductor (Non-Penetrable Surface) 114
Integral Equation Matrix Equation: Dirichlet Boundary Condition (EFIE for TE Case) Tapering of Incident Waves and Calculation of Scattered Waves Random Rough Surface Generation 1.4.1 Gaussian Rough Surface 114 116 118 124 124 132 134
1.4.2 Fractal Rough Surface
Neumann Boundary Condition (MFIE for TM Case)
Two-Media Problem
TE and TM Waves Absorptivity, Emissivity and Reflectivity Impedance Matrix Elements: Numerical Integrations Simulation Results 2.4.1 Gaussian Surface and Comparisons with Analytical Methods
139 141 143 145 145 150
2.1 2.2 2.3 2.4
2.4.2 Dirichlet Case of Gaussian Surface with Ocean Spectrum and Fractal Surface
2.4.3 Bistatic Scattering for Two Media Problem with Ocean Spectrum 151
Topics of Numerical Simulations
Periodic Boundary Condition MFIE for TE Case of PEC Impedance Boundary Condition
154 158 161
3.1 3.2 3.3
Microwave Emission of Rough Ocean Surfaces Waves Scattering from Real-Life Rough Surface Profiles
Introduction Rough Surface Generated by Three Methods
163 166
166 167
5.1 5.2
Numerical Results of the Three Methods References and Additional Readings
CHAPTER 5 FAST COMPUTATIONAL METHODS FOR SOLVING ROUGH SURFACE SCATTERING PROBLEMS 1 Banded Matrix Canonical Grid Method for Two-Dimensional Scattering for PEC Case Introduction Formulation and Computational Procedure Product of a Weak Matrix and a Surface Unknown Column Vector Convergence and Neighborhood Distance Results of Composite Surfaces and Grazing Angle Problems
177 179
1.1 1.2 1.3 1.4 1.5
179 180 187 188 189
Physics-Based Two-Grid Method for Lossy Dielectric Surfaces 196 Introduction Formulation and Single-Grid Implementation Physics-Based Two-Grid Method Combined with Banded Matrix Iterative Approach/Canonical Grid Method Bistatic Scattering Coefficient and Emissivity Steepest Descent Fast Multipole Method Steepest Descent Path for Green's Function Multi-Level Discretization of Angles and Interpolation Steepest Descent Expression of Multi-Level Impedance Matrix Elements SDFMM Algorithm Numerical Results Method of Ordered Multiple Interactions (MOMI) Matrix Equations Based on MFIE for TE and TM Waves for PEC
2.1 2.2 2.3 2.4
196 198 200 203
3.1 3.2 3.3 3.4 3.5 3.6
213 222 226 235 242
Multi-Level Impedance Matrix Decomposition and Grouping 216
4.2 4.3
Iterative Approach Numerical Results
245 247
5.1 5.2 5.3 5.4
Physics-Based Two-Grid Method Combined with the Multilevel Fast Multipole Method
Single Grid and PBTG Computational Complexity of the Combined Algorithm of the PBTG with the MLFMM Gaussian Rough Surfaces and CPU Comparison Non-Gaussian Surfaces
249 252 254 257
1 1.1
267 270
270 270 273 277 278 278 281 286
Scattering by Non-Penetrable Media
Scalar Wave Scattering 1.1.1 Formulation and Numerical Method 1.1.2 Results and Discussion
1.1.3 Convergence of SMFSIA
1.2 Electromagnetic Wave Scattering by Perfectly Conducting Surfaces 1.2.1 Surface Integral Equation
1.2.2 Surface Integral Equation for Rough Surface Scattering 280
1.2.3 Computation Methods
1.2.4 Numerical Simulation Results
2.1 2.2
Integral Equations for Dielectric Surfaces
Electromagnetic Fields with Electric and Magnetic Sources Physical Problem and Equivalent Exterior and Interior Problems
293 296 296
2.2.1 Equivalent Exterior Problem, Equivalent Currents and Integral Equations
2.2.2 Equivalent Interior Problem, Equivalent Currents and Integral Equations
298 300
Surface Integral Equations for Equivalent Surface Currents, Tangential and Normal Components of Fields
Two-Dimensional Rough Dielectric Surfaces with Sparse Matrix Canonical Grid Method
Integral Equation and SMCG Method Numerical Results of Bistatic Scattering Coefficient
3.1 3.2
304 318
Scattering by Lossy Dielectric Surfaces with PBTG Method
Introduction Formulation and Single Grid Implementation Physics-Based Two-Grid Method Numerical Results and Comparison with Second Order Perturbation Method Numerical Simulations of Emissivity of Soils with Rough Surfaces at Microwave Frequencies
4.1 4.2 4.3 4.4 4.5
326 328 329 334 343
Four Stokes Parameters Based on Tangential Surface Fields 350 Parallel Implementation of SMCG on Low Cost Beowulf System
Introduction Low-Cost Beowulf Cluster
6.1 6.2 6.3 6.4
354 355
Parallel Implementation of the SMCG Method and the PBTG Method 356 Numerical Results
References and Additional Readings CHAPTER 7 VOLUME SCATTERING SIMULATIONS
371 373
Combining Simulations of Collective Volume Scattering Effects with Radiative Transfer Theory
2 2.1 2.2 2.3 3 3.1 3.2 4
Foldy-Lax Self-Consistent Multiple Scattering Equations Final Exciting Field and Multiple Scattering Equation Foldy-Lax Equations for Point Scatterers The N-Particle Scattering Amplitude Analytical Solutions of Point Scatterers Phase Function and Extinction Coefficient for Uniformly Distributed Point Scatterers Scattering by Collection of Clusters Monte Carlo Simulation Results of Point Scatterers References and Additional Readings
376 376 379 382 382 382 389 392
403 404
404 406 409
1.1 1.2 1.3
Pair Distribution Functions and Structure Factors Introduction Percus-Yevick Equation and Pair Distribution Function for Hard Spheres Calculation of Structure Factor and Pair Distribution Function Percus-Yevick Pair Distribution Functions for Multiple Sizes Monte Carlo Simulations of Particle Positions Metropolis Monte Carlo Technique Sequential Addition Method Numerical Results Sticky Particles Percus-Yevick Pair Distribution Function for Sticky Spheres Pair Distribution Function of Adhesive Sphere Mixture Monte Carlo Simulation of Adhesive Spheres
411 414
415 418 418
3.1 3.2 3.3
4.1 4.2 4.3
424 429 434