This book presents a comprehensive treatment of ground penetrating radar using both forward and inverse scattering mathematical techniques. Use of field data instead of laboratory data enables readers to envision real-life underground imaging; a full color insert further clarifies understanding. Along with considering the practical problem of achieving interpretable underground images, this book also features significant coverage of the problem's mathematical background. This twofold approach provides a resource that will appeal both to application oriented geologists and testing specialists, and to more research-oriented physicists and engineers
Foreword xiii Acknowledgments xvii About the Author xix Contributors xxi 1 INTRODUCTION TO GPR PROSPECTING 1
1.1 What Is a GPR? 1
1.2 GPR Systems and GPR Signals 4
1.3 GPR Application Fields 5
1.4 Measurement Configurations, Bands, and Polarizations 6
1.5 GPR Data Processing 8 2 CHARACTERIZATION OF THE HOST MEDIUM 10
2.1 The Characteristics of the Host Medium 10
2.2 The Measure of the Propagation Velocity in a Masonry 11
2.3 The Measure of the Propagation Velocity in a Homogeneous Soil 13
2.3.1 Interfacial Data in Common Offset Mode with a Null Offset: The Case of a Point-like Target 13
2.3.2 Interfacial Data in Common Offset Mode with a Null Offset: The Case of a Circular Target 17
2.3.3 Interfacial Data in Common Offset Mode with a Non-null Offset: The Case of a Point-like Target 18
2.3.4 Noninterfacial Data in Common Offset Mode with a Null Offset: The Case of a Point-like Target 22
2.3.5 Interfacial Data in Common Midpoint (CMP) Mode 25
2.4 Lossy, Magnetic, and Dispersive Media 27 Questions 31 3 GPR DATA SAMPLING: FREQUENCY AND TIME STEPS 32
3.1 Stepped Frequency GPR Systems: The Problem of the Aliasing and the Frequency Step 32
3.2 Shape and Thickness of the GPR Pulses 36
3.3 Stepped Frequency GPR Systems: The Problem of the Demodulation and the Frequency Step 40
3.4 Aliasing and Time Step for Pulsed GPR Systems 45 Questions 47 4 THE 2D SCATTERING EQUATIONS FOR DIELECTRIC TARGETS 48
4.1 Preliminary Remarks 48
4.2 Derivation of the Scattering Equations Without Considering the Effect of the Antennas 51
4.3 Calculation of the Incident Field Radiated by a Filamentary Current 61
4.4 The Plane Wave Spectrum of an Electromagnetic Source in a Homogeneous Space 61
4.5 The Insertion of the Source Characteristics in the Scattering Equations 65
4.6 The Far Field in a Homogeneous Lossless Space in Terms of Plane Wave Spectrum 69
4.7 The Effective Length of an Electromagnetic Source in a Homogeneous Space 73
4.8 The Insertion of the Receiver Characteristics in the Scattering Equations 75 Questions 77 5 THE 2D SCATTERING EQUATIONS FOR MAGNETIC TARGETS 79
5.1 The Scattering Equations with Only Magnetic Anomalies 79
5.2 The Contribution of the x-Component of the Fitzgerald Vector 83
5.3 The Contribution of the z-Component of the Fitzgerald Vector 88
5.4 The Joined Contribution of Both the x- and z-Components of the Fitzgerald Vector 93
5.5 The Case with Both Dielectric and Magnetic Anomalies 94 Questions 95 6 ILL-POSEDNESS AND NONLINEARITY 96
6.1 Electromagnetic Inverse Scattering 96
6.2 Ill-Posedness 97
6.3 Nonlinearity 97
6.4 The Ill-Posedness of the Inverse Scattering Problem 100
6.5 The Nonlinearity of the Inverse Scattering Problem 103 Questions 103 7 EXTRACTION OF THE SCATTERED FIELD DATA FROM THE GPR DATA 105
7.1 Zero Timing 105
7.2 Muting of Interface Contributions 106
7.3 The Differential Configuration 110
7.4 The Background Removal 111 Questions 115 8 THE BORN APPROXIMATION 116
8.1 The Classical Born Approximation 116
8.2 The Born Approximation in the Presence of Magnetic Targets 119
8.3 Weak and Nonweak Scattering Objects 120 Questions 121 9 DIFFRACTION TOMOGRAPHY 122
9.1 Introduction to Diffraction Tomography 122
9.2 Diffraction Tomography for Dielectric Targets 123
9.3 Diffraction Tomography for Dielectric Targets Seen Under a Limited View Angle 130
9.4 The Effective Maximum and Minimum View Angle 140
9.5 Horizontal Resolution 142
9.6 Vertical Resolution 145
9.7 Spatial Step 147
9.8 Frequency Step 148
9.9 Time Step 149
9.10 The Effect of a Non-null Height of the Observation Line 150
9.11 The Effect of the Radiation Characteristics of the Antennas 156
9.12 DT Relationship in the Presence of Magnetic Targets 158
9.13 DT Relationship for a Differential Configuration 160
9.14 DT Relationship in the Presence of Background Removal 163 Questions 168 10 TWO-DIMENSIONAL MIGRATION ALGORITHMS 169
10.1 Migration in the Frequency Domain 169
10.2 Migration in the Time Domain (Raffaele Persico and Raffaele Solimene) 175 Questions 181 11 THREE-DIMENSIONAL SCATTERING EQUATIONS 182 Lorenzo Lo Monte, Raffaele Persico, and Raffaele Solimene
11.1 Scattering in Three Dimensions: Redefinition of the Main Symbols 182
11.2 The Scattering Equations in 3D 184
11.3 Three-Dimensional Green s Functions 184
11.4 The Incident Field 185
11.5 Homogeneous 3D Green s Functions 187
11.6 The Plane Wave Spectrum of a 3D Homogeneous Green s Fucntion 192
11.7 Half-Space Green s Functions 197 Questions 204 12 THREE-DIMENSIONAL DIFFRACTION TOMOGRAPHY 205
12.1 Born Approximation and DT in 3D 205
12.2 Ideal and Limited-View-Angle 3D Retrievable Spectral Sets 210
12.3 Spatial Step and Transect 212
12.4 Horizontal Resolution (Raffaele Persico and Raffaele Solimene) 213
12.5 Vertical Resolution, Frequency and Time Steps 217 Questions 218 13 THREE-DIMENSIONAL MIGRATION ALGORITHMS 219
13.1 3D Migration Formulas in the Frequency Domain 219
13.2 3D Migration Formulas in the Time Domain 222
13.3 3D Versus 2D Migration Formulas in the Time Domain 226 Questions 228 14 THE SINGULAR VALUE DECOMPOSITION 229
14.1 The Method of Moments 229
14.2 Reminders About Eigenvalues and Eigenvectors 231
14.3 The Singular Value Decomposition 234
14.4 The Study of the Inverse Scattering Relationship by Means of the SVD 238 Questions 241 15 NUMERICAL AND EXPERIMENTAL EXAMPLES 242
15.1 Examples with Regard to the Measure of the Propagation Velocity 242
15.1.1 Common Offset Interfacial Data with Null Offset on a Homogeneous Soil 242
15.1.2 Common Offset Interfacial Data on a Wall, Neglecting the Offset Between the Antennas 245
15.1.3 Interfacial Common Offset Data on a Homogeneous Soil: The Effect on the Offset Between the Antennas 247
15.1.4 Noninterfacial Common Offset Data with a Null Offset Between the Antennas 249
15.1.5 Common Midpoint Data 250
15.2 Exercises on Spatial Step and Horizontal Resolution 252
15.3 Exercises on Frequency Step and Vertical Resolution 264
15.4 Exercises on the Number of Trial Unknowns 271
15.5 Exercises on Spectral and Spatial Contents 274
15.6 Exercises on the Effect of the Height of the Observation Line 280
15.7 Exercises on the Effect of the Extent of the Investigation Domain 284
15.8 Exercises on the Effects of the Background Removal 295
15.9 2D and 3D Migration Examples with a Single Set and Two Crossed Sets of B-Scans (Marcello Ciminale, Giovanni Leucci, Loredana Matera, and Raffaele Persico) 304
15.10 2D and 3D Inversion Examples (Ilaria Catapano and Raffaele Persico) 311
APPENDICES 327
APPENDIX A (Raffaele Persico and Raffaele Solimene) 329
APPENDIX B 334
APPENDIX C 335
APPENDIX D 337
APPENDIX E 340
APPENDIX F (Raffaele Persico and Raffaele Solimene) 346
APPENDIX G: ANSWERS TO QUESTIONS 349
References 358
Index 365
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