Geophysical electromagnetic theory and methods

Cover......Page 1
Title page......Page 2
Copyright page......Page 3
Contents......Page 4
Preface......Page 11
1.1.1 Concept of the physical field......Page 16
1.1.2 Dot (scalar) and cross (vector) products of vectors......Page 19
1.1.3 Vector differential operators......Page 20
1.1.4 Differentiation of the products of scalar and vector fields......Page 24
1.2.1 Concept of work and flux of a field......Page 25
1.2.2 Gauss's theorem and its vector formulations......Page 28
1.2.3 Stokes's theorem and its vector formulations......Page 30
1.2.4 Green's formulas......Page 31
1.3.1 Concept of the differential form......Page 32
1.3.2 Exterior (wedge) product of the linear forms......Page 36
1.3.3 Canonical representations of the differential forms in three-dimensional Euclidean space......Page 37
1.3.4 The exterior derivative......Page 38
References and Recommended Reading......Page 41
CHAPTER 2 Foundations of Field Theory......Page 42
2.1.1 Harmonic functions; Liouville's theorem......Page 43
2.1.2 Uniqueness of determination of the scalar field by its gradient and the vector field by its divergence and curl......Page 45
2.1.3 Field generation conditions......Page 47
2.1.4 Sources of the field and their physical meaning......Page 49
2.1.5 Vortices of the field and their physical meaning......Page 51
2.1.6 Source field and vortex field......Page 54
2.2.1 Poisson's equations for scalar and vector fields......Page 55
2.2.2 Point source; Dirac singular function......Page 57
2.2.3 Fundamental Green's function for the Laplace equation......Page 59
2.3.1 Scalar potential of the source field......Page 62
2.3.2 Vector potential of the vortex field......Page 63
2.3.3 Helmholtz theorem and classification of the vector fields......Page 65
2.4.1 Nonstationary vector fields and differential forms in four-dimensional space E4......Page 67
2.4.2 Differential form equations......Page 68
2.4.3 Ampere-type differential forms and a continuity equation......Page 71
2.4.4 Faraday-type differential forms and the four-potential......Page 72
2.4.5 Nonstationary vector field equations......Page 73
References and Recommended Reading......Page 74
CHAPTER 3 Electromagnetic Field Equations......Page 78
3.1.1 Basic equations in the theory of electromagnetic fields......Page 80
3.1.2 Physical interpretation of Maxwell's equations......Page 84
3.1.3 Boundary conditions for the vector field......Page 90
3.1.4 The field in a homogeneous medium......Page 95
3.2 Time-Harmonic Electromagnetic Field......Page 96
3.3 Electromagnetic Energy and Poynting's Theorem......Page 99
3.3.2 Poynting's theorem in the time domain......Page 100
3.3.3 Energy inequality in the time domain......Page 102
3.3.4 Poynting's theorem in the frequency domain......Page 104
3.4.1 Green's tensors in the frequency domain......Page 106
3.4.2 Green's tensors in the time domain......Page 108
3.5.1 Lorentz lemma......Page 109
3.5.2 Reciprocity relations for the Green's tensors and electromagnetic fields......Page 111
3.5.3 Electromagnetic Green's tensor representation theorems......Page 113
References and Recommended Reading......Page 116
CHAPTER 4 Models of Electromagnetic Induction in the Earth......Page 118
4.1 Models of Electromagnetic Fields......Page 119
4.2 Static Electromagnetic Fields......Page 120
4.2.1 Electrostatic fields and electrostatic potentials......Page 121
4.2.2 Boundary conditions for electrostatic potential......Page 124
4.2.3 Calculation of the electrostatic field of a specified charge distribution......Page 125
4.2.4 Analogy between constant current fields and electrostatic fields......Page 126
4.2.5 Direct current flow, associated magnetic field, and the Biot-Savart law......Page 130
4.2.6 Point and dipole sources on a uniform earth......Page 133
4.2.7 DC potential in an anisotropic earth......Page 139
4.3 Electromagnetic Field Diffusion in Conductive Media......Page 143
4.3.1 Monochromatic quasi-static EM fields......Page 144
4.3.2 Plane electromagnetic waves in a homogeneous medium......Page 146
4.3.3 Electromagnetic potentials......Page 153
4.3.4 Quasi-stationary field of a dipole source in a homogeneous medium......Page 155
4.3.5 Spherical electromagnetic waves......Page 159
4.4 Electromagnetic Waves......Page 163
References and Recommended Reading......Page 164
CHAPTER 5 Electromagnetic Fields in Horizontally Stratified Media......Page 166
5.1.1 Plane electromagnetic wave in a horizontally stratified medium......Page 167
5.1.2 Low-frequency behavior of wave impedance......Page 175
5.1.3 Definition of frequency windows......Page 178
5.2.1 Fourier transform in the spatial domain......Page 181
5.2.2 Point source of the DC field in horizontally stratified medium......Page 185
5.2.3 Electric field of the point source in a layered earth......Page 196
5.2.4 Electrical dipole source of the DC field in a horizontally layered medium......Page 200
5.2.5 Expressions for electric fields in a horizontally layered medium using the Hankel transform......Page 201
5.3 Electromagnetic Field of an Arbitrary System of Magnetospheric Currents in a Horizontally Homogeneous Medium......Page 204
5.3.1 Spatial frequency-domain (SFD) representation of the electromagnetic field in a horizontally layered medium......Page 205
5.3.2 Lipskaya-Vanyan formulas concerning impedance ratios......Page 207
5.3.3 Horizontal polarization of the electric field in a horizontally homogeneous earth, and the reduced spatial wave number spectrum......Page 210
5.4.1 Spectral representation of the field of a horizontal current dipole on the surface of a horizontally layered medium......Page 213
5.4.2 Electromagnetic field of a horizontal current dipole at the surface of a homogeneous half-space......Page 219
5.4.3 Frequency domain representation of the field of a vertical magnetic dipole above a horizontally stratified medium......Page 223
5.4.4 The magnetic field of a vertical magnetic dipole on the surface of a uniform half-space......Page 226
5.4.5 Near and far fields......Page 227
5.4.6 Frequency domain method for computing transient fields......Page 230
5.4.7 Transient fields of a dipole source observed in a homogeneous medium and on the surface of a homogeneous conducting half-space fields in the near and far zones......Page 233
References and Recommended Reading......Page 243
CHAPTER 6 Electromagnetic Fields in Inhomogeneous Media......Page 246
6.1.1 Background (normal) and anomalous parts of the electromagnetic field......Page 248
6.1.2 Poynting's theorem and energy inequality for an anomalous field......Page 249
6.1.3 Integral equation method in two dimensions......Page 250
6.1.4 Calculation of the first variation (Frechet derivative) of the electromagnetic field for 2-D models......Page 253
6.1.5 Integral equation method in three dimensions......Page 256
6.1.6 Calculation of the first variation (Frechet derivative) of the electromagnetic field for 3-D models......Page 257
6.2 Integral Equation Method in Models with Inhomogeneous Background Conductivity......Page 260
6.2.1 Model with inhomogeneous background conductivity......Page 261
6.2.2 Accuracy control of the IBC IE method......Page 265
6.3 Family of Linear and Nonlinear Integral Approximations of the Electromagnetic Field......Page 267
6.3.1 Born and extended Born approximations......Page 268
6.3.2 Quasi-linear approximation and tensor quasi-linear equation......Page 269
6.3.3 QL approximation using a multigrid approach......Page 270
6.3.4 Quasi-analytical solutions for a 3-D electromagnetic field......Page 271
6.3.5 Quasi-analytical approximation for a model with variable background (QAVB)......Page 275
6.3.7 Localized nonlinear approximation......Page 278
6.3.8 Localized quasi-linear approximation......Page 280
6.4.1 Field equations and boundary conditions......Page 283
6.4.2 Electromagnetic potential equations and boundary conditions......Page 287
6.4.3 Finite difference approximation of boundary-value problem......Page 288
6.4.4 Discretization of Maxwell's equations using a staggered grid......Page 289
6.4.5 Discretization of the second order differential equations using the balance method......Page 293
6.4.6 Discretization of the electromagnetic potential differential equations......Page 298
6.4.7 Finite element solution of boundary-value problems......Page 301
References and Recommended Reading......Page 305
CHAPTER 7 Principles of Ill-Posed Inverse Problem Solution......Page 312
7.1.1 Formulation of well-posed and Ill-posed problems......Page 313
7.1.2 Correctness set......Page 314
7.1.3 Quasi-solution of the Ill-posed problem......Page 315
7.2.1 Definition of misfit functional......Page 316
7.2.2 Regularizing operators......Page 319
7.2.3 Stabilizing functionals......Page 320
7.2.4 Tikhonov parametric functional......Page 325
7.3.1 Tikhonov method of regularization parameter selection......Page 326
7.3.2 L-curve method of regularization parameter selection......Page 329
References and Recommended Reading......Page 332
CHAPTER 8 Electromagnetic Inversion......Page 334
8.1.1 Born inversion......Page 335
8.1.2 Discrete linear EM inverse problem......Page 336
8.1.3 The Tikhonov regularization method of linear inversion......Page 338
8.1.4 Definition of the weighting matrices for model parameters and data......Page 339
8.1.5 Approximate regularized solution of linear inverse problem......Page 341
8.1.7 Conductivity imaging by the Born approximation......Page 344
8.1.8 Iterative Born inversions......Page 349
8.2.1 Formulation of the nonlinear EM inverse problem......Page 350
8.2.2 Regularized solution of nonlinear discrete EM inverse problem......Page 351
8.2.3 The steepest descent method for nonlinear regularized least-squares inversion......Page 352
8.2.4 The Newton method for nonlinear regularized least-squares inversion......Page 353
8.2.5 Numerical schemes of the Newton method for nonlinear regularized least-squares inversion......Page 354
8.2.6 Nonlinear least-squares inversion by the conjugate gradient method......Page 355
8.2.7 The numerical scheme of the regularized conjugate gradient method for nonlinear least-squares inversion......Page 356
8.2.8 Frechet derivative calculation......Page 357
8.3.1 Principles of quasi-linear inversion......Page 360
8.4 Quasi-Analytical Inversion......Page 361
8.4.1 Frechet derivative calculation......Page 362
8.4.2 Inversion based on the quasi-analytical method......Page 363
References and Recommended Reading......Page 364
CHAPTER 9 Electromagnetic Migration......Page 366
9.1 Electromagnetic Migration in the Time Domain......Page 367
9.1.1 Physical principles of electromagnetic migration......Page 368
9.1.2 Migration in a model with homogeneous background conductivity......Page 369
9.1.3 Migration using integral transformation......Page 370
9.2.1 Analytic continuation of the EM field......Page 372
9.2.2 Migration as a spectral transformation......Page 374
9.2.3 Convolution form of migration operator......Page 376
9.2.4 Constructing a digital filter for EM migration......Page 377
9.2.5 Spectral characteristic of the digital filter......Page 380
9.3.1 2-D Finite difference migration......Page 383
9.3.2 Finite difference migration of a 3-D EM field......Page 387
9.4.1 Migration imaging condition in the frequency domain......Page 390
9.4.2 Migration imaging condition in the time domain......Page 392
9.5.1 Formulation of the inverse problem......Page 394
9.5.2 General concept of the migration anomalous field......Page 395
9.5.3 General migration imaging conditions......Page 397
9.5.4 Regularized iterative migration......Page 400
References and Recommended Reading......Page 403
CHAPTER 10 Electromagnetic Properties of Rocks and Minerals......Page 408
10.1.1 Electrical conductivity and resistivity......Page 409
10.1.2 Dielectric permittivity......Page 411
10.1.3 Magnetic permeability......Page 412
10.1.4 Wave number......Page 413
10.2.1 Electric properties of rock-forming minerals and rocks......Page 415
10.2.2 Induced polarization......Page 429
10.2.3 Dielectric properties of rock-forming minerals......Page 433
10.2.4 Magnetic properties of minerals......Page 438
10.3.2 Principles of the effective-medium theory......Page 440
10.3.3 Effective conductivity of heterogeneous medium......Page 446
10.4.1 Concepts of a geoelectric structure and a geoelectric section......Page 448
10.4.2 Longitudinal conductance and transverse resistance of the horizontally layered geoelectric section......Page 450
10.5.1 Geoelectric mesostructures and megastructures......Page 453
10.5.2 The oceans......Page 455
10.5.3 The atmosphere......Page 457
References and Recommended Reading......Page 459
CHAPTER 11 Generation and Measurement of Electromagnetic Fields in Geophysical Applications......Page 462
11.1.1 Sources of EM fields......Page 463
11.1.2 Cables......Page 466
11.1.3 Grounding structures......Page 467
11.2.1 Voltage, potential, and electric field......Page 472
11.2.2 Sensing the magnetic field......Page 478
11.3.1 Sampling in time......Page 491
11.3.2 Analog-to-digital conversion......Page 492
11.3.3 Filtering......Page 494
11.3.4 Stacking......Page 499
11.3.5 Deconvolution......Page 500
References and Recommended Reading......Page 502
CHAPTER 12 Direct Current and Induced Polarization Methods......Page 504
12.1 Vertical Electric Sounding and Apparent Resistivity......Page 506
12.1.1 Techniques for vertical electric sounding......Page 507
12.1.2 Three point electrode array......Page 515
12.1.3 Dipole electric sounding......Page 516
12.2.1 Induced polarization phenomena......Page 521
12.2.2 IP method in the frequency and time domains......Page 522
12.2.3 Resistivity/IP model of a typical porphyry copper system in the Southwestern U. S......Page 525
12.3 Physical and Mathematical Models of the IP Phenomenon......Page 528
12.3.1 IP phenomenon in the context of effective-medium theory......Page 529
12.3.2 Effective conductivity of a heterogeneous polarizable medium......Page 534
12.3.3 Self-consistent approximation for effective conductivity......Page 536
12.3.4 Anisotropy effect in IP data......Page 537
12.3.5 Fundamental IP model: effective resistivity of the isotropic multiphase heterogeneous medium filled with spherical inclusions......Page 538
12.4 Nonlinear Regularized Inversion of IP Data Based on the Cole-Cole Model......Page 543
12.4.1 Forward modeling of induced polarization based on the LQL approximation......Page 544
12.4.2 Inversion based on the LQL approximation......Page 545
12.4.3 Regularized solution of the material property equation......Page 547
12.4.4 Quantitative interpretation of IP data ΠThe road ahead......Page 550
References and Recommended Reading......Page 551
CHAPTER 13 Magnetotelluric and Magnetovariational Methods......Page 556
13.1 Earth EM Field of External Origin......Page 558
13.1.1 Quiet-time magnetic field variations......Page 560
13.1.2 Micropulsations......Page 562
13.1.3 Magnetic storms......Page 565
13.1.4 Substorms......Page 566
13.2.2 Concepts of apparent resistivity and sounding......Page 567
13.2.3 Relationships between the MT sounding curve and the actual 1-D resistivity model......Page 569
13.3 Theory of the MT and MV Transfer Functions......Page 577
13.3.1 Magnetotelluric operators: impedance and admittance, telluric and magnetic......Page 578
13.3.2 Induction vectors and magnetic and electric tippers......Page 581
13.3.3 Spectral magnetotelluric impedances......Page 582
13.4.1 Concepts of external and internal, normal and anomalous parts of an electromagnetic field......Page 587
13.4.2 Anomalous electromagnetic fields and their classification......Page 589
13.4.3 Fields in two-dimensionally inhomogeneous media and the concepts of E and H polarization......Page 590
13.5.1 The MTS, MTP, and TCM methods......Page 592
13.5.2 MVS and MVP survey methods......Page 595
13.6 Processing and Analysis of MT and MV Data......Page 596
13.6.1 The least-squares method......Page 597
13.6.2 Remote reference method......Page 605
13.6.3 Robust estimation of magnetotelluric and induction matrices......Page 606
13.6.4 Graphical presentation of magnetotelluric and induction matrices......Page 610
13.7 One-Dimensional Interpretation of MT Data......Page 612
13.7.1 Analysis of distorted MTS curves......Page 615
13.7.2 Quick and dirty MTS analysis......Page 621
13.8 Interpretation of MVP and GDS Data......Page 625
13.8.1 Separation of fields into internal and external parts......Page 627
13.8.2 Separation of fields into normal and anomalous parts......Page 631
13.9 Rapid Three-Dimensional Magnetotelluric Inversion Based on Linear and Quasi-Linear Approximations......Page 632
13.9.1 Iterative Born inversion of magnetotelluric data......Page 633
13.9.2 MT inversion based on the quasi-analytical method......Page 634
13.9.3 Regularized smooth and focusing inversion of MT data......Page 636
13.9.4 Principles of the re-weighted regularized inversion......Page 637
13.9.5 Minimum support nonlinear parameterization......Page 640
13.9.6 Case study 1: inversion of the Voisey's Bay MT data......Page 644
13.9.7 Case study 2: 3-D inversion of MT data collected by Phoenix Geophysics in Ontario, Canada......Page 647
13.10 Rigorous 3-D Magnetotelluric Inversion......Page 650
13.10.1 Tikhonov regularization in the full mt impedance tensor inversion......Page 651
13.10.2 Frechet operator and its adjoint for two-component impedance inversion......Page 652
13.10.3 Frechet operator for the full magnetotelluric impedance tensor inversion......Page 653
13.10.4 Frechet derivative calculation using quasi-analytical approximation for a variable background (QAVB)......Page 656
References and Recommended Reading......Page 658
CHAPTER 14 Electromagnetic Methods in the Frequency and Time Domains......Page 662
14.1 Electromagnetic Sounding in the Frequency and Time Domains......Page 663
14.1.1 Mutual coupling......Page 667
14.1.2 Theoretical curves for EM sounding in the frequency domain......Page 671
14.1.3 Time-domain electromagnetic sounding......Page 675
14.1.4 Properties of TDEM sounding curves......Page 685
14.2.1 The Price-Sheinman and Tikhonov-Dmitriev thin-film models with laterally varying conductance......Page 690
14.2.2 Transient field of a magnetic dipole above a conducting thin sheet......Page 693
14.2.3 S-Inversion method......Page 699
14.3.2 Profiling with large fixed sources......Page 702
14.3.3 Transient electromagnetic techniques: UTEM, LOTEM, and MTEM methods......Page 703
References and Recommended Reading......Page 704
CHAPTER 15 Marine Electromagnetic Methods......Page 708
15.1 Marine Magnetotelluric Method......Page 709
15.1.1 Main characteristic of seafloor EM equipment......Page 710
15.1.2 Comparison between land and sea-bottom electromagnetic anomalies......Page 713
15.1.3 Case study: marine magnetotellurics in the Gulf of Mexico......Page 714
15.2 Marine Controlled-Source Electromagnetic Methods......Page 717
15.2.1 Electrical exploration in shallow water......Page 718
15.2.2 Electrical exploration beneath deep oceans......Page 720
15.2.3 MCSEM method for offshore petroleum exploration......Page 726
15.2.4 Interpretation of MCSEM data......Page 729
15.2.5 Case study: iterative migration of Troll Gas Province MCSEM data......Page 740
References and Recommended Reading......Page 744
CHAPTER 16 Other Platforms, Other Methodologies......Page 748
16.1 Airborne Electromagnetic Methods......Page 749
16.1.1 Frequency domain airborne surveys......Page 750
16.1.2 Airborne transient electromagnetic systems (ATEM)......Page 764
16.1.3 Far field AEM methods......Page 765
16.2 Ground Penetrating Radar (GPR)......Page 769
16.3.1 Borehole-to-surface techniques......Page 777
16.3.2 Cross-well electromagnetic tomography......Page 780
16.4.1 Piezoelectric method......Page 787
16.4.2 Spontaneous polarization (SP) method......Page 789
References and Recommended Reading......Page 793
APPENDIX A Algebra of Differential Forms......Page 796
A.1.1 1-, 2-, and 3-forms......Page 797
A.1.2 Exterior product of the differential forms......Page 798
A.1.3 Basis of differential forms......Page 799
A.2.1 Euclidean space......Page 803
A.2.2 Differential forms in Euclidean space En......Page 806
A.2.3 Differential forms in Minkowskian space M4......Page 807
B.1 Exterior Differentiation of the Forms......Page 812
B.1.1 Exterior differential operator in multidimensional space En......Page 813
B.1.2 Exterior differential operator in four-dimensional space M4......Page 815
B.2.1 Three-dimensional space E3......Page 819
B.2.2 Beyond three-dimensional space......Page 821
APPENDIX C Mathematical Notations......Page 824
APPENDIX D Definition of Fields and Units......Page 828
APPENDIX E Linear Operators and Their Matrices......Page 832
Bibliography......Page 836
Index......Page 858