Potential Theory in Applied Geophysics (eBook)

Preface 6
Contents 12
1 Elements of Vector Analysis 23
1.1 Scalar & Vector
1.2 Properties of Vectors 23
1.3 Gradient of a Scalar 26
1.4 Divergence of a Vector 28
1.5 Surface Integral 29
1.6 Gauss’s Divergence Theorem 30
1.7 Line Integral 32
1.8 Curl of a Vector 33
1.9 Line Integral in a Plane (Stoke’s Theorem) 34
1.10 Successive Application of the Operator . 36
1.11 Important Relations in Vector Algebra 37
2 Introductory Remarks 39
2.1 Field of Force 39
2.2 Classification of Fields 41
2.3 Concept of Potential 47
2.4 Field Mapping 49
2.5 Nature of a Solid Medium 53
2.6 Tensors 54
2.7 Boundary Value Problems 56
2.8 Dimension of a Problem and its Solvability 58
2.9 Equations 60
2.10 Domain of Geophysics in Potential Theory 63
3 Gravitational Potential and Field 65
3.1 Introduction 65
3.2 Newton’s Law of Gravitation 66
3.3 Gravity Field at a Point due to Number of Point Sources 68
3.4 Gravitational Field for a Large Body 69
3.5 Gravitational Field due to a Line Source 70
3.6 Gravitational Potential due to a Finite Line Source 72
3.7 Gravitational Attraction due to a Buried Cylinder 75
3.8 Gravitational Field due to a Plane Sheet 76
3.9 Gravitational Field due to a Circular Plate 77
3.10 Gravity Field at a Point Outside on the Axis of a Vertical Cylinder 78
3.11 Gravitational Potential at a Point due to a Spherical Body 80
3.12 Gravitational Attraction on the Surface due to a Buried Sphere 84
3.13 Gravitational Anomaly due to a Body of Trapezoidal Cross Section 85
3.14 Gravity Field of the Earth 91
3.15 Units 94
3.16 Basic Equation 94
4 Electrostatics 96
4.1 Introduction 96
4.2 Coulomb’s Law 97
4.3 Electrostatic Potential 97
4.4 Electrical Permittivity and Electrical Force Field 98
4.5 Electric Flux 100
4.6 Electric Displacement 100
and the Displacement 100
Vector D 100
4.7 Gauss’s Theorem 101
4.8 Field due to an Electrostatic Dipole 103
4.9 Poisson and Laplace Equations 106
4.10 Electrostatic Energy 107
4.11 Boundary Conditions 108
4.12 Basic Equations in Electrostatic Field 109
5 Magnetostatics 111
5.1 Introduction 111
5.2 Coulomb’s Law 118
5.3 Magnetic Properties 118
5.4 Magnetic Induction B 122
5.5 Magnetic Field Intensity H 124
5.6 Faraday’s Law 124
5.7 Biot and Savart’s Law 126
5.8 Lorentz Force 128
5.9 Ampere’s Force Law 129
5.10 Magnetic Field on the Axis of a Magnetic Dipole 130
5.11 Magnetomotive Force (MMF) 132
5.12 Ampere’s Law 132
5.13 Div B = 0 133
5.14 Magnetic Vector Potential 134
5.15 Magnetic Scalar Potential 135
5.16 Poisson’s Relation 136
5.17 Magnetostatic Energy 137
5.18 Geomagnetic Field 138
5.19 Application of Magnetic Field Measurement in Geophysics 143
5.20 Units 144
5.21 Basic Equations in Magnetostatics 144
6 Direct Current Flow Field 146
6.1 Introduction 146
6.2 Direct Current Flow 150
6.3 Differential form of the Ohm’s Law 150
6.4 Equation of Continuity 151
6.5 Anisotropy in Electrical Conductivity 152
6.6 Potential at a Point due to a Point Source 153
6.7 Potential for Line Electrode Configuration 155
6.8 Current Flow Inside the Earth 158
6.9 Refraction of Current Lines 162
6.10 Dipole Field 163
6.11 Basic Equations in Direct Current Flow Field 168
6.12 Units 169
7 Solution of Laplace Equation 170
7.1 Equations of Poisson and Laplace 170
7.2 Laplace Equation in Direct Current Flow Domain 171
7.3 Laplace Equation in Generalised Curvilinear Coordinates 172
7.4 Laplace Equation in Cartesian Coordinates 175
7.5 Laplace Equation in Cylindrical Polar Coordinates 181
7.6 Solution of Laplace Equation in Spherical Polar Co- ordinates 202
7.7 Spherical Harmonics 220
8 Direct Current Field Related Potential Problems 225
8.1 Layered Earth Problem in a Direct Current Domain 225
8.2 Potential due to a Point Source in a Borehole with Cylindrical Coaxial Boundaries 241
8.3 Potential for a Transitional Earth 250
8.4 Geoelectrical Potential for a Dipping Interface 271
8.5 Geoelectrical Potentials for an Anisotropic Medium 275
9 Complex Variables and Conformal Transformation in Potential Theory 281
9.1 Definition of Analytic Function 281
9.2 Complex Functions and their Derivatives 282
9.3 Conformal Mapping 285
9.4 Transformations 287
9.5 Schwarz Christoffel Transformation 292
9.6 Geophysical Problems on S-C Transformation 296
9.7 Elliptic Integrals and Elliptic Functions 315
10 Green’s Theorem in Potential Theory 324
10.1 Green’s First Identity 324
10.2 Harmonic Function 325
10.3 Corollaries of Green’s Theorem 326
10.4 Regular Function 328
10.5 Green’s Formula 329
10.6 Some Special Cases in Green’s Formula 332
10.7 Poisson’s Equation from Green’s Theorem 333
10.8 Gauss’s Theorem of Total Normal Induction in Gravity Field 333
10.9 Estimation of Mass in Gravity Field 334
10.10 Green’s Theorem for Analytical Continuation 335
10.11 Green’s Theorem for Two Dimensional Problems 337
10.12 Three to Two Dimensional Conversion 338
10.13 Green’s Equivalent Layers 339
10.14 Unique Surface Distribution 341
10.15 Vector Green’s Theorem 343
11 Electrical Images in Potential Theory 345
11.1 Introduction 345
11.2 Computation of Potential Using Images ( Two Media) 345
11.3 Computation of Potential Using Images ( for Three Media) 348
11.4 General Expressions for Potentials Using Images 350
11.5 Expressions for Potentials for Two Electrode Configuration 352
11.6 Expressions for Potentials for Three Electrode Configuration 354
11.7 Expression for Potentials for Seven Electrode Configurations 357
List of Symbols 361
12 Electromagnetic Theory (Vector Potentials) 364
12.1 Introduction 364
12.2 Elementary Wavelet 369
12.3 Elliptic Polarisation of Electromagnetic Waves 371
12.4 Mutual Inductance 373
12.5 Maxwell’s Equations 378
12.6 Helmholtz Electromagnetic Wave Equations 381
12.7 Hertz and Fitzerald Vectors 384
12.8 Boundary Conditions in Electromagnetics 386
12.9 Poynting Vector 391
13 Electromagnetic Wave Propagation Problems Related to Geophysics 395
13.1 Plane Wave Propagation 395
13.2 Skin Depth 401
13.3 Perturbation Centroid Frequency 402
13.4 Magnetotelluric Response for a Layered Earth Model 403
13.5 Electromagnetic Field due to a Vertical Oscillating Electric Dipole 408
13.6 Electromagnetic Field due to an Oscillating Vertical Magnetic Dipole Placed on the Surface of the Earth 413
13.7 Electromagnetic Field due to an Oscillating Horizontal Magnetic Dipole Placed on the Surface of the Earth 422
13.8 Electromagnetic Field due to a Long Line Cable Placed in an Infinite and Homogenous Medium 430
13.9 Electromagnetic Field due to a Long Cable on the Surface of a Homogeneous Earth 435
13.10 Electromagnetic Induction due to an Infinite Cylinder in an Uniform Field 442
13.11 Electromagnetic Response due to a Sphere in the Field of a Vertically Oscillating Magnetic Dipole 448
13.12 Principle of Electrodynamic Similitude 455
14 Green’s Function 458
14.1 Introduction 458
14.2 Delta Function 460
14.3 Operators 461
14.4 Adjoint and Self Adjoint Operator 462
14.5 Definition of a Green’s Function 462
14.6 Free Space Green’s Function 464
14.7 Green’s Function is a Potential due to a Charge of Unit Strength in Electrostatics 465
14.8 Green’s Function can Reduce the Number of unknowns to be Determined in a Potential Problem 466
14.9 Green’s Function has Some Relation with the Concept of Image in Potential Theory 467
14.10 Reciprocity Relation of Green’s Function 469
14.11 Green’s Function as a Kernel Function in an Integral Equation 470
14.12 Poisson’s Equation and Green’s Function 473
14.13 Problem 1 474
14.14 Problem 2 476
14.15 Problem 3 478
14.16 Dyadics 479
15 Numerical Methods in Potential Theory 483
15.1 Introduction 483
15.2 Finite Difference Formulation/Direct Current Domain ( Surface Geophysics) 485
15.3 Finite Difference Formulation Domain with Cylindrical Symmetry DC Field Borehole Geophysics 494
15.4 Finite Difference Formulation Plane Wave Electromagnetics Magnetotellurics 502
15.5 Finite Element Formulation Direct Current Resistivity Domain 508
15.6 3D Model 519
15.7 Finite Element Formulation Galerkin’s Approach Magnetotellurics 521
15.8 Finite Element Formulation Galerkin’s Approach Isoparametric Elements Magnetotellurics 527
15.9 Integral Equation Method 540
16 Analytical Continuation of Potential Field 547
16.1 Introduction 547
16.2 Downward Continuation by Harmonic Analysis of Gravity Field 548
16.3 Taylor’s Series Expansion and Finite Difference Approach for Downward Continuation 549
16.4 Green’s Theorem and Integral Equations for Analytical Continuation 553
16.5 Analytical Continuation using Integral Equation and Taking Areal Averages 556
16.6 Upward and Downward Continuation using Integral Equation and Lagrange Interpolation Formula 562
16.7 Downward Continuation of Telluric Current Data 563
16.8 Upward and Downward Continuation of Electromagnetic Field Data 564
16.9 Downward Continuation of Electromagnetic Field 568
17 Inversion of Potential Field Data 573
17.1 Introduction 573
17.2 Wellposed and Illposed Problems 582
17.3 Tikhnov’s Regularisation 583
17.4 Abstract Spaces 583
17.5 Some Properties of a Matrix 587
17.6 Lagrange Multiplier 590
17.7 Singular Value Decomposition (SVD) 590
17.8 Least Squares Estimator 596
17.9 Ridge Regression Estimator 598
17.10 Weighted Ridge Regression 599
17.11 Minimum Norm Algorithm for an Under Determined Problem 601
17.12 Bachus – Gilbert Inversion 604
17.13 Stochastic Inversion 609
17.14 Occam’s Inversion 614
17.15 Global Optimization 615
17.16 Neural Network 628
17.17 Joint Inversion 633
References 637
List of Symbols 652
A. Symbols used to Represent more than one Physical Entity in Different Chapters 652
B. Symbols used to Represent one Parameter 653
Index 657