Roy Potential Theory in Applied Geophysics

1. Auflage 2007
ISBN: 978-3-540-72334-9
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 651 Seiten

This book introduces the principles of gravitational, magnetic, electrostatic, direct current electrical and electromagnetic fields, with detailed solutions of Laplace and electromagnetic wave equations by the method of separation of variables. Discussion includes behaviours of the scalar and vector potential and the nature of the solutions of these boundary value problems, along with the use of complex variables and conformal transformation, Green's theorem, Green's formula and Green's functions.

Autoren/Hrsg.

Roy, Kalyan Kumar

Inhaltsverzeichnis

1;Preface;6
2;Contents;12
3;1 Elements of Vector Analysis;23
3.1;1.1 Scalar & Vector;23
3.2;1.2 Properties of Vectors;23
3.3;1.3 Gradient of a Scalar;26
3.4;1.4 Divergence of a Vector;28
3.5;1.5 Surface Integral;29
3.6;1.6 Gauss’s Divergence Theorem;30
3.7;1.7 Line Integral;32
3.8;1.8 Curl of a Vector;33
3.9;1.9 Line Integral in a Plane (Stoke’s Theorem);34
3.10;1.10 Successive Application of the Operator .;36
3.11;1.11 Important Relations in Vector Algebra;37
4;2 Introductory Remarks;39
4.1;2.1 Field of Force;39
4.2;2.2 Classification of Fields;41
4.3;2.3 Concept of Potential;47
4.4;2.4 Field Mapping;49
4.5;2.5 Nature of a Solid Medium;53
4.6;2.6 Tensors;54
4.7;2.7 Boundary Value Problems;56
4.8;2.8 Dimension of a Problem and its Solvability;58
4.9;2.9 Equations;60
4.10;2.10 Domain of Geophysics in Potential Theory;63
5;3 Gravitational Potential and Field;65
5.1;3.1 Introduction;65
5.2;3.2 Newton’s Law of Gravitation;66
5.3;3.3 Gravity Field at a Point due to Number of Point Sources;68
5.4;3.4 Gravitational Field for a Large Body;69
5.5;3.5 Gravitational Field due to a Line Source;70
5.6;3.6 Gravitational Potential due to a Finite Line Source;72
5.7;3.7 Gravitational Attraction due to a Buried Cylinder;75
5.8;3.8 Gravitational Field due to a Plane Sheet;76
5.9;3.9 Gravitational Field due to a Circular Plate;77
5.10;3.10 Gravity Field at a Point Outside on the Axis of a Vertical Cylinder;78
5.11;3.11 Gravitational Potential at a Point due to a Spherical Body;80
5.12;3.12 Gravitational Attraction on the Surface due to a Buried Sphere;84
5.13;3.13 Gravitational Anomaly due to a Body of Trapezoidal Cross Section;85
5.14;3.14 Gravity Field of the Earth;91
5.15;3.15 Units;94
5.16;3.16 Basic Equation;94
6;4 Electrostatics;96
6.1;4.1 Introduction;96
6.2;4.2 Coulomb’s Law;97
6.3;4.3 Electrostatic Potential;97
6.4;4.4 Electrical Permittivity and Electrical Force Field;98
6.5;4.5 Electric Flux;100
6.6;4.6 Electric Displacement;100
6.7;and the Displacement;100
6.8;Vector D;100
6.9;4.7 Gauss’s Theorem;101
6.10;4.8 Field due to an Electrostatic Dipole;103
6.11;4.9 Poisson and Laplace Equations;106
6.12;4.10 Electrostatic Energy;107
6.13;4.11 Boundary Conditions;108
6.14;4.12 Basic Equations in Electrostatic Field;109
7;5 Magnetostatics;111
7.1;5.1 Introduction;111
7.2;5.2 Coulomb’s Law;118
7.3;5.3 Magnetic Properties;118
7.4;5.4 Magnetic Induction B;122
7.5;5.5 Magnetic Field Intensity H;124
7.6;5.6 Faraday’s Law;124
7.7;5.7 Biot and Savart’s Law;126
7.8;5.8 Lorentz Force;128
7.9;5.9 Ampere’s Force Law;129
7.10;5.10 Magnetic Field on the Axis of a Magnetic Dipole;130
7.11;5.11 Magnetomotive Force (MMF);132
7.12;5.12 Ampere’s Law;132
7.13;5.13 Div B = 0;133
7.14;5.14 Magnetic Vector Potential;134
7.15;5.15 Magnetic Scalar Potential;135
7.16;5.16 Poisson’s Relation;136
7.17;5.17 Magnetostatic Energy;137
7.18;5.18 Geomagnetic Field;138
7.19;5.19 Application of Magnetic Field Measurement in Geophysics;143
7.20;5.20 Units;144
7.21;5.21 Basic Equations in Magnetostatics;144
8;6 Direct Current Flow Field;146
8.1;6.1 Introduction;146
8.2;6.2 Direct Current Flow;150
8.3;6.3 Differential form of the Ohm’s Law;150
8.4;6.4 Equation of Continuity;151
8.5;6.5 Anisotropy in Electrical Conductivity;152
8.6;6.6 Potential at a Point due to a Point Source;153
8.7;6.7 Potential for Line Electrode Configuration;155
8.8;6.8 Current Flow Inside the Earth;158
8.9;6.9 Refraction of Current Lines;162
8.10;6.10 Dipole Field;163
8.11;6.11 Basic Equations in Direct Current Flow Field;168
8.12;6.12 Units;169
9;7 Solution of Laplace Equation;170
9.1;7.1 Equations of Poisson and Laplace;170
9.2;7.2 Laplace Equation in Direct Current Flow Domain;171
9.3;7.3 Laplace Equation in Generalised Curvilinear Coordinates;172
9.4;7.4 Laplace Equation in Cartesian Coordinates;175
9.5;7.5 Laplace Equation in Cylindrical Polar Coordinates;181
9.6;7.6 Solution of Laplace Equation in Spherical Polar Co- ordinates;202
9.7;7.7 Spherical Harmonics;220
10;8 Direct Current Field Related Potential Problems;225
10.1;8.1 Layered Earth Problem in a Direct Current Domain;225
10.2;8.2 Potential due to a Point Source in a Borehole with Cylindrical Coaxial Boundaries;241
10.3;8.3 Potential for a Transitional Earth;250
10.4;8.4 Geoelectrical Potential for a Dipping Interface;271
10.5;8.5 Geoelectrical Potentials for an Anisotropic Medium;275
11;9 Complex Variables and Conformal Transformation in Potential Theory;281
11.1;9.1 Definition of Analytic Function;281
11.2;9.2 Complex Functions and their Derivatives;282
11.3;9.3 Conformal Mapping;285
11.4;9.4 Transformations;287
11.5;9.5 Schwarz Christoffel Transformation;292
11.6;9.6 Geophysical Problems on S-C Transformation;296
11.7;9.7 Elliptic Integrals and Elliptic Functions;315
12;10 Green’s Theorem in Potential Theory;324
12.1;10.1 Green’s First Identity;324
12.2;10.2 Harmonic Function;325
12.3;10.3 Corollaries of Green’s Theorem;326
12.4;10.4 Regular Function;328
12.5;10.5 Green’s Formula;329
12.6;10.6 Some Special Cases in Green’s Formula;332
12.7;10.7 Poisson’s Equation from Green’s Theorem;333
12.8;10.8 Gauss’s Theorem of Total Normal Induction in Gravity Field;333
12.9;10.9 Estimation of Mass in Gravity Field;334
12.10;10.10 Green’s Theorem for Analytical Continuation;335
12.11;10.11 Green’s Theorem for Two Dimensional Problems;337
12.12;10.12 Three to Two Dimensional Conversion;338
12.13;10.13 Green’s Equivalent Layers;339
12.14;10.14 Unique Surface Distribution;341
12.15;10.15 Vector Green’s Theorem;343
13;11 Electrical Images in Potential Theory;345
13.1;11.1 Introduction;345
13.2;11.2 Computation of Potential Using Images ( Two Media);345
13.3;11.3 Computation of Potential Using Images ( for Three Media);348
13.4;11.4 General Expressions for Potentials Using Images;350
13.5;11.5 Expressions for Potentials for Two Electrode Configuration;352
13.6;11.6 Expressions for Potentials for Three Electrode Configuration;354
13.7;11.7 Expression for Potentials for Seven Electrode Configurations;357
13.8;List of Symbols;361
14;12 Electromagnetic Theory (Vector Potentials);364
14.1;12.1 Introduction;364
14.2;12.2 Elementary Wavelet;369
14.3;12.3 Elliptic Polarisation of Electromagnetic Waves;371
14.4;12.4 Mutual Inductance;373
14.5;12.5 Maxwell’s Equations;378
14.6;12.6 Helmholtz Electromagnetic Wave Equations;381
14.7;12.7 Hertz and Fitzerald Vectors;384
14.8;12.8 Boundary Conditions in Electromagnetics;386
14.9;12.9 Poynting Vector;391
15;13 Electromagnetic Wave Propagation Problems Related to Geophysics;395
15.1;13.1 Plane Wave Propagation;395
15.2;13.2 Skin Depth;401
15.3;13.3 Perturbation Centroid Frequency;402
15.4;13.4 Magnetotelluric Response for a Layered Earth Model;403
15.5;13.5 Electromagnetic Field due to a Vertical Oscillating Electric Dipole;408
15.6;13.6 Electromagnetic Field due to an Oscillating Vertical Magnetic Dipole Placed on the Surface of the Earth;413
15.7;13.7 Electromagnetic Field due to an Oscillating Horizontal Magnetic Dipole Placed on the Surface of the Earth;422
15.8;13.8 Electromagnetic Field due to a Long Line Cable Placed in an Infinite and Homogenous Medium;430
15.9;13.9 Electromagnetic Field due to a Long Cable on the Surface of a Homogeneous Earth;435
15.10;13.10 Electromagnetic Induction due to an Infinite Cylinder in an Uniform Field;442
15.11;13.11 Electromagnetic Response due to a Sphere in the Field of a Vertically Oscillating Magnetic Dipole;448
15.12;13.12 Principle of Electrodynamic Similitude;455
16;14 Green’s Function;458
16.1;14.1 Introduction;458
16.2;14.2 Delta Function;460
16.3;14.3 Operators;461
16.4;14.4 Adjoint and Self Adjoint Operator;462
16.5;14.5 Definition of a Green’s Function;462
16.6;14.6 Free Space Green’s Function;464
16.7;14.7 Green’s Function is a Potential due to a Charge of Unit Strength in Electrostatics;465
16.8;14.8 Green’s Function can Reduce the Number of unknowns to be Determined in a Potential Problem;466
16.9;14.9 Green’s Function has Some Relation with the Concept of Image in Potential Theory;467
16.10;14.10 Reciprocity Relation of Green’s Function;469
16.11;14.11 Green’s Function as a Kernel Function in an Integral Equation;470
16.12;14.12 Poisson’s Equation and Green’s Function;473
16.13;14.13 Problem 1;474
16.14;14.14 Problem 2;476
16.15;14.15 Problem 3;478
16.16;14.16 Dyadics;479
17;15 Numerical Methods in Potential Theory;483
17.1;15.1 Introduction;483
17.2;15.2 Finite Difference Formulation/Direct Current Domain ( Surface Geophysics);485
17.3;15.3 Finite Difference Formulation Domain with Cylindrical Symmetry DC Field Borehole Geophysics;494
17.4;15.4 Finite Difference Formulation Plane Wave Electromagnetics Magnetotellurics;502
17.5;15.5 Finite Element Formulation Direct Current Resistivity Domain;508
17.6;15.6 3D Model;519
17.7;15.7 Finite Element Formulation Galerkin’s Approach Magnetotellurics;521
17.8;15.8 Finite Element Formulation Galerkin’s Approach Isoparametric Elements Magnetotellurics;527
17.9;15.9 Integral Equation Method;540
18;16 Analytical Continuation of Potential Field;547
18.1;16.1 Introduction;547
18.2;16.2 Downward Continuation by Harmonic Analysis of Gravity Field;548
18.3;16.3 Taylor’s Series Expansion and Finite Difference Approach for Downward Continuation;549
18.4;16.4 Green’s Theorem and Integral Equations for Analytical Continuation;553
18.5;16.5 Analytical Continuation using Integral Equation and Taking Areal Averages;556
18.6;16.6 Upward and Downward Continuation using Integral Equation and Lagrange Interpolation Formula;562
18.7;16.7 Downward Continuation of Telluric Current Data;563
18.8;16.8 Upward and Downward Continuation of Electromagnetic Field Data;564
18.9;16.9 Downward Continuation of Electromagnetic Field;568
19;17 Inversion of Potential Field Data;573
19.1;17.1 Introduction;573
19.2;17.2 Wellposed and Illposed Problems;582
19.3;17.3 Tikhnov’s Regularisation;583
19.4;17.4 Abstract Spaces;583
19.5;17.5 Some Properties of a Matrix;587
19.6;17.6 Lagrange Multiplier;590
19.7;17.7 Singular Value Decomposition (SVD);590
19.8;17.8 Least Squares Estimator;596
19.9;17.9 Ridge Regression Estimator;598
19.10;17.10 Weighted Ridge Regression;599
19.11;17.11 Minimum Norm Algorithm for an Under Determined Problem;601
19.12;17.12 Bachus – Gilbert Inversion;604
19.13;17.13 Stochastic Inversion;609
19.14;17.14 Occam’s Inversion;614
19.15;17.15 Global Optimization;615
19.16;17.16 Neural Network;628
19.17;17.17 Joint Inversion;633
20;References;637
21;List of Symbols;652
21.1;A. Symbols used to Represent more than one Physical Entity in Different Chapters;652
21.2;B. Symbols used to Represent one Parameter;653
22;Index;657