Roy (2008) Potential Theory in Applied Geophysics

2024-05-29

#Geodesy #Book

Roy KK (2008) Potential Theory in Applied Geophysics. https://doi.org/10.1007/978-3-540-72334-9

1 Elements of Vector Analysis

1.1 Scalar&Vector

1.2 Properties of Vectors

1.3 Gradient of a Scalar

1.4 Divergence of a Vector

1.5 Surface Integral

1.6 Gauss’s Divergence Theorem

1.7 Line Integral

1.8 Curl of a Vector

1.9 Line Integral in a Plane (Stoke’s Theorem)

1.10 Successive Application of the Operator ∇

1.11 Important Relations in Vector Algebra

2 Introductory Remarks

2.1 Field of Force

2.2 Classification of Fields

2.2.1 Type A Classification
2.2.2 Type B Classification
2.2.3 Type C Classification
2.2.4 Type D Classification
2.2.5 Type E Classification
2.2.6 Type F Classification
2.2.7 Type G Classification
2.2.8 Type H Classification
2.2.9 Type I Classification
2.2.10 Type J Classification
2.2.11 Type K Classification

2.3 Concept of Potential

2.4 Field Mapping

2.5 Nature of a Solid Medium

2.6 Tensors

2.7 Boundary Value Problems

2.7.1 Dirichlet’s Problem
2.7.2 Neumann Problem
2.7.3 Mixed Problem

2.8 Dimension of a Problem and its Solvability

2.9 Equations

2.9.1 Differential Equations
2.9.2 Integral Equations

2.10 Domain of Geophysics in Potential Theory

3 Gravitational Potential and Field

3.1 Introduction

3.2 Newton’s Law of Gravitation

3.3 Gravity Field at a Point due to Number of Point Sources

3.4 Gravitational Field for a Large Body

3.5 Gravitational Field due to a Line Source

3.6 Gravitational Potential due to a Finite Line Source

3.7 Gravitational Attraction due to a Buried Cylinder

3.8 Gravitational Field due to a Plane Sheet

3.9 Gravitational Field due to a Circular Plate

3.10 Gravity Field at a Point Outside on the Axis of a Vertical Cylinder

3.11 Gravitational Potential at a Point due to a Spherical Body

3.12 Gravitational Attraction on the Surface due to a Buried Sphere

3.13 Gravitational Anomaly due to a Body of Trapezoidal Cross Section

3.13.1 Special Cases

3.14 Gravity Field of the Earth

3.14.1 Free Air Correction
3.14.2 Bouguer Correction
3.14.3 Terrain Correction
3.14.4 Latitude Correction
3.14.5 Tidal Correction
3.14.6 Isostatic Correction

3.15 Units

3.16 Basic Equation

4 Electrostatics

4.1 Introduction

4.2 Coulomb’s Law

4.3 Electrostatic Potential

4.4 Electrical Permittivity and Electrical Force Field

4.5 Electric Flux

4.6 Electric Displacement ψ and the Displacement Vector D

4.7 Gauss’s Theorem

4.8 Field due to an Electrostatic Dipole

4.9 Poisson and Laplace Equations

4.10 Electrostatic Energy

4.11 Boundary Conditions

4.12 Basic Equations in Electrostatic Field

5 Magnetostatics

5.1 Introduction

5.2 Coulomb’s Law

5.3 Magnetic Properties

5.3.1 Magnetic Dipole Moment
5.3.2 Intensity of Magnetisation
5.3.3 Magnetic Susceptibility (Induced Magnetism)
5.3.4 Ferromagnetic Paramagnetic and Diamagnetic Substances

5.4 Magnetic Induction B

5.5 Magnetic Field Intensity H

5.6 Faraday’s Law

5.7 Biot and Savart’s Law

5.8 Lorentz Force

5.9 Ampere’s Force Law

5.10 Magnetic Field on the Axis of a Magnetic Dipole

5.11 Magnetomotive Force (MMF)

5.12 Ampere’s Law

5.13 Div B = 0

5.14 Magnetic Vector Potential

5.15 Magnetic Scalar Potential

5.16 Poisson’s Relation

5.17 Magnetostatic Energy

5.18 Geomagnetic Field

5.18.1 Geomagnetic Field Variations

5.19 Application of Magnetic Field Measurement in Geophysics

5.20 Units

5.21 Basic Equations in Magnetostatics

6 Direct Current Flow Field

6.1 Introduction

6.2 Direct Current Flow

6.3 Differential form of the Ohm’s Law

6.4 Equation of Continuity

6.5 Anisotropy in Electrical Conductivity

6.6 Potential at a Point due to a Point Source

6.7 Potential for Line Electrode Configuration

6.7.1 Potential due to a Finite Line Electrode

6.8 Current Flow Inside the Earth

6.9 Refraction of Current Lines

6.10 Dipole Field

6.11 Basic Equations in Direct Current Flow Field

6.12 Units

7 Solution of Laplace Equation

7.1 Equations of Poisson and Laplace

7.2 Laplace Equation in Direct Current Flow Domain

7.3 Laplace Equation in Generalised Curvilinear Coordinates

7.4 Laplace Equation in Cartesian Coordinates

7.4.1 When Potential is a Function of Vertical Axis z i.e. φ=f(z)
7.4.2 When Potential is a Function of Both x and y i.e. φ=f(xy)
7.4.3 Solution of Boundary Value Problems in Cartisian Coordinates by the Method of Separation of Variables

7.5 Laplace Equation in Cylindrical Polar Coordinates

7.5.1 When Potentialisa Function of zi.e.φ=f(z)
7.5.2 When Potential is a Function of Azimuthal Angle Only i.e. φ = f(ψ)
7.5.3 When the Potential is a Function of Radial Distance i.e. Φ = f(ρ)
7.5.4 When Potential is a Function of Both ρ and ψ i.e. φ = f(ρ ψ)
7.5.5 When Potential is a Function of all the Three Coordinates i.e. φ=f(ρψz)
7.5.6 Bessel Equation and Bessel’s Functions
7.5.7 Modified Bessel’s Functions
7.5.8 Some Relation of Bessel’s Function

7.6 Solution of Laplace Equation in Spherical Polar Co-ordinates

7.6.1 When Potential is a Function of Radial Distance r i.e. φ=f(r)
7.6.2 When Potential is a Function of Polar Angle i.e. φ = f(θ)
7.6.3 When Potential is a Function of Azimuthal Angle i.e. φ = f(ψ)
7.6.4 When Potential is a Function of Both the Radial Distanceand Polar Angle i.e. φ=f(rθ)
7.6.5 Legender’s Equation and Legender’s polynomial
7.6.6 When Potential is a Function of all the Three Coordinates Viz Radial Distance Polar Angle and Azimuthal Angle i.e. φ = f(r θ ψ)
7.6.7 Associated Legendre Polynomial

7.7 Spherical Harmonics

7.7.1 Zonal Sectoral and Tesseral Harmonics

8.1 Layered Earth Problem in a Direct Current Domain

8.1.1 Cramer’s Rule
8.1.2 Two Layered Earth Model
8.1.3 Three Layered Earth Model
8.1.4 General Expressions for the Surface and Subsurface Kernels for an N-Layered Earth
8.1.5 Kernels in Different Layers for a Five Layered Earth
8.1.6 Potentials in Different Media

8.2 Potential due to a Point Source in a Borehole with Cylindrical Coaxial Boundaries

8.3 Potential for a Transitional Earth

8.3.1 Potential for a Medium Where Physical Property Varies Continuously with Distance
8.3.2 Potential for a Layered Earth with a Sandwitched Transitional Layer
8.3.3 Potential with Media Having Coaxial Cylindrical Symmetry with a Transitional Layer in Between

8.4 Geoelectrical Potential for a Dipping Interface

8.5 Geoelectrical Potentials for an Anisotropic Medium

8.5.1 General Nature of the Basic Equations
8.5.2 General Solution of Laplace Equation for an Anisotropic

9 Complex Variables and Conformal Transformation in Potential Theory

9.1 DefinitionofAnalyticFunction

9.2 Complex Functions and their Derivatives

9.3 Conformal Mapping

9.4 Transformations

9.4.1 Simple Transformations

9.5 Schwarz Christoffel Transformation

9.5.1 Introduction
9.5.2 Schwarz-Christoffel Transformation of the Interior of a Polygon
9.5.3 Determination of Unknown Constants
9.5.4 S-C Transformation Theorem

9.6 Geophysical Problems on S-C Transformation

9.6.1 Problem 1 Conformal Transformation for a Substratum of Finite Thickness
9.6.2 Problem 2 Telluric Field over a Vertical Basement Fault
9.6.3 Problem 3 Telluric Field and Apparent Resistivity Over an Anticline
9.6.4 Problem 4 Telluric Field Over a Faulted Basement (Horst)

9.7 Elliptic Integrals and Elliptic Functions

9.7.1 Legendre’s Equation
9.7.2 Complete Integrals
9.7.3 Elliptic Functions
9.7.4 Jacobi’s Zeta Function
9.7.5 Jacobi’s Theta Function
9.7.6 Jacobi’s Elliptic Integral of the Third Kind

10 Green’s Theorem in Potential Theory

10.1 Green’s First Identity

10.2 HarmonicFunction

10.3 Corollaries of Green’s Theorem

10.4 RegularFunction

10.5 Green’s Formula

10.6 Some Special Cases in Green’s Formula

10.7 Poisson’s Equation from Green’s Theorem

10.8 Gauss’s Theorem of Total Normal Induction in Gravity Field

10.9 Estimation of Mass in Gravity Field

10.10 Green’s Theorem for Analytical Continuation

10.11 Green’s Theorem for Two Dimensional Problems

10.12 Three to Two Dimensional Conversion

10.13 Green’s Equivalent Layers

10.14 Unique Surface Distribution

10.15 Vector Green’s Theorem

11 Electrical Images in Potential Theory

11.1 Introduction

11.2 Computation of Potential Using Images (Two Media)

11.3 Computation of Potential Using Images (for Three Media)

11.4 General Expressions for Potentials Using Images

11.5 Expressions for Potentials for Two Electrode Configuration

11.6 Expressions for Potentials for Three Electrode Configuration

11.7 Expression for Potentials for Seven Electrode Configurations

12 Electromagnetic Theory (Vector Potentials)

12.1 Introduction

12.2 Elementary Wavelet

12.3 Elliptic Polarisation of Electromagnetic Waves

12.4 Mutual Inductance

12.4.1 Mutual Inductance Between any Two Arbitrary Coils
12.4.2 Simple Mutual Inductance Model in Geophysics

12.5 Maxwell’s Equations

12.5.1 Integral form of Maxwell’s Equations

12.6 Helmholtz Electromagnetic Wave Equations

12.7 Hertz and Fitzerald Vectors

12.8 Boundary Conditions in Electromagnetics

12.8.1 Normal Component of the Magnetic Induction B is Continuous Across the Boundary in a Conductor
12.8.2 Normal Component of the Electric Displacement is Continuous Across the Boundary
12.8.3 Tangential Component of E is Continuous Across the Boundary
12.8.4 Tangential Component of H is Continuous Across the Boundary
12.8.5 Normal Component of the Current Density is Continuous Across the Boundary
12.8.6 Scalar Potentials are Continuous Across the Boundary

12.9 Poynting Vector

13.1 Plane Wave Propagation

13.1.1 Advancing Electromagnetic Wave
13.1.2 Plane Wave Incidence on the Surface of the Earth

13.2 Skin Depth

13.3 PerturbationCentroidFrequency

13.4 Magnetotelluric Response for a Layered Earth Model

13.5 Electromagnetic Field due to a Vertical Oscillating Electric Dipole

13.6 Electromagnetic Field due to an Oscillating Vertical Magnetic Dipole Placed on the Surface of the Earth

13.7 Electromagnetic Field due to an Oscillating Horizontal Magnetic Dipole Placed on the Surface of the Earth

13.8 Electromagnetic Field due to a Long Line Cable Placed in an Infinite and Homogenous Medium

13.9 Electromagnetic Field due to a Long Cable on the Surface of a Homogeneous Earth

13.10 Electromagnetic Induction due to an Infinite Cylinder in an Uniform Field

13.10.1 Effect of Change in Frequency on the Response Parameter

13.11 Electromagnetic Response due to a Sphere in the Field of a Vertically Oscillating Magnetic Dipole

13.12 Principle of Electrodynamic Similitude

14 Green’s Function

14.1 Introduction

14.2 Delta Function

14.3 Operators

14.4 Adjoint and Self Adjoint Operator

14.5 Definition of a Green’s Function

14.6 FreeSpaceGreen’sFunction

14.7 Green’s Function is a Potential due to a Charge of Unit Strength in Electrostatics

14.8 Green’s Function can Reduce the Number of unknowns to be Determined in a Potential Problem

14.9 Green’s Function has Some Relation with the Concept of Image in Potential Theory

14.10 Reciprocity Relation of Green’s Function

14.11 Green’s Function as a Kernel Function in an Integral Equation

14.12 Poisson’s Equation and Green’s Function

14.13 Problem1

14.14 Problem2

14.15 Problem3

14.16 Dyadics

15 Numerical Methods in Potential Theory

15.1 Introduction

15.2 Finite Difference Formulation/Direct Current Domain (Surface Geophysics)

15.2.1 Introduction
15.2.2 Formulation of the Problem
15.2.3 Boundary Conditions
15.2.4 StructureoftheFDBoundaryValueProblem
15.2.5 Inverse Fourier Cosine Transform
15.2.6 Calibration

15.3 Finite Difference Formulation Domain with Cylindrical Symmetry DC Field Borehole Geophysics

15.3.1 Introduction
15.3.2 Formulation of the Problem
15.3.3 Boundary Conditions
15.3.4 Grid Generation for Discretization
15.3.5 Finite Difference Equations
15.3.6 Current Density Factor q at the Source
15.3.7 Evaluation of the Potential

15.4 Finite Difference Formulation Plane Wave Electromagnetics Magnetotellurics

15.4.1 Boundary Conditions

15.5 Finite Element Formulation Direct Current Resistivity Domain

15.5.1 Introduction
15.5.2 Derivation of the Functional from Power Considerations
15.5.3 Equivalence between Poisson’s Equation and the Minimization of Power
15.5.4 Finite Element Formulation
15.5.5 Minimisation of the Power

15.6 3D Model

15.7 Finite Element Formulation Galerkin’s Approach Magnetotellurics

15.7.1 Introduction
15.7.2 Finite Element Formulation for Helmholtz Wave Equations
15.7.3 Element Equations

15.8 Finite Element Formulation Galerkin’s Approach Isoparametric Elements Magnetotellurics

15.8.1 Introduction
15.8.2 Finite Element Formulation
15.8.3 Shape Functions Using Natural Coordinates (ξ η)
15.8.4 Coordinate Transformation

15.9 Integral Equation Method

15.9.1 Introduction
15.9.2 Formulation of an Electromagnetic Boundary Value Problem

16 Analytical Continuation of Potential Field

16.1 Introduction

16.2 Downward Continuation by Harmonic Analysis of Gravity Field

16.3 Taylor’s Series Expansion and Finite Difference Approach for Downward Continuation

16.3.1 Approach A
16.3.2 Approach B
16.3.3 An Example of Analytical Continuation Based on Synthetic Data

16.4 Green’s Theorem and Integral Equations for Analytical Continuation

16.5 Analytical Continuation using Integral Equation and Taking Areal Averages

16.5.1 Upward Continuation of Potential Field
16.5.2 Downward Continuation of Potential Field (Peters Approach)

16.6 Upward and Downward Continuation using Integral Equation and Lagrange Interpolation Formula

16.7 Downward Continuation of Telluric Current Data

16.8 Upward and Downward Continuation of Electromagnetic Field Data

16.9 Downward Continuation of Electromagnetic Field

16.9.1 Downward Continuation of Hz

17 Inversion of Potential Field Data

17.1 Introduction

17.2 Wellposedand Illposed Problems

17.3 Tikhnov’s Regularisation

17.4 Abstract Spaces

17.4.1 N–Dimensional Vector Space
17.4.2 NormofaVector
17.4.3 Metric Space
17.4.4 Linear System
17.4.5 Normed Space
17.4.6 Linear Dependence and Independence
17.4.7 Inner Product Space
17.4.8 Hilbert Space

17.5 Some Properties of a Matrix

17.5.1 Rank of a Matrix
17.5.2 Eigen Values and EigenVectors
17.5.3 Properties of the Eigen Values

17.6 Lagrange Multiplier

17.7 Singular Value Decomposition (SVD)

17.8 LeastSquaresEstimator

17.9 Ridge Regression Estimator

17.10 Weighted Ridge Regression

17.11 Minimum Norm Algorithm for an Under Determined Problem

17.11.1 Norm
17.11.2 Minimum Norm Estimator

17.12 Bachus – Gilbert Inversion

17.12.1 Introduction
17.12.2 B-G Formulation

17.13 Stochastic Inversion

17.13.1 Introduction
17.13.2 Conjunction of the State of Information
17.13.3 Maximum Likelyhood Point

17.14 Occam’s Inversion

17.15 Global Optimization

17.15.1 Introduction
17.15.2 Monte Carlo Inversion
17.15.3 Simulated Annealing
17.15.4 Genetic Algorithm

17.16 Neural Network

17.16.1 Introduction
17.16.2 Optimization Problem

17.17 Joint Inversion