Physical Geodesy: A Theoretical Introduction

Preface
Contents
List of Figures
List of Tables
1 The Earth's Gravity Field and Some of Its Properties
1.1 Gravitational Attraction, Centrifugal Acceleration, and Gravity
1.1.1 Gravitational Attraction
Gravitational Attraction of a Point Mass
Gravitational Attraction of a System of Point Masses
Gravitational Attraction of a Material Surface
Gravitational Attraction of a Solid Body
1.1.2 Centrifugal Acceleration and Gravity
Centrifugal Acceleration
Gravity
Unit
The Concept of Field
1.2 Gravitational Potential, Centrifugal Potential,and Gravity Potential
1.2.1 Definition of Potential of a Vector
1.2.2 Gravitational Potential
1.2.3 Gravitational Potential of a Dipoleor a Double Layer
Gravitational Potential of a Dipole
Gravitational Potential of a Double Layer
1.2.4 Centrifugal Potential and Gravity Potential
1.3 Equipotential Surface of Gravity, Geoid, and Orthometric Height
1.3.1 Equipotential Surface of Gravity
1.3.2 Geoid, Dynamic, and Orthometric Heights
1.4 Gravitational Potential and Attraction of Some Simple Configurations
1.4.1 Homogeneous Spherical Surface
1.4.2 Homogeneous Sphere
1.4.3 Homogeneous Disk
1.4.4 Homogeneous Cylinder
1.5 Some Properties of the Gravitational Potential and Attraction
1.5.1 Regularity at Infinity
1.5.2 Material Surface: Continuity of Potential and Discontinuity of Attraction
1.5.3 Double Layer: Discontinuity of Potential
1.5.4 Solid Body: Continuity of Potential and Attraction
1.5.5 Laplace's Equation of the Gravitational Potential Outside the Attracting Mass
1.5.6 Poisson's Equation of the Gravitational Potential Inside a Solid Body
1.6 Curvature of Equipotential Surfaces of Gravity and Plumb Lines
1.6.1 Definition of Curvature
1.6.2 Curvature of an Equipotential Surface of Gravity
1.6.3 Curvature of a Plumb Line
Further reading
2 Elementary Potential Theory
2.1 Green's Identities
2.1.1 Green's First and Second Identities for anInternal Domain
2.1.2 Green's First and Second Identities for anExternal Domain
2.1.3 Green's Third Identity
When P is Outside
When P Is Inside
When P Is on the Surface
Summary
2.2 Some Applications of Green's Identities
2.2.1 Gauss' Law of Gravity
2.2.2 Determination of the Earth's Mass Using Gravity
2.2.3 Expression of the External Gravitational Potential as Surface Integrals
2.2.4 Stokes' Theorem
2.3 Boundary Value Problems and Uniqueness of Solution
2.3.1 Three Types of Boundary Value Problems
2.3.2 Uniqueness of Solution
2.4 Solution of the First-Type Boundary Value Problem
2.4.1 Method of Green's Function
2.4.2 Poisson Integral
2.5 Gradient of the Gravitational Attraction inSpherical Coordinates
2.5.1 Gradients of Scalars and Vectors and Their Coordinate Transformations
2.5.2 Spherical Coordinates
2.5.3 Gradients of Scalars and Vectors inSpherical Coordinates
2.5.4 Divergence and Curl of a Vector
2.5.5 Application to the Gravitational Field
Reference
Further Reading
3 Spherical Harmonics
3.1 Separation of Variables of Laplace's Equation in Spherical Coordinates
3.1.1 Laplace's Equation in Spherical Coordinates
3.1.2 Method of Solution by Separating Variables
3.1.3 The Concept of Eigenvalue Problem
3.2 Legendre Function
3.2.1 Power Series Solutions of the Legendre Equation
3.2.2 Convergence of the Power Series Solutions
3.2.3 Legendre Function
3.2.4 Rodrigues' Formula
3.2.5 A Generating Function
3.2.6 Some Recurrence Formulae
3.3 Associated Legendre Function
3.3.1 A Relation Between the Legendre and Associated Legendre Equations
3.3.2 Associated Legendre Function
3.3.3 Zeros of an Associated Legendre Function
3.3.4 Recurrence Formulae
3.3.5 Orthogonality
3.4 Spherical Harmonics
3.4.1 Expansion of a Harmonic Function as Spherical Harmonics Series
3.4.2 Geometrical Properties
3.4.3 Orthogonality
3.4.4 Addition Theorem and Azimuthal Average of Spherical Harmonics
3.5 Spherical Harmonic Series of the Gravitational Potential
3.5.1 Spherical Harmonic Series of the Gravitational Potential of a Solid Body
3.5.2 Properties of Some Lower Degree and Order Potential Coefficients
3.5.3 The Center of Mass and Principal Moment of Inertia Coordinate System
3.5.4 MacCullagh's Formula
3.5.5 Gravitational Attraction and Its Gradient in Spherical Harmonic Series
3.5.6 Fully Normalized Associated Legendre Function and Its Computation
3.6 Spherical Harmonic Series of a Functions on a Sphere
3.6.1 Definition of the Series
3.6.2 Partial Sum of the Series
3.6.3 An Auxiliary Formula
3.6.4 Convergence of the Series
3.6.5 Expression of a Function as a Spherical Harmonic Series
3.6.6 Application: An Alternative Derivation of the Poisson Integral
Reference
Further Reading
4 The Normal Gravity Field and Reference Earth Ellipsoid
4.1 Basic Concepts
4.1.1 The Normal Gravity Field and Reference Earth Ellipsoid
4.1.2 Geocentric and Geodetic Coordinates
4.2 Internal Gravitational Field of a Homogeneous Ellipsoid of Revolution
4.2.1 The Gravitational Attraction Components Fx and Fy
4.2.2 The Gravitational Attraction Component Fz
4.2.3 Gravitational Potential in the Interior
4.3 External Gravitational Field of a Homogeneous Ellipsoid of Revolution
4.4 Gravitational Potential of a Homogeneous Ellipsoidal Homoeoid of Revolution
4.4.1 Gravitational Potential in the Interior
4.4.2 Gravitational Potential in the Exterior
4.4.3 Surface Density
4.5 The Earth's Normal Gravity Field
4.5.1 Maclaurin Ellipsoid
4.5.2 Normal Gravitational Potential
4.5.3 The Reference Earth Ellipsoid
4.5.4 Normal Gravity on the Reference Earth Ellipsoid
4.5.5 Normal Gravity Above the Reference Earth Ellipsoid
4.5.6 Graphical Representation
4.6 Second Order Approximate Formulae
4.6.1 The Normal Potential
4.6.2 Normal Gravity on the Reference Earth Ellipsoid
4.6.3 Normal Gravity Above the Reference Earth Ellipsoid
4.7 Parameters of the Reference Earth Ellipsoid
References
5 Stokes' Theory and Beyond
5.1 Stokes' Boundary Value Problem
5.1.1 Disturbing Potential and Gravity Disturbance
5.1.2 Geoidal Height and Deflection of the Vertical
5.1.3 Gravity Anomaly and Fundamental GravimetricEquation
5.1.4 Stokes' Boundary Value Problem
5.2 Solution of Stokes' Boundary Value Problem
5.2.1 Solution of the Disturbing Potential and Geoidal Height
5.2.2 Solution of the Gravity Disturbance and Deflection of the Vertical
5.2.3 Gravity Anomaly Outside the Geoid
5.3 Boundary Value Theories Beyond That of Stokes
5.3.1 Formulation When the Gravity Disturbance Is Known
5.3.2 Formulation When the Deflection of the VerticalIs Known
5.3.3 Formulation When the Geoidal Height Is Known
5.4 Inclusion of Errors of the Reference Earth Ellipsoid in Stokes' Theory
5.4.1 Fundamental Relations
5.4.2 Solution of Disturbing Potential, Geoidal Height, and Deflection of the Vertical
5.4.3 Relation Between the Reference Earth Ellipsoid and the Geoid
5.5 Some Characteristics of the Earth's Gravitational Field
5.5.1 A Global Model of the Gravitational Potential
5.5.2 Degree Power Spectrum of the Gravitational Potential and Geoidal Height
5.5.3 Degree Power Spectrum of the Gravitational Attraction and Deflection of the Vertical
References
6 Gravity Reduction
6.1 Basic Corrections and Gravity Anomalies
6.1.1 Free Air Correction and Free Air Anomaly
6.1.2 Plate Correction and Incomplete Bouguer Anomaly
6.1.3 Terrain Correction and Complete Bouguer Anomaly
6.1.4 A Practical Formula for Computing Orthometric Height Using Spirit Leveling Data
6.1.5 Helmert Condensation and Helmert Anomaly
6.2 Isostasy, Isostatic Correction, and Isostatic Gravity Anomaly
6.2.1 Background Fact and the Pratt–Hayford Model
6.2.2 The Airy–Heiskanen Model
6.2.3 The Vening Meinesz Model
6.2.4 Isostatic Correction and Isostatic Anomaly
6.2.5 Determination of Isostatic Models: The Concept
6.3 Gravitational Field of a Layer of Mass Around theEarth's Surface
6.3.1 A Laterally Heterogeneous Layer of Mass Around the Earth's Surface
Basic Equations
Gravitational Potential in Spherical Harmonic Series
Gravitational Potential in Integral Form
Gravitational Attraction in Integral Form
6.3.2 A Homogeneous Cap-Shaped Shell Around the Earth's Surface
Exact Formulae
Approximate Formulae
6.4 Gravity Reduction for the Spherical Earth Model
6.4.1 Some Relations Between the Formulations for the Flat and Spherical Earth Models
The Bouguer Plate Corrections
Density of the Helmert Condensation Layer
Isostatic Compensation Depth
6.4.2 The Complete Bouguer Anomaly
6.4.3 The Helmert Anomaly
6.4.4 The Isostatic Anomaly
6.5 Indirect Effect of Gravity Reduction on the Gravity Field
6.5.1 Indirect Effect and Computational Procedure of the Earth's Gravity Field
6.5.2 Indirect Effect of the Global Gravitational Potential, the Topographic–Isostatic Model
The Helmert Anomaly
The Isostatic Anomaly
The Topographic–Isostatic Model
6.5.3 Indirect Effect of the Geoidal Height
The Helmert Anomaly
The Isostatic Anomaly
6.5.4 Indirect Effect of the Deflection of the Vertical
Preparatory Formulations
The Helmert Anomaly
The Isostatic Anomaly
References
Further Reading
7 Molodensky's Theory and Beyond
7.1 Molodensky's Boundary Value Problem
7.1.1 Normal Height, Height Anomaly, Telluroid,and Quasigeoid
7.1.2 Molodensky's Gravity Anomaly and Gravimetric Boundary Value Problem
7.1.3 A Generalized Definition of Gravity Anomaly in Space
7.2 Molodensky's Solution
7.2.1 Molodensky's Integral Equation
7.2.2 Molodensky's Shrinking
7.2.3 Solution of the Height Anomaly
7.2.4 Solution of the Deflection of the Vertical
7.2.5 Reduction of the Solution to a Practical Form
7.3 Solution by Downward Continuation
7.3.1 A Property of Molodensky's Solution
7.3.2 Analytical Downward Continuation to a Surface Through the Point of Interest
7.3.3 A Discussion on the Possible Analytical Downward Continuation to the Quasigeoid
7.3.4 An Iterative Approach of Harmonic Downward Continuation
7.4 Gravity Reduction
7.4.1 General Description
7.4.2 Indirect Effect of the Height Anomaly
The Helmert Anomaly
The Isostatic Anomaly
7.4.3 Indirect Effect of the Deflection of the Vertical
The Helmert Anomaly
The Isostatic Anomaly
7.4.4 Downward Continuation After Gravity Reduction: A Conceptual Procedure
7.5 The GNSS-Gravimetric Boundary Value Theory
7.5.1 The Gravity Disturbance and the GNSS-Gravimetric Boundary Value Problem
7.5.2 Solution of Disturbing Potential by Molodensky'sShrinking
7.5.3 Simplification of the Solution of the Disturbing Potential
7.5.4 Solutions of the Height Anomaly and the Deflection of the Vertical
7.5.5 Analytical Downward Continuation and GravityReduction
7.6 Combination of Different Types of Data
7.6.1 Conversion of Different Types of Data into Gravity Anomaly
7.6.2 Conversion of Different Types of Data into Gravity Disturbance
Appendix: Some Remarks on the Use of Runge's Theorem
References
Further Reading
8 Fundamentals of Computation and Determination
8.1 Preparation of Gravity Data
8.1.1 Transformation Between Different Geodetic Reference Frames
Transformation of Geodetic Coordinates and Gravity Disturbance
Transformation of Orthometric or Normal Height and Gravity Anomaly
8.1.2 Evaluation of Surface Integrals in Gravity Reduction
8.1.3 Interpolation of Point-Wise Gravity Anomaly
8.1.4 Computation of Grid Mean Gravity Anomaly
8.2 Computation of the Global Model of the Gravitational Potential
8.2.1 Computation by Evaluating Integrals
8.2.2 Computation of Integrals of the Associated Legendre Function
8.2.3 Pellinen and Gaussian Smoothing
General Formulation of Isotropic Smoothing
Pellinen Smoothing
Gaussian Smoothing
The Ideal Filter
8.2.4 De-smoothing of the Grid Mean Gravity Anomaly
8.2.5 Computation of Gravitational Potential Based on Least Squares Fitting
8.2.6 Geoidal Height, Height Anomaly, and Deflection of the Vertical Based on a Global Model of the Gravitational Potential
8.3 Computation of the Geoidal Height, Height Anomaly, and Deflection of the Vertical Using Gravity Data
8.3.1 Geoidal Height and Height Anomaly
8.3.2 The Deflection of the Vertical
8.4 Astro-geodetic Determination of the Deflection of the Vertical
8.4.1 Astro-geodetic Measurement of the Deflection of the Vertical
8.4.2 Interpolation Based on the Gravimetric Deflection of the Vertical
8.4.3 Interpolation Based on Isostatic Deflectionof the Vertical
8.5 Astro-geodetic Determination of Geoidal Height and Height Anomaly
8.5.1 Astronomical Leveling
Determination of Geoidal Height
Determination of Height Anomaly
8.5.2 Astro-gravimetric Leveling
8.5.3 GNSS and GNSS-Gravimetric Leveling
References
Further Reading
9 Flattening and Gravity Inside the Earth
9.1 Hydrostatic Equilibrium and the Spherical Earth Model
9.2 Gravitational Potential of a Homogeneous Aspherical Shell
9.3 Gravitational Potential of a Heterogeneous Aspherical Body
9.4 Clairaut's Equation of the Earth's Internal Flattening
9.5 An Analytic Approximate Solution of Clairaut's Equation
9.6 Earth's Internal Flattening and Gravity
Reference
Further Reading
A Supplementary Materials
A.1 Spherical Trigonometry
A.2 Some Elementary Properties of the Reference Earth Ellipsoid
A.3 Radii of Curvature of Meridians and Prime Verticals
A.4 Transformation Between Geodetic and Cartesian Coordinates
A.5 Coordinate Transformation Between CartesianCoordinate Systems
B General Literature
Index