Theory of the Earth's Shape, Volume 13

Preface

First Part Physical Geodesy

1. Origin, Structure and Form of the Earth

1.1 The Earth as a Planet of the Solar System

1.2 The Earth's Physical Structure

1.3 Brief Historical Survey of the Ideas Concerning the Earth's Form and Size

2 Elements of Potential Theory

2.1 Principles of Field Theory

2.2 Harmonic Functions

2.3 Newtonian Potential

2.4 Boundary-Value Problems of Potential Theory

3 Reference Surfaces. Coordinate Systems

3.1 Level Surfaces

3.2 The Level Spheroid

3.3 The Level Ellipsoid

3.4 Coordinate Systems

3.5 Altitude Systems

4 The Perturbing Potential

4.1 Bruns' Formula. The Third Boundary-Value Problem for the Geoid

4.2 The Third Boundary-Value Problem for the Earth's Physical Surface

4.3 Expansion of the Perturbing Potential in Spherical Harmonics. The Spherical Approximation of the Boundary Condition for the Geoid

4.4 The Gravity Anomaly in External Space

5 The Determination of the Geoid by Gravity Methods

5.1 Stokes' Formula

5.2 Generalization of Stokes' Formula

5.3 Determination of the Gravimetric Deflection of the Vertical

5.4 The Gravity Reduction

5.5 Types of Reductions Used in Physical Geodesy

5.6 Practical Determination of the Components of the Deflection of the Vertical and of the Geoid Undulations

6 Determination of the Geoid by Astro-Gravimetric Methods

6.1 The Astro-Geodetic Levelling

6.2 The Astro-Gravimetric Levelling

7 Determination of the Earth's Physical Surface

7.1 Molodenski's Problem

7.2 Determination of the Perturbing Potential by Using Green's Formulae

7.3 Solving the Main Integral Equation

7.4 Geometrical Interpretation of the Solution of Molodenski's Equation

7.5 The Deflection of the Vertical on the Earth's Physical Surface

Second Part Ellipsoidal Geodesy

8 The Rotation Ellipsoid as Reference Surface in Geodesy

8.1 The Parameters of the Reference Ellipsoid

8.2 The Parametric Equations of the Rotation Ellipsoid

9 Curves on the Surface of the Reference Ellipsoid

9.1 The Coordinate Lines

9.2 Normal Sections

9.3 The Geodetic Line

10 The Reduction of the Geodetic Observations on the Reference Ellipsoid's Surface

10.1 The Reduction of the Astronomical Observations

10.2 The Reduction of the Azimuthal Observations

10.3 The Reduction of the Zenithal Observations to the Normal to the Ellipsoid

10.4 The Reduction of the Distances on the Ellipsoid

11 Solving the Geodetic Triangles on the Reference Ellipsoid

11.1 The Spherical Excess

11.2 Solving Small Geodetic Triangles by Means of Legendre's Theorem

11.3 Solving Small Geodetic Triangles by the Additament Method

12 Calculation of the Geodetic Coordinates on the Reference Ellipsoid

12.1 Solving the Basic Geodetic Problems for Small Geodetic Distances

12.2 Gauss' Method (Method of the Mean Arguments) for Solving the Basic Geodetic Problems for Medium Geodetic Distances

12.3 Solving the Basic Geodetic Problems for Large Geodetic Distances

12.4 Differential Formulae

Third Part Astro-Geodetic Triangulation

13 The Notion of Performing Astro-Geodetic Triangulation

13.1 General Considerations

13.2 Error Propagation in the Astro-Geodetic Triangulation

14 Measurements in Astro-Geodetic Triangulation

14.1 Angle Measurements

14.2 Distance Measurements

14.3 Relative Measurements of Gravity

14.4 Levelling Measurements

14.5 Determinations of Geodetic Astronomy

15 Adjustment of the Astro-Geodetic Triangulation

15.1 General Considerations

15.2 Adjustment of the Triangulation Network by Means of the Method of Conditioned Observations

15.3 Adjustment of the Triangulation Network by Means of the Method of Indirect Observations

15.4 Weight Determination After Adjustment

15.5 Processing of the Observations in Free Geodetic Networks

16 The 1st-Order Astro-Geodetic Triangulation Network of the Socialist Republic of Romania

16.1 The Network Layout

16.2 Marking and Constructing the Station Points of the Network

16.3 Angle and Distance Measurements as well as Astro-Geodetic Determinations Carried out at the Station Points of the Network

16.4 The Network Adjustment

Fourth Part Three-Dimensional Geodesy

17 Basic Equations of Three-Dimensional Geodesy

17.1 Coordinate Systems Used in Three-Dimensional Geodesy

17.2 Origin, Methodology and Purpose of Three-Dimensional Geodesy

17.3 Basic Equations of Three-Dimensional Geodesy

18. Elementary Calculations in Three-Dimensional Geodesy

18.1 The Calculation Principle in the Local System

18.2 Choice of the Local Geodetic Trihedron

18.3 Calculating the Coordinates of a Point

18.4 Determining the Local Astronomical Vertical

18.5 Orientation Calculation in the Local System

18.6 Calculation of a Geodetic Traverse

18.7 Triangulation Calculations

19 Basic Methods for Developing the Geodetic Networks by Means of the Earth's Artificial Satellites

19.1 General Considerantions

19.2 Elements of an Artificial Satellite's Orbit

19.3 Geodetic Artificial Satellites

19.4 Methods of Observing Artificial Satellites

19.5 Spatial Triangulation

19.6 Spatial Trilateration

19.7 Vector Network

20 Triangulation Calculations Using Artificial Satellites

20.1 Identification of the Recorded Stars

20.2 Calculation of the Coordinates on the Photographic Plate of the Recorded Stars

20.3 Comparison of the Measured and Calculated Values

20.4 Linearization of the Satellite's Position on the Photographic Plate

20.5 Returning to the Global System

20.6 General Adjustment of a Spatial Triangulation by Means of Artificial Satellites

21 Some Examples of Spatial Triangulation Operations

21.1 The France — Algeria Connexion

21.2 The European Continent — Azores Islands Connexion

21.3 The World Network of Triangulation by Means of Artificial Satellites

21.4 Western European Triangulation by Means of Satellites (West)

21.5 Brief Survey of Other Triangulation Operations by Means of Artificial Satellites

22 General Survey of Other Geometrical Methods of Spatial Geodesy by Means of Artificial Satellites

22.1 The 4-Laser Method

22.2 The Optical-Telemetric Method

22.3 The Orbital Method

22.4 The Method of the Simultaneity Circle

22.5 Experiment of Using Laser Measurements

Fifth Part Methods for Determining the Terrestrial Ellipsoid and the Geoid

23 The Determination of the Reference Ellipsoid by Using Astro-Geodetic Methods

23.1 Determining the Reference Ellipsoid from Measurements of Meridian and Parallel Arcs

23.2 Determining the Reference Ellipsoid by Using the Method of Surfaces

23.3 Determining the Reference Ellipsoid by Using Differential Relations

24 The Determination of the Earth's Form from Astro-Geodetic and Gravity Measurements

24.1 Global Determination of the Geoid

24.2 Determining the Earth's Physical Constants

24.3 Determining the Gravity Field and the Parameters of the General Terrestrial Ellipsoid

25 Dynamic Geodesy

25.1 Non-Perturbed Orbits

25.2 Relationships Between the Elements of the Satellite Orbit and the Geocentric, Topocentric and Rectangular Coordinate Systems

25.3 Perturbed Orbits

25.4 The Luni — Solar Perturbation

25.5 Perturbations of the Artificial Satellites' Orbits Caused by the Resistance of the Atmosphere and by Radiation Pressure

25.6 Determining the Parameters of the Earth's Gravitational Field by Using Artificial Satellites

25.7 The Geoid Determination

25.8 Satellite Altimetry

25.9 Interpretation of the Results of Satellite Geodesy

26 The Geodetic Reference System

Sixth Part Determination of the Recent Movements of the Earth's Crust

27 Geodetic Methods for Determining the Recent Movements of the Earth's Crust

27.1 Introductory Remarks

27.2 Repeated Geometrical Levelling

27.3 Triangulation-Trilateration Networks for Determining the Horizontal Movements of the Earth's Crust

27.4 Astro-Geodetic Methods for Determining the Movements of the Continents

27.5 Other Methods Used for Determining the Crustal Movements

28 Processing the Geodetic Observations Carried Out for Determining the Recent Crustal Movements

28.1 A Functional-Stochastic Model for the Three-Dimensional Determination of the Recent Movements of the Earth's Crust within Areas of Limited Size

28.2 Particular Functional-Stochastic Models for Determining the Recent Crustal Movements

28.3 Possibilities of Adjusting the Repeated Geodetic Observations

28.4 Accuracy Estimation and Statistical Analysis of the Determinations of Recent Crustal Movements

29 Examples of Daterminations of Recent Movements of the Earth's Crust

29.1 Map of the Recent Crustal Movements in Eastern Europe

29.2 Map of the Vertical Crustal Movements on the Territory of the Socialist Republic of Romania

29.3 Utilization of Some Improved Functional-Stochastic Models for Determining the Vertical Crustal Movements

29.4 Other Examples of Determining Recent Movements of the Earth's Crust

Future Prospects in the Light of Present Geodetic Achievements

References

Abbreviations

Index

Theory of the Earth's Shape considers the physical-mathematical problems raised by the determination of the form of the planet, thereby making a significant contribution to the technological scientific literature in this field. This book is organized into six parts encompassing 29 chapters. The first part, entitled Physical Geodesy, presents the theory of the determination of the gravitational field, in the definition of which preference was given to the method of expansion in spherical harmonics recommended by the International Union of Geodesy and Geophysics in establishing the international "Geodetic Reference System 1967". Part II deals with the principal aspects of Ellipsoidal Geodesy, such as the methods of solving the geodetic problems on the reference ellipsoid. Part III considers the main problems associated with Astro-geodetic Triangulation, particularly with the conception of materialization and the necessary measurements as the required adjustment procedures. This part also provides approaches regarding the controlled analysis of angular measurements and the description of some original calculation and measurement methods. Part IV concerns one of the methods of determining the spatial coordinates of the geodetic points in a unitary system, such as the three-dimensional geodesy, which has had more concrete applications since the launching of the Earth's first artificial satellites. Part V describes the methods for determining the terrestrial ellipsoid and the geoid, as well as the conventional methods and the methods of Dynamical Geodesy. Part VI discusses the geodetic methods for the determination of the movements of the Earth's crust, along with an overall examination of the theoretical and practical aspects which in principle constitute the object of such activities. This book will prove useful to geophysicists, astronomers, Earth scientists, and researchers.