Vermeer M. (2020). Physical geodesy. Aalto University publication series. http://urn.fi/URN:ISBN:978-952-60-8940-9

Content

1. Fundamentals of the theory of gravitation

1.1 General

1.2 Gravitation between two masses

1.3 The potential of a point mass

1.4 Potential of a spherical shell

1.5 Computing the attraction from the potential

1.6 Potential of a solid body

1.7 Example: The potential of a line of mass

1.8 The equations of Laplace and Poisson

1.9 Gauge invariance

1.10 Single mass-density layer

1.11 Double mass-density layer

1.12 The Gauss divergence theorem

1.13 Green’s theorems

1.14 The Chasles theorem

1.15 Boundary-value problems

Self-test questions

Exercise 1–1: Core of the Earth

Exercise 1–2: Atmosphere

Exercise 1–3: The Gauss divergence theorem

2. The Laplace equation and its solutions

2.1 The nature of the Laplace equation

2.2 The Laplace equation in rectangular co-ordinates

2.3 The Laplace equation in polar co-ordinates

2.4 Spherical, geodetic, ellipsoidal co-ordinates

2.5 The Laplace equation in spherical co-ordinates

2.6 Dependence on height

Self-test questions

3. Legendre functions and spherical harmonics

3.1 Legendre functions

3.2 Symmetry properties of spherical harmonics

3.3 Orthogonality of Legendre functions

3.4 Low-degree spherical harmonics

3.5 Splitting a function into degree constituents

3.6 Spectral representations of various quantities

3.7 Often-used spherical-harmonic expansions

3.8 Ellipsoidal harmonics

Self-test questions

Exercise 3–1: Attenuation with height of a spherical-harmonic expansion

Exercise 3–2: Symmetries of spherical harmonics

Exercise 3–3: Algebraic-sign domains of spherical harmonics

Exercise 3–4: Escapevelocity

4. The normal gravity field

4.1 The basic idea of a normal field

4.2 The centrifugal force and its potential

4.3 Level surfaces and plumb lines

4.4 Natural co-ordinates

4.5 The normal potential in ellipsoidal co-ordinates

4.6 Normal gravity on the reference ellipsoid

4.7 Numerical values and calculation formulas

4.8 The normal potential as a spherical-harmonic expansion

4.9 The disturbing potential

Self-test questions

Exercise 4–1: The Somigliana–Pizzetti equation

Exercise 4–2: Centrifugal force

5. Anomalous quantities of the gravity field

5.1 Disturbing potential, geoid height, deflections of the plumb line

5.2 Gravity disturbances

5.3 Gravity anomalies

5.4 Units used for gravity anomalies

5.5 The boundary-value problem of physical geodesy

5.6 The telluroid mapping and the “quasi-geoid”

5.7 Free-air anomalies

Self-test questions

Exercise 5–1: The spectrum of gravity anomalies

Exercise 5–2: Deflections of the plumb line and geoid tilt

Exercise 5–3: Gravity anomaly, geoid height

6. Geophysical reductions

6.1 General

6.2 Bouguer anomalies

6.3 Terrain effect and terrain correction

6.4 Spherical Bouguer anomalies

6.5 Helmert condensation

6.6 Isostasy

6.7 Isostatic reductions

6.8 The “isostatic geoid”

Self-test questions

Exercise 6–1: Gravity anomaly

Exercise 6–2: Bouguer reduction

Exercise 6–3: Terrain correction and Bouguer reduction

Exercise 6–4: Isostasy

7. Vertical reference systems

7.1 Levelling, orthometric heights and the geoid

7.2 Orthometric heights

7.3 Normal heights

7.4 Difference between geoid height and height anomaly

7.5 Difference between orthometric and normal heights

7.6 Calculating orthometric heights precisely

7.7 Calculating normal heights precisely

7.8 Calculation example for heights

7.9 Orthometric and normal corrections

7.10 A vision for the future: relativistic levelling

Self-test questions

Exercise 7–1: Calculating orthometric heights

Exercise 7–2: Calculatingnormalheights

Exercise 7–3: Difference between orthometric and normal height

8. The Stokes equation and other integral equations

8.1 The Stokes equation and the Stokes integral kernel

8.2 Example: The Stokes equation in polar co-ordinates

8.3 Plumb-line deflections and Vening Meinesz equations

8.4 The Poisson integral equation

8.5 Gravity anomalies in the exterior space

8.6 The vertical gradient of the gravity anomaly

8.7 Gravity reductions in geoid determination

8.8 The remove–restore method

8.9 Kernel modification

8.10 Advanced kernel modifications

8.11 Block integration

8.12 Effect of the local zone

Self-test questions

Exercise 8–1: The Stokes equation in the near zone

9. Spectral techniques, FFT

9.1 The Stokes equation as a convolution

9.2 Integration by FFT

9.3 Solution in latitude and longitude

9.4 Bordering and tapering of the data area

9.5 Computing a geoid model with FFT

9.6 Use of FFT in other contexts

9.7 Computing terrain corrections with FFT

Self-test questions

10. Statistical methods

10.1 The role of uncertainty in geophysics

10.2 Linear functionals

10.3 Statistics on the Earth’s surface

10.4 The covariance function of the gravity field

10.5 Least-squares collocation

10.6 Prediction of gravity anomalies

10.7 Covariance function and degree variances

10.8 Propagation of covariances between various quantities

10.9 Global covariance functions

10.10 Collocation and the spectral viewpoint

Self-test questions

Exercise 10–1: Variance of prediction

Exercise 10–2: Hirvonen’s covariance equation and prediction

Exercise 10–3: Predicting gravity anomalies

Exercise 10–4: Predicting gravity anomalies (2)

Exercise 10–5: Propagation of covariances

Exercise 10–6: Kaula’s rule for gravity gradients

Exercise 10–7: Underground mass points

11. Gravimetric measurement devices

11.1 History

11.2 The relative or spring gravimeter

11.3 The absolute or ballistic gravimeter

11.4 Network hierarchy in gravimetry

11.5 The superconducting gravimeter

11.6 Gravity measurement and the atmosphere

11.7 Airborne gravimetry and GNSS

11.8 Measuring the gravity gradient

Self-test questions

Exercise 11–1: Absolute gravimeter

Exercise 11–2: Spring gravimeter

Exercise 11–3: Air pressure and gravity

12. The geoid, mean sea level, and sea-surface topography

12.1 Basic concepts

12.2 Geoid models and national height datums

12.3 The geoid and post-glacial land uplift

12.4 Determining the sea-surface topography

12.5 Global sea-surface topography and heat transport

12.6 The global behaviour of the sea level

12.7 The sea-level equation

Self-test questions

Exercise 12–1: Coriolis force, ocean current

Exercise 12–2: Land subsidence and the mechanism of land uplift

13. Satellite altimetry and satellite gravity missions

13.1 Satellite altimetry

13.2 Crossover adjustment

13.3 Choice of satellite orbit

13.4 In-flight calibration

13.5 Retracking

13.6 Oceanographic research using satellite altimetry

13.7 Satellite gravity missions

Self-test questions

Exercise 13–1: Altimetry, crossover adjustment

Exercise 13–2: Satellite orbit

Exercise 13–3: Kepler’s third law

14. Tides, the atmosphere, and Earth crustal movements

14.1 The theoretical tide

14.2 Deformation caused by the tidal potential

14.3 The permanent part of the tide

14.4 Tidal corrections between height systems

14.5 Loading of the Earth’s crust by sea and atmosphere

Self-test questions

Exercise 14–1: The permanent tide

15. Earth gravity field research

15.1 Internationally

15.2 Europe

15.3 The Nordic countries

15.4 Finland

15.5 Textbooks

A. Field theory and vector calculus — core knowledge

A.1 Vector calculus

A.2 Scalar and vector fields

A.3 Integrals

A.4 The continuity of matter

B. Function spaces

B.1 An abstract vector space

B.2 The Fourier function space

B.3 Sturm–Liouville differential equations

B.4 Legendre polynomials

B.5 Spherical harmonics

Self-test questions

Exercise B–1: Orthonormality of the Fourier basis functions

C. Why does FFT work?

D. Helmert condensation

D.1 The exterior potential of the topography

D.2 The interior potential of the topography

D.3 The exterior potential of the condensation layer

D.4 Total potential of Helmert condensation

D.5 The dipole method

E. The Laplace equation in spherical co-ordinates

E.1 Derivation

E.2 Solution

Bibliography

Index