Hofmann-Wellenhof B., Moritz H. Physical Geodesy

Springer, Wien, 2005, 403 pages"Physical Geodesy" by Heiskanen and Moritz, published in 1967, has for a long time been considered as the standard introduction to its field. The enormous progress since then, however, required a complete reworking. While basic material could be retained other parts required a complete update. This concerns, above all, the adaptation to the fact that the geometry can now be precisely determined by methods such as GPS, and that new satellite methods, combined with terrestrial methods, also make a detailed determination of the earth's gravitational field a possibility and a necessity.Fundamentals of potential theory.
Attraction and potential.
Potential of a solid body.
Harmonic function.
Laplace's equation in spherical coordinates.
Spherical harmonics.
Surface spherical harmonics.
Legendre's functions.
Legendre's functions of the second kind.
Fully normalized spherical harmonics.
Expansion of the reciprocal distance into zonal harmonics and decomposition formula.
Solution of Dirichlet's problem by means of spherical harmonics and Poisson's integral.
Other boundary-value problems.
The radial derivative of a harmonics function.
Laplace's equation in ellipsoidal-harmonic coordinates.
Ellipsoidal harmonics.
Gravity field of the earth.
Gravity.
Level surfaces and plump lines.
Curvature of level surfaces and plump lines.
Natural coordinates.
The potential of the earth in terms of spherical harmonics.
Harmonics of lower degree.
The gravity fields of the level ellipsoid.
Normal gravity.
Expansion of the normal potential in spherical harmonics.
Series expansions for the gravity field.
Reference ellipsoid numerical values.
Anomalous gravity field, geoidal undulations, and deflections of the vertical.
Spherical approximation and expansion of the disturbing potential in spherical harmonics.
Gravity anomalies outside the earth.
Stoke's formula.
Explicit form of Stoke's integral and Stoke's function in spherical harmonics.
Generalization to an arbitrary reference ellipsoid.
Gravity disturbances and Koch's formula.
Deflections of the vertical and formula of Vening Meinesz.
The vertical gradient of gravity.
Practical evaluation of the integral formulas.
Gravity reduction.
Introduction.
Auxiliary formulas.
Free-air reduction.
Bouguer reduction.
Poincare and Prey reduction.
Isostatic reduction.
The indirect reduction of Rudzki.
The condensation of Helmert.
Heights.
Spirit leveling.
Geopotential numbers and dynamic heights.
Orthometric heights.
Normal heights.
Comparision of different heights.
GPS leveling.
The geometry of the earth.
Overview.
Introduction.
Global reference system after GPS.
The Global Positioning System.
From GPS to coordinates.
Projection onto the ellipsoid.
Coordinate transformations.
Geodetic datum transformations.
Three-dimensional geodesy: a transition.
The three-dimensional geodesy of Bruns and Hotine.
Global coordinates and local level coordinates.
Combining terrestrial and GPS.
Local geodetic datums.
Formulation of the problem.
Reduction of the astronomical measurements to the ellipsoid.
Reduction of horizontal and vertical angles and of distances.
The astrogeodetic determination of the geoid.
Reduction for the curvature of the plumb line.
Best-fitting ellipsoids and the mean earth ellipsoid.
Gravity field outside the earth.
Introduction.
Normal gravity vector.
Gravity disturbance vector from gravity anomalies.
Gravity disturbances by upward continuation.
Additional considerations.
Gravity anomalies and disturbances compared.
Space methods.
Introduction.
Satellite orbits.
Determination of zonal harmonics.
Rectangular coordinates of the satellite and perturbations.
Determination of tesseral harmonics and station positions.
New satellite gravity missions.
Modern views on the determination of the figure of the earth.
Introduction.
Gravimetric methods.
Gravity reductions and the geoid.
Geodetic boundary-value problems.
Molodensky's approach and linearization.
The spectral's case.
Solution by analytical continuation.
Deflections of the vertical.
Gravity disturbances: the GPS case.
Gravity reduction in the modern theory.
Determination of the geoid from ground-level anomalies.
A first balance.
Astrogeodetic methods according to Molodensky.
Some background.
Astronomical leveling revisited.
Topographic-isostatis reduction of vertical deflections.
The meaning of the geoid.
Statistical methods in physical geodesy.
Introduction.
The covariance function.
Expansion of the covariance function in spherical harmonics.
Interpolation and extrapolation of gravity anomalies.
Accuracy of prediction methods.
Least-squares prediction.
Correlation with height.
Least-squares collocation.
Principles of least-squares collocation.
Application of collocation to geoid determination.
Computational methods.
The remove-restore principle.
Geoid in Austria by collocation.
Molodensky corrections.
The geoid on the internet.
References.
Subject index.