Geodetic Heights

1 Introduction.- 2 General coordinates in R³.- 2.1 Outline.- 2.2 Definitions and reminds.- 2.3 Relevant examples.- 2.3.1 Caresian coordinates.- 2.3.2 Spherical corrdinates.- 2.3.3 Ellipsoidal coordinates.- 3 The Earth gravity field: basics.- 3.1 Outline.- 3.2 Basic definitions of gravity and gravity potential.- 3.3 Plumb lines and equipotential surfaces.- 3.4 The gravity field outside a Brillouin.- 3.5 The normal gravity field.- 3.6 Definition of the geoid.- 4 The anomalous potential and its determination.- 4.1 Outline.- 4.2 The anomalous potential.- 4.3 The Runge-Krarup theorem: a mathematical intermezzo.- 4.4 Optimal degree of global models, or smoothing by truncation.- 4.5 Collocation theory, or smoothing by prior information.- 4.6 On the relation between potential and the surface gravimetric observables.- 4.6.1 Gravimetry.- 4.6.2 Levelling combined with gravimetry.- 4.6.3 Radar-altimetry on the oceans.- 4.7 The Geodetic Boundary Value Problem (GBVP).- 5 Geodetic coordinate systems.- 5.1 Outline.- 5.2 On the contination of gravity into the topographic layer.- 5.3 The Hotine-Marussi triad ( ,  , W).- 5.4 The Helmert triad ( ,  , H).- 5.5 The Molodensky triad ( ,  , h)- 6 The relation between levelling, geodetic and other unholonomic heights.- 6.1 Outline.- 6.2 The observation equation of DeltaL in terms of dynamic heights. 6.3 The observation equation of DeltaL in terms of normal heights.- 6.4 The observation equation of DeltaL in terms of orthometric heights.- 6.5 Levelling and normal orthometric heights: an unholonomic coordinate.- 6.6 Conclusions.- 7 The height datum problem.- 7.1 Outline. 7.2 Formulation of the global unification of the height datum.- 7.3 On the solution of the unification problem by a suitable global model.- 7.4 On local solutions of the height datum problem.