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This book provides the necessary background of geometry, mathematics and physical geodesy, useful to a rigorous approach to geodetic heights. The concept of height seems to be intuitive and immediate, but on the contrary it requires a good deal of scientific sharpness in the definition and use. As a matter of fact the geodetic, geographic and engineering practice has introduced many different heights to describe our Earth physical reality in terms of spatial position of points and surfaces. This has urged us to achieve a standard capability of transforming one system into the other. Often this is done in an approximate and clumsy way. This book solves the above practical problems in a rigorous way, showing what degree of approximation is used in approximate formulas. In addition the book gives a sound view on a matter that is presently occupying scientific associations, namely the unification of the global and regional height reference systems. It provides the mathematical background as well as the state of the art of its implementation. It will be particularly useful for professionals and national agencies.
1 Introduction.- 2 General coordinates in R(3).- 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 (V, , W).- 5.4 The Helmert triad (V, , 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 L in terms of dynamic heights. 6.3 The observation equation of L in terms of normal heights.- 6.4 The observation equation of L 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.