1 Introduction

1.1 Definition of geodesy

According to the classical definition of Friedrich Robert Helmert (1880), “geodesy (γη = Earth, δαιω = I divide) is the science of the measurement and mapping of the Earth’s surface”. Helmert’s definition is fundamental to geodesy, even today. The surface of the Earth, to a large extent, is shaped by the Earth’s gravity, and most geodetic observations are referenced to the Earth’s gravity field. Consequently, the above definition of geodesy includes the determination of the Earth’s external gravity field. Since ancient times, the reference system for the survey of the Earth has been provided by extraterrestrial sources (stars). This demands that the Earth’s orientation in space be implied into the focus of geodesy. In recent times, the objective of geodesy has expanded to include applications in ocean and space research. Geodesy, in collaboration with other sciences, is also now involved in the determination of the surfaces and gravity fields of other celestial bodies, such as the moon (lunar geodesy) and planets (planetary geodesy). Finally, the classical definition has to be extended to include temporal variations of the Earth’s figure, its orientation, and its gravity field.

With this extended definition, geodesy is part of the geosciences and engineering sciences, including navigation and geomatics (e.g., Nat. Acad. Sciences, 1978; Plag and Pearlman, 2009; Herring, 2015). Geodesy may be divided into the areas of global geodesy, geodetic surveys (national and supranational), and plane surveying. Global geodesy includes the determination of the shape and size of the Earth, its orientation in space, and its external gravity field. A geodetic survey deals with the determination of the Earth’s surface and gravity field over a region that typically spans a country or a group of countries. The Earth’s curvature and gravity field must be considered in geodetic surveys. In plane surveying (topographic surveying, cadastral surveying, and engineering surveying), the details of the Earth’s surface are determined at a local level, and thus, curvature and gravity effects are most often ignored.

There is a close relation between global geodesy, geodetic surveying, and plane surveying. Geodetic surveys are linked to reference frames (networks) established by global geodesy, and they adopt the parameters for the figure of the Earth and its gravity field. On the other hand, the results of geodetic surveys contribute to global geodesy. Plane surveys, in turn, are generally referenced to control points established by geodetic surveys. They are used extensively in the development of national and state map-series, cadastral and geoinformation systems, and in civil engineering projects. The measurement and data evaluation methods applied in national geodetic surveys, nowadays, are mostly similar to those used in global geodetic work. In particular, space methods (satellite geodesy), which have long been a dominant technique in global geodesy, are now also commonly employed in regional and local surveys. This also requires a more detailed knowledge of the gravity field at regional and local scales.

With the corresponding classification in the English and French languages, the concept of “geodesy” (la géodésie, “höhere Geodäsie” after Helmert) in this text refers only to global geodesy and geodetic surveying. The concept of “surveying” (la topométrie, Vermessungskunde or “niedere Geodäsie” after Helmert) shall encompass plane surveying.

In this volume, geodesy is treated only in the more restrictive sense as explained above (excluding plane surveying), and is limited to the planet Earth. Among the numerous textbooks and handbooks on surveying, we mention Brinker and Minnick (2012), Kahmen (2006), Nadolinets et al. (2017), and Ghilani (2022). For lunar and planetary geodesy, see Nothnagel et al. (2010) and Wiezorek (2015), and also Petit and Luzum (2010) and Luzum et al. (2011).

1.2 The objective of geodesy

Based on the concept of geodesy defined in [1.1], the objective of geodesy with respect to the planet Earth may be described as follows:

The objective of geodesy is to determine the figure and external gravity field of the Earth, as well as its orientation in space, as a function of time, from measurements on and exterior to the Earth’s surface.

This geodetic boundary-value problem incorporates a geometric (figure of the Earth) and a physical (gravity field) part; both are closely related.

By the figure of the Earth, we mean the physical and the mathematical surface of the Earth as well as a geodetic reference model (e.g., Moritz, 1990).

The physical surface of the Earth is the border between the solid or fluid masses and the atmosphere. The ocean floor may be included in this definition, being the bounding surface between the solid terrestrial body and the oceanic water masses. The irregular surface of the solid Earth (continental and ocean floor topography) cannot be represented by a simple mathematical (analytical) function. Continental topography is therefore described pointwise by coordinates of control (reference) points. Given an adequately dense control network, the detailed structure of this surface can be determined by interpolation of data from spatial and terrestrial topographic and photogrammetric surveying and from hydrographic surveys (e.g., Hake et al., 2001; McGlone, 2013; Konecny, 2014; Luhmann et al., 2019). On the other hand, the ocean surface (70 % of the Earth’s surface) is easier to represent. If we neglect the effects of ocean currents and other “disturbances” like ocean tides [3.4.2], it forms a part of a level or equipotential surface of the Earth’s gravity field (surface of constant gravity potential). We may think of this surface as being extended under the continents and identify it as the mathematical figure of the Earth, which can be described by a condition of equilibrium (Helmert, 1880/1884). J. B. Listing (1873) designated this level surface as geoid [3.4.1].

The great mathematician, physicist, astronomer, and geodesist Carl Friedrich Gauss (1777–1855) had already referred to this surface: “Was wir im geometrischen Sinn Oberfläche der Erde nennen, ist nichts anderes als diejenige Fläche, welche überall die Richtung der Schwere senkrecht schneidet, und von der die Oberfläche des Weltmeers einen Theil ausmacht … ”, which reads in English translation: “What we call surface of the Earth in the geometrical sense is nothing more than that surface which intersects everywhere the direction of gravity at right angles, and part of which coincides with the surface of the oceans” (C. F. Gauss: Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona, Göttingen 1828. C. F. Gauss Werke, Band IX, Leipzig 1903, p. 49, see also Moritz, 1977).

The description of the external gravity field including the geoid represents the physical aspect of the problem of geodesy. In solving this problem, the Earth’s surface and the geoid are considered as bounding surfaces in the Earth’s gravity field. Based on the law of gravitation and the centrifugal force (due to the Earth’s rotation), the external gravity field of the Earth can be modeled analytically and described by a large number of model parameters. A geometric description is given by the infinite number of level surfaces extending completely or partially exterior to the Earth’s surface. The geoid as a physically defined Earth’s figure plays a special role in this respect.

Reference systems are introduced in order to describe the orientation of the Earth in space (celestial reference system) as well as its surface geometry and gravity field (terrestrial reference system). The definition and realization of these systems has become a major part of global geodesy; the use of three-dimensional Cartesian coordinates in Euclidean space is adequate in this context. However, due to the demands of users, reference surfaces are introduced. We distinguish between curvilinear surface coordinates for horizontal positioning and heights above some zero-height surface for vertical positioning. Due to its simple mathematical structure, a rotational ellipsoid, flattened at the poles, is well suited for describing horizontal positions, and consequently, it is used as a reference surface in geodetic surveying. In plane surveying, the horizontal plane is generally a sufficient reference surface. Because of the physical meaning of the geoid, this equipotential surface is well suited as a reference for heights. For many applications, a geodetic reference Earth (Earth model, normal Earth) is needed. It is realized through a mean-Earth ellipsoid that optimally approximates the geometry (geoid) and the gravity field of the Earth. Figure 1.1 shows the mutual location of the surfaces to be determined in geodesy.

The body of the Earth, its gravity field, and its...