Keywords

1 Introduction

Any height system requires two constituents: A horizontal reference surface upon which all heights are equal to zero, and a prescription for how observed heights and height differences will be related to that surface. This is consistent with standard works such as Heiskanen and Moritz (1967) who define heights as path lengths from points of interest to reference surfaces and prescribe methods for using levelling observations to define heights of points. In the modern world, satellite positioning observations of heights above a superficial surface, an ellipsoid, must also be related to a reference surface, which is done using models of geoid-ellipsoid separation (geoidal heights) or of height anomaly. While traditional approaches often focus on heights of points on the topographical surface, we here intentionally discuss how height systems define heights more generally for points anywhere in a three-dimensional space near Earth’s surface.

The traditional interpretation of height systems and of relationships between them applies concepts of potential fields and their corresponding gradient vector fields to describe the form of reference surfaces and the paths along which heights are measured. The mathematical apparatus of this approach has been well-developed over many years by a progression of authors (e.g. Gauss 1828; Molodensky et al. 1962; Hotine 1969; Marussi 1985; Sansò and Vanìček 2006; Sansò et al. 2019). In this paper, we apply a mathematically equivalent interpretation whereby for almost all common height systems a 3-manifold (a 3-dimensional space that is locally Euclidian) can be defined such that the height of a point is the length in a Euclidean space of the line of steepest descent following the gradient of the manifold from the point to a reference surface.

Each 3-manifold, if it is to be used in this way, should be Riemannian. This implies a metric space, where a distance function can be defined. The level surfaces of the manifold comprise a family of 2D subspaces, called horizontal in our case. The 1-dimensional subspaces, orthogonal to the family of 2D horizontal subspaced are identical to what Sansò et al. (2019) call the “lines of the vertical”, and we will adopt that terminology here. Precisely one line of the vertical goes through every point in the region of interest, and the height of a point is the length measured in this 1-dimensional subspace from that point to the reference surface. While several different “lines of the vertical” arise in different height systems, their variations in curvature and torsion are not very significant. Their differing lengths, associated with differing “height metrics” of the 1-dimensional subspaces, are their important characteristic and are part of the 3-dimensional metric of the manifold.

The idea of describing heights based on the characteristics of metric spaces is not new. The physical space and approximate physical spaces discussed here, for example, can be understood mathematically according to Hotine’s discussion of N surfaces as coordinates (Hotine 1969, Chap. 12), reflected in Sansò et al.’s discussion of the Hotine-Marussi coordinate triad (2019, Sect. 5.3). Here we invoke the interpretation of heights via three-dimensional Riemannian manifolds only to explore differences between height systems. Similar comparisons of the geometry implied of different height systems have been undertaken before (e.g. Heiskanen and Moritz 1967; Vaníček and Krakiwsky 1986; Featherstone and Kuhn 2006; Sansò et al. 2019), but not with the same framework of manifolds and mappings between them. We will not in this paper present a mathematical framework for these systems’ implementation; this work has already been done in the references cited above. The intent is to use the concept of metric spaces as a lens to assess existing height systems.

2 Common Height Systems Defined by Their Metric Space and Reference Surface

We will next use the context of metric spaces to discuss some of the common height systems. We will exclude geopotential numbers, focusing instead on systems that deal with heights in linear units. We will also mostly exclude dynamic heights. While these systems fit within the understanding of height systems outlined above, and are important for some practical engineering applications, they have been excluded in the interest of space.

We begin with geodetic height, also sometimes called the ellipsoidal height (wouldn’t these be heights of the ellipsoid above the ellipsoid?), which is used here to mean the vertical coordinate h in the triplet of geodetic coordinates (φ, λ, h). These heights are associated with a space we call the geometric space, G, which is a Euclidean manifold. Since the space is flat, no single line of steepest descent can be defined, and lines minimizing Euclidean distance (straight lines) are used instead. The reference surface is a reference ellipsoid, and the geodetic height h of a point is simply the shortest distance from the ellipsoid to the point. The system is illustrated in Fig. 1.

Fig. 1
figure 1

Geodetic height system, showing normal to the ellipsoid through point A, along which height is measured, as medium dashed line; and other lines to the ellipsoid from A as faint narrower dashed lines

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The geometric space is so called because it has a role in relating any other space to geometrical measurements on Earth’s surface. In particular, the gometrical space is used for describing the position of points, which is the ultimate concern in geodesy. Other quantities, such as gravity potential, may be used as coordinates to represent the curved intrinsic geometry of the other spaces involved, but they must ultimately be related back to positions. The geopotential numbers, which do not use linear units, still operate by specifying a geometric location along a line of the vertical – in particular, that point where a equipotential surface having a specific potential intersects the line. We may thus consider all other spaces to be embedded in the geometric space.

Next, we discuss. The orthometric height system. In this system, the metric space used for defining the lines of height is the physical space, which we will call P, and the reference surface is the geoid. The physical space is a Riemannian manifold with a shape defined by the gravity potential function W = W(x), where x is a coordinate triad representing the position of any point in the geometrical space. In P, the family of equipotential surfaces are the level surfaces, and the plumblines, which are everywhere orthogonal to those surfaces, will be lines of the vertical. In the orthometric heights, the geoid, which is one of the level surfaces of P, serves as the reference surface. The system is illustrated in Fig. 2.

Fig. 2
figure 2

A point A in the physical space, with plumblines and equipotential surfaces represented by thin grey lines, the geoid by a thick grey line, topography by a medium black line, and orthometric height of point A by a medium dotted black line

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The formula for orthometric height H O above the geoid induced by this definition is exactly the plumbline length given by the standard formula (Heiskanen and Moritz 1967, Eqs. 4–21):

$$ {H}^O=\raisebox{1ex}{$C$}\!\left/ \!\raisebox{-1ex}{$\overline{g}$}\right., $$
(1)

where C is the geopotential number and \( \overline{g} \) is the integral mean of the vertical gradient of gravity potential along the plumbline between the geoid and the point of interest. The denominator in Eq. (1) arises from the choice of the plumbline as the line of the vertical, and uniquely transforms the geopotential numbers associated with the physical space into Euclidean lengths in the geometrical space. Any other denominator, unless it is just a scaling of the denominator in Eq. (1) by a constant value, would transform the geopotential number to a non-Euclidean space. A practical challenge exists in calculating \( \overline{g} \), because this requires sufficiently accurate topographical density models. Current methods for calculating \( \overline{g} \) (Santos et al. 2006) allow orthometric heights accurate to 1 cm or better except in high mountains.

Next we turn to the normal height systems, which we discuss as the Vignal or Molodensky type. Vignal heights, first called altitudes orthodynamiques were developed in the 1940s and 1950s (e.g. Eremeev 1965; Simonsen 1965), roughly parallel to Molodensky’s (Molodensky 1945) system. The goal was to replace dynamic heights with a system more closely matched to Earth’s gravity field. In Vignal’s system, the mean gravity value in the denominator of Eq. (1) is replaced with the integral mean normal gravity \( \overline{\gamma} \) between the ellipsoid and a point displaced above the ellipsoid by an amount equal to the height of the point of interest (Eremeev 1965). Vignal’s method takes \( \overline{\gamma} \) as a better approximation of \( \overline{g} \) than the arbitrary value used in dynamic heights, and thus an improved approximation of orthometric height in an era when real gravity not known as well as it is now. The heights arising from this system are given by the equation for the length of the normal plumbline from the reference ellipsoid to the telluroid (Heiskanen and Moritz 1967, Eqs. 4–55):

$$ {H}^N=\raisebox{1ex}{$C$}\!\left/ \!\raisebox{-1ex}{$\overline{\gamma}$}\right.. $$
(2)

From this formula, it is immediately clear that the geoid must be the zero-height surface for Vignal normal heights because for C = 0 (on the geoid), H N = 0. However, is it also clear that the Vignal normal heights of points will be different from their orthometric heights, because the denominator in Eq. (2) is different from that in Eq. (1). If we attempt to understand Vignal heights as lengths of a line of the vertical from the geoid to a point, this difference in denominator implies that those lengths are defined using a non-Euclidean metric. Interpreted in the context presented in this paper, this is a consequence of Vignal heights being defined using a space N that is not equivalent to the physical space P, but is only an approximation of it.

N is of a category that we will call approximate physical spaces, which comprise simplifications of the physical space. Such spaces are common in geodesy and surveying: Widespread examples include the planar and spherical approximations, where the shape of Earth’s gravity field and topography is distorted by simplification of Earth’s gravity field.

Like the spaces used in the planar and spherical approximations, the normal space N is also an approximate physical space. It is a bit closer to the real space, and improves upon the spherical approximation by using an ellipsoidal approximation of Earth’s gravity field. In N, the real plumblines map to the normal plumblines, which are the lines of the vertical in the Vignal system. The equipotential surfaces map to the ellipsoidal equipotential surfaces of the normal gravity field, of which the reference ellipsoid serves as the reference surface for Vignal heights. The shape of the space is described by the function U = U(x) in the same way that the function W = W(x) describes the physical space. The physical space P maps to N by a smooth mapping M N, as shown in Fig. 3, and based on Eqs. (1) and (2) the mapping of plumbline lengths is given by \( \overline{g}/\overline{\gamma} \). The approximation error of the normal space is distributed globally, and is on the order of no more than about 150 m for the region of interest (several kilometres above or below the topographical surface). The error in normal heights arising from this approximation is less than 4 m globally (Foroughi and Tenzer 2017), and much smaller than that in most areas.

Fig. 3
figure 3

The physical space mapped to the Normal approximation space N by mapping M N, with Vignal normal height indicated by the medium dotted black line in the Normal space

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Approximate physical spaces are important for simplifying the mathematical formulations involved in positioning. For example, the “reduction” of an observed vector between two points to a horizontal distance becomes more simple if a planar or spherical approximation is used. Likewise, the normal space provides a very important “reference space” for positioning and gravity field computations in geodesy, where it is often used as a first approximation of Earth’s gravity field. Differences between quantities in the physical space and the normal space can then be treated as small anomalies that can be dealt with in subsequent refinements more easily than if the full quantities were used. More detailed “spheroidal” approximations are also frequently made in geoid modelling computations. However, such simplification comes at the expense of geometry and should be applied with care. In the case of Vignal heights, while the normal space is a sufficient approximation of P for the accuracy goals and data availability of the 1950s, nowadays far better estimates of orthometric heights are available, like the rigorous orthometric heights (Santos et al. 2006).

We now turn to what we call the Molodensky normal heights. Molodensky, in his creation of a comprehensive normal height system (Molodensky et al. 1962) applies a definition similar to Vignal’s but more exactly calculated, and also applies two other definitions that were meant to be equivalent to it:

  1. 1.

    The height along a normal to the ellipsoid of a point displaced by some amount ζ (the height anomaly) from the topographical surface, i.e., the telluroid; and

  2. 2.

    The height of a point above the quasigeoid, along a normal to the ellipsoid.

Because the height anomaly is the vertical displacement between a point of potential W and a point of normal potential U = W, it represents the mapping of vertical point positions from P to N. Thus Definition 1 should theoretically produce heights identical to Vignal’s, apart from the difference between the normal plumbline and the ellipsoidal normal. Furthermore, since ζ can be calculated not only at the topographical surface but anywhere in the space near it, Definition 1 can be extended from Molodensky’s usage to provide a viable route for those wishing to transform geodetic heights to Vignal normal heights. However, because Molodensky’s approach to calculating ζ requires a regularized Earth surface (Molodensky 1945) it does not yield heights exactly equivalent to Vignal’s: An additional layer of approximation is added, as well as some ambiguity unless the choice of regularized Earth surface is specified.

Definition 2 is more problematic. As highlighted at the 2018 Hotine-Marussi symposium and discussed in subsequent publications (e.g. Kingdon et al. 2022), the quasigeoid is a folded and creased surface not well-suited as a reference surface for heights. Thus, a regularized topography must again be used to construct a usable quasigeoid. The quasigeoid constructured in this way will still not be a physically meaningful surface however: A marble placed on the geoid would not roll; a marble placed on the quasigeoid would navigate a path among the variations of topography and gravity anomalies. Furthermore, the quasigeoid is associated with the values of ζ at the (regularized) surface of topography only, and so Definition 2 will provide incorrect heights for points not situated at the topographical surface.

Because the use of the height anomalies was uniquely associated with the Molodensky system, herein we use the term “Molodensky heights” to refer to heights defined using height anomalies. There heights may be defined according to Definition 1, in which case they are equivalent to Vignal heights. However, if defined according to Definition 2 as shown in Fig. 4, they are defined with the quasigeoid as the reference surface and the ellipsoidal normals as the lines of the vertical. Notably, the lines of the vertical are not perpendicular to the reference surface in this definition, and all of the problems with the quasigeoid listed above are inherited. Any attempt to define an approximate physical space associated with the Definition 2 Molodensky heights would be quite complex, given their association with the irregular quasigeoid.

Fig. 4
figure 4

The Molodensky normal height system, showing normal to the ellipsoid through point A, along which height is measured, as a medium dashed line, and the quasigeoid as a thin black line

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3 Conclusions

Consistent interpretations of each height system are possible in the context of vertical lines in selected metric spaces from points of interest to defined reference surfaces. Such interpretation reveals that for most systems of heights, including the Vignal normal heights, the geoid is the reference surface. These systems all have the property that height is a member of a 1-dimensional space perpendicular to the level surfaces of the manifold in which they are defined, and the height dimension extends perpendicular to the reference surface. The only system that does not cleanly fit into this framework is the Molodensky system that uses the quasigeoid. The Vignal normal heights present a more consistent system, and Molodensky’s use of the height anomaly to define his normal heights presents a path to their use.

In the longer term, we have seen that any variety of normal height relies on an approximation of Earth’s gravity field, and this gives rise to certain challenges. The differences in accuracy between normal and orthometric heights are small, but as measurement precision increases small differences may become significant. At the same time, as better density models and gravity observation allow improved characterization of Earth’s gravity field, the arguments for using an approximate physical space for defining heights wane. Like the planar and spherical approximation, the normal approximation will always have a role in geodetic computation, generating reference ellipsoids and gravity anomalies, and for other purposes where a simplified or approximate gravity field is called for. However, the time to set it aside in the ultimate definition of height systems is near.