Keywords

1 Review

Heights are the vertical coordinates in a 3D curvilinear coordinate system and as such can have different metric for different height systems. The role of the Earth gravity field is much more important in the vertical dimension than in horizontal dimensions. [But look at the error in the length of a metre where the effect of gravity overwhelms the contribution of measurement error 50-times (Vaníček and Foroughi 2019)]. It is clearly a greater intellectual challenge to work with heights then to work with horizontal positions. Also the term “vertical” associated with heights implies that a horizontal, a.k.a., level surface should serve as the reference surface for heights.

Horizontality brings in the notion of a gravity equipotential surface (by definition) and a natural extension of this notion, the Geopotential numbers C. These are defined as

$$ C\left(\varOmega, H\right)={W}_0-W\left(\varOmega, H\right),\vspace*{3pt} $$
(1)

where W stands for real gravity potential, Ω for the couple of horizontal coordinates (ϑ, λ) and H for the height above the height reference surface. As W 0 is the potential of the reference surface, C can be used as a height indicator (a pseudo-height) if we were willing to use physical units of potential (Gal m) as “units of heights”.

Table 1 Overview of height systems discussed in this paper
Full size table

There are three kinds of terrestrial heights H used in practice: Orthometric H O, Dynamic H D and Normal H N, all referred to the geoid W(Ω) = W 0. They are all defined by a similar equation

$$ H\left(\varOmega \right)=C\ \left(\varOmega, H\right)/\mid \overline{\nabla V\left(\varOmega \right)}\mid, $$
(2)

where | \( \nabla \overline{V\left(\varOmega \right)} \) | is the integral mean absolute value of the gradient of potential V between the reference surface and the point of interest on the Earth surface. The value of the denominator in Eq. (2) for the three varieties of terrestrial heights H are:

  • H O … real gravity (V = W),

  • H D … agreed constant value of gravity and

  • H N … normal gravity (V = U).

We understand the normal height HN was introduced by Vignal in France in 1957 although the term “normal” might have been used already by Molodensky a few years earlier. In any case, there is a slight difference between Vignal’s and Molodensky’s versions of Normal heights in the way the mean normal gravity is computed. As a reference, the various height systems, including some to be discussed later, are summarized in Table 1.

The three classical height systems are described, for example, in Vaníček and Krakiwsky (1986). Orthometric height is the only height system to use Euclidean metric, i.e., H O is the real height above the geoid as measured by a constant metre. Note that we are here talking about rigorous Orthometric height and not the approximate Helmert variety (Tenzer et al. 2005) and the term “metric” is taken in the mathematical sense as the way of measuring distances. The Dynamic height H D is the only physically meaningful height (fluid does not flow between two points of equal dynamic heights), its Riemannian metric being dictated by the shape of gravity field. The Normal height H N is comparable in some ways to the Orthometric height, as it was originally meant as an approximation of Orthometric height, but like Dynamic height, is measured by a “rubber metre”. Its metric has no real meaning.

2 Problems with Molodensky’s Approach

In the mid-twentieth century, Russian physicist M.S. Molodensky observed that the lack of information on topographical density at that time caused too large an error in H to satisfy the practical accuracy requirements. To eliminate this problem, Molodensky devised an interesting alternative theory of heights and of the external gravity field, which does not require any knowledge of topographical density. Molodensky’s practical heights use the quasigeoid as the reference surface and are numerically quite close to Vignal’s Normal heights. The metric of Molodensky’s (practical) heights is Euclidean. We distinguish between Normal and Molodensky’s heights because while Molodensky was writing mostly about Normal heights, geodetic practise uses what we call here Molodensky’s heights; the commonly used terminology follows Heiskanen and Moritz (1967) explanations that is much the same as ours.

It has been known for some time that Molodensky’s theory has two flaws: (1) It calls for integration over a surface (Telluroid) which is reflective of topography and is thus not integra-ble and (2) the reference surface it uses, the quasigeoid, is not globally continuous. Being reflective of topography, it has folds, reflecting the folds of topographical surface, which cause the discontinuity and other funny features reflecting topography (Vaníček and Santos 2019).

How large can quasigeoid’s folds be? The size of the fold is given as

$$ \begin{array}{l}\Delta {\zeta}_{AB}={\zeta}_A-{\zeta}_B=\left({W}_A\hbox{--} {U}_A\right)/\gamma -\left({W}_B\hbox{--} {U}_B\right)/\gamma\\ \noalign{\vskip4pt} \qquad\;\;\, =\left({W}_A\hbox{--} {W}_B\right)/\gamma -\left({U}_A\hbox{--} {U}_B\right)/\gamma .\end{array} $$
(3)

Now,

$$ \left({W}_A\hbox{--} {W}_B\right)=-{\overline{g}}_{AB}\Delta {H}_{AB} $$
(4)

and similarly

$$ \left({U}_A\hbox{--} {U}_B\right)=-{\overline{\upgamma}}_{AB}\Delta {H}_{AB}. $$
(5)

Thus,

$$ \Delta {\zeta}_{AB}=\left(-{\overline{g}}_{AB}+{\overline{\upgamma}}_{AB}\right)\Delta {H}_{AB}/\gamma =-{\overline{\delta g}}_{AB}\Delta {H}_{AB}/\gamma, $$
(6)

where \( {\overline{\delta g}}_{AB} \) stands for mean gravity disturbance between points A and B. Let us assume \( {\overline{\delta g}}_{AB} \) to be 100 mGal and ∆ H AB to be 1,000 m. For this situation, the magnitude of the fold ∆ζ AB would be about 10 cm, but larger folds plausibly exist in the real world. There are also other problems with the quasigeoid, which is not a well mathematically behaving surface at all and neither has it any physical meaning even where it is defined.

The basic requirement of a height system is holonomity which, in brief really means a uniqueness of definition: for each horizontal position Ω there must be only one value of height H referred to the point of interest on the Earth surface (Sansò and Vaníček 2006). In other words, heights must be true coordinates. This property is also demanded by the classical treatment of levelled differences so that height differences between any two points on the Earth surface are the same whichever levelling route between them is followed. Clearly, if the terrain has overhangs then in the areas of overhangs we shall have to use some non-standard tool to vertically locate points that are underneath the upper topographical surface. Consequently, if the topographical surface is not a one-valued mathematical function; the uniqueness can be guaranteed if the reference surface is continuous. This is not the case with the quasigeoid as discussed above. Hence Molodensky’s heights cannot be used in a global height system.

3 Arrival of Satellites and the Problem of Height Congruency

Since the late 1960s satellite positioning techniques have become widespread. For the first time in history there appears an alternative approach to height determination – but the heights are of a different kind. They are Geodetic heights referred to the reference ellipsoid. These should not be called ellipsoidal heights as this would imply heights of an ellipsoid surface above some other reference surface, per common English usage (e.g., “sea surface heights”, “topographical heights”, “geoidal heights”, etc.). They are obtained through a geometrical transformation of the 3D positions derived from observations to satellites such as GNSS. Geodetic heights have the Euclidean metric but are not referred to a horizontal surface as the reference ellipsoid is not a horizontal surface. To assure the congruency between the existing Orthometric and the Geodetic systems of heights, we must introduce the height of the geoid above the reference ellipsoid N, a.k.a., Geoidal height and, forgetting about the negligeable differences among the lengths of different plumblines, we get the following simple relation

$$ \forall \varOmega \in {\varOmega}_0:h\left(\varOmega \right)={H}^O\left(\varOmega \right)+N\left(\varOmega \right). $$
(7)

The congruency with the other existing terrestrial height systems, Dynamic and Normal, is a-chieved simply through multiplication by the appropriate ratio of potential gradients, c.f., Eq. (2).

Satellite techniques have become so accurate that the accuracy of Geodetic heights h is now equivalent to that of standard Orthometric heights H O. It is thought that the standard deviation of Geodetic heights and the standard deviations of individual, or point terrestrial height is about the same, 2–3 cm. Since the standard deviation, as a measure of error of the computed Geoidal height N, can be about five-times smaller if the data available for geoid determination are of good quality and quantity (Foroughi et al. 2019) and the topographical heights are reasonably low. It is now more economical than using the classical levelling technique to determine Orthometric height as a difference of Geodetic height h and Geoidal height N.

As an aside, there is a very often asked question that makes sense as the ease with which we can measure and calculate Geodetic heights h for almost any point on the surface of the Earth with an accuracy good enough for most applications becomes obvious: Why not use Geodetic heights as practical heights? The problem is its datum: in the Geodetic height system the height of the sea shore height varies between −100 m and +100 m which makes it difficult to work with in a technically meaningful way. Clearly, Geodetic heights must be transformed to Orthometric, Dynamic or Normal heights which all have the same reference surface, the geoid, selected so that it approximates the mean sea level and are thus useful in practice.

It is impossible to use a simple equation like Eq. (7) to make the Molodensky heights congruent with geodetic heights since it is non-holonomic due to using the quasigeoid as its reference surface. The difference between the geoid and quasigeoid can be evaluated only approximately. Hence as a consequence of the quasigeoid being a discontinuous surface, the Molodensky height system cannot be made globally congruent with Geodetic height system.

4 Conclusions

The maintenance of the congruency of a terrestrial height system with the Geodetic system is an ongoing process. This process includes, of course, the increasing accuracy of geoidal height determination. Thus the congruency of the terrestrial and Geodetic height systems should be assured as much as possible at any stage of height densification process. The height systems are developed in successive iterations as the understanding of the involved problems improves and more and increasingly denser observations of the real world become available. To ensure that these iterations converge to a correct result, i.e., to the true congruency, the individual parts have to be formulated correctly in the physical sense.

As an aside, let us take, for illustration, the downward continuation of gravity. It is a well posed problem for reasonably low topography and reasonably large steps [more than 1 or 2 arc-minutes (Martinec 1996)] in the description of topography. When the step gets too small and/or topography gets too high the process becomes unstable and can be solved only by some artificial means (like regularization or, preferably, by Moore-Penrose generalized matrix inversion). But, if the combination of step size and topography yields a regular matrix, there exists a unique and physically correct solution. It seems that the existing regular geoid solutions can be accurate to 1 cm (Foroughi et al. 2019) and a more detailed solution will probably not give any real improvement. This would be true if the geoidal deflection of vertical does not change more than 1 second-of-arc in a horizontal distance of 1 km which may plausibly be the case worldwide. Hence, if the combination of step size and topographical heights yields a regular downward continuation system of equations then this process gives an example of a well posed problem that satisfies the requirement: each successive iteration will bring the real congruency closer and closer to the ideal.

On the other hand, Molodensky’s system suffers from flaws that come from the theoretical formulation rather than practical implementation. To use this system requires the user to evaluate integrals over a surface that does not allow integration over it. Similarly, the heights are defined so that the reference surface reflects the folds that exist on the topographical surface and no successive iterations are going to get rid of this problem. This behaviour makes the quasigeoid as a global reference surface for heights needed for the United Nations’ resolution on the “Global Geodetic Reference Frame (UN-GGRF) for Sustainable Development” unacceptable.

In the real world, when solving a practical problem, some people use an approach that is known to be somewhat suspect just because it is easier to use, it works to the presently required accuracy and it “satisfies the need” of the instant. We have written this paper to address specifically these geodesists who are doing a disservice to their profession by preparing the ground for future problems arising from using Molodensky’s height system that are bound to appear. Please abandon Molodensky’s height system, i.e., forget about using the quasigeoid and make sure that the height system you are working with has a chance to have the necessary congruency with the Geodetic height system, that it is physically rigorous and correct and has the potential to improve the height control with future iterations.