Keywords

1 Introduction

Geodetic heights are tools used in Geodesy as coordinates to place in 3D space points that belong to the surface of the earth S or to a layer extending dozens of kilometres above or below it, a set that we will call Ω.

There are four principal height systems in use in Geodesy: the gravity potential W(P) or the geopotential number C(P) = W 0 − W(P), or the dynamic height H D(P) = C(P)/γ 0, which are linear functions of it (for these definitions see e.g. § 4.2 of Heiskanen and Moritz 1967); the orthometric height, which is mostly preferred by surveyors as it is considered much akin to the levelling observations; the normal height, introduced as a tool for the linearization of the Geodetic Boundary Value Problem (Sansò and Sideris 2013; Yanushauskas 1989); the ellipsoidal height, which is a purely geometric concept, though nowadays accessible by direct observations from GNSS satellites.

The first and the last systems are, so to say, naturally global and adapt to build unambiguously a world-wide system (height datum).

On the contrary, orthometric and normal heights, may be for their closeness to the levelling increments

$$ \delta L=- dW/g\left(g\ \mathrm{Earth}\ \mathrm{gravity}\right) $$
(1.1)

have been adopted as local systems where corrections to the integral along levelling lines of the differential form (1.1) could be neglected.

Therefore, a practical idea was to assign a height 0 conventionally to some origin point and then give a height to other points connected by levelling lines by integrating δL along them. In this way, many local levelling height systems have been created.

Yet (1.1) is not an exact differential form, as it is proved in e.g. Sansò and Vaniček (2006). So, the integral of δL over a large, closed loop is significantly different from zero. In other words, levelled height systems brought to a global (or even only large) scale clearly show their non-holonomity.

Is such a property inherited by orthometric or by normal heights? As for orthometric heights, there has been a long-lasting discussion in the geodetic community. Yet, the question has been completely and satisfactorily solved in Sansò and Vaniček (2006), where it is shown that the orthometric height is a continuously differentiable single-valued function and, as such, it is a regular holonomic coordinate in the layer around S described above.

Now, the question seems to arise again in relation to the normal heights, if the quasi-geoid is claimed to be a reference surface and to display potentially such irregular features that make normal heights a non-holonomic system (Sansò and Vaniček 2006).

In the paper, the authors consider once again the question of holonomity and show that the mentioned proposition descends from a misinterpretation of the quasi-geoid as reference surface of normal heights.

2 What Is a Geodetic Heigh?

A regular system of coordinates in a domain Ω ⊆ ℝ 3 is a triple of points functions (q 1, q 2, q 3).

Such that, the correspondence

$$ P\in \Omega \leftrightarrow \left({q}_1,{q}_2,{q}_3\right) $$
(2.1)

is biunivocal and continuously differentiable, i.e. ∇P q i(P) should be continuous vector fields in Ω (Marussi 1985; Sansò et al. 2019). The above condition implies that, when Ω is simply connected, the integral of dq i on any closed rectifiable curve in Ω is zero (Sansò and Vaniček 2006).

Since the idea of height is related to represent something that is “above” or “below” an observer, one of the 3 coordinates q i (i = 1, 2, 3), say q 3 for the sake of definiteness, will be considered a “height” if it has a relation to the direction of the physical vertical, namely to the unit vector

$$ {\underline{n}}_P=-\frac{\underline{g}(P)}{g(P)} $$
(2.2)

where \( \underline{g}(P) \) is the vector of the Earth gravity field.

In Sansò et al. (2019), we have stipulated that q 3 is a geodetic height if the tangent \( {\underline{t}}_3 \) to the q 3 line at P makes an acute angle with \( \underline{n}(P) \), i.e.

$$ {\underline{n}}_P\cdotp {\underline{t}}_3(P)\ge 1-\varepsilon $$
(2.3)

for some value ε, 0 ≤ ε < 1.

In fact, an observer moving from P in the direction of \( {\underline{t}}_3 \) will see the gravity potential value W(P) decrease which is a distinctive property of being higher, at least up to some distance from the centre of the Earth, at which the centrifugal term of W(P) is still a perturbation of the main gravitational term.

Yet, such a definition has a logical limit in that the q 3 line is, as a matter of fact, defined by the other two coordinates, which have to be consistent along it. Indeed, q 3 has to change monotonously along its coordinate line, otherwise the one-to-one correspondence (2.1) can fail, so that condition (2.3) makes sense. However, there are many directions in which both q 3 and W(P) can change, one increasing and the other decreasing. So, we prefer to make a slightly generalization of (2.3) that could be verified on the basis of the function q 3(P) alone.

2.1 A New Definition of a General Geodetic Height H G

H G is defined in a layer Ω, a closed set as in Fig. 1, by means of the following three elements:

Fig. 1
figure 1

The ellipsoid E and the set Ω

Full size image
  1. a)

    a surface RS ⊂ Ω (Reference Surface) defined as a Lipschitz function of the ellipsoidal coordinates of its point

$$ RS=\left\{P;{\underline{r}}_P={\underline{r}}_{P_e}+{h}_P{\underline{\nu}}_{P_e}\right\} $$
(2.4)

h P = ellipsoidal height of P; P e = projection of P on the Ellipsoid; \( {\underline{\nu}}_{P_e} \) = normal to the Ellipsoid passing through P

  1. b)

    a family \( \mathfrak{I} \) of lines in Ω

$$ \mathfrak{I}=\left\{{L}_P,P\in \Omega \right\} $$
(2.5)

with the following characteristics: for every P ∈ Ω passes one and only one line

$$ {L}_P=\left\{{\underline{r}}_Q={\underline{\xi}}_P(Q)\right\} $$
(2.6)

each L P has a continuous tangent field, pointing upward

$$ {\underline{t}}_P(Q)=\left\{\frac{d{\underline{\xi}}_P(Q)}{dl};Q\in {L}_P\right\}. $$
(2.7)

As such, each L P is then rectifiable. We also require that each L P pins the RS at one point only P RS; the correspondence P → P RS is called the projection of P on RS along \( \mathfrak{I} \). By pointing upward, we mean that at any point P in Ω, direction of the vertical \( {\underline{n}}_P \) and upward tangent of L P, \( {\underline{t}}_P \), form and acute angle (see Fig. 2), namely

Fig. 2
figure 2

Ω, ellipsoid E, Reference Surface RS, cosθ = 1 − η, P RS projection of P on RS, \( {P}_{RS_e} \) projection of P RS on E

Full size image
$$ {\underline{n}}_P\cdotp {\underline{t}}_P=1-\eta\ \left(0\le \eta \le 1\right) $$
(2.8)
  1. c)

    let us first define a linear coordinate on L P, namely the arclength \( {l}_{P_{RS}}^P={l}^P \), counted positively outside RS and negatively inside; now fix a functional of the arc \( {L}_{P_{RS}}^P={L}^P \), a function F(l P), which is monotonic in the sense that

$$ {l}^{P^{\prime }}>{l}^P\Rightarrow F\left({l}^P\right)>F\left({l}^{P^{\prime }}\right) $$
(2.9)

and such that

$$ {l}^P=0\ \left(P={P}_{RS}\right)\Rightarrow F\left({L}^{P_{RS}}\right)=0 $$
(2.10)

Then we define the general geodetic height H G as

$$ {H}_G(P)=F\left({l}^P\right) $$
(2.11)

As it is obvious by the above definitions, the RS surface corresponds exactly to the points P where H G(P) = 0.

We will verify in the next section that all the four height systems, mentioned in the Introduction, comply with such a definition.

In doing so, we will clarify a small incongruence, which is present even in classical textbooks, in the definition of normal height.

Remark 2.1

It might seem that the new definition of H G is on the one hand too complicated and on the other hand too generic since it depends on the ambiguous constant η, as it was previously from ε. Yet, we must underline that our definition is as a matter of fact tailored on that of orthometric height and that, when we consider practical height systems, ε or η are in fact very small quantities making such systems numerically not very different from one another.

This is typically of the Earth gravity fields which seems a “black night where all cows are black” (F. Hegel) in the sense that variables related to it, though defined in different ways, appear often not well numerically distinguishable one from the other (particularly in small areas, although globally they display a different behaviour).

3 The Four Height Systems Are Geodetic Heights

We want to verify that: 1) the dynamic height H D; 2) the ellipsoidal height h; 3) the orthometric height H o; 4) the normal height H ∗, are in fact geodetic heights according to the definition of the previous section.

  1. 1)

    Let us recall that

$$ {H}_D(P)=\frac{W_0-W(P)}{\gamma_0} $$
(3.1)

where, W 0 is the potential assigned to the geoid as an equipotential surface, satisfying also

$$ {W}_0={U}_0 $$
(3.2)

with U 0 the value of the normal equipotential on the Earth Ellipsoid E; γ 0 on the contrary is a reference constant value for the normal gravity, typically taken as γ on E at latitude φ = 45° (see e.g. Heiskanen and Moritz 1967). The choice of γ 0 is just to give to H D the dimension of a length and a numerical value not too distant from other height systems.

For H D we have:

  1. a)

    RS is the geoid

$$ P\in RS,\kern0.5em W(P)={W}_0={U}_0 $$
(3.3)

In fact, this corresponds also to H D = 0.

  1. b)

    The family \( \mathfrak{I} \) for H D is that of plumb lines

$$ \mathfrak{I}=\left\{{L}_{Pb}\right\} $$
(3.4)

Therefore, the tangent field to L Pb is directly \( \underline{n} \) and the condition (2.8) is satisfied with

$$ \eta =0 \vspace*{-16pt}$$
(3.5)
  1. c)

    Taking an integral along the plumbline from P RS and P, we can clearly write

$$ {H}_D(P)=\frac{1}{\gamma_0}{\int}_{P_{RS}}^Pg(Q) dl=F\left({l}_{Pb}^P\right) $$
(3.6)

And since g(Q) is positive everywhere on L Pb ⋂ Ω, H D is clearly monotonous.

  1. 2)

    The ellipsoidal height h has its own geometric definition by means of the normal to the ellipsoid, \( {\underline{\nu}}_P; \)

$$ h(P)={\int}_{P_e}^P dh=F\left({l}_{P_e}^P\right) $$
(3.7)
  1. a)

    RS for h is the ellipsoid E

  2. b)

    The family \( \mathfrak{I} \) for h is the family of straight lines

$$ \mathfrak{I}=\left\{h{\underline{\nu}}_P;{P}_e+h{\underline{\nu}}_P\in \Omega \right\} $$
(3.8)

We have therefore that the tangent field to L P is

$$ {\underline{t}}_P={\underline{\nu}}_P $$
(3.9)

And

$$ {\underline{t}}_P\cdotp {\underline{n}}_P={\underline{\nu}}_P\cdotp {\underline{n}}_P=\mathit{\cos}{\delta}_p $$
(3.10)

where δ p is the deflection of the vertical at P. Since

$$ 1-{\underline{\nu}}_P\cdotp {\underline{n}}_P\cong \frac{1}{2}{\delta}_P^2 $$
(3.11)

Even assuming for δ p the overwhelming upper bound

$$ {\delta}_p<{10}^{-2} $$
(3.12)

we see that, from (2.8)

$$ \eta <0.5 \,\cdotp {10}^{-4} $$
(3.13)
  1. c)

    It is clear from the definition (3.7) that

$$ {h}_P={l}^P $$
(3.14)

The length of the ellipsoidal normal between P e and P. So, \( F\left({l}_{P_e}^P\right) \) is indeed tautologically monotonous.

  1. 3)

    The orthometric height H o is defined as the arclength of the plumbline between the geoid and the point P. Therefore, we have

  1. a)

    The reference surface RS in this case is the geoid

$$ RS=\left\{P;W(P)={W}_0\right\} $$
(3.15)
$$ P\in RS\Longrightarrow {H}_o(P)=0\vspace*{-8pt} $$
(3.16)
  1. b)

    The family of \( \mathfrak{I} \) is in this case the family of plumblines again

$$ \mathfrak{I}=\left\{{L}_{Pb}\right\} $$
(3.17)

and, once more, one has

$$ {\underline{t}}_P={\underline{n}}_P $$
(3.18)

and

$$ \eta =1-\underline{t}\cdotp \underline{n}=0 $$
(3.19)
  1. c)

    Since H o is directly the length of the plumbline, one has that

$$ F\left({L}_P\right)={H}_0(P)={l}^P $$
(3.20)

which is indeed monotonous.

  1. 4)

    The normal height H ∗ is defined as the ellipsoidal height of the point P ∗ such that the geodetic coordinates σ = (θ, λ), the normal potential U and the actual potential W satisfy the following conditions

$$ {\sigma}_{P^{\ast }}={\sigma}_P,U\left({P}^{\ast}\right)=W(P)\ \mathrm{or}\ {H}_P^{\ast }={h}_{P^{\ast }} $$
(3.21)

i.e. P ∗ is on the same normal to E as P and the second of (3.21) is verified.Footnote 1

A little thought shows that, calling P e the orthogonal projection of P on E, this definition can be written analytically as

$$ U\left({P}_e+{H}_P^{\ast }{\underline{\nu}}_P\right)=W(P)\vspace*{-8pt} $$
(3.22)
  1. a)

    from (3.22) it is clear that

$$ {H}_P^{\ast }=0\Longleftrightarrow W(P)=U\left({P}_e\right)={U}_0={W}_0 $$
(3.23)

namely P is on the geoid and P e is its projection on E. In other words, this means that

$$ RS\equiv Geoid \vspace*{-8pt}$$
(3.24)
  1. b)

    the family \( \mathfrak{I} \) in this case is

    $$ \mathfrak{I}=\left\{{P}_e+{H}_P^{\ast }{\underline{\nu}}_P\right\} $$

i.e. the family of ellipsoidal normal lines.

So,

$$ {\underline{t}}_P={\underline{\nu}}_P $$
(3.25)

and, again, we have what we have seen in (3.10) and (3.18), namely a very small η.

  1. c)

    Let us write (3.22) for a generic point \( Q={P}_e+{h}_Q{\underline{\nu}}_P \) on the ellipsoidal normal \( {\underline{\nu}}_P \), as

$$ U\left({P}_e+{H}_Q^{\ast }{\underline{\nu}}_P\right)=W\left({P}_e+{h}_Q{\underline{\nu}}_P\right) $$
(3.26)

We fix P and hence P e and \( {\underline{\nu}}_P \), move only Q along the normal and differentiate, getting the relation

$$ {\underline{\nu}}_{P^{\ast }}\cdotp \underline{\gamma}\left({H}_Q^{\ast}\right)d{H}^{\ast }={\underline{\nu}}_P\cdotp \underline{g}\left({h}_Q\right) dh $$
(3.27)

We exploit the fact that \( {\underline{\nu}}_{P^{\ast }}={\underline{\nu}}_P \), namely constant along L P, and

$$ {\underline{\nu}}_{P^{\ast }}\cdotp \underline{\gamma}\left({H}_Q^{\ast}\right)\cong -\gamma \left({H}_Q^{\ast}\right) $$
(3.28)
$$ {\underline{\nu}}_P\cdotp \underline{g}\left({h}_Q\right)=-g\left({h}_Q\right)\ {\underline{\nu}}_P\cdotp {\underline{n}}_Q $$
(3.29)
$$ g\left({h}_Q\right)=\gamma \left({H}_Q^{\ast}\right)+\Delta g $$
(3.30)

write (3.27) in the form

$$ d{H}_Q^{\ast }=\left(1+\frac{\Delta g}{\gamma}\right){\underline{\nu}}_P\cdotp {\underline{n}}_Q dh $$
(3.31)

or

$$ {H}_Q^{\ast }={\int}_{P_{RS}}^Q\left(1+\frac{\Delta g}{\gamma}\right){\underline{\nu}}_P\cdotp {\underline{n}}_Q dh $$
(3.32)

We stress once more that the integral in (3.32) is along the ellipsoidal normal and that P RS is just the projection of P along the normal on the geoid so that

$$ {l}^Q={\int}_{P_{RS}}^Q dh={h}_Q-{N}_{P_e} $$
(3.33)

with \( {N}_{P_e} \) the geoid undulation, i.e. it is the same as the orthometric height of Q.

Since both \( \left(1+\frac{\Delta g}{\gamma}\right) \) and \( {\underline{\nu}}_P\cdotp {\underline{n}}_Q \) are quite close to 1, hence positive, we see that

$$ {H}^{\ast }=F\left({l}_P\right) $$
(3.34)

is an increasing functional of l P according to (2.9). Therefore, also H ∗ is complying with our definition of general geodetic height.

Remark 3.1

It might be worth here to amend an excusable imprecision present in the definition of the normal height in classical books as Heiskanen and Moritz (1967).

In fact, instead of (3.32), as it is defined H ∗ in Heiskanen and Moritz (1967), § 8.3, one finds often the following alternative definition:

$$ RS= Geoid\vspace*{-18pt} $$
(3.35)
$$ \mathfrak{I}=\left\{{\overset{\sim }{L}}_P= force\ lines\ of\ \underline{\gamma}\right\}\vspace*{-18pt} $$
(3.36)
$$ {H}^{\ast \prime }={\int}_{P_e^{\prime}\left({\overset{\sim }{L}}_P\right)}^{P^{\ast \prime }} dl $$
(3.37)

which corresponds to integrating along the normal plumb line, instead of the ellipsoidal normal, until

$$ U\left({P}^{\ast^{\prime }}\right)=W(P) $$
(3.38)

The situation is illustrated in Fig. 3.

Fig. 3
figure 3

A comparison between H∗and H∗′

Full size image

Since the curvature of the normal plumbline is very small in Ω and the corresponding normal deflection of the vertical \( \overset{\sim }{\delta } \) is of the order of

$$ O\left(\overset{\sim }{\delta}\right)=5\times {10}^{-3}\ \frac{h}{R} $$
(3.39)

with R the mean radius of the Earth, one can easily see that

$$ O\left(\left|{H}^{\ast \prime }-{H}^{\ast}\right|\right)=O\left(\frac{1}{2}{\overset{\sim }{\delta}}^2h\right) $$
(3.40)

which is well below the mm level in our set Ω.

So, this alternative definition which leads to the other form of widespread use (see Heiskanen and Moritz 1967, § 4.5)

$$ \left\{\begin{array}{c}{H}^{\ast \prime }=\frac{W_0-W(P)}{\overline{\gamma}}\ \\ {}\overline{\gamma}=\frac{1}{H^{\ast \prime }}{\int}_0^{H^{\ast \prime }}\gamma (z) dz\end{array}\right. $$
(3.41)

has no relevant numerical difference with H ∗, although one could comment that this change of integration path has a logical relation also to the difference between the vector and the scalar Boundary Value Problem in linearized form (Sansò 1995).

4 Holonomity of the Geodetic Heights

We define a regular holonomic coordinate in a simply connected set Ω, a function q(P), P ∈ Ω, such that its differential

$$ dq=\nabla q(P)\,\cdotp d\underline{P} \vspace*{-3pt}$$
(4.1)

is a continuous function of P. So, holonomity is a matter of regularity and of the set Ω. In this sense, if q(P) is directly defined, its holonomity is guaranteed by inspecting its regularity in Ω and, as a consequence, for any closed rectifiable curve C in Ω, one has

$$ {\int}_C dq\equiv 0 \vspace*{-3pt}$$
(4.2)

Different might be the conclusion if q(P) was not directly defined but we rather define a 1-differentiable form ω = ω(P, dP), we assign a conventional value to q at some point P o internal to Ω and we put

$$ q(P)=q\left({P}_0\right)+{\int}_{P_o}^P\omega \vspace*{-3pt}$$
(4.3)

the integral being along some line joining P o to P.

In this sense, q in general is not a function of P only but also of the path L. This can be a non-holonomic coordinate. Only when ω(P, dP) is an exact differential form, namely

$$ \omega \left(P, dP\right)= df\left(P, dP\right)=\nabla f(P)\cdot d\underline{P},\vspace*{-3pt} $$
(4.4)

then we can say that

$$ q(P)=q\left({P}_0\right)+\left[f(P)-f\left({P}_0\right)\right] \vspace*{-3pt}$$
(4.5)

i.e. q is holonomic.

If we write ω, for instance, in 3D cartesian coordinates,

$$ \begin{array}{ll}\displaystyle \omega \left(P, dP\right)&\displaystyle =\underline{v}\,\cdotp d\underline{r}= Adx+ Bdy+ Cdz\\&\displaystyle \quad \left(\underline{v}=A{\underline{e}}_x+B{\underline{e}}_y+C{\underline{e}}_z\right)\end{array} $$
(4.6)

and we further assume regularity of A, B, C, we know that ω is exact iff

$$ \nabla \wedge \underline{v}=0 $$
(4.7)

This is the universally known Stokes theorem. So, the question of non-holomomity is posed only if q is defined starting from a differential form. Alternatively, we might have a non-holonomity problem when q is not a proper coordinate because it is multivalued. Since holonomic coordinates are common, we give a counter example for a coordinate in 2D that is not regular holonomic over the whole ℝ 2.

Example 4.1

Let us take the angular coordinate θ in ℝ 2 (see Fig. 4).

Fig. 4
figure 4

The angular coordinates θ P = s P in ℝ 2

Full size image

It is clear that θ is singular at P=0 where it is not defined. Therefore, we can say that θ is holonomic in Ω1 but not in Ω2 (see Fig. 5), because in Ω1 every closed curve can be shrunk to a point without exiting from the set, while it is not in the Ω2 because this circular crown is not simply connected. In fact, the integral of dθ along the curve L in Ω2 is non-zero

$$ {\int}_L d\theta =2\pi $$
(4.8)

Clearly, the line L in Fig. 5 cannot be shrunk to a point remaining in Ω2.

Fig. 5
figure 5

The Ω1 and the Ω2 domains

Full size image

Another way to look at the problem is to say that indeed θ is a multivalued function of P

$$ {\theta}_P={\overline{\theta}}_P+2 n\pi $$

The only way to avoid that and return to a single valued function is to cut the plane along a straight line issued from the origin. Therefore, Ω1 in Fig. 5 is acceptable for θ while Ω2 is not because it includes one part of the forbidden cut.

So, returning to geodetic heights, we need only to verify whether our definition of regular holonomity is satisfied by them.

First of all, we underline that our set Ω in ℝ 3, which is similar to a spherical crown, is simply connected. In fact, every loop in Ω can b continuously shrunk to a point remaining in Ω. So, the only thing to be verified is that H G is continuously differentiable.

Proposition 4.1

The geodetic heights H D, H o, H ∗, h are regular holonomic variables in Ω.

Proof

As for h(P) it is enough to observe that

$$ dh\left(P, dP\right)=\underline{\nu }(P)\cdotp d\underline{P} $$
(4.9)

i.e.

$$ \nabla h=\underline{\nu }(P) $$
(4.10)

which is certainly a continuous function of P in a layer around the Earth ellipsoid.

The proof for H D, H o, H ∗ is just a check on the regularity of W(P) and \( \underline{g}(P)=\nabla W(P) \) as they are all defined by means of the gravity field. The result comes if we consider that \( W(P)=V(P)+\frac{1}{2}{\omega}^2\left({x}^2+{y}^2\right) \), where the centrifugal part is clearly continuously differentiable in ℝ 3.

As for the Newtonian potential V(P) we need to remember that the mass density ρ(P) generating V(P) is bounded on the compact set B, the body of the Earth, and zero outside. Therefore, we have too ρ(P) ∈ L P(ℝ 3), ∀ p > 3. Then, as we know from potential theory (e.g. see Miranda 1970, § 13), we have that the potential V(P) generated by ρ(P) in any bounded domain (like a sphere with a large but fixed radius) is in C 1, α with \( \alpha <1-\frac{3}{p}. \) We recall that C 1, α functions are not only continuous with their first derivatives but even Hölder continuous with exponent α. So, \( V(P),\underline{g}(P) \) are α-Hölder continuous for any exponent α < 1.

Therefore, that H D is regular holonomic, is just tautological. That H o is regular holonomic has been proven in Sansò and Vaniček (2006) with a detailed geometric analysis of dH o.

That H ∗ is regular holonomic descends from its definition (3.26). In fact, differentiating such relation in the direction of \( {\underline{\nu}}_P \) we get

$$ \underline{\gamma}\left({P}^{\ast}\right)\cdotp {\underline{\nu}}_Pd{H}^{\ast }=\underline{g}(P)\,\cdotp {\underline{\nu}}_P dh $$
(4.11)

since P ∗ is a continuous function of P, according to the implicit function theorem, \( {\underline{\nu}}_P \) and \( \underline{g}(P) \) are continuous functions of P, because \( \underline{\gamma}\left({P}^{\ast}\right)\cdotp {\underline{\nu}}_P\ne 0 \) in Ω (in fact, \( \underline{\gamma}\left({P}^{\ast}\right)\cdotp {\underline{\nu}}_P\cong -\gamma \left({P}^{\ast}\right) \)); so we see that dH ∗ is a continuous function of P too, and the proof is complete.

Remark 4.1

In geodetic practice there is in use a true non-holonomic height, namely the so called normal orthometric height H no defined as (Sansò et al. 2019, §6.5)

$$ {H}_{no}(P)=-\frac{1}{\overline{\gamma}}{\int}_{P_0}^P\frac{\gamma (Q)}{g(Q)} dW(Q) $$
(4.12)

where \( \overline{P_0P} \) is a levelling line, \( -\frac{dW(Q)}{g(Q)} \) is the levelling increment and \( \overline{\gamma} \) is the average of γ(Q) on the ellipsoidal normal between the geoid and H no, under the point P. Such a height system is applied in the Australian continent (see Featherstone and Kuhn 2006).

As a matter of fact

$$ \omega =-\frac{\gamma }{g} dW $$
(4.13)

is not an exact differential form, as otherwise \( \frac{\gamma }{g} \) should be constant on equipotential surfaces, which is not.

5 Comparisons and Conclusions

The paper has examined four fundamental geodetic height systems with the purpose of clarifying that they are proper regular coordinates and none of them is non-holonomic. Moreover, the reference surface of the four height systems have been identified: they are the geoid for dynamic, orthometric and normal heights, the ellipsoid for ellipsoidal heights.

The question then arises on which height should be utilized in practice. The answer of the authors is that it depends on the use we want to make of it.

Certainly, the ellipsoidal height is the neatest concept from the geometrical point of view, and it is the natural to use it when GNSS observations play a main role; as an example, consider aerial navigation. On the contrary, the other height systems enter more naturally where the gravity fields play a role, for instance throughout levelling lines observations as it happens in many engineering applications. In this respect, it seems useful to us to evaluate the order of magnitude of the corrections to be applied to the integral of the levelling increment ∆L PQ along the levelling line, namely

$$ \Delta {L}_{PQ}={\int}_P^Q\delta L={\int}_P^Q\underline{n}\,\cdotp d\underline{r} $$
(5.1)

In this sense it is interesting to compare such corrections that we will call

$$ C\left({H}_D\right)=\Delta {L}_{PQ}-\Delta {H}_{D_{PQ}} $$
(5.2)
$$ C(h)=\Delta {L}_{PQ}-\Delta {h}_{PQ} $$
(5.3)
$$ C\left({H}^{\ast}\right)=\Delta {L}_{PQ}-\Delta {H}_{PQ}^{\ast } $$
(5.4)
$$ C\left({H}_o\right)=\Delta {L}_{PQ}-\Delta {H}_{o_{PQ}} $$
(5.5)

Such corrections are a metric index of the difference between the levelling increment added along a line, and the specific height difference of the two extreme points. The larger the correction, the larger are errors entering in its computation.

Based on formulas by Heiskanen and Moritz (1967), section 4.4, further elaborated in Sansò et al. (2019), section 6, it is easy to derive the following rough estimates of the four corrections, all referring to the worst case:

$$ O\left(C\left({H}_D\right)\right)\sim {10}^{-3}\Delta {H}_{PQ} \vspace*{-16pt}$$
(5.6)
$$ O\left(C(h)\right)\sim {10}^{-4}{l}_{PQ}\vspace*{-16pt} $$
(5.7)
$$ O\Big(C\left({H}^{\ast}\Big)\right)\sim {10}^{-4}\Delta {H}_{PQ}\vspace*{-16pt} $$
(5.8)
$$ O\left(C\left({H}_o\right)\right)\sim {10}^{-4}\Delta {H}_{PQ} $$
(5.9)

where ∆H PQ is the height difference between the end points of a levelling line and l PQ its horizontal length.

Yet it is necessary to recall that the computation of C(H o), contrary to that of C(H ∗), requires some knowledge of the density of the topographic masses, so introducing one further uncertainty in its computation.

We conclude then that H ∗ is better suited than H o to treat levelling observations, without introducing unnecessary uncertainties due to poor knowledge of the mass density. Of course, H ∗ is also the natural coordinate to be used when tackling the solution of the GBVP.