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1 Introductory Comparison Between Geoid and Quasi-Geoid

The geoid is simple to explain to the layman but not so the quasi-geoid. Also, only the geoid is an equipotential surface of interest in geophysics. Mathematically, to determine the geoid is an inverse problem while to determine the quasi-geoid is a forward problem, and the geoid problem is also a free boundary value problem (bvp) in the sense that the boundary itself is unknown (over dry land). On the contrary the quasi-geoid, or rather the height anomaly (ζ), can be achieved by solving one of the following problems. (See also Heiskanen and Moritz 1967, Chap. 8 and Sjöberg and Bagherbandi 2017, Sect. 6.3.)

Problem 1 (Molodensky’s problem)

Derive ζ from the known gravity and potential at the unknown topographic surface. This is a free bvp.

Problem 2 (a modern problem)

Derive ζ from the known gravity at the known topographic surface and orthometric height. This is a fixed bvp.

Problem 3 (A second modern problem)

Derive ζ from the known potential at the topographic surface and geodetic height (h). This is a fixed bvp.

In the sequel we will denote the solution of Problem 1 by Stokes’ formula, Method 1, the solution of Problem 2, Method 2, and the solution of Problem 3, Method 3. In the rest of the paper we will mainly refer to Methods 2 and 3. Geoid problems are caused by the partly unknown topographic density distribution and its extension down to the (unknown) geoid, problems that do not occur in quasi-geoid determination. On the other hand, in rough topography with over-hangs and vertical topography the quasi-geoid height is as ambiguous as the topography (e.g., Sjöberg 2018a).

The geoid height N is given by Bruns’ formula:

$$ N={T}_g/{\gamma}_0,\vspace*{-4pt} $$
(1)

where T g is the disturbing potential at the geoid and γ 0 is normal gravity at the reference ellipsoid.

On the contrary, the height anomaly ζ is given by the disturbing potential at point P on the Earth’s surface and normal gravity at the point Q on the telluroid (see Fig. 1):

$$ \zeta ={T}_P/{\gamma}_Q.\vspace*{-4pt} $$
(2)
Fig. 1
figure 1

Illustrations of the geodetic height (h), geoid height (N), orthometric height (H), height anomaly (ς) and normal height (H N)

Full size image

Note 1

The telluroid is defined as the surface, where the normal potential at each point Q equals the Earth’s surface potential at point P along the normal to reference ellipsoid (see Fig. 1).

Note 2

The height anomaly is the height from the telluroid to the Earth’s surface. The end user of the height anomaly usually places ζ at the reference ellipsoid under the name quasigeoid height, so that the height from this surface to the Earth’s surface becomes the normal height (H N).

Note 3

The fact that the orthometric height (between points P and P 0) is slightly curved along the plumbline is practically irrelevant.

2 Computational Steps for Geoid and Quasi-Geoid Determination

The determination of the geoid by Stokes’ formula requires downward continuation (DWC; e.g., by remove-compute restore technique) of gravimetric data to sea-level, while quasi-geoid determination in modern methods either requires DWC of gravity to sea-level or point level in Method 2 or direct employment of surface potentials in Method 3. These methods are illustrated below (see also Sjöberg and Bagherbandi 2017, Sects. 6.2–6.3.):

2.1 Geoid Determination by Stokes’ Formula

$$\vspace*{-4pt} {\displaystyle \begin{array}{l}\tilde{N}=\frac{R}{4{\pi \gamma}_0}\underset{\sigma }{\iint}\mathrm{S}\left(\psi \right){\left[\Delta \mathrm{g}+\Delta {\mathrm{g}}_{\mathrm{dir}}^{\mathrm{T}}\right]}^{\ast}\mathrm{d}\sigma +\mathrm{d}{\mathrm{N}}_{\mathrm{I}}^{\mathrm{T}}\\ {}\end{array}}\vspace*{-6pt} $$
(3)

where R is sea-level radius, σ is the unit sphere, S(ψ) is Stokes function with argument ψ being the geocentric radius between computation and integration points, \( \Delta {g}_{dir}^T \) is the direct topographic effect on the gravity anomaly Δg, []∗= DWC to sea-level, and \( \mathrm{d}{\mathrm{N}}_{\mathrm{I}}^{\mathrm{T}} \) is the indirect topographic effect.

Note 4

In Eq. (3) and below for quasigeoid determination Δg is the “modern” Earth surface gravity anomaly (introduced by M.S. Molodensky in 1945), i.e., surface gravity minus normal gravity at the telluroid/normal height (Fig. 1). However, geoid determination is traditionally conducted from a more approximate gravity anomaly determined by applying its free-air reduction to mean sea-level minus normal gravity at the reference ellipsoid.

2.2 Quasi-Geoid Determination

Method 1 (M.S. Molodensky 1945)

Here the disturbing potential is introduced as a surface integral of an unknown surface density (μ) on the telluroid (Σ). A Fredholm integral equation of the second kind relates μ to the known surface gravity anomaly. Assuming that normal heights are known all over the Earth and introducing several approximations, the latter integral can be solved for μ and ζ by successive iterations. However, the series will hardly converge for terrain slopes larger than 45°, which calls for a solution of low resolution or accuracy. (For more details, see Heiskanen and Moritz 1967, Chap. 8.)

Method 2 (Stokes Formula)

Approach 1 (Remove-Compute-Restore)

$$ \vspace*{-2pt}\zeta =\frac{R}{4{\pi \gamma}_Q}\underset{\sigma }{\iint}\mathrm{S}\left(\psi, {r}_P\right){\left[\Delta \mathrm{g}+\Delta {\mathrm{g}}_{\mathrm{dir}}^{\mathrm{T}}\right]}^{\ast}\mathrm{d}\sigma +\mathrm{d}{\zeta}_{\mathrm{I}}^{\mathrm{T}},\vspace*{-3pt} $$
(4)

where (S(ψ, r P) is the extended Stokes’ formula, \( \mathrm{d}{\zeta}_{\mathrm{I}}^{\mathrm{T}} \) is the indirect topographic effect and r P is the geocentric radius at point P.

or

Approach 2 (Direct DWC according to Bjerhammar 1962 )

$$ \zeta =\frac{R}{4{\pi \gamma}_Q}\underset{\mathrm{s}}{\iint}\mathrm{S}\left(\psi, {r}_P\right){\left[\Delta \mathrm{g}\right]}^{\ast}\mathrm{d}\sigma. $$
(5)

Approach 3 (DWC to point level of radius r P )

$$ \zeta =\frac{r_P}{4{\pi \gamma}_Q}\underset{\mathrm{s}}{\iint}\mathrm{S}\left(\psi \right){\left[\Delta \mathrm{g}\right]}^{\ast \ast}\mathrm{d}\sigma, \vspace*{-2pt} $$
(6)

where []∗∗ = DWC to point level.

Method 3 (Direct Determination from Known Surface Potential and Geodetic Height)

Assuming that the topographic surface is known from satellite geodetic positioning and the Earth’s potential at the surface from Earth Gravitational Models or in the future from direct determination by atomic clocks (e.g., Bjerhammar 1975, 1985), one can determine the surface disturbing potential and normal gravity at normal height (by iteration). Then the height anomaly/quasi-geoid follows from Eq. (2).

3 A Geoid Validation Problem

Let us assume that the geodetic height (h) is known from GNSS-leveling, and that the geoid and orthometric heights N and H are functions of the topographic density μ. Then it holds that

$$ h=N\left(\mu \right)+H\left(\mu \right)\iff N\left(\mu \right)=h-H\left(\mu \right). $$
(7)

(Note that Eq. (7) is an approximation, as the orthometric height is slightly curved along the plumbline, but that will not significantly affect the following result.)

If μ is in error, it follows that the errors of N and H are related by:

$$ dN\left(\upmu \right)=\hbox{-} dH\left(\upmu \right), $$
(8)

so that the erroneous density provides equal errors with opposite signs for geoid and orthometric heights. Hence, validating a gravimetric geoid model by GNSS-levelling ignores the error in topographic density. (See also Sjöberg 2018b.) One can show that this problem occurs also in validating a gravimetric geoid model by astro-gravimetric leveling, e.g., by using a zenith camera (Sjöberg 2022). That is, the true topographic density distribution cannot be verified by this validation process.

One may assume that Eq. (8) does not hold except for the true density μ, arguing that estimated geoid and orthometric heights are affected in different ways by the erroneous mass density, such that the estimated geodetic height \( \hat{h}\left(\mu \right) \) by Eq. (7) would disagree with its true value h (obtained by accurate geodetic positioning, e.g. GNSS). Then one could think of adjusting the density in \( \hat{h}\left(\mu \right) \) such that it matches h. However, this procedure would not be realistic, as one cannot solve the inverse gravimetric problem of the density of mass of the Earth by exterior gravity and geometric data. Hence, Eq. (8) must hold for any assumed density distribution.

On the other hand, if the gravimetric geoid and orthometric height models use different topographic density models, Eq. (8) does not hold.

Sometimes one justifies the accuracy of a gravimetric geoid model determined by adjusting overdetermined gravimetric data in a least-squares procedure (e.g., Foroughi et al. 2019). However, then the reported standard error can only estimate the internal accuracy, while any remaining topographic DWC problem is missing. See Sjöberg (2022).

The above verification problems do not occur in quasi-geoid determination.

4 Orthometric Height vs. Normal Height

Using the geoid as the reference surface, the natural height system is based on the orthometric height. However, determining the true orthometric height is an inverse problem just as the geoid problem. In practice one frequently introduces a topographic density model both for the geoid and the orthometric height to get a consistent system. Typical hereby is Helmert orthometric heights with a constant topographic density of 2,670 kg/m3. One should also remember that orthometric heights are curved along the plumbline, but the curvature can usually be ignored.

The normal height is the normal to the reference ellipsoid between the reference ellipsoid and telluroid. Assuming that the topographic height h (e.g., from GNSS positioning), normal gravity at the reference ellipsoid (γ 0), its vertical gradient (a = 0.3086 mGal/m) as well as the disturbing potential T at the point of computation are known (not necessarily at the Earth’s surface), the normal height H N can be determined in an iterative procedure simultaneously with normal gravity at normal height (γ Q) by using the start values γ Q ≈ γ 0 − ha and H N ≈ h. Then one iterates the solution by alternating between the following solutions until numerical convergence:

$$\vspace*{-3pt} {H}^N=h-T/{\gamma}_Q\vspace*{-3pt} $$
(9a)

and

$$ \vspace*{-3pt}{\gamma}_Q={\gamma}_0-{aH}^N.\vspace*{-3pt} $$
(9b)

In this way the reference ellipsoid is the zero-level of the normal height system. On the contrary, using the quasi-geoid as the zero-level of the normal height system will cause ambiguity problems in rough topography of a non-star shaped Earth model.

5 Concluding Remarks

  • In geophysics the quasi-geoid does not make sense, while the geoid is useful as a reference surface.

  • Geoid and orthometric heights depend on topographic density. This is not the case for quasi-geoid and normal heights.

  • No topographic reduction is needed for quasi-geoid and normal height determination.

  • Both the geoid and quasi-geoid can be evaluated by Stokes formulas after DWC of gravity anomalies to mean sea-level and point level, respectively.

  • The geoid validation problem in high topography due to the density uncertainty does not apply to quasi-geoid determination.

  • The topographic height, surface disturbing potential and normal gravity at the reference ellipsoid are sufficient information to determine the quasi-geoid height.

  • The quasi-geoid will be ambiguous in rough topography for a non-star shaped Earth model. In practice this problem can be solved by a mean quasi-geoid model corresponding to a specified resolution and mean topographic model.