Keywords

1 Introduction

Regional gravimetric geoid models are solutions to the Geodetic Boundary Value Problems (GBVPs). The GBVPs are solved in a fictitious harmonic space where there is no topographic mass above the boundary surface (geoid or reference ellipsoid) on which various gravity functionals (e.g., gravity anomalies or gravity disturbances) furnish the boundary values. Consequently, the solutions of the GBVP require that the topographic mass elevation and density be available to compute its gravitational attraction on the points where gravity is observed, prior to attempting their removal and the calculation of the boundary values (Heiskanen and Moritz 1967, Ch. 3). The Helmert second condensation method provides a mechanism to remove the mass above the boundary surface by condensing the topographic mass to a very thin layer (Vaníček and Martinec 1994). The gravitational attraction of the topography at the observation points is used to reduce the gravity measurements down to the boundary surface through an empty (harmonic) space. This downward continuation is possible via the calculation of the Direct Topographic Effect (DTE) and the Secondary Indirect Topographic Effect (SITE). To transfer back to the real space in presence of the topography, the Primary Indirect Topographic Effect (PITE) is used.

In the past, due to lack of knowledge of the variable topographic mass density, a constant density value of \( 2670\frac{kg}{m^3} \) has exclusively been used to compute the topographic effects mentioned above. The departure of the actual density value from its constant value, hereafter called anomalous density, and its impact on the geoid heights was first investigated at the University of New Brunswick by Martinec (1993) and then implemented over the years by Fraser et al. (1998), Pagiatakis et al. (1999), and Huang et al. (2001). It was shown that this effect may reach the decimetre level, which is far from the ever growing need for a sub-centimetre accurate geoid (Huang et al. 2001; Foroughi et al. 2017; Janák et al. 2018; Tenzer et al. 2021). The anomalous density is a 3D function, however, it has been shown that the laterally anomalous density is dominant compared to its radial counterpart in the topographic reduction calculations, e.g., DTE, SITE, PITE, and consequently, it is important in the geoid modelling (Kingdon et al. 2009). When the topographic reductions are computed using the anomalous density, they are called direct density effect (DDE), secondary indirect density effect (SIDE), and primary indirect density effect (PIDE), respectively (Huang et al. 2001).

The UNB_TopoDensT_2v01 is a global model of the variable topographic density considered as a function of the horizontal coordinates, and it provides three different resolutions namely, 30″ × 30″, 5′ × 5′, and 1° × 1° along with their corresponding standard deviations (STDs). This model provides valuable information in the computation of the contribution of the anomalous density on the geoid heights and the estimation of their uncertainties (internal error). The contribution of the anomalous density in regional geoid modelling has been well studied before, see for example (Hunegnaw 2001; Kuhn 2002; Sjöberg 2004; Kiamehr 2006; Sjöberg and Bagherbandi 2011; Chaves and Ussami 2013; Janák et al. 2018; Albarici et al. 2019; Vajda et al. 2020; Lin and Li 2022). However, the effect of the standard deviation of the anomalous density on the internal error of the geoid heights has not (or just partially) been addressed in the literature, see e.g., Huang et al. (2001) for Canada, Foroughi et al. (2019) for the Auvergne test case, and Foroughi et al. (2023) for the Colorado 1-cm geoid experiment.

The internal error estimate of the geoid heights in the Auvergne test area was performed by Foroughi et al. (2019). In their study, the STD of the DDE on the geoid heights was disregarded and only the Bouguer shell for the PIDE was considered for error propagation. In this contribution we aim to address two missing parts namely, (a) investigate whether our internal error estimate of the geoid heights in Auvergne still stays below one centimetre when considering the complete terms of the PIDE for error propagation and (b) include the errors of the DDE on the geoid heights. In this study, we use the one-step integration method for the geoid determination, which is a combination of the inverse Poisson integral and Stokes or Hotine integral transform (Novák 2003; Goli et al. 2019b). This method provides equivalent results to the two-step geoid determination method developed by the gravity research group at the University of New Brunswick (Vaníček and Martinec 1994). We apply our formulations in the Auvergne test area which has been used by other researchers for geoid determination using different methodologies (Duquenne 2007; Goyal et al. 2021).

2 Theory

Using the one-step integration method (Novák 2003), the regional geoid heights are computed as:

$$ N{\,=\,}\mathrm{D}\left[\delta g{\,+\,}{\delta g}^t{\,-\,}\left(\delta {g}_L{\,+\,}\delta g^t_L\right)\right]+\left({N}_L+\delta N^t_L\right)+\delta N^t, \vspace*{-3pt}$$
(1)

where δg is the observed gravity disturbance (usually predicted at grid points of various grid sizes for numerical simplicity), δg t and δN t are the DTE and PITE respectively; δg L and N L are the long wavelength (reference) of the gravity disturbances and the geoid heights computed using an Earth gravity model (EGM) up to a maximum degree/order L; \( \delta g^t_L \) and \( \delta N^t_L \) are the computed long-wavelength components of the DTE and PITE, using the spherical harmonic coefficients of the topography or by numerical integration over a reference topography. D stands for the one-step inverse operator. Our focus in this study is on the contribution of topographic reductions (N t) on the geoid heights (and their STDs) when the anomalous density rather than a constant mean density is used. Neglecting the density variations in the reference DTE and PITE (\( \delta g^t_L \) and \( \delta N^t_L \)) we limit our formulations to:

$$ {N}^t=\mathrm{D}\left[\delta g^t\right]+\delta N^t. $$
(2)

Note that, there are other small terms in Eq. (1) that we neglected here, such as the ellipsoidal corrections, and the atmospheric effects because of their small contributions and irrelevance to the topic of this study. We also did not include the SITE here since we are using gravity disturbances as opposed to gravity anomalies, therefore the SITE term is not required; please see Vaníček and Martinec (1994) and Janák et al. (2018) for further details.

As mentioned above, according to the Helmert second condensation method, the topography above the boundary surface (here spherical approximation of the reference ellipsoid) is condensed to a thin layer on the same surface. The topographic reductions in Helmert space are applied by using the residual gravitational potential (δV) as the difference between the gravitational potential of the real (V T) and of the condensed topography (V C), i.e.,

$$ \delta V={V}^T-{V}^C. $$
(3)

Using Bruns formula in Eq. (3), the PITE reads:

$$ \delta N^t=\frac{\delta V}{\gamma_0}, $$
(4)

where γ 0 is the normal gravity computed on the reference ellipsoid. The DTE on the gravity disturbances is computed by the radial derivative of the δV, i.e.,

$$ \delta {g}^t=\frac{\partial \delta V}{\partial r}.\vspace*{-4pt} $$
(5)

Computation of the δV is achieved by using the following equation (Martinec 1998, Eqs. (3.3) and (3.4)):

$$ \begin{array}{ll}\displaystyle\delta V\left(r,\Omega \right)&=G\underset{\Omega^{\prime }}{\iint}\overline{\rho}\!\left({\Omega}^{\prime}\right){\int}_{{\mathrm{r}}^{\prime }=\mathrm{R}}^{\mathrm{R}+\mathrm{H}\left({\Omega}^{\prime}\right)}\frac{r^{\prime 2}}{\mathcal{L}\left(r,\psi, {r}^{\prime}\right)}\mathrm{d}{r}^{\prime}\mathrm{d}{\Omega}^{\prime }\\&\quad \displaystyle -{R}^2\ \underset{\Omega^{\prime }}{\iint}\frac{\sigma\!\left({\Omega}^{\prime}\right)}{\mathcal{L}\left(r,\psi, R\right)}\mathrm{d}{\Omega}^{\prime },\end{array}\vspace*{-4pt} $$
(6)

where, R is the mean radius of the Earth; Ω and Ω′ are indicators of the spherical latitude(φ) and longitude (λ); r and r ′ are the radii at the computation and integration points, respectively; G is the gravitational constant; ψ and \( \mathcal{L} \) are the angular and spatial distance between the two points; and ρ and σ are the indicators of topographic and condensed mass density, respectively. As mentioned above, the constant density value of \( {\rho}_0=2670\ \frac{kg}{m^3} \) is used when computing topographic reductions in Helmert space and the anomalous topographic density is limited to only the lateral variations (δρ(Ω)), i.e.,

$$ \rho\!\left(\Omega \right)={\rho}_0+\delta \rho\!\left(\Omega \right).\vspace*{-4pt} $$
(7)

Inserting Eq. (7) into Eq. (2) and only keeping the terms with density variation will result in the total effect of the anomalous density on the geoid heights:

$$ {N}_{\delta \rho}^t=D\left[{\delta g}_{\delta \rho}^t\right]+\delta {N}_{\delta \rho}^t. \vspace*{-4pt}$$
(8)

The DDE on gravity measurements can be written as (Martinec 1998, Eq. (6.7))

$$ \begin{array}{ll}\displaystyle{\delta g}_{\delta \rho}^t\left(r,\Omega \right)&\displaystyle =\mathrm{G}\underset{\Omega^{\prime }}{\iint }\ \left[\delta \rho\!\left({\Omega}^{\prime}\right){\left.\frac{\widetilde{\partial {\mathcal{L}}^{-1}}\left(r,\psi, {r}^{\prime}\right)}{\partial r}\right|}_{r^{\prime }=R}^{R+H\left({\Omega}^{\prime}\right)}\right.\\&\displaystyle \quad \left.-\delta \rho \left(\Omega \right){\left.\frac{\widetilde{\partial {\mathcal{L}}^{-1}}\left(r,\psi, {r}^{\prime}\right)}{\partial r}\right|}_{r^{\prime }=R}^{R+H\left(\Omega \right)}\right.\\&\displaystyle \quad\left.-{R}^2\left[\delta \sigma\!\left({\Omega}^{\prime}\right)-\delta \sigma \left(\Omega \right)\right]\ \frac{\partial {\mathcal{L}}^{-1}\left(r,\psi, R\right)}{\partial r}\ \right]\mathrm{d}{\Omega}^{\prime }, \end{array}\vspace*{-3pt}$$
(9)

and the PIDE on the geoid is (ibid, Eq. (6.10)):

$$\begin{array}{ll} \displaystyle\delta {N}_{\delta \rho}^t\left(\mathrm{r},\Omega \right)&\displaystyle=-\frac{2\pi G}{\gamma_0}\delta \rho \left(\Omega \right){H}^2\left(\Omega \right)\left[1+\frac{2}{3}\frac{H\left(\Omega \right)}{R}\right]\\&\quad \displaystyle +\frac{G}{\gamma_0}\underset{\Omega^{\prime }}{\iint }\ \left[\delta \rho\!\left({\Omega}^{\prime}\right){\left.\widetilde{\ {\mathcal{L}}^{-1}}\left(R,\psi, {r}^{\prime}\right)\right|}_{r^{\prime }=R}^{R+H\left({\Omega}^{\prime}\right)}\right.\\&\quad \left. \displaystyle -\delta \rho \left(\Omega \right){\left.\widetilde{\ {\mathcal{L}}^{-1}}\left(R,\psi, {r}^{\prime}\right)\right|}_{r^{\prime }=R}^{R+H\left(\Omega \right)}\right.\\&\quad \left. \displaystyle -{R}^2\frac{\delta\!\sigma \left({\Omega}^{\prime}\right)-\delta \sigma \left(\Omega \right)}{\mathcal{L}\left(R,\psi \right)}\ \right]\mathrm{d}{\Omega}^{\prime }, \end{array}$$
(10)

where δσ(Ω) is the anomalous mass density of the condensed layer (ibid, Eq. (6.4)):

$$ \delta \sigma \left(\Omega \right)=\delta \rho \left(\Omega \right)\left[H\left(\Omega \right)\left(1+\frac{H\left(\Omega \right)}{R}+\frac{H^2\left(\Omega \right)}{3{R}^2}\right)\right], $$
(11)

where H is the height of the points and the kernels and spatial distances are defined in Table 1.

Table 1 Definition of the kernels and spatial distances in the Helmert topographical corrections
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By propagating the anomalous topographic density (δρ) error in Eq. (9) and by neglecting the correlation between the two terms, the uncertainties of DDE are

$$ \begin{array}{ll}\displaystyle {s}_{{\delta g}_{\delta {\rho}}^t}^2\left(r,\Omega \right)&\displaystyle ={G}^2\underset{\Omega^{\prime }}{\iint }\ \left[{\left(\mathcal{M}\left(r,\psi, {r}^{\prime}\right)\right)}^2\ {\mathrm{s}}_{\delta \rho}^2\left({\Omega}^{\prime}\right)\right.\\&\displaystyle \left.+{\left(\mathcal{N}\left(r,\psi, {r}^{\prime}\right)\right)}^2{\mathrm{s}}_{\delta \rho}^2\left(\Omega \right)\right]\ \mathrm{d}{\Omega^{\prime}}^2, \end{array}$$
(12)

where

$$ \begin{array}{ll}\displaystyle\mathcal{M}\left(r,\psi, {r}^{\prime}\right)& \displaystyle ={\left.\frac{\widetilde{\partial {\mathcal{L}}^{-1}}\left(r,\psi, {r}^{\prime}\right)}{\partial r}\right|}_{r^{\prime }=R}^{R+H\left({\Omega}^{\prime}\right)}\\&\quad \displaystyle -{R}^2\delta \tau\!\left({\Omega}^{\prime}\right)\ \frac{\partial {\mathcal{L}}^{-1}\left(r,\psi, R\right)}{\partial r},\end{array} \vspace*{-12pt}$$
$$ \begin{array}{ll}\displaystyle {\mathcal{N}}\left({r},{\psi}, {{r}}^{\prime}\right)&\displaystyle =-{\left.\frac{\widetilde{{\partial}{{\mathcal{L}}}^{-{1}}}\left({r},{\psi}, {{r}}^{\prime}\right)}{{\partial r}}\right|}_{{{r}}^{\prime }={R}}^{{R}+{H}\left({\varOmega} \right)}\\&\quad \displaystyle +{{R}}^{{2}}{\delta} {\tau} \left({\varOmega} \right)\ \frac{{\partial}{{\mathcal{L}}}^{-{1}}\left({r},{\psi}, {R}\right)}{{\partial r}},\end{array} $$

where \( \delta \tau \left(\Omega \right)=\left[H\left(\Omega \right)\left(1+\frac{H\left(\Omega \right)}{R}+\frac{H^2\left(\Omega \right)}{3{R}^2}\right)\right] \), and s is the standard deviation. Similar to DDE, the error of PIDE can be obtained by applying the error of propagation to Eq. (10) :

$$ \vspace*{-3pt}\begin{array}{ll}\displaystyle {s}_{{\delta N}_{\delta \rho}^t}^2\left(r,\Omega \right)&\displaystyle ={\left(\frac{G}{\gamma_0}\right)}^2\ \underset{\Omega^{\prime }}{\iint }\ \left[{\left(\mathcal{P}\left(r,\psi, {r}^{\prime}\right)\right)}^2\ {s}_{\delta \rho}^2\left({\Omega}^{\prime}\right)\right.\\&\displaystyle \quad \left. +{\left(\mathcal{R}\left(r,\psi, {r}^{\prime}\right)\right)}^2\ {s}_{\delta \rho}^2\left(\Omega \right)\right]\ \mathrm{d}{\Omega^{\prime}}^2,\end{array}\vspace*{-3pt} $$
(13)

where

$$ \vspace*{-3pt}\mathcal{P}\left(r,\psi, {r}^{\prime}\right)={\left.\widetilde{\ {\mathcal{L}}^{-1}}\left(R,\psi, {r}^{\prime}\right)\right|}_{r^{\prime }=R}^{R+H\left({\Omega}^{\prime}\right)}-{R}^2\frac{\delta \tau\!\left({\Omega}^{\prime}\right)}{\mathcal{L}\left(R,\psi \right)},\vspace*{-3pt} $$
$$ \vspace*{-3pt}\begin{array}{ll}\displaystyle\mathcal{R}\left(r,\psi, {r}^{\prime}\right)&\displaystyle =-2\pi G\ {H}^2\left(\Omega \right)\left[1+\frac{2}{3}\frac{H\left(\Omega \right)}{R}\right]\\&\displaystyle \quad -{\left.\widetilde{\ {\mathcal{L}}^{-1}}\left(R,\psi, {r}^{\prime}\right)\right|}_{r^{\prime }=R}^{R+H\left(\Omega \right)}+{R}^2\frac{\delta \tau \left(\Omega \right)}{\mathcal{L}\left(R,\psi \right)}\ .\end{array}\vspace*{-3pt} $$

The one-step integration method is a combination of the Hotine integral transforms and the inverse of the Poisson integral equation (Novák 2003). The integration kernel of the one-step method relates the disturbing potential at the boundary level to the gravity disturbances above this surface. Given its inverse operator and the numerical instability of the downward continuation, the solution to this method is achieved iteratively(Goli et al. 2019b). Returning to our main attempt, we intend to propagate the anomalous topographical density error in Eq. (8). The error propagation in the iterative techniques is not a simple task, therefore, we seek an alternative regularized solution using classic Tikhonov regularization. Neglecting the correlation between two terms of Eq. (8), the generalized formula of the geoid internal error due to the errors of the topographical density variation reads:

$$\vspace*{-3pt} {\mathbf{C}}_{N_{\delta \rho}^t}={\mathbf{B}}^T{\mathbf{C}}_{{\delta g}_{\delta \rho}^t}\mathbf{B}+{\mathbf{C}}_{{\delta N}_{\delta \rho}^t},\vspace*{-13pt} $$
$$ \mathbf{B}={\left({\mathbf{D}}^{\boldsymbol{T}}\mathbf{D}+\upmu \mathbf{I}\right)}^{-1}{\mathbf{D}}^{\boldsymbol{T}} $$
(14)

where, D is a coefficient matrix containing discretized values of the one-step integration kernel (Novák 2003; Goli et al. 2019b), \( {\mathbf{C}}_{{\delta g}_{\delta \rho}^t} \) and \( {\mathbf{C}}_{{\delta N}_{\delta \rho}^t} \) are the covariance matrices of DDE and PIDE, respectively, and μ is a regularisation parameter that is estimated to provide an equivalent solution to the that of the iterative approch. Please see Foroughi et al. (2023) for further details on the estimation of the regularization parameter in the one-step integration method. In the digital density models, only the STDs of the laterally varying density models are provided and the correlation between these values is complicated to estimate and therefore not available. Consequently, we assume that the \( {\mathbf{C}}_{{\delta g}_{\delta \rho}^t} \) and \( {\mathbf{C}}_{{\delta N}_{\delta \rho}^t} \)are diagonal matrices with their diagonal elements computed by discretization of the integrals in Eqs. (12) and (13), therefore, the STDs of the anomalous density on the geoid heights read:

$$ {s}_{N_{\delta \rho}^t}^2=\left(\mathit{{diag}}\left({\mathbf{C}}_{N_{\delta \rho}^t}\right)\right). $$
(15)

3 Numerical Results

We consider the Auvergne test area to evaluate our formulations because this area has been used for international collaborations of different geoid determination methods (Valty et al. 2012; Foroughi et al. 2017; Mahbuby et al. 2017; Janák et al. 2018; Foroughi et al. 2019; Goyal et al. 2021; Abbak et al. 2022; Klees et al. 2022). Besides, we already have an internal error estimate of the geoid heights in this area where the contribution of the anomalous topographic density was missing so we can simply include this new contribution here. The Auvergne test case was introduced by Duquenne (2007) for comparing different geoid determination methods and determining whether the effort should be on methodological improvement or more gravity observations for more accurate geoid models. The Auvergne gravity data coverage is limited by −1° < λ < 7° and 43° < φ < 49° and the geoid computation area is the 3° × 2°centre block between −1.5° < λ < 4.5° and 45° < φ < 47° with medium topography (maximum height of 2000m). The geoid heights have been sought with a resolution grid of 1′ × 1′ this area. We extract the lateral density variations and their STDs from the global model UNB_TopoDensT_2v01 (Sheng et al. 2019) with two different resolutions of 30″ × 30″ and 5′ × 5′. Table 2 provides the statistics of the anomalous topographic mass density and their standard deviation in the study area.

Table 2 Laterally anomalous density values and their standard deviations in the Auvergne test case
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Fig. 1
figure 1

Lateral anomalous density variations (a) and their STDs in Auvergne (b) [kg/m3]

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Fig. 2
figure 2

The effect of the anomalous density on the DTE (i.e., DDE) (a) and the DDE STDs (b) in Auvergne [mGal]

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Fig. 3
figure 3

The effect of anomalous density on PITE (i.e., PIDE) (a) and the PIDE STD (b) in Auvergne [cm]

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Please note that the STDs of the UNB_TopoDensT_2v01 are estimated using the range of density values suggested for the 15 rock types included in the global lithospheric model of GLiM (Hartmann and Moosdorf 2012) and further incorporated into UNB_TopoDensT_2v01. The range of STDs of the lateral density variation in this model is large and it will improve with a better estimate of the rock density structure. Both DDE and PIDE are computed by integration over the inner zone, near zone, and far zone areas. The inner zone covers an area of 5′ × 5′ containing grid values of 3″ × 3″ around each computation point. The finer 3″ × 3″ anomalous density grid was interpolated from the 30″ × 30″ grid to match the resolution of the existing DEMs used for computing DTE and PITE when integrating over the inner zone. The near and far zones cover 10′ × 10′ and 3° × 3° comprising grid values on 30″ × 30″ and 5′ × 5′ spacing, respectively. At first, the point values of the DDE and PIDE (and their STDs) were computed on a 15″ × 15″ computation grid and then the mean values on a 1′ × 1′ grid was estimated. The calculation of the mean values is recommended (especially in the rough topography area) since the mean gravity values are typically used for the calculation of the local gravimetric geoid (Vaníček and Martinec 1994; Janák and Vaníček 2005; Afrasteh et al. 2019; Goli et al. 2019a). We have computed the DDE for the whole data coverage and the PIDE for the geoid computation area.

The anomalous lateral density is shown in Fig. 1(a) and their STDs are shown in Fig. 1(b). Figures 2 and 3 show DDE and PIDE along with their STDs respectively with a summary of their statistics provided in Table 3.

Table 3 DDE, PIDE and their STDs in Auvergne
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Please note that the minimum STD of zero in Table 3 is due to the Mediterranean Sea in the southeast part of the Auvergne area where its density was set to \( 1027\frac{kg}{m^3} \) and the uncertainty was set to zero in the UNB_TopoDensT_2v01.

Figure 2 represents the effect of DDE on the gravity observations and its estimated STDs. Using the one-step integration method we also compute the effect of DDE on the geoid heights that is shown in Fig. 4(a) and estimate its STDs (i.e., the first term on the right-hand side of Eq. (14)) that is shown in Fig. 4(b) and are mostly below 1mm in this area. Statistics are provided in Table 4.

Fig. 4
figure 4

The effect of DDE on the geoid heights and STD

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The total effect of the anomalous density on the geoid heights (i.e., Eq. (8)) is displayed in Fig. 5(a) and the STDs (\( {s}_{{\delta N}_{\delta \rho}^t} \)) are displayed in Fig. 5(b) and statistics are provided in Table 4.

Table 4 Statistics of the total effects of DDE and PIDE on the geoid heights in Auvergne
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Fig. 5
figure 5

Total effect of density anomalous on the geoid (a) and STDs (b)

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By adding the \( {s}_{\delta {N}_{\delta \rho}^t} \) to the total error estimate of the geoid heights reported by Foroughi et al. (2019), the error estimate of the geoid heights including the errors of the anomalous density is shown in Fig. 6 with statistics provided in Table 5.

Fig. 6
figure 6

Total error estimate of the geoid heights in Auvergne including all error sources (Foroughi et al. 2019) plus the STDs of the anomalous density

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Table 5 Statistics of the uncertainties of the geoid heights including all error sources (Foroughi et al. 2019) plus DDE and PIDE
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4 Conclusion and Remarks

With the existence of the global models of the topographical density variations, the topographic reductions for geoid determination can be computed using the actual density instead of a constant density value which may be far from reality, up to 20% (Kuhn 2002). The knowledge of the anomalous topographic density variation is increasing and with a better understanding of the rock types and their density structure, we will be able to compute the topographic reductions more accurately. We used the recently available global laterally varying topographic mass density model UNB_TopoDensT_2v01 to compute the direct and indirect density effects (DDE and PIDE respectively) and their uncertainties on the gravity and geoid heights using the Helmert second condensation method. Due to numerical complexity and lack of information on the covariance of the topographical density variations, we only considered the variances and neglected the correlations between error terms. Given the structure of the UNB_TopoDensT_2v01, finding the correlation between density variations is not an easy task. All we can say is that the correlations are predominantly positive and neglecting them will give a larger error estimate of the geoid heights and vice versa. Since a negative correlation is unlikely, our results are an overly pessimistic estimate of the geoid error, i.e., the internal error estimate of this study can be well trusted.

We tested our formulations in the Auvergne test area and showed that PIDE is the dominant source of uncertainty in the geoid heights. In a medium topography area, like the Auvergne region, the maximum uncertainty of the geoid heights due to the errors in the anomalous density is less than 2 cm with a mean value of only 1.5 mm which is below the target sub-centimetre threshold for internal error of geoid heights. We also added our DDE and PIDE error estimates to those computed by Foroughi et al. (2019) for the same region and confirmed that even using a comprehensive error propagation of the STDs of the anomalous density in the geoid determination, the mean value of the internal error is still below one-centimetre threshold. The STDs of the geoid heights including the uncertainties caused by the inclusion of anomalous density are larger than 1 cm in higher topography where the mean STD of the geoid heights is higher than 1 cm anyway. The results of this contribution confirm that topographical density information is now known well enough (errors in the density variation values are small enough) to make the resulting geoid more accurate than when the geoid is computed with assumed constant topographical density. We showed that a geoid with an internal error estimate of better than one centimetre is achievable considering the density variation for most of the globe where the topography is lower than 2000 m.