The use of gravity data to determine orthometric heights at the Hong Kong territories

Albertini Nsiah Ababio; Robert Tenzer

doi:10.1515/jag-2022-0012

Article

The use of gravity data to determine orthometric heights at the Hong Kong territories

Albertini Nsiah Ababio and Robert Tenzer

Published/Copyright: August 4, 2022

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From the journal Journal of Applied Geodesy Volume 16 Issue 4

Abstract

The Hong Kong Principal Datum (HKPD) is the currently adopted official geodetic vertical datum at the Hong Kong territories. The HKPD is practically realized by heights of levelling benchmarks. The HKPD heights are, however, neither normal nor orthometric. The reason is that heights of levelling benchmarks were determined from precise levelling measurements, but without involving gravity observations along levelling lines. To reduce systematic errors due to disregarding the gravity information along levelling lines, we used terrestrial and marine gravity data to interpolate gravity values at levelling benchmarks in order to compute and apply the orthometric correction to measured levelling height differences. Our results demonstrate the importance of incorporating the gravity information even for a relatively small region but characterized by a rough topography with heights of levelling benchmarks exceeding several hundreds of meters. According to our estimates, the orthometric correction reaches (and even slightly exceeds) ±2 cm, with maxima along levelling lines crossing mountain chains.

Keywords: gravity; levelling; orthometric height; vertical geodetic control

Funding source: Research Grants Council, University Grants Committee

Award Identifier / Grant number: 15218819

Funding statement: The work presented in this article was supported by the by the Hong Kong GRF RGC project 15218819: “The modernization of height datum in the Hong Kong territories”.

Appendix A The error analysis

The relation between the (actual) error of the orthometric correction ε O C and the errors ε Δ h and ε H of measured height differences and heights of levelling benchmarks can be derived from Eq. (10) in the following form

(A.1) ε O C = 1 g ‾ i + 1 g i + g i + 1 2 − g ‾ i + 1 ε Δ h + g ‾ i g ‾ i + 1 − 1 ε H .

Assuming that for relatively small distances between levelling benchmarks g ‾ i ≈ g ‾ i + 1 and g i ≈ g i + 1 , the expression in Eq. (A.1) simplifies to

(A.2) ε O C ≈ 1 g ‾ g − g ‾ ε Δ h .

Substituting from Eq. (3) for the mean gravity g ‾, the term g − g ‾ in Eq. (A.2) is further rearranged into the following form

(A.3) g − g ‾ ≅ g − g + 1 2 ∂ g ∂ h H ≅ 1 2 ∂ γ ∂ h H + 2 π G ρ H ,

where ∂ γ / ∂ h ≅ 3 × 10 − 6 m s − 2 .

Inserting from Eq. (A.3) back to Eq. (A.2), we arrive at

(A.4) ε O C ≈ H g ‾ 1 2 ∂ γ ∂ h + 2 π G ρ ε Δ h .

The mean actual gravity g ‾ in Eq. (A.4) can be replaced by the mean normal gravity γ ‾, so that

(A.5) ε O C ≈ H γ ‾ 1 2 ∂ γ ∂ h + 2 π G ρ ε Δ h .

For the values 2 − 1 ∂ γ / ∂ h ≅ 1.5 × 10 − 6 m s − 2 and 2 π G ρ ≈ 1.1 × 10 − 6 and maximum heights of levelling benchmarks in Hong Kong H ≈ 500 m, the uncertainties in height differences ε Δ h ≤ 10 cm cause errors in computed values of the orthometric correction that are completely negligible.

Similarly, the errors of orthometric correction ε O C due to errors ε Δ g F A of interpolated free-air gravity anomalies can be estimated. From Eq. (10), the relation between these errors is found to be

(A.6) ε O C ≈ Δ h g ‾ ε Δ g F A ≅ Δ h γ ‾ ε Δ g F A .

For maximum height differences Δ h ≈ ±100 m along levelling lines in Hong Kong (see Fig. 12), the error ε Δ g F A ≈ ± 10 mGal = ± 1 × 10 − 4 m s − 2 introduces the error in computed values of the orthometric correction of about ±1 mm.

Appendix B Comparison of different gravity interpolation techniques

We used the kriging, natural neighbour, least-squares collocation, nearest neighbour, and radial basis functions to interpolate the simple planar Bouguer gravity anomalies at levelling benchmarks and compared them with the interpolated values obtained by applying the inverse distance weighting for the weight w = s i , i + 1 − 2 . The statistics of differences is summarized in Table B.1.

Table B.1

Statistics of differences (in mGal) between the interpolation results obtained by applying the kriging, natural neighbour, least-squares collocation, nearest neighbour, and radial basis functions and the interpolated values of the simple planar Bouguer gravity anomalies at levelling benchmarks (Table 3) computed by applying the inverse distance weighting (for the weight w = s i , i + 1 − 2 ).

Method	Min	Max	Mean	STD
Kriging	−3.04	2.58	0.10	0.74
Natural Neighbor	−3.25	2.50	0.09	0.67
Nearest Neighbor	−4.04	4.12	0.27	0.82
Radial Basis function	−3.501	3.92	0.21	0.88
Least-squares collocation	−3.23	2.87	0.12	0.75

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Received: 2022-04-18

Accepted: 2022-07-18

Published Online: 2022-08-04

Published in Print: 2022-10-26

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Articles in the same Issue

https://doi.org/10.1515/jag-2022-0012

Keywords for this article

gravity; levelling; orthometric height; vertical geodetic control