Comparison of Helmert and rigorous orthometric heights over Japan
- Patroba Achola Odera^{1}Email author and
- Yoichi Fukuda^{2}
https://doi.org/10.1186/s40623-015-0194-2
© Odera and Fukuda; licensee Springer. 2015
Received: 13 November 2014
Accepted: 26 January 2015
Published: 24 February 2015
Abstract
The local vertical datum in Japan is based on the Helmert’s approximation of mean gravity along the plumbline. However, determination of a rigorous orthometric height requires that the integral-mean value of gravity along the plumbline between the geoid and the Earth surface has to be known precisely. An attempt has been made to obtain rigorous orthometric heights at 816 GPS/levelling points distributed over four main islands of Japan (Hokkaido, Honshu, Shikoku and Kyushu) by applying corrections to the Helmert orthometric heights. The corrections to Helmert orthometric heights are evaluated from the differences between the integral-mean gravity and the approximate mean gravity along the plumbline. The corrections to Helmert orthometric heights vary from −30.9 cm to 0.0 cm with a mean value of −0.4 cm and standard deviation of ±1.7 cm. An improved high-resolution gravimetric geoid model covering four main islands of Japan from our previous study is used to compare the consistency of the two height systems to a regionally defined gravimetric geoid model. The standard deviation of the differences between gravimetric and GPS/levelling geoid undulations is ±7.5 cm, when Helmert orthometric heights are used. The standard deviation of the differences between gravimetric and GPS/levelling geoid undulations reduces to ±7.3 cm, when rigorous orthometric heights are used. This indicates that rigorous orthometric heights are more consistent with the gravimetric geoid model than Helmert orthometric heights.
Keywords
Background
There are basically two different categories of height systems used in geodetic positioning. These are gravity field related (gravimetric) height system and ellipsoidal height system. The gravity field related heights are based on spirit levelling and gravity data along the levelling lines while the ellipsoidal heights are realised through satellite techniques. Gravimetric heights are obtained by dividing geopotential number by mean gravity between the geoid and the Earth’s surface. The way in which mean gravity is defined therefore determines the type of gravimetric height system. Orthometric height is one of the gravimetric height systems used in mapping, engineering works, navigation and other geophysical applications.
Orthometric height in the rigorous sense can be defined as the curved distance between the geoid and the Earth’s surface along the plumbline. The determination of a rigorous orthometric height therefore requires that the integral-mean value of gravity along the plumbline between the geoid and the Earth’s surface be known precisely. This is indeed an uphill task complicated further by the need to account for both lateral and radial mass-density variations.
It is important to note that there is no vertical datum in the world today that is based on a rigorous orthometric height system due to difficulty in obtaining integral-mean value of gravity along the plumbline between the geoid and the Earth surface. However, significant efforts have been made recently towards the realization of a rigorous orthometric height system (e.g. Santos et al. 2006; Tenzer et al. 2005; Kingdon et al. 2005; Dennis and Featherstone 2003; Tenzer and Vaníček 2003; Hwang and Hsiao 2003; Allister and Featherstone 2001; Kao et al. 2000).
Three techniques have been proposed and used in practice for the approximation of the integral-mean value of gravity between the geoid and the Earth’s surface along the plumbline. These include the Helmert method (Helmert 1890; Heiskanen and Moritz 1967), Mader method (Mader 1954) and Niethammer method (Niethammer 1932). Although Mader and Niethammer orthometric heights seem to be slightly more accurate than the Helmert orthometric heights, they are rarely used in practice; probably because of difficulty in the computations of the terrain correction, instead Helmert orthometric heights are in common use.
The Helmert orthometric heights are currently being used to assess the accuracy of local, regional and global gravimetric geoid models. For example, the difference between ellipsoidal height (obtained from GPS) and Helmert orthometric height (obtained from spirit levelling and gravity data) at a control point is compared with the gravimetric geoid undulation in the assessment of a geoid model. This comparison is affected by inherent errors in the gravimetric geoid model, ellipsoidal heights and orthometric heights.
This paper describes the procedure for determining corrections to Helmert orthometric heights at 816 GPS/levelling points over Japan. Comparisons between GPS/levelling and gravimetric geoid undulations (Odera and Fukuda 2014) are also presented. In this case, both Helmert and rigorous orthometric heights have been used for the comparisons. To avoid unnecessary repetitions in the subsequent sections, rigorous and Helmert orthometric heights are generally represented as H ^{ O } and H respectively.
Methods
Description of orthometric height systems
Due to the difficulty in the determination of integral-mean value of gravity along the plumbline, approximations are normally made for practical determination of orthometric heights. This gives rise to various orthometric heights depending on the approximation method used for the determination of the mean gravity along the plumbline. Helmert orthometric height system is one of the orthometric height systems that is widely used. Other orthometric height systems include; Mader orthometric heights (Mader 1954; Krakiwsky 1965) and Niethammer orthometric heights (Niethammer 1932; Krakiwsky 1965). Although, normal heights (Molodensky et al. 1960), and normal-orthometric heights (Rapp 1961; Heck 2003) are used for the establishment of vertical datum in many countries, they are not referred to the geoid.
where the units of g and H are mGal and m respectively.
Practical determination of orthometric height is always achieved through precise spirit levelling (geodetic levelling). Helmert orthometric heights can be considered as the sufficient approximation of the rigorous orthometric heights for most practical applications. However, it should be noted that this approximation embeds a constant topographic mass-density for the Bouguer shell and completely neglects terrain roughness residual to the Bouguer shell (Santos et al. 2006).
Determination of rigorous orthometric heights over Japan
where g ^{ NT } is the gravity generated by masses contained within the geoid (e.g. Vaníček et al. 2004) and g ^{ T } is the gravitational attraction generated by the topography.
where \( {\varepsilon}_{H^H} \) is the correction to Helmert orthometric height and \( {\varepsilon}_{\overline{g}} \) is the difference between the integral-mean gravity along the plumbline and the approximate value (i.e. \( {\varepsilon}_{\overline{g}}=\overline{g}-{\overline{g}}^H \)).
The establishment of the Japanese vertical datum can be traced to the levelling survey carried out by the Army Land Survey in 1883 (Imakiire and Hakoiwa 2004). The vertical datum was obtained through tidal observations from 1873 to 1879 at Reigan-jima in Tokyo Bay (e.g. Matsumura et al. 2004). Initially, normal-orthometric height system was used in Japan before conversion to the current Helmert orthometric height system obtained by incorporating measured gravity data. The first set of Helmert orthometric heights was published in 2002 by Geospatial Information Authority of Japan.
where Ω is a dummy argument representing the spatial position (φ and λ), and \( \overline{K} \) represents the intermediary integration kernel (averaged Poisson’s kernel).
where \( \varDelta {g}_{cg}^H \) is the Helmert’s gravity anomaly evaluated at the co-geoid, V _{ g } ^{ CT } is the gravitational potential of condensed topographical masses on the geoid, V _{ g } ^{ T } is the gravitational potential of topographical masses computed at the geoid, the disturbing potential at the geoid (T _{ g }) in the second term and the third term [in the bracket] are related to the geoid undulation (N) and indirect effect (N _{ ind }) respectively through Bruns’s formula, and the last term is the gravitational attraction of the condensed layer on the geoid. The geoid related terms are obtained from the recent geoid model for Japan (Odera and Fukuda 2014).
Results and discussions
Statistics of components of rigorous mean gravity along the plumbline (units in mGal)
Min. | Max. | Mean | Std | |
---|---|---|---|---|
Mean normal gravity | 979,388.4 | 980,637.2 | 979,888.4 | 329.0 |
Mean geoid-generated gravity disturbance | −0.3 | 33.6 | 1.6 | 3.0 |
Mean gravitational attraction of spherical bouguer shell | 0.1 | 187.3 | 11.6 | 19.2 |
Mean gravitational attraction of terrain roughness | −5.5 | 13.5 | 1.0 | 1.8 |
Rigorous mean gravity along the plumbline | 979,396.1 | 980,640.3 | 979,902.5 | 326.6 |
It is interesting to note that the biggest problem in geoid determination is also encountered in the mountainous areas. This brings to question the use of Helmert orthometric heights in validating gravimetric geoid models in general and specifically in mountainous areas, in view of the increasing accuracy in geoid modelling. It may be interesting to find out the contribution of the differences between rigorous and Helmert’s orthometric heights on downward continuation of gravity anomalies onto the geoid, especially in mountainous areas, where there are significant differences.
Statistics of the differences between gravimetric and GPS/levelling geoid undulations using rigorous orthometric heights, Bracketed values represent the differences using Helmert orthometric heights (units in cm)
Region | Points | Minimum | Maximum | Mean | Std |
---|---|---|---|---|---|
Hokkaido | 163 | −7.88 (−7.88) | 27.29 (27.54) | 6.73 (6.82) | 6.19 (6.22) |
North Honshu | 171 | −14.19 (−14.19) | 22.18 (22.18) | 5.95 (6.07) | 6.88 (6.90) |
Central Honshu | 163 | −7.81 (−7.81) | 36.67 (36.67) | 6.64 (7.70) | 6.86 (7.14) |
West Honshu | 158 | −8.97 (−8.98) | 21.16 (21.16) | −1.05 (−1.04) | 4.92 (4.93) |
Shikoku | 56 | −14.27 (−14.27) | 16.98 (16.98) | −0.36 (−0.34) | 6.53 (6.55) |
Kyushu | 105 | −21.57 (−21.58) | 9.37 (9.37) | −2.87 (−2.77) | 5.07 (5.18) |
Whole | 816 | −21.57 (−21.58) | 36.67 (36.67) | 3.32 (3.59) | 7.30 (7.48) |
Table 2 shows that there is a slight improvement in the standard deviation in all the regions (Hokkaido, North Honshu, Central Honshu, West Honshu, Shikoku and Kyushu), with Central Honshu (a mountainous region) having the largest improvement (from ±7.1 to ±6.9 cm), when rigorous orthometric heights are used. The standard deviation improves from ±7.5 to 7.3 cm in the whole area of study. This shows that the rigorous orthometric heights are more consistent with the determined gravimetric geoid model than Helmert orthometric heights over Japan. We would like to note that, this observation is only indicative given the scope of this study.
Conclusions
An attempt has been made to determine rigorous orthometric height system in Japan. Corrections to the existing Helmert orthometric heights at 816 GPS/levelling points have been computed. These corrections vary from −30.9 cm to 0.0 cm with a mean value of −0.4 cm and a standard deviation of ±1.7 cm. The gravitational attraction due to the laterally varying topographical density is ignored in this work because of lack of actual topographical density model in the area of study. Considering the heights of the GPS/levelling points, the magnitude of the gravitational attraction due to the laterally varying topographical density is small, hence may not affect the determined corrections significantly in this case. However, this effect requires a careful investigation.
An improved high-resolution gravimetric geoid model covering four main islands of Japan from our previous study is used to compare the consistency of the two height systems to a regionally defined gravimetric geoid model. The standard deviation of the differences between gravimetric and GPS/levelling geoid undulations is ±7.5 cm, when Helmert orthometric heights are used. The standard deviation of the differences reduces to ±7.3 cm, when rigorous orthometric heights are used. This indicates that rigorous orthometric heights are more consistent with the gravimetric geoid model than Helmert orthometric heights. Therefore rigorous orthometric height system should be used in validating gravimetric geoid models in general and specifically in mountainous areas, in view of the increasing accuracy in geoid modelling.
Although lateral topographic density model has been ignored in this study, partly because of the effective height range (0.3 to 903.4 m), its contribution is significant for more accurate determination of orthometric heights over Japan, if the full height range (−4 to 3,776 m) is considered. Observed gravity data at the benchmark points would improve the accuracy of rigorous orthometric height determination. Finally we recommend a determination and inclusion of lateral topographical density model in similar future studies covering the full height range over Japan to validate our initial proposal for the adoption of a rigorous orthometric height system in Japan.
Declarations
Acknowledgements
We would like to thank the Geospatial Information Authority of Japan for providing GPS/levelling and other additional data sets covering the study area. Nagoya University and other organizations in Japan provided a detailed gravity database, covering south western parts of Japan. We are grateful to the anonymous reviewers for their constructive comments and suggestions that have helped improve the paper.
Authors’ Affiliations
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