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# Recovery of orthometric heights from ellipsoidal heights using offsets method over Japan

- Patroba Achola Odera
^{1}Email author and - Yoichi Fukuda
^{2}

**Received:**21 May 2015**Accepted:**11 August 2015**Published:**21 August 2015

## Abstract

One of the most important applications of a geoid model is a recovery of orthometric heights from ellipsoidal heights (normally obtained from GNSS). The application of the geoid model for recovering orthometric heights from ellipsoidal heights is normally achieved by fitting the geoid model to a local vertical datum. The fitting procedure is usually accomplished by least squares collocation (LSC), using planar or spherical covariance functions. This procedure warps the gravimetric geoid model onto the local vertical datum, hence the local geoid model derived by this procedure, though convenient for local applications, it is not an equipotential surface. We propose offsets method for practical orthometric height recovery from a geoid model. The proposed procedure is more realistic because it does not constrain the local geoid to be coincident to the local vertical datum. We compare the performance of plannar fitting and offsets methods over Japan using a cross-validation procedure. Results show that offsets method performs better than the normally used planar fitting in the recovery of orthometric heights from ellipsoidal heights using a geoid model. The standard deviations of the differences between established and converted orthometric heights at randomly selected GPS/levelling test points over Japan are ±4 and ±3 cm for planar fitting and offsets methods, respectively. The offsets method is therefore more appropriate for converting ellipsoidal heights to orthometric heights than the planar fitting in the area of study.

## Keywords

- Orthometric height
- Ellipsoidal height
- Geoid model
- Planar fitting
- Offsets method
- Least squares collocation
- Kriging

## Background

One of the challenges to geodesists today is how to determine a consistent and functional vertical component of the geodetic datum. The existing height systems are defined in different ways but basically referred to the local mean sea level. The natural height datum is the geoid, which is part of the integrated geodetic datum. In geodetic positioning, the geoid is normally approximated with a rotational reference ellipsoid (with semi-minor axis perpendicular to the equatorial plane) as a conventional reference surface.

The precise geoid model not only enables us to convert ellipsoidal heights to levelled heights but also plays an important role in combining levelling data with GPS measurements to study vertical crustal movements for a longer period of time (Kuroishi et al. 2002). Amos and Featherstone (2009) investigated the use of quasi-geoid in the unification of New Zealand’s local vertical datums. A precise geoid model is necessary for the establishment of a rigorous orthometric height system and unification of vertical datums. An accuracy of ±1 cm for the global geoid model is considered sufficient for these purposes. However, the current high-resolution gravitational model (EGM2008) approximates the global geoid at an accuracy of ±15 cm (e.g. Pavlis et al. 2012).

Satellite positioning is gaining a lot of applications in Earth sciences today. One of the most extensively used satellite positioning in Earth sciences is the Global Positioning System (GPS). It is fast and efficient in determination of positions based on the World Geodetic System of 1984 (WGS84). It measures heights above WGS84 reference ellipsoid. These heights are called ellipsoidal heights (*h*). However, orthometric heights (*H*) are the functional heights for mapping, engineering works, navigation and other geophysical applications (Ayhan 1993).

The orthometric heights are normally obtained through spirit levelling, which is a very tedious and expensive process. The application of GNSS for height determination using a precise geoid model is desired. The current application involves warping the gravimetric geoid model to fit onto the local vertical datum. The problem with this method is that the surface realised after fitting is not an equipotential surface hence its physical applications are limited.

This paper describes the procedure of vertical datum establishment and related practical challenges. From the practical challenges, the usual straightforward relationship between the vertical datum (obtained through spirit levelling) and the reference ellipsoid (obtained through GPS levelling) is modified by adding another variable called an offset (distance between the geoid and a local vertical datum). It then compares two methods (planar fitting and offsets) for orthometric height recovery from a regionally defined gravimetric geoid model over Japan. An improved geoid model over Japan (Odera and Fukuda 2014) and rigorous orthometric heights over Japan (Odera and Fukuda 2015) have been used in this study. To avoid unnecessary repetitions of rigorous orthometric heights, they are simply referred to as orthometric heights in the subsequent sections.

## Methods

### Establishment of a vertical datum

The difference in height between the RBM and the conventional zero of the tide gauge (CZTG), *ΔH*
_{BM − TG}, is precisely determined. Observations at the tide gauge are made over a period of time (preferably 18.6 years, but rarely met in practice) to determine the local mean sea level, *H*
_{LMSL}. The height of the RBM above LMSL is then obtained as *H*
_{LMSL} + *ΔH*
_{BM − TG}. The heights of all other points in the network are subsequently obtained from the height of the RBM by accumulating the height differences along the interconnecting levelling lines (Cannon 1929).

The heights derived in this manner are generally referred to as orthometric heights (*H*). It should be noted that the position of the instantaneous sea level (ISL) in Fig. 1 is only indicative (the exact position may vary). In this approach, each country defines its own mean sea level, hence the existence of numerous vertical datums in the world today. The assumption that the geoid coincides with the mean sea level ignores the presence of sea surface topography (SST) as shown in Fig. 1. Rizos (1980) observed that the mean value of LMSL observed at the tide gauge stations cannot be considered to coincide with the geoid. Similar studies also confirmed this fact (Rapp and Balasubramania 1992; Pan and Sjöberg 1998).

### Planar fitting and offsets methods

*θ*) for practical applications, is normally given as

where *N* is the gravimetric geoid undulation, *H* is orthometric height based on the local vertical datum and *h* is ellipsoidal height at a point *P*, on the topographical surface. Equation (1) shows a perfect scenario which deviates from the practical reality because of the way local vertical datums are established, orthometric height system adopted and related systematic and non-systematic errors.

It should be noted that there is no vertical datum in the world today that is based on a rigorous orthometric height system. This is because of the difficulty in determining rigorous mean gravity along the plumbline between the geoid and a point at the topographical surface. This set-up may work after making a number of assumptions (not necessarily valid), where the local vertical datum is to be maintained and the problem is only how to convert ellipsoidal heights into the local vertical datum. However, it cannot be used in a country where there are many local height datums or systems. It is also not applicable for unification of regional height datums. This is the usual technique used in the conversion of ellipsoidal heights into orthometric heights. Planar or spherical covariance functions are normally used to fit a gravimetric geoid model onto a local vertical datum. The interpolation of geoid undulations is done using least squares collocation (LSC). Development and applications of LSC techniques in geodesy and related fields have been studied by several authors (e.g. Krarup 1969; Moritz 1973; Tscherning 1976; Kearsley 1977; Moritz 1980; Knudsen 1987; Forsberg 1987; Schaffrin 1989).

where *O*
_{LVD} is the offset of the existing LVD with respect to the local geoid, the orthometric height *H* is obtained from the local levelling network, ellipsoidal height *h* is obtained from GPS data and the geoid undulation *N* is obtained from a precise geoid model.

The above procedure is more realistic because it does not constrain the local geoid to be coincident to the local vertical datum. It can be used for the establishment of a geoid consistent vertical datum and regional unification of height datums using a precise regional geoid model. The set-up can also be applied where the local vertical datum is to be maintained and the problem is only how to convert ellipsoidal heights into the local orthometric heights. In this procedure, the geoid undulation and offset are interpolated using Kriging technique (Krige 1951; Schaffrin 2001). This technique is referred to as offsets method in this study. A comparative study of the two techniques (LSC and Kriging) in interpolation of gravity anomalies can be found in Odera et al. (2012). Amos and Featherstone (2009) used a similar model for unification of vertical datums based on a quasi-geoid model in New Zealand. They used an iterative gravimetric quasi-geoid computation procedure which is different from our proposal. A new method used over Japan decomposes differences between GPS/levelling and gravimetric geoid undulations into two components, namely ramp and residual geoid height but applies LSC for fitting (Miyahara et al. 2014).

## Results and discussion

where the gravimetric geoid undulation (*N*) and offset (*O*
_{LVD}) are interpolated from the neighbouring grids by inverse distance weighting (IDW) for easier application. IDW is a practical approximation that would not need extra painstaking computation at the application level.

Statistics of the differences between established and converted orthometric heights at 20 test points, using planar fitting and offsets methods (units in centimeters)

Method | Min | Max | Mean | SD |
---|---|---|---|---|

Planar fitting | −7.75 | 5.34 | −1.19 | 3.99 |

Offsets | −6.72 | 5.81 | −0.55 | 3.28 |

The standard deviations of the differences between established and converted orthometric heights at the test points using planar fitting and offsets methods are ±4.0 and ±3.3 cm, respectively. The offsets method is therefore more appropriate for accurate conversion of ellipsoidal heights to orthometric heights over Japan than the planar fitting. The actual height differences (Fig. 6) show that the orthometric heights of most points (16 out of 20) would be determined within 4 cm. Although offsets method is being applied for the first time over Japan, LSC has been used in the past with similar results as obtained in this study (e.g. Fukuda et al. 1997; Kuroishi et al. 2002; Kuroishi, 2009).

Apart from orthometric height determination from ellipsoidal height, local vertical offsets can be used to study deformations related to a local vertical datum although only approximately. The offsets around Aburatsubo tidal station in Tokyo Bay (origin of levelling network) is nearly 0 cm (Fig. 5), indicating a near coincidence between the geoid model and the Japanese local vertical datum at the origin. However, two distinct trends are observed with reference to Aburatsubo tidal station. The local vertical datum is generally below the gravimetric geoid model towards north while the local vertical datum is generally above the gravimetric geoid model towards south. This interpretation is based on the fact that gravimetric geoid model in the study area is above the reference ellipsoid. The two trends are not uniform indicating the presence of systematic errors (error propagation with distance from the origin) and non-systematic errors. The non-systematic errors may be due to ground movements caused by crustal deformations. Large offsets (>15 and < −25 cm) are generally occurring in known tectonically active areas over Japan. Crustal deformations over Japan caused by earthquakes, tsunamis, volcanic eruptions and related forces have led to a number of revisions of geodetic control values due to displacement of physical survey marks (e.g. Doi et al. 2005; Tobita 2009; Hiyama et al. 2011).

The boundary between vertical datum and a precise geoid model cannot be explained purely from this study. The said boundary is a result of various errors associated with geoid determination, local vertical datum establishment, crustal deformations and related datasets among others. An understanding of this boundary would be a key to a realization of a geoid consistent vertical datum in Japan.

## Conclusions

Recovery of orthometric heights from ellipsoidal heights using offsets method is found to perform better than planar fitting. In addition, this method does not constrain the gravimetric geoid model to fit onto the local vertical datum. The standard deviations of the differences between established and converted orthometric heights at randomly selected GPS/levelling test points over Japan are ±4 and ±3 cm for planar fitting and offsets methods, respectively. The offsets method is therefore more appropriate for converting ellipsoidal heights to orthometric heights over Japan than planar fitting. It is expected that this accuracy would improve with improvements in geoid modelling and modernisation of local vertical datum.

## 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. We are also grateful to the anonymous reviewers for their valuable comments and suggestions that have improved the paper. A part of this study was conducted when the first author was a PhD. candidate at Kyoto University, Geodesy Laboratory.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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