Orthometric heights from GPS-levelling observations

# Orthometric heights from GPS-levelling observations

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B Nagarajan
Director, Geodetic and Research Branch
[email protected]

S K Singh,
Geodetic & Research Branch,

It is necessary to transform the ellipsoidal heights to orthometric heights and this procedure is managed with the fundamental mathematical relationship between the two height systems. The application of GPS for providing height control in an area with existing leveling data may combine the tasks of local geoid determination and its bias with the local leveling datum

1. Introduction:
The development of satellite based Global Positioning System (GPS) has facilitated the more accurate, more practical and more economic use in geodesy and surveying professions. Today GPS serves the very wide range of geodetic applications ranging from monitoring earth crustal deformation to building the basis for GIS (Geographical Information System). An established reliable geodetic infrastructure in the working area is the prerequisite requirement for making the most efficient use of GPS. Geoid is one of the most important part of a geodetic infrastructure. It is well known that globally, mean sea level best fits, in a least square sense, geoid which represents equipotential surface of earth’s gravity field and used as reference for physical height system like orthometric heights. With the advancement of theory and methodology during the past few years the accurate geoid modeling has been made possible in many parts of the world (Denker et. al. 2000; Kuroishi 2001; Smith and Roman 2001). A Geoid determined by gravimetric method by making use of sufficiently dense gravity and terrain informations along with a global geopotential model fits well on a regional / local scale at short wavelength. In most of the applications distance of a point above the geoid is expressed with orthometric height which is a physical concept through which vertical position of a point is uniquely defined. On the other hand, vertical position of a point derived by GPS is defined in geometrical sense which does not have any practical meaning. Therefore it is necessary to transform the ellipsoidal heights to orthometric heights and this procedure is managed with the fundamental mathematical relationship between the two height systems. The application of GPS for providing height control in an area with existing leveling data may combine the tasks of local geoid determination and its bias with the local leveling datum. Even if the sparse gravity are available the height of new points can be determined by making use of GPS ellipsoidal heights at these points (Tscherning et. al., 2001).

Present study was taken up as a part of project work of providing Ground Control Points (GCP’s) for initiating a comprehensive development plan for Bangalore metropolitan area. The area in question was moderate to densely populated with heavy traffic plying on the roads. The topography of the area is best described by gentle slopes and low hilly terrain. Though it was some what easy to provide planimetric coordinates of Ground Control Points using GPS but the main problem was to connect them by conventional levelling in order to form the reference network for producing large scale maps of the area with 1 m contour interval. It was almost impossible to run the levelling lines all around the city due to heavy traffic and public movement through thickly populated and busy market places. Hence it was considered necessary to devise an alternative technique through which the orthometric heights could be provided by making use of GPS ellipsoidal heights. For the last few years GPS, along with a suitable geoid model have been becoming a progressive tool in establishing a vertical network for engineering applications. GPS measurements made at the levelling benchmarks give pointwise geoid undulations and contain a complete range of geoid signals. The geoid heights given by GPS / levelling observation is only an approximations since GPS refers to global geocentric reference frame while levelling is related to the local vertical datum. The most accurate and reliable technique of determination of high spatial resolution geoid could be to combine local gravimetric models with GPS / levelling geoid heights (Milbert 1995; Kuroishi et. al 2002). GPS / levelling is especially useful for obtaining orthometric heights in areas which are either inaccessible by conventional methods or where transnormal conditions disrupt the procedural requirements for levelling.

Therefore, it was decided to use GPS levelling as an alternative to conventional levelling in the area. It was quite hard to expect some optimistic results in the absence of an accurate gravimetric geoid for the region and only choice left was to model the geometrical differences between the GPS ellipsoidal and orthometric heights observed on the common points and subsequent interpolation at the desired locations.

2. GPS/ Levelling Geoid Heights.

2.1 Height Relationship:
The procedure of geodetic levelling provides a height that is commonly known as a height above mean sea level. The process gives level differences between t wo consecutive benchmarks, which is expressed by aligning the level bubble with the graded values on forward and backward level staves. The orthometric heights so derived reflect local variations in gravity as well as topographic gradients.

The reference datum for orthometric height, the geoid, is approximated by Mean Sea Level (MSL) in this case. Basically one has to establish a relationship between the orthometric heights obtained from geodetic levelling and GPS derived ellipsoidal heights using a common reference datum. The technique is often called geometrical approach for ‘height bias’ estimation.

The basic equation which relates the orthometric and ellipsoidal heights is
h=N+H ……………….. (1)
or
N = h – H ———————-(2)
Where H= Orthometric height, measured along curved plumb line
h= Ellipsoidal height, measured along the ellipsoidal normal.
N= Geoid height, the separation between geoid and ellipsoid. Ellipsoidal height is purely geometric value whereas orthometric height has a physical meaning and depends upon the gravity field of the earth. The relationship has been shown in fig. 1. Theoretically, since the ellipsoidal height and orthometric heights are measured along the normal to ellipsoid and along the direction of plumb line respectively the relationship defined in equation (1) is only an approximation but serve the purpose for most of the engineering applications.

Fig 1: Relationship between Ellipsoidal height,Orthometric height,Geoidal height and Deflection of vertical (e)