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A STUDY OF SOME TRANSFORMATION PROCEDURES FOR NIGERIAN GEODETIC NETWORK
P.C. Nwilo, F.A. Fajemirokun, C.U. Ezeigbo, A. M. Oyewusi & E.G. Ayodele Department of Surveying & Geoinformatics Faculty of Engineering University of Lagos, Akoka-Lagos, Nigeria. email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org & email@example.com
ABSTRACTThe connection between global and local datums is usually established by transformation parameters. In this study, two sets of transformation parameters relating the Nigerian geodetic datum (Clarke 1880) and the global datum (WGS84) were investigated using the coordinates of fourteen common points on both datums. The first set of transformation parameters (model1) are given in terms of variation in the
x0 , y0 , z0 ) of the center (shift parameters) of the ellipsoid, semi-major axis ( a ) and flattening
of the ellipsoid, while the second set (model 2) are given in terms of the changes in the geodetic
( 0 , 0 , h0 )
of the initial point, semi-major axis
( a ) and flattening ( f ) . Each set of
transformation parameters was used to transform the coordinates referred to WGS84 ellipsoid into the coordinates referred to Clarke 1880 ellipsoid. The effects of the different configurations of the data on the estimated transformation parameters were investigated. From this study, it was found that, using model 2, configuration 5, which consists of points centrally located in the geodetic network, gave the best set of transformation parameters. The best set of transformation parameters are
0 = 1.70 0.51" ,
0 = 1.93 1.16" , h0 = 17.83 1.0m .Keywords: Datum, transformation parameters, geodetic coordinates.
INTRODUCTION The Clarke 1880 ellipsoid was adopted as the reference ellipsoid for the Nigerian Geodetic Datum to meet the requirements for mapping and engineering projects. With the advent of the Navy Navigational Satellite System (NNSS) (Doppler), and later Global Positioning System (GPS) as tools for geodetic positioning, there is often the need to transform coordinates on the geocentric systems to coordinates on the local systems, and vice versa. 1
Efforts have been made by both Nigerian and Foreign agencies to determine transformation parameters for the Nigerian Geodetic Datum (Fubara, 1995). There are various methods by which the transformations can be executed. No conclusion has been reached as to which of them is the preferred approach and no doubt this is an area where discussions will continue for some time to come (Smith, 1997). In this paper, efforts were made to investigate two types of transformation procedures relating the Nigerian Geodetic Datum (Minna Datum) and the global datum, the World Geodetic system (WGS 84), on which the GPS is based.
BASIC CONCEPT OF GEODETIC DATUM A geodetic datum is a set of parameters that defines the size and shape of a given reference ellipsoid, as well as its position and orientation with respect to the real earth (geoid). A geodetic datum is often defined by a set of five parameters, namely (Seeber, 2003): a f Semi major axis of the reference ellipsoid Flattening of the ellipsoid and
x0 , y0 , z0 Change in the coordinates of the origin of the coordinatesystem of the reference ellipsoid. A geodetic datum can also be defined in terms of a, f , 0 , 0 , h0 (Fubara, 1995): Where, a and f are as defined above are changes in the latitude and longitude of the initial point of the geodetic network
0 , 0 h0
is the change in ellipsoidal height at the initial point of the geodetic network 2
Two types of unrelated datums were in the past considered in geodesy: A horizontal Datum, which forms the basis for the computation of horizontal coordiates and a vertical datum, to which heights refer. The horizontal and vertical datums were kept strictly separate, but today, because of satellite techniques, a unified global datum is often adopted for the realization of integrated geodetic procedures. In the Nigerian geodetic network, the Clarke 1880 ellipsoid was adopted as the horizontal geodetic datum, while the Lagos datum (physical location unknown), close to the East mole, was adopted as the vertical datum. With the evolution of space technology, a geocentric datum, which is used all over the world, is also in use in Nigerian geodetic network. The connection between global and local datums is established by
transformation parameters. To determine these transformation parameters, points whose coordinates are known on both datums are chosen.
THE NIGERIAN GEODETIC DATUM Several attempts have been made to determine transformation parameters for the Nigerian Geodetic Datum, using either Doppler or WGS 84 coordinates and Clarke 1880 coordinates (Oyeneye, 1985; Fajemirokun & Orupabo, 1986; Ezeigbo, 1990a, b). Some work was done by Campagine Generalie de Geophysique (Nigeria) Limited (Fubara, 1995). The report provided coordinates of data points without relevant background information for the assessment of the data quality. The values of the datum shifts were also published without any indication on how they were derived (Fubara, 1995). Geodetic Positioning Service, in collaboration with oil producing companies in Nigeria, has also determined transformation parameters for the Nigerian National
Petroleum Corporation (NNPC) using the combination of GPS and the Transit Satellite System (Doppler) data (Fubara, 1995).
Geodetic Surveys Nigeria also determined seven transformation parameters for Shell Petroleum Development Company (SDPC), for use in the Southern Nigeria (Fubara, 1995). In most of these investigations, it was observed that the transformation parameters were determined using either Molodensky-Badekas or Bursa-Wolf Model. In this study, two transformation procedures (models) are investigated. In the first model, the set of transformation parameters are the shift in the origin of the coordinate axes ( x0 , y0 , z0 ) , changes in semi-major axis ( a ) and flattening of the ellipsoid ( f ) . In the second model, the set of transformation parameters
are the changes in geodetic coordinates
( 0 , 0 , h0 )
at the network origin,
change in semi-major axis ( a ) and flattening ( f ) . The basic transformation equations used are presented in the next section.
DATA USED FOR THE INVESTIGATIONS The data used for this investigation were obtained from the Office of the Surveyor General of the Federation (OSGOF). They consist of the geodetic (curvilinear) coordinates of sixteen coincident points on both Minna datum and WGS84. Only fourteen out of the sixteen data points were actually used for the investigation. At an average of approximately one data point per 64,000 square km, the available data are far from being adequate. The data points are equally poorly distributed over the geodetic network. Furthermore, only scanty information on the accuracy and the reliability of those data is available.
MATHEMATICAL FORMULATIONS The relevant transformation equations in terms of x0 , y0 , z0 , a, f are given by (Heiskanen & Moritz, 1967, equation (5.55)): a = sin cos x0 + sin sin y0 cos z0 + 2a sin cos f a cos = sin x0 cos y0 (2.1a) (2.1b) (2.1c)
h = cos cos x0 cos sin y0 sin z0 a + a sin 2 fwhere x0 , y0 , z0 , a, f are as defined in section 1.2.
Similarly, the transformation equations in terms of variations 0 , 0 , h0 , a, f are given by (Heiskanen & Moritz, 1967, equation (5.57)):
= ( cos 0 cos + sin 0 sin cos ) 0 sin sin cos 0 0h a + ( sin 0 cos cos 0 sin cos ) 0 + + sin 2 0 a a + 2 cos ( sin sin 0 ) f cos = sin 0 sin 0 + cos .cos 0 0 h a cos 0 sin 0 + + sin 2 0 f a a (2.2b) f (2.2a)
h = ( cos 0 sin + sin 0 cos cos ) 0 + cos sin .cos 0 0 a h a + ( sin 0 sin + cos 0 cos cos ) 0 + + sin 2 0 f a a a + ( sin 2 2sin 0 sin ) f awhere = 0
0 , 0 , h0
changes in the geodetic coordinates of an initial point 5
( x0 , y0 , z0 ) parallel displacement or shift component of the origin of thecoordinate system of the reference ellipsoid
, , h a, f
changes in the geodetic coordinates at an arbitrary point changes in the parameters of the reference ellipsoid
Equations (2.2a, b & c) express the , , h at an arbitrary point in terms of
0 , 0 , h0 at the initial point of the network and a, f .There are some similarities between equations (2.1a, b & c) and (2.2a, b & c). They are infinitesimal transformations of geodetic coordinates. However, they differ in the parameters used for the transformation (Heiskanen & Moritz, 1967).
Equations (2.2a, b & c) can also be expressed in terms of variations of deflection components o and o and of geoidal undulation N o at the initial point of the network using Vening Meinesz formulae (Heiskanen & Moritz, 1967, equation (5.58); Torge, 1980; Musa, 2003):
0 = 0 0 cos = 0 h0 = N0Substituting equation (2.3) is into equations (2.2a, b & c), we obtain: (2.3)
= ( cos 0 cos + sin 0 sin cos ) 0 sin sin .0 N0 a ( sin 0 cos cos 0 sin cos ) + + sin 2 0 a a 2 cos ( sin sin 0 ) f f (2.4a)
= sin 0 sin 0 + cos h0 N0 a + cos 0 sin + + sin 2 0 a a f (2.4b)
N = ( cos 0 sin sin 0 cos cos ) 0 cos sin 0 a N0 a + ( sin 0 sin + cos 0 cos cos ) + + sin 2 0 f a a a + ( sin 2 2sin 0 sin ) f a 0 = 0 0' 0 = 0 0' N 0 = N 0 N0' a = a a' f = ff'where
( a ', f '; , , N )' 0 ' 0 ' 0
refer to a local geode