Coordinate Transformation between Everest and Wgs – 84 Datums

    Coordinate Transformation between Everest and Wgs – 84 Datums

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    S. K. Singh
    S K Singh
    Geodetic & Research Branch
    Survey of India, Dehradun, India
    grb82p@yahoo.com.in

    Survey of India (SOI) has been recognized as one of the most important national agencies for research and develop ment activity in many diverse fields such as space geodesy, astronomy, gravity and magnetism, seismo–tectonics etc besides its paramount importance in the field of surveying and mapping. Though department has been supplying topographic maps to users, organizations and individuals since a long time, there have been some problems in the distribution of maps due to security reasons. There has been consistent demand from several quarters that the topographical data should be treated as national assets and to be made available to general public without much restriction. The GIS industry has also been raising the issue on various platforms to speedily evolve a long-term policy on data sharing between government and non-government organizations. The government realized the genuineness of the problem and set-up a committee of intellectuals and experts to review the existing map restriction policies. The committee have gone through the various aspects of data sharing and its implications national security and suggested dual maps series in India.

    As per the recommendations of the committee a new series of maps based upon WGS-84 datum will be introduced in SOI apart from existing Everest datum maps. The WGS-84 series maps will be openly made available to civil users whereas existing maps will continue to meet defence requirements. The main task before the surveying community was how to produce the new maps. The easiest approach is to convert the existing maps through a set of datum transformation parameters. These transformations parameters were required to be determined immediately which involved tremendous amount of field work. The job was taken up by Geodetic & Research Branch of Survey of India under the financial support from Department of Science and Technology ( DST ), Government. of India.

    Everest and wgs-84 datums
    The Everest Ellipsoid has been used for India and several adjacent countries for mapping activities. Named after Sir George Everest the ellipsoid was derived in 1830 and since then it has been used as basis for all types of control surveys. Sir George Everest paid careful attention to the measurement of bases and astronomical latitudes and longitudes were measured throughout the arc of the meridian especially at Kalianpur (240 7) where many observation were made (Cook, A, 1990). Dimensions of Everest ellipsoid and its orientation at origin were carried out one by one at a number of times. Semi major axis (a), flattening (f) and North – South component of Deflection of Vertical (Meridonal) was defined by Everest in 1840 whereas East – West (Prime – Vertical) component was defined by Walker in 1878. Though Everest ellipsoid has been the best fitting mathematical surface for India and adjacent countries but it cannot be extended too far from the origin and hence its applications are rather limited. For this reason the Indian Datum is probably the least satisfactory of the major datums (DMA, 1983).

    The terrestrial reference system used by the U.S. Department of Defence (DoD) for GPS positioning is the World Geodetic System 1984 (WGS-84). The navigation message broadcast by GPS satellites are expressed in this system. WGS-84 is a global geocentric coordinate system defined originally by DoD based on Doppler observations of the TRANSIT satellite system (the predecessor of GPS). WGS-84 was first determined by aligning as closely as possible, using a similarity transformation, the DoD reference frame NSWC–92Z and Bureau International de l’Heure (BIH) Conventional Terrestrial System (BTS) at the epoch 1984.0 (BIH is the predecessor of the IERS and BTS is the predecessor of ITRF). It was realized by the adopted coordinates of a globally distributed set of tracking stations with an estimated accuracy of 1-2 meters (compare to 1-2 cm accuracy of ITRF) (Teunissen P.J.G., Kleusberg, A., 1998).

    Datum transformation parameters
    The difference between positions in terms of an individual local datum and positions in terms of a global datum may be of the order of several hundred meters, and may vary considerably even for a single local datum. If the local survey network has variable quality or does not have a continuous landmass, a country may effectively have a number of local datums, requiring a number of different transformations to the global datum.

    With the increasing exchange of geographic information local and globally, positions need to be available in terns of both a local and global datums. The process of mathematically converting positions from one datum to the other is known as datum transformation.

    Datum transformation parameters define functional relationship between two reference frames. The GPS derived coordinates and local terrestrial coordinates of collocated points may be processed together using an appropriate transformation model. The outcome of the processing would be a set of quantities termed as transformation parameters which could be used for converting coordinates from one datum to the other and vice-versa. The combination of data sets from two or more different sources is strictly an adjustment problem and must be solved by applying appropriate technique.

    Transformation procedure
    There are a number of ways to mathematically transform positions from one datum to another, but they all require “common points”. Common points are surveyed points that have known positions in terms of both the local and the global datum. The achievable accuracy of the datum transformation will be determined by the number, distribution and accuracy of these common points and the transformation technique adopted. Generally speaking, the greater the accuracy required, the more common points are needed.

    Selection of common points
    The present study was focused on estimation of transformation parameters between Everest & WGS-84 datum in order to convert the existing topo sheets to WGS-84 series maps without loss of generality. To accomplish the job 300 existing Great Trigonometrical Series (GT) stations, evenly distributed throughout the country were selected for GPS observations. The points were chosen in such a manner that it represents a good sample of the true relationship between the local and global datums. Since Indian triangulation network comprised of independently adjusted series with different levels of accuracy i.e. primary and secondary, it was highly unlikely to get a single set of transformation parameters with desirable accuracy for the entire country.

    Tranformation models
    Several mathematical models have been developed which describe the functional relationship between pairs of three dimensional coordinates. Three mathematical models, namely Bursa-Wolf (Bursa, 1962, Wolf, 1963), Molodensky (Molodensky et.al., 1962) and Veis (Veis, 1960) are noted as standard models due to their extensive use around the world over a number of years. The models differ from each other in several ways including a priori conditions, the type of coordinate used and the interpretation of results.

    The GPS provides new and independent source of data. Whether or not this data will yield a satisfactory solution of terrestrial network transformation problems is dependent on the accuracy and homogeneity of local geodetic coordinates and the mathematical model employed in the estimation procedure (Singh, S K 1994). In our case the accuracy of GPS derived coordinates of each stations has been assured by processing data with Bernese scientific software making use of precise ephemerides and keeping IGS stations as fixed sites. The Bursa-Wolf transformation model is the most popular and effective one and it has been used by several countries around the world. A typical example is South Africa where this transformation model was used to determine the relationships between various local datums and Conventional Terrestrial System (CTS) (Rens, J, Merry, C L 1990). The simplicity of the Bursa-Wolf Transformation Model is another reason for applying it for our transformation problem.

    Bursa–wolf transformation model
    The Bursa-Wolf method assumes a similarity three dimensioned relationship between two consistent sets of Cartesian coordinate through seven parameters :

    • three translations (DX, DY, DZ)
    • three rotations around X, Y, Z axis respectively (Î,y, w)
    • a scale change ( DL )

    Fig. 1 shows graphic representation of the transformation model.


    Fig.1 Graphic representation of a 7 Parameter similarity Transformation
    (a: Translations b: Rotations. c: Scale and d: Total)

    If U, V and W represent the Cartesian components of a station in reference frame 1 say EVEREST and X, Y, Z represent the Cartesian component of same stations in reference frame number 2 say WGS – 84, the transformation can be expressed as:

    where R represents a 3 x 3 rotation matrix and defined as

    R= R1 (e) R2 (y) R3 (w)

    If all three angles are small the above rotation matrix can be written in its simplified form by setting sine of an angle equal to the angle itself, cosine of the angle equal to 1 and the product of sines equal to zero.

    Thus after simplification the above matrix will appear as


    The transformation equation (4-1) can now be written as


    Method of estimation of transformation parameters
    A point physically identifiable on the surface of Earth which has been assigned coordinates in at least two separate systems of coordinates is termed as collocated station. The Cartesian coordinates of sufficient number of collocated stations (U, V, W, X, Y, Z) can be used as observations in a least square adjustment for the seven transformation parameters. The model in symbolical form can be written as :

    F ( L, X ) = 0 ……………………………………………….(4-4)
    Where
    L = observations (U, V, W, X, Y, Z)
    X = parameters ( DX, DY, DZ, DL, w, y, e)

    By arranging the equations (4-3) into this form will result in

    DX + U + w V – yW + DL U + w DL V – y DL W – X = 0 ……………………….. (4-5a)

    DY + V – w U + eW + DLV – wDL U + eDLW – Y = 0 …………………………………..(4-5b)

    DZ + W + yU – eV + DL W + y DL W – eDL V – Z = 0 ……………………………….(4-5c)

    These three equations represent the functional relationship between any two closely oriented, closely scaled, ortho normal cartesian coordinates systems. Since the observations (6 cartesian components per station) have systematic and other errors with them, the usual combined least squares procedure of minimizing the weighted sum of residuals squared (VTPV) is followed.

    Analysis of results
    From the very beginning it was suspected that a single set of transformation parameters for the entire country may not give the level of accuracy required for topographical applications due to inconsistency and irregularity in Indian triangulation network. However to form a definite opinion about it, initially the model was applied to all the 300 points but 20% of the points which were kept reserved for checking the quality of transformation process. The strategy did not work and as expected the transformation parameters derived from the model when applied on check points resulted in large positional errors.

    In order to minimize errors in transformed coordinates resulting due to non – homogeneity of Everest coordinates it was decided to produce separate 7-parameter sets for sub-regions. The entire region covered by GPS observations was divided into 10 Zones (Refer Fig. 2) to meet the requirement of consistency in the GT C oordinates. The transformation parameters, computed zone-wise, have brought out the significance of improvement in accuracy of transformed coordinates. For most of the stations the the transformation accuracy was improved to the level of +3m, which is well within permissible limit on 1:50,000 scale and would serve the purpose of converting existing topo maps to WGS-84 maps.

    The various levels of positional errors in transformed coordinates has been shown zone wise in Figures 3-12. The effect of consistency in Everest Coordinates on accuracy of transformation is clearly visible in Figure-9, which represents zone-5. This is the only zone in which most of the stations belong to primary series eminating from Calcutta longitudinal series. G.T. Stations falling in all other zones are either of secondary order accuracy or from a cluster of independently adjusted series. However, the transformation derived from this exercise are accurate enough to accomplish the goal of producing WGS-84 series maps on 1:50,000 or smaller scales.


    Fig.3 Positional Error of Transformed Coordinates in Zone – 1


    Fig.4 Positional Error of Transformed Coordinates in Zone – 2


    Fig.5 Positional Error of Transformed Coordinates in Zone – 3A


    Fig.6 Positional Error of Transformed Coordinates in Zone – 3B


    Fig.7 Positional Error of Transformed Coordinates in Zone – 4A


    Fig.8 Positional Error of Transformed Coordinates in Zone – 4B


    Fig.9 Positional Error of Transformed Coordinates in Zone – 5


    Fig.10 Positional Error of Transformed Coordinates in Zone – 6


    Fig.11 Positional Error of Transformed Coordinates in Zone – 7


    Fig.12 Positional Error of Transformed Coordinates in Zone – 8

    Conclusion
    The present study was focused on various aspects of Transformation Parameters between Everest and WGS-84 datums. It turns out that if the local terrestrial network is consistent or well-adjusted within a framework, few points would be sufficient to determine datums (local and global) relationship. However, this is a highly unlikely scenario and may be feasible in countries where extent of local network is confined to a smaller region. But in India, where the geodetic network is more irregular and coverage is large it would be difficult to select the few points to represent the characteristics of the entire network. Even after the division into ten parts only meter level transformation accuracy could be achieved. Therefore, it is highly desirable to readjust the existing network, so that every point would represent the actual quality of network. Also there is need to apply more sophisticated techniques of transformation such as surface fitting which caters for distortion in local survey network, providing improved transformation for derived data sets. Where sufficient quality data set has been used, transformation with an accuracy of better than 10 cm has been achieved (Collier, 2002).

    Acknowledgements
    This study was financially supported by the Department of Science and Technology (DST), Ministry of Human Resources and Development (HRD), New Delhi. We gratefully acknowledge the consistent support and encouragement provided by Dr Prithvish Nag, Sur–veyor General of India, in writing the report. The authors appreciate the indispensable assistance provided by Dr G D Gupta, Head, Earth Science Division, DST, Government of India. The authors are thankful to officers and staff of No. 82 Party (G&RB) Survey of India for conducting field activities, data processing and other miscellaneous work connected with the study.

    References :

    • Bursa, M. , 1962, The theory of the Determination of the Non-parallelism of the Minor axis of the Earth and Initial Astronomical and Geodetic meridians from observation of Artificial Satellites, Translation From Studia Geophysica et Geodetica, Vol. 6 , No. 2.
    • Collier P., 2002, Development of Australia’s National GDA Transformation Grids.
    • Cook, A., 1990, The Achievements of Sir George Everest in Geodesy, Survey Review, 30, 368-374.
    • DMA Technical Report, December 1983, Geodesy for the Layman, DMA-TR80-003.
    • Teunissen, P.J.G., Kleusberg, A., 1998, GPS for Geodesy, Springer-Verlag Berlin Heidelberg, New York.
    • Malys, Stephen, 1988, Dispersion and Correlation among Transformation Parameters relating two Satellite Reference Frames, Department of Geodetic Science, O.S.U. Report No. 392, The Ohio State University , Columbus, Ohio.
    • Molodensky, M. , Yeremeyev, V. and Yurkina, M. , 1962, Methods for Study of the External Gravitational field and Figure of the Earth, Israel Program for Scientific Translations, Jerusalem.
    • Rens, J. and Merry, C.L., Datum Transformation Parameters in South Africa, Survey Review, 30, 281-293.
    • Singh, S.K., 1994, Determination of Transformation Parameters between Everest Spheroid and WGS-84 datum using GPS, M.Tech. thesis, Centre for Remote Sensing, Institute of Post Graduate Studies & Research, Jawaharlal Nehru Technological University, Hyderabad, India.