Coordinate Systems and Frames

Professor Terry MooreInstitute of Engineering Surveying and Space Geodesy

The University of NottinghamNottingham

United Kingdom

1 Introduction

Until the advent of satellite positioning and navigation systems, and in particular the Global Positioning System (GPS), geodetic coordinate systems were of little interest to many of the users of coordinate position information. Indeed, many of today's problems and mistakes stem from this historical misunderstanding of the true complexity of systems of coordinates. This paper will describe a number of common types of coordinate representation, their implementation in the definition of coordinate systems, and the realisation of these systems as coordinate frames and datums. Examples of typical local and geocentric datums are outlined as illustrations of the general principals. In order to combine coordinates based on differing systems it is also necessary to understand the methods of transforming coordinates from one system to another.

2 Astronomical Coordinates

In order to define astronomical latitude and longitude (φA and λA) it is necessary to specify an equator and a zero meridian. The equator is a plane perpendicular to the spin axis of the Earth and passing through the Earth’s centre of mass. The zero meridian is an arbitrary reference plane which contains the spin axis. In simple terms, the astronomical latitude of a point is given by the angle between the vertical (ie the direction of the gravity vector) at that point and the equatorial plane. The astronomical meridian is defined as the plane passing through the vertical at a point and (a line parallel to) the spin axis of the Earth (Fig 1). Conventionally, the zero meridian is accepted to be the astronomical meridian through the transit telescope at the old Greenwich Observatory. It follows that the astronomical longitude of a point is the angle between two planes, one of which is the local meridian (or meridional plane) and the other the zero meridian at Greenwich.

λ A

Vertical

Solid Earth

Equator

Spin Axis

zero longitude

φA

Figure 1Astronomical Latitude and Longitude

The above definitions, however complex in appearance, are gross over-simplifications on three grounds. Firstly, the instantaneous axis of rotation of the Earth is not fixed with respect to the solid mass of the Earth, but is in a state of continuous motion known as polar motion. Although this effect was predicted by Euler in 1765, it was not observed and determined until about a hundred

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years ago. As a result the position of the ‘north pole’, the intersection of the spin axis of the Earth with the surface of the Earth, can move by between 5 and 10 metres per year. It is therefore usual to define astronomical coordinates with respect to an internationally agreed mean axis and not the true, or instantaneous, spin axis. Secondly, the zero meridian does not pass through a particular point at Greenwich, but is defined as the mean value of the adopted longitudes of a number of observatories around the world. Since 1988, the International Earth Rotation Service (IERS), based in Paris, has defined the mean spin axis, the IERS Reference Pole (IRP), and the zero meridian, the IERS Reference Meridian (IRM).

However, the third and most significant drawback of geographical (or astronomical) latitude and longitude is that, unlike geodetic latitude and longitude, they are not proper measures of position on Earth, but indications of the inclination of the direction of the local vertical with respect to the instantaneous axis of rotation of the Earth and the IERS zero longitude (IRM) respectively. More explicitly, if the directions of the vertical at two or more points on Earth are (very conceivably) parallel, then these points will have identical astronomical coordinates. Therefore astronomical coordinates do not constitute a coordinate system in the geometrical sense.

3 Geodetic (Ellipsoidal) Coordinates

The geodetic position of a point on Earth is defined by the ellipsoidal coordinates of the projection of this point on to the surface of a reference ellipsoid, along the normal to that ellipsoid (Fig 2). Geodetic latitude (φG) is defined as the inclination of the normal to the ellipsoidal equatorial plane, and geodetic meridian as the plane through the normal and the minor axis of the reference ellipsoid. Lastly, the geodetic longitude (λG) of a point is the angle between its geodetic meridional plane and an arbitrary zero meridian (typically the IRM, IERS Reference Meridian).

Normal

Solid Earth

Zero Longitude

φGλG

Figure 2Geodetic Latitude and Longitude

Unlike astronomical coordinates of points on the surface of the Earth, no two projection points on the reference ellipsoid may have identical geodetic coordinates. By implication, the same applies to the geodetic coordinates of any two points on the Earth, unless they happen to lie on the same normal to the ellipsoid. Moreover, as with the plane and the surface of the sphere, one can develop formulae and compute the (ellipsoidal) distance and azimuth between two points whose geodetic (ellipsoidal) coordinates are known. The reason for using an ellipsoid of rotation rather than a plane, a sphere or a cube as the geodetic reference surface is one of convenience (and not theoretical rigour), because the ellipsoid of rotation is the closest regular geometrical shape to the figure of the Earth. The size (ie the semi-major axis, a, and the flattening, f) of the reference ellipsoid can be chosen arbitrarily, but the positioning of the ellipsoid with respect to the solid Earth is of crucial

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importance. Generally, the ellipsoid is positioned so that it is a close fit to the geoid (the equipotential surface closest to mean sea level) in the particular area of interest (Fig 3). In classical geodetic practice, this is done by arbitrarily adopting (convenient) values of geodetic latitude, longitude and height above the ellipsoid for a station of origin, and by using mathematical formulae which maintain parallelism between the minor axis of the ellipsoid and the mean axis of rotation of the Earth (Bomford, 1980).

G

LLocalEllipsoid

GeocentricEllipsoid

Normals

Solid Earth

φG

φ L

Figure 3Local and Geocentric Ellipsoids

4 Cartesian Coordinates

A more useful (and popular) alternative to using the angular measurements of latitude and longitude, is to describe the position of a point on, or indeed above or below, the Earth’s surface in terms of cartesian coordinates. Having specified the equator and zero meridian of either an astronomical or geodetic coordinate system, it is possible to define an associated cartesian coordinate system, X, Y and Z. Conventionally, the axes form a left handed triad with the Z axis in the direction perpendicular to the equator, the X axis in the direction of the zero meridian, and the Y axis perpendicular to the other two (see Fig 4). The origin of such a cartesian system may either be the centre of the associated reference ellipsoid or the implied mass centre of the Earth (geocentric cartesian coordinates).

O

Z

Y

XEquator

GreenwichMeridian

ReferenceEllipsoid

Figure 4Cartesian Coordinate Systems

Theoretically, the axes of all properly defined geodetic cartesian coordinate systems must be parallel to one another. It therefore follows that the only coordinate transformations which are necessary to convert from one geodetic cartesian coordinate system to another are three translations of the origin, ∆X, ∆Y and ∆Z. However, in practice, some of these transformations may also involve other parameters, such as rotations about the axes (see § 6). A set of 3-d cartesian coordinates, X, Y and Z, can also be converted into corresponding sets of latitude, longitude and

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height above an ellipsoid of given dimensions (a and e), whose origin and axes coincide with the cartesian X, Y, Z system, by using the cartesian to ellipsoidal conversion formulae.

tan λG =YX

tan φG = ( )

Z + e sin

X + Y G

2 2

2 ν φ

h =X

cos cos φ λν

−

where ν = a

(1 - e sin )2 2Gφ

a = semi major axis of ellipsoide = eccentricity of ellipsoidh = ellipsoidal height of point (see § 5)

Although direct formulae do exist the simple equations stated here involve an iteration for the latitude term. One can also use a corresponding set of reverse (ellipsoidal to cartesian) formulae to convert latitude, longitude and height above the ellipsoid back to X, Y, Z.

X = (ν + h) cos φG cos λGY = (ν + h) cos φG sin λGZ = (ν (1-e2) + h) sin φG

The greatest advantage of a cartesian coordinate system is that it is completely defined by the direction of the three axes and the position of the origin, with none of the complications of the reference ellipsoid, projection grids, etc. It is unfortunate that this simplicity of concept is not matched by user convenience. With the exception of aircraft, satellite and space navigation, geodetic cartesian coordinates would be very inconvenient for most users, notably cartographers, surveyors, engineers, sea and land navigators. With geodetic cartesian coordinates, ‘going-up’ by h meters would not involve an identical increase of the value of Z, except at the North Pole. On the equator, the value of Z would remain unchanged however large the value of h. Nevertheless, all satellite based navigation and positioning systems measure, compute and (in the first instance) output coordinates in 3-d X, Y, Z terms. It is therefore important to understand the concepts involved, and the relationship between 3-d systems and the more familiar 2-d geodetic and projection grid coordinates.

5 Height

In a geodetic coordinate system, as defined earlier, the third dimension is given by the ellipsoidal height, h, of the point, which is defined as the linear distance of the point above, (or below), the ellipsoid, measured along the normal to the ellipsoid. Ellipsoidal heights are generally unsuitable for practical use, as they are often difficult to determine and, more importantly, bear no relation to the Earth’s gravitational field which, for example, determines the direction of the flow of water. Orthometric heights or, as they are more generally called, heights above mean sea level, are much more useful. The orthometric height, H, of a point is defined as the linear distance from that point to a reference equipotential surface of the Earth’s gravity field, measured along the gravity vector. Typically, the geoid is specified as the reference surface, being the equipotential surface of the Earth that best fits mean sea level (Bomford, 1980). The difference between ellipsoidal (h) and orthometric (H) heights is denoted by N which is referred to as the geoid-ellipsoid separation or the

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geoid height, (see Fig 6 where, for clarity the differentiation between the measurement along the normal and the vertical has been omitted as it has negligible effect in height terms) and is given by

h = N + H

H

N

h

Earth's Surface

Geoid

Ellipsoid

Figure 6Ellipsoid, Geoid and Height

There are other definitions of height systems in addition to Orthometric and Ellipsoidal heights. In particular Dynamic heights are defined in terms of the potential (gravitational) difference between a point and the geoid. This value, scaled to units of length, is not the same as Orthometric height due to the lack of parallelism between equipotential surfaces. Dynamic heights can be useful when dealing with such problems as irrigation schemes where it is gravitational potential that is important, rather than linear separation, but they are otherwise seldom used.

Finally, more recently, Normal heights have been defined and are growing in use in gravimetric and geodetic surveying, most notably in the definition of national height networks, for their theoretical and computational advantages. They are defined as the linear separation between the ellipsoid and a surface called the Telluroid. This surface, which is not an equipotential, is the surface having the same Normal (ie reference ellipsoidal) potential as the actual gravitational potential at the corresponding point (along the plumb line) on the Earth’s surface. Extending Normal heights down from the Earth’s surface yields the so called Quasigeoid, which is not an equipotential, but deviates only by small amounts from the geoid (cm to dm) and is coincident with it over the oceans.

The use of satellite positioning systems, such as GPS, which produce the geometric ellipsoidal height, requires the corresponding geoid-ellipsoid separation to be known before the traditionally more commonly used orthometric height can be obtained. As a result the development of high precision geoidal models has been undertaken in many countries.

Global and Local Geoid Models

The computation of the complex geoid surface in order to provide geoid heights, thus enabling the direct conversion between ellipsoidal and orthometric heights, is a demanding mathematical problem. Several techniques for such computation have been developed based around the original formulation by Stokes in the 19th century. The methods include the use of Least Squares Collocation and Fast Fourier Transformations the details of which are outside the scope of this book. All such computations rely on the availability of gravity data (either directly measured or in combination with related quantities such as the deviation of the vertical, potential or geoid height), which theoretically needs to be global and continuous. These last conditions obviously restrict the ability to compute and exact geoid surface, and geoid models (both global and local) are therefore not perfect. However, recent satellite altimetry missions have considerably improved the coverage of data over the oceans, where previously data was sadly lacking, and this has significantly improved the accuracy of global geoid models.

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Global Geoid Models

Until the advent of satellite geodesy it was almost impossible to collect sufficient global gravity data to compute global geoid models. However in the last 25 years or so there has been a continued development and refinement of such models as more and better data has become available. Probably the three most important models produced have been the WGS 84 Earth Gravitational Model (not to be confused with the WGS 84 Ellipsoid), the Ohio State University’s OSU 91 model, and most recently the Earth Gravity Model EGM 96 (joint project of NASA/GSFC and NIMA). These models all express the complex geoid surface as a Spherical Harmonic expansion with coefficients complete to degree and order 360 (EGM 96 and OSU 91A) or 180 (WGS 84). Despite the very large number of coefficients this requires (105 for degree and order 360), these models only provide a smooth, long wavelength, view of the geoid with a spatial resolution of, at best, 1° in latitude and longitude. This limits their accuracy, as geoid variations within this resolution are smoothed. However EGM 96 is claimed to have a global accuracy of between ±0.5m and ±1.0m. The accuracy of global models relies on the availability of the input data, and EGM 96 has been computed with much better global data coverage than previous models. However there are still areas of sparse data coverage (eg central Africa) and the model’s accuracy will inevitably be lower in the vicinity of such regions. Figure 7 shows the EGM 96 global geoid model, with geoid heights plotted with respect to the EGM 96 defined ellipsoid (which differs slightly from WGS 84). It should be noted that at the scale of the figure only the major geoid features are discernible, and all global models would look similar.

Figure 7EGM 96 Global Geoid Model

Local Geoid Models

Many countries have recently realised the need for an accurate local geoid (or quasigeoid) model to enable the conversion of satellite derived ellipsoidal heights to their local Orthometric or Normal height system. There has therefore been a rapid increase in the production of local or regional models to satisfy this need. Such local computations are based on global models (such as OSU 91A or EGM 96), which provide the long wavelength information, and as such have their absolute

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accuracy limited by that of the global model. However, the local model is derived from local gravity data of a (usually much) better spatial resolution, eg 2km surface gravity observations, and often higher accuracy, and can thus produce relative geoid accuracy of the order of a few centimetres over distances of the order of 100km. Examples of such local/regional models are the Ordnance Survey (GB)’s OSGM91, the European (quasi)geoid EGG 97 (computed by the University of Hannover), and the conterminous US GEOID96/G96SSS models (NOAA-NGS).

6 Datum Transformations

6.1 Introduction

With so many geodetic datums in current use, it is becoming almost common practise to transform coordinates from one datum to another. Although this problem always existed, the advent of the TRANSIT and GPS satellite positioning systems have added increased interest. For example, a navigator on board a ship approaching a United Kingdom port, using GPS, will obtain the WGS84 (see § 7) geocentric, geodetic or cartesian, coordinates of the vessel. In order to plot the position on the corresponding local chart, these WGS84 coordinates must be first transformed into the local datum on which the chart is based, which in the UK would most probably be OSGB36. A number of different procedures are available for performing coordinate transformations. The merits and drawbacks of the available techniques should be considered with respect to accuracy requirements and computational simplicity. Of the many geodetic techniques only three, the Helmert, Molodensky and multiple regression transformation formulae, will be described here.

Differences of coordinates on two different datums may be expressed by a number of parameters. The semi major axis (a) and flattening (f), which define the shape of the ellipsoid associated with each datum, must be known and any differences accounted for in the transformation. The difference of origin (centre of the ellipsoid) between the two datums may be described by three translations (∆X, ∆Y, ∆Z) of the origin. Similarly, any small differences in the directions of the axes are allowed for by small rotations (θx, θy, θz) about the axes, to ensure the axes of both systems are parallel. A θz rotation expresses the difference in the definition of the zero meridian, and θx and θy the definition of the adopted equator (or pole). Finally, any differences in realisation of scale between the two coordinate systems may be described by a single scale factor (µ), usually expressed in parts-per-million.

In practice, very few transformations would include all these 9 parameters, and more generally only a subset are of any significance. It must also be recognised that, if there are local regional variations in the realisation of a geodetic datum, these are not accounted for by the adopted ‘average’ transformations. Specialised techniques have been evolved to deal with such situations, such as the ‘multiple regression formulae’ (see § 6.4).

6.2 Helmert Transformations

The most general method of transforming coordinates from one geodetic datum to another uses all 7 geometrical transformation parameters to convert cartesian coordinates (X, Y and Z). Therefore, any translations of the origin, rotations of the axes and a scale change may be accommodated. The general formulation of this transformation is usually attributed to the German geodesist ‘Helmert’ and expressed as,

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XYZ

2

2

2

=

XYZ

1

1

1

+

µ θ θθ µ θθ θ µ

--

-

z y

z x

y x

XYZ

1

1

1

+

∆∆∆

XYZ

where X1, Y1, Z1 cartesian coordinates in first datumX2, Y2, Z2 cartesian coordinates in second datum.

The rotation angles θx, θy and θz, expressed in radians in the matrix, are assumed to be ‘small’ angles, ie of the order of a few seconds-of-arc. The complete formulation would include a full rotation matrix including sine and cosine expressions of the angles. However, provided there are only small differences in the definition of the directions of the axes the above formula is adequate. Given the geodetic coordinates (φ, λ, h) of a point it is first necessary to convert these to cartesian coordinates, by using the expressions in §4, before the transformation can be applied. Similarly, the resulting coordinates in the new datum could also be converted to the corresponding geodetic latitude and longitude using the parameters of the ellipsoid associated with this new datum.

In order to determine the 7 parameters which describe the transformations between two geodetic datums, it is necessary to know the positions of a number of common points in both systems. Theoretically the 3-d coordinates of three points would allow the seven parameters to be determined. However, in practice it is usual to use as many common points as possible and obtain the best estimate of the parameters by using a least squares computation (Ashkenazi et al, 1984).

The necessity to use a three-stage computational procedure is a clearly a minor drawback. As a result, although these are the most generalised form of the simple coordinate transformations, there are shorter and more efficient procedures which may be used when only a small number of parameters are involved.

6.3 Molodensky Formulae

As described in § 4, the transformation between two different, but well defined, geodetic datums usually involves only the translation of the origin (the centre of the ellipsoid). In such instances, the Molodensky datum transformation formulae provide a simple one-stage procedure to convert the geodetic latitude and longitude (and ellipsoidal height) of a point in one datum directly to the second datum. Two versions of the formulae are commonly used, the so-called ‘standard’ and ‘abridged’ formulae. Traditionally the abridged versions have been used more extensively than their standard counterparts, however, due to their increased accuracy, the standard formulae are now being recommended for most applications. On the surface of the Earth, the differences between coordinates transformed by using the two versions are less than 0.6 m, and at an altitude of 12 km (40,000ft) the difference is of the order of 1 metre.

Standard Formulae

∆φ = ( ) ( ) ( )( )1

1 1 1 1ρ ν φ φ ρ ν φ φ+ + +

h a f a b b a

- X sin cos - Y sin sin + Z cos +

a e

1 1 1 1 12

∆ φ λ ∆ φ λ ∆ φ

∆ sin cos sin cos∆

∆λ = ( )- X sin + Y cos

h cos 1 1∆ λ ∆ λ

ν + φ

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∆h = ( ) ( )∆ ∆ ∆ ∆ ∆X Y Z a a f b acos cos cos sin sin sinφ λ φ λ φ ν ν φ1 1 1 1 12

1+ + − +

Abridged Formulae

∆φ = 1ρ

- X sin cos - Y sin sin + Z cos + (a f + f a)sin 21 1 1 1 1 1 1 1∆ φ λ ∆ φ λ ∆ φ ∆ ∆ φ

∆λ = - X sin + Y cos

cos 1 1∆ λ ∆ λ

φν

∆h = ∆X cos φ1 cos λ1 +∆Y cos φ1 sin λ1 + ∆Z sin φ1 + (a1 ∆f + f1∆a) sin2 φ1 -∆a

with φ2 = φ1 + ∆φλ2 = λ1 + ∆λh2 = h1 + ∆h

where φ1, λ1, h1 geodetic coordinates in first datumφ2, λ2, h2 corresponding coordinates in second datuma1, f1, ellipsoidal parameters (first datum)∆a, ∆f differences between the ellipsoidal parameters of the two datums∆X, ∆Y, ∆Z Translation of the origin

ν radius of curvature in the prime vertical, ν = a

(1 - e sin )2 2 1/2φ

ρ radius of curvature in the meridian, ρ = a (1 - e )

(1 - e sin )

2

2 2 3/2φ

e eccentricity of the ellipsoid, e2 = 2f - f2

It should be noted that, although the ellipsoidal heights may be transformed by using these equations, the latitude and longitude of a point may be transformed even if the corresponding height is not known. The 'abridged' expressions for latitude and longitude do not involve heights, however the 'standard' formulae do treat the height of the point more rigorously.

This method of transforming coordinates only allows the origin to be translated (ie ∆X, ∆Y, ∆Z) and the shape of the ellipsoid to be changed (∆a and ∆f). A number of modifications have been applied to these basic formulae, to allow for the effect of a scale change, µ, and a rotation about the z axis θz. Basically, a scale is represented by a uniform radial change, ∆r, which effects ellipsoidal height by the same amount, but does not affect latitude or longitude. Similarly, a rotation about the Z axis is equivalent to a change of the zero meridian and a consequent change of all the longitudes by the same amount ∆λ = -θz. Various specific transformations have adopted these ad-hoc modifications, but they should not be generalised. If more than the basic three translations are necessary, the more rigorous Helmert approach would generally be more appropriate.

6.4 Multiple Regression Formulae

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The above methods only accommodate coordinate transformations relating to two geodetic homogeneous datums. In many instances, particularly for the many classical local datums, there are known to be regional changes of local scale and orientation around the region over which coordinates are realised in the datum. Consequently the simple 'global' changes of scale and orientation of the axes cannot effectively allow for the variations. Various methods have been proposed to address this problem, and one of the most popular approaches are the multiple regression formulae (DMA 1987). In simple terms these are polynomial functions which represent the variations, as a function of position, of the differences of latitude, longitude and height (or X, Y and Z coordinates). They are initially computed by fitting a number of coefficients of the polynomial function to a large number of points, with a good geographical distribution, at which coordinates are known in both systems. There application simply involves the evaluation of the function at a particular location to give the transformation as changes in coordinates (ie ∆φ, ∆λ, ∆h or ∆X, ∆Y, ∆Z). By evaluating these function at a large number of points it is possible to produce a contour chart representing the variations in coordinate difference (DMA 1987). These provide a quick and easy method of transforming coordinates with no computation. Although the regional variation are being allowed for there is clearly only a limited accuracy achievable by this approach.

As an example the DMA local geodetic system to WGS84 datum transformation multiple regression equation is of the form

∆ φ = + + + + + + + + + +A A U A V A U A UV A V A U V A U V A U V0 1 2 22

4 52

559

568 2

999 9... ...

where A0, A1,...A99 100 possible coefficientsφ, λ latitude and longitude of pointk scale factor and degree to radian conversionφm, λm mid-latitude and mid-longitudeU = k(φ-φm) Normalised latitudeV = k(λ-λm) Normalised longitude

7 Major National and International Datums

7.1 Ordnance Survey of Great Britain

The first triangulation of Great Britain, known as the Principal Triangulation, was started in 1784, and was adjusted by least squares using logarithmic tables. As a result the British Isles were divided into 21 computing blocks, with each block adjusted by holding the boundary points of previously computed blocks as fixed. The astrogeodetic origin of the datum was defined at the old Greenwich Observatory in London. By the early years of this century, the lack of consistency of the mapping became apparent. Moreover, as most of the original triangulation points had been destroyed, it was decided that a new triangulation was required. The Re-triangulation of Great Britain was observed between 1935 and 1951, and became the Ordnance Survey of Great Britain 1936 (OSGB36) Datum. The triangulation was computed by using logarithmic tables and mechanical calculators. In an attempt to minimise the disturbance to the existing triangulation coordinates, a mean fit was used between the coordinates of 11 triangulation stations of their previous Principal Triangulation coordinates. This provided scale and orientation and an ‘implied’ mean origin at Greenwich.

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With advances in technology in the 1950’s and 1960’s, and principally the ease with which EDM distances could now be measured, it was apparent that there were large variations of scale throughout Great Britain. Consequently, the triangulation was readjusted by the Ordnance Survey and the University of Nottingham. The resulting geodetic datum is known as OSGB70(SN), the OSGB Scientific Network (Ashkenazi et al, 1972). The coordinates of the origin pillar at the Royal Greenwich Observatory at Herstmonceux were held fixed at the OSGB36 values, so providing a consistent transfer of origin. The advent of satellite positioning in the 1970’s highlighted systematic scale and orientation errors in the OSGB70(SN) coordinates. The triangulation was again re-computed, by the Ordnance Survey and the University of Nottingham, and the resulting datum which included Ireland, is known as OS(SN)80. This latter datum combined the traditional triangulation network with positions determined from the TRANSIT doppler system. It is interesting to note that the length of the British Isles (from the south coast to the Shetland Islands) became shorter by 25 m between OSGB36 and OSGB70(SN), but bounced back by 4.5 m between OSGB70(SN) and OS(SN)80. (Ashkenazi et al, 1984). Although the OS(SN)80 coordinates are used for scientific purposes and offshore, the national mapping is still based on the OSGB36 datum, with all its inherent inaccuracies.

With the advent of GPS, and the subsequent change in the practises of land and geodetic surveyors, it was decided that a new National GPS Network, to effectively replace the first and second order triangulation networks. The EUREF network, as described in the following section, provided six sites within Great Britain of the highest accuracy, with coordinates in ETRS89, at intervals of approximately 500km. The Scientific Network of Great Britain (SciNet92) is defined by 22 stations at 100-150km intervals, and the National GPS Network (OSGPS93) has a station distribution of the order of 25km in urban areas and 50km in rural areas. The observation of the National GPS Network was started in 1989, and the Scientific GPS Network was observed in 1992. In addition, transformation parameters and methods been developed to convert existing OSGB36 coordinates to either OS(SN)80 or WGS84 (Christie 1992). The Ordnance Survey Terrestrial Network (OS(TN)94) has recently been adjusted. This comprises 3500 stations, and all the existing terrestrial observations, of the first and second order triangulation networks and is constrained to the SciNet stations. It is considered (Calvert, 1995) that the terrestrial network OS(TN)94 is not as accurate as the GPS network OS(GPS)93, but it provides a consistent, and very dense, network of ETRS89 coordinates throughout Great Britain.

A new map projection OSGRS80 was adopted in 1995 (OS, 1995a). This is based on the geocentric GRS 80 ellipsoid, but with the same Transverse Mercator projection parameters as the existing National Projection. An alternative method of transforming coordinates between ETRS 89 (ie WGS 84) and OS GB 36 was also introduced, using a gridded two dimensional transformation. This is designed to accommodate the regional variations in scale and orientation known to exist in the OS GB 36 datum, and is an alternative approach to the multiple regression formulae. ETRS89 coordinates are first projected to give OSGRS80 coordinates (Ordnance Survey, 1995a). These Eastings and Northings are then transformed to OSGB36 National Grid coordinates by applying a simple shift in Easting and Northing. The shifts are derived by bi-linear interpolation between a grid of values of transformations. The density of this grid determines the accuracy of the transformation. The Ordnance Survey published a coarse grid, with a value of ∆E and ∆N for every 350 km of Easting and Northing (Ordnance Survey, 1995b). This transformation had a stated accuracy of better than 2m. A more accurate transformation (at around 0.2 m) was also available from the Ordnance Survey (OS, 1995b), but on a commercial basis.

A revised version of this new transformation was released for free use in 2000, and is referred to as OSTN97. The national geoid was also published, as OSGM91. The OSTN97 transformation is based on a Grid Look up Table, similar to the previous ‘2m’ transformation, but the grid density is

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much greater, and 1km grid squares are used in this case. The stated accuracy of the OSTN97 transformation is 0.2m rmse. The OSGM91 geoid model is distributed in a similar manner, but with a 2km grid spacing. These two models form the basis of the standard OS transformation between ETRS 89 and OSGB36. This transformation is now freely available on CD or through the Ordnance Survey's web site (www.gps.gov.uk).

Since October 1999 the Ordnance Survey have undertaken an extensive GPS observation campaign throughout Great Britain. Over 3000 existing triangulation points have been observed, which will have their existing OSGB36 coordinates and new coordinates in ETRS89. A total of over 6000 points will form the basis of a new, and final, transformation set known as OSTN02. The main aim of this transformation will be to improve on the existing OSTN97, by increasing the density of points and providing a transformation with a rms accuracy of 0.1m. Once completed this transformation will become the Definitive Transformation, and as such will define the National Grid in conjunction with the ETSR89 positions of the National GPS Network stations. The format of the OSTN02 transformation will be the same as OSTN97, and so software developed for OSTN97 will also work on the values of OSTN02.

7.2 European Datum

In the period following the Second World War survey data covering the central area of mainland Europe, acquired by the then US Army Map Service, was used to provide a common datum for the allied military mapping of Europe. The first adjustment of Central Europe was computed in 1947. There was considerable pressure to extend the network. The south-western block and the Scandinavian block were soon added, with the assistance of the International Association of Geodesy (IAG). The completed datum was known as European Datum 1950 (ED50). Great Britain was connected to the ED50 later by two links, one in the Dover-Calais area and the second in the area of the Isle of Wight and the Cotentin Peninsula. This allowed for ED50 coordinates to be obtained in Great Britain, through a transformation to the national datum OSGB36. There are no actual ED50 coordinates in the British Isles resulting from the adjustment of the triangulation network of the European Datum but only ED50 coordinates which can be obtained via OSGB36. When the exploitation of the North Sea area for oil and gas in the 60’s became imminent, it was necessary to define internationally agreed median lines between the nations bordering the North Sea. As the military ED50 datum was the only suitable unified datum at the time, it was soon de-classified and became freely available for commercial and public usage.

Not long after ED50 was first computed, it was realised that only some of the available data had been used. In 1954 the IAG established a Sub-Commission to investigate a new Triangulation of Europe, to be known as RETrig (Réseau Européen de Triangulation). After 25 years, a new European datum was provisionally defined to give the European Datum 1979, and finally the European Datum 1987 (ED87). At last, this was a rigorously adjusted datum for the whole of Europe, with scale and orientation control provided by space geodetic techniques, and an origin near Munich consistent with ED50. Subsequently, transformation parameters were published between ED87 and the WGS84 global datum. The latter is used by both TRANSIT and GPS.

In 1988, a new IAG Sub-Commission was established to investigate a new high precision European Three Dimensional Reference Frame, EUREF. This will be consistent with the latest space geodetic techniques of Satellite Laser Ranging, Very Long Baseline Interferometry and the Global Positioning system (GPS). A dense network of 93 fundamental points were observed by GPS, during May of 1989. The conclusion of EUREF, is a single unified datum throughout Europe which is consistent with the GPS datum, WGS84 and the International Terrestrial Reference Frame

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(ITRF) of the International Earth Rotation Service (IERS). The resulting coordinate system is known as ETRS89. This is a well defined and stable system in Europe, which is fully compatible with WGS84. Many countries have adopted the EUREF sites as a 'zero order' network from which they are increasing the densification of the realisation of the coordinates with national GPS networks.

7.3 Global Datums

Global Systems are used by all the major satellite positioning systems and a number of worldwide terrestrial navigation systems. In very general terms, their origins are as close as possible to the centre of mass of the Earth, and their axes oriented to the corresponding axes of the International Earth Rotation Service (IERS) system. Prior to 1988, the axes were oriented to the BIH Conventional Zero Meridian and Conventional International Origin (pole). An ellipsoid is also chosen to provide the best fit to the geoid throughout the world.

Although it is easy to define a global system in such terms, it is very difficult to actually realise such a system as coordinates on the Earth. Global systems are defined by assigning cartesian coordinates to a number of points around the world, and these implicitly define the origin and orientation of the axes. Clearly, unless the definition of these coordinates is based on very precise observations the resulting datum will be distorted due to measurement errors. Consequently, the datums used for satellite positioning have evolved from the US DoD World Geodetic System 1960, WGS60, through WGS66, WGS72 to the current definition of WGS84. This progression has resulted from a gradual increase in the accuracy of observations and led to more precise definitions of the datums.

The World Geodetic Datum 1984 (WGS84) of the United States Department of Defense is now becoming an internationally adopted standard datum for positioning and navigation. The datum is used for both the two principal satellite navigation systems, TRANSIT and GPS. Although it is often viewed as an evolution from WGS72, they are almost independent. Whereas WGS72 was defined by a number of different observation types, WGS84 is based almost exclusively on TRANSIT positions. A total of 1591 TRANSIT positions were determined by the US Defense Mapping Agency, and these coordinates define the WGS84 system. By co-locating TRANSIT receivers with Satellite Laser Ranging and Very Long Baseline Interferometry stations, systematic errors of the TRANSIT precise ephemeris coordinate system (NSWC-9Z2) were identified. As a result the positions were scaled, rotated (about the Z axis) and shifted (along the Z axis) to give WGS84. Because WGS84 is defined by TRANSIT positions, which are inherently accurate to 1-2 metres, the WGS84 datum is only realised to this ‘absolute’ accuracy. Nevertheless, WGS84 is the best global datum available for worldwide positioning and navigation. During 1994 the Defense Mapping Agency (DMA) redefined the basic coordinates and gravity field of WGS 84, using GPS. GPS pseudo range and carrier phase observations, taken during 1992, were used to determine the coordinates of the five stations of the GPS control segment and the additional five DMA sites. This revision of WGS 84 is known as G730, which refers of the GPS week number of its introduction. A second revision took place in 1996, and was introduced to the GPS in January 1997. It was known as WGS 84 (G873). The five GPS control segment sites, and an additional seven NIMA sites were re-coordinated using carrier phase GPS to the network of IGS stations (see below). The IGS coordinates were referenced to ITRF 94 (see below), and the resulting WGS 84 coordinates have been attributed an accuracy of about 5cm. Therefore, for all practical purposes WGS84 and ITRF 94 could be considered to be identical (at the centimetre level). This revision of WGS 84 also introduced the new gravity field of EGM96 and its associated geoid model (NIMA, 1997). In January 2002 a third revision of WGS 84 took place, and is currently implemented as WGS 84 (G1150). The 17 OCS and NIMA stations were coordinated, along with 10 other sites in a 15 day

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GPS campaign held in February 2001. The coordinates of 49 IGS stations were constrained to tie the coordinates to the International Terrestrial Reference Frame 2000 (ITRF 2000). The overall stated accuracy is better than one centimetre in each component (Merrigan et al, 2002).

The International Earth Rotation Service maintain a precise global frame, known originally as the IERS Terrestrial Reference Frame (ITRF), but now more commonly known as the International Terrestrial Reference Frame. This is realisation of the International Terrestrial Reference System (ITRS) and is updated periodically. It is defined by the coordinates of Satellite Laser Ranging, Lunar Laser Ranging, Very Long Baseline Interferometry facilities, and more recently DORIS and GPS sites. This reference frame provides the fundamental definition of the position of the centre of mass of the Earth to within a few centimetres, and the orientation of the axes to correspondingly high accuracies. The maintenance of a frame at this level of accuracy requires constant monitoring of the rotation of the Earth, the motion of the pole and the movements of the plates of the crust of the Earth. The current definition of the ITRF system is known as ITRF 2000 (lareg.ensg.ign.fr/ITRF/ITRF2000). The scale, orientation and origin of the frame are defined by a core of 95 SLR and VLBI sites, with a reference epoch of 1997.0. In total around 385 geodetic sites are included in ITRF2000.

On the 1st of January 1994, the International GPS Geodynamics Service (IGS) was formally introduced, although the service has now been in operation for some time. A permanent 'global' network of around 250 GPS receivers, provide precise GPS measurements on a daily basis to three data collection and archiving centres. The aims of this service include the maintenance of a precise global datum based on GPS, and the dissemination of the tracking data, orbital information and other products to the scientific community (igscb.jpl.nasa.gov).

8 References

V Ashkenazi, 1986. 'Coordinate Systems: How to get your Position Very Precise and Completely Wrong'. The Journal of Navigation, Volume 39, No 2.

V Ashkenazi, S A Crane, W J Preiss and J Williams, 1985, 'The 1980 Readjustment of the Triangulation of the United Kingdom and the Republic of Ireland - OS(SN)80'. Ordnance Survey Professional Papers, New Series, No 31.

V Ashkenazi, P A Cross, M J K Davies and D W Procter, 1972. 'The Adjustment of the Retriangulation of Great Britain, and its Relationship to the European Terrestrial and Satellite Networks'. Ordnance Survey Professional Papers, New Series No 24.

G Bomford, 1980. 'Geodesy (4th Edition)', Clarendon Press, London.

C Calvert, 1995. 'Ordnance Survey Policy and Procedures for WGS84 to National Grid (OSGB36 Datum) Transformations'. Proc RIN 95, Personal Navigation, Royal Institute of Navigation, London

P A Cross, 1989. 'Position: Just What Does it Mean'. Proc NAV 89, Satellite Navigation, Royal Institute of Navigation, London

R C Christie , 1992. 'The Establishment of a New 3-D Control Framework for GB Based on Modern Space Techniques. Surveying World, November 1992.

Defense Mapping Agency, 1987. Department of Defense World Geodetic System 1984. Technical Report (and supplements), DMA TR-8350.2.

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M Greaves and P Cruddace, 2001. ‘A New Coordinate Transformation for Great Britain, The Ordnance Survey’s New GPS to OSGB36 National Grid Transformation’, Geomatics World, Volume 10, Issue 1, November / December 2001

International Earth Rotation Service, 1990. Annual Report for 1989, IERS, Paris.

International Earth Rotation Service, 1997. Annual Report for 1996, IERS, Paris.

S Malys and J A Slater, 1994. 'Maintenance and Enhancement of the World Geodetic System 1984'. ION GPS 94, Salt Lake City, September 1994.

Ordnance Survey, 1995a. 'The Ellipsoid and the Transverse Mercator Projection'. Version 1.1, January 1995.

Ordnance Survey, 1995b. 'National Grid / ETRF89 Transformation Parameters'. Version 1.1, January 1995.

Ordnance Survey, 2000. ‘Coordinate Positioning, Ordnance Survey Policy and Strategy’, Information Paper 1/2000.

E R Swift, 1994. 'Improved WGS 84 Coordinates for the DMA and Air Force GPS Tracking Sites'. ION GPS 94, Salt Lake City, September 1994.

National Imagery and Mapping Agency, 1997. Department of Defense World Geodetic System 1984. Technical Report, NIMA TR-8350.2, Third Edition.

M J Merrigan, E R Swift, R F Wong, J F Saffel, 2002. A Refinement to the World Geodetic System 1984 Reference Frame. ION GPS 2002, Portland, Oregon, USA. September 2002.

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APPENDIX A

Example of a single point transformed from WGS 84 to OS GB 36 by various different transformations. All cartesian coordinates are in metres. All latitude and longitude coordinates are in degrees, minutes and seconds. All heights are ellipsoidal heights and are in meters. All projection coordinates (E, N) are OS National Grid, except for the OS GRS 80 projection coordinates.

WGS 84 / ETRS89

X, Y, Z 3850817.000 -79943.000 5066910.000φ, λ, h 52 56 40.94188 -1 11 21.44447 90.334

OS GRS 80

E, N 454476.097 338971.924

OS GB 36 (via OSTN97 grid transformation from ETRS89 and OSGM91 geoid model)

E, N 454574.554 338896.191φ, λ, h 52 56 39.69756 -1 11 15.90755 41.770 OS GB 36 (via OS 2m grid transformation from OS GRS 80)

E, N 454574.755 338895.880φ, λ, h 52 56 39.68742 -1 11 15.89697

OS GB 36 (Helmert, ∆X = -315 m, ∆Y = 111 m, ∆Z = -431 m)

X, Y, Z 3850442.000 -79832.000 5066479.000φ, λ, h 52 56 39.84353 -1 11 15.91771 42.924E, N 454574.313 338900.699

OS GB 36 (Abridged Molodensky, ∆X = -315 m, ∆Y = 111 m, ∆Z = -431 m)

X, Y, Z 3850441.845 -79832.005 5066478.907φ, λ, h 52 56 39.84570 -1 11 15.91817 42.756E, N 454574.304 338900.766

OS GB 36 (Standard Molodensky, ∆X = -315 m, ∆Y = 111 m, ∆Z = -431 m)

X, Y, Z 3850441.935 -79832.009 5066478.914φ, λ, h 52 56 39.84351 -1 11 15.91825 42.816E, N 454574.303 338900.698

OS GB 36 (NIMA Multiple Regression)

X, Y, Z 3850445.502 -79830.729 5066476.012φ, λ, h 52 56 39.69558 -1 11 15.84576 42.633E, N 454575.708 338896.143

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