Application of Geodetic Datums in Georeferencing EUMETNET/OPERA 1999-2006 WD 2005/18 Anton Zgonc Environmental Agency of the Republic of Slovenia E-mail: [email protected] Revision 1.3: December 2006 Abstract The aim of this work is to encourage OPERA forum to include at least a minimal set of new georeferencing parameters which describe geodetic datum, i.e. position and orientation of a chosen ellipsoid in space. The need for more accurate description of position on the Earth surface appears as soon as we deal with horizontal resolution below 1 km. Detailed aspects about georeferencing are not widely known within meteorological commu- nity. Geodetic datum is briefly explained, and difference between geodetic and geocentric latitude is given as well. Luckily, all the proposed inovations are already available in newer releases of the PROJ pack- age. Guidelines for usage of command-line utilities, as well as some remarks on the libproj library, are presented. Some attention to key parameters which influence the position of radar pixels is shown, too. It is evident that a reasonable compromise between desired accuracy and natural limitations has to be taken. Finally, some proposals on how to change the current practice in projectional parameters, are given. 1 EUMETNET/OPERA 1999-2006 WD 2005/18 Contents 1 Introduction 3 2 Geodetic Background 3 2.1 Ellipsoid and Geodetic Datum . 3 2.2 Geodetic Coordinate System . 5 2.3 Geodetic and Geocentric Latitude . 6 2.4 Geocentric anf Geodetic latitudes in Projectional Parameters . 8 3 Datum Transformations 9 3.1 Helmert Transformation . 9 3.2 Sources of datum parameters . 14 3.3 Usage withing the PROJ4 package . 16 3.4 Support in the libproj library . 17 4 Errors in positions of radar pixels 18 4.1 Effects due to the geoid shape . 18 4.2 Effects due to deviations of refractivity gradient . 23 5 Conclusions 25 References 28 List of Figures 1 Geodetic coordinates . 6 2 Geodetic vs. geocentric latitude . 8 3 Datum transformation . 10 4 Parameter errors . 11 5 Datum shifts in Westere Europe . 12 6 Position shifts . 13 7 Coastline shifts . 13 8 Curvature radii . 20 9 World geoid . 21 10 Influences of curvature radius and refractivity gradient . 24 List of Tables 1 Geocentric and geodetic latitude . 7 2 Intermixing of geodetic and geocentric latitudinal parameters . 9 3 Forward and backward parameters . 11 4 Builtin datums in PROJ . 14 5 Datums in Europe . 15 6 Builtin ellipsoids in PROJ . 22 2 EUMETNET/OPERA 1999-2006 WD 2005/18 1 Introduction Horizontal location of weather radar data, as well as other meteorological data on the Earth’s surface, is at present described by latitude and longitude (λ;ϕ) on a chosen ellipsoid with specified semi-major and semi-minor axis. In fact, in many meteorological applications the sphere approxima- tion is sufficient. As soon as we cross the border of 1 km horizontal accuracy, things become more complicated, but not just because of more decimals in location parameters. The first reason is that cartographical products apply the geodetic latitude ϕ, not the geocentric latitude φ as commonly believed. Geodetic latitude has an unpleasant property of being dependent on the ellipsoid axes ratio (eq. 5). As shown in table 1, difference between geodetic latitudes on different ellipsoids reaches up to 20-30m, but if one of them is a sphere (we have the geocentric latitude then), a difference of almost 20 km appears. The second reason is that ellipsoids, which are used to describe the local or global geoid, are slightly displaced and rotated between each other. Position and orientation of ellipsoid in space is referred as geodetic datum. As stated in [16], nearly 1000 datums are or have been used around the world. The absolute datum used as a general reference is the WGS84 datum. The subject begun to spread outside geodetic community in 1990’s and became popular by avail- ability of satellite measurements of Earth’s surface, especially by GPS. The GPS users were one of the first to realize that GPS λ;ϕ is not necessarily identical to the one on maps. To get acquainted with theoretical background, we warmly recommend to read the WGS84 Implementation Manual [5], especially chapters 1, 3, 4.1 and appendices B-E. Although radar data are burdened with horizontal location error of about 100m/1km, we cannot simply ignore the datum they depend on. As shown in equation 9 or in figure 5, longitude and latitude shifts on different datums may be approximated as average shifts of few 10" for an area covered by a single weather, that is several 100 m within an area, covered by a single weather radar. In more extreme cases shifts may be near 1 km (or even more), as shown in figure 6. We have to decide to which level we are to support the additional georeferencing parameters, to satisfy the accuracy requirements we need. There are several possible approaches, which are discussed in the Conclusions. There is a good news that geodetic datums are supported in the PROJ package since the release 4.4 and newer. 2 Geodetic Background 2.1 Ellipsoid and Geodetic Datum Earth’s surface is usually described by a chosen value of it’s geopotential, which includes the gravitational and centrifugal acceleration due to Earth’s rotation. Sea water tends to minimize its potential energy, therefore the mean ocean level represents one equipotential surface, which is then extrapolated towards continental regions. This surface is referred as geoid. The geoid is of irregular shape. For most applications, it is approximated by an ellipsoid of revo- lution, i.e an oblate ellipsoid or even a sphere An oblate ellipsoid has semi-major axis in the equatorial plane and semi-minor axis coincides with the rotational axis. It is described by two parameters, either the semi-major and semi-minor axis a;b or more often, the semi-major axis and one of the following 3 EUMETNET/OPERA 1999-2006 WD 2005/18 derived parameters: a b f = − 1=298 flattening a ≈ 2 2 b 2 st e = 1 8:18 10− 1 eccentricity − a2 ≈ · Location and orientation of an ellipsoid is important, too. Before the space age, ellipsoids have been obtained by longer-term astronomical observations and ground gravimetric measurements. They usually fit best to the local geoid in the area of interest. Ellipsoid position and orientation in space is referred as geodetic datum 1. It is defined by a set of 8 parameters which describe dimensions, position and orientation of a given ellipsoid in space. Position and orientation is given relatively to an agreed absolute geocentric datum (mostly WGS84), so we speak of an relative or astrogeodetic datum: a; f major semi-axis and flattening ∆x;∆y;∆z displacement of ellipsoid center from the Earth’s mass center (1) α;β;γ rotation of the ellipsoid frame around X,Y,Z axes ∆S scale correction The scale correction parameter ∆S is usually added to the list, although strictly speaking it is not the part of the datum parameters. Before the space age datums were initialized by their anchor or control point, whose coordinates were assigned from longer-term astronomical observations. For exact location of ellipsoid in space, the coordinates of another reference point are needed. Therefore datums which have different second reference points should have both of them in their identification. Swiss datum CH_CH1903 (see table 5), for example, is based on the Hermannskogel datum, but the second reference points differ from the one of the original datum. British datum GB_OSGB36, on the other hand, has a set of 12 anchor points. The absolute WGS84 datum has following special features: 1. its minor semi-axis coincides with the mean Earth’s rotation axis 2. it is geocentric – its center coincides with the Earth’s mass center 3. The prime meridian is Greenwich Year in the datum’s name signify the time, when Earth’s tectonical and mechanical properties were incorporated into datum’s definition. WGS84 thus describes the Earth’s state at 1984-01-01. If you read geodetic literature, you’ll find much more precise definitions, which are far beyond our scope. Briefly, when horizontal accuracy around 1 m or below is required, tectonics has to be taken into account. For these reasons, a high-accuracy version of WGS84, International Terrestrial Reference System (ITRS), has been created in a number of versions since 1989 for global purposes. For the European continent, the European Terrestrial Reference System 1989 (ETRS89) has been designed. It is based on ITRS, however anchor points are bound to European continent which is in motion with respect to the WGS84 coordinate system at a rate of about 2.5 centimeters per year. For our purposes, we will not distinguish among absolute datums WGS84, ITRS and ETRS89. As it can be imagined, one can easily find out that datum transformations (see sec. 3.1) between each 1According to [1], datum (pl. datums) is set of quantities which serve as a referent for calculation of other quantities. Geodetic datum is also named as terrestrial reference frame or system (acronyms TRS and TRF, respectively). 4 EUMETNET/OPERA 1999-2006 WD 2005/18 pair of these refined absolute datums are of 2-3 orders of magnitude less than those between relative and absolute datums (eq. 8). See e.g. [17] for details. Relative motion between ETRS89 ans WGS84 is of that magnitude that we might consider both of them static for decades, within our perhaps finest accuracy of 10 meters. We may therefore ignore any refinements of absolute datums which occur from time to time, too.