Application of Geodetic Datums in...

Application of Geodetic Datums in GeoreferencingEUMETNET/OPERA 1999-2006

WD 2005/18

Anton ZgoncEnvironmental 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 ofnew georeferencing parameters which describe geodetic datum, i.e. position and orientation of achosen ellipsoid in space. The need for more accurate description of position on the Earth surfaceappears 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 latitudeis 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 libprojlibrary, 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 hasto be taken.

Finally, some proposals on how to change the current practice in projectional parameters, aregiven.

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EUMETNET/OPERA 1999-2006 WD 2005/18

Contents

1 Introduction 3

2 Geodetic Background 32.1 Ellipsoid and Geodetic Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Geodetic Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Geodetic and Geocentric Latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Geocentric anf Geodetic latitudes in Projectional Parameters . . . . . . . . . . . . . 8

3 Datum Transformations 93.1 Helmert Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Sources of datum parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Usage withing the PROJ4 package . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Support in the libproj library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Errors in positions of radar pixels 184.1 Effects due to the geoid shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Effects due to deviations of refractivity gradient . . . . . . . . . . . . . . . . . . . . 23

5 Conclusions 25

References 28

List of Figures

1 Geodetic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Geodetic vs. geocentric latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Datum transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Parameter errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Datum shifts in Westere Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Position shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Coastline shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Curvature radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 World geoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110 Influences of curvature radius and refractivity gradient . . . . . . . . . . . . . . . . 24

List of Tables

1 Geocentric and geodetic latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Intermixing of geodetic and geocentric latitudinal parameters . . . . . . . . . . . . . 93 Forward and backward parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Builtin datums in PROJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Datums in Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Builtin ellipsoids in PROJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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1 Introduction

Horizontal location of weather radar data, as well as other meteorological data on the Earth’ssurface, is at present described by latitude and longitude (λ,ϕ) on a chosen ellipsoid with specifiedsemi-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 morecomplicated, but not just because of more decimals in location parameters.

The first reason is that cartographical products apply the geodetic latitude ϕ, not the geocentriclatitude φ as commonly believed. Geodetic latitude has an unpleasant property of being dependent onthe ellipsoid axes ratio (eq. 5). As shown in table 1, difference between geodetic latitudes on differentellipsoids 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, areslightly displaced and rotated between each other. Position and orientation of ellipsoid in space isreferred as geodetic datum. As stated in [16], nearly 1000 datums are or have been used around theworld. 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 ofthe first to realize that GPS λ,ϕ is not necessarily identical to the one on maps. To get acquaintedwith 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 cannotsimply ignore the datum they depend on. As shown in equation 9 or in figure 5, longitude and latitudeshifts on different datums may be approximated as average shifts of few 10" for an area covered bya single weather, that is several 100 m within an area, covered by a single weather radar. In moreextreme 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, tosatisfy the accuracy requirements we need. There are several possible approaches, which are discussedin the Conclusions. There is a good news that geodetic datums are supported in the PROJ packagesince 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 thegravitational and centrifugal acceleration due to Earth’s rotation. Sea water tends to minimize itspotential energy, therefore the mean ocean level represents one equipotential surface, which is thenextrapolated 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 equatorialplane and semi-minor axis coincides with the rotational axis. It is described by two parameters, eitherthe semi-major and semi-minor axis a,b or more often, the semi-major axis and one of the following

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derived parameters:

f =a−b

a≈ 1/298 flattening

e2 = 1−b2

a2 ≈ 8.18·10−2 1st eccentricity

Location and orientation of an ellipsoid is important, too. Before the space age, ellipsoids havebeen obtained by longer-term astronomical observations and ground gravimetric measurements. Theyusually fit best to the local geoid in the area of interest. Ellipsoid position and orientation in spaceis 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 toan agreed absolute geocentric datum (mostly WGS84), so we speak of an relative or astrogeodeticdatum:

a, f major semi-axis and flattening

∆x,∆y,∆z displacement of ellipsoid center from the Earth’s mass center

α,β,γ rotation of the ellipsoid frame around X,Y,Z axes

∆S scale correction

(1)

The scale correction parameter ∆S is usually added to the list, although strictly speaking it is not thepart of the datum parameters.

Before the space age datums were initialized by their anchor or control point, whose coordinateswere 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 secondreference points should have both of them in their identification. Swiss datum CH_CH1903 (see table5), for example, is based on the Hermannskogel datum, but the second reference points differ fromthe one of the original datum. British datum GB_OSGB36, on the other hand, has a set of 12 anchorpoints.

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 propertieswere 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 beyondour scope. Briefly, when horizontal accuracy around 1 m or below is required, tectonics has to betaken into account. For these reasons, a high-accuracy version of WGS84, International TerrestrialReference 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 beendesigned. It is based on ITRS, however anchor points are bound to European continent which is inmotion 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. Asit 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).

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pair of these refined absolute datums are of 2-3 orders of magnitude less than those between relativeand absolute datums (eq. 8). See e.g. [17] for details. Relative motion between ETRS89 ans WGS84is of that magnitude that we might consider both of them static for decades, within our perhaps finestaccuracy of 10 meters. We may therefore ignore any refinements of absolute datums which occurfrom time to time, too.

2.2 Geodetic Coordinate System

A location on an ellipsoid surface can be described by 3D cartesian coordinates relative to theellipsoid center (X,Y,Z), referred as absolute cartesian coordinates throughout this document, or byone of at least 4 sets of spherical coordinates. An excellent overview about the subject can be foundin [3].

The classical spherical coordinates, called geocentric coordinates, consist of a longitude λ , (geo-centric) latitude φ and distance r from the center of an ellipsoid:

X = r cosφcos λ φ = arctanZ√

X2 +Y 2

Y = r cosφsin λ λ = arctanYX

Z = r sinφ r =√

X2 +Y 2 +Z2

Geocentric latitude φ may be considered as an elevation above the equatorial plane. These coordinatesare commonly known and mistakenly thought to be used worldwide in cartographic products.

In geodesy, geodetic coordinates are used instead. If longitude is represented on an interval of[−180oW,+180oE], we talk about geographic coordinates.

Geodetic longitude λ is the same as the geocentric one. The striking difference is in the definitionof geodetic latitude ϕ, which is the elevation of the ellipsoid normal at the current point above theequatorial plane:

X = N(ϕ) cos ϕ cosλ ϕ = arctanZ/(1− e2)√

X2 +Y 2

Y = N(ϕ) cos ϕ sinλ λ = arctanYX

(2)

Z = (1− e2)N(ϕ) sin ϕ N(ϕ) =a

√

1− e2 sin2 ϕ

where N(ϕ) is curvature radius of the prime vertical. As not too obvious on the figure 1, the planewhere the circle of radius N lies is perpendicular to the meridional plane and the tangent plane. Thechoice is obvious from the geodetic point of view: normal to the ellipsoid surface is parallel to thelocal plumb line, if we neglect deviations from the local geoid.

The slantwise quasi-cartesian coordinates, often used in meteorology, thus differ on the sphericaland ellipsoidal surface. We well need only the differential form:

dx = R cosφ dλ dy = R dφ sphere (3)

dx = N(ϕ) cosϕ dλ dy = N(ϕ)dϕ ellipsoid (4)

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2.3 Geodetic and Geocentric Latitude

Comparing left and right set of equations in (2), we obtain the relationship between the geocentricand geodetic latitude:

tanφ = (1− e2) tanϕ =b2

a2 tan ϕ (5)

As can be seen on the figure 1, if normal to the ellipsoid is prolongated towards the rotation axis, itintersects it below the equatorial plane. Therefore, geodetic latitude is not an universal one like thegeocentric one. It depends on the ellipsoid axes ratio. In a case of sphere R = a = b, φ = ϕ and allcurvature radii are equal to R.

Figure 1: Geodetic coordinates on an oblate ellipsoid. Note that geodetic latitude ϕ is an elevation ofa normale to the ellipsoid and intersects the Z-axis below the center of ellipsoid.

An easy way to demonstrate the difference is to convert chosen latitudes from a sphere to a givenellipsoid. We used the WGS84 ellipsoid and its normal sphere 14. The same experiment was per-formed on two ellipsoids: Bessel 1841 and WGS84. The following sh script does just that:

#!/bin/bash2## A demo script for geocentric/geodetic latitude

echo "# sphere vs. WGS84"for lat in $(seq -f "%+7.4f" 0 10 90); do

echo " 0.0 $lat"done | \

cs2cs -E -f ’%7.4f’ +ellps=WGS84 +R_V +proj=longlat \+to +ellps=WGS84 +proj=longlat

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echoecho "# bessel vs. WGS84"for lat in $(seq -f "%+7.4f" 0 10 90); do

echo " 0.0 $lat"done | \

cs2cs -E -f ’%7.4f’ +ellps=bessel +proj=longlat \+to +ellps=WGS84 +proj=longlat

Note that we had to use cs2cs because proj does not support +proj=longlat. We couldn’tapply the option for geocentric latitude +geoc, because it affects both input and output coordinates.

The results are very instructive in the table 1 and in the figure 2. As we can see,

ϕ ≥ φ (6)

as expected. In the midlatitudes, difference goes up to 0.2o (around 20 km!!). Difference betweengeodetic latitudes on two ellipsoids is around 3 orders of magnitude smaller, but still around 20 m.

The height difference clearly shows that if Bessel ellipsoid were positioned on the same coordinateframe as WGS84, it is smaller for around 700m, so it is nonsense to use it for global georeferencingpurposes. However, in it’s correct Hermannskogel datum position, difference is around +45 m overthe area of Slovenia.

Table 1: Demonstration of differences between geocentric vs. geodetic latitude and two geodeticlatitudes. The last column is height of normal sphere above WGS84 ellipsoid.

# lon1 lat1 lon2 lat2 dh[m]

# sphere vs. WGS840.0 +0.0000 0.0000 0.0000 -7136.20960.0 +10.0000 0.0000 10.0661 -6488.21930.0 +20.0000 0.0000 20.1241 -4623.51410.0 +30.0000 0.0000 30.1670 -1769.80320.0 +40.0000 0.0000 40.1896 1725.55070.0 +50.0000 0.0000 50.1892 5438.91680.0 +60.0000 0.0000 60.1661 8922.43800.0 +70.0000 0.0000 70.1231 11758.01980.0 +80.0000 0.0000 80.0655 13606.78230.0 +90.0000 0.0000 90.0000 14248.4762

# bessel vs. WGS840.0 +0.0000 0.0000 0.0000 -739.84500.0 +10.0000 0.0000 10.0002 -737.84630.0 +20.0000 0.0000 20.0004 -732.08920.0 +30.0000 0.0000 30.0005 -723.26240.0 +40.0000 0.0000 40.0006 -712.42420.0 +50.0000 0.0000 50.0006 -700.87780.0 +60.0000 0.0000 60.0005 -690.01570.0 +70.0000 0.0000 70.0004 -681.15220.0 +80.0000 0.0000 80.0002 -675.36280.0 +90.0000 0.0000 90.0000 -673.3514

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15 ° 30 ° 45 ° 60 ° 75 ° 90 °Φ

0.05 °

0.10 °

0.15 °

0.20 °j-Φ

Figure 2: Deviation of geodetic latitude ϕ from the geocentric latitude φ on the WGS84 ellipsoid.

2.4 Geocentric anf Geodetic latitudes in Projectional Parameters

A word should be said about intermixing of geocentric and geodetic latitudes in projectional pa-rameters. Strictly by mathematical rules, we should apply geocentric latitudes as soon as we applythe spherical Earth model. The common practice is that only the geodetic latitudes are used, by whichwe introduce a sort of nearly linear shift of geocentric latitudes towards north because of ineq. 6.This shift retains the nearly linear form also when we transform to cartesian coordinates x,y. As itwill be shown later, nonlinear contribution is really small and explains us the reason why this type ofmisusage of projectional parameters is so frequent. Usually we don’t run into troubles because wecheat the other georeferencing data, such as rain gages, river basins etc., in the same way.

We made numerical experiments with the native single-weather-radar projection, namely az-imuthal equidistant projection, and a typical composite projection, namely Lambert conical com-formant projection. We took the Lisca radar domain of (400× 400km2) and the CERAD compositedomain of (1560×1280km2). The Lisca projectional parameters are

# geodetic latitudes+proj=aeqd +lon_0=15.28972 +lat_0=46.06806 +R=6371000 +units=km# geocentric latitudes+proj=aeqd +lon_0=15.28972 +lat_0=45.87632 +R=6371000 +units=km

and the CERAD projectional parameters are

# geodetic latitudes+proj=lcc +lon_0=13.33333 +lat_0=47.00000 +R=6379000 +units=km \

+lat_1=46.00000 +lat_2=47.00000 +x_0=-20000 +y_0=-360000# geocentric latitudes+proj=lcc +lon_0=13.33333 +lat_0=46.80800 +R=6379000 +units=km \

+lat_1=45.80767 +lat_2=46.808001 +x_0=-20000 +y_0=-360000

The Cartesian origin (x∗,y∗) of a domain is bound to the middle pixel of the domain (i∗, j∗) =(nx div2+1,ny div2+1). Pixel coordinate is represented by the pixel’s center coordinate.

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We calculated shifts between Cartesian coordinates of all 4 domain corners, obtained with geocen-tric and geodetic latitudinal parameters. For demonstration purposes, we performed calculations fora double-sized CERAD domain, too. We subtracted the main contribution, i.e. the difference of bothlatitudes at the central pixel (i∗, j∗) of the domain ϕ∗−φ∗, which is around +0.2o in midlatitudes, asalready mentioned before.

Table 2: Nonlinear parts of positional shifts (in km’s) in the domain corners when we intermix geo-centric vs. geodetic latitudal projection parameters

# Lisca # CERAD # Doubled CERAD-0.0074 +0.0070 -0.2790 +0.0169 -1.0295 +0.0728+0.0074 +0.0070 -0.2376 +0.0885 +1.0294 +0.0728+0.0066 +0.0069 +0.2375 +0.0885 +1.0234 +0.3755-0.0066 +0.0069 +0.2790 +0.0169 -1.0234 +0.3755

The results in the table 2 show that what remains is a nonlinear shift, which can be neglected for asingle radar domain (< 75 m) and the CERAD domain (< 300 m), but becomes evident on a domainover whole Europe (≈ 1 km).

3 Datum Transformations

There are several approaches how to transform between two datums. In a linear approximationyou have to describe the translation and rotation. This is done by the 7-parameters transformation,named Helmert or Bursa-Wolff transformation. If only translations are applied, we talk about the3-parameter transformation then.

There are other approaches, namely Molodensky (7-parameter) transformation and 5-parametertransformation (3 translations and 2 differences between ellipsoid dimensions). Another approach isto use numerical shift grids. The latter is also supported within the PROJ package. When dealing withnonlinear distortions, one may apply multiple regression equations. We will not go into further detailshere.

3.1 Helmert Transformation

Helmert transformation between source (S) datum to target datum (T) consists of three steps:

1. transform (λ,ϕ)S to absolute cartesian coordinates (X ,Y,Z)S

2. transform (X ,Y,Z)S to the cartesian coordinated relative to te target datum (X ,Y,Z)T

3. transform (X ,Y,Z)T to the target geodetic coordinates (λ,ϕ)T

The key part is the 2nd step. We have to convert 8 source datum parameters to the target 8 ones.The core part of Helmert transformation consists of translation from one ellipsoid center towardsanother (∆X ,∆Y,∆Z), rotation around all three axes X,Y,Z (α,β,γ) and scale correction 1+∆S (figure3):

XYZ

T

=

∆X∆Y∆Z

+(1+∆S)R(α,β,γ)

XYZ

S

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Figure 3: Datum transformation (eq. 7) scheme between the source (S) and target (T) coordinateframe. Translation terms are represented by ~T .

The rotation matrix is in a form of a Cardan matrix, i.e. a product of rotation matrices around allthree axes of the source coordinate frame. The order is important, but not any more if the matrix canbe linearized:

R(α,β,γ) = Rz(γ)Ry(β)Rx(α) ≈

1 α −β−α 1 γβ −γ 1

Linearization is acceptable when angles do not exceed 30" . In this case the horizontal accuracyreaches about 1 dm, thus far beyond of our desired limits. The PROJ 4.4 package supports onlythe linearized Cardan matrix, as probably the most of georeferencing software does. The completelinearized transformation

XYZ

T

=

∆X∆Y∆Z

+(1+∆S)

1 α −β−α 1 γβ −γ 1

XYZ

S

Helmert transformation (7)

is directly reversible. The backward transformation is obtained by changing signs of the all 7 param-eters. This is exceptable because contributions to (X ,Y,Z) in eq. 7 are small enough. A practicalproof for that can be seen in table 3, where parameters for both transformations are compared. As fol-lows, parameters of both transformations have opposite signs, and their absolute values are equal upto relative accuracy of 10−3 -10−4 . Following the scale analysis, shown in the figure 4, the minimalaccuracy of translation terms should be 10m and 0.1" for rotations. We however suggest to stick tothe practice in the CRS table (table 5), where translations are given per 0.1 m and rotations per 0.01" .

The rotational parameters for a chosen datum should be verified for the sign and notation, becausethree interpretation conventions are in use. If counterclockwise angles are considered positive, likein our case, the transformation (7 ) is referred as coordinate frame transformation. If clockwise

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Figure 4: Position errors caused by errors in 7-parameter transformation (from [5]).

Table 3: Comparison of datum parameters for forward and backward transformation between Slove-nian datum SI_D48 and WGS84. Obtained from GPS measurement campaigns EUREF 1994 andSLOVENIJA 1995.

SI_D48 to WGS84 SI_D48 from WGS84

∆X [m] +426.9466 -426.9206∆Y [m] +142.5844 -142.6186∆Z [m] +460.0891 -460.0871α [" ] -4.907900 +4.908060β [" ] -4.488389 +4.488093γ [" ] -12.423059 +12.423166

∆S [ppm] +17.1131 17.1128

angles are considered positive, it is then referred as position vector transformation2. In some casesthe angle-axis notation is reversed, which leads to interchange between α vs. γ . According to severalsources, the forward 7-parameter coordinate frame transformation, as used throughout this document,is standardized in ISO/CD 19111, Annex D, D3 Datum transformations.

It is worthwhile to mention that all numerical computations with datum transformations must beimplemented in at least double precision arithmetics (64-bit double type in C language). A refinednumerical approach might reduce the required precision arithmetics down to 32-bit float type in Clanguage.

Orders of magnitude of single terms in eq. 7 can be estimated from the table 5:

∆(X ,Y,Z) ∼ 500m

(α,β,γ) ∼ 10” ≈ 10−5rad (8)

∆S ∼ 10ppm = 10−5

2Terminology comes from treatment of inertial systems in mechanics, see [11] for details.

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Typical magnitude of cartesian coordinates is 107m. Translation and rotation terms therefore con-tribute several 100m and the scale correction usually at least an order of magnitude less.

For most applications the 3 translation parameters are sufficient. In this case we talk about the3-parameter transformation. But, following the scale analysis, one shouldn’t simply cut off the rest5 parameters, but recalculate them via statistical or empirical means from the 7-parameter set. Youare inivited to compare the builtin PROJ datums (mostly in 3-parameter version) and those from theCRS (table 5) and WGS84 Impl. Manual [5, p. 85].

Figure 5: Position shifts between absolute WGS72 and relative (i.e. local) datums in Western Europe(from [5]).

Positional shifts due to different datums can be approximately expressed as linear translations inan area of coverage of a typical weather radar (see figure 5). As an illustrative example, Slovenia isa small country with a shape of a feetless hen, nearly covered by a circle of 100 km radius and thuscomparable to a coverage of a single 100 km weather radar. The average shift between λ,ϕ in localSI_D48 and WGS84 datum is

λloc −λWGS84 ≈ 17” ±1” (≈ 520 m)

ϕloc −ϕWGS84 ≈ 1” ±1” (≈ 20 m)(9)

where we applied the geodetic slant coordinates (4) for distance estimations.In the most approximative approach, a simple linear shift of λ,ϕ still satisfies an accuracy of

1” ≈ 25m. However, usage of all 7 parameters from table 5 should results in horizontal accuracy of1m over Slovenia.

A misuse or neglectance of geodetic data my result in severe position errors. A good illustrativeexample is on the figure 6. Sometimes the correct datum parameters are not known. An example ison the figure 7.

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Figure 6: Position shifts of the Texas Capitol Dome from datum differences (from [2]).

Figure 7: Coastline shifts of the Northern Adriatic coastline. The DEM is from [18] (WGS84 datum),the shorelines and political boundaries being from two separate databases [13] . The average shift issome 2 km towards SSE. It has been discovered during the revision of this document that the error isin visualization. However, this is most likely how a typical misuse of datums might look like: a nearlylinear shift of some 100 meters or a kilometer or two.

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3.2 Sources of datum parameters

As already mentioned, datum parameters are supposed to conform to ISO standard, therefore weexpect the values are to be interpreted as throughout this document. There are at least 3 useful sources,namely the PROJ 4.4 package (table 4), WGS84 Impl. Manual [5, p. 85] and EU-CRS (European Co-ordinate Reference Systems) (table 5). One can find there informations about all European countries.Unfortunately, the list is not available in one piece, so we extracted examples only from few countries.You will find out that major coutries use or have used many relative datums.

Table 4: Datums built in the PROJ package rel. 4.4.9. Output produced by cs2cs -ld. 3-parametersets are obviously not in correlation with 7-parameter sets in table 5.

__datum_id__ __ellipse___ __definition/comments______________________________WGS84 WGS84 towgs84=0,0,0

GGRS87 GRS80 towgs84=-199.87,74.79,246.62Greek_Geodetic_Reference_System_1987

NAD83 GRS80 towgs84=0,0,0North_American_Datum_1983

NAD27 clrk66 nadgrids=@conus,@alaska,@ntv2_0.gsb,@ntv1_can.datNorth_American_Datum_1927

potsdam bessel towgs84=606.0,23.0,413.0Potsdam Rauenberg 1950 DHDN

carthage clark80 towgs84=-263.0,6.0,431.0Carthage 1934 Tunisia

hermannskogel bessel towgs84=653.0,-212.0,449.0Hermannskogel

ire65 mod_airy towgs84=482.530,-130.596,564.557,\-1.042,-0.214,-0.631,8.15

Ireland 1965nzgd49 intl towgs84=59.47,-5.04,187.44,0.47,-0.1,1.024,-4.5993

New Zealand Geodetic Datum 1949

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Tabl

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Dat

umtr

ansf

orm

atio

nsbe

twee

nlo

cald

atum

san

dE

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15


3.3 Usage withing the PROJ4 package

Geodetic datums are available at least from the PROJ package rel. 4.4.5 on. They are still verypoorly documented in [15]. At present, as the release 4.5.0 is available, there is still no sign of it in anupdated documentation either in the Users Manual [6] or in the LibProj4 Manual [8].

We all have used extensively the proj program in the past. proj has a minor deficiency, i.e. atransformation between two projections in cartesian units x,y has to be work-arounded via the inter-mediate λ,ϕ.

From the PROJ package rel. 4.4.5 on, an additional program cs2cs is available. It has almostthe same arguments syntax with the following extensions:

• The source and target projection may be in cartesian (x,y) or geodetic (λ,ϕ) coordinates+proj=longlat; that proj doesn’t recognize it). The parameters ot the target projectionstarts with a descriptor +to.

• The source and target projection may have the 3 or 7 datum transformation parameters included.If used, they should be applied for both the source and target projection. If not used, bothprojections are considered on the same datum and no datum transformation is applied at all(like proj).

• There is another extension - the output set of points include the height of the transformed pointabove the actual ellipsoid, too. It may be also used in input points. Warning: ellipsoidal heightsare usually found only in satellite raw terrain measurements, however in most cartographicalproducts, heights over the local geoid are applied.

There are three ways how to describe a chosen datum:

1. A built-in datum and ellipsoid: +datum=hermannskogel +ellps=bessel. A list ofbuilt-in datums is increasing, print it out with cs2cs -ld.

2. An explicit 3- or 7-parameter set and ellipsoid: 653.0,-212.0,449.0 +ellps=bessel.Ellipsoid parameters can be given explicitely, of course.

3. A list of files with local datum shift grids:nadgrids=conus,alaska,ntv2_0.gsb,ntv1_can.dat.This is a replacement of the former nad2nad program. It is not of our detailed interest.

Here is an example for Slovenia state coordinate system SI_D48 (see table 5):

# A conversion between long,lat on absolute WGS84 datum and Slovenian# Gauss-Krueger coordinates. For inverse operation include thwe switch# ’-I’ (as in proj)

echo ’14.81527778 46.12000000’ | \cs2cs +proj=longlat +datum=WGS84 \

+to +proj=tmerc +lon_0=15E +x_0=500000 +y_0=-5000000 +k=0.99 \+ellps=Bessel \+towgs84=426.94,142.58,460.09,4.91,4.49,-12.42,17.11’

16


3.4 Support in the libproj library

The best advice is to inspect source code of the program cs2cs and the C modules of thelibproj library. Key routines are in the module pj_transform.c:

• pj_geocentric_to_wgs84(): (X ,Y,Z)local to (X ,Y,Z)WGS84 (forward Helmert transfor-mation)

• pj_geocentric_from_wgs84(): (X ,Y,Z)WGS84 to (X ,Y,Z)local (backward Helmert trans-formation)

Three fields are added to PJconsts structure (see projects.h):

int datum_type; /* PJD_UNKNOWN/3PARAM/7PARAM/GRIDSHIFT/WGS84 */double datum_params[7];

A structure PJ_DATUMS is defined, analogous to PJ_ELLPS.I didn’t try it myself, but it seem’s that routine pj_init(n_params, params) is able to

parse also datum transformation parameters in the params array of strings, created from command-line arguments:

+proj=... +x_0=... +datum=.. +to ...

That means that datum parameters may be freely added to the existing PROJ parameters, providedthat they are parsed by the PROJ programs or library routines from versin 4.4.5 or newer.

In the cs2cs.cmodule source and target projection+datum parameters are split to from_argvand to_argv string arrays, respectively. Separate calls of pj_init() on both string arrays producePJconsts structures fromProj and toProj, respectively.

17


4 Errors in positions of radar pixels

Thhe main stream of this document is to reveal some geodetic issues which are not well knownwithin meteorological community, but do affect positioning of radar pixels. This chapter however, isthe other side of a coin. If one is to deliver weather radar products in more fine resolution (below 1km)with otherwise consistent projectional parameters, the physical limitations that affect the positioningaccuracy should not be ignored.

4.1 Effects due to the geoid shape

Height and slant range of a given radar polar volume, measured at distance r along the ray path ina given azimuth, strongly depend on the so called equivalent Earth radius [4, p. 21]:

ae =R

1+Rdn(h)

dh

(10)

where dn(h)/dh is the refractive index vertical gradient of refractive index and R is the spherical Earthradius, or more precisely, the local curvature R(h) of the Earth geopotential at the height h from themean sea level. Usually, one assumes for R the normal sphere RN (eq. 13), added by the height of theradar antenna above sea level. For the refractive index vertical gradient, value of the lateral standardatmosphere is applied and the equivalent Earth radius becomes:

dn0

dh≈−39·10−6/km ≈−

14 R

(11)

ae0 ≈ 8.5·106 km (12)

Let us consider the spherical Earth radius approximation. The simplest approach being used is totake the normal sphere of the WGS84 ellipsoid:

RN =3√

a2 b normal sphere (13)

RN(WGS84) = 6371000.79 m (14)

The normal sphere 3 has the same volume as the corresponding ellipsoid.To estimate the effects of spherical approximation, let us scroll over different curvature radii on

an ellipsoid of revolution, used in geodesy. The meridional radius of curvature

M(ϕ) =a(1− e2)

(1− e2 sin2 ϕ)3/2(15)

belongs to the circle of the meridional plane and is thus perpendicular to the plane of the circle withradius N(ϕ). Both of them are also the principal curvature radii, which means that a curvature radiusin any direction (azimuth α)

Rα(α,ϕ) =N(ϕ)M(ϕ)

M(ϕ)sin2 α+N(ϕ)cos2 α(16)

3The PROJ package uses the value of the authalic sphere of the Clarke 1866 ellipsoid: 6370997 m (+ellps=sphere).Authalic sphere has the same surface as the corresponding ellipsoid of revolution. The correct PROJ syntax for the WGS84normal sphere is +ellps=WGS84 +R_V.

18


is limited byM(ϕ) ≤ Rα(α,ϕ) ≤ N(ϕ)

As seen on the figure 8(a), a weather radar experiences most variations of the curvature radius atequator (≈ 0.7%) and none at poles.

The best local spherical approximation based on an ellipsoid of curvature is radius of curvature,averaged over all azimuths (figure 8(b)):

RG(ϕ) = Rα(α,ϕ) =a√

1− e2

1− e2 sin2 ϕGauss sphere (17)

The real geopotential has a shape of geoid (fig. 9) which deflects from the WGS84 ellipsoid formostly ±100 m. Calculation of its local curvature radii is far beyond scope of this document. Wecan only assume that geoid contributes half more to the deflection of curvature radii from the normalsphere, within typical range of weather radar (meso-α scale). The total deflection of real azimuthalcurvature radii from a normal sphere would therefore sum up to ≈ 1%. This is not related neither withdeflections of accurately designed local ellipsoids (0.01%; see table 6) nor scale corrections (10 ppm;see table 5) because the latter two are obtained from data of wider areas.

We can ignore deflections of geoid in scales finer than meso-α scale. This is even more valid as welift from the mean sea level, because geopotential is becoming more spherical away from the planet.

19


90 ° 180 ° 270 ° 360 °Α

0.996

0.998

1.002

1.004

R�Rn

j=90°j=60°j=45°j=30°j=0°

(a) Relative azimuthal radii at chosen latitudes

15 °30 °45 °60 °75 °90 °j

0.996

0.998

1.002

1.004

RG�RNNHjL�RNMHjL�RN

(b) Relative principal radii and Gauss sphere at chosenlatitudes

15 ° 30 ° 45 ° 60 ° 75 ° 90 °j

0.001

0.002

0.003

0.004

0.005

0.006

HRmax-RminL�Rn

(c) Difference between minimal and maximal azimuthalcurvature radius at all latitudes: (N(ϕ)−M(ϕ))/RN

Figure 8: Curvature radii on an ellipsoid of revolution, northern hemisphere. All radii are relativeto the normal sphere RN of the ellipsoid (eq. 13).; a wheather radar experiences most anomalies atequator (≈ 0.7%) and none at poles.

20


−80

−60

−40

−40

−40

−40

−20

−20

−20

−20

0

0

0

0

20

20

20 20

40

40

60

60

Figure 9: Geoid undulations (anomalies) from the WGS84 ellipsoid in meters; Positive anomalies inred, negative in blue; Majority of Europe lies westerly from the saddle between local maximum innorthern Atlantic Ocean (+60 m) and local minimum in the Tibet plateau (-50 m). Data are from theOSU91A1F 1o ×1o geoid model of Ohio University.

21


Table 6: Ellipsoids built in the PROJ package rel. 4.4.9. Deviations from the WGS84 normal sphere(eq. 14) may reflect the local curvature of the Earth geopotential. See the note on different normalspheres on the page 18.

name a [m] b [m] 1/ f RN [m] e[·10−2] 1− RNRN(W GS84)

MERIT 6378137.00 6356752.30 298.2570 63771000.78 8.1819 -0.000000%SGS85 6378136.00 6356751.30 298.2570 63770999.79 8.1819 -0.000016%GRS80 6378137.00 6356752.31 298.2572 63771000.79 8.1819 -0.000000%IAU76 6378140.00 6356755.29 298.2570 63771003.78 8.1819 0.000047%

airy 6377563.40 6356256.91 299.3250 63770453.31 8.1673 -0.008593%APL4.9 6378137.00 6356751.80 298.2500 63771000.62 8.1820 -0.000003%NWL9D 6378145.00 6356759.77 298.2500 63771008.61 8.1820 0.000123%mod_airy 6377340.19 6356034.45 299.3249 63770230.35 8.1673 -0.012093%

andrae 6377104.43 6355847.42 300.0000 63770010.87 8.1582 -0.015538%aust_SA 6378160.00 6356774.72 298.2500 63771023.59 8.1820 0.000358%GRS67 6378160.00 6356774.52 298.2472 63771023.52 8.1821 0.000357%bessel 6377397.16 6356078.96 299.1528 63770283.16 8.1697 -0.011264%

bess_nam 6377483.87 6356165.38 299.1528 63770369.77 8.1697 -0.009905%clrk66 6378206.40 6356583.80 294.9787 63770990.71 8.2272 -0.000158%clrk80 6378249.14 6356514.97 293.4663 63770996.17 8.2483 -0.000072%CPM 6375738.70 6356666.22 334.2900 63769374.86 7.7291 -0.025521%

delmbr 6376428.00 6355957.93 311.5000 63769597.33 8.0064 -0.022029%engelis 6378136.05 6356751.32 298.2566 63770999.83 8.1819 -0.000015%evrst30 6377276.34 6356075.41 300.8017 63770201.52 8.1473 -0.012545%evrst48 6377304.06 6356103.04 300.8017 63770229.21 8.1473 -0.012111%evrst56 6377301.24 6356100.23 300.8017 63770226.39 8.1473 -0.012155%evrst69 6377295.66 6356094.67 300.8017 63770220.82 8.1473 -0.012243%evrstSS 6377298.56 6356097.55 300.8017 63770223.71 8.1473 -0.012197%fschr60 6378166.00 6356784.28 298.3000 63771030.78 8.1813 0.000471%

fschr60m 6378155.00 6356773.32 298.3000 63771019.79 8.1813 0.000298%fschr68 6378150.00 6356768.34 298.3000 63771014.80 8.1813 0.000220%helmert 6378200.00 6356818.17 298.3000 63771064.74 8.1813 0.001004%hough 6378270.00 6356794.34 297.0000 63771103.40 8.1992 0.001611%

intl 6378388.00 6356911.95 297.0000 63771221.27 8.1992 0.003461%krass 6378245.00 6356863.02 298.3000 63771109.69 8.1813 0.001709%kaula 6378163.00 6356776.99 298.2400 63771026.35 8.1822 0.000401%lerch 6378139.00 6356754.29 298.2570 63771002.78 8.1819 0.000031%mprts 6397300.00 6363806.28 191.0000 63786115.89 10.219 0.237248%

new_intl 6378157.50 6356772.20 298.2496 63771021.08 8.1820 0.000319%plessis 6376523.00 6355863.00 308.6410 63769628.88 8.0433 -0.021534%SEasia 6378155.00 6356773.32 298.3000 63771019.79 8.1813 0.000298%

walbeck 6376896.00 6355834.85 302.7800 63769867.87 8.1207 -0.017782%WGS60 6378165.00 6356783.29 298.3000 63771029.78 8.1813 0.000455%WGS66 6378145.00 6356759.77 298.2500 63771008.61 8.1820 0.000123%WGS72 6378135.00 6356750.52 298.2600 63770998.86 8.1819 -0.000030%WGS84 6378137.00 6356752.31 298.2572 63771000.79 8.1819 0.000000%sphere 6370997.00 6370997.00 — 63770997.00 — -0.000059%

22


4.2 Effects due to deviations of refractivity gradient

Vertical gradient of the refractive index is usually expressed in more convenient way via refractiv-ity:

N = (n−1) ·106 (18)

The lateral vertical gradient −1/4R thus becomes dN0/dh≈−39/km. It is well known that the refrac-tivity gradients exhibit severe anomalies inside inversion layers. We can estimate from [4] and [10]that deflections of ±20/km are frequent, however extreme anomalies span over −300/km,+20/km.Extreme anomalies occur within layers of some 100 meters depth. We will skip the discussion onthe sub- and superrefraction phenomena which are well known within the weather radar community(see [4], for instance). It is worthwhile to mention that equivalent radius (eq. 10) has a singularityat around dN/dh ≈−157/km, which can be well seen on figures 10. At that point the beam is benttowards the Earth surface and the sign of curvature becomes negative.

We used the standard equations for the arc distance s and beam height h, obtained by the radar-measurable parameters range r and elevation ε ([4, p. 21]):

h(r,ε) =√

r +a2e +2 r ae sinε−ae (19)

s(h,r,ε) = ae arcsin(r cos ε

ae +h

)

(20)

We calculated the arc distance (slant range) and ray height at typical range of 200 km. We variedthe Earth radius R∈ [0.994,1.004]RN (WGS84) in equation for the equivalent Earth radius ae (eq. 10).We also varied the vertical refractivity gradient dN/dh ∈ [−200/km,+20/km]. We applied constantvalues for dN/dh over all troposphere which is not realistic. However, we are mainly focused to theweight factors of both key parameters, i.e. (local) Earth radius and refractivity gradient. The figures10 clearly show that refractivity gradient is the strongly dominant parameter.

One can see on fig. 10(b) and 10(c) that the arc distance s is only moderately affected for several100 m, which is the same order of magnitude as caused by false geodetic datum. As expected, devi-ations are increasing with higher elevations, but as beams reach the tropopause at closer ranges, thisproblem is not that important.

The difference becomes more obvious in ray height calculations (fig. 10(d) and 10(e)). As alreadymentioned, we are not that focused to height problems in this document.

The conclusion would be that atmospheric fluctuations degrade the estimation of local Earth cur-vature radius for an order of magnitude, in average. One should keep this in mind when deliveringradar products in fine horizontal and vertical resolution. There is often a transition between volumet-ric processing, which is done by the weather radar software, and radar products, which may be doneby national meteo services. The detailed treatment of pixel locations may well differ and should beinspected.

23


0.9950.9975

11.0025

R�RN-200-150-100

-50

0

dN�dh @�kmD -4

-2

0

2

4

ae�ae00.995

0.99751

1.0025R�RN

(a) Equivalent Earth radius

Elevation 1°

0.9981

1.002

1.004R�RN

-200-150

-100-50

0

dN�dh @�kmD

0

0.05

0.1@kmD

0.9981

1.002

1.004R�RN

-200-150

-100-50

0

dN�dh @�kmD

(b) Deviations of arc distances at 1o

elevation

Elevation 10°

0.9981

1.002

1.004R�RN

-200-150

-100-50

0

dN�dh @�kmD

-0.2

0

0.2

0.4

0.6

@kmD

0.9981

1.002

1.004R�RN

-200-150

-100-50

0

dN�dh @�kmD

(c) Deviations of arc distances at 10o

elevation

Elevation 1°

0.9981

1.002

1.004R�RN

-200-150

-100-50

0

dN�dh @�kmD

-2

-1

0

1

@kmD

0.9981

1.002

1.004R�RN

-200-150

-100-50

0

dN�dh @�kmD

(d) Deviations of ray heights at 1o el-evation

Elevation 10°

0.9981

1.002

1.004R�RN

-200-150

-100-50

0

dN�dh @�kmD

-2

-1

0

1

@kmD

0.9981

1.002

1.004R�RN

-200-150

-100-50

0

dN�dh @�kmD

(e) Deviations of ray heights at 10o el-evation

Figure 10: Influences of curvature radius and refractivity gradient on the equivalent Earth radius, arcdistance and ray heights. Refractivity gradient is obviously heavily dominating.

24


5 Conclusions

Typical desired and realistic, thus minimalistic position accuracy in weather radar imagery isaround 100 m/100 km (i.e. 0.1%), considering radial and azimuthal errors. Therefore ellipsoid axesare given per 1 km, geographic coordinates in 2-3 decimals. Position shifts due to geodetic datumsbehave differently from radar-induced positional errors. They can be treated in average as several100 m horizontal shifts, i.e. 0.005o(eq. 9). Even in minimal approach, datum shifts affect geographiccoordinates. From geodetic point of view, it is inconsistent to use a non-global ellipsoid and discardingthe datum it is bound to.

There are certain applications where radar location is required in higher precision, like urbanhydrology (250 m) or occultations effects in mountaneous terrain near radar site. A good practice is toassure that georeferencing parameters don’t induce positional errors above half of the grid resolution.

Therefore, geodetic datums must be taken into account already in 1 km resolution imagery. Theellipsoid axes [km] and projection reference points [deg] should contain 2 and 4 digits, respectively.

The requirement seems controversal because a single weather radar experiences up to 1% of localdeviations of local Earth radius. However, the atmospheric anomalies contribute much more. Surpris-ingly, the overall horizontal error in positioning is moderate and has the same order of magnitude asdatum-induced errors. So, there is no need to complicate with local Earth radius.

There are several proposals, how to incorporate geodetic datum transformation in radar imagery.The following proposals are valid for a single weather radar:

1. Reproject all products, intended for international exchange, onto the WGS84 datum. In thiscase, the present set of georeferencing parameters may remain intact.

2. If projection is centered on the radar site (like the native azimuthal equidistant projection) andthe spherical Earth is used, only the following metadata are to be changed:

• transform radar geographic coordinates to the WGS84 datum. Be careful to use the geo-centric latitude.

• apply the WGS84 normal sphere radius

No reprojection is needed at all. This way we cure the largest part of the false datum - linearshift of geographic coordinates. Single radar data with such a set of projectional parameters arerecommended to deliver to other users.

3. Include only the 3 translational parameters relatively to the WGS84 datum, which assure thehorizontal accuracy of around 10 m over the valid area. If taken from the 7-parameters set,it should be done via some statistical means. This implies some extra effort outside our mainscope. We should rely on the standardized framework provided by geodetic community.

4. Include all the 7 parameters relatively to the WGS84, as used throughout this document anddescribed in the ISO standard. The provided accuracy of few meters is certainly far beyondour needs, at present. But, it is widely supported in software tools and thus requires minimaladaption of present production and research frameworks.

As one may have guess already, the proposal no. 2 is preferred by the author.There is only one proposal for the radar composites - use the WGS84 datum. Be careful to use the

geocentric latitude if you use the spherical Earth radius instead of ellipsoid.

25


To avoid users of treating radar pixel positions too accurately (especially those far away from theradar), we should have our radar data equipped with per-pixel weight factors which would be based onpositional errors, among other factors which contribute to quality of each pixel. This approach wouldwork fine for radar composites, too.

26


References

[1] M. Balridge, editor. Geodetic Glossary. National Geodetic Survey, 1986.

[2] Peter H. Dana. Geodetic Datum Overview. The University of Colorado, Boulder, 1999. [Online]http://www.colorado.edu/geography/gcraft/notes/datum/datum.html.

[3] Department of Oceanography at the Naval Postgraduate School, USA. Ellipse glossary, Aug1997. [Online] http://www.oc.nps.navy.mil/~garfield/ellipse_app2.pdf.

[4] R.J. Doviak and D. Zrnic. Doppler Radar and Weather Observations. Academic Press, 2.edition, 1992.

[5] Eurocontrol and IfEN. WGS 84 IMPLEMENTATION MANUAL, February 1998. [Online]http://www.wgs84.com/files/wgsman24.pdf.

[6] G. I. Evenden. Cartographic Projection Procedures for the UNIX Environment – A User’sManual. US Dept. of the Interior Geological Survey, Jan 2003. [Online] ftp://ftp.remotesensing.org/pub/proj/new_docs/OF90-284.pdf.

[7] G. I. Evenden. Cartographic Projection Procedures. Release 4. Interim Report. US Dept. of theInterior Geological Survey, Jan 2003. [Online] ftp://ftp.remotesensing.org/pub/proj/new_docs/proj.4.3.pdf.

[8] G. I. Evenden. libproj4: A Comprehensive Library of Cartographic Projection Functions, Mar2004. [Online] http://members.verizon.net/~vze2hc4d/proj4/manual.pdf.

[9] Federal Agency for Cartography and Geodesy, Germany. Information and Service System for Eu-ropean Coordinate Reference Systems - CRS, Jun 2004. [Online] http://crs.bkg.bund.de/crs-eu/.

[10] K. Browser J. Gao and M. Xue. A comparison of the radar path equations and approximations foruse in radar data assimilation. Advance in Atmospheric Sciences, June 2005. Revised manuscript.

[11] Young-Hoo Kwon. Mechanical Basis of Motion Analysis. Theoretical Foundation. VISLo, Inc.,Korea, 1998. [Online] http://kwon3d.com/theory/basis.html.

[12] Nasa, NIMA. EGM96 - The NASA GSFC and NIMA Joint Geopotential Model, November 2004.[Online] http://cddis.gsfc.nasa.gov/926/egm96/egm96.html.

[13] NGDC/WDC MGG Boulder, Boulder. Coastline Extractor (online coastlines and bound-aries data), December 2004. [Online] http://rimmer.ngdc.noaa.gov/mgg/coast/getcoast.html.

[14] Ordnance Survey. A guide to coordinate systems in Great Britain, 2000. [Online] http://www.gps.gov.uk/additionalInfo/images/A_guide_to_coord.pdf.

[15] Remotesensing.org. PROJ.4 - General Parameters, Mar 2003. [Online] http://www.remotesensing.org/proj/gen_parms.html.

[16] Irvin Schollar. Where on Earth is it?, Sep 2002. [Online] http://aarg.univie.ac.at/aerarch/airphoto/airphoto.html.

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[17] SNAP Laboratory, University of New South Wales. Principles and Practice of GPS Survey-ing, December 2004. [Online] http://www.gmat.unsw.edu.au/snap/gps/gps_survey/principles_gps.htm.

[18] U.S. Geological Survey’s EROS Data Center. GTOPO30 - Global topographic Data (onlineworld DEM in 30" resolution), 1996. [Online] http://edcdaac.usgs.gov/gtopo30/gtopo30.asp.

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