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The Geodesy Corner

DATUMS AND ELLIPSOIDS- ARE THEY THE SAME?

BY J AMES FERGUSON

Datums, ellipsoids, spheroids, refer-ence surfaces, surfaces of revolution,oblate spheroids of revolution, thegeoid do these terms sound familiar?Along with a host of others, the forego-ing terms have been used to describethe basis for relating survey measure-ments to the earth, and compare meas-urements to each other. But do all these

terms mean the same? In short, theanswer is no, but some of the terminol-ogy does describe the same thing. As wewill discover, the distinct elements areintrinsically linked, and can be relatedto each other using some fundamentalprinciples.

Before we get too far, let us simplifythe terminology, and decide on threeitems that will guide us through thisdiscussion a datum, an ellipsoid andthe geoid.

A datum is a system defined andused to relate and present coordinateinformation in a logical and repeatablefashion. I t has an origin, orientation,and is propagated or extended throughmany types of survey methods. Ingeneral, datums are used to orientlarge systems of geodetic control suchas triangulation and trilaterationchains, and in a more modern context,satellite survey networks.

A simple example of a "user defined"datum is the land surveyors use of alocal coordinate system to control asubdivision. It will have an origin with

a known coordinate (10,000, 10,000 forexample), and will have orientation inthe form of a baseline with a knownbearing or azimuth. I t will probablyalso have some type of vertical informa-tion, with the top of a fire hydrant ormanhole cover as the initial referenceelevation. Hence we can speak abouthorizontal datums and verticaldatums.

An ellipsoid is a three dimensionalmathematical figure, used to best rep-resent the figure of the earth over a

specific geographic area. Some ellip-

soids attempt to be a "best fit" over whole earth, while others are bsuited to an area, say the size location of North America. Throughthe world there are many definitionthese "football" shaped figures, some of the more common ones are Clarke 1866 ellipsoid, the WGS84 esoid and the GRS80 ellipsoid. E

soids also have an origin aorientation, and several parametersdescribe their size and shape.

The geoid is described as the bapproximation of mean sea level othe whole earth [Vanicek aKrakiwsky 1982]. Since mean sea leis not a smooth regular surface asellipsoid is, neither is the geoid. I t mathematical figure however, and an undulating pattern related to terrain of a particular geographic a

In order to further our discussionus take one particular datum, relatto an ellipsoid, and discover how we work with it in surveying. For the mpart, all control surveys performedCanada (and until recently NoAmerica) have been referred to two tinct datums the North AmericDatum 1927 or NAD27, and the NoAmerican Vertical Datum 1929NAVD29. As one might guess, NADis a horizontal datum, and NAVD2a vertical datum. We will talk abNAVD29 and height, in a future arti

NAD27 is related mathematicall

the Clarke 1866 ellipsoid. That is, are using a mathematical fig(Clarke 1866) to approximate the face of the geoid so that we can reearth based measurements to a cmon system. It is important to note tin some places t}ie ellipsoid will below the geoid, and in other placewill be above the geoid. Furthermthe earths surface topography unlates above and below the geoid.

The origin of the Clarke 1866 elsoid is somewhere near the centre

mass of the earth (theoretically kno

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DATUMS AND ELLIPSOIDS contd

but unnecessary here), and its orienta-ion can be described by a cartesian

coordinate system (X,Y,Z). The Z axis isvery nearly parallel to the axis of rota-ion of the earth, the X axis is pointed

n the direction of the Greenwichmeridian, and the Y axis is perpen-dicular to the X axis. The two principalparameters used to describe the ellip-soid are its semimajor and semiminoraxes, designated by "a" and "b" respec-ively. These two parameters are sub-

sequently used to determine thelattening of the ellipsoid "f", its ec-centricity "e", and other desired quan-ities.The NAD27 datum has its origin at

a place known as Meades Ranch inKansas, U.S.A. An observed latitudeand longitude at this point furnish theorigin, and an azimuth mark from theorigin supplies the orientation for thesystem. The actual NAD27 frameworkwas the culmination of an adjustmentdone in 1927, utilizing all appropriateriangulation and trilateration net-

works surveyed up to that time. Sub-sequent adjustments have been donenationally and internationally usingadditional survey information, but thesystem is still referred to as NAD27.

Any measurement of distance,

azimuth, height, latitude or longitudeperformed on the earths surface is af-ected by the earths gravity field. Thats, the levelling of the instrument usinga level bubble of vial is directly relatedo the force and direction of gravity athe survey station. I t is common prac-ice to call these earth based measure-

ments astronomical quantities. Thene perpendicular to the level of thenstrument is known as the plumb line,and plumb line is naturally parallel tohe gravity vector at the station. I f the

plumb line is extended through the ref-erence ellipsoid, one would find that its not perpendicular to the survey of theellipsoid. Hence we discover angulardifferences known as the deflections ofhe vertical. Practically speaking,hese deflections translate into the dif-erence between astronomic and

geodetic positions where astronomicmeasurements are actual earth basedmeasurements, and geodetic quan-tities are those that are related to thereference ellipsoid. The difference is

known as the Laplace correction.Using the basic parameters of thereference ellipsoid ("a" and "b"), twoother important quantities must bedetermined in order to interrelate dif-ferent forms of coordinates; the radiusof curvature of the ellipse in the primevertical (along a line of longitude), andthe radius of curvature along themeridian (along a parallel of latitude).Both of these quantities change as afunction of the latitude of the station inquestion, and are fundamental in relat-ing earth based measurements to theellipsoid (and viceversa), via complexmathematical formulae.

How can the above theory be appliedto a common survey scenario? Since alarge proportion of the work a landsurveyor does is over a relatively smallarea, it can be performed locally in auser defined plane system. In thisscenario the survey area is treated as aflat surface, and all measurements arerelated to a local origin. When the coor-dinate system is required, such as theUniversal Transverse Mercator (UTM)map projection, a common referencesystem is needed. In fact, our UTMexample is convenient because it isbased on the Clark 1866 reference el-lipsoid. This means that any new sur-vey can be transformed into UTMcoordinates as long as several existingstations having UTM coordinates areoccupied in the same survey. An adjust-ment procedure then produces coor-dinate information in a commonsystem, relative to the existing controldatum (UTM on NAD27). A word of

caution here, if the existing control hasbeen integrated into a later adjustment(ie. MAY 1976 adjustment used byOntarios MNR) then it is essential thatall the existing coordinate informationbe taken from the same adjustment.Disastrous results would occur i f con-trol coordinates were mixed between

different adjustments, even thouthey are referenced to the same esoid.

There are many applications which the knowledge of a surv

datum, either horizontal or verticand its corresponding reference esoid are essential to the productionuseful coordinate information. In adtion, there are many aids to the sveyor for the computation of susystems. Software has been writtenallow a surveyor to: traverse on anlipse using a known starting codinate, perform transformatiobetween mapping plane coordinaand ellipsoidal coordinates, adjust nworks using a combination of existcontrol and new survey data, and perform a variety of other computions too numerous to mention here

Further reading on the topicdatums and ellipsoids can be foundseveral sources Vanicek aKrakiwsky [1982] is a good one, asIvan Muellers Spherical and PractiAstronomy [1969]. Additional informtion can be found in a number of trational surveying textbooks. Actucoordinate information can be obtainfrom the Geodetic Survey of Cana(Data Services Branch), as well as tMinistry of Natural Resources in Otario.

Next time in the Geodesy Corne'Azimuths Grid, Astronomic aGeodetic".

The Geodesy Comer -Author Profile

J ames Ferguson graduated from tUniversity of Toronto with a B.Sc.Survey Science. He is a member of tA.O.L .S., holding a CertificateRegistration in the field of Geode

J ames works with the Ottawa bafirm GEOsurv Inc.

The Ontario Land Surveyor, Fall 1990