L1 Geodesy - Lecture notes 1 PDF

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Summary

Earth and Ocean Observation – Geodesy

Geodesy is the science of measuring the Earth to determine its shape, size and to locate features of the Earth’s surface.
Includes the definition of “terrestrial coordinate systems.”
calculations for the shape of the earth and diffe...

Description

Earth and Ocean Observation – Geodesy Geodesy is the science of measuring the Earth to determine its shape, size and to locate features of the Earth’s surface. Includes the definition of “terrestrial coordinate systems.” Without extra information, from 53.505° North, 4.150° West, 50m Altitude, you wont be able to tell where you are. 53.505° - the first decimal is 100m difference on the ground Different reference systems can affect where you think the location is Datum = terrestrial reference system NEED TO KNOW WHAT SYSTEM TO KNOW WHAT ELLIPSOID IS USED TO AVOID MISS DIRECTON Same coordinate but different system Even mm GPS could be 700m out due to lack of information of reference system Assuming or using the wrong TRS (datum) can have consequences. Land-mine field example of differing TRS’s. When using GIS, ‘layers’ are superiposed onto one another but if they are from different reference systmes, it can lead to many issues The lack of understanding of reference systmes can have huge consequences, many are finantial due to the waste of time

Miss haps on land, more chance of noticing if anything is wrong or incorrect due to more land marks etc

Ocean = no reference points to see if correct or incorrect Radar to other known points to see if they’re correct Can be expensive to use the incorrect reference system – re move etc BP yellow sea – china (looking for oil and gas but wrong location) The shape and size of the Earth  Possible to simply take the read surface  Mathematically complicated, rather changeable  It approximates to a flattened sphere- an oblate spheroid Ellipse: The equation for an ellipse is

An ellipsoid is a 3D version of this so has another axis length, C But for the earth, A = C. if an ellipsoid has A = C, called an oblate spheroid a = 6378000 m and b = 6357000 m 21 km of difference (a longer than b) – C = a for the Earth Often, you see the semi-major axis length, a, quoted with the flattening f: And on occasion, the eccentricity is quoted: For the Earth, a = 6378000 m and b = 6357000 m 1/f = 1/298

e = 0.08182 – so small that it is almost spherical (1/300 exaggeration) Unfortunately, the terms oblate spheroid and ellipsoid are used inter-changeably in many geodesy references A and 1/f value you can calculate b from b = a – a/f The ellipsoid that best-fits the ‘real’ shape of the Earth – WGS84 WGS = World Geodetic System / 84 = year in which it was defined WGS84 = a = 6378137.0 m and 1/f = 1/298.257223563

In geodesy, geocentric co-ordinate systems are sometimes used. They can use the familiar x, y, z Cartesian system X-axis: from a defined Earth centre, through the Equator at the defined prime meridian Y-axis: perpendicular to x-axis, through Equator Z-axis: from a defined Earth centre, through the Earth’s North Pole. Sometimes referred to as an Earth Centred, Earth Fixed system – ECFF X, y, z = underground or in space, no elevation reference

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Positioning systems for a sphere – geocentric or ECEF systems t ( Poin

They can use the less familiar ECEF

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Positioning system for a sphere Onasphere,la tudeandlongitudeseemlikeastraightforward wayofposi oningyourself: La tude: anglefromEquator

Longitude: anglerela vetoarbitrary primemeridian

arc (60’) minutes of 0 6 = (60”) e re g 1 de conds of arc e s 0 6 = rc a 1 minute of be quoted in ngitude can lo d n a e d tu Lati . nt ways e.g many differe utes) es and min re g e (d ’ 3 1 ° 53 as Is the same es) ecimal degre 53.21666° (d th a degree d 13/60 of n a 3 5 s a it ersion) (think of as the conv ° 6 6 1 .2 3 5 to get to

Positioning systems for an oblate spheroid - GEODETIC system

Butforanoblatespheroid,it’snot quitethatsimple. Seeinthe diagramhowla tudeismadecomplicatedcomparedtoasphere.

Geode cLa tude: anglefromEquator,butperpendiculartotheoblatespheroid’s surface Geode cLongitude: s llokasananglerela vetoarbitraryprimemeridian Ellipsoidal height: distancebetweentheperpendicularto theellipsoidand thepoint in ques on

The mathematics of geodesy Converting between different systems – 3-D polar and cartesian

Yourproblemsheettasksyouwithconver ngbetweenthesecoordinatesystemtypes

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The mathematics of geodesy Converting between different systems – ECEF and geodetic

Problem3stretchesyoufurther:

Mass will always fall perpendicular to surface

The shape and size of the Earth – equipotential surfaces WeretheEarthanaquaplanet (noland)andthewaterwasn’tmoving, thesurfaceofthatoceanwouldbean equipoten alsurface.With con nents,imaginecu ngcanalsthroughthem. Abuildermakingawall‘level’usingabubblelevelisconstruc ngthe wallalonganequipoten al.

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Shape and size of the Earth – equipotential surfaces  These equipotential surface are caused by variations in the density of subsurface materials  Close to Earth, close to the source of the gravitational force and to the source of these density variations, the equipotential surface is quite, lumpy  Orbital heights of satellites vary because of the lumpiness of the equipotential caused by the Earth’s gravity field  GPS satellites, at 20,200 km, are affected by around 10 m. The Geoid  Further away, in space, the equipotential surface around the Earth is much smoother. The density variations are not noticeable  The geoid is the equipotential surface that best fits around the WGS84 ellipsoid  G e o i d

     

is close to natural undulating surface Geoid is pretty much sea level (minus 100 to + 80 meters) Sea level is closer to ellipsoid Sea level/geoid is 54-60 meters above the ellipsoid At Bangor, the Geoid (= sea level) is about 54m above the ellipsoid Not really a link to plate tectonics – pacific plate, the mid Atlantic ridge – Andies etc Whatever controls Geoidal pattern, does not control plate tectonics but will control the deep mantle plumes

Relating the Geoid and ellipsoid  Outdated but shows the lumpy Geoid and the smooth ellipsoid differ  Notice how the ellipsoid fits some parts of the Earth better than others in this schematic example.  In the UK, being approximately at sea level has you around +55m above the WGS ellipsoid! It transpires you need different ellipsoids – of different a and b values and in different orientations – to provide a good fit to the Geoid in different parts of the globe 

Notice how ellipsoids aren’t necessarily ECEF

Tailored best fit to their part of the Earth/Geoid No longer earth centred – change shape and orientation to best fit a country etc

Relating the Geoid and ellipsoid Herearesome(therearedozens)

O S gr

id

id U T M gr GPS geo d e c

Notethattheseareellipsoids– thedataonlytellsyou aboutthe shapeandsize,nottheorienta on. Whentheorienta ondatais alsoincluded,it’scalledadatumor,morerecently,aTerrestrial ReferenceSysteme.g.OrdnanceSurveymapsintheUKusethe Airy1830spheroidandangleitinacertainwaysuchthatthe datumiscalledOSGB36.

Example ellipoids Ordnand Survey – 1830 ellipsoid (best estimate that fitted Britain) Define ellipsoid and then where to position it – airy also used by Irish (a and b value) but position it slightly differently Called this a datum – gives info of what spheroid and positioning To best fit geoid in your area Airy = OSGB36

Positions for different TRS’s (datums) Itisclearthatthesamela tude,longitudeandheightrepresent differentpointsinspacedependingonwhichTRS(datum)isused

Thatisthe effectseenintheopeningexamples Depends on ellipsoid used = radically different places

EIGHT parameters required: • 3-D location of spheroid origin (3) • 3-D orientation of spheroid axes (3) • Size of the spheroid ‘a’ semi-major axis (1) • Shape of the spheroid ‘ 1/f ’ (1) (the flattening) • • •

Terrestrial Reference System is the now the accepted name for what was once called a datum though they’re both used in practice A TRS is still only a mathematical surface Later, we’ll see how a TRS is made real physically

GPS – raw data, give you height above ellipsoid (Beach at sea level, 54m elevation) Don’t know orthometric value until you know the Geoidal separation h = height above sea level

true Geoid = undisturbed sea level (- the tides, pressure, storm surges etc) can never know the true Geoid, only get so close, depends on the model used etc Ordinance Survey measure it for ever km grid square

The full definition of the WGS84 datum (TRS) is: • WGS84 Cartesian coordinates and ellipsoid are geocentric (centre of mass) • The axes of orientation coincide with the equator and prime meridian of the Bureau International de l’Heure at the moment in time 1984. (Midnight new year’s eve 1983). • Since 1984 the orientation of the axes and ellipsoid have changed such that the average motion of the crustal plates relative to the ellipsoid is zero. • UK not moving at the ‘average’ so WGS84 has been moving around 15mm per year NE. • Shape and size of the WGS84 spheroid defined by a=6378137.0m and 1/f = 298.257223563 • WGS84 moves with the plates, (different directions) – mean drift • Fixed system drifts with the plates • Europe struggles – different to global average and direction – so move it with European plate (called ETRS89)

Where are the GPS’s ??? How do we know where the satellites are? – radar Tracking positions – modern triangulation pyramids

To overcome the problem with local rates of tectonic drift, in Europe, WGS is realised as a slightly different TRS – same size and shape as WGS84, different orientation. Called ETRS89: • Fix the datum “epoch” as 1989 • Coincided with WGS84 in 1989 but progressively moving steadily away from the axes and ellipsoid of WGS84 because it follows the European Plate, not global mean plate velocity • Associated TRF is ETRF89 • In the UK, the primary physical definition of positions is in the form of active GPS station network. ETRF89 for the UK is given a specific name, OSNet.

Defined, ridged points of GPS

Before satelites…  National mapping agencies pre-date satellite systems  Their ETRS’s often established in the early to mid 19th century  OS depends on a modified version of the Airy (1830) spheroid.  The OSGB36 TRF are trig points measured by triangulation between 1783 and 1853, and later between 1936 and 1953.  Nowadays, trig points and OSGB36 aren’t definitive. Instead, the OSNet GPS stations are the ETRF of ETRS89, and these are converted to OSGB36. Converting between TRS’s (or datums) HELMERT transformations  2 ellipsoids can differ by the position of their origin, orientation of axes and the size and shape of the reference ellipsoid.  By working in 3-D Cartesian coordinates we use 6 parameters to describe the difference between 2 datums (3 translation of origin and 3 rotations of axes)  It is traditional to add a 7th parameter the “scale factor” which allows the scale of axes to vary between systems 7-parameter, transformations  This transformation can be thought of as the TRF of points remaining constant while the Cartesian coordinates are rotated, translated and rescaled  The maths uses linear formula that assumes that the rotation parameters are small.  Any small transformation can be reversed by simply reversing the signs of all the parameters and applying the equations...