Coordinate Systems ig g

Earth Ellipsoid, ellipsoidal coordinates, satellite coordinate systems

(Part I)

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MSc Geodetic Engineering

Module Coordinate Systems 1st Semester, 2020/21

Jürgen Kusche Coordinate Systems ig g

Today • Earth as a flattened body, Earth ellipsoid • Ellipsoidal coordinates and transformations • Earth’s rotation and inertial coordinate system • Kepler angles and satellite coordinate systems 2

References • Kusche J, Lecture Notes -> ecampus • Jekeli C., Geometric Reference Systems in Geodesy, Ohio State University (-> https://kb.osu.edu/handle/1811/77986) Coordinate Systems ig g

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Earth as a flattened body Coordinate Systems ig g

Some observations • Earth is not a perfect sphere. • In first approximation, it resembles a sphere flattened at the poles • Best-fitting polar radius 20 km less than equatorial radius (~ 6380km) • Therefore, in geodesy we use an ellipsoid of revolution as a reference surface → latitude, longitude and height 4

How do we know this? • History of geodesy: arc measurements, 17th and 18th century • World-wide geodetic networks, geodetic datums • Satellite observations of the (dynamical) flattening of the Earth: precession of satellite orbital plane observed already with Sputnik-1 Coordinate Systems ig g

History of geodesy: arc measurements, 17th Cassini’s Ellipsoid and 18th century

Semimajor axis

Rotation axis 5

Eccentricity parameter

Modern geodesy: Huygens’s Ellipsoid precession of satellite orbital plane Coordinate Systems ig g

Geodetic measurements enable to determine 3D positions with mm – accuracy

GPS installation for monitoring potential volcanic deformation (Japan, Photo: J. Kusche)

6 Coordinate Systems ig g

Reference surfaces in geodesy require a definition precise at mm-level. → ellipsoid of revolution with • semi-major axis a • semi-minor axis b < a 7 a, b defined at sub-mm-level. Coordinate Systems ig g

Ellipsoid of revolution

Parametric description

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For an ellipsoid of revolution it is sufficient to consider the meridional ellipse, with the p-coordinate Coordinate Systems ig g

Ellipsoid of revolution

Use of other ellipsoid parameters

9 Linear eccentricity

2nd numerical eccentricity Geometrical flattening 1st numerical eccentricity Coordinate Systems ig g

Ellipsoid parameters in practice (+ derived from other parameters)

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There are several relations between the parameters, e.g. Coordinate Systems ig g

Important

• For a given ellipsoid only two parameters are independent 11 • All others are “derived” and can (and must) be computed from the two given ones with maximum precision • “Maximum precision” means all parameters must be given or computed with sufficient precision (digits) that allow transformation of coordinates with 0.1 mm accuracy. • This must be considered in particular for numerical eccentricy or flattening parameters Coordinate Systems ig g

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Earth ellipsoid

(introducing the ellipsoidal latitude) Coordinate Systems ig g

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Ellipsoidal latitude  and the slope of the tangent Coordinate Systems ig g

(all derivations can be found

in lecture notes J. Kusche) 14

 → p, z

(“radius equation”)  → r Coordinate Systems ig g

Ellipsoidal longitude and latitude

15 Coordinate Systems ig g

N: transverse radius of curvature (radius of prime vertical)

p b 16

a Sphere

N R N = p cos B

R Coordinate Systems ig g

K N: transverse radius of curvature (radius of prime vertical) e3

Surface normal

P 17

N N

K e2

K e1 Coordinate Systems ig g

K N: transverse radius of curvature (radius of prime vertical) e 3 M: meridional radius of curvature

P 18

N M

K e2

K e1 Coordinate Systems ig g

M: meridional radius of curvature

19 Coordinate Systems ig g

6400000 m c

6380000 m N

6360000 m 20 K e3 6340000 m

6320000 m 0 45 90 P

M N Equator North Pole N = semi-major axis a N = maximum

K e1 Coordinate Systems ig g

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Ellipsoidal coordinates Coordinate Systems ig g

Ellipsoidal (also called geographical) longitude and latitude

22 Coordinate Systems ig g

Exterior to the ellipsoid: Ellipsoidal longitude, latitude, and height

23 Coordinate Systems ig g

Ellipsoidal longitude, latitude, and height

24 Coordinate Systems ig g

Ellipsoidal longitude and latitude (“GPS-coordinates”) (e.g. ITRF)

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Always need to know • Ellipsoid parameter • Location & orientation of ellipsoid w.r.t. solid Earth = reference frame (e.g. ITRF) → lecture MGE-05 Google maps Coordinate Systems ig g

Ellipsoidal heights (GNSS) and physical heights

Ellipsoidal heights Physical heights - Geometrical reference surface - Reference surface = Geoid - GNSS-determined (i.e. from x,y,z) - Spirit levelling (+ gravity) 26

Geoid

Ellipsoid Coordinate Systems ig g

Ellipsoidal heights (GNSS) and physical heights: Geoid undulations

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Ellipsoid

Geoid Coordinate Systems ig g

Geoid (Geoid undulations = height of the Geoid above the Ellipsoid)

GFZ Potsdam

28 Coordinate Systems ig g

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Coordinate transformations

(for given ellipsoid parameters) Coordinate Systems ig g

Ellipsoidal coordinates → Cartesian coordinates

1. Compute N

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2. Compute x, y, z

or √ Coordinate Systems ig g

Cartesian coordinates → Ellipsoidal coordinates (I)

1. compute

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2. Since N depends on , computing latitude and height…

…requires iterative solution.

E.g. start values: Coordinate Systems ig g

Cartesian coordinates → Ellipsoidal coordinates (II)

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Terminate e.g. when |h[k+1]-h[k]|<

and R|[k+1]-[k]|<

(e.g. for =0.1 mm)

Typically 4-5 iterations sufficient. √