Height systems

Rudi Gens Alaska Satellite Facility Outline

y Why bother about height systems? y Relevant terms y Coordinate systems y Reference surfaces

Height systems Height y Geopotential number y Height systems

2 Why bother about height systems?

y give a meaning to a value defined for height y combination of measurements from different sources x GPS measurements vs. leveling measurements y three-dimensional calculations Height systems Height x SAR interferometry

3 Relevant terms

y spheriod x any surface resembling a sphere x an ellipsoid of revolution y ellipsoid x defined by axes, flattening and eccentricity

Height systems Height y flattening and eccentricity x characterize the deviation from a sphere

4 Geographical and geodetic coordinates

f

b

a Height systems Height Geographic Geodetic latitude latitude

5 Geographical and geodetic coordinates y geographical coordinates x implying spherical Earth model y geodetic coordinates x implying ellipsoidal Earth model Height systems Height

6 Cartesian coordinates

y geodetic coordinates inappropriate for satellite imagery Æ cartesian coordinates Height systems Height

7 Approximation vs. Reality

y ellipsoid is a good approximation to the shape of the Earth but not an exact representation y Earth surface is everywhere perpendicular to the direction of gravity Æ equipotential surface

Height systems Height y true shape of the Earth is known as geoid

8 Reference surfaces

Height systems Height y three reference surfaces x topography x geoid x ellipsoid

9 Reference surfaces

Height systems Height y topography represents the physical surface of the Earth

10 Reference surfaces

Height systems Height y geoid defined as level surface of gravity field with best fit to mean sea level x maximum difference between geoid and mean sea level about 1 m

11 Reference surfaces

Height systems Height y ellipsoid defines mathematical surface approximating the physical reality while simplifying the geometry

12 Reference surfaces

Height systems Height y geoid undulation: vertical separation between geoid and reference ellipsoid x differences between ± 100 m x global root mean square of around 30 m

13 Reference surfaces

Height systems Height y vertical deflection: angle between the ellipsoid normal and the plumb line x usually resolved in a north-south component ξ and an east-west component η x angles usually amount to a few arc seconds

14 Global earth model

y geoid defined by a set of coefficients of a spherical harmonic expansion Æ global earth model y several models available x OSU91

Height systems Height x Earth Geopotential Model 1996 (EGM96)

15 Geopotential number

y different height systems can be related to each other by the geopotential number C = − = point C W0 W g dn ∫geoid

x W and W0: the potentials of gravity of a point and Height systems Height the geoid x g: gravity value x dn: leveling increment

16 Geopotential number

y different heights calculated by dividing the geopotential number by a gravity value Height systems Height

17 Heights

y dynamic height γ x constant normal gravity 0 for an arbitrary standard latitude (usually 45 degrees) x no geometrical meaning y orthometric height x natural “height above sea level” Height systems Height x measured along the current plumb line from the foot point on the geoid and the point on the surface x gravity value: mean gravity

18 Heights

y normal height x vertical distance from terrain surface to the ellipsoid reduced by the height anomaly x measured along the ellipsoidal normal x gravity value: mean normal gravity Height systems Height

19 Solution

y ellipsoid is convenient reference frame x mathematical figure x provides good approximation to the geoid y geoid better height reference system x reference to mean sea level allows to use tide gauges as height reference points Height systems Height x physical significance: ensures horizontal representation of water surfaces such lakes and seas

20 Questions Datums

y datum x describes the relationship between a particular local ellipsoid and a global geodetic reference system y coordinate system x shape and size given by the ellipsoid x position given by the fixing of the origin Height systems Height x fixing of the origin defines a datum

22 Datums

y global datums (geocentric) x Geodetic Reference System 1980 (GRS 80) x World Geodetic System 1984 (WGS 84) y local datums x North American 1927 Height systems Height

23