Download Height systems

Height systems
Rudi Gens
Alaska Satellite Facility
Outline
y Why bother about height systems?
Height systems
y Relevant terms
y Coordinate systems
y Reference surfaces
y Geopotential number
y Height systems
2
Why bother about height systems?
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
x SAR interferometry
3
Relevant terms
y spheriod
Height systems
x any surface resembling a sphere
x an ellipsoid of revolution
y ellipsoid
x defined by axes, flattening and eccentricity
y flattening and eccentricity
x characterize the deviation from a sphere
4
Geographical and geodetic
coordinates
Height systems
f
b
a
Geographic
latitude
Geodetic
latitude
5
Geographical and geodetic
coordinates
y geographical coordinates
Height systems
x implying spherical Earth model
y geodetic coordinates
x implying ellipsoidal Earth model
6
Height systems
Cartesian coordinates
y geodetic coordinates
inappropriate for
satellite imagery
Æ cartesian
coordinates
7
Approximation vs. Reality
Height systems
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
y true shape of the Earth is known as geoid
8
Height systems
Reference surfaces
y three reference surfaces
x topography
x geoid
x ellipsoid
9
Height systems
Reference surfaces
y topography represents the physical surface
of the Earth
10
Height systems
Reference surfaces
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
Height systems
Reference surfaces
y ellipsoid defines mathematical surface
approximating the physical reality while
simplifying the geometry
12
Height systems
Reference surfaces
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
Height systems
Reference surfaces
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
Height systems
y geoid defined by a set of coefficients of a
spherical harmonic expansion
Æ global earth model
y several models available
x OSU91
x Earth Geopotential Model 1996 (EGM96)
15
Geopotential number
Height systems
y different height systems can be related to each
other by the geopotential number C
C = W0 − W =
∫
point
geoid
g dn
x W and W0: the potentials of gravity of a point and
the geoid
x g: gravity value
x dn: leveling increment
16
Geopotential number
Height systems
y different heights calculated by dividing the
geopotential number by a gravity value
17
Heights
Height systems
y dynamic height
x constant normal gravity γ0 for an arbitrary standard
latitude (usually 45 degrees)
x no geometrical meaning
y orthometric height
x natural “height above sea level”
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
Height systems
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
19
Solution
y ellipsoid is convenient reference frame
Height systems
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
x physical significance: ensures horizontal
representation of water surfaces such lakes and
seas
20
Questions
Datums
y datum
Height systems
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
x fixing of the origin defines a datum
22
Datums
y global datums (geocentric)
Height systems
x Geodetic Reference System 1980 (GRS 80)
x World Geodetic System 1984 (WGS 84)
y local datums
x North American 1927
23