Transcript
Satellite geodesy. Basic concepts. I1.1a
Z
b
a
r
Meridian plane
h
φ
X-Y
= geocentric latitude
φ = geodetic latitude
r = radial distance, h = ellipsoidal height
a = semi-major axis, b = semi-minor axis
z = axis of rotation, 1900. flattening = (a-b)/a.
C.C.Tscherning, University of Copenhagen, 2007-10-25
1
Coordinate-systems
Example:
Frederiksværk
φ=560, λ=120, h= 50 m
C.C.Tscherning, 2007-10-25.
Geoid and mean sea level
H
N
Earth surface
Geoid: gravity potential constant
Ellipsoid
h=H+N=Orthometric height + geoid height
along plumb-line
=HN+ζ=Normal height + height anomaly,
along plumb-line of gravity normal field
C.C.Tscherning, 2007-10-25.
Gravity potential, Kaula Chap. 1.
• Attraction (force):
mM
F k 2
r
• Direction from gravity center of m to M.
• With m = 1 (unitless), then acceleration
kM
a 2 2
r
r
C.C.Tscherning, 2007-10-25.
Gradient of scalar potential, V,
V
x
V
kM
a V , V ( x, y, z )
point - mass
r
y
V
z
C.C.Tscherning, 2007-10-25.
Volume distribution, ρ(x,y,z)
V ( x' , y ' , z ' )
( x, y , z )
k
( x x' ) 2 ( y y ' ) 2 ( z z ' ) 2
Earth
dx dy dz
k d
r
Earth
• V fulfills Laplace equation
2
2
2
V 2 V 2 V 2 V 0, outside masses
x
y
z
C.C.Tscherning, 2007-10-25.
Spherical coordinates
• Geocentric latitude
• Longitude, λ, r = distance to origin.
x r cos cos
y r cos sin
z r sin
Volume - measure : d dxdydz cos r 2 d ddr
C.C.Tscherning, 2007-10-25.
Laplace in spherical coordinates
V
1
2 V
)
(cos
r
r r cos
2
1 V
cos 2 2
Solution : V R(r ) ( ) ( ),
1
V 2
r
or r , ( ) cos m or sin m
R(r ) r
( ) Pnm (sin ), n degree, m order.
n 1
n
C.C.Tscherning, 2007-10-25.
Spherical harmonics
• Define:
cos m , m 0
an
Vnm (r , , ) n 1 Pn|m| (sin )
r
sin | m | , m 0
then
V (r , , ) kM
n 0
n
C
m n
V (r , , )
nm nm
C.C.Tscherning, 2007-10-25.
Orthogonal basis functions
• Generalizes Fourier-series from the plane
90 180
V
nm
90
(r , , )Vij (r , , ) cos d d
0
kMCnm for n i, m j
for n i or m j
0
Functions may be (fully)nor malized, then
C nm Cnm and Pnm Pnm
C.C.Tscherning, 2007-10-25.
Centrifugal potential
• On the surface of the Earth we also measure the
centrifugál acceleration,
1 2 2
2
W V r cos
r
2
1 2 2
2
V (x y )
2
rotational velocity (in inertial space).
C.C.Tscherning, 2007-10-25.
Normal potential, U
•
•
•
•
Good approximation to potential of ideal Earth
Reference ellipsoid is equipotential surface, U=U0, ideal geoid.
It has correct total mass, M.
It has correct centrifugal potential
2
a 2
a 4
2
U 1 J 2 ( ) P20 (sin ) J 4 ( ) P40 (sin ) ....
r cos 2
r
r
2
2
3
2
J 2 f (1 f / 2) m / 3(1 m f ) ......
3
2
7
m 2 a / g e , (gravity at Equator)
• Knowledge of the series development of the gravity potential
can be used to derive the flattening of the Earth !
C.C.Tscherning, 2007-10-25.
Satellite Geodesy: distances or ranges
• If we know the orbit of the satellite and
measure distances (or distance differences) to
the satellite, then we may determine our own
position: GPS, Doppler.
S2
S1
Us
C.C.Tscherning, 2007-10-25.
S3
Satellite Geodesy: ranging to the Earth
• Known orbit. One or two way ranging.
Envisat
• Satellite altimetry
Ocean or ice
ERS1
• PRARE
• Synthetic Aperture Radar
C.C.Tscherning, 2007-10-25.
Satellite Geodesy: from ground to satelite
• Laser ranging
• Optical directions
• Enables orbit determination
• Enables gravity potential estimation
C.C.Tscherning, 2007-10-25.
LAGEOS
Satellite to satellite tracking
• We measure the position of a satellite from
other satellites: SST
• Enables calculation of position and velocity
• Enables calculation of Kinetic Energy
S2
S1
S4
C.C.Tscherning, 2007-10-25.
S3
Satellite to satellite tracking, Low-Low
• We measure range rates between two satellites
• Gives (approximately) difference in Kinetic
energy
S1
GRACE
S2
S4
Range-rate
C.C.Tscherning, 2007-10-25.
S3
S4
SST and gradiometry
• Measurement of acceleration differences inside
satellite (GOCE).
• Determines second-order derivatives of
potential.
Mass1
Mass2
C.C.Tscherning, 2007-10-25.
Task of Geodesy: to determine
• Points on the surface of the Earth (and Planets) and
changes (geodynamics, climate changes, sea-level)
• Geoid/quasi-geoid determines state of sea-level if no
currents – tides – salinity or temperature variations
• Determines tides, (geostrophic) currents etc. If geoid
known
• Position of the Earth in Inertial system, Earth
rotation, pole-position, Earth elastic parameters.
• Changes of gravity (due to crustal uplift or
hydrosphere changes)
C.C.Tscherning, 2007-10-25.