satellite geophysics. basic concepts. i1.1a
DESCRIPTION
Satellite geophysics. Basic concepts. I1.1a. Z. Meridian plane. h. r. = 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. b. φ. X-Y. a. - PowerPoint PPT PresentationTRANSCRIPT
Satellite geophysics. Basic concepts. I1.1a
= 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, 2011-10-25 1
Meridian planehrb
a
Z
X-Yφ
Coordinate-systems
Example:
Frederiksværk
φ=560, λ=120, h= 50 m
C.C.Tscherning, 2011-10-25.
h=H+N=Orthometric height + geoid height
along plumb-line
=HN+ζ=Normal height + height anomaly,
along plumb-line of gravity normal field
Geoid and mean sea level
C.C.Tscherning, 2011-10-25.
Ellipsoid
Earth surface
N
HGeoid: gravity potential constant
4
Coordinate-systems and time.
NON INERTIAL SYSTEM
CTS:Conventional
Terrestrial System
Mean-rotationaxis1900.
Greenwich
X
Y- Rotates withthe Earth
Z
Gravity-centre
5
POLAR MOTION
• Approximatively circular
• Period 430 days (Chandler period)
• Main reason: Axis of Inertia does not co-inside with axis of rotation.
• Rigid Earth: 305 days: Euler-period.
6
POLBEVÆGELSEN
• .
• http://aiuws.unibe.ch/code/erp_pp.gif
7
Ch. 3, Transformation CIS - CTS
• Precession• Nutation• Rotation+• Polar movement
Sun+Moon
CISCTS rSNPr
Gravity potential, Kaula Chap. 1.
• Attraction (force):
• Direction from gravity center of m to M.• With m = 1 (unitless), then acceleration
2r
mMkF
C.C.Tscherning, 2011-10-25.
22 rr
kMa
Gradient of scalar potential, V,
C.C.Tscherning, 2011-10-25.
mass-point),,(,r
MkzyxV
z
Vy
Vx
V
Va
Volume distribution, ρ(x,y,z)
• V fulfills Laplace equation
dr
k
dzdydxzzyyxx
zyxk
zyxV
Earth
Earth
222 )'()'()'(
),,(
)',','(
C.C.Tscherning, 2011-10-25.
masses outside,02
2
2
2
2
2
Vz
Vy
Vx
V
Spherical coordinates
• Geocentric latitude• Longitude, λ, r = distance to origin.
C.C.Tscherning, 2011-10-25.
drddrdxdydzd
rz
ry
rx
2cos :measure-Volume
sin
sincos
coscos
Laplace in spherical coordinates
C.C.Tscherning, 2011-10-25.
order. m degree, n ),(sin)(
sinor cos)( ,or )(
),()()( :Solution
cos
1
)(coscos
11
1
2
2
2
22
nm
nn
P
mmrrrR
rRV
V
V
r
Vrrr
V
Spherical harmonics
• Define:
C.C.Tscherning, 2011-10-25.
n
nmnmnm
n
mnn
n
nm
rVCkMrV
mm
mmP
r
arV
),,(),,(
then
0,||sin
0,cos)(sin),,(
0
||1
Orthogonal basis functions
• Generalizes Fourier-series from the plane
C.C.Tscherning, 2011-10-25.
nmnm
nm
ijnm
PP
jmin
kMC
ddrVrV
and CC
thenmalized,(fully)nor bemay Functions
or for 0
jm i,nfor
cos),,(),,(
nmnm
180
0
90
90
Centrifugal potential
• On the surface of the Earth we also measure the centrifugál acceleration,
space). inertial(in velocity rotational
)(2
1
cos2
1
222
222
yxV
rVW
C.C.Tscherning, 2011-10-25.
r
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
• Knowledge of the series development of the gravity potential can be used to derive the flattening of the Earth !
Equator)at (gravity ,/
......)7
2
2
31(3/)2/1(
3
2
cos2
....)(sin)()(sin)(1
2
2
222
404
4202
2
egam
fmmffJ
rPr
aJP
r
aJU
C.C.Tscherning, 2011-10-25.
Anomalous potential,T• T=W-U,
• same mass and gravity center.
• Makes all quantities small,gives base for linearisation.
C.C.Tscherning, 2011-10-25.
ocean. on theanomaly height height Geoid
/ :anomalyHeight
.height point within gravity normalr
2- :anomalyGravity
.height lellipsoida with P,in gravity normal minusgravity
:edisturbancGravity
T
H
Tr
Tgg
hr
Tgg
QP
PP