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, 2013-10-25 1

Meridian planehrb

a

Z

X-Yφ

Coordinate-systems

Example:Frederiksværk φ=560, λ=120, h= 50 m

C.C.Tscherning, 2013-10-25.

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, 2013-10-25.

Ellipsoid

Earth surface

NH Geoid: gravity potential constant

4

GEOID

5

Coordinate-systems and time.

NON INERTIAL SYSTEM

CTS:Conventional

Terrestrial System

Mean-rotationaxis1900.

Greenwich

X

Y- Rotates withthe Earth

Z

Gravity-centre

6

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.

7

POLBEVÆGELSEN

• .

http://aiuws.unibe.ch/code/erp_pp.gif

8

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

2rmMkF

C.C.Tscherning, 2013-10-25.

22 rrkMa

Gradient of scalar potential, V,

C.C.Tscherning, 2013-10-25.

mass-point),,(,rMkzyxV

zVyVxV

Va

Volume distribution, ρ(x,y,z)

• V fulfills Laplace equation

dr

k

dzdydxzzyyxx

zyxk

zyxV

Earth

Earth

222 )'()'()'(

),,()',','(

C.C.Tscherning, 2013-10-25.

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, 2013-10-25.

drddrdxdydzd

rzryrx

2cos :measure-Volume

sinsincoscoscos

Laplace in spherical coordinates

C.C.Tscherning, 2013-10-25.

order. m degree, n ),(sin)( sinor cos)( ,or )(

),()()( :Solutioncos

1

)(coscos

11

1

2

2

2

22

nm

nn

PmmrrrR

rRV

V

VrVr

rrV

Spherical harmonics

• Define:

C.C.Tscherning, 2013-10-25.

n

nmnmnm

n

mnn

n

nm

rVCkMrV

mmmm

PrarV

),,(),,(

then0,||sin

0,cos)(sin),,(

0

||1

Orthogonal basis functions

• Generalizes Fourier-series from the plane

C.C.Tscherning, 2013-10-25.

nmnm

nm

ijnm

PP

jminkMC

ddrVrV

and CC

thenmalized,(fully)nor bemay Functionsor for 0

jm i,nfor

cos),,(),,(

nmnm

180

0

90

90

16

Gravity model database.

Spherical harmonic coefficients:http://icgem.gfz-potsdam.de/ICGEM/

CCT, Nov. 2013. (CCT)

Centrifugal potential

• On the surface of the Earth we also measure the centrifugál acceleration,

space). inertial(in velocity rotational

)(21

cos21

222

222

yxV

rVW

C.C.Tscherning, 2013-10-25.

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 ,/

......)72

231(3/)2/1(

32

cos2

....)(sin)()(sin)(1

2

2

222

404

4202

2

egam

fmmffJ

rPraJP

raJU

C.C.Tscherning, 2013-10-25.

Anomalous potential,T• T=W-U, • same mass and gravity center.• Makes all quantities small,gives base for linearisation.

C.C.Tscherning, 2013-10-25.

ocean. on theanomaly height height Geoid/ :anomalyHeight

.height point within gravity normalr2- :anomalyGravity

.height lellipsoida with P,in gravity normal minusgravity

:edisturbancGravity

TH

TrTgg

hrTgg

QP

PP

20

Gravity.

Source: DTU-Space. Ole Andersen.

CCT, Nov. 2013. (CCT)