Do we need a new definition of gravity anomalies? · Do we need a new definition of gravity...

Institute of Geodesy

Torsten Mayer-Gürr 1

Do we need a new definition of gravity anomalies?

Torsten Mayer-Gürr and Christian Pock

WG Theoretical Geodesy and Satellite Geodesy Institute of Geodesy, NAWI Graz

Graz University of Technology



Goal: Geoid determination in Austria with cm accuracy

GARFIELD



Input data: Gravimetric observations

72.723 observations in and around Austria

Accuracy for most data: << 0.1 mGal



Fundamental equation of physical geodesy

“This error is [on a global average] on the order of ±20 cm in the geoidal height. This is one order of magnitude smaller than the accuracy implied by the present gravimetric and satellite data.”

Helmut Moritz (1980), Advanced physical geodesy: “This spherical approximation causes an error which is negligible in most practical applications.”

r

T

r

Tg 2

Linearized with spherical approximation

Connection

Is the fundamental equation of physical geodesy in linearized form (with spherical approximation) accurate enough to model the observations?



Observation equations

Linearization of non-linear observation equations

)()()( 00

0

xxx

fxfxf

x

ellipsoidal height = orthometric height + geoid height

)(WNHh

),,( hBLWx Unknown parameter Gravity potential

),,(0 hBLUx Taylor point Normal potential

),,(0 hBLTxx Difference Disturbance potential

),,()( hBLgxf Observed Observed absolute gravity

),,()( 0 hBLxf Computed Normal gravity

with 00 N



with



)()()( 00

0

xxx

fxfxf

x

),,( hBLWx

),,(0 hBLUx

),,(0 hBLTxx

Unknown parameter

Taylor point

Difference

Observed ),,()( hBLgxf

Gravity potential

Normal potential

Disturbance potential

Observed absolute gravity

Normal gravity Computed

00 N

)( 0

0

xxx

f

x

Linear model

r

T

r

T2

Fundamental equation in physical geodesy

(In spherical approximation)

)()( 0xfxf Reduced observations gNHBL

NHBLg

),,(

),,(

0

(Free air) gravity anomalies

),,()( 00 hBLxf

r

T

r

Tg 2





)()()( 00

0

xxx

fxfxf

x

x

)(xf

0x

)( 0xf

Possible improvements of accuracy:

1. Better linearization (without spherical approximation)





)()()( 00

0

xxx

fxfxf

x

x

)(xf

0x

)( 0xf



2. Inlcude quadratic terms





)()()( 00

0

xxx

fxfxf

x

x

)(xf

0x

)( 0xf



2. Include quadratic terms





)()()( 00

0

xxx

fxfxf

x

x

)(xf

0x

)( 0xf




3. Better Taylor point



Geoid Satellite model GOCO05s

Taylor point

Approximate values (Taylor point)

Classic ),,()( 00 NHBLxf

),,()( 0 satsat NHBLgxf New approach

with 00 N



Geoid Satellite model GOCO05s ),,(),,(),,( rZrVrW Gravity potential:

Taylor point

Approximate values (Taylor point)

Classic ),,()( 00 NHBLxf

),,()( 0 satsat NHBLgxf satsat W gNew approach

with 00 N

12

0

1

0

),(),(),,(max n

m

nmnmnmnm

nn

n

SsCcr

R

R

GMrV Gravitational potential:

sin2

1),( 2rrZ Centrifugal potential:

rW

rW

rW

zW

xW

xW

Wg

sinGravity:



Reduced observations

gg

Classic approach

satgg g

New approach




satggg

Classic approach

satgg g

New approach



satggg


Classic approach

satgg g

New approach

Difference




satggg

Classic approach

satgg g

New approach



Topography

Gravitational potential from topographic masses

)()(),(

1)( QQ

QP

P dl

GT rrrr

r

Satellite model includes topographic effect

Topography without satellite model part

N

n

topo

ntopotopo TTT0

Spherical harmonic expansion of topography

0

1

)()(n

n

nm

Pnm

topo

nm

n

Ptopo Yar

R

R

GMT rr

)()()(

'

)12(

1QQnmQ

n

topo

nm dYR

r

nMa rrr




satggg

Classic approach

satgg g

New approach




Classic approach New approach

toposat gggg toposatgg gg






Difference



Estimated Geoid



Residual geoid Residual geoid



Estimated Geoid



Residual geoid Residual geoid Difference



Summary

gg

Not accurate enough for geoid computation

satgg g

Generalized definition: Gravity anomalies = Observed absolute gravity – Computed gravity from an approximate model

Classical definition: Gravity anomalies = Observed absolute gravity - Normal gravity

Do we need a new definition of gravity anomalies? · Do we need a new definition of gravity...

Documents

Transcript of Do we need a new definition of gravity anomalies? · Do we need a new definition of gravity...