do we need a new definition of gravity anomalies? · do we need a new definition of gravity...
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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
Institute of Geodesy
Torsten Mayer-Gürr 2
Goal: Geoid determination in Austria with cm accuracy
GARFIELD
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Input data: Gravimetric observations
72.723 observations in and around Austria
Accuracy for most data: << 0.1 mGal
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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?
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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
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with
Observation equations
Linearization of non-linear observation equations
)()()( 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
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Observation equations
Linearization of non-linear observation equations
)()()( 00
0
xxx
fxfxf
x
x
)(xf
0x
)( 0xf
Possible improvements of accuracy:
1. Better linearization (without spherical approximation)
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Observation equations
Linearization of non-linear observation equations
)()()( 00
0
xxx
fxfxf
x
x
)(xf
0x
)( 0xf
Possible improvements of accuracy:
1. Better linearization (without spherical approximation)
2. Inlcude quadratic terms
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Observation equations
Linearization of non-linear observation equations
)()()( 00
0
xxx
fxfxf
x
x
)(xf
0x
)( 0xf
Possible improvements of accuracy:
1. Better linearization (without spherical approximation)
2. Include quadratic terms
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Observation equations
Linearization of non-linear observation equations
)()()( 00
0
xxx
fxfxf
x
x
)(xf
0x
)( 0xf
Possible improvements of accuracy:
1. Better linearization (without spherical approximation)
2. Include quadratic terms
3. Better Taylor point
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Observation equations
Linearization of non-linear observation equations
)()()( 00
0
xxx
fxfxf
x
x
)(xf
0x
)( 0xf
Possible improvements of accuracy:
1. Better linearization (without spherical approximation)
2. Include quadratic terms
3. Better Taylor point
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Geoid Satellite model GOCO05s
Taylor point
Approximate values (Taylor point)
Classic ),,()( 00 NHBLxf
),,()( 0 satsat NHBLgxf New approach
with 00 N
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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:
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Reduced observations
gg
Classic approach
satgg g
New approach
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Reduced observations
satggg
Classic approach
satgg g
New approach
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satggg
Reduced observations
Classic approach
satgg g
New approach
Difference
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Reduced observations
satggg
Classic approach
satgg g
New approach
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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
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Reduced observations
satggg
Classic approach
satgg g
New approach
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Reduced observations
Classic approach New approach
toposat gggg toposatgg gg
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Reduced observations
Classic approach New approach
toposat gggg toposatgg gg
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Reduced observations
Classic approach New approach
toposat gggg toposatgg gg
Difference
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Reduced observations
Classic approach New approach
toposat gggg toposatgg gg
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Estimated Geoid
Classic approach New approach
toposat gggg toposatgg gg
Residual geoid Residual geoid
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Estimated Geoid
Classic approach New approach
toposat gggg toposatgg gg
Residual geoid Residual geoid Difference
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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