Fundamental Equation of Physical Geodesy

Home , Physical geodesy

Institute of Geodesy

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

Torsten Mayer-Gürr 1 Institute of Geodesy Goal: Geoid determination in Austria with cm accuracy

GARFIELD

Torsten Mayer-Gürr 2 Institute of Geodesy Input data: Gravimetric observations

72.723 observations in and around Austria

Accuracy for most data: << 0.1 mGal

Torsten Mayer-Gürr 3 Institute of Geodesy Fundamental equation of physical geodesy

Connection T T g   2 r r

Linearized with spherical approximation

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

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

Torsten Mayer-Gürr 4 Institute of Geodesy Observation equations Linearization of non-linear observation equations f xfxf  xx )()()(   0 x 0 x0

Unknown parameter  hBLWx ),,( Gravity potential

Taylor point 0  hBLUx ),,( Normal potential with N0  0  hBLTxx ),,( Difference 0 Disturbance potential

Observed  hBLgxf ),,()( Observed absolute gravity Normal gravity Computed 0   hBLxf ),,()(

ellipsoidal height = orthometric height + geoid height  WNHh )(

Torsten Mayer-Gürr 5 Institute of Geodesy Observation equations Linearization of non-linear observation equations f T T xfxf  xx )()()(   g   2 0 x 0 r r x0

Unknown parameter  hBLWx ),,( Gravity potential

Taylor point 0  hBLUx ),,( Normal potential with N0  0  hBLTxx ),,( Difference 0 Disturbance potential

Observed  hBLgxf ),,()( Observed absolute gravity Normal gravity Computed 0   hBLxf 0 ),,()(

Reduced  xfxf 0 )()(   NHBLg ),,( (Free air) gravity anomalies observations   0 ),,(  gNHBL f T T  xx )(   2 Fundamental equation in physical geodesy Linear model x 0 r r x0 (In spherical approximation)

Torsten Mayer-Gürr 6 Institute of Geodesy Observation equations Linearization of non-linear observation equations f xfxf  xx )()()(   0 x 0 x0

xf )( Possible improvements of accuracy:

1. Better linearization (without spherical approximation)

xf 0 )(

x0 x

Torsten Mayer-Gürr 7 Institute of Geodesy Observation equations Linearization of non-linear observation equations f xfxf  xx )()()(   0 x 0 x0

xf )( Possible improvements of accuracy:

1. Better linearization (without spherical approximation) xf )( 0 2. Inlcude quadratic terms

x0 x

Torsten Mayer-Gürr 8 Institute of Geodesy Observation equations Linearization of non-linear observation equations f xfxf  xx )()()(   0 x 0 x0

xf )( Possible improvements of accuracy:

1. Better linearization (without spherical approximation) xf )( 0 2. Include quadratic terms

x0 x

Torsten Mayer-Gürr 9 Institute of Geodesy Observation equations Linearization of non-linear observation equations f xfxf  xx )()()(   0 x 0 x0

xf )( Possible improvements of accuracy:

1. Better linearization (without spherical approximation) xf )( 0 2. Include quadratic terms

3. Better Taylor point

x0 x

Torsten Mayer-Gürr 10 Institute of Geodesy Observation equations Linearization of non-linear observation equations f xfxf  xx )()()(   0 x 0 x0

xf )( Possible improvements of accuracy:

1. Better linearization xf 0 )( (without spherical approximation)

2. Include quadratic terms

3. Better Taylor point

x0 x

Torsten Mayer-Gürr 11 Institute of Geodesy Taylor point Approximate values (Taylor point)

Classic 0    NHBLxf 0 ),,()( with N0  0

New approach 0  sat  NHBLgxf sat ),,()(

Geoid Satellite model GOCO05s

Torsten Mayer-Gürr 12 Institute of Geodesy Taylor point Approximate values (Taylor point)

Classic 0    NHBLxf 0 ),,()( with N0  0

New approach 0  sat  NHBLgxf sat ),,()( g sat  Wsat

Gravity potential:    rZrVrW ),,(),,(),,( Geoid Satellite model GOCO05s n1 GM nmax  R  n12 Gravitational potential:  rV ),,(     Cc nmnm   Ss nmnm  ),(),( R n0  r  m0 1 Centrifugal potential: ),(  2rrZ sin 2

 xW   rW        Gravity: Wg   xW    rW sin         zW    rW 

Torsten Mayer-Gürr 13 Institute of Geodesy Reduced observations

Classic approach New approach

gg   gg  g sat

Torsten Mayer-Gürr 14 Institute of Geodesy Reduced observations

Classic approach New approach

  ggg sat gg  g sat

Torsten Mayer-Gürr 15 Institute of Geodesy Reduced observations

Classic approach New approach

  ggg sat gg  g sat

Difference

Torsten Mayer-Gürr 16 Institute of Geodesy Reduced observations

Classic approach New approach

  ggg sat gg  g sat

Torsten Mayer-Gürr 17 Institute of Geodesy Topography

Gravitational potential from topographic masses 1 rP )(  GT   Q d rr Q )()(  l rr QP ),(

Satellite model includes topographic effect

 Spherical harmonic expansion of topography

 n1 n GM  R  topo T rPtopo )(     nm Ya rPnm )( R n0  r   nm

n topo 1  r'  a   dY  rrr )()()( nm   QnmQ Q nM  )12(   R 

Topography without satellite model part

N topo  topo topo  TTT n n0

Torsten Mayer-Gürr 18 Institute of Geodesy Reduced observations

Classic approach New approach

  ggg sat gg  g sat

Torsten Mayer-Gürr 19 Institute of Geodesy Reduced observations

Classic approach New approach