To apply the block-diagonal technique for generating and solving the normal equation
system, each data set has to be prepared in a special way, so that some conditions are
fulfilled (Colombo, 1981). They must be complete (globally defmed), equally spaced along
each parallel and uncorrelated with longitude independent weights. In addition the
coefficients of the spherical harmonic series must be arranged primarily with respect to
increasing order and secondary with respect to increasing degree. Generating normal
equations from such data sets a block-diagonal structure appears, which can easily be
solved order by order. There are no correlations between coefficients of different orders.
Terrestrial gravity anomalies are corrected for the attraction of the atmospheric and
topographic masses. The topographic correction is computed on a. global base using
spherical harmonic expansions of the TUG87 (Wieser M., 1987) topographic height data
set. Before, appropriate mean height values are generated from the original 5',0' cells
(30' or 1 degree depending on degree of expansion). The topographic masses above the
geoid are removed and then condensed on the geoid. The attraction of these shifted masses
then is computed for each block-mean value and applied as a correction to the observed
free-air mean anomalies. Where only 1 degree information is available, all 4 half degree
blocks are set to the 1 degree block value. Long wavelength effects, which are introduced
by this method are eliminated by a smoothing algorithm, applied to these blocks. Then
gaps in the mean anomalies data set are filled with gravity anomalies derived from the apriori gravity field spherical harmonic series (GRIM4-C4B). The satellite-only GRIM4-S4
solution was used as reference field and subtracted from mean gravity anomalies.
From the high resolution mean sea surface model, 30' x 30' block mean values are
generated. To take into account aliasing effects, which would happen when generating
normals from the original data set, a smoothing algorithm was applied to fit the data
resolution to the fmal gravity field resolution. A tidal correction term is added to remove
the permanent influences of sun and moon to the sea surface (permanent tidal correction)
(Rapp, 1992). This is necessary, because the shape of the mean sea surface adjusts to all
effective forces, whereas only the attraction forces of the Earth shall ble determined. Other
forces, like wind, which are not related to the gravity field are eliminated by using a long
period mean sea surface. For taking into account the permanent part of the sea surface
topography (deviation of the mean sea surface from the geoid), which is caused by ocean
currents, wind, salinity and others, a hydrodynamic model, which is nearly independent
from altimetry is used. The model is used with the assumption that the global mean value
of the permanent sea surface topography is zero. This centered model then is simply
subtracted from the corrected sea surface heights to get geoid heights above the reference
ellipsoid. Finally, before normal equations are generated, gaps are filled with geoid heights
from an intermediate high resolution gravity model, which is based on the gravity
anomalies data sets and the a-priori long wavelength solution.
All normal equation systems are added and the system is successively solved order by
order. Due to the relative small matrices to be inverted (maximum 361 columns and rows
for order 0), also the accuracy estimates for all coefficients can be delived. Mter adding
the coefficients of the reference potential, the final 360 gravity solution is available. For
the GFZ95A model a calibration of the error estimates was not performed. This means all
error estimates are determined only from the least squares approach, and can only be
regarded as internal error estimates.
64
system, each data set has to be prepared in a special way, so that some conditions are
fulfilled (Colombo, 1981). They must be complete (globally defmed), equally spaced along
each parallel and uncorrelated with longitude independent weights. In addition the
coefficients of the spherical harmonic series must be arranged primarily with respect to
increasing order and secondary with respect to increasing degree. Generating normal
equations from such data sets a block-diagonal structure appears, which can easily be
solved order by order. There are no correlations between coefficients of different orders.
Terrestrial gravity anomalies are corrected for the attraction of the atmospheric and
topographic masses. The topographic correction is computed on a. global base using
spherical harmonic expansions of the TUG87 (Wieser M., 1987) topographic height data
set. Before, appropriate mean height values are generated from the original 5',0' cells
(30' or 1 degree depending on degree of expansion). The topographic masses above the
geoid are removed and then condensed on the geoid. The attraction of these shifted masses
then is computed for each block-mean value and applied as a correction to the observed
free-air mean anomalies. Where only 1 degree information is available, all 4 half degree
blocks are set to the 1 degree block value. Long wavelength effects, which are introduced
by this method are eliminated by a smoothing algorithm, applied to these blocks. Then
gaps in the mean anomalies data set are filled with gravity anomalies derived from the apriori gravity field spherical harmonic series (GRIM4-C4B). The satellite-only GRIM4-S4
solution was used as reference field and subtracted from mean gravity anomalies.
From the high resolution mean sea surface model, 30' x 30' block mean values are
generated. To take into account aliasing effects, which would happen when generating
normals from the original data set, a smoothing algorithm was applied to fit the data
resolution to the fmal gravity field resolution. A tidal correction term is added to remove
the permanent influences of sun and moon to the sea surface (permanent tidal correction)
(Rapp, 1992). This is necessary, because the shape of the mean sea surface adjusts to all
effective forces, whereas only the attraction forces of the Earth shall ble determined. Other
forces, like wind, which are not related to the gravity field are eliminated by using a long
period mean sea surface. For taking into account the permanent part of the sea surface
topography (deviation of the mean sea surface from the geoid), which is caused by ocean
currents, wind, salinity and others, a hydrodynamic model, which is nearly independent
from altimetry is used. The model is used with the assumption that the global mean value
of the permanent sea surface topography is zero. This centered model then is simply
subtracted from the corrected sea surface heights to get geoid heights above the reference
ellipsoid. Finally, before normal equations are generated, gaps are filled with geoid heights
from an intermediate high resolution gravity model, which is based on the gravity
anomalies data sets and the a-priori long wavelength solution.
All normal equation systems are added and the system is successively solved order by
order. Due to the relative small matrices to be inverted (maximum 361 columns and rows
for order 0), also the accuracy estimates for all coefficients can be delived. Mter adding
the coefficients of the reference potential, the final 360 gravity solution is available. For
the GFZ95A model a calibration of the error estimates was not performed. This means all
error estimates are determined only from the least squares approach, and can only be
regarded as internal error estimates.
64