is merged together in the upper left, while the block-diagonal parts form sparse matrices
in the off-diagonal parts of the system. This system can be solved by block-matrices
techniques. By dividing the system into 9 sub-matrices we get on the upper left a full
system, in the middle and on the lower right block-diagonal systems and on the middle left
a sparse matrix. Four matrices are completely filled with zero elements. The system then
can be solved step by step as it is described in detail in Bosch (1993).
An even more efficient solution for the normal equation system is described in Stinkel
(1995). In this study the solution strategies for a combination of satellite-to-satellite
tracking (SST) and satellite gravity gradiometry (SGG) spherical harmonic series normal
equations are investigated. Because the SST normal equation system is represented by a
full matrix, and the SGG system is strictly block-diagonal for an idealized homogeneous
and full coverage of the sphere, the same problem has to be solved. In this study a new
ordering scheme for normal equations is described, which avoids completely the fill-in
during the factorization for the case of an ideal data distribution based on gridded data.
This numbering scheme simply is the reverse of the numbering scheme shown in figure
8. This means that by mirroring the normal equation system in figure 8 at a line from the
lower left to the upper right, we get a normal equation system, which can be solved in a
easy way, even for very large systems.
CONCLUSIONS
A new high resolution gravity field model GFZ95A, complete to degree and order 360 was
computed. The model is based on a combination of the long-wavelength GRIM4-C4B
solution, of half and one degree surface gravity data and of a ERS-l mean sea surface
model. The combination is done by an iterative least squares harmonic analysis approach,
as it was also used for the previous solutions. For GFZ95A an improved weighting scheme
for the relative weighting of the different data sets was introduced. Also completely new
data sets, with respect to the previous solutions, were used. Over land newly available data
over South America and CIS (Central Asia) are included instead of predicted gravity
anomalies. Over the oceans a new mean sea surface, based on 2 years of ERS-l altimeter
data is used. For reducing the sea surface heights to geoid heights, an improved dynamic
topography model, based on the Levitus model is taken. For this model 3 years of Geosat
altimeter data were used to perform an eddy statistic in ocean areas with strong mesoscale
variability. With respect to the original Levitus climatological dynamic topography, the
new model provides better results in regions with large ocean variability.
Different comparisons of the new model with the OSU91A model are performed. For the
lower frequencies the new GFZ95A model shows slightly better results than OSU91A. The
complete model is tested by geoid height differences on Doppler/GPS stations, by geoid
height differences between the models and by analyzing the power spectra. The largest
differences occur in regions, where new data is available (South America, CIS) and in
polar regions. The power spectrum of GFZ95A is above the theoretical Kaula spectrum for
degrees up to 180 and below for the higher degrees. The error power spectrum shows that
all data sets in the GFZ95A model are proper weighted, while in OSU91A a jump between
the long-wavelength a-priori model and the new estimated coefficients occurs.
Finally a new strategy for combination of full normal equations and block-diagonal
69
in the off-diagonal parts of the system. This system can be solved by block-matrices
techniques. By dividing the system into 9 sub-matrices we get on the upper left a full
system, in the middle and on the lower right block-diagonal systems and on the middle left
a sparse matrix. Four matrices are completely filled with zero elements. The system then
can be solved step by step as it is described in detail in Bosch (1993).
An even more efficient solution for the normal equation system is described in Stinkel
(1995). In this study the solution strategies for a combination of satellite-to-satellite
tracking (SST) and satellite gravity gradiometry (SGG) spherical harmonic series normal
equations are investigated. Because the SST normal equation system is represented by a
full matrix, and the SGG system is strictly block-diagonal for an idealized homogeneous
and full coverage of the sphere, the same problem has to be solved. In this study a new
ordering scheme for normal equations is described, which avoids completely the fill-in
during the factorization for the case of an ideal data distribution based on gridded data.
This numbering scheme simply is the reverse of the numbering scheme shown in figure
8. This means that by mirroring the normal equation system in figure 8 at a line from the
lower left to the upper right, we get a normal equation system, which can be solved in a
easy way, even for very large systems.
CONCLUSIONS
A new high resolution gravity field model GFZ95A, complete to degree and order 360 was
computed. The model is based on a combination of the long-wavelength GRIM4-C4B
solution, of half and one degree surface gravity data and of a ERS-l mean sea surface
model. The combination is done by an iterative least squares harmonic analysis approach,
as it was also used for the previous solutions. For GFZ95A an improved weighting scheme
for the relative weighting of the different data sets was introduced. Also completely new
data sets, with respect to the previous solutions, were used. Over land newly available data
over South America and CIS (Central Asia) are included instead of predicted gravity
anomalies. Over the oceans a new mean sea surface, based on 2 years of ERS-l altimeter
data is used. For reducing the sea surface heights to geoid heights, an improved dynamic
topography model, based on the Levitus model is taken. For this model 3 years of Geosat
altimeter data were used to perform an eddy statistic in ocean areas with strong mesoscale
variability. With respect to the original Levitus climatological dynamic topography, the
new model provides better results in regions with large ocean variability.
Different comparisons of the new model with the OSU91A model are performed. For the
lower frequencies the new GFZ95A model shows slightly better results than OSU91A. The
complete model is tested by geoid height differences on Doppler/GPS stations, by geoid
height differences between the models and by analyzing the power spectra. The largest
differences occur in regions, where new data is available (South America, CIS) and in
polar regions. The power spectrum of GFZ95A is above the theoretical Kaula spectrum for
degrees up to 180 and below for the higher degrees. The error power spectrum shows that
all data sets in the GFZ95A model are proper weighted, while in OSU91A a jump between
the long-wavelength a-priori model and the new estimated coefficients occurs.
Finally a new strategy for combination of full normal equations and block-diagonal
69