The results presented so far indicate that the two techniques (NQ and BD) yield solutions
(V022 and HDM036) which appear to be of the same quality when compared to certain
independent data. Comparison of V022 to HDM036 in terms of 30' mean values of geoid
undulations (to Nmax.=360) yields 0.20 m global std. deviation difference (0.33 mover
land and 0.10 m over ocean cells). The extreme differences are 5.2 m (land) and 2.9 m
(ocean) and occur in areas of poor gravity anomaly data.
It is instructive to compare the coefficients of V022 and HDM036, to the coefficients
obtained from a harmonic analysis (NQ) of the unadjusted global 30' ~g file (input). In
terms of percentage differences (see Rapp, 1986, eq. 53), this comparison is shown in
Fig. 1. As expected, V022 above degree 70 is practically identical to the harmonics
obtained from the unadjusted ~g. In contrast, due to the existing correlations between low
and high degree harmonics of a given order, HDM036 above degree 70 differs
substantially from the harmonics of the unadjusted ~g. These differences represent the
influence of the satellite information on the higher-degree terms. This information passes
to the high-degree coefficients through the correlations that are represented in the "wing"
part of the normals.
Fig. 2 shows the anomaly degree variances implied by V022, HDM036, their difference
as well as the errors associated with each model. The difference between the two
solutions is roughly an order of magnitude smaller than the error variance of either model
for the same degree. One should also notice the excellent agreement between the error
variances implied by the two models. This is rather remarkable since the error variances
of V022, above degree 70, are computed (sampling error part) based on a semi-empirical
formula devised by Jekeli (Colombo, 1981). The signal degree variances of HDM036
imply a rougher field above degree approximately 180, as compared to V022. Relative
undulation comparisons over GPS/Leveling traverses (see also Rapp and Pavlis, 1990,
Table 7) show V022 to be in slightly better agreement with the independent data than
HDM036. These comparisons are most sensitive to the high-degree part of the field and
may indicate a problem with the BD solutions related to aliased power (above degree
360). Such power, which may be present in the ~g data, needs to be filtered out prior to
the least-squares adjustment (Pavlis, 1988). Further study is needed to clarify this issue.
SUMMARY
A satellite-only gravitational model and its full covariance matrix were used in
combination with a global set of 30' mean gravity anomalies, to produce two high-degree
combination solutions. The first was developed using numerical quadratures for the
analysis of the anomalies and the adjustment technique described by Rapp and Pavlis
(1990). The second used observation equations and a block-diagonal least-squares
adjustment, which for the first time incorporated the full normal equations of the
satellite-only model. Intercomparison of the resulting high-degree models yields a global
geoid height difference (to Nmax=360) of ± 20 cm. The error spectra from the two
solutions are in excellent agreement. The two techniques perform equally well in orbit fit
tests and in absolute comparisons with GPS/Leveling-derived undulations over British
Columbia and US. The higher degree harmonics obtained from the block-diagonal
technique appear to overestimate the power in the field. This problem requires further
study. Provided that this effect is understood and corrected, a number of experiments can
be attempted using the block-diagonal technique. These include the removal of the a
priori constraints (modified Kaula rule) from the satellite-only normals and the
incorporation of normal equations obtained from direct tracking altimeter data.
119
(V022 and HDM036) which appear to be of the same quality when compared to certain
independent data. Comparison of V022 to HDM036 in terms of 30' mean values of geoid
undulations (to Nmax.=360) yields 0.20 m global std. deviation difference (0.33 mover
land and 0.10 m over ocean cells). The extreme differences are 5.2 m (land) and 2.9 m
(ocean) and occur in areas of poor gravity anomaly data.
It is instructive to compare the coefficients of V022 and HDM036, to the coefficients
obtained from a harmonic analysis (NQ) of the unadjusted global 30' ~g file (input). In
terms of percentage differences (see Rapp, 1986, eq. 53), this comparison is shown in
Fig. 1. As expected, V022 above degree 70 is practically identical to the harmonics
obtained from the unadjusted ~g. In contrast, due to the existing correlations between low
and high degree harmonics of a given order, HDM036 above degree 70 differs
substantially from the harmonics of the unadjusted ~g. These differences represent the
influence of the satellite information on the higher-degree terms. This information passes
to the high-degree coefficients through the correlations that are represented in the "wing"
part of the normals.
Fig. 2 shows the anomaly degree variances implied by V022, HDM036, their difference
as well as the errors associated with each model. The difference between the two
solutions is roughly an order of magnitude smaller than the error variance of either model
for the same degree. One should also notice the excellent agreement between the error
variances implied by the two models. This is rather remarkable since the error variances
of V022, above degree 70, are computed (sampling error part) based on a semi-empirical
formula devised by Jekeli (Colombo, 1981). The signal degree variances of HDM036
imply a rougher field above degree approximately 180, as compared to V022. Relative
undulation comparisons over GPS/Leveling traverses (see also Rapp and Pavlis, 1990,
Table 7) show V022 to be in slightly better agreement with the independent data than
HDM036. These comparisons are most sensitive to the high-degree part of the field and
may indicate a problem with the BD solutions related to aliased power (above degree
360). Such power, which may be present in the ~g data, needs to be filtered out prior to
the least-squares adjustment (Pavlis, 1988). Further study is needed to clarify this issue.
SUMMARY
A satellite-only gravitational model and its full covariance matrix were used in
combination with a global set of 30' mean gravity anomalies, to produce two high-degree
combination solutions. The first was developed using numerical quadratures for the
analysis of the anomalies and the adjustment technique described by Rapp and Pavlis
(1990). The second used observation equations and a block-diagonal least-squares
adjustment, which for the first time incorporated the full normal equations of the
satellite-only model. Intercomparison of the resulting high-degree models yields a global
geoid height difference (to Nmax=360) of ± 20 cm. The error spectra from the two
solutions are in excellent agreement. The two techniques perform equally well in orbit fit
tests and in absolute comparisons with GPS/Leveling-derived undulations over British
Columbia and US. The higher degree harmonics obtained from the block-diagonal
technique appear to overestimate the power in the field. This problem requires further
study. Provided that this effect is understood and corrected, a number of experiments can
be attempted using the block-diagonal technique. These include the removal of the a
priori constraints (modified Kaula rule) from the satellite-only normals and the
incorporation of normal equations obtained from direct tracking altimeter data.
119