must be emphasized however that these techniques pertain to the problem of harmonic
analysis, which is only part of the problem of developing combination solutions. Once a
complete grid of data is analyzed harmonically with anyone of the above techniques,
there is still an adjustment to be performed, that will combine the satellite-derived
coefficients with those obtained from the analysis of the gridded data.
Global coverage techniques are only applicable to grids of data of uniform type, and do
not allow for data gaps or arbitrary data weights, if strict compliance with the anticipated
sparsity patterns of normal or covariance matrices is expected. Therefore data types such
as direct altimetry cannot be incorporated to the combination solution in this manner.
One may use altimeter-derived gravity anomalies along with land values to form a
complete anomaly grid that can be input to the above algorithms, as it was done in the
development of the OSU-89A/B models (Rapp and Pavlis, 1990).
The numerical quadrature technique and the remove-restore technique can be used as
parts of an iterative process: A complete grid of gravity anomalies is analyzed using
quadratures to define a high-degree model. The higher degree harmonics of this model
are used to filter out the high-degree contribution of the field from surface gravity and
direct altimetry data. A combination solution is performed that yields the lower degree
harmonics, improved orbits of the altimeter satellites and a model of the SSDT. Using the
SSDT model and the improved orbits, one may iterate the prediction of altimeter-derived
anomalies, which along with surface data define an improved global anomaly grid. A new
quadrature solution can be obtained based on this global grid, and the whole process may
be iterated. This has been the philosophy behind the development of OSU-91A and
JGM-l and 2. Rapp (1993) gives a detailed description of the analytical formulation
underlying such an estimation strategy. One disadvantage of this strategy is the fact that
the high-degree model is estimated in a piece wise fashion. The higher degree coefficients
of the model are determined solely on the basis of the global anomaly grid, therefore the
least-squares combination solution pertains only to the lower degree part of the model. It
is therefore desirable to investigate techniques which may be capable of performing the
combination solution and estimating both low and high degree coefficients in a single
step. Least-squares, using block-diagonal normal equations, is such an estimation
technique. It can be applied to the same input data as the quadrature technique, while it
possesses additional features that may be preferable to quadratures. A comparison of the
two techniques, given the same input data, is a logical starting point of investigating the
applicability of the block-diagonal technique in potential coefficient determination.
QUADRATURE AND BLOCK-DIAGONAL LEAST -SQUARES ESTIMATION
The analytical and numerical differences between the numerical quadrature (NQ) and the
least-squares (LS) estimation techniques can be studied from both the harmonic analysis
and the combination solution perspectives. Such studies have been reported by, e.g., Rapp
(1969; 1986), Colombo (1981), Pavlis (1988) and Sneeuw (1994). A brief review of the
conclusions reached by these investigations follows. NQ is used here to identify the
simple quadratures formula with the "composite" set of quadrature weights proposed by
Colombo (1981, p. 76).
Harmonic Analysis. Given a complete set of mean gravity anomalies llgij on an
equiangular grid on the reference ellipsoid, NQ estimates a set of spherical harmonic
potential coefficients C nm by:
114
analysis, which is only part of the problem of developing combination solutions. Once a
complete grid of data is analyzed harmonically with anyone of the above techniques,
there is still an adjustment to be performed, that will combine the satellite-derived
coefficients with those obtained from the analysis of the gridded data.
Global coverage techniques are only applicable to grids of data of uniform type, and do
not allow for data gaps or arbitrary data weights, if strict compliance with the anticipated
sparsity patterns of normal or covariance matrices is expected. Therefore data types such
as direct altimetry cannot be incorporated to the combination solution in this manner.
One may use altimeter-derived gravity anomalies along with land values to form a
complete anomaly grid that can be input to the above algorithms, as it was done in the
development of the OSU-89A/B models (Rapp and Pavlis, 1990).
The numerical quadrature technique and the remove-restore technique can be used as
parts of an iterative process: A complete grid of gravity anomalies is analyzed using
quadratures to define a high-degree model. The higher degree harmonics of this model
are used to filter out the high-degree contribution of the field from surface gravity and
direct altimetry data. A combination solution is performed that yields the lower degree
harmonics, improved orbits of the altimeter satellites and a model of the SSDT. Using the
SSDT model and the improved orbits, one may iterate the prediction of altimeter-derived
anomalies, which along with surface data define an improved global anomaly grid. A new
quadrature solution can be obtained based on this global grid, and the whole process may
be iterated. This has been the philosophy behind the development of OSU-91A and
JGM-l and 2. Rapp (1993) gives a detailed description of the analytical formulation
underlying such an estimation strategy. One disadvantage of this strategy is the fact that
the high-degree model is estimated in a piece wise fashion. The higher degree coefficients
of the model are determined solely on the basis of the global anomaly grid, therefore the
least-squares combination solution pertains only to the lower degree part of the model. It
is therefore desirable to investigate techniques which may be capable of performing the
combination solution and estimating both low and high degree coefficients in a single
step. Least-squares, using block-diagonal normal equations, is such an estimation
technique. It can be applied to the same input data as the quadrature technique, while it
possesses additional features that may be preferable to quadratures. A comparison of the
two techniques, given the same input data, is a logical starting point of investigating the
applicability of the block-diagonal technique in potential coefficient determination.
QUADRATURE AND BLOCK-DIAGONAL LEAST -SQUARES ESTIMATION
The analytical and numerical differences between the numerical quadrature (NQ) and the
least-squares (LS) estimation techniques can be studied from both the harmonic analysis
and the combination solution perspectives. Such studies have been reported by, e.g., Rapp
(1969; 1986), Colombo (1981), Pavlis (1988) and Sneeuw (1994). A brief review of the
conclusions reached by these investigations follows. NQ is used here to identify the
simple quadratures formula with the "composite" set of quadrature weights proposed by
Colombo (1981, p. 76).
Harmonic Analysis. Given a complete set of mean gravity anomalies llgij on an
equiangular grid on the reference ellipsoid, NQ estimates a set of spherical harmonic
potential coefficients C nm by:
114