The Quadrature Technique
The technique estimates a complete 360 x 360 model using a two-step process. First, a
global set of gravity anomalies and orthogonality relations are used to determine a set of
"terrestrial" hannonic coefficients, for those hannonics that are included in the satellite-only
model. In addition, through the orthogonality relations, the errors of the gravity anomalies
are propagated to form a full error covariance matrix for the "terrestrial" harmonics. Based
on the satellite-only model and its associated error covariance matrix, and their "terrestrial"
counterparts, a least-squares adjustment is performed that yields a unique set of coefficients
complete to degree 70, essentially as a weighted average of the two estimates. This
adjustment yields also a set of adjusted anomalies which, in a second step, are harmonically
analyzed to yield the higher-degree coefficients of the model. This procedure was
originally proposed by Kaula (1966) and has been used in a number of models developed
at the Ohio State University (Rapp, 1981; Rapp and Cruz, 1986; Rapp and Pavlis, 1990).
Rapp and Pavlis (1990) give a detailed description of the analytical formulation of the
technique.
The Block-Diagonal Technique
This single-step technique is based on the fact that the normal equations developed for
gravity anomalies given on a complete equiangular grid on a surface of revolution (e.g., a
rotational ellipsoid) are block-diagonal, if the coefficients are ordered by hannonic order
(Colombo, 1981). This is exactly true provided certain conditions hold for the error
variances of the anomalies, but only approximately true for present quality data.
Reordering these "terrestrial" normals in a specific way, makes their combination with the
complete set of satellite-derived normals, and the subsequent inversion of the combined
system, very efficient computationally. Details on the implementation of this technique can
be found in Pavlis et al. (this issue).
PRELIMINARY RESULTS
A number of preliminary gravity solutions have been developed as intermediate results
from this project. They may be described as follows:
• "Sat" - a model complete to degree 70 based only on the available satellite tracking data.
• "Comb" - a model complete to degree 70 based on the satellite tracking data, direct
altimetry from Geos-3, Seasat, Geosat, and TIP, and a lOx 1 0 surface gravity data set
from Rapp et al. (1991)
• "Quad" - a model complete to degree 360 computed using the quadrature solution
technique, the "Sat" normal equations, and the merged 30' x 30' gravity anomaly file.
• "Block" - identical to "Quad", except the block diagonal solution technique was
employed.
For reference, Figure 2 shows gravity anomalies complete to degree 360 computed using
the "Quad" model. The residual differences between these anomalies and the raw 30' x 30'
anomalies are one indicator of how well the satellite data and the gravity anomalies are
being combined. The residuals almost entirely reflect disagreements in the observed
gravity between the satellite data and the 70 x 70 part of the 30' x 30' anomalies. Although
not shown here, we have seen considerable reduction in these residuals versus what was
observed in the development of the OSU91A model. However, there are a number of
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The technique estimates a complete 360 x 360 model using a two-step process. First, a
global set of gravity anomalies and orthogonality relations are used to determine a set of
"terrestrial" hannonic coefficients, for those hannonics that are included in the satellite-only
model. In addition, through the orthogonality relations, the errors of the gravity anomalies
are propagated to form a full error covariance matrix for the "terrestrial" harmonics. Based
on the satellite-only model and its associated error covariance matrix, and their "terrestrial"
counterparts, a least-squares adjustment is performed that yields a unique set of coefficients
complete to degree 70, essentially as a weighted average of the two estimates. This
adjustment yields also a set of adjusted anomalies which, in a second step, are harmonically
analyzed to yield the higher-degree coefficients of the model. This procedure was
originally proposed by Kaula (1966) and has been used in a number of models developed
at the Ohio State University (Rapp, 1981; Rapp and Cruz, 1986; Rapp and Pavlis, 1990).
Rapp and Pavlis (1990) give a detailed description of the analytical formulation of the
technique.
The Block-Diagonal Technique
This single-step technique is based on the fact that the normal equations developed for
gravity anomalies given on a complete equiangular grid on a surface of revolution (e.g., a
rotational ellipsoid) are block-diagonal, if the coefficients are ordered by hannonic order
(Colombo, 1981). This is exactly true provided certain conditions hold for the error
variances of the anomalies, but only approximately true for present quality data.
Reordering these "terrestrial" normals in a specific way, makes their combination with the
complete set of satellite-derived normals, and the subsequent inversion of the combined
system, very efficient computationally. Details on the implementation of this technique can
be found in Pavlis et al. (this issue).
PRELIMINARY RESULTS
A number of preliminary gravity solutions have been developed as intermediate results
from this project. They may be described as follows:
• "Sat" - a model complete to degree 70 based only on the available satellite tracking data.
• "Comb" - a model complete to degree 70 based on the satellite tracking data, direct
altimetry from Geos-3, Seasat, Geosat, and TIP, and a lOx 1 0 surface gravity data set
from Rapp et al. (1991)
• "Quad" - a model complete to degree 360 computed using the quadrature solution
technique, the "Sat" normal equations, and the merged 30' x 30' gravity anomaly file.
• "Block" - identical to "Quad", except the block diagonal solution technique was
employed.
For reference, Figure 2 shows gravity anomalies complete to degree 360 computed using
the "Quad" model. The residual differences between these anomalies and the raw 30' x 30'
anomalies are one indicator of how well the satellite data and the gravity anomalies are
being combined. The residuals almost entirely reflect disagreements in the observed
gravity between the satellite data and the 70 x 70 part of the 30' x 30' anomalies. Although
not shown here, we have seen considerable reduction in these residuals versus what was
observed in the development of the OSU91A model. However, there are a number of
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