a) Form separate normal equations from each individual data type, to a maximum degree
and order that corresponds to the resolution of the available data and their sensitivity to
the gravitational signal.
b) Treat satellite altimeter data as direct tracking observations, i.e. ranges from the
spacecraft to the ocean surface whose upper endpoint senses (through the orbit
dynamics) attenuated gravitational signals (static and time-varying), while their lower
endpoint senses the combined effects of geoid undulation, SSDT as well as tides and
other time varying effects without any attenuation. Extensive discussion on this topic
can be found in (Rapp, 1993).
c) Combine the various normal equations (with appropriate relative weights) and invert
the resulting system, to estimate the combination solution to a high degree (360) along
with its full error covariance matrix.
The main advantages of such an ideal estimation strategy are:
Non-global and overlapping data (e.g., marine gravimetry and satellite altimetry) are
optimally incorporated to the solution.
Each observation can be weighted individually, according to its estimated error variance.
The geographic location of each observation can be arbitrary.
Treatment of satellite altimetry enables the simultaneous estimation of the gravitational
and SSDT signals (as well as ocean tides) and the improvement of the radial orbit
accuracy.
A combination solution to degree 360, if performed as outlined above, would require the
formation of full normal matrices (from satellite altimetry and surface gravimetry) for
more than 130000 parameters. This task is beyond our present computational capabilities.
Two alternative strategies can be followed:
Remove-Restore Technique. The principle here is to apply the ideal estimation
technique for coefficients of only the low degree part of the gravitational spectrum.
Present computational resources permit solutions to maximum degree and order 70 or 90.
This enables accurate modeling of the gravitational signal sensed by all the presently
available tracking data. Furthermore, since SSDT and orbit errors are of long wavelength
character, the benefits of direct tracking altimetry are almost entirely retained. However,
to minimize aliasing effects, the high frequency gravitational component present in
altimeter and surface gravity data has to be removed prior to the combination solution
adjustment. This implies that a high-degree model needs to be available beforehand. The
higher degree coefficients of this model are used afterwards to augment the lower degree
part that is solved for in the combination solution. Solutions that have been developed in
this manner include OSU-91A (Rapp et at, 1991) and JGM-l and 2 (Nerem et al., 1994).
Global Coverage Techniques. Given a complete ~lobal ~rid of observations of a
functional of the field (e.g., ~g), certain symmetries of the grid's geometry and of the
error properties of the data, lead to highly efficient estimators for high-degree coefficient
recovery. These techniques have been studied and put forward by Colombo (1981).
Estimators falling in this category are: (1) Numerical quadrature, i.e. the discrete
counterpart of orthogonality relations [Examples: OSU-86E/F (Rapp and Cruz, 1986a),
OSU-89A/B (Rapp and Pavlis, 1990)], (2) Least-squares adjustment using block-diagonal
normal equations [Example: DGFI-92A (Gruber and Bosch, 1992)] and (3) Optimal
estimation [Examples: OSU-86C/D (Rapp and Cruz, 1986b)].
The main advantage of these techniques is their extreme computational efficiency, due
to the sparsity of the normal matrices involved and the applicability of FFT algorithms. It
113
and order that corresponds to the resolution of the available data and their sensitivity to
the gravitational signal.
b) Treat satellite altimeter data as direct tracking observations, i.e. ranges from the
spacecraft to the ocean surface whose upper endpoint senses (through the orbit
dynamics) attenuated gravitational signals (static and time-varying), while their lower
endpoint senses the combined effects of geoid undulation, SSDT as well as tides and
other time varying effects without any attenuation. Extensive discussion on this topic
can be found in (Rapp, 1993).
c) Combine the various normal equations (with appropriate relative weights) and invert
the resulting system, to estimate the combination solution to a high degree (360) along
with its full error covariance matrix.
The main advantages of such an ideal estimation strategy are:
Non-global and overlapping data (e.g., marine gravimetry and satellite altimetry) are
optimally incorporated to the solution.
Each observation can be weighted individually, according to its estimated error variance.
The geographic location of each observation can be arbitrary.
Treatment of satellite altimetry enables the simultaneous estimation of the gravitational
and SSDT signals (as well as ocean tides) and the improvement of the radial orbit
accuracy.
A combination solution to degree 360, if performed as outlined above, would require the
formation of full normal matrices (from satellite altimetry and surface gravimetry) for
more than 130000 parameters. This task is beyond our present computational capabilities.
Two alternative strategies can be followed:
Remove-Restore Technique. The principle here is to apply the ideal estimation
technique for coefficients of only the low degree part of the gravitational spectrum.
Present computational resources permit solutions to maximum degree and order 70 or 90.
This enables accurate modeling of the gravitational signal sensed by all the presently
available tracking data. Furthermore, since SSDT and orbit errors are of long wavelength
character, the benefits of direct tracking altimetry are almost entirely retained. However,
to minimize aliasing effects, the high frequency gravitational component present in
altimeter and surface gravity data has to be removed prior to the combination solution
adjustment. This implies that a high-degree model needs to be available beforehand. The
higher degree coefficients of this model are used afterwards to augment the lower degree
part that is solved for in the combination solution. Solutions that have been developed in
this manner include OSU-91A (Rapp et at, 1991) and JGM-l and 2 (Nerem et al., 1994).
Global Coverage Techniques. Given a complete ~lobal ~rid of observations of a
functional of the field (e.g., ~g), certain symmetries of the grid's geometry and of the
error properties of the data, lead to highly efficient estimators for high-degree coefficient
recovery. These techniques have been studied and put forward by Colombo (1981).
Estimators falling in this category are: (1) Numerical quadrature, i.e. the discrete
counterpart of orthogonality relations [Examples: OSU-86E/F (Rapp and Cruz, 1986a),
OSU-89A/B (Rapp and Pavlis, 1990)], (2) Least-squares adjustment using block-diagonal
normal equations [Example: DGFI-92A (Gruber and Bosch, 1992)] and (3) Optimal
estimation [Examples: OSU-86C/D (Rapp and Cruz, 1986b)].
The main advantage of these techniques is their extreme computational efficiency, due
to the sparsity of the normal matrices involved and the applicability of FFT algorithms. It
113