University of Copenhagen
Department of Geophysics.
C.C.Tscherning. 15 April 1999
Contribution to: WP 2 progress report "Detailed scientific data processing" from UCPH.
(1) Identification of unknowns.
Spherical harmonic coefficients (SHC), GM*Cij, up to a degree NMAX (e.g. equal to
240) and their standard errors and error-correlations.
Mean gravity anomalies and geoid heights at the surface of the Earth (not at height 0
!) of equiangular or equal area blocks with side-length expressed in latitude
difference BSIZE (e.g. 0.25 degrees).
The (mean) gravity disturbance vector components and vertical gravity gradient, Tzz,
in pre-defined normal points or in normal areas, at a fixed altitude, e.g. equal to the
mean satellite altitude.
Instrument bias and drift terms in specific time intervals.
(2) There will be regional and global stochastic models, if a suitable method for the
representation of non-isotropic functions has not been found meanwhile. (Work on this is in
Both the global and the regional models will be defines using the following parameters
Depth to the Bjerhammar-sphere
Mean gravity variance
Scale-factor on the error-degree-variances of the spherical harmonic model used as
The mathematical model used to represent the stochastic behaviour will be the so-called T/R-
model, see Tscherning & Rapp, 1974.
(3) The observation equations are either related to functionals associated with a point or with a mean value. SST observations are also supposed to be of this kind, which requires that we can consider the GPS satellites as having a known orbit. The functionals are then as described in several publications: linear combinations of first and second derivatives of the anomalous gravity potential, T. Since the original data are associated with the gravity potential, V, the precise calculation of the functionals applied on T, requires that the position of the satellite and the attitude of the SGG instrument is given. The position must be given with a vertical position better than 1 m. (Tzz changes 1.44 E -3 EU pr. m at satellite altitude)
The attitude should be known better than 10", so that an error in Tzz does not project into
errors in other gravity gradient components.
(4) One of the coefficient estimation methods will be Least Squares Collocation (LSC). The
coefficient estimation method is based on an estimate
T(P) = SUM (i=1,n) b(i)*cov(obs(i),T(P)).
From this can be predicted coefficients and their error estimates using that the covariance
between an coefficient GM*Cij and an observation is equal to L(Yij), where Yij is the i,j-th
Spherical Harmonic Function (SHF), L is the linear functional associated with the
observation (e.g. twice the radial derivative of Tzz), i.e.
GM*Cij = SUM (i=1,n) b(i)*cov(GM*Cij, obs(i)).
The programming of the calculation of L(Yij) and cov(GM*Cij, obs(i)) has started, cf. WP 3.
(5) In the general LSC procedure, SSG and SST observations are treated in the same way.
SST observations, however, must first be converted to range rate rates, by forming
differences twice and dividing by the time interval twice (see Tscherning et al. 1990, section
2). This results in correlated observations, the correlation of which must be taken into
account when determining the quantities b(i).
(6) According to earlier simulation studies, the most advantageous combination is the brute
force one which use all observations simultaneously, including ground data. This will be
possible in regional solutions using LSC. For the integration method, the data must be pre-processed to form means of blocks of e.g. Tzz at satellite altitude. LSC can also be used to
handle such data. The advantage of this is that error-correlations can be taken into account,
and that error-estimates can be calculated.
(7) If blocks are used, symmetries are created for data blocks having the same latitude, on the
condition that the errors are identical. This will probably be so, if data have been
preprocessed in an optimal way (high frequency information removed a-priori).
(8) The ground tracks of the satellite will cross. This has as a consequence that data close to
the cross-over points are strongly (spatially) correlated. In LSC there is implicitly taken
advantage of this. Since the integration method is expected to have mean block values
formed by LSC, this means that the blocks will have a smaller error than if the calculation of
the means is made with simpler methods.
(9) Simulations of the influence of having to assume identical errors for mean values in the
same latitude band are waiting.
(10) In LSC the local and global approaches are seen from the theoretical standpoint
identical. The difference arises because the use of LSC in the global case will require that a
very large system of equations have to be solved if the basic observations are to be used. If
mean values of e.g. Tzz are used as normal points, also the global case can be handled by
LSC. (Work on improved methods for solving the equations are progressing, cf. WP 3).
(11) It has not been possible to investigate how or to which extend long wavelength
information is lost in a regional approach.
(12) Investigations on the use of preconditioning for the solution of very large systems are in
Tscherning, C.C., R.Forsberg and M.Vermeer: Methods for regional
gravity field modelling from SST and SGG data. Reports of the Finnish
Geodetic Institute, No. 90:2, Helsinki, 1990.
Tscherning, C.C. and R.H.Rapp: Closed Covariance Expressions for
Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical
Implied by Anomaly Degree-Variance Models. Reports of the
Department of Geodetic Science No. 208, The Ohio State University,
Columbus, Ohio, 1974.
Last update 1999-04-16 by cct.