**University of Copenhagen**

**Department of Geophysics.**

C.C.Tscherning.

**Draft WP 2 report "Detailed scientific data processing using the
space-wise approach".**

Prepared for the E2M Mid-term Review, Sept. 1999.

In the following is described some elements of the data processing process which must be
implemented if the space-wise approach is to be used.

**(1) Identification of unknowns and of stochastic parameters.**

The following quantities are the unknowns to be determined by the space-wise approach:

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 the 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. Other components like Txx and Tyy could also be
considered.

Instrument bias and drift terms in specific time intervals.

In principle, these unknowns are the same as those which must be determined by the time-wise approach.

Stochastic models must be defined when using estimation methods such as Least Squares
Collocation. There will be regional and global stochastic models, if a suitable method for the
representation of (global) non-isotropic functions has not been found meanwhile. (Work on
this is pending, theoretical proposal is published).

Both the global and the regional models will be defined 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
reference.

The mathematical model used to represent the stochastic behaviour will be the so-called T/R-
model, see Tscherning & Rapp (1974).

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.

**(2) Recovery of unknowns using integration or collocation.**

Suppose block mean values of gravity anomalies or disturbances or of Tzz, covering a sphere
which includes the Earth, have been calculated by a local prediction method like least-squares collocation (LSC). In this case numerical integration procedure can be used to
recover the spherical harmonic coefficients. Since the mean-values will have errors (also
calculated as a by-product of the local prediction process if LSC is used), the numerical
integration must take this into account. This leads again to the use of LSC, contingently in a
finite-dimensional subspace, exactly as used when calculating standard global models like
the OSU91 or the EGM96 model.

The use of LSC for coefficient prediction has been implemented in the GRAVSOFT program
GEOCOL as a part of WP3, see Tscherning (1999). The program is now undergoing detailed
testing.

**(3) Use of alternative representations.**

The use of one alternative, namely Slepian functions, has been described earlier in the literature. At present no further investigations have been made.

**(4) Simultaneous use of SGG and SST data.**

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
using LSC.

The two data types will have a high physical correlation because they will be related to the
same point or segment of the satellite orbit. This may cause singularities. If this is the case
only one data-type should be used. Simulations can be made to see which type of data is the
most advantageous to be used.

**(5) Normal-point values or normal mean values as the basic observation ?**

When forming "normal" values such as gradiometer data referring to a specific point on the
satellite orbit or mean values for a certain segment, it is still an open question which quantity
is the most advantageous to use. Point values are easy to represent in a numerical procedure,
while track-segment means require a larger numerical effort.

**(6) Brute force or reduction of the number of data by using "normal" values.**

According to earlier simulation studies, the most advantageous combination is the brute force
one which use all observations simultaneously, including ground data. This will only 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) Use of symmetries.**

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 not be so, since if data have been
preprocessed in an optimal way, and even if high frequency information has been removed a-priori, the signal variance will vary. This may force the use of a uniform noise in latitude
bands as done when estimating the EGM96 coefficients of higher degree.

The influence of having to assume an uniform noise can be investigated by determining a
low degree solution with and without uniform noise. The treatment of a non-uniform signal
variance is still waiting a numerical implementation.

**(8) Use of data-redundancies.**

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 as compared to a
situation where the calculation of the means is made with simpler methods. The
redundancies will also be used in data screening, a subject to be treated in WP 4.

**(9) Global and regional (local) data processing approaches.**

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 in progress and will be reported in WP 3.

When determining "normal" values a local approach is clearly of advantage. One will be able
to use a stochastic model which represents the local signal variation. However, the
local/regional solutions should not be merged to a global solution (by e.g. defining a solution
for each block of a certain size covering the Earth). This opens up for a step-wise approach,
where the regional solutions have as their only purpose the use in the calculation of suitable
"normal" values and their error-estimates.

**(10) Loss of information in regional solutions.**

Using the developments in the use of LSC for coefficient prediction as described in
Tscherning (1999) it is now possible to investigate to which extend long wavelength
information is lost in a regional approach. One example is described in the paper.

**(11) Use of preconditioning when solving very large systems of equations.**

Theoretical investigations on the use of preconditioning for the solution of very large systems
have been completed (Moreaux, 1999). Numerical tests with larger data sets are in progress.

**References**:

Moreaux, G.: IUGG99 paper.

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.: Prediction of spherical harmonic coefficients using Least-Squares Collocation. Department of Geophysics, University of Copenhagen, September 1999.

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.