This is a list of research problems which I consider important, but not necessarily are able to solve myself.

1. Effect of erroneous bathymetry.

The spherical harmonic expansions of the rock-equivalent topography (ret)
has shown that there must be very large errors in the bathymetry.
The mass-center of the topography is many hundreds of meters
from the Earth center. The correlation between the spherical
harmonic expansion of the ret and the potential is strange for low degrees.

How can we use the existing topographic information if it is so erroneous ?
It must for example give large errors in the topographic isostatic
reductions.

2007-07-14: One proposal is not to use it, but to use a spherical harmonic
expansion, which anyway will represent most of the potential of the topography.
Paper with POLIMI is in preparation for Porto, 2004.

Here the rtm-method avoids the problem, because it only considers
the local FIXED topography.

2. Error-covariances are computed e.g. when computing a spherical harmonic model using least-squares. We know there are scale errors in the variances. But how do we calibrate the covariances ?

3. How do we construct a covariance function/reproducing kernel on closed form., which is homogeneous on the reference ellipsoid ? (I.e. has constant variance of geoid, gravity anomalies etc. at this surface). Added 2004-07-14: Ellipsoidal spherical harmonics could in principle be used, but how do we find closed expressions ?

4. When GPS/levelling is compared to a gravimetriq quasi-geoid
there are frequently large biases and tilts in the differences.
What is the cause ? Long-wavelength errors in the reference potential
(EGM96), GPS-reference system not geocentric ? Systematic errors
in the levelling ? Geodynamic phenomena (land uplift/subsidence) ?
2007-07-14: Or: errors caused by not considering the difference between
orthometric and normal heights.

Today (2012) the main difference seems to be due to the sea-surface topogaphy.

5. How do we update the deflections of the vertical to a geocentric system ? (If we know the datum-shift, the problem should easily be solved.).

6. What is the correct definition of a mean-value: mean over
an equi-angular block in a fixed ellipsoidal height ?

See for a proposal.

7. When using residual terrain modelling (rtm) the total mass change is zero, so the zero order term is still zero. But what about the earths Center. Will it move ?

8. Least-squares is used for estimating spherical harmonic coefficients.

But has it been proved that one may use least-squares ? Do there
exist convergence proofs ? (When one get more data one get closer to the
solution).

9. Under which circumstances will the normal equation matrix in collocation be singular if different data-types associated with the same point are used ? One example is if all three diagonal components of the gravity gradient matrix are used. 2007-07-14: It seems that this does not cause a problem. At least not when noise is added to the diagonal components, see Sapporo paper, 2003.

10. How does one calibrate a variance co-variance matrix ?

11. How does one simulate correlated noise ? This seems meanwhile that a solution has been found. The discrete PSD values are used to generate normally distributed noise for each frequency where the noise variance is equal to the associated PSD value. The program ecf.f can be used.

12. The foundation for a theory for analytic covariance function modelling is needed.

13. A study of aliazing effects arizing when using LSC or least-squares methods for estimating spherical harmonic coefficients. (see paper on alizing and the program of the IAG Study Group 2.5 ).

14. The use of least-squares collocation requires that as many equations
as observations are solved.

How do we reduce the number of equations without degrading the solution:

- Compute mean-values along orbits or tracks ?

- leave out data from areas with a very smooth gravity field ?

15. What is the best strategy for multi-processing a Cholesky-reduction ?

Several OMP implementations have been investigated using different blocking of the normal-equation matrix. MPI is now beeing investigated, so that different computers
and not only several processors can be used. (2012-11-16).

Last update 2012-11-16.