File:H:\CCTWP\e2m9934.wpd, last change: 17 March 2000
The polar gap problem arises due to the orbit inclination of GOCE. However, GOCE-only solutions may be valuable. Such solutions may be determined in an optimal manner using Least Squares Collocation (LSC) and the Slepian approach.
LSC is a general procedure, which permit the direct use of ground data (both inside the gaps and outside). In the general integration approach, data has to be predicted covering the gaps and converted to the same data type as otherwise used for the integration. This prediction may be made using regional LSC solutions.
The inclusion of e.g. gravity anomaly data from inside the gaps will have no important consequence for the sparseness of the matrices used in the conjugent gradient method with sparse preconditioners. This is because these preconditioners have been generated from a general kernel with compact support.
Finally an overview is given of current and planned gravity surveys in the polar areas.
The orbit inclination of GOCE leaves the polar areas north of 82 degrees and south of -82 degrees - the polar gaps - without data. These gaps may be filled in two ways
We will in the following concentrate on how ground data can be utilized in the space-wise approach.
The new satellite missions, GOCE, GRACE and CHAMP have spurred an interest in having the polar gaps filled with data. We will give a (unfortunately incomplete) overview of the available data and of the at this time planned activities.
5.2 GOCE-only solutions.
It may be of interest to have solutions (spherical harmonic coefficients, ground geoid height
and gravity values) based on only data from GOCE. Such solutions will be internally
consistent. When including other data types, we may not know whether these are
contaminated with systematic errors - or whether GOCE data contains such errors. A
comparison of independent solutions will be important for the mission calibration and
Both of the two space-wise methods, LSC and integration, may be used to obtain GOCE-only
solutions. For LSC the solution is obtained by only using GOCE data. Data inside the gaps
are not needed in order to obtain a solution, but using such data will obviously improve the
For the integration method, the Slepian approach (Albertella et al., 1999, Albertella and
Sneeuw, 1999) only used the data outside the gaps, and gives an improved solution compared
to using the "global" harmonic function approach.
5.3 Combining GOCE data with gravity data in the gaps.
The integration approach is based on the use of only one datatype, with all data located on
the same sphere. Let us suppose the sphere is having a radius equal to the mean radius of
GOCE for the whole lifetime of GOCE. The data will be either gravity disturbances, Tz, or
anomalous vertical gravity gradients, Tzz . Such quantities will be normal mean area values
over blocks typically of size 0.3 degree * 0.3 degree, corresponding to a 4 second sampling of
the GOCE data.
In the gaps such values may be predicted from ground data using the upward continuity
capability of LSC with harmonic kernels (GEOCOL) or FFT based upward continuation from
gridded ground data (GEOFOUR), see Tscherning et al. (1992, 1994). Both ground and
airborne gravity anomalies as well as the GOCE data close to the cap boundary should be
Using LSC, data in the gaps may readily be combined due to general purpose character of the
method. One could here worry about the structure of the sparseness of the banded matrix
arising when using sparse preconditioners. However these preconditioners have been
developed from general sparse kernels (Moreaux et al. 1999, Moreaux, 1999). Only if the
correlation is larger for the ground data than the GOCE data will the bandwidth be larger,
probably a factor 2 in the worst case.
5.4 Data in the polar gaps.
Earlier gravity data has been collected on the ground using traditional gravimetry. These data
are available through the Bureau Gravimetric Internationale, Toulouse, see Fig. 1 and 2.
There is one problem connected with the land data from Antarctica, namely the errors in the
associated altitude. Heights must be determined using levelling, which is a very tedious
procedure. In polar areas it is often substituted by barometric levelling, which may have
errors larger than 2 m, if not done very carefully.
Submarine gravity tracks.
Today GPS is used to calculate the
ellipsoidal height so the height problem
is solved. Furthermore airborne gravity
techniques have been developed, and are
the only reasonable method to use on
In the Arctic ocean submarines collect
gravity data, see Fig. 3.
The international scientific community is aware of the importance of collecting gravity data
for the gravity satellite missions. The International Association of Geodesy (IAG) did at its
General Assembly in Birmingham, July 1999 adopt the following resolution:
The International Association of Geodesy
1.the need for terrestrial and airborne gravity measurements due to the lack of gravity coverage over the polar caps by the planned satellite missions, and
2.the need for improved geoid models in the polar regions,
a concerted international effort to compile existing available gravity data and to encourage new gravity surveys in the
Besides passing the resolution, the association established a project (formally a working
group under the IAG Gravity and Geoid Commission), the Arctic Gravity Project. The home-page of the project is
where information about the project can be found. In the project the aim is to be able to
produce a free-air gravity map of the Arctic at the end of 2000. It is planned to release a final
grid product with 5' spacing in 2001.
On Antarctica, the Scientific Committee for Antarctic Research (SCAR) has a geodesy and
geographic information working group (SCAR-GGI), which also deals with the collection of
gravity data, see
A gravity reference network has been established and several groups are working here to collect airborne gravity data. The aim is to produce a much needed geoid map of Antarctica,
see e.g. Capra and Gandolfi (1999).
The Alfred Wegener Institute for Polar and Marine Research (AWI), Bremerhafen, has in
January 2000 completed its third airborne gravity survey. AWI aims at mapping the off-shore area in the Eastern Weddell Sea. The British Antarctic Survey files regularly over the
Antarctic Continent. The US also collects airborne data (U.Meyer, GFZ Potsdam, personal
information, January 2000). However, the Antarctic is very large and not easy to access. But
a stronger effort is needed here.
Methods exist for a GOCE-only solution, which does not include data from the polar gaps.
But obviously the inclusion of gravity data will improve the estimates of the spherical
harmonic coefficients, for which the quality depends on the availability of global data.
Software enabling the use of such data in combination with GOCE data are available.
However a large international projects are aiming at providing new gravity data. The Arctic
will be covered at the end of 2000 with a 5' grid. The Antarctic is a very difficult area due to
its size and inaccessibility.
Albertella, A., F.Sanso' and N.Sneeuw: Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere. In print Journal of Geodesy, 1999.
Albertella, A. and N.Sneeuw: The analysis of gradiometric data with Slepian functions. Presented EGS General Assembly, The Hague, 1999.
Capra, A. and S.Gandolfi: A project for archiving and managing physical geodesy data in Antarctica. Presented Arctic Geodesy Symposium 99, 14-16 July 1999.
Forsberg, R.: Report of the Arctic Gravity project - IAG SSG 3.178. Travaux de l'Assciation Internationale de Geodesie, (in print), 2000.
Moreaux, G.: Harmonic Spherical Splines with Locally Supported Kernels. Proceedings IAG General Assembly, Birmingham, 1999 (in print).
Moreaux, G., C.C.Tscherning & F.Sanso': Approximation of Harmonic Covariance Functions by non Harmonic Locally Supported Ones. Journal of Geodesy, Vol. 73, pp. 555 - 567, 1999.
Tscherning, C.C., P.Knudsen and R.Forsberg: Description of the GRAVSOFT package. Geophysical Institute, University of Copenhagen, Technical Report, 1991, 2. Ed. 1992, 3. Ed. 1993, 4. ed, 1994.
Tscherning, C.C., R.Forsberg and P.Knudsen: The GRAVSOFT package for geoid determination. Proc. 1. Continental Workshop on the Geoid in Europe, Prague, May 1992, pp. 327-334, Prague, 1992.
Last change 2000-01-30 by cct.