**ESA Project: Refinement of observation requirements for GOCE.**

Draft 1998-11-15.

Note on error estimates of Spherical Harmonic Coefficients.

**by C.C.Tscherning & D.Arabelos.**

The error of estimation of a gravity field quantity (like a point or mean geoid height or
gravity anomaly) from space-measurement like those planned for GOCE will (or should)
depend on the magnitude of the gravity field variation in the vicinity of the quantity being
estimated: The number of measurements needed for the estimation of a point or mean value
is higher in an area with high gravity field variation than in an area with low gravity field
variation.

However, the error estimates available from EGM96 seems not to depend on the location.
The error estimates from a related ESA project seems only to depend on latitude. The error
estimates reflect the existence of the polar gaps, and the fact that the orbits becomes
"closer" when the latitude increases numerically.

This has initially caused problems for the in this
project planned "Monte-Carlo" simulation
of the errors in the estimated ground gravity quantities.

But how can we solve this problem ? In Least-Squares-Collocation procedures (see e.g.
Knudsen (1987)) the problem is solved by using a local covariance function, where the low-degree-variances are equal to scaled error-degree-variances, obtained from the Spherical
Harmonic Expansion (SHE) used in a remove-restore procedure. The scale factor is larger
than 1 if the local gravity field varies more than the global and it is less than one if we are in
a smooth region. This has worked quite satisfactory, except that is seams that the errors in the
low harmonics have been too low, when all error-degree-variances are multiplied with the
same constant. Probably not a constant, but a function dependent on the degree should have
been used.

We therefore propose to use a similar procedure for the error-variances for the SHM
coefficient error-estimates. The scale factor (scf) is computed as the regional variance
divided by the global gravity variance as computed from the coefficients of the used SHM
(EGM96 or GPM98A). It is used then either as a constant scale factor for all degrees or as a
linear factor being equal to 1 at degree 0 and equal to scf at the maximal degree used. This
we be our first try.

There is however a problem. If the data in EGM96 or GPM98A have been computed from
mean-values of blocks with size larger than 0.5 degree for EGM96 and 0.1 degree for
GPM98A, then the local variation is erroneous. It should therefore be compared to the
variation of the topography. If the topography has a large variation, then an error in the
gravity data may have been detected.

Any other suggestions are welcome.

Reference:

Knudsen, P.: Estimation and Modelling of the Local Empirical Covariance Function using gravity and satellite altimeter data. Bulletin Geodesique, Vol. 61, pp. 145-160, 1987.

Last update 1998-11-14 by cct.