24 
which afterwards allow to derive the unique weighted 
estimate represented by 
Differences 
àg-d ~ àg$ 
fig \ - fig\ 
fig r - fig r 
mean (mGal) 
r.m.s. (mGal) 
-0.049 
0.589 
0.066 
0.317 
-0.204 
0.470 
Table 2: Comparison between estimated and simulated 
values. 
figr 
figo 
figx 
dT 
dr 
1 dT 
r d'à 
1 dT 
r sin 7? d\ 
(7) 
where: 
dT GM1 (R\ t+ \ n 
= + 1) [Tim cosraA-t- 
+ T£ ) _ m sinmA] P* TO (cos$)} 
- T lm sin mA] Pim (cos i?)} 
The results of the estimation procedure are summa 
rized in Table 2. 
These results quantify the noise produced for each com 
ponent by the applied collocation procedure. Therefore 
three sets of noise data were synthetized, with mean 
square values equal to those computed from the differ 
ences between simulated and estimated values. 
3 Recovery of the gravity field 
coefficients 
The final step consisted in computing two sets of har 
monic coefficients by discretizing the harmonic analysis 
formulas for the radial and horizontal components of 
the gravity field 
*-(s) 
r \ ^+2 
R* + 1) 47T 
Tim 
(8) 
where 
°h = °HTL) = (iŸ e+1 
»I = = (*f +4 
4* (&)'(* + !)’ 
4* (&) *(*+!) 
(9) 
of, a\ being the Wiener variance densities of radial and 
horizontal components. 
An index showing the highest estimable degree is the 
signal-to-noise ratio, obtained by comparison of the 
curve of the degree variances given by Kaula’s rule 
with the curve of the degree variances of the estimated 
model. This gives a maximum attainable degree equal 
to i max = 49, as shown in Fig. 3. 
We took this result as provisory, probably due to a too 
rough approximation in quadrature formulas. We say 
that, because a theoretical prediction of i max from a 
uniform noise of 0.5 mGal resulted in ¿ max ~ 62, which 
by the way is in agreement with the results obtained by 
the research team working with SAGE simulated data 
in the framework of the timewise approach [ASI, 1998]. 
References 
[ASI, 1998] ASI: 1998, SAGE Phase A Final Report, 
Agenzia Spaziale Italiana, 1998. 
[Bassanino et al., 1992] Bassanino M., Migliaccio F., 
Sacerdote F.: 1992, A BVP approach to the re 
duction of GPS and accelerometric observations, 
in: Proc. Symp. IAG 110 From Mars to Green 
land: charting gravity with space and airborne in 
struments, ed. O.L.Colombo, Springer-Verlag New 
York. 
[ESA, 1998] ESA: 1998, European Views on Dedicated 
Gravity Field Missions: GRACE and GOCE ESD- 
MAG-REP-CON-001, ESA, May 1998. 
[Reigber et al., 1996] Reigber Ch., Bock R., Forste 
Ch., Grunwaldt L., Jakowski N., Liihr H., 
Schwintzer P. and Tilgner C.: 1996, CHAMP Phase 
B Executive Summary, GFZ, STR96/13.