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.
