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International cooperation and technology transfer
Mussio, Luigi

r £ ha = D - ZAt — 2B sin nt + 2C cos nt
2 a
£ hr = —B cos nt + C sin nt (5)
, £/io = E cos nt + F sin nt
- J [ [1 - cosn(t - T)]6g r {T)dr+
+ £ [ sin n(i - r) Jo
-sf {t - r)5g a {r)dT
J o
£ [ sin7i(i T)5g r {j)dT+
+ n[ i 1 - cosn(i - r)](5p a (r)dr
£ [ sinn(t- T)8g 0 [r)dT
So what we have to do is basically to invert (4), (5),
(6) for 6g a ,6g r ,6g 0 .
This can be done by a spacewise approach, which
amounts to directly inverting equations (4), (5), (6)
with a suitable stochastic inverse method; this step
provides the vector {6g a ,6g r ,6g 0 ) on a sphere at satel
lite altitude; the vector is then used to estimate the
coefficients Ti m via integration with spherical harmon
2 Numerical tests with the
spacewise approach
In order to verify the theoretical procedure described
in Section 1, it was decided to produce simulated data
to be treated in the spacewise approach. Using the
EGM96 gravity model, the three components of the
residual gravity accelerations were computed at points
spanning a quarter of an orbit 1 , with the initial condi
tion that latitude ip = 0 when t = 0 both for ascending
and for descending arcs. Other parameters of the sim
ulation can be found in Table 1. The interval between
two subsequent points along the axe is At = 5s.
1 One choice is fundamental, namely to use a short arc ap
proach, inverting half a cycle at a time so that the central point
of the arc has a maximum distance from the ends equal to a quar
ter of a cycle. First of all this approach shows that data could be
treated even in case of (relatively) frequent interruptions with
out degrading the mission. Moreover this choice is done to be
sure that the noise in f (t) is limited to 1 4- 2 cm.
R (cm)
GM (Sj£)
3986004.415- 10 14
n (¡2d)
7292115-10“ 11
r (cm)
Table 1: Parameters used for the simulation of data to
be treated in the spacewise approach.
To obtain the orbit anomalies according to (3), only the
particular solution was used, in fact the homogeneous
part of the integral is irrelevant to our reasoning. This
happens because in the subsequent estimation proce
dure of 6g (obtained by applying the Hill operator) the
contribution of the homogeneus solution (5) is equal to
By using a simple numerical integration algorithm, the
three components £ pa ,€pr,€po were computed.
Afterwards, from the data as represented in Fig. 1
two contributions were subtracted: one represents the
average value of the data themselves, while the other
is the trend produced by the homogeneous solution of
(3). Before applying the estimation procedure, a white
noise with mean square value equal to 2 cm was added
to the "observations” (cf. Fig. 2).
Starting from the simulated observations f, we now
had to study the behaviour of the estimated values of
6g over a regular grid covering the Earth surface. In
order to do this, we decided to choose a sample area
with dimensions 2° x 2°: over this area data were simu
lated (according to the previously described procedure)
for a mission lifetime of one year, corresponding to 15
ascending and 15 descending arcs.
The idea was to derive the signal Sg from f and obser
vation equations (3) by applying a collocation approach
along the arc.
To optimize the collocation procedure, the empirical
covariance function was estimated after grouping the
arcs in sets of five each and referring the averaged data
(five by five) to the points of the medium arc. However,
the collocation estimate of the functionals needed to
invert Hill’s equations was subsequently performed at
the original points of each arc.
The inversion of Hill’s equations gave the values of the
components Sg a ,Sg r ,Sg 0 at all the observation points
of the sample block: simply averaging the single com
ponents, the mean values were obtained. These val
ues, after undergoing a suitable rotation to a Earth
fixed reference system (and assuming no attitude er
ror), were referred to the center point of the block: we
call them Sg^,6g x ,6g r . The same quantities were also
directly simulated using the EGM96 gravity model: