r £ ha = D - ZAt — 2B sin nt + 2C cos nt 
2 a 
£ hr = —B cos nt + C sin nt (5) 
r 
, £/io = E cos nt + F sin nt 
- J [ [1 - cosn(t - T)]6g r {T)dr+ 
Jo 
+ £ [ sin n(i - r)<fy a (r)cir+ 
Jo 
-sf {t - r)5g a {r)dT 
J o 
£ [ sin7i(i T)5g r {j)dT+ 
Jo 
+ n[ i 1 - cosn(i - r)](5p a (r)dr 
io 
£ [ sinn(t- T)8g 0 [r)dT 
Jo 
(6) 
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 
ics. 
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. 
£min 
11 
max 
90 
R (cm) 
637813630 
GM (Sj£) 
3986004.415- 10 14 
n (¡2d) 
7292115-10“ 11 
r (cm) 
680813630 
I 
87° 
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 
zero. 
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: