DIIAR - Sez. ... Now since SK is a small rotation, it can be written as. SK = I + A. (1.3) with A an antisymmetric matrix. In reality we first rotate to a locai System ...
ON THE EFFECTS OF THE ATTITUDE RECONSTITUTION ERROR ON THE GRAVITY FIELD RECOVERY IN THE ARISTOTELES MISSION
M. Bassanino, F. Migliaccio, F. Sanso' DIIAR - Sez. Rilevamento - Politecnico di Milano - Italy
§ 1 - THE MIS-ATTITUDE ERROR 2
*
In thè Aristoteles misslon we measure thè tensor r = V + fl + n (V = gravity gradients tensor, n = satellite's spin tensor)' along certain instrumentai axes and we transforra such measurements to a locai geographical frame ( z radiai, x to north and y to cast) ; to perform this computation we need to know thè attitude of thè satellite i.e. thè rotation matrix ft between thè two systems. The Information on R is given by startrackers measurements and we assume that thè rotation angles in 3? are known with an accuracy er (angles) = 6-10 rad . This means that after rotating f there is a residuai mis-attitude error which is a small rotation 6K. Assuming n2 to be known and O to be eliminated, we remain thus with an equation which reads
V = ÓK X [T - B2] 3?* 63?*
(1.1)
where
[r - n2ls = I ter - n2) + (r* - n2)] .
(1.2)
Now since SK is a small rotation, it can be written as
SK = I + A
(1.3)
with A an antisymmetric matrix
In reality we first rotate to a locai System oriented on thè orbitai motion and then by averaging these observations we obtain quantities referred to a geographical frame. 55
O
a
0
-a
O
y
-p
-y
o
(A
= -A)
(1.4)
and oc,/3,y representing rotations around Neglecting quadratic terms in
a, (3,y
z.y.x axes.
we see that thè error that we
commit by using only
u = K [r - n2Js 3?
(i.s)
instead of thè correct (1.1), is exactly
6U = U - V = -AU - UÀ
= UÀ - AU
(1.6)
Remark 1 : it is important to reraark here that thè potential we are 1coking for is already thè residuai potential that we obtain from thè earth's potential
by subtracting
thè
best
known model.
This
is
already u, of which U is thè matrix of second derivaiives appearing in (1.6).
Remark 2: thè rotation 3? in (1.5) is essentially a rotation through thè angle I (orbit inclination) around thè axis z; therefore thè only components in U whìch are of interest are u
yy
, u
yz
, u
zz
and thè same
is true for SU. Explicitely computing (1.6) we get
Su
yy
= 2(au
xy
- yu ) zy
5u yz = au xz + /3u yz - y(u zz - uyy ) Su
zz
= 2(/3u
zx
+ yu
zy
)
(1.7)
To tackle thè problem of thè propagaiion of thè errors a (3 y into thè recovered gravity field we must make hypotheses on thè statistlcal behaviour of such errors. To this aim we shall first of ali assume that a |3 y are independent and wlth thè same r. m. s. O-Q = 6-10 rad. What is more criticai however is thè assumption of thè statistical behaviour in time: in fact it is not reasonable to assume that thè attitude error is incorrelated from one observation epoch to thè next after few seconds. So we shall stipulate for instance that a |3 y are Constant over thè time span of 4 measurements, i. e. as many as fall in thè orbit are corresponding to thè variation of latitude of 1°. With this specification in mind when we will try to compare thè effect of this error on block averages computed for further gravity analyses, we must consider thè 4 measurements along thè piece of track falling into thè (e.g. ) 1° x 1° block as having thè same error so that thè variance is not decreased by averaging. On thè contrary we shall assume that after thè time span of 1 cycle thè correlation of a /3 y has become zero, so that thè error of any other track falling in thè same block is completely incorrelated. Remark 3: from thè above assumptions it follows that in order to compare thè mis-attitude error with thè measurement noise of thè gradiometer, cr
=
10
_2
E. U. , we must first multiply by a faeton 2
thè r.m.s. of thè former because in this way they are put on an equal basis in front of thè subsequent over 1 x 1 blocks.
averaging procedure, supposingly
From (1.7) we can therefore compute formai variances
