The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
30 
The disturbances dynamic calibration is carried out 
systematically during in-flight commissioning and is 
monitored after. This calibration is not part of the chain but 
these results are used as parameters for the second step. 
So Ax(f, Tj ) = p(co, Tj Y ■ v(a,(O,ç,t) (3) 
Because 
Complementary processes are possible but are largely specific: 
they consist in deducting the attitude correction signal for the 
other retinas from those calculated by local integration. 
3.2 Method of local integration 
= ¿^r[(l - e~ io,Tj ) e i{eo ‘ ,+,p ‘ ) - (l - e“° iTj ) e ~ i( ^ ,+<p ‘ ] 
The goal of this method is to compute correction attitude 
profiles dx(t) from measurements of several differential attitude. 
It works independently on roll and pitch. 
The mains issues are : 
==> The differential are sampled with gaps and noise. 
■=> The sought signal is complex: for example with 4 noisy 
exciting frequencies, 2 noisy main harmonics and 6 
secondary 
■=> Instability of the signal models: the exciting frequencies 
are slightly variable in time, and the drift is not linear. 
So a local calculation , combining several differentials is very 
efficient to overcome this issues and to obtain accurate results. 
Also for several differentials, the linear formula between 
differential and sinus base is : 
Ax(t,r 2 ) 
= 
ß(to,r 2 Y 
(4) 
3.2.1 Principles of the method 
On a stationary harmonic signal 
At a time t, a signal composed of n harmonic stationary pulses 
can be expressed as a linear combination of 2 x n differential. 
Indeed the disruptive signal and the couple xj differential can be 
written 
x(t) = 
i=l 2i (2) 
= a H • \{a,(o,(p,t) 
Where sin(/) = ~j(e u ~ e “).i 2 = -1 et a" =(!...!) 
r a \ 
1 c '(tO|Ti+<pl) 
2 i 
l n /(CO„T„+(p„) 
v(a,ö>,cp,i) = 
2 i 
a 
e 
2 i 
1 +q>i) 
V 
g-i(w n r„+(p„) 
2/ ) 
So if we have m differentials with m=2*n and B invertible 
we obtain equation : | det(B(fi>,x)) | » 0 
x(/) = a H \{a,(o,(p,t)= a // B(co,T)” i 
x(t,T m ) y 
2xn . . 
x(l) = w^t)" • Ax(/,t) = 'YjWj - Ax\f,Tj) 
(5) 
and the Wj coefficients depend on the frequency and time lags, 
not magnitudes nor phases. 
On a quasi-stationary quasi-harmonic noisy signal 
The methodology frees itself from previous assumptions by 
least squares approximation and regularization of the solution. 
First, we eliminate the secondary differentials according to 
weight values wj, and secondly, we introduce into the 
simulation the knowledge of differential noise and all hidden 
variables (non-observable frequencies) and information on the 
harmonics noise, frequencies uncertainties, and on frequencies 
temporal variations. The sought signals may modelized 
following: 
Kt k )=JJ a , +e a )- sinfe (t k )xt k +(p) 
i-\ 
(6)