The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
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Where\ Q i + £ a ) is the magnitude of the frequency coi, cp; represents 
phase, ©,(**) = 2n '{ft + V',(/*) + e / ) 
Ax(i, Tj ) = Ax(/, T t ) + £ a , where e a , s f and e A are random 
variables with zero mean and Gaussian distribution, with 
standard deviation a f , a h and a respectively, randomly chosen 
at each moment t k , and where V j (t k ) is the temporal drift of 
each frequency fj and is not linear. 
So we can compute the wj coefficients by least squares: 
can have random magnitudes, the process only restores 
magnitudes observed locally. Removing 3 exciter frequencies 
increases errors by a factor of 1.7. It is important to note that 
the noise is an important factor of the problem. The calibration 
coefficients Wj must take into account the measurement noise 
that are calculated for a minimum of 20 points per line 
correlation. The accuracy are decreased only with a factor of 
1.16 on rms compared to a case without noise . 
We can also increase the number of retrieved HF frequencies 
by decreasing the correlation step, like on Figure 5 right. 
w = (7) 
with R ^ = E t [ax(/, t) • \x(t, t) h ] and 
r Ax* =£,[Ax(/,t) •*(/)] 
The implemented method returns the main frequencies and 
takes into account the measurement noise. It does not require 
pre-processing of signals (such as Fourier transform). The 
sampling of the signal may be irregular or have some gaps but 
all the sampling differential must be synchronous. 
Figure 6. FFT on disturbance signal and retrieved signal only 
with three differentials and 7 local sampling, left: without 
measurement noise, right: with measurement noise. 
It has no blind frequencies if independent differentials are 
combined. It requires, however, a pre-determination of the 
frequencies of disturbance, their maximum magnitudes, and an 
estimation of the differentials measurement noise. But this 
analysis is part of current assessments performed during in 
flight commissioning. The accuracy can be increased using 
several sampling of the differential around the processed time 
step, like 2 or 3 for example. The results are better and more 
stable if the secondary differentials are removed from the 
equation 7 (according to the coefficients w, weight). 
3.2.2 Theoretical results 
As example, with magnitudes of almost one pixel and the 
overlay of 8 disruptive harmonics signals of 4 stimulating very 
closed frequencies, with a non linear frequency drift (1%), with 
noises at 1 sigma of 1% on frequency and 1% on harmonics, 
10% on amplitude, two main sought harmonics in the range of 
50 to 78 Hz, with noise associated to the correlation of images, 
using three retina couples measurements sampled at 4 ms and 
only one local time sample, we get errors less than 0.064 pixel 
rms on the sought frequencies. 
cam t/»H> «H-tteaxHxi <»i nMft vs 
Figure 5. Example of results with MF sought on the left and 
HH on the right. 
With a sampling of 0.4 ms and only three differentials Figure 6 
presents the sought signal and the retrieved signal after only one 
local integration. The local integration gives the signal with all 
the frequencies until 600 Hz. 
3.2.3 Application to PHR data 
In the PHR focal plane XS retina are not registered and almost 
parallel to each other and synchronized. The sampling time for 
XS retina is Te XS = 0.4 ms (fech = 2500 Hz). The XS 
maximum B/H ratio between Bl et B2 is 2.10-4 so DTM errors 
are under the correlation noise (altitude errors of 500m give 
displacement of about O.lpixels Pan=0.025pXS). The chosen 
couples for PHR is B2-B1, B0-B1, B2-B0 , the most accurate 
for subpixel level image matching. This 3 couples give 
differential data about the 3 different delays, usable on PHR 
focal plane. 
The Pan retina is a TDI detector with complex geometry, more 
distant from XS retina so the B/H ratio requires to use DTM for 
colocalisation and B2 is the better for image matching but not 
very efficient. So Pan is not useful for the correction 
computation heart. 
The 4 gyroscopic actuators CMG are source of disruptive 
signals. The speed of AG will be adjusted to avoid resonating 
patterns of the satellite . But in-orbit before this adjustment, we 
will observe disruptive signals. They will be the sum of 8 
harmonics (out of phase and noisy) of 4 excitatory frequencies, 
slightly variables over time. The PHR localisation budget is 
very accurate so the process have to correct the frequency 
between 16Hz and 110Hz . Figure 5 presents the disruptive 
simulated signal and the results . 
A lot of checking tests and sensibility analysis were conducted. 
The main results are very interesting. For a given set of Wj 
coefficients, bias on exciter frequency of 13% has no effect on 
performance. An error of + -33% on the values of harmonics 
has only little effect. The main characteristics of frequencies 
As we see in chapter 4.1, the correlation cut-off sampling 
frequency is around 830 Hz (3 pixel XS)1. With a matching 
step of 1 pixel XS the frequencies could be retrieved until this 
cut off frequency; with 3 pixel XS step, the retrieved 
frequencies are less than 410 Hz and with aliasing