Speckle statistics 
We will now consider higher order statistical properties (pixel to pixel 
correlation) of the speckle noise. We will start from the hypotheses of the 
Rayleigh speckle model, and we add the following assumption : 
- the phase of a scatterer @#(X) has a correlation radius much smaller 
than the pixel size of the processed SAR image. 
Another form of this assumption is to say the phases of the scatterers 
are statistically independent, except to a microscopic scale. This assumption is 
supported by a fact the surface is rough compared to the wavelength (and this 
hypothesis of roughness is fundamental in the Rayleigh speckle model), and 
therefore the phase of a scatterer bears no relation with the phase of another 
scatterer, except for a neighboring one. 
In that case : 
- since the phase and the amplitude of the scatterer responses are 
statistically independent of each other, they are decorrelated[3]and 
therefore the autocorrelation of the scatterer complex response can 
be written as : 
: 6) 
E[ 5(t). 5(6+7)]= E[ 5,(0 Sane 
E(&t9.&( 9) e[e a) 
C 5» (cec) . S(T) 
= SC) 
and this autocorrelation is equivalent to an impulse whose width equals 
342) | 
the correlation radius of the phase of the scatterers. 
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