\thens, 
re based 
matrix. 
t. If the 
| uses in 
sets: the 
speckle 
, forest, 
images 
tent” of 
of each 
ich time 
ceuracy 
rding to 
, 1994), 
se/cover 
1). An 
Wishart 
mming, 
lassifier 
veighed 
(Smith, 
rmation 
tives of 
our by 
"ding to 
, and b) 
position 
| on the 
rimetric 
position 
e filter 
»d with 
he area 
L band 
  
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
4. POLARIZATION AND PAULI DECOMPOSITION 
THEORY 
2.] Wave Polarization 
For a plane transverse electromagnetic wave, the Æ vector of 
the electric field oscillates in a plane perpendicular to the 
propagation direction. The vectorial nature of these waves is 
called polarization and is independent of the chosen coordinate 
system. In the case that the trace of the tip of the field vector 
E within a plane perpendicular to the propagation direction is 
an ellipse, the wave is elliptically polarized. Special cases of the 
elliplically polarization are linear or circular polarization. For 
the description of polarization, a coordinate system and a 
reference direction of propagation are needed. For compatibility 
with fully polarimetric radar systems that use two orthogonal 
linear polarized antennas, a Cartesian coordinate system is 
introduced, where +k is the propagation direction of the 
plane transverse electromagnetic wave, and À and V are the 
horizontal and vertical directions of the plane of the electric 
field. The equation of a transverse electromagnetic wave as a 
function of its position 7^ is: 
E(F)- E exp (ik - 7) (1) 
— 
The vector E of the complex electric field consists of a hE, 
component and a VE, component, which are perpendicular to 
the propagation direction. 
EhE, -VE, (2) 
These components can be expressed on the basis of their real 
amplitude a, =| E;| and phase exp(iô; ): 
  
Ë,=j-Ë=a,expl iô,) (3) 
where  j-horv. The component E; of the electric field can 
also be written as a function of time t and position r: 
| 
Ens Rel£, exp i(k -F ~ at)}= Rea, expi(k F-ot+d,)j=a, cos(k -F — et *ó,) 
7d,cos(r-Ó,) (4) 
whete r2 £.rF-ot.By defining angle à as the relative 
phase difference between the two components, ö = à, — 6p, it 
results that: 
E 
—— = cos( 7 + & + À, ) = cos( 7 + À, ) cos à — sin( 7 + &, ) sin 6 
a 
=cos(r +6, ) cosó —4/1— cos" (z - , ) sinó (5) 
Substituting cos( 64) by equation (4) we obtain: 
281 
  
  
E (Pt) E (F.f) : ; 
= cos à — sin à 
a, a h 
E (Ry EX.) SERIE, Tt > (6) 
m UE EIU en o EU) us 
a a, dud, 
which is the equation of an ellipse (figure 1) with an orientation 
angle y such that 
tan (Qu )= ses i cos ó : 
qu S 
Consequently, polarization may also be described by the 
geometrical properties of the ellipse which are the orientation 
angle y and the ellipticity angle y 
tan Gs: (8) 
) 
where 2a and 2b are the minor and major axes of the ellipse, 
and x specifies the shape of the ellipse as well as the sense of 
rotation of the vector £ . The polarization is left handed for 
0<y<n/4 and right handed for —/4<y<0 for an observer looking 
in the direction of the propagation. The polarization angles y 
and x are related to the wave parameters a,, aj, and 6 by 
sin(2y) 7 sin(2a) sin(8) 
tan(2y) 7 tan(2a) cos(8) (9) 
where the angle a is defined as 
tan(a) = _v_ (10) 
a, 
(2) 
  
  
  
Figure 1. Wave elliptically polarized in (h,v ) plane with 
propagation direction + À 
If both amplitudes are equal a, = a, and x = F4, the 
polarization is circular. For x 7 7/4 the polarization is left hand 
circular and for y = -n/4 the polarization is right hand circular.