normalised coherent sum of all contributions, can be formulated 
for a random distribution of scatterers, as shown in (1), where s; 
and s; are the complex signals received at either end of the 
interferometric baseline, * represents the complex conjugate 
and <...> is the expectation, in practise replaced by an 
averaging operation. 
  
  
h, 
; ' Jr t 
js (ss) iis | (ze dz = eh gern, (1) 
(sis) (sas) Í f(z)dz' 
with 0 <7] <1 : 
In practice, the upper bound of integration is at z=z, +4, 
where the lower bound is at the reference datum, z= z,. The 
interferometer is then measuring a composite complex signal 
given by the weighted sum of contributions (taking into account 
phase). Before estimating the height using the phase to height 
relation ¢ = k,.h, , the sensitivity of the interferometers to height 
variation has to be defined using the vertical wavenumber factor 
k_ (Bamler, 1997): 
_ 4740 
ios A 4zB, 
AsinO  AsinQ' 
  
  
radians / meter (2) 
where A8is the incidence angle difference between the two 
antennas as seen from the target, Bis the spatial baseline 
between the two antennas, Ais the model forest height, B, is 
the normal baseline, A is wavelength of the radar system, and 6 
is the incidence angle from one of the antennas. Equation (1) 
shows that there is a direct relationship between the observed 
coherence and the structural properties of the scattering. In the 
most widely used representation of the RVoG model, the 
structure function is assumed to have an exponential shape 
(Papathanassiou and Cloude, 1997). However in the PCT 
approach, a Fourier-Legendre polynomial decomposition is 
proposed to describe the structure function, f(z’), in terms of 
known basis functions with unknown real coefficients. 
It is important to notice that the structure function is bounded 
by the underlying ground phase and the vegetation height. 
Thus, it is critical to obtain good estimates of those elements 
prior to the profile estimation through the expanded Fourier- 
Legendre series. In the Legendre series development, n 
baselines provide 2n-1 terms of the series. Therefore, in the 
simplest practical application, a single baseline solution can 
provide the second order of the scattering profile.. In this 
special single baseline case (n=1), it allows determination of up 
to three terms of the series. The unknown coefficients a, (for 
the single baseline case) can then be estimated using a simple 
matrix inversion of equation (3): 
1:0 0 || ay 1 
0 —-ji 0a - Im(7,) |, (3) 
0 0 fle Re(7,) - f, 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B8, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
    
   
    
   
   
   
  
   
    
  
   
   
  
   
  
  
  
  
   
    
    
    
  
   
   
     
    
   
   
   
    
   
   
   
  
      
   
  
   
    
   
     
   
      
where f represents the Legendre polynomial parameter at order 
i, Ÿ,represents the polarimetrically optimised complex 
coherence for baseline-dependent Æ, and a,, are the (real) 
Legendre coefficients. It should be noted that the f£. terms are 
functions of the known quantities &,'h,. The evaluation of these 
coefficients is solved as a matrix inversion problem. The 
equation can be inverted in a simple manner as shown in 4): 
Ig» 
Il 
[Fla=g => ä=[F]'g (4) 
In the instance of two baselines, for instance, an improved 
vertical profile description at the second order is obtained by an 
expanded structure function now containing four unknown 
coefficients as shown in (5), where P; are the Legendre 
polynomials 
f) - Và Rz) * àP ) à, P (2) * à P). ) 
The main purpose of this work is to determine the structure 
function using this PCT approach applied to data from a 
‘single-pass’ polarimetric InSAR system. Single-pass in this 
context refers to a single baseline with rigidly connected 
antennas such that temporal decorrelation effects are absent 
from the data. Therefore (3) and (4) are relevant in this case. 
2.2 Tree Height and Ground Phase Estimation 
In this methodology, we use a ‘boot-strapping’ approach to 
obtain two of the unknown parameters, h, and à for the initial 
canopy boundary estimates. The height and ground retrievals 
are obtained by the use of a modified RVoG model, where 
another parameter is added to compensate for the 
"underestimation" of the RVoG. For that purpose, (Treuhaft et 
al., 2000) considered a fixed extinction assigned empirically to 
the radar. The mean extinction parameter allows computation of 
the canopy thickness. The fixed extinction coefficient 
compensates for both density and structure variations 
(Papathanassiou et al, 2003). In this two-layer model, the 
observed coherence is given by the formula: 
uL UA fet ums cO. An 6 
di ge dicun -e [o pol (6) 
where ¢, is the ground phase relative to the reference datum, 
/,is the complex volume coherence and wis the ratio of 
effective ground surface to volume scattering. 7(w) refers to 
the normalized Fourier transform of the attenuated response 
from the vertical distribution of scatterers (z) and w represents 
the observed polarisation state. Equation (6) shows that by 
isolating the polarisation dependent terms (w), the resulting 
coherence lies in a straight line inside the complex coherence 
plane (Cloude et al., 2003). 
In our study, the results of the inversion coming from the RVoG 
were used as bounds for the vertical profiles reconstruction, 
  
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