In: Wagner W., Székely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Voi. XXXVIII, Part 7B 
where i = y/—l. 
2.2 Phase delay computing 
It was shown in (Oguchi, 1983) that for the case of coherent prop 
agation in rain the propagation phase delay, 5 in mm/km, can be 
calculated as: 
Ö = 10 
-a 2-7T 
k 2 
/ 
Re(f(D))N(D)dD, 
(6) 
where k is the wave number (corresponding to a wavelength of 56 
mm for the case of ERS1/2) and f(D) is the forward scattering 
amplitude in mm and N(D) is in mm^'m" 3 . 
In this case f(D) can be evaluated using Rayleigh approximation 
(Bringi and Chandrasekar, 2001). The scattered amplitude for h- 
polarized incidence wave can be then calculated as: 
Figure 1 : Fitting Rayleigh approximation to a powerless one. 
}(D) = tl) 3( e - 1)A , (7) 
where e is the relative permittivity of water and A is given by: 
1 
A = 
1 + Ai (e — 1) 
(8) 
where 
A = and A 2 = 1 - jfan -1 (/)) , (9) 
where 
/ = e -2 — 1. (10) 
As we can conclude the expression of Re(f(D)) is very com 
plicated and cannot be used directly for deriving an analytical 
expression for the phase delay related to the rain. Thus more 
suitable expressions are needed. 
In order to overcome this problem, we have simulated the for 
ward scattering amplitude for D ranges between 1 and 8 mm. we 
have fixed the temperature at 10 oc and the wavelength at 5.6 cm. 
In these conditions, the value of e can be approximated by 70 + 
30j. Then, for a given value of D, the corresponding axis ratio e is 
computed using equation 2. Then, f{D) is computed using equa 
tions 3 and 5-11. From the resulting curve of the real part we can 
derive a powerless relation between Re(/(Z))) and D that have 
the forme aD 13 . Coefficients a and /3 are estimated by the non 
linear regression algorithm. For our simulation a = 3.7 x 10~ 4 
and /3 = 3.02. Figure 1 shows the good agreement between the 
Rayleigh approximation and the powerless one. 
After replacing Re(/(D)) by its powerless expression and per 
forming some simple mathematical manipulations equation 6 be 
comes: 
6 = J D 13 exp(—-D)cfD 
, (11) 
= 
where T(-) is the gamma function. 
Finally a simple expression relating the phase delay to the rain 
rate can be derived using equation 4 as: 
6 = 
8\ 2 a4.1~ ß R°- 21ß r 
(12) 
mm, since particles with a larger diameter are unstable. In Figure 
2 the result of these calculations is given. It can be seen that 
for high rain intensities the path delay can be as high as several 
centimeters. 
Figure 2: Path delay as a function of the rain rate. 
3 SIMULATED INTERFEROGRAMS 
Our simulation for this work consists in generating two interfero- 
grams for a perfectly flat area. The first one was generated with 
out taking into consideration any rain effects and the second one 
takes into a consideration a layer of rain that covers the imaged 
area. Figure 3 and 4 show the considered computation geometry. 
3.1 Computing geometry 
In figure 3, designates the baseline vector, 0 the incidence 
angle, T the radar swath and n and r 2 are ranges between a 
fixed point in the imaged area and the position of the radar for 
the first and the second acquisition respectively. In the absence of 
rain ATp P rop depends only in n and r 2 (Abdelfattah and Nicolas, 
2002): 
4tt 
Alpprop = ^“( r 2 - »*l). (13) 
Thus, rain free interferogram can be directly generated using n 
and r 2 . If we consider the configuration of figure 3 and after 
performing some geometrical manipulations we can deduce that: 
The path delay for different rain intensities were calculated. It 
should be noted that raindrop diameter is usually truncated at 8 
ri = yjx 2 + y 2 + H 2 
r 2 = y/(x - B x ) 2 + (y - B y ) 2 + (H + B z ) 2