In: Wagner W., Szekely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010,1 APRS, Vol. XXXVIII, Part 7B 
for 
x e [H tan(0) — y, H tan(0) + y ] 
y€[- ?,?] 
Now, let consider the configuration with the presence of a rain 
layer of figure 4. h designates the hight of the rain layer and n r 
and r2r are ranges between a fixed point in the imaged area and 
the intersection points of n and r2 with the top of the rain layer 
respectively. r\ r and r2 r corresponds exactly to the path crossed 
by the radar signal throw the rain medium. Also, r ir and r2r 
can be computed using some simple geometrical manipulations. 
Here we avoided to give expressions of n r and r2r for reasons 
of clarity. Let R\ and R2 the rain rates corresponding to the first 
and the second acquisition respectively. The phase difference due 
to propagation throw the rain is then : 
Alpprop—r “ r*2r X ^(^2) tTr X &(Rl')‘ (15) 
and the total phase difference, from which the rain contaminated 
interferogram is computed, is: 
A'ljj /\'lj)prop T dsljjprop— r* (lb) 
It should be noted that these calculations do not include estima 
tions of the delay due to the melting layer of precipitations and 
due to the precipitating cloud above the melting layer. However, 
we expect that contribution of the melting layer is rather limited 
since it occupies the limited height range (usually in the order of 
few hundred meters). The precipitating cloud on the other hand 
can have a rather large height range, but due to the fact that the 
relative permittivity of ice is much smaller than the one of wa 
ter the contribution of the precipitating cloud to the signal delay 
would be negligible. 
3.2 Interferograms generation and discussion 
In order to simulate interferograms we have fixed the configura 
tion parameters to be equal to the ERS 1/2 radar ones. Table 1 
shows their values. 
Parameter 
Simulation value 
H 
785 Km 
6 
23° 
A 
56 mm 
T 
10 Km 
~1T~ 
[10 10 100]m 
Pixel Resolution 
50x50m 
h 
10 Km 
Table 1: simulation parameters 
Note that the rain was considered to be uniform, with the same 
rain rate, for the hole area. Figure 5 shows resulting phase de 
lay (in cm) for different rain rates. From these results, we can 
conclude that there is a strong increase in a propagation delay as 
sociated with rain rate and that, the rain can induce a considerable 
propagation delay of several centimeters. Also we can verify by 
these results that the rain induced phase delay presents very small 
variation between the nearest and the most far point in the imaged 
area. Therefor, in these conditions, the rain induced phase delay 
can be considered as constant all over the imaged zone. Thus 
its effect on the interferogram will be a simple translation of the 
fringes. This fact is clear in figure 6 which shows the rain-free 
and some rain-contaminated interferograms. As expected, inter 
ferograms for the area, considered to be perfectly flat, consists 
in a succession of parallel fringes. The amount of the fringes 
Figure 3: Geometry used for Insar computations for a flat area in 
the absence of rain. 
transition can be deduced by dividing the rain induced delay by 
A/2. For example, light rain with 5mm/hr induce a transition 
of approximately half a fringe. We can easily verify this result 
by comparing the rain-free and the rain-contaminated interfero 
grams. 
4 CONCLUSION 
In this paper we discussed the influence of the rain rate on SAR 
interferograms. Calculations of the path delay for different rain 
rate intensities was made by considering a physical model for the 
rain drops and applying the Rayleight approximation. The calcu 
lations do not include estimations of the delay due to the melting 
layer of precipitations and due to the precipitating cloud above the 
melting layer since they could be neglected. Simulated examples 
of interferograms considering perfectly flat areas were generated 
and showed that there is a strong increase in a propagation delay 
associated with rain rate and that, the rain can induce a consider 
able propagation delay of several centimeters causing translation 
of interferogram’s fringes. 
REFERENCES 
Abdelfattah, R. and Nicolas, J.M., 2002. Topographic SAR inter 
ferometry formulation for high-precision DEM generation, IEEE 
Transaction on Geoscience and Remote Sensing, vol. 40, NO. 11, 
pp. 2415-2426. 
Bringi, V.N. and Chandrasekar, V., 2001. Polarimetric Doppler 
Weather Radar. Principles and Applications, 1st ed. Cambridge, 
U.K.: Cambridge Univ. Press.