tSPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS", Bangkok, May 23-25, 2001 
179 
relies on a Lambertian assumption for the terrain backscatter 
model. Then a single-line integral process is applied to calculate 
each pixel altitude. But the result is still badly contaminated with 
noise. Finally the multi-line integral processing with various 
directions and the simulated annealing algorithm are orderly 
carried out to improve the single-line integral processing result. 
The experiment results are promising in many geographic 
applications. 
2. System parameter of SAR image 
Shading is the variation of brightness/gray exhibited in 
images. SAR image is very sensitive to the terrain shape. The 
tittle undulation of the terrain may induce the change of the 
image gray distribution and/or the texture characteristics. The 
backscatter describes the relationship of the shading and the 
interaction between SAR and the earth's surface. For the SAR 
image, the backscatter of the surface depends not only on the 
imaged ground orientation, but also on its cover type. The exact 
mathematical equation of the backscatter model is generally 
unknown. Many backscatter models and estimation methods 
have been proposed. The selection of suitable one from these 
models or the estimation of the relevant parameters of the model 
is quite difficult while no extra data, such as a digital elevation 
model (DEM), is available. 
Although more sophisticated models should take into 
account the SAR and surface interaction, the Lambertian model 
for homogeneous surfaces is usually selected for simplification. 
Without taking into account all the details of the SAR system, we 
consider the system and ground parameters as followings. With 
regard to the radar system, we selected three parametersas 
shown as Fig. 1, d^R^Rg. e is the angle of incidence of the 
electromagnetic wave emitted by the radar, and calculated from 
the nadir, R^ and R a is the distance resolution and the azimuthal 
resolution of a flat ground pixel respectively. For simplification, 
we approximate the electromagnetic waves emitted by the radar, 
as plane wave. 
With regard to the geometric parameters of the ground, we 
must define the motion of “parcel”. The “ parcel ” is a piece of 
the ground surface, which is intercepted by a resolution cell of 
the radar, and imaged in a pixel. This parcel is considered as a 
plane surface (Fig. 2). 
Then the expressions of the geometric relationship are 
shown as 
L(a) = 
Rj sin(6>) 
sin|# - a 
(1) 
L(ß) = 
cos(ß) 
(2) 
A(a, P) = L(a) x L(j3) ^ 
Using the assumption of a Lambertian diffusion, the 
backscattered intensity (l r ) may be written as 
I r (a ß)= Kcr Q cos 2 (#-a)cos 2 (ß)A(a ß) 
(4) 
Here K is the calibration constant of the radar. 
3. Reconstruction Procedures 
3.1 the integral of the elevation increments along the image 
line 
In order to solve Equation (4), some assumptions are 
necessary because there are too many unkown variables for 
each pixel. At first, we assume the area homogeneous enough 
so that oo is a constant in the test area. This is indeed a 
realistic assumption in the tropical forest like Amazonian area 
where there is a quite regular and dense cover. Then, we 
suppose that the terrain is mostly flat and without noticeable 
skew. Again this is a realistic assumption in the area of interest 
for our application. We deduce Q(a, /3) as Equation (5): 
ß(« 
p) P) 
h 
sin#cos 2 (# -a) 0 
J r COS ß 
cos <9 sin# 
(5) 
Equation 5 is still depended on two unknowns and cannot be 
solved exactly in a and /3 for each pixel. We will proceed in two 
steps. At first, we supposed angle /3=0 and obtain a simplified 
Equation (6): 
«» 
COS #Sin|6'-Of (6) 
a could be derived from Equation (6). Considering equation (1), 
the elevation increment 5h for each pixel is expressed as 
Equation (7). 
, t \ . R d sin 6 sin a 
oh = L(a Jsin a = 
sin 
a 
(7) 
The elevation may be derived from the single line integral of Sh.