2 2 2) 
E(f(X * AX) - fps elax[" Gs) C 
Here H and c are two important parameters to control the 
model, and the meanings of them can be illustrated in 1D case 
[Zhu Q., 1995]. 
Let 
Y - f(0- /(-X) 
Z - f(X)- f(0) 
be the past and future increment of f, respectively, thus the 
correlation coefficient of Y and Z is described as 
E(YZ) 
| = 970 1 (3) 
GG 
pY.Z)- 
It indicates that A is a measure of correlation between 
increments of f£. The smaller His, the rougher f shows in 
shape. 
In addition, let AX =1, then (2) yields 
Elf + = 0° (4) 
(4) indicates that c is an index of the average difference of f 
at unit distance, or is regarded as average slope. 
Generalizing this to 2D case, the terrain surface can be modeled 
as a 2D fBm. Then H reflects the roughness of the surface, 
and c reflects the average slope of the surface. In general, the 
flat region is with big H and small c. The contrary case is 
with the high mountain region. And the middle mountain or hill 
region is between the former two cases. 
Therefore, the different types of the terrain can be generated by 
adjusting parameters 7 and c , and used to validate the 
interferogram simulation algorithm. 
2.2 Algorithm 
Here we use the midpoint displacement algorithm [Zhu Q., 
1995] to simulate terrain data. The main idea of this algorithm 
is to form the final regular elevation grid by recursive 
subdivision. In each subdivision, the elevation of a point h is 
obtained by interpolating its four neighbors /;(k 21,2,3,4) and 
a random stationary displacement A, , where the displacement 
is determined by H , & and the recursive number i. 
    
  
  
  
     
  
   
   
  
  
   
   
  
  
    
    
     
    
   
  
  
  
   
  
   
   
  
   
   
   
  
  
    
   
   
   
  
  
  
   
    
   
   
  
   
    
   
     
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004 
x et EEG pnm 
Tio ON | (b) (c) 
Figure 1. (a) The ith subdivision, 
(b) The (i+ /)th subdivision d. 4 d, 
(c) The (i+2)th subdivision, d. id, 
  
In Figure 1, the ith subdivision is illustrated in (a) with the grid 
interval d, ; in (b), the (i*7)th subdivision is completed by 
interpolating each four neighbors to get the red points and then 
form the red grid that rotate 45 degrees against the ith one; (c) 
is the (i+2)th subdivision, with the blue points interpolated, the 
grid becomes denser with the interval of d,,, as half as d, . 
During the subdivision, the interpolation is formulated as 
follows, 
  
1 4 
hl HA, (5) 
k=l 
A, ~ N(0,v?) (6) 
vedo us (7) 
dz #4, (8) 
So repeating this process can reach a small enough grid interval 
d 
! 
3. INTERFEROGRAM SIMULATION 
3.1 Principle 
It is well-known that the distance between the satellite and the 
target on the ground can be determined accurately by the phase 
information of the SAR image. Based on the observation, the 
phase difference of two SAR images is defined as interferogram 
@ given by 
4 
$-y cy mom on) (9) 
where 7,, r; = the distances between a target point P and satellite 
S, S», respectively. 
A = wavelength of radar. 
The simulation of interferogram relies therefore on the accurate 
determination of the range difference between two imagery 
distances. 
3.2 Target point position in Cartesian coordinates system 
The DEM data (latitude 9 , longitude v , height /) are generally 
defined in geodetic coordinates. For the simulation, the data 
should be firstly converted to the corresponding (P,, P,, P.) in 
the Earth Center Cartesian coordinates system. The 
  
   
International 
transformatic 
coordinates i 
where P. P 
Cart: 
a t 
b= il 
e = - 
R= 
3.3 Calcul: 
For two sate 
to be calcul: 
image, the 1 
numbers) ca 
P.) by solvi 
equation. 
Assume tha 
and has co 
position is a 
the satellite 
(row, col) o 
equation ca 
independent 
Let the time 
For the sing 
image rows 
Frequency, 
where dt = 
PRF 
row 
The satelli 
function of 
and 
Vs 
where (S, 
position ar 
vectors giv 
at the time