ul 2004 
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transformation from geodetic coordinates to Cartesian 
coordinates is given as 
P=(R+ h)cos(ÿ)cos(w ) 
P, = (R + h)cos(¢)sin(y) ao 
P. =[R(1-e")+ h]sin(4) 
where P,, P,, P. — coordinates of target point P in Earth center 
Cartesian coordinates system 
a = the semi-major axe of the Earth 
b = the semi-minor axe of the Earth 
e = AJ(a? * b?)/a? ,the earth numeric eccentricity 
R= JA — a^ sin (9) , local Earth's radius of curvature 
3.3 Calculating satellite positions 
For two satellite imaging the target point P, their positions need 
to be calculated. According to the geolocation principle of SAR 
image, the image coordinates of a target point (row and column 
numbers) can be calculated from its Cartesian coordinates (Py, Py, 
P.) by solving a system of the range equation and the Doppler 
equation. 
Assume that the target point P =(P,, P,, P.) is in a SAR image 
and has coordinates (row, col). We know that the satellite 
position is a function of azimuth time, namely rows number, and 
the satellite state vectors can be expressed by image coordinates 
(row, col) of the target P. So the range equation and the Doppler 
equation can be converted to an equation system with two 
independent variables: row & col. 
Let the time of the first state vector be the reference time origin. 
For the single look complex (SLC) data, the absolute time of the 
image rows is known by means of the PRF (Pulse Repeat 
Frequency, given in SLC header file) 
t(row ) = JOUE 4 (11) 
PRF 
where df — the difference of time of the first image row with 
respect to the time of the first state vector 
PRF = Pulse Repeat Frequency 
row = index of SAR image in azimuth direction 
The satellite position and its velocity can be expressed as 
function of the image rows: 
S (row) 7 [Sx (t (row)), Sy ( (row)), Sz (t (row))] (12) 
and 
Vs (row) = [Vx (1 (row)), Vy G (row)), Vz (t (row))] (13) 
where (S, (),S, (),S. ()) and (V. (0. V, (1), V. (1)) are satellite 
position and velocity respectively interpolated from the state 
vectors given in SLC header file with a cubic spline interpolation 
at the time /. 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004 
The slant range of target point P can be expressed by target 
column index col: 
r{col) =r, + col “spc (14) 
where r, 7 slant range of first column, 
spc = range resolution 
In order to achieve image coordinates (row, col) of the target 
point P, the following equations system must be solved: 
IS(row) - P. =r(col) range equation 
=f, Doppler equation 
A S(row) — P| 1 TE 7 
  
where the symbol - stands by the inner product of two vectors. 
By assuming the Doppler frequency is equal to zero, the image 
coordinates (row, col) of the point are retrieved using a 
minimization method with the pair of image center coordinates 
(rowc, colc) as an initial guess. 
So satellite position (S,, S, S.) in the Earth center Cartesian 
coordinates system can be obtained by the imagery time which 
is decided by row numbers of the target point. 
Here we calculate the slave statellite position with respect to the 
targe point by the coregistration warp function of master and 
slave images. This approach not only can decrease the 
geolocation errors in synthetic interferogram but also can 
eliminate the phase errors caused by coregistration when 
synthetic interferogram is used in D-InSAR processing. For the 
simulation example (see the next section), the geolocation 
errors are about in a resolution cell of ERS-1/2 image, namely 
3.9m in azimuth direction and 7.9m in range direction. Because 
the accuracy of co-registration is sub-pixel level, the position 
accuracy of the target point in slave SAR image is better than 
that of geolocation. 
The index of the target point P in slave SAR image equals to 
] 
row, a U, d, 
= :| row,, (16) 
col h oh 5 
cal, 
where row,,, col, = index of target point P in master image 
row,, col, = index of target point P in slave image 
dy. 4 O5 i ; 
* | warp matrix of master to slave image 
b, ois 
From the known positions of a target and the satellites in the 
Cartesian coordinates system, the distance of the target to each 
of the two satellites can be derived. Thus, the interferogram can 
be simulated. 
In radar image coordinates, row and col index of the target 
points form an irregular grid. In order to get a phase image in