The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
264 
where R s and R ( are the satellite and target position vectors, 
respectively, and “ • ” is the vector dot product. 
2.2 Doppler equation 
the slant range R , the satellite position R s , the target 
position Rt, the Doppler frequency f D , the satellite velocity 
V and the mean normal height h, yields 
The return echo data from a point target is shifted in frequency 
by an amount proportional to the relative velocity between the 
satellite and target. The equation describing the Doppler shift is 
given by 
fo = 
2 
1R 
(K-V,).(R s -R,) 
(2) 
where f D is the Doppler frequency associated with the return 
echo data, /1 is the radar wavelength, V % is the satellite 
velocity vector, V is the target velocity vector. The WGS84 
coordinate system is adopted in this paper, so the V t equals to 
zero. 
The Doppler equation given by Eq. (2) defines the plane of the 
center of the radar beam at a specific instant in time. The 
intersection of this plane with the assumed earth model is the 
locus of potential targets as shown in Fig. 1. 
2.3 Earth model equation 
The equation describing the approximate shape of earth is 
given by 
+ 
y, 
+ 
t _ 
(a + hr (a + hy 
= 1 
a = 6378.137£m 
b = 6356.152km 
(3) 
where X t ,y t ,Z ( are the 3-d coordinate components of the 
RdR = Ax s ,A dx s , + Ay sl Ady sl + Az s ,Adz s , (4) 
? d fo = hc,A, + 4V,Ay + Az s ,dv 2 
2 (5) 
+ v Adx„ + v,A dy„ + v z Adz„ 
(a + h)b 2 dh = x t b 2 dx+ y,b 2 dy f 
. , 2 («) 
+ z t (a + h) dz t 
Ax„ =x,-x, 
A dx st = dx s - dx t 
where X s , y s , Z s and V x , V y , V_ are the components of the 
satellite position R s and velocity V s in the x, y and z 
direction, respectively. AjF ç/ and AZ s[ are the same as 
Ax st ,but for y and z direction. dx t , dy t and dz ( are pixel 
location uncertainties in x, y and z direction, respectively. 
dx s , dy s , dz s , dv x , dv y , dv : , dR, dh and df d are 
the uncertainties of the satellite orbit measurements, 
range measurements caused by atmospheric delay, the 
normal height measure and the Doppler information, 
respectively. Ady st and Adz st are the same as Adx sl ,but 
for y and z direction. Using equations (4)~(6), the 
dx t , dy t , dz t can be expressed as the linear functions of dx s , 
dy s , dz s , dv x , dv y , dv z , dR , dh and df d whose 
coefficients, symboled as ,h,i, can be computed 
when the initial values, such as h , R ,V s , Rs and Rt, are 
known. 
target position vector Rt in the x, y and z direction, 
respectively, a is the semi-major axis, b is the semi-minor axis, 
and h is the mean normal height. 
3. ERROR ANALYSIS 
3.1 Error models 
3.2 Calculation and analysis 
Table 1 lists three scenes test data of the ERS-2 C-band SAR 
from Tibet Zhang autonomous region, Nanjing city and Taiwan 
province. Based on the three test data, three groups of error 
coefficients are computed, respectively. Only the first group 
for Tibet Zhang are given in table 2 due to space constraint in 
this paper. 
Base on the error propagation theory, the error models can be 
derived by differentiating the equations (1 )~(3) with respect to 
Site R(m) h(m) R s (mx 10 6 ) F s (mxl0 3 /s) R, (mx 10 6 ) 
TZ 
NJ 
TW 
84927 
84928 
84718 
4827 
18 
759 
-0.441 5.808 4.162 
-3.293 5.152 3.725 
-3.616 5.491 2.838 
1.459 4.398 -5.963 
-0.594 4.168 -6.271 
-0.201 3.373 -6.758 
-0.093 5.146 3.760 
-2.664 4.707 3.370 
-2.958 5.021 2.584 
Table 1. The parameters of SAR images in three sites