International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
  
polynomial parameters. Secondly, we add height displacement 
and finally get true image coordinates. 
2. METHODS 
It is usual to adopt following expressions for quadratic 
polynomial rectification. 
x-agtaj Xa; Yay X ^ ta4X Ya, Y? 
y=botb; X+b,Y+b1X"+b4XY+b;Y" ( 1 ) 
Where x,y lis image point coordinates, and correspondingly 
X,Y | is its ground point coordinates. a; and b; are unknowns. 
In this paper, we introduce height displacement caused by 
elevation which is corrected in advance and make use of (2) for 
polynomial orthorectification. 
x+dx=a,+a; X+a, Y+a;X"+a,XY+asY" 
y=bytb,X+b,Y+b,X"+b4XY+b;Y” (2) 
Where dx is height displacement caused by elevation. 
Displacement of radar image caused by terrain can be displayed 
in Fig 1. 
  
  
> ™ ; : X 
! 
e R i 1 
^ 
= i 1 
^ 
~ I 1 
~ 
- 1 ! 
^ J 4 
x 
x 1 
^ / 
> / 
x 
^ / 
x 
x / 
x / 
> / z 
S 7 
A 
/ SN 
Z SN 
  
Figure 1. Terrain influence to radar image 
c 
We presume that h is elevation of ground point P', and its image 
coordinate is X’= A R', where 4 is imaging scale and P is 
projective point in ground datum plane. The slant range of P can 
be approximately describes as following equation: 
R^ R-hcos Q 
Where 9 is the imaging angle of P'. Accordingly, displacement 
caused by elevation can be showed as: 
dx-X'-X* — Ahcos 
Suppose s is ground resolution, and we can realize following 
transform: 
dx * —hicosO is 
where 
COS Ü | (H-h)R 
SO 
dx = -h(H-h)/R/s (3) 
Equation 3 is approximate expression of height displacement, 
If it needs strict computation, the equation as follows can be 
used: 
dy z J(U - (0H -hy)e H^ -R 
JE TH Men (4) 
When there is a biggish Gand ground point is far away from 
ground nadir point(for ERS-2 image, when 0is 25 degree, the 
distance from ground point to ground nadir point is 323 
kilometers), height displacement may be affected by ellipticity 
of the earth. We can reduce such affection by correcting datum 
plane which includes the correction of slant angle 0 and flight 
height. In this way, according to normal of central tangential 
plane in cach frame, We can calculate slant angle 0, and change 
flight height into distance from imaging center to tangential 
plane. 
  
  
  
  
  
  
  
  
  
  
Figure 2. slope correction 
Showing as Fig 2: where the earth level surface is regarded as a 
sphere O with a semi diameter Re |A is projection of ground 
point toward image point in the level surface, and N is ground 
nadir point. 
We regard 3 as [1AOS. Then, in the surface of AOS, AN' is 
tangent of the sphere O across point A, and SN' pass through 
point S which is vertical line according to AN'. So N' and. 0' 
respectively express the new ground nadir point and slant angle. 
Where OB SALB is. intersection of OB and SA . because of 
SN'/AOUthen: 
EUN 
BAO=0' 
0'=arcsin[(Re+H)sin0/Re] 
H'=ResinBcos(0 B)/sin0 
Corrected H' replaces H in formula(3)or (4). It is modified 
height displacement corrective formula and expresses as (3) or 
(4) in order to convenient for description. 
If flight height is unknown, then dx is also unknown, and 
expression (2) must be modified. 
Thinking of expression (3), the new expression (2)is the follow: 
x+h"/R=agt a X+taY+ta; X +a4XY+ a; Y^-H-h/R 
y=bg +b; X+b,Y +b; X +b, XY +b Y? ( 
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