The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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3 3 3 
yyya t P‘U n H k n 
n ( n r TT \ Inn n 
r - P\ ’ L n ’ n n) _ «-=0 J=0 *=0 
Pi (p„,k,h„) ~ ±±± biPX H: 
/=0 7=0 *=0 
„ / p 7 LJ \ ¿mM LmU Inn n 
_ Pl\ r n’ L n’ M n) _ /=0 y=0 *=« _ 
c_ - 
/=0 7=0 ¿=0 
(1) 
The image coordinate and ground coordinate 
, whose value are within [-1, +1], are both the 
standardization coordinates through translation and scale, for 
the purpose of reduce the rounding errors in calculation because 
of quantitative difference. The unit of is pixel; 
P n , L n are the coordinates of WGS84 with unit degree; H is 
the geodetic height with unit meter. (P n ,L n ,H n ) can be 
expressed as equation (2): 
p x =a x + a 2 L n + a 2 P n +a 4 H n +a 5 L n P n 
+a 6 L n H n + a 1 P n H n + a % L 2 + a 9 P n 2 + a w H 2 + a xx P n L n H n + a n L 2 
+a Xi L n P 2 +a lA L n H 2 + a xs L n 2 P n + a X6 P n 2 + a xl P n H 2 + a xs L 2 H n 
+a X9 P 2 H n +a 20 H n i 
(2) 
Where: 
a t is the polynomial coefficients 
(/ = 1,2,...20\i 2 j,k = 0,1,2,3) , other polynomials have 
the similar expression. 
The terrain dependent approach is to calculate the 80 
parameters of RPC using GCP from field survey. For 
connivance in the following expression, the (P n , L n , H n ) are 
shorted for (P,L,H) , and for (/% c) , so equation 
(1) and (2) can be expressed as (3): 
1 
L 
P 
H 
P 2 H H 5 
rL 
rP 
rP 2 H 
rH i 1 
B 
B 
B 
B 
B B 
B 
B 
B 
» 1 
1 
L 
P 
H 
P 2 H H 5 
cL 
cP 
cP 2 H 
cH 1 
v<r 
D 
D 
D 
D 
D D 
D 
D 
D 
D 
(4) 
Where: 
M 1 
Z Y 
X •• 
• Y 3 ^T 3 )C(1 b x •• 
• K) T 
J =i a o 
a x •• 
0| 9 
b x b 2 ••• 6,9 ) 
D = { ■ 
Z Y 
X •• 
• Y 3 X 3 )^l d, 
■: dj 
II 
* 
c> •• 
C, 9 
d x d 2 ••• <^ 19 ) 
(5) 
Suppose there are n GCP, the error equation is: 
V = WTI-WG 
Where: 
" 1 
w = 
0 
0 
0 0 
D, 
0 
0 
0 
o ... 0 0 ••• — 
L aJ 
"l ••• -r x Hl 0 ••• 0 ■ 
1 ••• -r n Hl 0 ••• 0 
rin ^ n n 
' 0 ••• 0 1 ••• -c,Hf 
0 ••• 0 1 ••• -cHl 
_ W n _ 
/=[û^ ••• t\ ••• bfl q CjQ ••• djq] 
{\ L P H •• 
• P 2 H 
H'fa 
°2 * 
- “J 
(l L P H ■ 
•• P 2 H 
H'W 
6, •• 
■ h ni 
(l L P H ■ 
• P 2 H 
^2 * 
■■ cj 
(\ L P H ■ 
• P 2 H 
"’ft 1 
d, - 
■ dj 
(3) 
If the n observations are unit weight observations, then the 
normal equation is: 
Linearization of equation (3) (Tao and Hu, 2001) we could get 
equation (4): 
T t W 2 TI-T t W 2 G = 0 
Then we get the coefficient matrix I: 
(7)