1 
< 
X 
1 
x s 
u 
a x" 
Y a 
= 
Y s 
+ R- 
V 
+ 
a Y 
1 
< 
N 
i 
Z s 
w 
a z 
where,t 0 is a reference time; a x , a Y , a z , b x , b Y , b z 
are a set of correcting parameters of the linear drift error. 
Equation 2 represents a strict coordinate relationship 
between the camera perspective center and the airborne 
GPS antenna phase center. The unknown parameters in 
this equation contain three components of camera 
position, three orientation angles ((p,o),K ) implicit in the 
matrix R, three components of the antenna offset, and six 
correction parameters of the linear drift error. In order to 
combine equation 2 with the base observation equations 
of conventional bundle block adjustment, the above 
described equation must be linearized, which can be 
expressed as 
x A l 
pc 
. ^x a y a z a 
dtpcoK 
A(p 
[zlX s ~ 
Au 
zla Y 
zlb Y 
y a 
- 
y a 
Aco 
+ 
^Y s 
+ R- 
A\ 
+ 
zla Y 
+ (t — t o ) - 
zlb Y 
z A J 
[z A J 
Ak 
L^ZsJ 
Aw 
zla z 
o' 
N 
1 
computed observed 
2.2 Combined Bundle Adjustment with 3D 
Coordinates of Airborne GPS Antenna 
Phase Center 
As described in the previous section, the error equations 
of the combined bundle adjustment with the GPS- 
determined camera positions are the combination that of 
the conventional bundle adjustment and equation 3. It can 
be written as the following matrix form: 
V x = Bx+Af + Cc 
V C =E X * 
V s = E c c 
V G = At + Rr + Dd-L G , weight P G 
where, 
V x , V c , V s , V G are the residual vectors to measured 
coordinates of the image points, ground 
coordinates of the observed control points, 
fictitious observations of the self-calibration 
parameters and GPS-determined camera 
positions, respectively. 
x.= [zlX AY zlZ] T is the correction vector to the 
approximate values set for ground coordinates 
of the photogrammetric points. 
t = [A<p Aco Ak z!X s z!Y s Z1Z S ] T is the 
correction vector to the exterior orientation 
parameters of aerial photographs. 
c = [a, a 2 a 3 ] T is the vector to the self- 
calibration parameters selected. 
r = [Au A\ zlw] T is the correction vector to the 
measurements of airborne GPS antenna offset. 
d = [a x a Y a z b x b Y b z ] T is the vector to 
correcting parameters of the linear drift error. It 
should be noted that unknown d represents 
-L x , weight E 
-L c , weight P c 
-L s> weight P s 
actually the correction vector of 
[Aa x zla Y zla z zlb x Ab Y <db z ] T (please see 
equation 3). In general, the initial values of the 
correcting parameters are often set to zero, so 
the solution of unknown d equals the vector 
of [a x a Y a z b x b Y b z ] T in essence. In 
order to keep the identical expression as 
unknown c , the former is given here. 
A, B, C are three coefficient matrices whose 
elements are the partial derivatives of the 
collinearity condition equations with self 
calibration model with respect to the unknowns 
t, X, c , respectively. 
A, R, D are three coefficient matrices with respect 
to the unknowns t, r, d related to equation 3. 
In 1994, Yuan derived the partial derivatives 
from equation 2 [Yuan, 1994]. 
E, E x , E c are unit matrices, respectively. 
*-(*) 
y-(y) 
is the misclosure vector associated 
with image points, where, x, y are image 
coordinates; (x), (y) are image coordinates 
computed in collinearity equations. 
L c is the misclosure vector associated with the 
ground control points. 
L s is the misclosure vector associated with the 
selecting additional parameters. 
L g 
X a -(X a ) 
y a -(Y a ) 
_ Z a-(Z a ). 
is the misclosure 
vector 
associated with GPS camera stations, where, 
(X A ), (Y a ), (Z a ) are the coordinates of the 
antenna phase centers computed in equations 2. 
P c = —j-E is the weight matrix for the ground 
a c 
control points, where, cr 0 is the standard 
deviation of unit weight, which can be