794 
• The base vector b between the forward and aft views 
of any specific ground feature is given by: b = pk—Pj- 
• Anv specific ground feature (e.g. control point) is 
represented by the vector g. 
• The difference vector v between the sensor at some 
instant (e.g. p) and a specific ground feature g is 
given by v = g — p, where p may represent either 
the position of the forward view pj or that of the aft 
view pic. 
The photogrammetric condition equations are de 
scribed by: 
Collinearity: For ground control points, the pointing 
vector x (i.e. x is either x} or x*), must be collinear 
with the corresponding sensor to ground feature dif 
ference vector v. Another way of expressing this con 
dition is to say that the vector cross product of the 
two vectors is zero. This is a vector equation given 
by 
x x v = 0, (1) 
where 0 denotes the null vector. 
Coplanarity: For ground features of unknown coordi 
nates which appear in two views, (also known as 
stereo conjugate points) the coplanarity condition is 
a statement of the fact that the vector triple product 
of the two pointing vectors and the base vector be 
tween the views has a value of zero. This is a scalar 
equation given by 
xj x Xk • b = 0 
or d • b = 0, (2) 
where d = Xj x x^- 
Although Figure 1 illustrates the geometry for the case of 
forward and aft looking channels, it should be noted that 
any combination of forward, nadir and aft views may be 
used in the analysis. 
The objective of the photogrammetric solution which 
utilizes these condition equations is to determine time 
varying position correction terms identified as 6p(t) for 
the data contained in the vectors p(t) and angle correc 
tion terms identified as 6a(i) for the data in the vectors 
a(t). There are several excellent references describing in 
detail the nature of inertial navigation system errors, thus 
no attempt will be made in this paper to show the deriva 
tion of the error model (Britting, 1971, Broxmeyer, 1964). 
Since the position errors inherent in the inertial naviga 
tion system have very low rates of change and the time 
periods of typical flight lines are relatively short, the po 
sition correction term 8p(t) may be approximated by a 
low order time-dependent polynomial, that is, 
6p = 8po + <5pi< -|- 6p 2 i 2 + h 8p„t n , (3) 
where the variable t represents time. It has been found 
that for the majority of data sets evaluated to date at 
CCRS, a value of 1 for n in the above equation is sufficient 
to describe the position errors. 
The attitude errors represent a hardware misalignment 
between the inertial navigation system coordinate frame 
and the sensor coordinate frame plus any angular drift 
errors in the inertial system. The attitude errors could be 
approximated in a similar manner to the position errors, 
however that has not been necessary for the tests done to 
date; a simple constant term 6ao has been sufficient. 
Equations (1) and (2) contain non-linear functions of 
the attitude angles dj and a* and thus linear approxima 
tions to the equations must be derived in order to for 
mulate a least-squares solution to solve for the required 
correction terms. The linearization of the equations is 
achieved by taking the first term of the Taylor Series ex 
pansion of each equation. Since there are vector and ma 
trix elements in the equations, the definitions of the dif 
ferentials developed in the references (Gelb, 1974, p. 22 
and Britting, 1971, p. 27) are very useful. 
Note that in the equations that follow, an upper case 
letter represents a skew-symmetric matrix containing the 
elements of the vector defined by the same (lower case) 
letter. That is, 
’ v x 
0 
Vz 
~ v y 
for V = 
v y 
Vz 
, then V = 
-v z 
. v v 
0 
-v x 
Vx 
0 
The results from linearizing the Collinearity Condition, 
Equation (1), assuming a first order position error model 
from Equation (3) are given by 
[X Xt VX] 
<5 Pu 
6p\ 
6 do 
[x x v], 
(4) 
and for the Coplanarity Condition, Equation (2), 
[o r 
(t* 
-ij) b'iXjXk-XkXj)] 
8po 
<5Pi 
6ao 
[ b • cf]. 
( 5 ) 
In the equations above, the superscript symbol T on a 
vector represents the transpose of that vector. Although 
equations (4) and (5) above are derived for the case of a 
single flight line, it is a simple matter to identify the cor 
rection terms for each of several flight lines by means of an 
additional subscript and to expand the set of equations to 
accommodate the multiple flight lines in a simultaneous 
solution. That is, the terms 6po,6pi and 6a<j in equations 
(4) and (5) become 6po,-, Spu and Saoi where the subscript 
i indicates which flight line the terms are associated with. 
In this situation, the coefficients in the left-hand array in 
equation (5) change form slightly. As mentioned previ 
ously, these equations are then assembled into a larger 
array in a least squares solution which is initialized at 
some starting point and then iteratively adjusted using 
the values computed for the correction terms.