lan Dowman 
  
data a priori error covariance C, is critical for proper weighting, and hence, for an optimal solution X with reliable 
error propagation P. 
The new replacement sensor model error propagation information is based on the concept of an adjustable (rational) 
polynomial. There are two possible variations. 
The polynomial with an image space adjustment is of the following form: 
u =(a0+alx+a2y+a3z+a4xy+..)+(Aa0+ Aalx+ Aa2y) 
v=(b0+blx+h2y+b3z+b4xy+..)+ (ADO + Ablx + Ab2y), and 
[5] 
AT =[Aa0 Aal Aa2 AbO Abl Ab2], where 
Aa0, .., Ab2 are the additional adjustment parameters. 
A total of six low order corrections (adjustment parameters) are assumed and defined as the elements of the adjustment 
vector A. Although the above 6 corrections are typical, any set of low order corrections can be defined. (Note that 
low order corrections can also be similarly applied to a rational polynomial, i.e., the ratio of two polynomials. 
However, corrections are only applied to the numerator.) 
The polynomial with a ground space adjustment is of the following form: 
u = (a0+alx+a2y+a3z+a4x' y+..) 
v = (b0+blx'+b2y'+b3z'+b4x'y'+..) , where 
x Ax 1 y -fpix 
: 6 
y'i=|4y|+|-y 1 a|y|, with [6] 
z! Az Be-—X1:4z 
A =l4x Ap dz e B y] 
A six element affine transformation consisting of small ground space corrections to the polynomial's independent 
variables are assumed. These corrections are defined as the elements of the adjustment vector A. Although the above 
6 corrections are typical, other ground space adjustment parameters can be defined. 
For both the image space adjustment and the ground space adjustment, the error covariance is defined as 
EfeAeA” |= Ci, where ^E" stands for expected value and "€" error. Typically the polynomial is actually 
unadjusted, therefore A=0. However, its value (zero) is in error. This error is to represent the error in the original 
sensor parameters, hence, C, # 0. 
C, is generated from Cg, the sensor parameter error covariance, in such a manner as to include the effects of all 
intra-image and inter-image sensor parameter uncertainties and their correlations: 
C 4 is generated such that B ACA pT =BgCg pf ,1.e., the matrix norm 
T T 
B14C4BA-BsCsBS 
  
  
  
  
is minimized, where the partial derivatives B A and Bg 
are generated over a grid of ground point locations at different elevations and [7] 
within the image footprints. B A is the partial of the image measurements M 
with respect to the adjustment vector A. 
In general, C, is generated from C. as above for all images with correlated support data errors. If there are m 
images, r sensor parameters per image, and q polynomial adjustable parameters per image, C, is an mr x mr matrix 
and C, an mq x mq matrix. In this case, the adjustment vector A refers to the collection of individual adjustment 
vectors A corresponding to the m images. 
  
262 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.