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Ian Dowman 
5. IMPROVED ERROR PROPAGATION 
5.1 Background 
As mentioned earlier, error propagation information is extremely limited in current replacement sensor models, i.e., 
polynomials and rational polynomials fit to the original rigorous sensor model’s ground-to-image relationship. In 
particular, for a given image, the USM’s error propagation consists of a summary of expected ground point extraction 
accuracy assuming a monoscopic extraction or a stereoscopic extraction based on the image and a specified stereo 
mate. Thus, USM error propagation information is limited to a ground space based summary over the area within an 
image footprint. It does not provide image space based error propagation information required when performing 
extractions using this image with additional images besides the specified stereo mate. 
Rigorous and complete image space based error propagation information is critical for optimal geopositioning and 
reliable error propagation. It allows for minimum variance ground point solutions (extractions) and for reliable 
absolute and relative accuracy predictions accompanying those solutions. The original rigorous sensor model 
provides the required image space based error propagation information via the sensor support data error covariance 
(when available) for that image and any other images with correlated errors. Correlations can be introduced either by 
imaging on the same pass or by subsequent triangulation of the sensor support data, 
A new method has been developed by one of the authors (Dolloff) to provide image space based error propagation 
information to accompany the replacement sensor model’s (rational) polynomial. This information and the algorithm 
to generate it are summarized below. Initial testing results are also summarized. They indicate the method’s promise, 
although further analyses and testing are required and are on going. 
5.2 Description of the method 
In the new method, the error propagation information for a replacement sensor model is captured in an error 
covariance matrix C, associated with a pre-defined vector of adjustable parameters A of the polynomial. This error 
covariance is generated at the same time the polynomial is generated. The generation process requires the original 
rigorous sensor model and Cg, the a priori error covariance of the model’s support data parameters S (e.g., sensor 
position, attitude, etc.). 
Prior to providing specifics of the above process, a representative geopositioning solution technique is first presented 
in order to understand the role of the original rigorous sensor model and its support data error covariance. The 
following is a simultaneous Best Linear Unbiased Estimate for multiple 3-dimensional ground points using conjugate 
image measurements of the ground points from multiple images: 
1 
X - X9 * AX. , where. óX = PBLWZ , p-|^;! «BIwa,] 
1 
W= Ir + Bg Cs BI | , and Z-M-Mg ; iterate as necessary, i.e., [4] 
set X 0 to current solution X and redo solution. 
The various vectors and matrices are defined as follows: 
X — vector of 3-dimensional ground coordinates of multiple ground points 
P - a posteriori error covariance of solution vector X 
M — image measurement vector 
R — mensuration error covariance 
B — partial derivatives of measurements M with respect to X (subscript x) or with respect to S (subscript s), the sensor 
support data parameters for all images 
C, — a priori error covariance of S 
The subscript *o" on X, P, and M corresponds to a priori (initial) values. In particular, the a priori M is generated 
using the rigorous sensor model's ground-to-image transformation. The weight matrix W provides the mechanism to 
weight the various components of M in a manner inversely proportional to the combined measurement uncertainty due 
to mensuration errors and sensor support data errors propagated to image space. The rigorous sensor model's support 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 261