The full multi-state vector error covariance matrix: why needed and its practical representation

Abstract

Whether statistically representing the errors in the estimates of sensor metadata associated with a set of images, or statistically representing the errors in the estimates of 3D location associated with a set of ground points, the corresponding “full” multi-state vector error covariance matrix is critical to exploitation of the data. For sensor metadata, the individual state vectors typically correspond to sensor position and attitude of an image. These state vectors, along with their corresponding full error covariance matrix, are required for optimal down-stream exploitation of the image(s), such as for the stereo extraction of a 3D target location and its corresponding predicted accuracy. In this example, the full error covariance matrix statistically represents the sensor errors for each of the two images as well as the correlation (similarity) of errors between the two images. For ground locations, the individual state vectors typically correspond to 3D location. The corresponding full error covariance matrix statistically represents the location errors in each of the ground points as well as the correlation (similarity) of errors between any pair of the ground points. It is required in order to compute reliable estimates of relative accuracy between arbitrary ground point pairs, and for the proper weighting of the ground points when used as control, in for example, a fusion process. This paper details the above, and presents practical methods for the representation of the full error covariance matrix, ranging from direct representation with large bandwidth requirements, to high-fidelity approximation methods with small bandwidth requirements.