The theory of single sensor altitude determination

Beginning with the phenomenological equations of remote sensing and atmospheric transmission, we show that the altitude of a bright point-source target (such as a missile or afterburning aircraft) within the atmosphere can be determined with a single spaceborne multispectral sensor. Our calculation assumes that the target signal can be isolated from its background. And we do not consider noise. Essentially, we are measuring an attenuating molecules column density along the line of sight to determine range to the target. The range to the target which satisfies our equations is then transformed to altitude through our column density characterization. The column density measurement requires the sensor to simultaneously (or alternately) collect in two or more spectral bands associated with the attenuating molecule, and depends on the key assumptions that (1) we know the relative spectral emission of the target (i.e. we can classify the target) and (2) we know the attenuating molecule's vertical number density distribution. We evaluate the effects of uncertainty in our knowledge of these two key parameters by using a hydrocarbon-burning missile and carbon dioxide as examples. We demonstrate that the error in our determination of altitude is a small fraction of one scale height of the attenuating molecule (assuming an exponentially characterized atmosphere).