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 molecule's column density along the line of sight, and the value of the molecule's vertical number density which satisfies our equations to determine the target's altitude. 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). We suggest how this method could be applied to rapidly determine a missile's direction of flight.