Derives mathematical formulas for inferring the status of a hardened point target attacked by a nuclear weapon from the distance between the detonation and the target. With observational inaccuracy described by a circular normal distribution, the probability of the target being damaged is expressed by the target damage function used for perfect observations, with the weapon/target parameters suitably transformed. Based on the Defense Intelligence Agency damage function, the formula applies to others also. Results are extended to derive the probability of target damage as a function of the number of weapons shot at the target and the number of bomb damage assessment observations. Distributions of miss distance observations are constructed, and models are proposed for finding the distributions for single shoot-look-shoot campaigns with noisy observations and for multiple campaigns with exact observations. Results are combined with the damage probability formula to compute the expected fraction of targets damaged for any number of shoot-look sequences with a specified stopping rule.