Supersonic bullet trajectory estimation using ballistic shock wave arrivals at an acoustic sensor array

One approach to locate the point of fire of a supersonic bullet is to first estimate the trajectory of the bullet and then trace the trajectory back to topographic or man-made obstructions on a digital map. The supersonic flight of a bullet generates a ballistic shock wave, and the trajectory of the bullet can be estimated by measuring the time delay between the shock wave arrivals at each sensor pair of an acoustic array and using an exterior ballistics model for the bullet to account for its decreasing speed with the distance travelled. In this paper, the bullet trajectory estimation problem is formulated, followed by a Cramer-Rao lower bound error analysis. A nonlinear least-squares (NLS) solution to the bullet trajectory estimation problem is then described, which assumes the ballistic constant of the bullet is known a priori. Any uncertainty in the ballistic constant will degrade the accuracy of the bullet trajectory estimation and subsequently the localization accuracy for the point of fire. The performance of the NLS method when the ballistic constant is exactly known and the degrading effect of an erroneous ballistic constant are evaluated using both simulated data and real data.