Analytical and graphical determination of the trajectory of a fireball using seismic data

In this paper we discuss two methods, one analytical and the other graphical, to determine the trajectory of a fireball using the arrival times of atmospheric shock waves recorded by a seismic network. In the analytical method the trajectory and the raypaths are assumed to be straight and we solve for the fireball velocity, the azimuth (ϕϕ) and elevation angle (δδ) of the trajectory, the coordinates of the intersection of the trajectory with the earth's surface, and the corresponding intersection time (t0t0). Because the problem is nonlinear, we solve it iteratively. The fireball velocity cannot be determined uniquely, and trades off with t0t0. The graphical method is based on the drawing of contours of arrival times, which should be elliptical for fireball shock waves. If the distribution of seismic stations is appropriate, the horizontal projection of the fireball is given by the axis of symmetry of the contours, which allows the estimation of ϕϕ, while δδ can be estimated from the spacing between contours along the symmetry axis. Application of the two methods to data from four fireballs shows that the graphically derived parameters can be within a few degrees of the analytical parameters. In addition, a fireball recorded in the Czech Republic has reliable trajectory parameters derived from video recordings, which allows an independent assessment of the quality of the parameters determined analytically. In particular, ϕϕ and δδ have errors of 1.7∘ and 1.3∘, respectively, which are not particularly large considering that the station distribution was not favorable.