Justification of the TOA Method for Lightning Positioning and Algorithm of Application

The paper considers a mathematical problem underlying the problem of positioning a lightning discharge by the TOA method for three stations. The mathematical problem is formulated as follows. Three points are fixed on the unit sphere. Three spherical segments are introduced, coming out of the given points in the direction of some unknown point, whose position on the sphere must be determined. The requirement must be met: the distance from each point to an unknown point must be equal to the sum of the length of the segment coming out of this point, and some increment, the same for all three segments. In the article, the conditions for the solvability of the problem are established using the methods of spherical trigonometry and vector algebra. It is proved that this geometric problem is always solvable when they are fulfilled. The number of solutions is two, except in rare cases where there is only one solution. An algorithm for the numerical solution of the problem of positioning a lightning discharge has been developed.