Visualization of spherical tangency solutions for locating a source point from the clock time at four receiver locations

John C. Bancroft

Previous work (Bancroft and Du, 2006) derived the source point of an event from the first arrival times at four receiver locations. The 3D solution was based on the 2D problem in which the source was located from three receiver locations. That solution required the tangency of a source circle with three circles whose radii were proportional to the receiver clock-times. This 2D solution could be visualized, since the circles could be constructed and or drawn on a plane surface. However the 3D solution requires the tangency of a source sphere with four spheres whose radii are also proportional to the receiver clock-times. This paper demonstrates a method of visualizing the 3D solution. This work has numerous applications that range over well fraccing, the monitoring of hazardous sites, sniper locating, global positioning, and Kirchhoff depth migrations which require the transfer of traveltimes, that are estimated along raypaths, to those on a grid.