Geometry of locating sounds from differences in travel time: isodiachrons

Calling animals may be located from measurements of the differences in acoustic travel time at pairs of receivers. For inhomogeneous fields of speed, locations can be made with better accuracy when the location algorithm allows the speed to vary from path to path. A new geometrical shape, called an isodiachron, is described. It is the locus of points corresponding to a constant difference in travel time along straight paths between the animal and two receivers. Its properties allow an interpretation for locations when the speed differs from path to path. An algorithm has been developed for finding the location of calling animals by intersecting isodiachrons from data collected at pairs of receivers. When the sound speed field is spatially homogeneous, isodiachrons become hyperboloids. Unlike a hyperboloid that extends to infinity, an isodiachron is confined to a finite region of space when the speeds differ between the animal and each of two receivers. Its shape is significantly different than a hyperboloid for cases of practical interest. Isodiachrons can be used to better understand locations of calling animals and other sounds in the sea, Earth, and air.