Locating Objects with Signal Times Amongst Shadows and Black Holes in Two-Dimensional Models

Abstract

Calling mammals, ships, and many other objects have been commonly located during the last century with two-dimensional (2D) models from measurements of signal time even when the objects are not on the 2D surface. The overwhelmingly common method for locating signals with 2D models takes signal speed as constant. Distance is computed by multiplying this speed by signal time. For monostatic, bistatic, and Time Differences of Arrival (TDOA) measurements, the distances constrain locations to circles, ellipses, and hyperbolas respectively, whose intersections yield location. However, the speed needed to obtain correct locations depends on the horizontal separation between object and instrument. In fact, if their horizontal separation is zero the speed needed for correct location must also be zero. In light of this singularity, methods are derived for generating extremely reliable confidence intervals for location and identifying regions of the 2D model where a 3D model is needed. Because speeds needed for correct location are spatially in-homogeneous in the extreme, isosigmachrons and isodiachrons emerge as natural geometries for interpreting location instead of ellipses and hyperbolas. These issues are caused by choice of coordinates, and the same phenomena occur in general relativity regarding the speed of light and black holes.