Dimension Reduction in Location Estimation—the Need for Variable Propagation Speed

Abstract

Calling mammals, ships, and many other objects have been commonly located during the last century with two-dimensional (2D) models from measurements of a signal’s Time Differences of Arrivals (TDOA) when the objects are not on the 2D surface. The overwhelmingly common method for locating signals with 2D models takes signal speed as constant and location is derived by intersecting hyperbolas. However, when correct locations are required for 2D models, the speed used to derive location must depend on the geodesic distance along the 2D model surface between the object and the instrument. For example, when this distance is zero, the speed needed for correct location must also be zero. The dimension reduction from three to two introduces large errors in 2D models both near and far from the instruments unless the variable speeds induced by the dimensional reduction are explicitly accounted for. In light of these findings, methods are derived for generating extremely reliable confidence intervals for estimated locations in 2D models and identifying regions of the 2D model where a 3D model is needed. Because speeds needed for correct location are spatially inhomogeneous in the extreme, isodiachrons emerge as a natural geometry for interpreting location instead of hyperbolas. These issues are caused by choice of coordinates, and the same phenomena occur when coordinate transformations are applied in other fields of physics.