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.

中文翻译：

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位置估计中的降维-可变传播速度的需求

摘要

在上个世纪中，通常使用二维（2D）模型来定位哺乳动物，轮船和许多其他物体，这些模型是根据物体不在2D表面时的信号到达时间差（TDOA）的测量得出的。使用2D模型定位信号的绝大多数方法都以信号速度为常数，并且通过与双曲线相交得出位置。但是，当2D模型需要正确的位置时，用于得出位置的速度必须取决于对象和仪器之间沿着2D模型表面的测地距离。例如，当该距离为零时，正确定位所需的速度也必须为零。从3缩小到2的尺寸缩减会在2D模型中向远离或远离仪器的方向引入较大的误差，除非明确考虑了由尺寸缩减引起的可变速度。根据这些发现，得出了用于为2D模型中的估计位置生成极其可靠的置信区间并识别2D模型中需要3D模型的区域的方法。由于正确定位所需的速度在极端情况下在空间上是不均匀的，因此等时线作为解释位置而不是双曲线的自然几何出现。这些问题是由坐标的选择引起的，并且在物理的其他领域中应用坐标变换时也会发生相同的现象。推导了用于为2D模型中的估计位置生成极其可靠的置信区间并识别2D模型中需要3D模型的区域的方法。由于正确定位所需的速度在极端情况下在空间上是不均匀的，因此等时线作为解释位置而不是双曲线的自然几何出现。这些问题是由坐标的选择引起的，并且在物理的其他领域中应用坐标变换时也会发生相同的现象。推导出了用于为2D模型中的估计位置生成极其可靠的置信区间并识别2D模型中需要3D模型的区域的方法。由于正确定位所需的速度在极端情况下在空间上是不均匀的，因此等时线作为一种自然的几何图形出现，用于解释位置而不是双曲线。这些问题是由坐标选择引起的，当坐标变换应用于其他物理学领域时，也会发生相同的现象。