In the Initial Orbit Determination (IOD) context, Admissible Regions are subsets of parameter space of orbital elements that are reckoned as functions of the tracking measurement variables. Physically acceptable orbits, e.g., orbits with negative energies, constitute an Admissible Region. This paper defines sets of orbits that satisfy constraints imposed by the measurement variables. Using a probabilistic representation of constraints on some of the orbital parameters the Admissible Region can be further constrained to give a Probabilistic Admissible Region (PAR). PAR gives a particle cloud representation of the initial probability density function (pdf) of the state of a Resident Space Object (RSO). This paper presents a geometric solution to the Probabilistic Admissible Region (G-PAR). G-PAR is a set of algorithms sharing the same underlying template that geometrically maps postulated statistics on some orbital elements and statistics of the measurement process to the uncertainty in the states. The proposed scheme gives a simple closed-form solution for mapping particles to get the PAR pdf for the first time. This speeds up the PAR initial orbit determination with a single partial state measurement. The effectiveness of the proposed G-PAR is shown on diverse combinations of sensors and prior knowledge.

在初始轨道确定（IOD）环境中，可接受区域是轨道元素参数空间的子集，被视为跟踪测量变量的函数。物理上可接受的轨道，例如具有负能量的轨道，构成可接受区域。本文定义了满足测量变量施加的约束的轨道组。使用对某些轨道参数的约束的概率表示，可以进一步约束可接受区域以给出概率可接受区域（PAR）。 PAR 给出驻留空间物体 (RSO) 状态的初始概率密度函数 ( pdf )的粒子云表示。本文提出了概率可接受区域 (G-PAR) 的几何解决方案。 G-PAR 是一组共享相同基础模板的算法，该模板将某些轨道元素的假设统计数据和测量过程的统计数据几何映射到状态的不确定性。所提出的方案首次给出了映射粒子以获得 PAR pdf 的简单封闭式解决方案。这通过单个部分状态测量加速了 PAR 初始轨道确定。所提出的 G-PAR 的有效性在传感器和先验知识的不同组合上得到了证明。