The source localization problem, fundamental to applications like GPS, is typically approached as a minimization problem in the presence of various types of noise. Ensuring the uniqueness of solutions in GPS technology is vital for the reliability and accuracy of applications, from everyday navigation to critical military operations. In this paper, we examine two key minimization problems: one focused on distance error and the other on squared distance error. We explore these problems in both three-dimensional space, the standard scenario, and in two-dimensional space as a simplified case. Furthermore, we discuss the number of possible source solutions when the number of measurements is fewer than three.

翻译：源定位问题是GPS等应用中的基础问题，通常被建模为存在各类噪声时的最小化问题。在GPS技术中，确保解的唯一性对于从日常导航到关键军事行动等各种应用的可靠性与精度至关重要。本文研究了两个关键的最小化问题：一个聚焦于距离误差，另一个聚焦于平方距离误差。我们分别在标准的三维空间场景及作为简化情形的二维空间中探讨了这些问题。此外，我们还讨论了当测量数量少于三个时可能的源解的数量。