Abstract

PPP-RTK combines the advantages of both precise point positioning (PPP) and real-time kinematic (RTK) techniques. While constructing PPP-RTK models based on undifferenced and uncombined observations offers apparent benefits, these observation equations suffer from a rank deficiency issue. To address this problem, the Singularity-system (S-system) theory can be utilized. This theory imposes constraints on a minimal subset of parameters, known as the S-basis, by setting them to arbitrary values, typically zeros. Despite the existence of multiple options for the S-basis, prevailing research conventionally selects the parameters of one receiver—the pivot—as the S-basis. In this study, we depart from this practice by selecting the mean of receiver-related parameters as the S-basis. This departure prompts an exploration into how the S-basis choices influence PPP-RTK outcomes regarding network product precision and user positioning performance. Our comparative analysis of the mean receiver (MR) and pivot receiver (PR) models unveils distinctions in the combined product precision. These products include satellite clocks, satellite phase biases, and ionospheric delays (excluding tropospheric delays). The distinction emerges because the estimable satellite clocks in the PR model incorporate atmospheric delays specific to the pivot receiver, in contrast to the MR model, which integrates mean atmospheric delays from all receivers. Despite the distinction in the analytical form of combined product and its precision, both model results in similar positioning performance. This is because variations in product precision levels caused by selecting different atmospheric parameters as the S-basis can be nullified by the parameterized atmospheric delays on the user side. With the inclusion of tropospheric delays, the PR and MR models also demonstrate similar performance and yield more accurate user positioning when located near the pivot receiver compared to positions farther from the pivot receiver when employing ambiguity-float network products. This dependence on the pivot receiver stems from both models selecting the pivot receiver ambiguities as the S-basis, while opting for mean ambiguities across all receivers negates the integer nature of ambiguities. Our conclusion underscores that identical positioning outcomes in PR and MR PPP-RTK models rely on both models selecting the same ambiguities as the S-basis. This highlights the potential variability in PPP-RTK performance when different ambiguity parameters are selected as the S-basis, particularly in the absence of network integer ambiguity resolution.

中文翻译：

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两种PPP-RTK模型对比：S基选择、网络产品精度、用户定位性能

摘要

PPP-RTK结合了精密单点定位（PPP）和实时运动（RTK）技术的优点。虽然基于无差异和未组合观测构建 PPP-RTK 模型具有明显的好处，但这些观测方程存在秩不足问题。为了解决这个问题，可以利用奇点系统（S系统）理论。该理论通过将参数设置为任意值（通常为零），对参数的最小子集（称为 S 基）施加约束。尽管S基存在多种选择，但主流研究通常选择一个接收器——枢轴——的参数作为S基。在本研究中，我们通过选择接收器相关参数的平均值作为 S 基础来背离这种做法。这种偏离促使人们探索 S 基础选择如何影响 PPP-RTK 在网络产品精度和用户定位性能方面的结果。我们对平均接收器 (MR) 和枢轴接收器 (PR) 模型的比较分析揭示了组合产品精度的差异。这些产品包括卫星时钟、卫星相位偏差和电离层延迟（不包括对流层延迟）。之所以出现这种区别，是因为 PR 模型中的可估计卫星时钟包含了枢轴接收器特定的大气延迟，而 MR 模型则整合了所有接收器的平均大气延迟。尽管组合产品的分析形式及其精度有所不同，但这两种模型都产生相似的定位性能。这是因为由于选择不同的大气参数作为S基础而引起的产品精度水平的变化可以通过用户侧的参数化大气延迟来抵消。由于包含对流层延迟，PR 和 MR 模型也表现出相似的性能，并且当位于枢轴接收器附近时，与采用模糊浮动网络产品时距离枢轴接收器较远的位置相比，可以产生更准确的用户定位。这种对枢轴接收器的依赖性源于两个模型都选择枢轴接收器模糊度作为 S 基础，而选择所有接收器的平均模糊度则否定了模糊度的整数性质。我们的结论强调，PR 和 MR PPP-RTK 模型中相同的定位结果依赖于两个模型选择相同的模糊度作为 S 基础。这凸显了当选择不同的模糊度参数作为 S 基础时，尤其是在没有网络整数模糊度分辨率的情况下，PPP-RTK 性能的潜在变化。