The Kalman filter stands as one of the most widely used methods for recursive parameter estimation. However, its standard formulation typically assumes that all state parameters avail initial values and dynamic models, an assumption that may not always hold true in certain applications, particularly in global navigation satellite system (GNSS) data processing. To address this issue, Teunissen et al. (2021) introduced a generalized Kalman filter that eliminates the need for initial values and allows linear functions of parameters to have dynamic models. This work proposes a least-squares approach to reformulate the generalized Kalman filter, enhancing its applicability to GNSS data processing when the parameter dimension varies with satellite visibility changes. The reformulated filter, named generalized least-squares filter, is equivalent to the generalized Kalman filter when all state parameters are recursively estimated. In this case, we demonstrate how both the generalized Kalman filter and the generalized least-squares filter adaptatively manage newly introduced or removed parameters. Specifically, when estimation is limited to parameters with dynamic models, the generalized least-squares filter extends the generalized Kalman filter by allowing the dimension of estimated parameters to vary over time. Moreover, we introduce a new element of least-squares smoothing, creating a comprehensive system for prediction, filtering, and smoothing. To verify, we conduct simulated kinematic and vehicle-borne kinematic GNSS positioning using the proposed generalized least-squares filter and compare the results with those from the standard Kalman filter. Our findings show that the generalized least-squares filter delivers better results, maintaining the positioning errors at the centimeter level, whereas the Kalman-filter-based positioning errors exceed several decimeters in some epochs due to improper initial values and dynamic models. Moreover, the normal equation reduction strategy employed in the generalized least-squares filter improves computational efficiency by 23% and 32% in simulated kinematic and vehicle-borne kinematic positioning, respectively. The generalized least-squares filter also allows for the flexible adjustment of smoothing window lengths, facilitating successful ambiguity resolution in several epochs. In conclusion, the proposed generalized least-squares filter offers flexibility for various GNSS data processing scenarios, ensuring both theoretical rigor and computational efficiency.

卡尔曼滤波器是递归参数估计中使用最广泛的方法之一。然而，其标准公式通常假设所有状态参数都适用于初始值和动态模型，这一假设在某些应用中可能并不总是成立，尤其是在全球导航卫星系统 （GNSS） 数据处理中。为了解决这个问题，Teunissen 等人（2021 年）引入了一种广义卡尔曼滤波器，它消除了对初始值的需求，并允许参数的线性函数具有动态模型。这项工作提出了一种最小二乘方法来重新构建广义卡尔曼滤波器，当参数维度随卫星能见度变化而变化时，增强了其对 GNSS 数据处理的适用性。重新制定的滤波器称为广义最小二乘滤波器，当递归估计所有状态参数时，它等效于广义卡尔曼滤波器。在这种情况下，我们演示了广义卡尔曼滤波器和广义最小二乘滤波器如何自适应地管理新引入或删除的参数。具体来说，当估计仅限于动态模型的参数时，广义最小二乘滤波器通过允许估计参数的维度随时间变化来扩展广义卡尔曼滤波器。此外，我们还引入了最小二乘平滑的新元素，创建了一个用于预测、过滤和平滑的综合系统。为了验证，我们使用所提出的广义最小二乘滤波器进行模拟运动学和车载运动学 GNSS 定位，并将结果与标准卡尔曼滤波器的结果进行比较。 我们的研究结果表明，广义最小二乘滤波器提供了更好的结果，将定位误差保持在厘米级，而由于初始值和动态模型不正确，基于卡尔曼滤波器的定位误差在某些时期超过了几分米。此外，广义最小二乘滤波器中采用的法则方程约简策略在模拟运动学和车载运动学定位中的计算效率分别提高了 23% 和 32%。广义最小二乘滤波器还允许灵活调整平滑窗口长度，有助于在多个 epoch 中成功解决歧义。综上所述，所提出的广义最小二乘滤波器为各种GNSS数据处理场景提供了灵活性，确保了理论的严谨性和计算效率。