Efficient Variance Component Estimation (VCE) is significant to optimal data combination in large-scale least-squares problems as those encountered in satellite geodesy, where millions of observations are jointly processed to estimate a huge number of unknown parameters. In this paper, an efficient VCE algorithm with rigorous trace calculation is proposed based on the local–global parameters partition scheme in satellite geodesy, which is directly applicable to both the simplified yet common case where local parameters are unique to a single observation group and the generalized case where local parameters are shared by different groups of observations. Moreover, the Monte-Carlo VCE (MCVCE) algorithm, based on the stochastic trace estimation technique, is further extended in this paper to the generalized case. Two numerical simulation cases are investigated for gravity field model recovery to evaluate both the accuracy and efficiency of the proposed algorithm and the extended MCVCE algorithm in terms of trace calculation. Compared to the conventional algorithm, the relative trace calculation errors in the efficient algorithm are all negligibly below 10^–7%, while in the MCVCE algorithm they can vary from 0.6 to 37% depending on the number of adopted random vector realizations and the specific applications. The efficient algorithm can achieve computational time reduction rates above 96% compared to the conventional algorithm for all gravity field model sizes considered in the paper. In the MCVCE algorithm, however, the time reduction rates can change from 61 to 99% for different implementations.

有效方差分量估计（VCE）对于大规模最小二乘问题中的最佳数据组合具有重要意义，例如卫星大地测量中遇到的问题，其中数百万个观测值被联合处理以估计大量未知参数。在本文中，基于卫星大地测量中的局部-全局参数划分方案，提出了一种具有严格轨迹计算的高效VCE算法，该算法直接适用于局部参数对单个观测组唯一的简化但常见的情况和局部参数由不同的观察组共享的一般情况。此外，基于随机轨迹估计技术的蒙特卡洛VCE（MCVCE）算法在本文中被进一步扩展到广义情况。研究了重力场模型恢复的两个数值模拟案例，以评估所提算法和扩展 MCVCE 算法在迹计算方面的准确性和效率。与传统算法相比，高效算法的相对迹计算误差均在10以下可忽略不计^–7 %，而在 MCVCE 算法中，它们可以在 0.6 到 37% 之间变化，具体取决于采用的随机向量实现的数量和具体应用。对于论文中考虑的所有重力场模型大小，与传统算法相比，该高效算法可以实现 96% 以上的计算时间减少率。然而，在 MCVCE 算法中，对于不同的实现，时间减少率可以从 61% 变为 99%。