Efficient variance component estimation for large-scale least-squares problems in satellite geodesy

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.