Information-preserving maximum likelihood noise analysis of GNSS coordinate time series with missing values in the first-differenced domain

Abstract Global Navigation Satellite System‌ (GNSS) coordinate time series often exhibit temporally correlated colored noise and missing values. When non-stationary noise component is present, first-order differencing can be used to transform the noise into a stationary process. However, the standard first-differenced Maximum Likelihood Estimate (MLE) discards all differenced observations involving missing epochs, resulting in an additional loss of valid differenced information. To address these issues, this paper proposes two modified first-differenced MLE algorithms. Their performance is evaluated and compared with existing algorithms using, an experimental dataset comprising 10,800 simulated GNSS coordinate time series generated through Monte-Carlo simulation under different percentages of missing values, together with 180 real GNSS coordinate time series with different percentages of missing values from 60 global GNSS sites. The experimental results from both the simulated and real datasets demonstrate that the noise model parameters and velocity uncertainty estimated by the two modified first-differenced MLE algorithms are highly consistent. Under the experimental conditions of this study, these two algorithms exhibit different computational advantages depending on the percentages of missing values. Specifically, the algorithm based on the marginal likelihood function is more efficient when the percentage of missing values is below 60%, whereas the algorithm based on linear transformation matrix becomes more computationally efficient at higher missing values percentages. Compared with normal MLE, the modified first-differenced MLE algorithm provides more accurate estimates of noise model parameters and velocity uncertainty, particularly, in the presence of non-stationary noise and high percentage of missing values in GNSS coordinate time series.