High-precision Global Navigation Satellite System (GNSS) coordinate time series underpin studies of plate motion, earthquakes, terrestrial water storage, sea level change, and landslides. Their noise is often temporally correlated colored noise, and power-law (PL) noise with spectral index alpha ≤ -1 is non-stationary, making covariance matrices non-Toeplitz. First-order differencing can transform such noise into stationary processes, but missing values cause extra loss: one missing epoch removes two adjacent first differences. This loss can bias noise model parameters and velocity uncertainty, especially when non-stationary noise and many gaps coexist. Given these challenges, in-depth research is needed on information-preserving noise analysis for GNSS coordinate time series with missing values in the first-differenced domain.
Researchers from the China University of Mining and Technology, the State Key Laboratory of Spatial Datum, the Jiangxi University of Science and Technology, and the University of Beira Interior published (DOI: 10.1186/s43020-026-00220-0) their study in Satellite Navigation on 05 October 2026. The work develops two modified first-differenced maximum likelihood estimation (MLE) algorithms for GNSS coordinate time series with missing values. One uses a linear transformation matrix; the other uses a marginal likelihood function. Both aim to preserve valid differenced information that standard first-differenced MLE discards.
The first algorithm constructs free differenced observations from consecutive available epochs, even across missing segments, using a linear transformation matrix. The second treats missing original observations as latent variables and integrates them out through a marginal likelihood, while preserving the Toeplitz structure of the complete first-differenced covariance matrix. The authors evaluated the methods using 10,800 simulated GNSS coordinate time series covering stationary, weakly non-stationary, and strongly non-stationary power-law noise. Missing values percentages ranged from 0% to 50% in the main comparative experiments and were extended to 65% for the computational efficiency analysis. The evaluation also included 180 real GNSS coordinate time series from 60 global GNSS sites. The two modified algorithms produced highly consistent estimates, with Pearson correlation coefficients of 0.9999 for key noise parameters and velocity uncertainty and near-zero mean differences. Their main difference was speed. The marginal likelihood algorithm was generally faster when missing values were below about 60%, whereas the linear transformation algorithm became more efficient at higher missing percentages. In real data, 12% of series favored the linear transformation approach, close to the share with more than 60% missing values. Both modified algorithms outperformed normal MLE and standard first-differenced MLE under non-stationary noise and high missingness.
The authors said the key advance is not filling gaps but keeping valid differences that span them. They said that instead of treating every difference near a missing epoch as unusable, their methods preserve the information already present in available observations. They explained that the two algorithms are statistically equivalent in parameter estimates but differ in speed: the marginal likelihood approach is often faster with low-to-moderate missing data, while the linear transformation approach gains an edge when gaps are extensive. They added that this makes the framework practical for large global GNSS archives.
More reliable GNSS velocity uncertainty can improve interpretations of plate motion, earthquake deformation, sea level change, water storage, and landslide monitoring. The framework also offers a complementary first-differenced extension to existing fast Toeplitz-based MLE tools, rather than replacing them. Likelihood values from the proposed algorithms can support model selection criteria such as the Akaike information criterion (AIC) or Bayesian information criterion (BIC). The authors caution that applying the methods to other geophysical time series may require adapting the functional model, sampling interval, and covariance construction, especially for irregular sampling, non-Gaussian errors, or noise processes different from white noise plus power-law (WN+PL).
###
References
DOI
10.1186/s43020-026-00220-0
Original Source URL
https://doi.org/10.1186/s43020-026-00220-0
Funding information
This work is supported by the National Natural Science Foundation of China (42364002); Outstanding Youth Fund of Jiangxi Natural Science Foundation (20252BAC220015); the Graduate Innovation Program of China University of Mining and Technology (2026WLKXJ086); the Postgraduate Research & Practice Innovation Program of Jiangsu Province (26CXJH4871).
About Satellite Navigation
Satellite Navigation (ISSN: 2662-1363; ISSN: 2662-9291) Satellite Navigation is the official journal of the Aerospace Information Research Institute. The journal aims to report innovative ideas, new results, and progress in the theories, techniques, and applications of satellite navigation. The journal welcomes original articles, reviews and commentaries.