Accurate assessment of random walk noise on long-term GNSS time series with RMLE algorithm

Accurate characterization of stochastic variations in Global Navigation Satellite System (GNSS) time series is crucial for reliable velocity estimation and uncertainty analysis in geodesy. Among noise types, the low-frequency property of Random Walk (RW) noise leads to biased estimation of velocity and uncertainty. This study systematically evaluates the performance of Maximum Likelihood Estimation (MLE) and Restricted Maximum Likelihood Estimation (RMLE) in detecting RW noise, using both simulated and real GNSS time series spanning 6-30 years and 15-32.8 years, respectively. Moreover, we analyzed the impact of colored noise, in addition to white noise, on the estimations of velocity and uncertainty in real GNSS time series using the RMLE method. Simulations of GNSS time series demonstrate that RMLE is significantly superior to MLE in identifying RW noise. The greatest improvement in identification accuracy occurs at lower RW proportions (≤0.5%), where RMLE achieves a performance gain by a factor of 1.9-10 over MLE. Whereas at higher RW proportions (1%-10%), the RMLE maintains an advantage of more than 6%. The length of the GNSS time series is also a critical determinant for RW noise identification. The reliable RW identification in long-term observation spans (>15 years) is achieved when the noise proportion exceeds 1% for MLE and 0.5% for RMLE. Analyses of real GNSS time series reveal that RMLE increases the detection rate for Random Walk noise plus Flicker Noise plus White Noise (RWFNWN) by 10.52% compared to MLE, with a significant improvement observed in the horizontal components. Although the optimal noise model exerts negligible influence on velocity estimates derived via the RMLE method (variations<0.06 mm/yr across all components), it has a profound impact on uncertainty. The assumption of pure white noise underestimates the uncertainty by a factor of 11.82-61.31 relative to a combination of colored noise plus white noise.