GVMD-NLM: A Hybrid Denoising Method for GNSS Buoy Elevation Time Series Using Optimized VMD and Non-Local Means Filtering

GNSS buoys are essential for real-time elevation monitoring in coastal waterways, yet the vertical coordinate time series are frequently contaminated by complex non-stationary noise, and existing denoising methods often rely on empirical parameter settings that compromise reliability. This paper proposes GVMD-NLM, a hybrid denoising framework optimized by an improved Grey Wolf Optimizer (GWO). The method introduces an adaptive convergence factor decay function derived from the Sigmoid function to automatically determine the optimal parameters (<i>K</i> and α) for Variational Mode Decomposition (VMD). Sample Entropy (SE) is then employed to identify low-frequency effective signals, while the remaining high-frequency noise components are processed via Non-Local Means (NLM) filtering to recover residual information while suppressing stochastic disturbances. Experimental results from two datasets at the Dongguan Waterway Wharf demonstrate that GVMD-NLM consistently outperforms SSA, CEEMDAN, VMD, and GWO-VMD. In Dataset One, GVMD-NLM reduced the RMSE by 26.04% (vs. SSA), 17.87% (vs. CEEMDAN), 24.28% (vs. VMD), and 13.47% (vs. GWO-VMD), with corresponding SNR improvements of 11.13%, 7.00%, 10.18%, and 5.05%. In Dataset Two, the method achieved RMSE reductions of 28.87% (vs. SSA), 17.12% (vs. CEEMDAN), 18.45% (vs. VMD), and 10.26% (vs. GWO-VMD), with SNR improvements of 10.48%, 5.52%, 6.02%, and 3.11%, respectively. The denoised signal maintains high fidelity, with correlation coefficients (<i>R</i>) reaching 0.9798. This approach provides an objective and automated solution for GNSS data denoising, offering a more accurate data foundation for waterway hydrodynamics research and water level monitoring.