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

## TL;DR

This paper introduces a new hybrid method to clean up noisy GNSS buoy elevation data, improving accuracy for coastal water monitoring.

## Contribution

A novel hybrid denoising framework (GVMD-NLM) using an improved Grey Wolf Optimizer and Non-Local Means filtering for GNSS data.

## Key findings

- The improved Grey Wolf Optimizer automatically determines optimal VMD parameters, reducing reliance on empirical settings.
- GVMD-NLM outperforms existing methods like SSA, CEEMDAN, and VMD in denoising GNSS buoy elevation time series.
- The method achieves high fidelity with correlation coefficients up to 0.9798 in real-world datasets.

## Abstract

What are the main findings?
An improved Grey Wolf Optimizer was proposed to automatically and optimally determine the key parameters for VMD, overcoming the reliance on empirical presetting and enhancing its performance for GNSS signals.A hybrid denoising strategy was developed that combines the optimized VMD with Non-Local Means filtering, using Sample Entropy to effectively separate and process noise and signal components for superior denoising.

An improved Grey Wolf Optimizer was proposed to automatically and optimally determine the key parameters for VMD, overcoming the reliance on empirical presetting and enhancing its performance for GNSS signals.

A hybrid denoising strategy was developed that combines the optimized VMD with Non-Local Means filtering, using Sample Entropy to effectively separate and process noise and signal components for superior denoising.

What are the implications of the main findings?
The method provides an objective and automated solution for parameter selection and noise-signal separation in GNSS buoy data processing, improving its reliability.This approach achieves high-quality denoising for GNSS buoy elevation time series in coastal waterways, offering a more accurate data foundation for applications in waterway hydrodynamics and water level monitoring and research.

The method provides an objective and automated solution for parameter selection and noise-signal separation in GNSS buoy data processing, improving its reliability.

This approach achieves high-quality denoising for GNSS buoy elevation time series in coastal waterways, offering a more accurate data foundation for applications in waterway hydrodynamics and water level monitoring and research.

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 (K 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 (R) 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.