On the stochastic significance of peaks in the least-squares wavelet spectrogram and an application in GNSS time series analysis

In this paper, the mathematical derivation of the underlying probability distribution function for the normalized least-squares wavelet spectrogram is presented. The impact of empirical and statistical weights on the estimation of the spectral peaks and their significance are demonstrated from the statistical point of view both theoretically and practically. The simulation results show an improvement of approximately 0.02mm (RMSE) for annual signal estimation when statistical weights are considered in the least-squares wavelet analysis (LSWA). The weighted LSWA estimates the signals more accurately than the ordinary LSWA for different percentage amount of missing data. As a real-world application, Global Navigation Satellite Systems (GNSS) time series for a station in Rome, Italy are analyzed. The analyses of the GNSS time series provided by different agencies for the same station reveal statistically significant annual peaks, more significant in 2010 but less significant between 2018 and 2020, while the higher frequency components show different spectral patterns over time. A declining trend of approximately -0.42 mm/year since 2004 is estimated for the GNSS height time series, likely due to gradual land subsidence. The results not only highlight the advantages of LSWA but can also help to better understand the uncertainties involved in signal estimation.