Non-stationary time series (TS) analysis has gained an explosive interest over the recent decades in different applied sciences. In fact, several decomposition methods were developed in order to extract various components (e.g., seasonal, trend and abrupt components) from the non-stationary TS, which allows for an improved interpretation of the temporal variability. The wavelet transform (WT) has been successfully applied over an extraordinary range of fields in order to decompose the non-stationary TS into time-frequency domain. For this reason, the WT method is briefly introduced and reviewed in this paper. In addition, this latter includes different research and applications of the WT to non-stationary TS in seven different applied sciences fields, namely the geo-sciences and geophysics, remote sensing in vegetation analysis, engineering, hydrology, finance, medicine, and other fields, such as ecology, renewable energy, chemistry and history. Finally, five challenges and future works, such as the selection of the type of wavelet, selection of the adequate mother wavelet, selection of the scale, the combination between wavelet transform and machine learning algorithm and the interpretation of the obtained components, are also discussed.

Soil Moisture (SM) monitoring is crucial for various applications in agriculture, hydrology, and climate science. Remote Sensing (RS) offers a powerful tool for large-scale SM retrieval. This paper explores the advancements in RS techniques for SM estimation. We discuss the applications of these techniques, along with the advantages and limitations of traditional physical models and data-driven Machine Learning (ML) based approaches. The paper emphasizes the potential of combining ML and physical models to leverage the strengths of both approaches. We explore the challenges associated with this integration and future research directions to improve the accuracy, scalability, and robustness of RS-based SM retrieval. Finally, the paper also discusses a few issues such as input data selection, data availability, ML complexity, the need for public datasets for benchmarking, and analysis.

Satellite image time-series (SITS) methods have contributed notably to detection of global change over the last decades, for instance by tracking vegetation changes. Compared with multi-temporal change detection methods, temporally highly resolved SITS methods provide more information in a single analysis, for instance on the type and consistency of change. In particular, SITS decomposition methods show a great potential in extracting various components from non-stationary time series, which allows for an improved interpretation of the temporal variability. Even though many case studies have applied SITS decomposition methods, a systematic comparison of common algorithms is still missing. In this study, the seasonal trend loess (STL), breaks for additive season and trend (BFAST) and multi-resolution analysis-wavelet transform (MRA-WT) were explored in order to evaluate their performance in modelling, monitoring and detecting land-cover changes with pronounced seasonal variations from simulated normal difference vegetation index time series. The selected methods have all proven their ability to characterize the non-stationary vegetation dynamics along with different physical processes driving the vegetation dynamics. Our results indicated that BFAST is the most accurate method for the examined simulated dataset in terms of RMSE, whereas MRA-WT showed a great potential for the extraction of multi-level vegetation dynamics. Considering the computational efficiency, both STL and MRA-WT outperformed BFAST.