# Wavelet Transform Application for/in Non-Stationary Time-Series Analysis: A Review
## 3 Wavelet Transformation for Non-Stationary Time-Series: State-of-the-Art
### Application of WT in Hydrology  
Several approaches have been proposed to analyze the hydrology process due to its important impacts on the ecosystem and the varieties of TS. However, the observed hydrologic series are usually complex and show non-stationary and multi-temporal scale characteristics in daily, monthly, annual, inter-annual, decadal and larger scales [77; 78; 3]. To deal with this problem, the WT has been widely applied in several fields of hydrology which were reviewed by [PERSON] [46]. The paper classified the wavelet applications into six categories, namely multi-temporal scale characteristics (i.e., CWT), wavelet aided deterministic component identification, wavelet entropy, wavelet-based denoising, cross-correlation analysis, and forecasting. As for [PERSON] et al. [47], they have studied trend analysis using a Daubechies DWT (from db5 to db10) based on monthly, annual and monsoon precipitation data in seven areas in India to identify the seasonal variation and the long- and short-term fluctuation. Firstly, a linear regression test has been applied to detect the linear trend. Then, some statistical analyses along with the Sequential Mann-Kendall (SQMK) test for the seasonal and annual precipitation series have been calculated. The selection of the mother wavelet has been made using the MRE and RE criteria of the z value of the Mann-Kendall test. In addition, the maximum level of decomposition for the different mother waveletshas been calculated. Finally, the SQMK has been analyzed for the combination of approximations, detailed series and visualised events at different thresholds. In addition, [PERSON] et al. [48] have deployed a wavelet decomposition to evaluate the correlation between U.S. snow water equivalent and two major oceano-atmospheric indices. A Morlet CWT has been used to evaluate the variability of the data. The XWT and the WTC, which were respectively introduced in Sections 2.1.1 and 2.6 as two derivatives of CWT, have been also used to illustrate and quantify the associations between the different parameters o 56-year TS obtained from 323 Snow Telemetry (SNOTEL) sites. [PERSON] et al. [49] have presented an overview of the different wavelet applications for discharge and suspend sediment analysis. Four different objectives have been presented in their study: wavelet-based multi-temporal scale, trend analysis, forecasting, and wavelet-aided simulation of synthetic series. The wavelet-based hybrid black box model has been introduced for forecasting. Recently, [PERSON] et al. [50] have used WT for a forecasting task. The proposed approach is based on WT for denoising coupled with Rank-set Pair analysis (RSPA). The authors have affirmed that the quality of denoising is related to the selection of the threshold value (T) and the suitable mother wavelet. They have first compared the Coif3-RSPA (Coiflets wavelet), bior 2.4-RSPA (BiorSplines wavelet), Artificial Neural Network (ANN) and Autoregressive Method (AR) using annual runoff TS at the Huayuanku station and lower Yellow River. Then, the bior2.4-RSPA compared with RSPA has been tested for different thresholds (T = 4, 5, 6) for the same data. In addition, the RSPA, AR, ANN, db6-RSPA (Daubechies wavelet) and dmey-RSPA (DMeyer wavelet) have been compared for different thresholds (T = 4, 5, 6) in Sanmencia Station and Middle Yellow River annual runoff in the second case and at the Beijing station annual precipitation in the fourth case. Finally, in the third case, the RSPA, AR, ANN, db9-RSPA (Daubechies wavelet) and rbio3.5-RSPA (ReverseBior wavelet) have been applied for different thresholds (T = 4, 5, 6) in Zhengzhou Station and lower Yellow River annual precipitation time-series. To evaluate the performance of some models (e.g., the RE and the RMSE) in all cases, nine error metrics have been calculated. [PERSON] et al. [51] have recently studied the changes of streamflow times series in nine stations in the Black Sea region of Turkey.