# 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 Geo-Sciences and Geophysics  
The WT was investigated in several applications related to the climate, meteorology, oceanography, geo-sciences and geophysics [10; 11]. [PERSON] and [PERSON] [74] discussed the advantage of WT in geophysics due to their unique properties for the analysis of natural phenomena. In fact, the WT originated in geophysics in the early 1980s for the analysis of seismic signal. After a decade of significant mathematical formalism, they are now also being exploited for the analysis of several other geophysical processes such as atmospheric turbulence, space-time rainfall, ocean wind waves, seafloor bathymetry, geologic layered structures, climate change, among others. In addition, [PERSON] and [PERSON] [75] introduced the power of wavelets for analysis of non-stationary processes that contain multi-scale features, detection of singularities, analysis of transient phenomena, fractal and multi-fractal processes, and signal compression. [PERSON] [23] reviewed the applications for the turbulences analysis. In the first step, the CWT, DWT and their properties were introduced. They concluded that WT is a very suitable method for turbulence analysis. Additionally, [PERSON] [9] presented the CWT. The usefulness of the WT in oceanography and meteorology was discussed by examining the dispersion of Yanai waves. [PERSON] and [PERSON] [24] applicated the CWT in order to analyze two climate TS. The first TS was the 2.5 million year (Myr) deep-sea sediment record in the North Atlantic, which provides information on the variation of global ice volume. However, the second is the monthly NHST for the period January, 1854-July, 1993. The Morlet wavelet was used to study the trend for both TS. In addition, the basic concept of WT, compared with similar concepts used in music, was also provided. The WT was applied in [25] to analyze the relationship between two atmospheric phenomena interact: the MJO and the ENSO events. Their method was based on the cross-spectral analysis and MODWT to analyze the variance, covariance and the wavelet subseries of the daily measurements. Furthermore, [PERSON] et al. [26] applicated the CWT and XWT in order to analyze the coherency between two times series: the standardized TS of winter (DJF) and the standardized BMI percentile TS. In fact, the mechanistic models of physical relationships between the TS was tested. In addition, the phase angle statistics that can be used to gain confidence in causal relationships was demonstrated. Finally, the Monte Carlo method was used to assess the statistical significance against the red noise background. [PERSON] and [PERSON] [27] investigated the WT in order to analyze the Holocene climate variability. The [PERSON]'s wavelet analysis algorithm was applied in order to determine the different records in Greenland ice cores. Then, [PERSON] et al. [76] used the wavelet power spectra to solve the bias problem for atmospheric and oceanic data sets. In fact, the study was based on the definition of the energy of the wavelet power spectrum which make a substantial improvement in the spectral estimate, allowing for a comparison of the spectral peaks across scales. The results were validated with an artificial TS and a real coastal sea level record. Moreover, [PERSON] et al. [28] used the wavelet spectral analysis in order to analyze the SST and the NAO pattern. In fact, the XWT was applied to investigate the environmental effects (SST, NAO) on octopus abundance fluctuations measured as capture per unit of effort (CPUE) from 1989 to 2007 in the waters of the Canary Islands. Recently, [PERSON] et al. [29] reviewed the application of wavelet based spatial verification techniques in order to develop a high-resolution numerical weather prediction model. They concluded that the MRA was principally used in order to analyze the meteorological applications. Particularly, the Point-Measure Enhancements (PME), the Intensity-Scale Skill Score (ISS) and the Wavelet Coefficient Score (WCS) were developed.