2.5.1. Wavelet Transform

Recently, wavelet analysis has been widely accepted in a wide range of science and engineering applications. Some of the latest studies utilizing the wavelet analysis are [4,27–31]. The wavelet analysis technique has also been used in: data and image compression, partial differential equation solving, transient detection, pattern recognition, texture analysis, noise reduction, trend detection, etc. Wavelets have been identified as more effective tools than the Fourier transform (FT) in analyzing the non-stationary time series. Instead of FT, which analyses the data in two dimensions, i.e., time and frequency, wavelet transform was used, which analyses the data in three dimensions, i.e., time, space, and frequency. This provides a significant opportunity to examine the variation in the hydrological processes.

#### 2.5.2. Continuous and Discrete Wavelet Analysis

Wavelet transform (WT) breaks down/separates data series into logically-ordered wave-like oscillations (wavelets) analogous to data vis-à-vis time within a range of frequencies. The original time series can be depicted with regard to a wavelet expansion that uses the coefficients of the wavelet functions. Several wavelets can be made from a function *ψ*(*t*) known as a "mother wavelet", which is restricted in a finite/bound interval. That is, WT expresses/breaks a given signal into frequency bands and then analyses them in time. WT is widely categorized into the continuous wavelet transform (CWT) and discrete wavelet transform (DWT). CWT is defined as the sum over the whole time of the signal to be analyzed, multiplied by the scaled and shifted versions of the transforming function *ψ*. The CWT of a signal *f*(*t*) is expressed as follows:

$$\mathcal{W}\_{a,b} = \frac{1}{\sqrt{a}} \int\_{-\infty}^{\infty} f(t) \Psi^\* \left( \frac{t-b}{a} \right) dt \tag{3}$$

where "\*" denotes the complex conjugate. On the other hand, CWT looks for correlations/mutual relationships between the signal and wavelet function. This measurement is done at distinct scales of *a* and locally around the time of *b*. The result is a ripple/wavelet coefficient *Wa*,*<sup>b</sup>* outline sketch. However, enumerating the wavelet/ripple coefficients at every likely scale (resolution level) demands a huge amount of data and calculation time. DWT analyzes a given time series with distinct resolutions for a distinct range of frequencies. This is done by decaying the data into coarse approximation and detail coefficients. For this, the scaling and wavelet/ripple functions are utilized. Choosing the scales *a* and the positions *b* based on the powers of two (binary scales and positions), DWT for a discontinuous time series *fi*, becomes:

$$\mathcal{W}\_{m,n} = 2^{-\frac{m}{2}} \sum\_{i=0}^{N-1} f\_i \Psi^\* \left( 2^{-m} i - n \right) \tag{4}$$

where *<sup>i</sup>* is the integer time steps (*<sup>i</sup>* = 0, 1, 2, ..., *<sup>N</sup>* − 1 and *<sup>N</sup>* = 2*M*); *<sup>m</sup>* and *<sup>n</sup>* are integers that control, respectively, the scale and time; *Wm*,*<sup>n</sup>* is the wavelet coefficient for the scale factor *a* = 2*<sup>m</sup>* and the time factor *b* = 2*mn*. The original signal can be built back/recreated using the inverse discrete wavelet transform as follows:

$$f\_i = A\_{M,i} + \sum\_{m=1}^{M} \sum\_{n=0}^{\left(2^{M-m}-1\right)} \mathcal{W}\_{m,n} 2^{\frac{m}{2}} \Psi\left(2^{-m}i - n\right) \tag{5}$$

or in a simple form as:

$$f\_i = A\_{M,i} + \sum\_{m=1}^{M} D\_{m,i} \tag{6}$$

where *AM*,*<sup>i</sup>* is called an approximation sub-signal at level *M* and *Dm*,*<sup>i</sup>* are the detail sub-signals at levels *m* = 1, 2, ..., *M*. The approximation coefficient *AM*,*<sup>i</sup>* represents the high-scale, low-frequency component of the signal, while the detailed coefficients *Dm*,*<sup>i</sup>* represent the low-scale, high-frequency component of the signal.

There are a number of mother wavelets such as: Haar; Daubechies; Coiflet; and biorthogonal. Normally, Daubechies, belonging to the Haar wavelet, achieves improved results in sediment transport processes due to its inherent capacity to discover time localization information, such as dealing with the annual recurrence and hysteresis/lag phenomenon; the time localization information is beneficial in flow discharge and sediment processes. The different Daubechies wavelet families from [40] are shown in Figure 6. The Coiflet wavelet is more symmetrical than the Daubechies wavelet. Likewise, biorthogonal wavelets have the characteristic of the linear phase, which is required for signal rebuilding [29]. The appropriateness and selection of the mother wavelet are dependent on application type and characteristics of the data.

**Figure 6.** Daubechies wavelet families.
