## 3 Methods of Analysis
### Discrete Wavelet Transform (DWT)  
Wavelet transform separates a signal into multi-resolution components, so the fine and coarse resolution components capture different features in the signals. It has been referred to as a sort of mathematical microscope, with different parts of the signal examined by automatically adjusting the focus. The discrete wavelet transform (DWT) has been commonly used in multi-scale complex system resolution due to its inherent time-level locality characteristics [34]. The DWT is defined as:  
\[DWT(l,k)=\frac{1}{\sqrt{2^{l}}}\int X(t)\psi\Bigg{(}\frac{t-k\cdot 2^{l}}{2^{l}} \Bigg{)}dt \tag{8}\]  
In Equation (5), \(X(t)\) is a differential pressure time series, \(\psi\) is mother wavelet function, \(l\) and \(k\) represent the resolution level and time lag coefficient, respectively. The \(DWT\) decomposes the discrete signal, \(X(n)\), into an ordered set of orthogonal approximation and detailed functions, \(A^{j}(n)\) and \(D^{j}(n)\), as expressed by the following equations:  
\[A^{l+1}(n)=\sum_{m}h_{m-2n}A^{l}(m) \tag{9}\]  
\[D^{l+1}(n)=\sum_{m}g_{m-2n}D^{l}(m) \tag{10}\]  
where \(h_{n}\) and \(g_{n}\) represent a low-pass filter and a high-pass filter, respectively, and vary with the mother wavelets function used. The original data are considered to be the level function coefficients at the highest level, denoted as \(A_{0}(t)\). Thus, the original signal was decomposed into two components \(A_{1}(t)\)and \(D_{1}(t)\) at the first level, and the decomposition of \(A_{l}(t)\) continues until the desired decomposition level \(L\) was reached. Subsequently, the original signal can be recovered by summing the approximation and details as follows:  
\[X(t)\approx A_{L}(t)+D_{L}(t)+D_{L-1}(t)+\cdots D_{1}(t)=\sum_{1}^{L}D_{l}(t)+A_ {L}(t) \tag{11}\]  
The detailed and approximation components, \(D_{l}(t)\) and \(A_{L}(t)\), lie within a frequency band of \([2^{-(l+1)}f_{s},2^{-l}f_{s}]\) and \([0,2^{-(l+1)}f_{s}]\), respectively, where \(f_{s}\) is the sampling frequency. In the present study, Daubechies wavelets were used as mother wavelet functions for the resolution of differential pressure signals [30]. The decomposition error between the original signals and reconstructed signals reached its minimum at compact support 2 for the decomposition levels 1-10, so that a compact support 2 was employed in wavelet transform. With Daub2 as the mother wavelet function, the amount of entropy remaining in \(A_{L}\) approximated zero at decomposition level 8 for all signal studied, suggesting a resolution limit of 8 levels for this work.  
A very useful parameter derived from wavelet decomposition is wavelet energy or power, which can be calculated as:  
\[E_{l}^{A}=\sum_{t=1}^{N}\lvert A_{l}(t)\rvert^{2} \tag{12}\]  
\[E_{l}^{D}=\sum_{t=1}^{N}\lvert D_{l}(t)\rvert^{2} \tag{13}\]  
The total energy of the original signal \(X(t)\) is obtained by summing energies of the sub-signals, including \(A^{L}(t)\) and \(D^{l}(t)\) (\(l\) = 1, 2,..., \(L\)), and then the energy fraction of \(X(t)\) distributed in each frequency band (each level of the signal) can be written as:  
\[E_{D}^{l}=E_{l}^{D}/\left(\sum_{l=1}^{L}E_{l}^{D}+E_{L}^{A}\right) \tag{14}\]