# Digital Processing of Seismic Data from Open-Pit Mining Blasts
## 2 Method  
Blast waves have different characteristics that can be extracted using Fourier transform [27], wavelet transform, and fractal analysis [28; 29; 30]. The wavelet wave signal decomposition is used to derive the wavelet coefficients. These coefficients are linked with blast wave characteristics. \(S(t)\) decomposition is computed as [31]:  
\[S(t)=\sum_{j=0}^{2^{i}-1}f_{i,j}(t_{j})=f_{i,0}(t_{0})+f_{i,1}(t_{1})+\cdots+f _{i,j}(t_{j})\]  
where \(f_{i,j}(t_{j})\) is the reconstructed signal of the wavelet packet at the point \((i,j)\), \(j=1,\ 2,\ \ldots,2^{i}-1\ (i=1,2,3,4,5)\). According to wavelet theory [32], the primary function for spectral analysis of wavelet packet energy should be selected. This was the 'dB8' wavelet. The sampling frequency of the equipment is 1024 Hz, so the corresponding Nyquist frequency is 512 Hz. The wave is decomposed into five layers, with \(S_{5,j}\) energy denoted by \(E_{5,j}\) :  
\[E_{5,j}\ =\int\bigl{|}S_{5,j}\ (t)\bigr{|}^{2}dt=\sum_{k=1}^{m}\bigl{|}x_{j,k} \ \bigr{|}^{2}\]  
where \(x_{j,k}\ (j=0,\ 1,\ 2,\ \ldots,\ 2^{5}-1;k=1,2,\ldots,m)\) is the amplitude of discrete points of the reconstructed signal \(S_{5,j}\) and m is the number of discrete sampling points.  
The energy percentage in each frequency band is computed as:  
\[P_{j}=\frac{E_{5,j}}{\sum_{j=0}^{2^{5}-1}E_{5,j}}*100\%.\]  
The wavelet analysis results were estimated by means of correlation analysis. Within the fractal analysis, it is necessary to determine several characteristics, including Delay period, Delay time for the gap, Embedding dimension, Entropy, Predictability period, and Correlation dimensionality [33]. For the first time, the series analysis was described by [PERSON] [34]. Based on this approach, it is possible to predict the system dynamics by its time series if the following hold:  
1. The system is indeed dynamic (has a finite degree of freedom)
2. The forecast horizon of the system is clear (by the entropy of the time series)
3. The forecast accuracy can be determined.  
The first characteristic of the dynamical system description is the possibility of finite-dimensional description, namely, the determination of factors influencing the system dynamics. For this, the embedding value is determined, that is, the minimum number of dynamic variables that uniquely describes the system behavior. The C-C algorithm was chosen [35] for delay time calculation.  
According to the C-C algorithm, the correlation integral for the embedded time series is computed as:  
\[C(m,N,r,t)=\frac{2}{M(M-1)}\times\sum_{1\leq i\leq j\leq M}H(r-\|X_{i}-\ X_{j}\|),r>0,\]  
where H is the Heaviside step function, \(N\) is the size of the dataset, t is the index lag, and \(M=N\)-(\(m\)-1)\(t\) is the number of embedded points in \(m\)-dimensional space. The mathematical sign of \(\|\ldots\|\) represents the sup-norm, and \(C(m,N,r,t)\) measures the fraction of the pairs of points \(\{\ x_{i}\}\), \(i=1,2,\)\(\ldots,M\), whose sup-norm separation is no greater than the radius of neighborhood \(r\).  
If the stochastic process \(\{x_{i}\}\) is independent identically distributed, so  
\[C(m,r)=\ C^{m}(1,r).\]  
The Brock-Dechert-Scheinkman (BDS) statistic corresponding to equation  
\[S(m,N,r,t)=C(m,N,r,t)-C^{m}(m,N,r,t).\]  
According to [26], the best embedding dimension m is computed as:  
\[m=\frac{\tau_{W}}{\tau}+1.\]  
The entropy was calculated using the following standard formula:  
\[R_{q}(a)=\frac{1}{1-q}\ln\left(\sum_{i=1}^{N(a)}p_{i}^{q}\right),\textit{where }q\in(-\infty;+\infty),\]  
where \(q\) is the attractor dimensional, and p is the probability of finding an element of the attractor value in \(i\)th hypercube with edge \(\varepsilon\) covering part of the n-dimensional Euclidean space.