Method for studying the structure of blast-induced vibrations in open-cast mines

2.1. Fourier transform (FT)

Analysis in the frequency domain (i.e. spectral analysis) enables more accurate knowledge of the structure of vibrations [8, 9]. It involves decomposition of the original time series (signal) into simpler components, the sine and cosine functions of various frequencies, and determining which ones are most important in building the image of vibrations (Fig. 3).

The result of this process is also the so-called Fourier transform or the Fourier integral. It shows the amplitude spectrum, illustrating the dependency of amplitudes of all signal components on frequency, and enables transforming any analysis of a function described in the time domain into the frequency domain. The effect of Fig. 2 signal analysis is shown in Fig. 4.

Fig. 3The original recorded signal [7]

Fig. 4Running time amplitude spectrum shown in Fig. 3 [7]

Unfortunately, signal analysis using the Fourier transform has a significant drawback – it shows the signal in the form of harmonic components, providing information on frequencies, but also losing information about the time of occurrence of the frequencies in the signal. In order to include the time domain signal, an inverse Fourier transform may be performed. That, however, loses data about the frequencies in the signal. It is, therefore, a method that does not give entirely satisfactory results.

Deficiencies of the analyses carried out, either in the time domain or in the frequency domain, contributed to the search for methods that can analyze both parameters simultaneously, giving fuller information on the frequency of the signal components and their occurrence within a specified period of time.

2.2. 3rd octave band analysis – time signal filtering

One of the analyses providing information about the structure of vibrations both in the time domain and the frequency domain is filter analysis, carried out, among other things, by means of third octave filters [8, 17, 18]. As a result of the analysis, the measurement signal is distributed onto the components contained in certain frequency bands. The band-pass filter transmits only certain specific frequencies, suppressing those remaining.

Signal filtering by means of a set of band-pass filters enables division of the whole signal into smaller bands and analysis of the signal in a given frequency range. In the case of the 3rd octave band filtration, filters with a constant coefficient of bandwidth are used and the central frequency is a normalized value. As a result of such a filtration, one timeline results in a set of timelines, enabling a time analysis in the selected frequencies, which is not possible in the case of the Fourier transform (Fig. 5).

Fig. 5An example of 3rd octave band filtration effect for the timeline course shown in Fig. 3 [7]

2.3. Short-time Fourier transform (STFT)

Another way of describing the structure of signals in the time domain and the frequency domain, much more precise than the filtration analysis, is the short-time Fourier transform (STFT) developed in 1946 by Dennis Gabor. He attempted to correct the imperfections of the Fourier transform and adapted it to the analysis of only a small portion of the signal in time [19]. This process is called windowing the signal. The STFT analysis involves cutting out successive sections of the signal using the window function and calculating the Fourier transform.

It should be noted, however, that the short-time Fourier transform provides information about when and which frequencies occurred at a given signal with limited precision. This limitation is due to the inconvenience of the STFT analysis, the once-adjusted width of the analysis window, the same for all frequencies. A wide window means a high frequency resolution but little time resolution. What is more, high frequencies are characterized in the time domain with the same precision. On the other hand, a window being too narrow makes it impossible to detect lower frequencies present in the signal. Unfortunately, the selection of the analysis window width is done by trial and error.

The result of the STFT Fig. 3 signal analysis is shown in Fig. 6.

Non-stationery signals are of a very different nature and, therefore, require a more flexible approach, in order to obtain accurate information in either the time domain or the frequency domain. This is enabled by another method of signal processing, the wavelet transform.

Fig. 6The STFT analysis result for the time signal shown in Fig. 3 [7]

2.4. Wavelet transform

The wavelet transform is the next step in the analysis of signals, using the technique of windowing, but the analysis windows are of a different length and frequency [19, 20], which allows the use of long intervals when there is a need to capture more precise information about low frequencies and shorter areas where high frequency information is needed.

The wavelet transform represents a signal as a sum of the basic functions called wavelets, assuming a waveform of a limited duration, with the average value equal to zero. The wavelet feature is also a limited frequency band.

The wavelet is characterized by two basic parameters, i.e. the scale factor and the shift (translation). The scale factor changes the duration of wavelets, in other words, its compressions or tensions. In contrast, the translation factor provides the wavelet shift along the signal in the time domain, with a fixed scale factor, meaning a delay or acceleration of the beginning of the wavelet. Therefore, it is necessary to note that wavelet analysis does not include the time – frequency area, but the time – scale area.

As a result of matching the wavelet to the signal, we obtain a number of CWT wavelet factors which determine the degree of correlation of the wavelets with the signal used (Fig. 7).

Fig. 7Map of CWT wavelet factors for the time signal shown in Fig. 3 [7]

In the case of the wavelet transform, the important fact is that the real signal recorded by the measuring equipment is always matched to one wavelet of the whole set, with scale and translation factors changed, which results in unnecessary multiple descriptions of the same part of the signal.

2.5. Matching pursuit algorithm (MP)

Non-stationary signals are characterized by very high volatility, and concise representation of them can be achieved by describing them through the selection of a function from a very large redundant set, called a dictionary, from which functions best reflect the characteristics of the signal [21].

In order to capture the most relevant and reliable data on the structure of the recorded vibrations, the Matching Pursuit (MP) can be applied, a method of time – frequency analysis [21, 22], based on an iterative process of matching a whole set of functions to a signal. However, after each stage of approximation, the signal portion matched to a given function is subtracted, and the new functions are compared to the rest of the signal. This enables avoiding unnecessary multiple descriptions of the same part of the signal.

The MP algorithm involves the decomposition of a signal with time – frequency functions called atoms, selected from a dictionary $G=\left\{g_1\left(t\right),g_2\left(t\right),\dots ,g_n\left(t\right)\right\}$.

The concept of the atom should be defined as an elementary signal part whose aim is to best reflect the characteristics of the signal. The atom can be expressed as a single function window $g\left(t\right)$ by scaling a, translation b and frequency modulation $\xi$ (Eq. (1)) [21]:

where: $I$ – index defining the atom set of parameters, $I=\left(a,b,\xi \right)$, $a$ – scale factor, $b$ – shift (translation) factor, $\xi$ – frequency modulation, $e^{i\xi t}$ – complex exponential function.

In practice, the Gabor dictionary is used, containing Gabor atoms, sine wave signals modulated by a Gaussian envelope (Fig. 8). Multiplying these two functions by themselves can result in a whole range of shapes, depending on their characteristics. The amplitudes are tuned in such a manner that each function (atom) has equal (unit) energy, as the scale product of the timeline of unit energy and the signal will be a direct measure of the share of the given structure in the signal energy.

Fig. 8Gabor atom scheme [6, 7]

Regardless of the variety of shapes of Gabor atoms, the argument for their use is that they provide the best, optimal time – frequency resolution. In addition, the MP algorithm may be used with any dictionary containing a sufficient number of functions, so as to reproduce any signal.

With a dictionary comprising a variety of Gabor atoms, one should consider how to use them to make the process of approximation as effective as possible. One can try to use the products of all atoms and signals but, due to the large resources of the vocabulary, the most representative functions need to be selected.

For example, Fig. 9 shows four Gabor atoms, matched to specific fragments of the recorded signal [7, 17, 18]. Atoms numbered 0, 3, 6 and 7, shown in the Figure, successively explained 36.6 %, 4.61 %, 3.20 % and 2.93 % of the signal. Other atoms, matched to the signal (1, 2, 4, 5, 8 and 9) were deliberately omitted in order not to obscure the drawing.

As the signal decomposition progresses, the number of matching atoms is used and the residues contain less and less energy. The process of approximation of atoms to the signal does not last forever. The MP algorithm is stopped when approximately 95 % of the energy of the signal is explained (or after a set number of repetitions), and the remainder is treated as an error of approximation (e.g. the noise in the signal).

Fig. 9Gabor atoms approximated to the scenario presented in Fig. 3 [7]

Results of the analysis using the method of matching pursuit are presented in a table (Table 1), including characteristics of individual parameters of Gabor atoms. On this basis, one can make a very thorough time – frequency analysis of a signal and, of particular importance, describe the share of individual Gabor atoms in the construction of the vibration energy image.

For example, based on Table 1, we obtain the following information about parameters:

Gabor atom no. – determines subsequent atoms participating in the reconstruction of the signal and explaining certain percentage of its energy. The order of the atoms is determined by the energy value represented by the given atom.

Time – specifies the location of the Gabor atom center on the timeline.

Position – determines the position of the Gabor atom center expressed in the number of the signal samples.

Scale (scale factor) – results in the change of the duration of the Gabor atom, its compressions or tensions.

The length of the Gabor atom - specifies the width of the atom in the time domain and depends on the number of samples and sampling frequency.

Frequency – the Gabor function modulation frequency.

Gabor atom signal energy – using this parameter, the energy value represented by a window function (Gabor atom), as the square of its amplitude, is specified.

For a better illustration, some parameters are shown as exemplified by a zero atom in Table 1 (Fig. 10).

Fig. 10Graphical representation of certain Gabor atom parameters [7]

In addition, the results of the Matching Pursuit are illustrated as the distribution of the energy density obtained with the Wigner distribution, also called the Wigner map (Fig. 11).

Fig. 11The Wigner map for the timeline in Fig. 3 [7]

The Wigner map shows Gabor atoms (in the form of colored spots – “blobs”) that matched to the actual signal, reconstituted it and explained approximately 95 % of its energy. As seen in Fig. 11, Gabor atoms make it possible to timeline the characteristic frequencies, important in the signal construction. The color intensity of each atom is the energy density, as represented in various frequency bands. The sum of time – frequency atoms matched gives the signal energy, in accordance with the principle of energy conservation.

In order to demonstrate the precision of the reconstruction of the actual signal by Gabor atoms, Fig. 12 shows Fig. 3’s (blue) actual vibration waveform and the reconstructed signal (red) on it. As it can be seen, the differences are minor, especially in the initial phase of the signal excitation, which indicates a very precise matching of time – frequency atoms to the signal and exact reconstruction of its features. It should be noted that the recorded signals may be interfered with (e.g. with noise), resulting in the formation of certain differences between the real signal and the reconstructed signal.

Fig. 12A comparison of the Fig. 2 actual timeline and its reconstruction by Gabor atoms [7]

The structure analysis of signals can employ several methods, and selecting the most suitable one remains an open issue. However, you should consider which one is best from the perspective of a good characterization of the features of the recorded vibrations. In the case of the Fourier transform (FT), there is a possibility to analyze the signal in one of the two domains – the time domain or the frequency domain, but never in both, which does not enable obtaining information on the full vibration image. However, it precisely describes stationary signals in which there is a small range of frequencies. Using two new and popular methods of signal analysis, i.e. the short-time Fourier transform (STFT) and the wavelet transform (CWT), there is a possibility of describing signal features using a set of functions matched to them. The wavelet transform includes functions called wavelets, and the STFT includes window functions used for cutting signal fragments, and their Fourier transforms are calculated. Use of these methods results in a vibration image on a time – frequency plane, but they have their drawbacks, hindering the precise characterization of the structure of vibrations. In the case of the STFT, there is a problem of selecting the optimum width of the window, selected by trial and error, while, in the case of the wavelet transform, it may be difficult to accurately describe, on a selected wavelet, both lower and higher frequencies. In addition, the original signal is matched to the same wavelet, despite the change of its parameters, resulting in the possibility of some portions of the signal being unnecessarily repeatedly described.

With regard to vibrations recorded while blasting, the choice of the Matching Pursuit algorithm, providing good time – frequency resolution and, at the same time, characterizing the energetic image of signals, is very important. For example, based on the results obtained, it is possible to conduct analyses enabling identification of the differences in ground quality under buildings, at the source of vibrations or the quality of the center where blasting is done.