Spectral-Coherency Detection of Periodic and Nonperiodic Acoustic Radiation in Discrete Rock Medium

Abstract

This work is performed within laboratory-field studies begun in 2015 at the mines of Khibiny appatite deposit, Kola Peninsula, Russia. The studies are aimed to determine and implement seismic criteria of risky dynamic behavior in the areas of extreme mining pressure. We observe and fix different kinds of rock dynamics – from local scale (tunnel fracturing) to regional scale (induced earthquakes). The underlying factor of such a risky dynamic behavior is the tectonics acting in directions close to horizontal. We apply the acoustic emission (AE) technique to control local instabilities in rocks. For further development of the method, we need to search for the physically-proved AE parameters that deliver information about the focal nature of hazards in mines. I suggest an approach based on spectral-correlation analysis to study a selected number of AE events. This group of signals is characterized by a two-component spectral structure and manifests itself by a specific П-pattern coherence function. Relying on the long-term observations and considering the relation of these events to the fractured geological structure, I attribute this AE type to the stick-slip phenomenon. Detailed analysis of AE data revealed two natures of these AE-type sources: impulse (non-periodic) and regular (quasi-periodic) radiation.

1. Introduction

Classic seismology treats the problem of determination of natural seismic source parameters through the source time function (STF) [1]. STF is one of the main parameters of the kinematic model of extended source, such as geological fault or sharp boundary, between geological formations. Once occurred on fault surfaces under mechanical stress such rock discontinuity as a shear dislocation propagates with a certain velocity along an extended source. Shear dislocation is described by the slip vector ∆u corresponding to a relative displacement of the two sides of a fault. The time dependence of the slip ∆u derivative is called STF. Dislocations also appear on microscale due to crystalline lattice imperfections. Their dislocations move along the slip plane as a cross line of broken atomic bonds [2, 3]. Shearing can occur on the grain boundaries of different stiffness in polycrystalline material such as rocks or on micro-fracture surfaces [4]. The dislocation cumulative process manifests itself on the macroscale as irreversible deformation, progressive development of the latter is accompanied by discrete acoustic radiation. This phenomenon is called an acoustic emission (AE).

We intend to develop our study of the AE process in rocks of different lithology in the direction of physical insight on how the strength of material relates to the defect structure evolution from micro-fractures to the fault when subjected to stress or/and temperature. To take on this problem one should think about a way to identify correctly the scale and deformation mechanism of fracturing by means of AE parameters. This is an inversion problem and it derives the properties of AE source from the acoustic far-field observations. But geophysical medium that influences the origin wave propagation is strongly inhomogeneous, thus we should find a method to account for the transfer function of the system «medium and seismic sensor (AE transducer)». In the suggested chapter, I consider the transformation process, which is associated with normal modes resulting from acoustical impact. To study and interpret a series of AE events acquired in mines of Khibiny appatite deposit in the frequency range of 2–28 kHz the spectral-correlation analysis is applied. The coherence function calculations revealed a specific spectrum pattern, which is attributed to the stick-slip seismic source activated by tectonic forces. In terms of deformation mode, the stick-slip event represents an unstable sliding on fault surfaces [1, 5, 6] characterized by two phases—stick as elastic energy storage, and slip—as energy release. Calculations of statistically significant values of spectral densities [7, 8] allowed two types of AE signals to be distinguished—non-periodic and quasi-periodic.

The experiments on fracture generation in glass plates showed that the initial acoustic radiation produced by running fracture is characterized by two parts: the compressional part and subsequent dilatational part [9, 10]. Thus, the arriving pulse can be described as a bipolar pulse. The transfer function of linear system «medium and seismic sensor» we simulate as a superposition of exponentially attenuating normal modes on the corresponding frequencies. Then, as is shown in Ref. [11], the transformation of initial pulsed radiation to the resultant AE signal we derive from convolution integral producing in an output a transient process. Here I apply the normal mode approach to the analysis of AE transducer amplitude-frequency response characteristics. Whereas the normal mode phenomenon can be observed also in the discrete geophysical medium. For example, in Ref. [12] we describe the effect of waveguide propagation of the 1st-order normal mode that arose in a horizontal ore body of thickness 30–40 m at the depth of 1000 m (Norilsk ore field, Russia). A possible theoretical view of this observation is presented in Ref. [13]. The frequency band of this wave has amounted to 40–100 Hz. In [14] normal modes of the zero order ranging from 6 to 21 Hz have been identified by means of spectral analysis. These modes correspond to the vibrations of a part of limestone massif with a volume of about 1000 m³ in French Alps separated by an open fracture of about 15 m depth. Seismic monitoring of unstable rock volume (260,000 m³) in the Eastern Alps, the Hochvogel summit, includes modal analysis as well aiming to control variations of rock properties due to the influence of ambient thermal factors [15]. On the other hand, the authors observe pulse-like seismic activity of 1–2 day long periods, which they presumably link to an early stage of stick-slip evolution of the unstable rock mass at the summit induced by mass wasting processes on the catastrophic fracture of 60 m depth.

Finally, regarding the laboratory scale of rock investigations, we evidence the normal mode phenomenon as a stop-band behavior [16] in the spectra of registered waves when the modes related to the fracture spacing stop propagating through a periodic structure of fractures [17]. Such a behavior has been observed on a frequency of about 80 kHz [16] in the core of rock salt from the Asse mine, Germany, during its critical stage of deformation. This frequency is related to the mean size of crystalline segments, which is about 20 mm. The segments are formed by clear fractures that go through the grain boundaries.

In this work, I suggest the following problem to be approached:

• To find a method of STF inversion from observed AE parameters taking into account a complex spectral structure of AE signals;

• To find a robust numerical criterion for the wave pattern recognition classified as a certain focal mechanism.

2. Normal modes of AE instrument

Let us consider a general problem of eigenfrequencies of a solid with free faces. In our case, we deal with lead-zirconate-titanate piezoceramic disk, which is a sensor of AE waves. It is encapsulated in a special instrument housing and transforms alternating deformation into an electrical signal. In order to estimate eigenfrequencies, which are the frequencies of normal modes, let us consider the fundamental Rayleigh-Lamb dispersion equation for symmetric and antisymmetric Lamb waves in the non-zero and zero cut-off frequencies [18, 19]. In the nonzero cut-off frequencies f_n the running plate waves turn into the standing dilatation D waves or shear S waves through the thickness h of plate (wave indexes are adopted from [18]):

$$f_n=\frac{2n-1}{2h}{c}_D$$E1

and

$$f_n=\frac{n}{h}{c}_S$$E2

for symmetric mode, and

$$f_n=\frac{n}{h}{c}_D$$E3

and

$$f_n=\frac{2n-1}{2h}{c}_S$$E4

for antisymmetric mode, where c_D and c_S are dilatation and shear wave velocities, respectively, n is an order of wave.

The second limiting condition relates to the lowest modes in plate [20, 21], which exist as propagating at any frequency no matter how low it is (the cut-off frequencies are zero). In this case, we are looking for the frequencies of standing waves arising along the diameter d dimension of a disk—they are the fundamental longitudinal and bending waves. Consideration of Rayleigh-Lamb dispersion equation for symmetric waves upon conditions that the wavelength λ is significantly greater than the plate thickness h and that the fundamental frequency f_d of standing longitudinal wave is determined by the wavelength λ = 2d, leads us to the equation of longitudinal wave frequency along the diameter d:

$$f_d=\frac{c_S\sqrt{c_D^2-c_S^2}}{c_{D}d}$$E5

Applying the same conditions to the dispersion equation for antisymmetric wave, we obtain the fundamental frequency f_d of bending wave along d:

$$f_d=\frac{\pi }{\sqrt{3}}\frac{h}{2d^2}\frac{c_S}{c_D}\sqrt{c_D^2-c_S^2}$$E6

where d is a diameter of piezoceramic disk.

In order to visualize free oscillations, the numerical simulation has been carried out by means of ANSYS mechanical finite element analysis software. The models of three thickness h modes of vibrations induced in the glass rectangular prisms of 1/5 aspect ratio are shown in Figure 1 (left). I simplify the oscillation motions by picturing the displacement and stress distributions schematically for them (Figure 1 (right)). The modes are the following: the 1st-order symmetric mode formed by the standing dilatation D wave (Figure 1a), corresponding frequency is expressed by Eq. (1); the 1st-order antisymmetric mode formed by the standing shear S wave (Figure 1b), corresponding frequency is expressed by Eq. (4); the 1st-order antisymmetric mode formed by the standing dilatation D wave (Figure 1c), corresponding frequency is expressed by Eq. (3). The latter is described by the full wave (see Figure 1c, right) in contrast to the former, where in one-half wavelength h/2 the displacements produce dilatation stresses, and in another—compression stresses.

Figure 1.

Oscillation motions in standing waves through the lateral dimension h (left), and the schematic view of displacements (blue line) and stresses (red line) in waves (right): a—the standing dilatation D wave of the 1st-order symmetric mode; b—the standing shear S wave of the 1st-order antisymmetric mode; c—the standing dilatation D wave of the 1st-order antisymmetric mode. σ—dilatation-compression stresses, τ—shear stresses, U_D—longitudinal displacements in wave, and U_S—transverse displacements in wave.

AE waveforms apparently comprise the normal mode motions of the instrument, which are the matter of initial source transformation into a secondary acoustical process. In Figure 2, an example of high-frequency AE signal and its spectrum is performed. This example is extracted from AE data acquired during the deformation process of red granite from Blauenthal, Germany [22]. To study the nucleation process of brittle deformation, we conducted laboratory experiments with loading MTS frame on rock cores in GeoForschungsZentrum, Potsdam. To acquire AE data, the KRENZ PSO9070 transient recorder [23, 24, 25] has been used. This system digitized each AE waveform with a sampling rate of 5 MHz. The example in Figure 2a depicts a typical AE waveform occurring during the stage of brittle micro-fracture formation at the beginning of irreversible deformation. One can see that the informative frequencies range up to 2000 kHz (Figure 2b). Piezoceramic disk (Marco GmbH, Hermsdorf) of 1 mm thickness h and 5 mm diameter d has been used as a sensor of AE transducer. Let us estimate the normal mode frequencies of this sensor with the help of Eqs. (1)–(6), and see how they appear on the spectrum. Taking into account the wave velocities for that particular type of piezoceramics (c_D ≈ 4420 m/s, c_S ≈ 3125 m/s), we write the normal mode frequencies estimated by the Eqs. (1)–(6) in the order of increasing: zero-order antisymmetric mode along d (bending) 80 kHz; zero-order symmetric mode along d (longitudinal) 442 kHz; the standing shear S wave of the 1st-order antisymmetric mode (thickness h mode) 1563 kHz; the standing dilatation D wave of the 1st-order symmetric mode (thickness h mode) 2210 kHz; the standing shear S wave of the 1st-order symmetric mode (thickness h mode) 3125 kHz; and the standing dilatation D wave of the 1st-order antisymmetric mode (thickness h mode) 4420 kHz. I identify the dominant frequency on the spectrum in Figure 2b with longitudinal mode because it is close to the estimated value of 442 kHz. The reduced spectral value compared to the estimated one can result from exponential attenuation.

Figure 2.

Example of AE waveform, (a) and its spectrum and (b) selected from AE data of the red granite experiment (taken from [22]).

The next distinct peak on the spectrum (Figure 2b) appears in the vicinity of 1500 kHz and relates to the standing shear S wave of the 1st-order antisymmetric mode estimated as 1563 kHz. The lowest slight peak occurs in vicinity of 100 kHz and can be attributed to the bending mode characterized by estimated 80 kHz.

Thus, we can conclude that AE spectra contain two zero-order modes associated with diameter d of the sensor disk, and one evident mode associated with thickness h–the 1st-order antisymmetric mode formed by the standing shear S wave. Displacement and stress distributions of the latter are shown in Figure 1b. The energy of a wide-band AE source radiation partially transformed into a set of narrow-band modes.

Correspondence of estimated mode frequencies of piezo-disk to AE transducer eigenfrequencies we prove when analyzing the amplitude- and phase-frequency response characteristics performed in Figure 3. Calibration data of AE transducers was provided by the research team from GeoForschungsZentrum, Potsdam, in 2007. The scientists measured the response characteristic of each transducer with the network analyzer Advantest R3754A and performed in the form of decibel-frequency characteristics. Measurements are based on synthesizing of frequency sources, the frequency of which is swept to rapidly obtain amplitude and phase information of the receiver over a frequency band of interest. In these calibrations, a frequency band of 100–5000 kHz has been applied.

Figure 3.

Amplitude- and phase-frequency response of AE transducer type (GFZ, Potsdam): amplitude-frequency—red line, phase-frequency—blue line.

Looking at transducer characteristics (Figure 3), we find that close to the frequency 442 kHz of longitudinal d mode there is an explicit peak (red curve). At the same time, we observe the corresponding sharp drop of phase to the negative values (blue curve) which is an evidence of a strong resonance at this frequency. The group of peaks within the frequency range of 2000–3000 kHz can be presumably related to the 1st-order symmetric mode associated with thickness h. It is interesting to note that in the vicinity of 4420 kHz, which is a frequency of the standing dilatation D wave of the 1st-order antisymmetric mode, we observe a second drop of phase. This behavior of phase also reflects a strong resonance at this frequency. Oscillations for this certain mode are shown in Figure 1c.

In summary, the solutions of Rayleigh-Lamb equation in cut-off frequencies for an order n ≤ 1 bring the modes explicitly detectable after impulse excitation. The normal mode of a linear system is a specific space distribution of displacements that cannot be decomposed into elementary distributions. Each normal mode is energy isolated from the other. Thus, an energy transmitted to the system by an impact can be estimated as a sum of energies of each normal mode. In [26, 27] it is shown that when radiated impulse duration is equal to a period of sample natural vibrations the total transmitted energy is converted into normal mode energy.

The forced response of a complex linear system «medium and seismic sensor» to the wide-band AE radiation is a superposition of normal modes. The intensity of each mode of excitation characterizes the STF of AE radiation.

At the end of this section, I introduce the response frequency characteristic of AE transducer type we are using in mine measurements. We will need it in Section 4. This is an acceleration instrument AP2099–1000 manufactured by the company GlobalTest, Russia, with a piezoelectric sensor embedded into the instrument. Frequency characteristic is presented in Figure 4. We see a strong increase in sensitivity on a frequency of 14 kHz, up to 40% of sensitivity on the reference frequency of 200 Hz (according to the calibration data provided by GlobalTest). Definitely, it is a result of transducer eigenfrequency excitation.

Figure 4.

Frequency characteristic of AE acceleration instrument (GlobalTest, Russia) for mine measurements.

3. Spectral density and coherence function calculations

AE signal is a transient process that exists for a finite time T. The wave packet of such a process contains a sum of attenuating harmonic oscillations, the meaning of which is the normal modes of a system «medium and seismic sensor». Then it is reasonable to apply the Finite Fourier Transform and find Fourier coefficients for the function of T period [7, 8]. In executing this procedure one should take into account an additional scaling factor.

Calculations of spectral density and coherence function I perform by self-developed computer application. At first step, the basic Fast Fourier procedure produces on its output the real Re and imaginary Im components of discrete time series of Fourier transform pair:

$$\left.\begin{array}{l}{X}_k=\sum _{n=1}^N{x}_{n}exp\left(\left(-i\frac{2\pi }{N}\right)\left(n-1\right)\left(k-1\right)\right)\\ x_n=\sum _{k=1}^N{X}_{k}exp\left(\left(i\frac{2\pi }{N}\right)\left(n-1\right)\left(k-1\right)\right)\end{array}\right\}$$E7

where X_k: direct Fourier transform; x_n: inverse Fourier transform; n = 1, …, N; k = 1, …, N; n: time counter; k: frequency counter; N: the number of samples of transient process. The correspondence of the real ReX_k and imaginary ImX_k components to Fourier coefficients a_k и b_k is expressed as follows:

$$\left.\begin{array}{l}Re{X}_k=\frac{T}{2\mathrm{\Delta }t}{a}_k=\frac{N}{2}{a}_k=\sum _{n=1}^N{x}_{n}cos\left(\frac{2\pi }{N}\left(n-1\right)\left(k-1\right)\right)\\ Im{X}_k=\frac{T}{2\mathrm{\Delta }t}{b}_k=\frac{N}{2}{b}_k=\sum _{n=1}^N{x}_{n}sin\left(\frac{2\pi }{N}\left(n-1\right)\left(k-1\right)\right)\end{array}\right\}$$E8

where Δt: sampling period; T = NΔt: signal length. X_k is a Finite Fourier Transform on discrete frequencies f_k = k/T.

Further, it is necessary to obtain physically meaningful estimations of spectral densities. For discrete x_n transient process with the sampling period Δt the initial spectral density evaluation we derive as follows:

$${\tilde{G}}_x\left(f_k\right)=\frac{2}{\mathrm{\Delta }\mathit{tN}}\left(\mathrm{\Delta }{t}^2{\left(Re{X}_k\right)}^2+\mathrm{\Delta }{t}^2{\left(Im{X}_k\right)}^2\right)=\frac{2\mathrm{\Delta }t}{N}{\left|X_k\right|}^2$$E9

where ${\left|X_k\right|}^2={\left(\mathrm{Re},X_k\right)}^2+{\left(\mathrm{Im},X_k\right)}^2$: squared modulus of Finite Fourier Transform.

The scaling factor 2∆t/N introduces the Finite Fourier Transform as a weight of a frequency component f_k normalized to the full-scale range. Thus, the dimension of spectral density becomes «Amplitude²/Hz».

With the help of spectral density, it becomes suitable to make an estimate of transient energy in any frequency band defined by certain measurement conditions. For this purpose, we invoke the Parseval theorem [28], which poses the relation between the mean square of any transient process x_n and the spectral integral:

$$\frac{1}{N}\sum _{n=1}^N{x}_n^2=\frac{1}{2}\sum _{k=1}^{\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(a_k^2+b_k^2\right)$$E10

The computer application also calculates the frequency-dependent correlation of the signals x_n and y_n arising in the stream of AE events by means of the coherence function. Radiation sources are coherent if they radiate harmonic waves of time-constant phase shift, and their sum gives a wave of the same frequency. For that, we should satisfy a requirement for the phases of this certain frequency to be non-random when dividing each of the signals into n_d equal segments. Thus, to carry out coherence analysis, it is necessary to derive statistically significant values of spectral densities. For the discrete time series x_n we obtain:

$$G_{\mathit{XX}}\left(f_k\right)=\frac{2}{n_d{T}_j}\sum _{j=1}^{n_d}\left(\mathrm{\Delta }{t}^2{\left(Re{X}_{\mathit{kj}}\right)}^2+\mathrm{\Delta }{t}^2{\left(Im{X}_{\mathit{kj}}\right)}^2\right)=\frac{2\mathrm{\Delta }t}{n_d{N}_j}\sum _{j=1}^{n_d}{\left|X_{\mathit{kj}}\right|}^2$$E11

where j = 1, …, n_d; T_j = ΔtN_j: segment length. The same we obtain for the second time series y_n as a value G_YY(f_k).

Further, we should calculate the cross-spectral density G_XY(f_k), which is a complex value unlike the values of G_XX(f_k) and G_YY(f_k). The program implements these calculations in the following way. Firstly, the real ReXY_kj and imaginary ImXY_kj components of cross-spectrum Finite Fourier Transform are derived:

$$\left.\begin{array}{l}Re{\mathit{XY}}_{\mathit{kj}}=Re{X}_{\mathit{kj}}\cdot Re{Y}_{\mathit{kj}}+Im{X}_{\mathit{kj}}\cdot Im{Y}_{\mathit{kj}}\\ Im{\mathit{XY}}_{\mathit{kj}}=Re{X}_{\mathit{kj}}\cdot Im{Y}_{\mathit{kj}}-Im{X}_{\mathit{kj}}\cdot Re{Y}_{\mathit{kj}}\end{array}\right\}$$E12

where X_kj and Y_kj are the Finite Fourier Transforms on discrete frequencies f_k = k/T_j of the time series x_n and y_n for a segment j. Afterward, we find cross-spectrum modulus:

$$\left|G_{\mathit{XY}}\left(f_k\right)\right|=\frac{2\mathrm{\Delta }t}{n_d{N}_j}\sqrt{{\left({\sum }_{j=1}^{n_d}Re{\mathit{XY}}_{\mathit{kj}}\right)}^2+{\left({\sum }_{j=1}^{n_d}Im{\mathit{XY}}_{\mathit{kj}}\right)}^2}$$E13

Finally, we obtain the coherence function γ²_XY(f_k) on discrete frequencies f_k. To explicitly specify the coherence function, I suggest the following notations:

A = ReX_kj; B = ImX_kj; C = ReY_kj; D = ImY_kj

Then, it gives expression in the following form:

$${\gamma }_{\mathit{XY}}^2\left(f_k\right)=\frac{{\left(\sum \left(\mathit{AC}+\mathit{BD}\right)\right)}^2+{\left(\sum \left(\mathit{AD}-\mathit{BC}\right)\right)}^2}{\left(\sum \left(A^2+B^2\right)\right)\left(\sum \left(C^2+D^2\right)\right)}$$E14

Here, we mean a summation over a number n_d of segments on each frequency f_k. In such representation of the coherence function, we can assure that, in case two signals are coherent on a certain frequency f_k, that is A = C and B = D, the coherence function gets its maximum value at this frequency equal to 1.

4. Examples of stick-slip seismicity control in the mines of the unique Khibiny appatite deposit, Kola peninsula, Russia

Rock massif of Khibiny appatite deposit is characterized by a complex geological structure—this is a system of joints running in different directions. As a result of geological processes, a block-like structure was formed, furthermore, the faults are scale ranked as large-block (extension is hundreds of meters) and small-block (extension less than tens of meters). In terms of geological specifics, the faults can be filled with secondary penetration minerals (e.g., monchikite dike), or empty. In accordance with these geological features, we introduce the following terms of fault surface contact—the contact «rock-dyke» and «rock-rock». This we need for spectrum classification of stick-slip process.

To describe the conditions of mining excavation a few words should be mentioned. “Apatit” is a company that developed this unique apatite-nepheline ore deposit in the mountain region since 1929. It produces a high-grade phosphate rock. At the same time, conditions of mining become complicated because of high tectonic stress resulting in different kinds of dynamic rock activity, such as microcrack nucleation and stick-slip events, which in turn lead to failure incident. In general, this deposit is classified as a rock-burst hazardous one.

In 2015, during AE monitor procedure in one of the underground mining tunnels, we registered a sequence of specific events in the course of a week [29]. The main feature of the rock massif on site was a monchikite dike on the contact with appatite-nepheline ore. It should be noted that the same signature of AE signals has been revealed at the post-strength stage during the laboratory test of the red granite core [22] (we mentioned in Section 2). This stage is characterized as a frictional sliding or shearing on the fault surfaces. I show an example of such AE doublet in Figure 5a and b, as well as their cross-spectrum (black line), and coherence function (blue line), c).

Figure 5.

AE doublet acquired during the laboratory study of the deformation process of red granite core [22] on the post-strength stage of relative shearing on fault surfaces, (a) and (b); cross-spectrum (black line) and coherence function (blue line) derived for these signals, (c).

The spectral structure of highlighted AE signals is characterized by two well-defined dominant frequencies DFr—117 and 352 kHz (Figure 5b). The coherence function indicates high correlation values (close to 1) within the frequency range of cross-spectrum significant values—100–500 kHz (Figure 5c). It is interesting to point out that the upper limit of this range coincides with the frequency of zero-order symmetric mode 442 kHz of the piezoelectric sensor (see Section 2). Thus, we may conclude that the signals of this particular type are of similar radiation nature, which is qualified by two independent spectral components and П-pattern coherence function. According to the spectral pattern similarity of these laboratory AE signals to the seismicity registered in mine in the vicinity of the contact «rock-dyke» [29] I have designated the mining AE events to be attributed to stick-slip process. Tectonic activity in this region can generate frictional sliding on the contact «rock-dyke» the same as in laboratory rock cores pressure initiates the shearing on fault surfaces at the post-strength stage.

After this finding, we proceeded to monitor AE process in different places of mine close to dike occurrence. The result was the same selected type of signal. In this work, I perform the results of AE monitoring carried out by the special mining devices of Prognoz-L product line (Figure 6a). These devices are developing in the mining institute of the far-eastern division of the Russian Academy of Science [30]. The acceleration instrument AP2099–1000, GlobalTest, is used as AE transducer (see Section 2). The measurement frequency range is 0.5–30 kHz, the eigenfrequency is equal to 14 kHz (Figure 4). The nominal value of the electroacoustic transfer factor on the reference frequency is 100 mV/m/s².

Figure 6.

Mining devices Prognoz-L for seismicity control, (a); a photograph of rock massif, view from the tunnel, (b) the tunnel exhibits a great number of fractures and a trace of monchikite dyke, marked by the black arrow.

Let us consider the measurement series conducted in the period of 22.10.2021–12.11.2021 aimed to control the stability of rock massif after a strong seismic event, which was recorded by the seismic acquisition system of mine. The rocks of the massif are strongly fractured, and the monchikite dike of 0.6 m thickness occurs with an inclined angle of 60°–65° (Figure 6b).

Figures 7 and 8 perform the capture samples of AE process and their spectrum moduli. It is evident that both spectra consist of two independent spectral components: low-frequency narrow-band component corresponding to a quasi-stationary monochromatic process and high-frequency wide-band component described by the main lobe and two side lobes. According to the spectral theory [31], the latter (underlined by a red transparent line) is produced by a sine fragment of frequency which corresponds to the main lobe peak. Moreover, the spectral half-width of the main lobe is 2 kHz and it characterizes approximate value of the envelope length (about 0.5 μs). It means that this process has an apparent impulse (non-periodic) property with a basic frequency of 14 kHz related to the transducer eigenfrequency (Figure 4). In other words, the normal mode effect shifts the envelope function of sinc(ω) in the frequency ω domain by the value of normal mode.

Figure 7.

Two-component spectrum of a typical AE stick-slip signal presumably generated on the contact «rock-dyke»: signal—blue line; trapezium-like envelope—red line (upper part of the interface); spectrum—black line (lower part of the interface): the frequency range associated with the envelope and inserted frequency 14 kHz is underlined by red transparent line—a fast radiation; the mode of 4.89 kHz—a slow radiation.

Figure 8.

Two-component spectrum of a typical AE stick-slip signal presumably generated on the contact «rock-rock»: signal—blue line; trapezium-like envelope—red line (upper part of the interface); spectrum—black line (lower part of the interface): the frequency range associated with the envelope and inserted frequency 14.59 kHz is underlined by red transparent line—a fast radiation; the mode of 7.33 kHz—a slow radiation.

Thus, Figures 7 and 8 illustrate the same property of wave source. The only difference we can emphasize is that the two-component spectrum pattern in Figure 8 is shifted to the higher frequencies in comparison with the pattern in Figure 7. The frequency of monochromatic component moves from 4.89 kHz position to 7.33 kHz position, and the onset of the impulse spectrum moves from 7.184 to 9.862 kHz. Interpretation of two-component spectrum feature is based on the models of complex sources [1] one of which is the model of asperities on the fault surfaces. In the focal region of a stick-slip source, two different processes occur. When a force is applied to one of the rock blocks, its elastic strain energy increases until it exceeds the strength of asperities on the contact «rock-dyke» or «rock-rock». Once the strength is achieved and asperities are fractured, the tangential stress drop occurs on the fault surfaces resulting in the instantaneous slip of a block, which is accompanied by elastic wave radiation. Fast radiation (short pulse) corresponds to asperity fracturing, the spectrum pattern of which is designated by a shifted sinc-function. Whereas the slow radiation (low-frequency monochromatic) relates to quasi-stationary frictional sliding.

In Figure 9, I show the examples of AE stick-slip signals initiated on the contact «rock-dyke», (a) and (b), their cross-spectrum (black line) and coherence function (blue line), (c), and the examples of AE stick-slip signals initiated on the contact «rock-rock», (d) and (e), their cross-spectrum (black line) and coherence function (blue line), (f). Moreover, the signals are decomposed into two components: black color depicts the narrow-band process related to quasi-stationary frictional sliding and red line depicts impulse wide-band process related to asperity fracturing on fault surfaces. It appears that the coherence function is described by П-pattern, the same as observed during the red granite deformation process (Figure 5). It is worth noting that one can see the shift of the lower limit of П-pattern toward higher frequencies for the stick-slip process on the contact of «rock-rock» (Figure 9f, blue line) in comparison with «rock-dyke» (Figure 9c, blue line). This fact reflects the shifting of the left side lobe lower limit for «rock-rock» stick-slip (Figure 8) relative to «rock-dyke» stick-slip (Figure 7).

Figure 9.

AE doublet of impulse (non-periodic) signals of stick-slip on the contact «rock-dyke», (a) and (b); their cross-spectrum (black line) and coherence function (blue line), (c); AE doublet of impulse (non-periodic) signals of stick-slip on the contact «rock-rock», (d) and (e); their cross-spectrum (black line) and coherence function (blue line), (f).

The evidence of asperity influence on AE process during stick-slip is experimentally shown, for example, in Ref. [32]. There the authors observe several bursts of AE activity against the background of a gradual AE increase prior to the main slip event. These AE bursts explained by the failure of grain-scale asperities on rough surfaces of a fault.

Let us proceed to the quasi-periodic radiation of AE stick-slip source, which is characterized by multiple asperity failure. Evidently, in this case, the coherence function П-pattern will be destroyed (Figure 10d, blue line) because the phases of harmonics characterizing the impulse radiation become random resulting in the reduction of the values of coherence function. At the same time, the calculations of statistically significant values of spectral densities reveal the transformation of the continuous spectrum (Figures 7 and 8) into a discrete one (Figure 10b and d) in the area of transducer eigenfrequency 14 kHz. The spectrum discrete nature is defined by frequency increment ∆f, which is an inverse value of a period T which in turn characterizes a sequence of pulses generated by asperity fracturing (Figure 10a and c, the filtered impulse component is depicted by red color). Based on spectral theory [31], this transformation can be illustrated as it is shown in Figure 11. The spectrum (Figure 11b) of the synthetic signal is calculated, which is composed of attenuating bipolar pulses with a period 10 μs and length 2.5 μs (Figure 11a). As a result, the continuous spectrum of a single pulse (Figure 11b, gray line) turns into a discrete spectrum of periodic sequence of pulses (Figure 11b, black line).

Figure 10.

AE stick-slip signals associated with multiple asperity failure on the contact «rock-rock» recorded in rock massif of small-block faults, (a) and (с); their spectral densities presented by discrete spectra, (b) and (d); coherence function for these signals (blue line), (d).

Figure 11.

Illustration of continuous spectrum transformation into discrete spectrum due to pulse periodic repeat: (a) synthetic signal composed of attenuating bipolar pulses with period 10 μs and length 2.5 μs; (b) discrete spectrum of pulse periodic repeat (black line), and continuous spectrum of the single pulse (gray line).

In the end of this section, it should be noted that AE stick-slip signal classification according to the contact «rock-dyke» or «rock-rock» is based on two items: geological characteristic of rock massif on site, and spectral displacement to higher frequencies talking about the contact «rock-rock». I argue the latter by the fact that the higher the frequencies are, the shorter is STF, which in turn results from stick-slip radiation in small-block faults (on the contact «rock-rock»).

5. Conclusions

Based on spectral-coherency analysis of AE signals on laboratory scale and mining scale, we can conclude the following:

• AE sensor normal modes strongly influence a radiated source wave modifying it into a set of narrow-band processes;

• Thus, we may convolve an origin AE wave A(f) with normal modes NM(f) of a system «medium and seismic sensor» and derive a resultant output signal: S(f) = A(f) · NM(f);

• П-pattern of coherence function has been revealed, which presumably specifies AE doublet of nonperiodic signals of stick-slip type, which can be easily selected in AE data;

• Periodic behavior in stick-slip source is detected by discrete spectrum obtained by calculations of statistically significant values of spectral densities, which in turn is evidence of multiple asperity failure on fault surfaces.

The suggested approach of spectral-coherency analysis of AE control in mines can be applied to the development of new algorithms for critical-stage automatic detection.

Acknowledgments

I thank Prof. Dr. Georg Dresen from GeoForschungsZentrum, Potsdam (Germany) for providing me with the calibration data of AE transducers.

I appreciate the help and support of the team of Rock Hazard Prevention Service of JSC «Apatit» (Kola Peninsula, Russia) in conducting of AE measurements under mining conditions and interpretation.

I thank engineers from Mining Institute of the Russian Academy of Sciences (Far East, Russia) for consulting me on the technical issues of portable geoacoustic device «Prognoz-L» operation.

Conflict of interest

The author declares no conflict of interest.