# Energy Classification of Acoustic Events Using the Coda of a Signal

Tatiana I. Kartseva, Nikolai M Shapiro, Andrey V. Patonin, Vladimir A. Smirnov, Alexander Ponomarev

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Tatiana I. Kartseva, Nikolai M Shapiro, Andrey V. Patonin, Vladimir A. Smirnov, Alexander Ponomarev. Energy Classification of Acoustic Events Using the Coda of a Signal. Seismic Instruments, 2022, 58 (1), pp.18-25. 10.3103/S0747923922010054. hal-03612866

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# Energy Classification of Acoustic Events Using the Coda of a Signal

T. I. Kartseva<sup>a, b, *</sup>, N. M. Shapiro<sup>b, d</sup>, A. V. Patonin<sup>c</sup>, V. B. Smirnov<sup>a, b</sup>, and A. V. Ponomarev<sup>b</sup>

Received July 18, 2021; revised September 14, 2021; accepted September 16, 2021

**Abstract**—A new energy classification method is proposed for acoustic events recorded in laboratory experiments on rock destruction. The method analyzes the coda waves of acoustic emission (AE) events. Coda waves are considered as reverberation of the acoustic field in the test sample. The new classification was tested on two experiments carried out on different rocks: granites of the Voronezh massif and Berea sandstone, on an INOVA-1000 controlled hydraulic press at the Borok Geophysical Observatory (GO), Schmidt Institute of Physics of the Earth, Russian Academy of Sciences (IPE RAS). Comparison of the new classification with the one used at Borok GO showed that both methods give well consistent results in the middle range of energies of AE events. For strong events with the saturated initial parts of the signals due to the limitations of the recording equipment, the new technique demonstrated better results, leading to energy estimates of such events from the undistorted coda of the signal.

**Keywords:** acoustic emission, rock physics, coda of a signal, energy classification

**DOI:** 10.3103/S0747923922010054

## INTRODUCTION

One reason why it is possible to simulate seismic phenomena in a laboratory is the self-similarity of the fracturing processes process. On this basis, microcracks in rock samples in laboratory conditions are compared with earthquake sources in field conditions (see, e.g., (Smirnov and Ponomarev, 2020) and references therein). The energy distribution of AE events, like the distribution of earthquakes, can be described by the empirical Gutenberg–Richter power law: $log N = a - bM$, where *M* is the magnitude of the event, *a* and *b* are constants (Gutenberg and Richter, 1944), often referred to as the recurrence law. In Soviet and Russian seismological studies, instead of magnitude, it is often customary to use the earthquake energy class K (Rautian, 1960, 1964); the recurrence curve is described as $log N = a - bK$. The slope *b* of the recurrence law is an important parameter for studying fracturing modes, and its estimation can be strongly influenced by the method of determining magnitudes (magnitude scale) or classes (energy classification).

For natural earthquakes, there are numerous magnitude scales (energy classification schemes) that analyze direct waves (Gutenberg and Richter, 1956), scattered waves (described below), energy (Rautian, 1960, 1964; Choy and Boatwright, 1995), or the spectrum of

the displacement signal (Kanamori, 1977; Hanks and Kanamori, 1979). In laboratory conditions, the number of wavelengths that fit into the path of an elastic wave from a source to a receiver is always much less than in natural conditions, which makes it difficult to directly transfer seismological energy classification methods to laboratory conditions. The simplest and most common classification uses the maximum amplitudes of the wavetrain $A_{max}$, which are formed mainly at the time of the first wave reflections from the walls of a sample. The techniques are described in (Lockner et al., 1991; Zang et al., 1998). Smirnov et al. (Smirnov 1995) proposed using the quantity $K = 2 \log A_{max}$ in analogy to the seismic energy class.

An alternative classification method can be based on the decaying part of the signal. In seismology, this part is called the coda and is formed mainly due to waves being scattered by lithospheric inhomogeneities (Aki and Chouet, 1975; Gusev and Lemzikov, 1985; Abubakirov and Gusev, 1990). Absorption in the medium and geometrical spreading also make a significant contribution to the decay of the coda. The coda envelope has a fairly stable form for earthquakes of different classes and at different seismic stations; therefore, magnitude scales and classification schemes based on it were proposed (Rautian and Khalturin, 1978; Lemzikov and Gusev, 1989; Mayeda and Walter,

<footnote><sup>a</sup> Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991 Russia</footnote>

<footnote><sup>b</sup> Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, 123242 Russia</footnote>

<footnote><sup>c</sup> Geophysical Observatory Borok, Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Borok, Yaroslavl oblast, 152742 Russia</footnote>

<footnote><sup>d</sup> ISTERRE, CNRS, Université Grenoble-Alpes, Grenoble, 38058 France</footnote>

<footnote>*e-mail: ti.kartceva@physics.msu.ru</footnote>
1996; Mayeda et al., 2003; Shevchenko and Yakovenko, 2020).

In the laboratory, such a classification is also possible, but a different mechanism for the formation of coda-waves is assumed. Since the entire sample has dimensions, usually just a several times several times greater than the wavelength (and sometimes comparable to it), the main coda formation process is multiple reflections from the walls of the sample (reverberation). The nature of the decay in the coda is determined by reverberation and absorption losses in the medium. Note also that multiple reflections exclude the influence of the directivity pattern of the microcrack, which is a very significant argument in favor of the proposed method.

Taking into account inhomogeneities in the walls of sample, reverberation can be described as diffuse reflection (Kanev, 2011). After many reflections, the wave field is assumed to be uniformly distributed in space and attenuation of the amplitudes of this uniform (in other terminology, diffuse) field in the course of further reverberation occurs exponentially (Weaver, 1984, 1987; Kanev, 2011). The successful use of this approximation in experiments on the rebound of a ball from surfaces of samples of different sizes was demonstrated in (Farin et al., 2016). We accept that the decay of the AE signal is determined mainly by the reverberation and absorption of waves, and we also approximate it by an exponential dependence. After the logarithm is taken, it turns into a linear time dependence, which greatly simplifies analysis. The geometric divergence of acoustic waves in the sample, expressed as a function of time $1/t$, can be neglected, since by the time the coda starts, the decrease in acoustic signal amplitude due to divergence is two orders of magnitude smaller than that due to reverberation and absorption.

This article presents the developed method of energy classification of AE events based on the coda of a signal. As an example two experiments are used, with similar loading schemes performed at the Borok GO of IPE RAS on samples of granite from the Voronezh crystalline massif and of Berea sandstone, in order to compare the classification results by the proposed method with classification data used in laboratory studies based on the maximum amplitudes of the wave packet.

## EXPERIMENTAL

### *Recording and Energy Classification of Acoustic Events*

The data of experiments carried out on an INOVA-1000 controlled hydraulic press at the Borok GO of IPE RAS were studied. A detailed description of the hardware and test methodology is presented in (Patonin et al., 2014; Patonin et al., 2019). In the experiments, standard cylindrical rock samples with a diameter of 30 mm and height of 60 mm were used.

The test sample was sealed against oil penetration and installed in a special cell with a system of ultrasonic and strain gauge sensors. The cell was then mounted in a high-pressure chamber and placed on a test press. The ultrasonic sensors formed two independent recording systems.

The first system, consisting of 16 sensors located symmetrically on the lateral surface of the sample, is designed to record elastic wave propagation velocities and individual AE signal waveforms. The second system, consisting of two sensors located in the upper and lower bases of the cell, is used to continuously record the AE flux from the sample. AE signals are amplified and transferred to the inputs of a multichannel ADC with a sampling rate of 2.5 MHz per channel. The preamplifier gain is 40 dB for a bandwidth of 700 Hz–1.2 MHz.

The energy classification of all acoustic events recorded by both systems is based on the maximum effective amplitude of the continuous signal from the least noisy end sensor. The energy class of an event is determined as $K = 2 \log A_{max}$, where $A_{max}$ is the maximum amplitude of the envelope for this event (Smirnov et al., 1995).

Signals from the lateral sensor system start recording after the signal exceeds the discrimination threshold. In this case, not only the signal itself is recorded, but also its prehistory. A discrimination threshold 2.7 times higher than the noise level is selected. When a signal is detected, data with a length of 1500 samplings from 16 sensors are saved to disk. We call such a recording a “frame.” For a sampling step of 0.4 µs, the frame length is 600 µs, and the first arrival of the signal is on average 250 µs from the beginning of the frame (Figs. 1a and 1b).

Due to different degrees of pressing of the sensors against the sample, the lateral sensor system has a lower sensitivity compared to the end sensors pressed against the sample by the axial load of the press. As a result, for the energy classification by maximum amplitudes used at the Borok GO, the lateral system was not used: only the elastic wave arrival times at each of the sensors, their amplitudes, and signs were taken as the useful information. These were used to locate the AE sources and determine the types of source mechanisms based on the method of (Zang et al., 1998). Events whose signals are registered by the lateral sensors are assigned energy classes according to the end sensor data by synchronizing the signal arrival times at both sensor systems.

### *Methodology of Energy Classification of Acoustic Events by the Coda of a Signal*

If we consider not only the first arrivals of recordings of signals from the lateral sensor system, but also the entire waveform, then we can extract more infor-
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Signals of two classes of events: (a, c) *K* = 5.7 and (b, d) *K* = 9.4. (a, b) Waveforms for 3 of 16 sensors; (c, d) amplitude envelopes at 16 sensors (I) and averaged coda envelopes (II). 1–10, Sensor numbers.</figcaption></figure>

mation about the source itself and the medium in
which the acoustic wave propagates.

In these experiments, for the beginning of the
decaying part of the signal, the coda, for events of all
energies, we take t₁ = 320 µs of the recording frame
(Figs. 1c, 1d). If we assume that the average start of the
signal is at 250 µs, then by the time t₁ = 320 µs, about
ten wave reflections from the walls have time to occur,
and there is a sufficient signal duration until it merges
with noise even for weak events. Figures 1c and 1d
show examples of the envelopes of two signals on 16
sensors, obtained by convolution of a Henning win-
dow with the absolute values of signal records at
100 points (40 µs).

Curves II in this figure show the average values of
these envelopes at the signal intervals, which we attri-
bute to the coda. For most events, the coda amplitude

has time to decrease to the noise level (on average, up
to 10 units of the ADC) within the duration of a stan-
dard recording frame, as, for example, for an event
with class *K* = 5.7 in Fig. 1c. For the biggest events
with *K* = 7.5–9.6 the duration of a standard frame is
insufficient (Fig. 1d).

As mentioned above, the decay of the envelope of
the coda with time can be described by the exponential
dependence y = y₀e<sup>at</sup>. Taking the logarithm, we
obtain the linear dependence log y = log y₀ + at log e.
In Fig. 2, which shows the averaged coda envelopes for
several events of different classes; clearly, such an
approximation is really well applicable. The slope of
the straight line on a decimal logarithmic scale
A = a log e is negative and characterizes reverberation
and absorption losses, that is, the medium in which
<figure><img src="image_3.png" /><figcaption>Fig. 2. Amplitude envelopes for several events averaged over 16 sensors; dotted lines are their linear approximations. Numerals near curves are classes of events in maximum amplitude classification. Two weak events with nearly the same classes (4.56 and 4.57) in the coda classification will have a larger difference in class than in the maximum amplitude classification.</figcaption></figure>

the signal propagates, and the coda level log $y_0$ characterizes the signal source and the gain of the sensor by which it was recorded.

In what follows, we separate the contributions from the source and sensor: log $y_0 = B + C$. Coefficients B and C will be used in the new classification. It is important to note that in this formulation, the coda level is taken as the level of beginning of the record: 0 µs. It would be more correct to take this level for origin time, but due to the specifics of the recording system, the earliest arrival $t_0$ to one of 16 sensors may be a more suitable time. In this case, the maximum error will be $A\Delta t$, where $\Delta t$ is the one-way travel time of a wave in the sample. For an average value of $A = 0.005$ µs$^{-1}$ ($a = 0.01$), which we obtained when analyzing the data, and an average travel time of a wave of 10 µs, the error will affect the second decimal place in the energy class estimate.

Since the internal structure of the sample changes in the course of the experiment, it is better to estimate the coda parameters for groups of neighboring events under the assumption that the environment remains unchanged during the period of occurrence of these events, that is, with the parameter A common to this group. A change in the environment can also lead to a change in the boundary conditions, which in turn

affect the transfer characteristics of the sensors, expressed by the C parameters. We accept that the parameters C will be different for different sensors but the same for records obtained from one sensor for different events.

Joint estimation of the coda parameters can be achieved by solving a system of linear algebraic equations for sequential groups of N events. This system is written as follows:

$$Y = AT + \sum_{i=1}^{16} B_i X_i + \sum_{j=1}^{16} C_j S_j, \quad (1)$$

where Y is the vector of logarithms of the envelope values of the coda in time counts; T is the vector of time counts in microseconds; $B_i$ is a parameter characterizing the ith event; $X_i$ is a vector consisting of zeros and ones (ones are placed in the lines of the system in which records of the ith event are considered); $C_j$ is a parameter characterizing the jth sensor; $S_j$ is the vector of zeros and ones (ones are placed in lines in which records from the jth sensor are considered).

There are a total of 17+N unknowns in system (1) and 16$\sum_{i=1}^{N} D_i$ equations, where $D_i$ is the number of time counts, determined for each specific event as
$(t_2 - t_1)/dt$, where $dt = 0.4$ µs. We take as the start of the coda for all events $t_1 = 320$ µs and times $t_2$ that determine the duration of the coda; for events of different classes, these will be different, as mentioned above. The system of equations is strongly overdetermined, so the least squares method is used to solve it.

The coda class for the ith event is determined as follows:

$$K_{\text{coda}}^{i} = 2(B_{i} + \bar{C} - A t_{0}^{i}), \quad (2)$$

where $\bar{C}$ is average, for 16 $C_j$ values, signal gain in a given group of events; $A t_0^i$ is correction for the time of first arrival. Coefficient 2 is introduced analogously with seismic energy classes.

**Coda Classification Scheme:**

(1) The first *N* events adjacent in time in the experiment are selected.

(2) Smoothing of all 16*N* records yields the envelopes, the values of which will be used further in the system of equations (in this study, a Henning window was used, with smoothing over 100 points or 40 µs of the record).

(3) System of equations (1) is formed an example of a system string for the first sampling of the record of the first event from the first sensor will look as follows:

$$\log y_1^i(t_1) = At_1 + 1 \times B_1 + 0 \times B_2 + \dots + 0 \times B_N + 1 \times C_1 + 0 \times C_2 + \dots + 0 \times C_{16}.$$

(4) The system is solved by the least squares method using any standard function for multiple linear regression (in this study, we used the Linear Regression function of the Scikit-learn library, Python (Pedregosa et al., 2011)).

(5) The classes of *N* events are calculated by formula (2).

(6) The scheme is repeated for the next *N* events, and when all events recorded in the experiment are exhausted, a general catalog is compiled with a new energy classification of each event.

## RESULTS OF EXPERIMENTAL STUDIES OF THE DEVELOPED TECHNIQUE

To test the new technique, two experiments were chosen: on granite from the Voronezh crystalline massif and on Berea sandstone. The experimental scheme was similar: first, the uniform pressure was raised to 12 MPa, then axial loading of the samples brought them almost to the strength limit (about 120 MPa for sandstone and 270 MPa for granite); this was followed by a series of sawtooth changes in the axial load, and at the end, the uniform pressure and loads were reset to zero. The waveforms were recorded by the lateral sensors on granite for 38349 total events; on sandstone, 59830. All coda parameters were evaluated in sequential nonoverlapping groups of *N* = 100 events.

<table><caption>**Table 1.** Values of slopes of recurrence laws for different classifications in two experiments</caption><thead><tr><th>Experiment</th><th>Maximum amplitude classification</th><th>Coda classification</th></tr></thead><tbody><tr><td>Granite</td><td>b = 0.83 ± 0.01</td><td>b = 0.77 ± 0.01</td></tr><tr><td>Sandstone</td><td>b = 0.97 ± 0.01</td><td>b = 0.95 ± 0.01</td></tr></tbody></table>

Figure 3 shows the results from comparing the new coda classification and currently used maximum amplitude classification for the two experiments. Figures 3a and 3b show the scatter diagrams of classes determined by the two methods. The orthogonal regression method was used to obtain formulas for recalculating coda classes to the old classes. The recurrence laws in a unified class scale are shown in Figs. 3c and 3d (the coda classes were recalculated to the classes according to the maximum of the record using the obtained regression relations in Figs. 3a and 3b).

Table 1 shows the values of the parameters *b* determined by the maximum likelihood method (Aki, 1965) according to the recurrence curves in Figs. 3c and 3d in the field of representative classes $K> 6$. The higher values of slopes of the recurrence curves, constructed by the maximum amplitude classification, can probably be explained by underestimation of the classes of strong events due to the limitation of the ADC of the maximum amplitudes of strong events.

## DISCUSSION

From the recurrence laws (Figs. 3c and 3d), it follows that the class of completeness (about 6) is the same for both classifications; in the ranges of classes 6–7.5, the recurrence curves agree best; in the field of major events ($K> 7.5$), the results differ markedly: one can see the limitation of the classes determined from the maximum amplitudes, a value of about 9.6. This discrepancy is due to the limitation of the maximum signal amplitudes by the ADC range (see Fig. 1b), while the upper threshold results from 16 bits of the ADC and is $2\log(2^{16}) \approx 9.63$. The new classification, which uses the undistorted decaying part of the signal, makes it possible to reconstruct the classes of the strongest events and thereby expand the available scale by at least half a class.

The higher values of the slopes of the recurrence laws constructed by the maximum amplitude classification can probably be explained by underestimation of the classes of strong events due to the limitation of the ADC of the maximum amplitudes of strong events.

Note that the experiments from which the data were used in this study were carried out on dry sam-
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Comparison of two classification methodologies in two experiments: (a, c) on granite; (b, d) on sandstone. (a, b) Scatter diagrams of classes determined by two methods. (1) Straight lines of orthogonal regression (formulas for recalculating coda classes to current classes K: (a) K = 1.12K<sub>coda</sub> + 2.95, (b) K = 1.13K<sub>coda</sub> + 3.65); (c, d) recurrence curves according to maximum amplitude classification K (2) and coda classification (3), recalculated to K.</figcaption></figure>

ples. The exponential decay of the coda was related
more to the effect of frequency-independent reverber-
ation. We did not consider the contribution of absorp-
tion and its variability to the decay of the coda during
the experiments. When fluid is present in the samples,
especially when its concentration changes during the
experiment, the issue of taking absorption into
account will obviously require additional research. In
particular, the frequency dependence of the absorp-
tion coefficient will need to be taken into account and,
probably, the frequency range of the wave coda should
be properly selected for their use in energy classifica-
tion.

CONCLUSIONS

A new method for the energy classification of AE
events in laboratory experiments based on wave coda
data has been developed, implemented, and tested and
has shown its effectiveness in this configuration of the
experiment. The methodology makes it possible to
expand the range of classes by bypassing instrumental
limitations on the amplitudes of large event signals.

The wave coda classification excludes the influ-
ence of the radiation pattern, and in the conditions of
our experiment, several more advantages were found
that increase its reliability: instead of 2 sensors, 16 are
used; class estimation takes into account changes in
the environment; for energy classification, the same
lateral sensor system is used as for solving problems of
hypocenter localization and categorization of the
source mechanisms of AE events.

ACKNOWLEDGMENTS

The experimental part of the research was carried out on
equipment of the Shared Use Center of IPE RAS “Petro-
physics, Geomechanics, and Paleomagnetism.”

FUNDING

This study was supported by the Ministry of Education
and Science (grant no. 075-15-2021-628, Geophysical
research, monitoring, and prediction of the development of
destructive geodynamic processes in the Far East of the
Russian Federation) and the European Research Council
(ERC grant no. 787399-SEISMAZE).

CONFLICT OF INTEREST

The authors declare that they have no conflicts of inter-
est.

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