## 2 Materials and Methods
### Algorithm for Detecting Weak Signals (\(\phi\)-Method)  
The electric signal at the output of the AE amplifier depends on many factors. First, it depends strongly on the sensor frequency response and the media transfer function [61]. The electric signal \(U(t)\) at the output can be expressed as a convolution integral of the dynamic source function \(E(t)\), which is represented by the average stress drop (in a tensorial form) at the source location, with the total transfer function \(A(t)\), i.e.,  
\[U(t)=\int\limits_{-\infty}^{\infty}dt^{\prime}\cdot A(t-t^{\prime})\cdot E(t^{ \prime}), \tag{1}\]  
where \(A(t)\) combines the transfer function of the media, in which the elastic waves propagate, and of the apparatus. Using the transfer function formalism, the latter equation in the Fourier domain transforms into a simple product (c.f. [62])  
\[G(f,t)=\overline{A}(f)\cdot\overline{E}(f,t), \tag{2}\]with tilde denoting the respective spectral density functions. [PERSON] and [PERSON] [63] noticed that by far the most important consequence of the transfer function formalism is that information about the source is passed frequency by frequency to the output through the transfer function. The AE detector is constructed as:  
\[\varphi(t)=\frac{1}{\Delta f}\int\limits_{f_{\min}}^{f_{\max}}\left[\frac{ \partial\ln G(f,t)}{\partial t}\right]df, \tag{3}\]  
with \(\Delta f=f_{\max}-f_{\min}\) - the frequency band. Obviously, this parameter equals to zero when the sources are stationary, and their properties (the PSD, to be more specific) do not change with time, and \(\varphi(t)\) is nonzero when the sources evolve: the faster the spectral density of the sources \(G(f,t)\) changes, the higher the absolute value of \(\varphi(t)\). The logarithmic derivative \"amplifies\" the changes in the AE PSD when the stationary background noise changes to a mixture of noise and transient signal components. Besides, in constructing the function \(\varphi(t)\) in the form of Equation (3) we were motivated by the long-standing desire to reduce or eliminate the influence of the transfer function on the AE descriptive parameter. By plugging Equation (2) into Equation (3), one can see that the \(\phi\)-function is essentially free from this influence, provided \(\overline{A}(f)\) is a function of frequency, and this function does not change in time (the condition that is often met in laboratory experiments or in-field AE inspections):  
\[\varphi(t)=\frac{1}{\Delta f}\int\limits_{f_{\min}}^{f_{\max}}\left[\frac{ \partial\ln\left(\overline{A}(f)\cdot\overline{E}(f,t)\right)}{\partial t} \right]df=\frac{1}{\Delta f}\int\limits_{f_{\min}}^{f_{\max}}\left[\frac{1}{E (f,t)}\frac{\partial\overline{E}(f,t)}{\partial t}\right]df \tag{4}\]  
The beauty of the last expression is that it no longer involves the transfer function, and the introduced \(\phi\)-parameter depends primarily on the dynamics of the evolution of the source function \(\overline{E}(f,t)\). Using a sliding window with a user-defined size and step, Equation (3) transforms the original time series into a new train of pulses with a substantially enhanced SNR. Figure 1 illustrates the AE transient waveform (a) and the corresponding shape of the \(\phi\)-parameter (b). Figure 1: (**a**) A typical fragment of the AE signal. (**b**) The corresponding \(\phi\)-parameter before smoothing and normalisation procedure. (**c**) The corresponding \(\phi\)-parameter after smoothing and normalisation procedure. According to a proposed detection scheme, events are distinguished by a decision function \(g(m)\) defined at a pre-set constant level has:  
\[EVENT=\left\{ \begin{array}{l}True,g(m)>h\\ False,g(m)<h\.\end{array} \right. \tag{5}\]  
where \(h\) is a detection threshold. The procedure and guidelines for its determination are outlined in detail below. Hereinafter, the index m denotes discrete time. The beginning and the end of the \(i\)-th event, \(t_{on}\) and \(t_{off}\), respectively, are defined as:  
\[