*3.2. b-Value versus Natural Time Analyses of AE Time Series*

In order to investigate the entrance of the structural element to a critical state, we focus on the cluster of AE hits occurred in the interval 490–510 h. The temporal evolution of two AE parameters has been considered: the *b*-value of the Gutenberg-Richter (GR) law, and the variance *κ*<sup>1</sup> of the natural-time transformed time series [25–27].

The GR law, initially introduced in seismology [41] and then extended to the statistics of AE signals [38,42–46], is expressed by the relation

$$
\log\_{10} N = a - b \,\, \times M \tag{1}
$$

where *M* ≡ log10(*A*max/1 μV) is the magnitude [38,45], i.e., the logarithm of the AE peak amplitude, *N* is the incremental frequency, i.e., the number of AE hits with magnitude greater than the threshold *M*, and *a* and *b* (termed as *b*-value) are empirical constants to be fitted. The standard error of *b*-value is *b*/ <sup>√</sup>*<sup>N</sup>* for a population of *<sup>N</sup>* samples and 95% confidence limits are twice this value.

The *b*-value is the negative slope of the linear descending branch of the GR law, and it grossly represents the relative number of micro-fractures to macro-fractures. Generally, systematic decreasing *b*-values are indeed observed during laboratory loading tests, with a single minimum just prior to specimen failure [38,42–46].

In the present study, the *b*-value analysis has been carried out twice, by partitioning the data set in groups formed by 300 and 800 hits. The *b*-value of each subset has been obtained through Equation (1) and regression analysis in the linear range of GR graphs, from 2.0 to 3.8 or 4.1 magnitude (i.e., from 40 to 76 or 82 dB). The related time series of *b*-values exhibit similar trends, especially sharing a global minimum *b* = 0.9–1 at time *tb-min* = 501 h (marked in bottom diagram of Figure 4). It is worth remembering that *b*-values close to unity correspond to the growth of macro-fractures in the monitored element [38,39].

**Figure 4.** From top to bottom: accumulated number of AE signals; AE signals count rate; time series of AE signals frequencies (derived from signal duration and ring-down count) and amplitudes; *b*-value over time calculated using groups of, respectively, 300 (yellow squares), and 800 (blue squares) numbers of signals. The dashed line indicates a critical point revealed by the natural time analysis.

The results of the *b*-value analysis appear to be reliable, as they are substantially independent from the chosen partition. The final trend toward higher *b*-values suggests that the monitored masonry wall is substantially stable, with a damaging episode driven by the specific seismic event, and interpretable as a sign of enhanced structural sensitivity to light earthquakes.

In order to take into account variations in the statistical properties of the AE amplitude distributions, the "improved *b*-value" (*Ib*) was introduced and defined as follows [42,44,45]:

$$Ib = \left[\log\_{10} N(\mu - a\_1 \sigma) - \log\_{10} N(\mu + a\_2 \sigma)\right] / \left[(a\_1 + a\_2)\sigma\right] \tag{2}$$

where *μ* and *σ* are mean and standard deviation of each AE amplitude subset, found to be, respectively, ~45 and 8 dB, whereas *α*<sup>1</sup> and *α*<sup>2</sup> are user-defined coefficients representing lower and upper limits of the amplitude range in which the cumulative frequency–magnitude distribution properly fits a straight line. The *Ib*- and *b*-values formed by a 300-hit partitioning are plotted in Figure 5, showing a good accordance between the two analyses.

**Figure 5.** The *b*-value and *Ib*-value calculated using groups of 300 numbers of signals.

Besides the *b*-value, other synthetic parameters acting as potential fracture precursors can be extracted from a time series of *N* AE hits by the natural time analysis. This concept is introduced by ascribing the natural time *χ<sup>k</sup>* = *k/N* to the *k*-th event of energy *Qk* [25–27].

Regarding the normalized energies *pk* ≡ *Qk*/Σ*k*=1*NQi* as the probability distribution of the discrete variable *<sup>χ</sup>k*, the variance *<sup>κ</sup>*<sup>1</sup> ≡ *<sup>χ</sup>*<sup>2</sup> − *<sup>χ</sup>*2, is considered a key parameter for identifying the approach to a critical state, and defined as follows:

$$\kappa\_1 \equiv \sum\_{k=1}^N p\_k \chi\_k^2 - \left(\sum\_{k=1}^N p\_k \chi\_k\right)^2 \equiv <\chi^2> - <\chi^2>\tag{3}$$

As *χ<sup>k</sup>* and *pk* are rescaled upon the occurrence of any additional hit, *κ*<sup>1</sup> results to be an evolutionary parameter.

It has been successfully shown that a variety of dynamical systems (2D Ising model [47], Bak-Teng-Wiesenfeld sandpile model [12,47], and pre-seismic electric signals [25–27]) become critical when *κ*1, evolving hit by hit, approaches the value 0.07.

Two criteria were defined to identify the entrance of a system to true critical state [47–50]:


$$S \equiv <\chi \text{ } \ln \chi> - <\chi> \text{ } \ln < \chi> \tag{4}$$

where <sup>&</sup>lt; *<sup>χ</sup>* ln *<sup>χ</sup>* <sup>&</sup>gt;<sup>=</sup> *<sup>N</sup>* ∑ *k*=1 *pkχ<sup>k</sup>* ln *χ<sup>k</sup>* .

Here, the evolution of variance *κ*<sup>1</sup> and entropies *S* and *Srev* of the natural-time transformed AE time series {*χk*} has been studied, where the event energy *Qk* is derived from the amplitude *Ak* through the relation *Qk* = *cAk* 1.5, where *c* is a constant of proportionality [51,52]. Plotting all natural-time quantities as functions of the conventional time *t* provides a visual way to reveal the possible entrance point to "critical stage," corresponding to the fulfillment of criticality Conditions (1) and (2). An entrance point to critical stage has been identified at time *tcrit* = 492 h (criticality initiation time, marked by a vertical dashed line in Figure 4 and by a vertical dotted line in Figure 6), i.e., 9 h before the *b*-value reaches its minimum. Remarkably, this critical point corresponds to the middle-height peak in the AE rate (highlighted by the dashed line before the highest peak in Figure 4) and to the appearance of high-amplitude AE signals.

This analysis suggests that the variance *κ*<sup>1</sup> of the natural-time transformed AE series can be identified as a pre-failure indicator, before the onset of non-reversible damage—supposedly revealed by the minimum *b*-value—within the bulk of the structural element.

**Figure 6.** Time evolution of natural-time quantities *κ*1, *S* and *Srev*. Horizontal dashed lines represent the characteristics value *κ*<sup>1</sup> = 0.07 and *Su* = 0.0966 defining the criticality initiation time *tcrit*. Note the relative positions of *tcrit* and *tb-min* along the time-axis (the inset shows the approach of natural-time quantities to the critical point).

#### **4. Conclusions**

The stability of a masonry wall of the Asinelli Tower has been assessed by the Acoustic Emission (AE) technique for a four-month period. The trend of the whole AE time series suggests that the monitored structural element is substantially stable, albeit with an ongoing damage process revealed by the AE activity observed over the entire monitoring period.

The observed correlation between the AE time series and the sequence of local earthquakes suggests that the local structural response is driven by the nearby seismicity. In particular, the densest AE cluster has been recorded within a few hours after the occurrence of the main local earthquake (4.1 magnitude). The *b*-value analysis of the AE cluster has revealed a significant, though momentary, acceleration of aging and deterioration processes in the monitored element, whose mechanical stability appears to be affected by light earthquakes.

Another emerging pattern is the transition of the monitored element to a state of criticality according to the paradigm of the natural time analysis [25–27]. The entrance to the critical state before the *b*-value reaches its minimum leads to consider the natural time variance *κ*<sup>1</sup> [47–50], as a possible early failure precursor. Future investigations in similar surveys would hopefully include the use of a seismometer, installed together with the AE system to record possible small seismic events affecting the observations.

**Supplementary Materials:** Earthquake data were taken from the website of the Istituto Nazionale di Geofisica e Vulcanologia-INGV, http://www.ingv.it/it/.

**Author Contributions:** A.C. conceived the research study; G.L. designed and performed the experiments; G.N. analyzed the data and wrote the paper.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors wish to thank the Municipality of Bologna and Eng. R. Pisani for allowing this study on the Asinelli Tower. This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

**Conflicts of Interest:** The authors declare no conflict of interest.
