# Time Series Analysis of Acoustic Emissions in the Asinelli Tower during Local Seismic Activity
## 3 Results
### 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 \(\kappa_{1}\) of the natural-time transformed time series [25; 26; 27]. Figure 3: Map showing the epicenter of the 4.1-magnitude earthquake (yellow point) occurred on 13 October 2010 with epicenter 100 km far from the monitoring site in Bologna (red point). The GR law, initially introduced in seismology [41] and then extended to the statistics of AE signals [38; 42; 43; 44; 45; 46], is expressed by the relation  
\[\log_{10}N=a-b\ \times M \tag{1}\]  
where \(M\equiv\log_{10}(A_{\max}/1\ \upmu\mathrm{V})\) 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 \(b/\sqrt{N}\) for a population of \(N\) 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; 43; 44; 45; 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 \(t_{b\text{-}min}\) = 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]. 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\" (\(lb\)) was introduced and defined as follows [42; 44; 45]:  
\[lb=\big{[}\log_{10}N(\mu-\alpha_{1}\sigma)-\log_{10}N(\mu+\alpha_{2}\sigma) \big{]}/\big{[}(\alpha_{1}+\alpha_{2})\sigma\big{]} \tag{2}\]  
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. where \(\mu\) and \(\sigma\) are mean and standard deviation of each AE amplitude subset, found to be, respectively, \(\sim\)45 and 8 dB, whereas \(\alpha_{1}\) and \(\alpha_{2}\) 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 \(lb\)- and \(b\)-values formed by a 300-hit partitioning are plotted in Figure 5, showing a good accordance between the two analyses. 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.