# Acoustic Emission for Structural Monitoring of Historical Masonry: An In-Field Application
## 3 Theoretical Framework
### Signal Attenuation  
When AE waves propagate through a material, their amplitude decreases as they move away from the source. This phenomenon, known as attenuation, is directly related to diffusion, reflection, and absorption processes. Attenuation can be expressed as follows:  
\[\mathrm{A_{f}}=\mathrm{A_{0}e^{-\alpha d}} \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Source Information** \\ \hline Amplitude & Event intensity, source orientation \\ \hline Arrival time & Source localization \\ \hline Duration and counts & Event energy \\ \hline Waveform & Event type \\ \hline Signal energy & Damage type (e.g., initial cracking, propagation, etc.) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Correlation between time-domain AE parameters and source-related information.  
where A\({}_{\rm f}\) is the wave amplitude at the sensor location, A\({}_{0}\) is the wave amplitude at the source, \(\alpha\) is the attenuation coefficient (which varies depending on the material), and d is the distance travelled by the wave [22].  
Attenuation assessment is an essential aspect of any AE-based testing or monitoring process. Understanding the attenuation law not only enables the differentiation of background noise but also facilitates the localisation of the signal source detected by the sensor.  
The ASTM E976 standard provides a methodology for evaluating attenuation laws through source simulation. The majority of tests are conducted by simulating the source through the breakage of a calibrated lead (Hsu-Nielsen source) on the surface of the specimen under examination. This test is commonly referred to as the pencil test [25]. When the lead, pressed against the element's surface, breaks, it generates a short-duration localised impulse, closely resembling that produced by a natural AE source following crack formation. The Hsu-Nielsen lead break test must be repeated multiple times, progressively increasing the distance from the sensor and recording, for each break, the amplitude of the signal detected by the transducer.  
Signal attenuation plays a critical role in the application of AE monitoring to masonry structures. Due to the heterogeneous composition of masonry, characterised by irregular stone units, variable mortar joints, and inherent discontinuities, acoustic energy emitted from microstructural events undergoes significant amplitude reduction and frequency distortion during propagation. This directly impacts the accuracy of event localization, damage classification, and the reliability of AE parameters, making a quantitative understanding of material-specific attenuation indispensable. Proper characterisation of signal loss as a function of distance and propagation direction enables the calibration of detection thresholds, the optimisation of sensor placement, and more accurate interpretation of AE data. In historical structures, where invasive testing is limited and preservation is paramount, understanding attenuation behaviour is essential to ensure the effectiveness and precision of real-time structural monitoring.