## 2 Materials and Methods
### Anomalous Fading Correction  
Anomalous fading represents an inadvertent occurrence affecting thermoluminescence (TL). This phenomenon manifests in certain minerals, where luminescence intensity is contingent on the elapsed time since irradiation. The term \"anomalous\" is employed due to its propensity to unexpectedly release trapped electrons. Notably, electrons that should remain trapped at room temperature are liberated earlier than anticipated, often within a few days, rendering them unsuitable for precise dating.  
The significance of anomalous fading in TL dating was initially emphasized by [PERSON] in 1973, leading to the integration of anomalous fading tests into measurement protocols [29]. Subsequently, [PERSON] proposed a model based on tunneling mechanisms to elucidate this phenomenon in 1985 [21]. [PERSON]'s model introduced a logarithmic law to delineate the signal decay: \(ln\left(T/T_{C}\right)\).  
This relationship characterizes the luminescence intensity versus time:  
\[I=I_{C}\bigg{(}1-k\,ln\bigg{(}\frac{T}{T_{C}}\bigg{)}\bigg{)} \tag{1}\]  
Considering the intensity \(I_{C}\) at an arbitrary time (\(T_{C}\)), and \(\kappa\) is a constant, characteristic of the sample. In archaeological materials, continuous exposure to natural radiation, maintaining a proportional relationship with age, must be considered. Nevertheless, through the integration of Equation (1), it becomes possible to determine that \(T\) represents the true age, contrasting with \(T_{F}\), which denotes the fading age.  
\[\frac{T_{F}}{T}=1-k\,\bigg{[}ln\bigg{(}\frac{T}{T_{C}}\bigg{)}-1\bigg{]} \tag{2}\]  
The origin of the tunnelling model lies in the data derived from feldspar fading studies [30]. However, its applicability to different materials remains uncertain. On the other hand, the absence of a universal model can be addressed through the delineation of empirical data. [PERSON] and [PERSON] propose an ingenious solution by exploring the attenuation resulting from a single ionizing event [26]. Their research has evolved to include incident dosimetry for phone displays. They model the empirical behavior of TL intensity over time, fitting it perfectly to the outlines of a hyperbolic curve plotted directly on experimental data. Here \(A\), \(B\), and \(C\) are specific parameters of the sample.  
\[I=\frac{A}{\left(1+BT\right)^{C}} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Step** & **Treatment** & \\ \hline
1 & TL 450 \({}^{\circ}\)C, 2 \({}^{\circ}\)C/s & Natural TL \\ \hline \hline \end{tabular}  
* 5.5 Gy (50 s \(\beta\))
* 11 Gy (100 s \(\beta\))
* 16.5 Gy (150 s \(\beta\))
* 3 & TL 450 \({}^{\circ}\)C, 2 \({}^{\circ}\)C/s & TM Growth  
\end{table}
Table 4: Classical thermoluminescence (TL) protocol.  
The study extends [PERSON] and [PERSON]'s model to amorphous materials, as evident in the analysis of mosaic tiles, adhering to the superposition principle illustrated by [PERSON] and [PERSON]. The trend of fading should be examined for all samples over a fixed period of time and considering several aliquots. By integrating the normalized dose over time and replacing parameters \(A\), \(B\), and \(C\), the corrected age for fading is determined:  
\[\frac{T_{fad}}{T}=\frac{1}{T}\,\int_{0}^{T}\frac{I(T-t^{\prime})}{I_{0}}\,dt^{ \prime}=\frac{1}{T}\frac{A}{B\,(1-C)}\Big{[}(1+BT)^{1-C}-1\Big{]} \tag{4}\]  
It is worth noting that since age, adsorbed dose and thermoluminescence intensity are all proportionally related, there is no distinction in describing abnormal fading in terms of \(T_{fad}/T\), \(I_{fad}/I\) or \(D_{fad}/D\), as pointed out by [PERSON] and [PERSON] in 2001 [30].