# Pulsed Thermography Applied to the Study of Cultural Heritage
## 3 Results
### Analysis of Bronzes
#### 3.1.2 Evaluation of the She Wolf Thermal Diffusivity and Quantitative Studies  
As mentioned earlier on, in order to quantitatively characterize the geometrical aspects of some of the detected subsurface features, it is essential to first determine the bronze thermal diffusivity. In the case of the She Wolf, the thermal diffusivity has been determined at the right ear, using the rear face heating mode since both the ear sides could be accessed by the necessary instrumentation. The heating was generated by a single 3 kW flash lamp oriented perpendicularly to the illuminated surface. Figure 5A shows the thermogram obtained on the ear, where the average thickness of the bronze had been measured in zone 1 as \(L\) = 0.57 cm. This area was selected because both the surfaces are well finished there, the curvature is negligible, and it is close enough to the edge to allow researchers to measure its thickness by means of a micrometer. Figure 5B shows the time dependence of the thermographic signals originating from the area selected in the thermogram, normalized to the corresponding maximum stationary value. The continuous black line represents the best fit to  
Figure 4: **(A)** Photograph of the She Wolf fur; **(B)** thermogram of the original statue where inhomogeneities at the fur edges are highlighted by the dashed circles; **(C)** thermogram of the She Wolf copy where no inhomogeneity is detected [47]. the experimental data obtained, neglecting the heat losses to the surrounding air, using Equation (1) with \(x=L\), which has yielded a value of \(D=0.15\)\(\pm\) 0.02 cm\({}^{2}\)/s. It must be remarked that the thermal diffusivity can also be determined by evaluating the time \(t_{1/2}\) when the thermographic signal crosses a level corresponding to half of its maximum value, as expressed in Equation (7). Moreover, this latter procedure leads to a value of \(D=0.15\)\(\pm\) 0.02 cm\({}^{2}\)/s. The reported uncertainty was set at two times the standard deviation of the mean obtained from repeated measurements of \(L\) and \(t_{1/2}\) performed in zone 1. It is worth noting that the value of \(D\) has also been determined by using the front face heating mode in a completely different area of the She Wolf, namely in the right rear paw, giving values similar to those obtained in the correspondence of the ear [49]. Once the thermal diffusivity is known, it may be possible to analyze the structural and geometrical subsurface elements and casting faults, like the cavities which originate from the trapping of air bubbles in the molten metal, and to obtain information on their relative depth. Figure 6A,B show the thermograms, obtained by the front face heating mode, where different cavities are revealed in the rear left paw. The arrows in Figure 6B indicate the thermographic features of two cavities which can be viewed in the same thermogram, and the position of a third cavity which, being at a greater depth, presents a clear contrast when that of the shallower two cavities has pretty much faded away. Evaluations concerning the depth of the cavities can be performed by analyzing the time dependence of the so called contrast coefficient [63]:  
\[C_{r}(t)=\frac{T_{c}(t)-T_{0}(t)}{T_{0}(t)} \tag{8}\]  
where \(T_{c}\) represents the temperature evolution obtained over the area of the cavity, and \(T_{0}\) is that determined in a position where no subsurface faults are present. According to the results of the one dimensional heat diffusion theory obtained for a laterally semi-infinite discontinuity, the time dependence of \(C_{r}\) should display a bell shaped profile like those reported in Figure 6C for each of the cavities indicated in Figure 6B. The maxima of the different \(Cr\) profiles appear at different times, indicating the fact that the cavities are buried in the bronze volume at different depths. Within the limit of validity of such a theoretical model, the depth of the fault can then be determined by the expression:  
Figure 5: (**A**) Thermogram obtained at the She Wolf right ear.