## 4 Uncertainty of the Geometrical Acoustic Prediction Models
### General Procedure for the Implementation of the Models
The Odeon software allows giving as input value for the scattering coefficient the value as an average between 500 and 1000 Hz, and considers a frequency dependent scattering by using default interpolation curves as shown in the Manual. These curves have been used in CATT-Acoustic, i.e., a frequency dependent scattering coefficient, by inserting each value for each octave-band. The values given in Figure 4b refer to the mean values at 500 and 1000 Hz. 4. General settings: The following settings were considered for all the simulations: a 100 dB source sound power level, 1500 ms as the impulse response length, and 4 million rays. The Transition Order (TO) in Odeon was limited to 1, which better resembles the impulse response characteristics in the real condition with only one specular reflection from the stage floor. The third calculation algorithm in CATT-Acoustic, described above, was chosen as it is the most suitable for the simulation of open-air spaces. Scattering and diffraction settings were defined as in Table 2 in order to allow a more coherent comparison between the two software. The diffraction phenomenon occurs when a sound wave hits edges, i.e., intersections between surfaces, or when the surface dimensions are limited. Both these events are taken into account by the software Odeon and CATT-A when the Reflection-Based Scattering and Diffuse reflection method are enabled. In Odeon, the Lambert and the Oblique Lambert functions for scattering were disabled, as suggested in a previous paper [39]. The uniform scattering distribution was considered more suitable for the cavea which is made of steps that can be reassembled as periodic triangular section [39; 40; 41], as shown in Figure 3a. In CATT-Acoustic, the diffraction after 1 st order option was deactivated, even though it is usually suggested for ancient theatres [42], in order to take into account the current large amount of damage to the cavea steps in SR. In this way, it was possible to avoid the typical \"chirp\" echo due to diffraction phenomenon which has been attested to come from the regular stone steps in ancient theatres in empty conditions [43]. This phenomenon was not encountered during in situ measurements or recordings in SR. However, the first order diffraction has been taken into account since it occurs in coherence to the scattering phenomena. Moreover, based on the literature [44], it was found that higher orders and combinations of edge diffraction components were not usually as significant as first-order diffraction components when the receiver was visible to the source. The environmental data considered in both of the prediction tools were those obtained during the in situ measurements (t = 33 \({}^{\circ}\)C, RH = 65%). 5. Data analysis: the analysis algorithm has been taken into account. Odeon conducts an energy based analysis, while the CATT-Acoustic software conducts both energy and pressure-based analyses. The variation of different types of analysis algorithms can lead to different results, as pointed out in [PERSON] [45]. Thus, in order to avoid further uncertainty in the results, the simulated IRs have been exported and analysed by means of Aurora, version 4.4, in the same way as the measurements. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Phenomenon** & **Model** & **Odeon** & **CATT-Acoustic** \\ \hline \multirow{2}{*}{Scattering} & \multirow{2}{*}{Lambert} & \multirow{2}{*}{Disabled} & Late part of the IR (not \\  & & & manageable by the user) \\ \hline \multirow{2}{*}{Uniform} & \multirow{2}{*}{Disabled} & \multirow{2}{*}{Not managed by the software} \\  & & & of the IR \\ \hline Edge + surface & & Enabled (i.e., Reflection-Based \\ diffraction & & Scattering) & Diffuse reflection) \\ \hline Diffraction after 1 st order & Not managed by the software & Disabled (in TUCT) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Scattering and diffraction set-up in Odeon and CATT-Acoustic.