## 4 Uncertainty of the Geometrical Acoustic Prediction Models
### General Procedure for the Implementation of the Models  
In order to compare the two software packages and to obtain the best match with the measurement results, it was necessary to perform simulations with the same geometric model and source/receiver positions as in the measurements. To the best of the authors' knowledge, this preliminary benchmark procedure has never been performed before on ancient open-air theatres, although many studies on indoor environments have been conducted [13; 14; 15]. Both types of software used for the simulation, that is, Odeon and CATT-Acoustic, have been validated in Round Robin tests. One of the main findings of these tests was that precise knowledge of the characteristics of the surface material is an importantprerequisite for a reliable room simulation. Thus, a more detailed analysis on absorption and scattering coefficient changes was proposed. A preliminary benchmark test study was carried out on SR, whose model had previously been used in different investigations, e.g., simulations concerning its ancient conditions, during the European ERATO project, [3] and in investigations on its contemporary use [25]. Figure 3a shows the 3D model configuration of SR. The procedure applied for the comparison of the simulation tools was focused on solving the following issues:  
1. Geometrical model: MATLAB software, version R2015b, was used to create a parametric open-air theatre script. Two 3D cavea model script outputs were created, one suitable for Odeon (dxf file) and the other for CATT-Acoustic (.geo file). In order to reduce the simulation time, the theatre geometry was simplified and designed as symmetric. A few geometrical simplifications have been performed in both Odeon and CATT-A models. The number of surfaces was 1357 in CATT-Acoustic and 1362 in Odeon. As recommended previously [14], the steps were modelled. The higher number of surfaces in Odeon corresponds to an additional boundary box with totally absorbing walls and top which is required in Odeon to simulate open-air conditions [22]. CATT-A algorithms are implemented in order to detect lost rays, i.e., rays that escape from the geometrical model. In open cases, such as an open-air ancient theatre, rays disappear whenever they do not hit any surface during the calculation time. This principle is similar to the one used in Odeon, where the escaping rays disappear since they are totally absorbed by the boundary box. The circular geometry was modelled with 20 segments, as recommended by [PERSON] [40], who showed that a number between 12 and 24 segments is optimal with respect to the influence on the objective acoustic parameters. 2. Source-receivers: The source-receiver path was defined as in the measurement set-up, considering the theatre as unoccupied. For an easier comparison, only the source in position S1 was considered, as shown in Figure 3a. 3. Surface material properties: The main surface considered in the model was the cavea, the stone and steps of which are not well-conserved. In both types of software, 20 material alternatives were assigned to the cavea stone, that is, from the most reflective one, with \(\alpha_{w}=0.05\) and s = 0.25, to the most porous one, with \(\alpha_{w}=0.2\) and s = 0.85, and all the intermediate combinations of \(\alpha_{w}\) were tested in steps of 0.05, while s was tested in steps of 0.15, as explained in Figure 4b [14]. Other elements were then added: the remains of the ancient entrances to the orchestra area (aditi  
Figure 3: 3D model and source-receiver simulation set-up of SR (**a**) and scheme of the characteristics of the material chosen for the cavea (**b**). maximi), which was considered as an aperture (\(\alpha_{\rm w}=0.9\); s = 0), and the floor, which includes the ruins of the scaenae froms (\(\alpha_{\rm w}=0.8\); s = 0.8) and the better conserved orchestra area (\(\alpha_{\rm w}=0.1\); s = 0.2). Odeon and CATT-Acoustic software allow for frequency dependent absorption coefficients. The same absorption coefficients have been used for both software.