## 4 Uncertainty of the Geometrical Acoustic Prediction Models
### Number of Rays  
GA software usually distinguishes between deterministic and stochastic ray-tracing, depending on which algorithm is applied: The first algorithm is used to detect the image sources, while the second is used to estimate the reverberant tail. It is possible to select separately the number of early and late rays in O. Early rays are used in the deterministic ray-tracing, while late rays determine the ray density in the late part of the IR. The number of rays/cones in C only refers to the stochastic ray-tracing; that is the construction of the late rays. It becomes important to investigate the variation in results due to stochastic ray-tracing, which is a random source of uncertainty in GA.  
Stochastic ray-tracing was here investigated by comparing simulations with different numbers of rays (4000-40,000-400,000-4 million). A Normalized Error analysis revealed that the results for each receiver position and octave-band frequency were all within the upper and lower limits of the respective limit range. This investigation was performed in order to verify the stochastic fluctuation, which may result as numerical errors in the results due to the low number of rays. This has been extensively studied and validated in systematic experiments [48]. The number of rays is strictly related to the systematic uncertainty in the final results of the parameters, and independently on the used method of the ray tracing, the fluctuations can be reduced by increasing the number of rays or by averaging repeated simulations. The choice of the number of rays becomes important in cases where large environments with uneven distribution of the absorption are considered. Therefore, a compromise should be reached between a very large number of rays and a smaller one since it may affect significantly the computation time. In fact, the reverberant field in a simulated open-air theatre is spatially uneven. The absorbing area is concentrated on the ceiling of the boundary box (in the case of O), while the theatre itself is mostly reflective. Thus, despite the prolongation of the computation time, a number of rays above 1 million would be preferable for the correct estimation of the reverberation tails at different receiver positions [22]. It is assumed that at least one ray is received at the longest source-to-receiver distance, which in this case is about 40 m (R10). The receiving area is considered as a spherical receiver with a radius r\({}_{\rm d}\) of about 0.06 m, thus the area of the visibility cone per ray A(ray) was 0.01 m\({}^{2}\). Considering that the total surface covered by the emitted rays is a sphere of radius 40 m,whose surface A(sph) is equal to 20,096 m\({}^{2}\), it is possible to calculate the minimum required number of rays N\({}_{\rm min}\)(rays) by means of Equation (6), which was also indicated in Vorlander [13]:  
\[\rm N_{min}(rays)=\frac{A(sph)}{A(ray)}=\ \frac{4(ct)^{2}}{r_{d}^{2}} \tag{6}\]  
where c and t are the speed of sound in air and the max arrival time counted from source excitation, respectively.  
N\({}_{\rm min}\)(rays) is equal to 2 million rays. Thus, 4 million rays are necessary to ensure that at least two rays (instead of one) arrive at the receiver at a distance of 40 m from the source.