## 2 Methods
### Numerical Method
#### 2.2.3 Reflection-Based Scattering  
The scattered energy in Odeon is a combination of the surface roughness scattering coefficient \(\mathrm{s_{s}}\) with the scattering coefficient due to diffraction \(\mathrm{s_{d}}\) that is calculated individually for each reflection during the simulations. This combination leads to the \"reflection-based scattering coefficient\" \(\mathrm{s_{r}}\)[41; 46], which can be calculated from:  
\[\mathrm{s_{r}}=1-(1-\mathrm{s_{d}})(1-\mathrm{s_{s}}), \tag{1}\]  
Figure 4: Comparison between diffusion polar plots and different scattered sound models with directional distribution of sound scattered from the _periactes_ and a flat surface with sound incidence angle of \(-45^{\circ}\). (**a**) and (**b**) Polar representation of the pressure amplitude at 500 and 1000 Hz obtained using AFMG Reflex software; (**c**) and (**d**) vector mixing representation for high and low surface scattering value [44]; (**e**) and (**f**) RAPDFs for \(\mathrm{s}=0.75\) and \(\mathrm{s}=0.05\)[44]; (**g**) Simplified reflection model for 3D modeled _periactes_. where \(\mathrm{s_{d}}\) is the fraction of energy scattered due to diffraction and \(\mathrm{s_{s}}\) is the fraction of scattering caused by surface roughness. Surface scattering (\(\mathrm{s_{s}}\)) is assumed to appear due to random surface roughness, as defined in ISO17497-1. This type of situation gives rise to scattering that increases with frequency and can be inserted by the user based on scattering coefficient measurement databases, i.e., user-based scattering. In Odeon, typical measured scattering coefficient frequency functions [41; 46] are used to expand a mid-frequency scattering coefficient input by the user to all frequency bands. This means that only one input value for the scattering coefficients needs to be specified for a surface at a middle frequency around 707 Hz (average of 500-1000 Hz bands). These coefficients are then expanded into values for each octave band, using interpolation or extrapolation. Surface edge diffraction (\(\mathrm{s_{d}}\)) is scattering due to the surface limited size and surface edges. It is a frequency-dependent phenomenon that affects low frequencies, i.e., frequencies lower than a limiting frequency (\(\mathrm{f_{g}}\)). This limiting frequency is evaluated considering the reflection path lengths, surface dimensions and distance from edge of surface, angles between surfaces, offsets, angle of incidence, etc. [41; 46]. The \(\mathrm{s_{d}}\) is handled automatically by the software since most of the factors affecting its value are not known by the user, and is considered when \"reflection-based scattering\" is enabled. high-frequency sources, which were positioned at different heights, that is, at 0.40 m, 3.70 m, and 3.90 m, respectively. The position of the low-frequency source on the floor level has been considered acceptable, since humans are not able to locate the direction of such low frequencies. In the simulations, a single omnidirectional source has been used and located in the same position used in measurements for the mid-frequency omnidirectional loudspeaker. The main input settings to perform simulations, that is, the number of rays, the maximum reflection order, and the transition order (TO), have to be decided carefully according to the aim of the simulations. First, a reasonable number of rays could be estimated by considering that an expected error of less than 1 JND is regarded as sufficient when estimating the objective acoustic parameters [2]. However, the number of rays has been estimated following the software indication of the minimum number of rays, which is derived by taking into account the aspect ratio of the room as well as the size and number of surfaces in the geometry. This automatic number shown in the settings window was equal to 6161 rays, but a greater number was chosen to avoid artifacts and obtain higher quality in the auralizations used later in the listening tests [18; 48].