## 2 Methods
### Numerical Method
#### 2.2.2 Scattered Sound in Odeon  
Odeon considers scattered sound both in the early and in the late part of the calculations. The early scattering is emitted from surface sources that are simulated each time an image source is detected. In this way, at each reflection point of the early scattering rays, a secondary source is created. This process lasts from the current reflection order up to the TO.  
The late reflection method is used to treat all reflections that are not considered by the early reflection method. Every time a late ray is reflected on a surface, a small secondary source is generated, similar to a surface source. The secondary sources are assigned a frequency-dependent directionality, which may be a _Uniform, Lambert, or Oblique Lambert directivity_ depending on the properties of the reflection as well as the calculation settings chosen by the user. _Oblique Lambert directivity_ is a default setting. The Uniform directivity is a simpler model that redirects the scattered reflections uniformly over a hemi-sphere above the incident point. The Lambert directivity is based on the cosine law directivity. These first two options are based on the HRM. A revised version of the Lambert directivity is the oblique Lambert (Figure 3), which uses the concept of \"vector-based scattering,\" i.e., vector mixing. In this model, the orientation of the Lambert sources is obtained by taking into account the \"reflection-based scattering coefficient,\" which combines the scattering due to surface roughness and to surface edge diffraction. If s = 1, the reflected ray will propagate in a scattered direction conforming to the traditional [PERSON]'s law. If s = 0, the reflected ray will propagate in a specular direction, which is obtained from [PERSON]'s law, i.e., the angle of incidence is equal to the angle of reflection [2]. The \"vector-based scattering\" model is used when the scattering coefficient values are between 0 and 1. The resulting direction is determined using s as a weighting between the pure specular direction and the scattered direction, chosen randomly as in HRM (Figure 4a,b). A shadow zone is created between the surface and the incident ray, where no sound reflections can be directed. Intuitively, the shadow zone is small if the scattering is high or if the incident ray is perpendicular to the surface (Figure 3a). Conversely, it is big if scattering is low and the incident ray is oblique (Figure 4b,c). The method is corrected to take into account the fact that for radiation angles different from 0\({}^{\circ}\), the model leads to energy loss due to the fact that part of the Lambert balloon radiates energy outside the room. The correction factor varies from 1 for a radiation angle of 0\({}^{\circ}\) to 2 for radiation angles of 90\({}^{\circ}\).  
Figure 3: Oblique Lambert model. **(a)** Traditional Lambert directivity is generated when the angle of incidence is 0\({}^{\circ}\). Correction factor = 1; **(b)** Oblique Lambert directivity is generated when the angle of incidence is within the interval (0\({}^{\circ}\), \(\pm\)90\({}^{\circ}\)). Correction factor within the interval (1, 2); **(c)** Oblique Lambert directivity is generated when the angle of incidence is \(\pm\)90\({}^{\circ}\). Correction factor = 2 [41].