## 4 Uncertainty of the Geometrical Acoustic Prediction Models  
In the acoustic domain, it is important to recall that the parameters have the aim of evaluating the perception of the acoustic signal, namely the average capability of a \"conventional\" listener to notice sound variations. An important factor that correlates the subjective field to objective measures has been defined as the Just Noticeable Difference (JND), that is, the smallest perceivable change in a given acoustical parameter, which is specified for information in Annex A of ISO 3382-1 [2] for central frequencies (500 Hz and 1 kHz), but which is also acceptable for lower and higher frequencies [32; 33; 34; 35]. This issue will be further discussed, when analysing the accuracy of the acoustic prediction models.  
The uncertainty contribution of the input data, propagated to the results obtained from two different types of room acoustic software, Odeon version 13.02 and CATT-Acoustic version 9, was assessed and compared with the measurements values.  
Odeon version 13.02 [22] is based on a hybrid calculation method. Early reflections are calculated through a mixture of the Image Source Method and the Ray-Tracing Method (RTM), by means of a stochastic scattering process that uses secondary sources. Late reflections are calculated by means of a special RTM, where the secondary sources radiate energy locally from the surfaces and are assigned a frequency-dependant directionality, namely the reflection-based scattering coefficient. The secondary sources may have a Lambert, Lambert oblique, or Uniform directivity: this directivity depends on the properties of the reflections as well as on the calculation settings.  
CATT-Acoustic version 9 [23] is made up of two modules: CATT-A is the main programme, and it handles the modelling, surface properties, and directivity libraries, and TUCT (The Universal Cone Tracer), which is the main prediction and auralisation programme. TUCT can use three alternative cone-tracing algorithms: the first algorithm is based on stochastic diffuse rays, while the second and third algorithms are based on the split-up of the actual diffuse rays. The difference between these modules is that the latter handles two orders of diffuse split-up reflections in a deterministic way, thus resulting in lower random run-to-run variations.  
Both CATT-Acoustic and Odeon base their scattering algorithms on two main implementations, which are described in detail in a previous paper [36]. These two methods are the Hybrid Reflectance Model (HRM) and vector mixing (VM). The HRM method complies with the definition of the scattering coefficient based on ISO 17497-1 [20] which defines it in a quantitative way as the fraction of the non-symmetrically reflected energy. In the HRM method, a random number between 0 and 1 is used to determine whether the reflection is specular or scattered. This number is compared with the surface scattering coefficient (s) assigned to the surface. In case it exceeds the value of s, the scattered energy is assumed to be distributed according to Lambert's Law, i.e., the intensity of the reflected ray is independent on the angle of incidence but proportional to the cosine of the angle of reflection. This is the basic concept implemented in CATT-Acoustic [23] and in Odeon for the uniform and Lambert directivity scattering [22]. On the contrary, the VM is based on the linear interpolation of the specular and diffuse reflection [37]. In this way the direction of a reflection vector is calculated by adding the specular vector scaled by a factor (1-s) to a scattered vector following a certain direction that has been scaled by a factor s. This is the basic concept implemented in Odeon [22; 38; 39], named \"vector-based scattering\", where the scattered vector follows a random direction, generated according to the Lambert distribution named oblique Lambert directivity.