## 5 Discussion
### Early-to-Late Energy Ratio as a Distance Cue  
The envelope-based model and the rendering method based on measurements were shown to provide very similar auditory distance perceptions. This result was further confirmed by the post hoc analysis applied to each tested distance. However, they differ significantly by design in terms of their early-to-late energy ratio (See Figure 7). The envelope-based model strongly underestimates the early energy for offset times under 50 ms, especially with increasing distances. This perceptual similarity raises the question of the definition of a reverberation-related distance cue, and which definition could best explain the perceptual similarity between these two methods. Previously, [PERSON] and [PERSON] (2018) have suggested that an early-to-late power ratio is the best candidate to act as a reverberation-related distance cue. However, past studies that explored the importance of this ratio as an auditory distance cue present contrasting results on the temporal limit that should be considered between the early energy and late energy ([PERSON] and [PERSON], 2018; [PERSON], 2018; [PERSON], 2018; [PERSON], 2018). In our case, the differences in terms of late energy between the measurements and the extrapolated responses by the envelope-based model are not significant for any onset time considered superior to 10 ms (see Figure 7). Early energy differences tend to reach their minimum for a considered offset time of 50 ms. When considering an early-to-late ratio for a limit of 50 ms, consistent with the definition of speech clarity C50, the difference between measurements and generated impulse responses is around 1 dB. For speech signals,  
Figure 6: Value of the fitting coefficient \(a\) (green dots) estimated on the intensity-based model, for participants of the CW group as a function of the self-reported volume of the room. the just-noticeable difference of C50 is around 1.1 dB, and a significant change value is considered to be around 3 dB [33]. This value is superior to the minimal early energy difference observed between the measured impulse responses and those generated by the envelope-based model. Thus, this similarity, in terms of early energy, could explain the perceptual similarity observed in the current experiment between the envelope-based model and the rendering method based on measurements. As the experiment results are only related to a single acoustical environment, it is not sufficient to draw a general conclusion about what appropriate limit should be considered as an early-to-late energy ratio in order to compute a distance cue. It does, however, corroborate the assumption that early reflections have little to no effect on distance perception on their own, but are perceived as part of the direct sound [7]. Figure 7: Early and late energy differences of the BRIRs generated with the envelope–based model compared to measured BRIRs. \(T_{mix}=15\) ms corresponds to the transition time between the early and the late reverberation segments considered in the envelope-based model to generate the BRIRs. The design of the envelope-based model entails that numerous aspects of the generated impulse responses were inherited from the initial measurement. In this experiment, a measurement taken at a distance of 1 m was employed to extrapolate further distances. Several reverberation-related cues that were acknowledged as influential on auditory distance perception were not reproduced correctly with distance. Among these, the initial time delay gap (ITDG) [11], which denotes the arrival time of the first reflections, remains constant across all generated impulse responses. However, no significant perceptual differences were identified between the reference stimuli and those generated by the envelope-based model. Because Equation (3) was applied to the entire early part of the initial impulse response prior to \(T_{mix}\), generated BRIRs underestimate the energy contained in the first reflections (see Figure 7).