## 2 Psychoacoustic Principles of Auditory Distance Perception  
### Cues for Auditory Distance Perception  
Although human auditory distance estimation ability is generally inferior to that of visual distance estimation, preliminary but biased auditory distance estimation can still be formed [3]. Experiments demonstrated that the human auditory system tends to significantly underestimate distances to sound sources with a physical distance farther than a rough average of 1.6 m, and typically overestimates distances to nearby sound sources with a physical distance of less than a rough average of 1.6 m. This finding suggests that the perceived source distance is not always identical to the physical one. [PERSON] examined experimental data from a variety of studies and found that the relationship between the perceived distance \(r_{I}\) and the physical distance \(r_{S}\) can be well-approximated with a compressive power function by using a linear fit method [4]:  
\[r_{I}=\kappa r_{S}^{\delta}, \tag{1}\]  
where \(\kappa\) is a constant whose average is slightly greater than one (average of approximately 1.32); and \(\delta\) is a power-law exponent whose value is influenced by various factors, such as experimental conditions and subjects, so this value varies in a wide range with a rough average of 0.4. The just notice difference (JND) of the percentage variation in distance also varies in different experiments and depends on multiple factors. Auditory distance perception, which was thoroughly reviewed by [PERSON] et al. [5] and [PERSON] et al. [6], is the consequence of a complex and comprehensive process based on multiple cues. The known cues are outlined as follows:  
(1) Level or loudness cue  
In a free field, the sound pressure generated by a point source with constant power is inversely proportional to the distance between the sound source and the receiver (the \(1/r_{S}\) law); that is, the sound pressure level (SPL) decreases by \(-6\) dB for each doubling of the source distance. Subjective loudness is closely related to the SPL and considered an effective cue in distance perception. As the source distance increases, the variation in SPL with the source distance decreases. From the \(1/r_{S}\) law, we can derive that the percentage variation in the source distance is related to the variation in the sound level \(\Delta L\) in dB as:  
\[\frac{\Delta r_{S}}{r_{S}}\times 100\%=\frac{1}{\exp(\Delta L/20)-1}\times 100\%. \tag{2}\]  
Therefore, the level or loudness cue provides information on the percentage variation in distance. If a JND of SPL of 1 dB is assumed, the threshold of percentage distance discrimination on the basis of the level cue is about 20%. The experimental results of the JND of percentage variation in distance based on level cue vary from 5% to 25% [6]. The variation in the SPL with source distance derives from the \(1/r_{5}\) law with a rate of less than \(-6\) dB for each doubling of a source distance in a reflective room, depending on the acoustic characteristics of the room. The sound pressure and loudness at a listener's position are also closely related to source properties, such as source power and directivity. Prior knowledge of sound sources or stimuli also influences distance estimation performance when loudness-based cues are used. In general, loudness is regarded as a relative distance cue unless the listener is highly familiar with the pressure level of the sound source. (2) Spectral cue  
The high-frequency attenuation caused by air absorption is another cue in auditory distance perception. For a distant sound source, air absorption acts as a low-pass filter and modifies the spectra of the sound pressures at the receiving position. A change in the spectrum may cause perceivable timbre variation. This effect is important only for an extremely far sound source (farther than 15 m) and negligible in an ordinary-sized room [2]. However, this effect is valid for reflections in a room because the accumulative propagation distance for a sound is large after being reflected repeatedly.