## 1 Introduction  
The recent interest in the design of ancient theatres and in their acoustical characteristics has drawn attention to the lack of methodologies in metrology for historical acoustics [1]. The ISO 3382-1 standard [2] was used in the European ERATO project [3] to evaluate the acoustical apparatus of ancient theatres through room acoustic parameters, such as the Early Decay Time (EDT), Reverberation Time (RT), Clarity (C\({}_{80}\)), and Sound Strength (G). However, ISO 3382-1 basically refers to indoor environments and temporal decay parameters seem to be less suitable for open-air conditions [4; 5; 6; 7; 8]. [PERSON] et al. [4] reported that EDT is not a robust predictor of the acoustic quality of open-air theatres. The lack of robustness in EDT is due to a marked and intrinsic variability of this parameter, according to the source position, which defines the delay and incidence direction of the first reflections to the receivers. The same study asserted that RT behaviour in an open-air theatre is clearly different from that dealt with in the classical reverberation theory, which refers to a reference room volume. However, this parameter showed a limited variability. [PERSON] et al. [5] also suggested the use of RT when comparative studies are performed. However, it should be utilised with caution since it is usually used to evaluate enclosed spaces. [PERSON] et al. [6] conducted a listening test with monaural and binaural auralisations of an open-air space. They stated that the perceived reverberance in an unroofed space is not only affected by the temporal characteristics during the decay process, but also by the spatial characteristics, due to the distribution of the reflections. The results showed that the conventional RT described in ISO 3382-1, which only deals with the sound energy decay rate, is not suitable for evaluating the reverberance of an unroofed space. Thus, more insight is needed into the adoption of an indoor acoustic measurements standard for the investigation of the acoustic conditions of open-air theatres. These sites represent particular environments that have their own specific sound field, which is rather different from the ideal diffuse field. Besides the doubts about the applicability of the aforementioned indoor standard to outdoor case studies, other specific problems could arise when conducting measurements in ancient theatres. In fact, archaeological field measurements are also clearly influenced by the current conditions of the architecture of the theatres. Most ancient theatres have undergone damage of anthropogenic and atmospheric nature. It was attested in [PERSON] et al. [4] that the measured values of RT, G, and C\({}_{80}\) in ancient theatres are affected to a great extent by the state of conservation of the theatres themselves, with particular reference to the completeness of the architectural elements. Therefore, it is currently difficult to design acoustical correction guidelines for their contemporary reuse as performance spaces. Moreover, particular attention should be paid to the outdoor environmental conditions, such as temperature (t), relative humidity (RH), and air velocity, which could affect the variability of the measurement results, in the same way as for indoor measurements [9, 10]. The topic of acoustical characterisation has already been examined in detail for indoor spaces, through statistical analysis, in order to investigate the reproducibility of measurements, the accuracy of the parameter calculation, the influence of source-receiver position displacement, and the measurement chains of different systems [9, 11, 12]. An alternative to the experimental acoustical characterisation is the virtual reconstruction of the theatre, using room acoustics simulation software. Since they were introduced, geometrical acoustic (GA) software applications have been used as the standard room acoustics models [13].