## 2 Materials and Methods
### Key Studies on Rock Mass Disturbance
\[E_{m}=0.1\left(\frac{RMR}{10}\right)^{3} \tag{7}\]  
\[BQ=80.786+6.0943 RMR \tag{8}\]  
where \(E_{m}\) is the deformation modulus of the rock mass. The \(BQ\) is defined by Equations (9) and (10):  
\[BQ=100+3R_{c}+250K_{v} \tag{9}\]  
\[K_{v}=\left(\frac{v_{mp}}{v_{rp}}\right)^{2} \tag{10}\]  
where \(K_{v}\) is the rock mass integrity index (ranging from 0 to 1), \(v_{mp}\) is the wave velocity of the rock mass, \(v_{rp}\) is the wave velocity of the rock block, and \(R_{c}\) is the saturated uniaxial compressive strength. In the last stage, [PERSON] et al. [7] derived the following relation between \(D\) and \(BQ\) by using the three empirical Equations (6)-(8) within each other. \[D=2\left(1-\left(\frac{BQ_{d}-80.786}{BQ_{ud}-80.786}\right)^{3}\right) \tag{11}\]  
where \(BQ_{d}\) and \(BQ_{ud}\) are the disturbed and undisturbed basic quality index, respectively. Since the relation derived by [PERSON] et al. [7] is based on three empirical equations, due to the nature of empirical equations, there is a potential for the errors from each of them to combine and increase in the derived D relation. Moreover, as is well known, stress relief due to the removal of overburden--particularly in deep excavations--causes disturbance in the rock mass. In deep slope excavations, application of the derived continuous relation for \(D\) requires determination of \(BQ_{d}\) by accounting for changes in \(K_{v}\), which vary depending on the progress of the excavation phase. For disturbed rock mass conditions, determining \(K_{v}\) by measuring wave velocity with depth is only feasible once the excavation has reached the relevant depth. This represents a major limitation in the derived relation proposed by [PERSON] et al. [7], potentially reducing its practical applicability, especially for deep slope excavations. Furthermore, considering that each empirical criterion should be applied in line with its original recommendations, the compatibility of this relation with the lower limit values of \(D=0.5\) and \(D=0.7\)--suggested by the developers of the Hoek-Brown failure criterion for small- and large-scale slopes, respectively--is open to debate [6]. Therefore, despite its theoretical contribution, the study appears to fall short as a practical tool for estimating the disturbance factor (\(D\)) from surface to depth during the initial stages of deep slope design analysis. A recent study performed by [PERSON] et al. [8] was another attempt to define a continuous relationship of \(D\). As in the study performed by [PERSON] et al. [7], [PERSON] et al. [8] used Equation (6) as the starting point for the derivation of the continuous form relation of \(D\). However, the following empirical relation (Equation (12)) between \(E_{m}\) and \(RMR\) proposed by [PERSON] et al. [11] was preferred by [PERSON] et al. [8]. \[E_{m}=E_{i}exp\left(\frac{RMR_{89}-100}{36}\right) \tag{12}\]  
where \(E_{i}\) is the elastic modulus of intact rock, and \(RMR_{89}\) is the 1989 version of [PERSON]'s RMR. In addition, as available in manuscript by [PERSON] et al. [8], to transform Equations (12) and (13), in which the acoustic wave velocity (\(C_{p}\)) is an independent parameter, some empirical relations were considered by [PERSON] et al. [8], including [PERSON]'s \(Q\), \(P\) wave velocity (\(V_{p}\)), the 1979 version of [PERSON]'s RMR (\(RMR_{79}\)), and \(C_{p}\). \[E_{m}=E_{i}exp\left(\frac{13.64C_{p}-96.4}{36}\right) \tag{13}\]  
The following relation (Equation (14)) between \(D\) and \(C_{p}\) was derived by [PERSON] et al. [7]. \[D=2\left[1-exp\left(-\frac{13.64\Delta C_{p}}{36}\right)\right] \tag{14}\]  
However, it can be concluded that all the limitations in [PERSON] et al. 's [7] Equation (11) are valid for Equation (14) proposed by [PERSON] et al. [8], in which \(C_{p}\) was used instead of \(V_{p}\).