Comparative analysis of different calculation methods of the Geological Strength Index (GSI) based on qualitative and quantitative approaches

In this research, the dispersion of the Geological Strength Index (GSI) values obtained with quantitative and qualitative approaches has been evaluated in four rock outcrops of different quality. The subjective component associated with qualitative or visual methods has been studied by conducting a virtual survey in a group of forty participants constituted by civil engineers, geological engineers, and mining engineers from Peru, Spain, and Chile, who were given a data sheet with a photograph and a basic description of each rock mass. The results showed that the GSI values fit a normal distribution characterized by a mean value and a standard deviation, which in some cases could present moderate to high coefficients of variation (COVs). This paper also includes the study of the dispersion of the GSI values obtained with quantitative formulations that have been evaluated and incorporated into regional databases to assess trends, mainly in the GSI-RMR’ relationships. The results indicate that the average GSI values reported with both approaches are similar; however, with the quantitative methodologies, COV values were classified as low to moderate, which is better adjusted to the suggested COV values for the GSI. Despite this, quantitative methodologies must be used with caution, taking into account the characteristics of the rock masses on which the relationships have been defined.


Introduction
The Geological Strength Index (GSI) (Hoek, 1994;Hoek et al., 1995) was conceived as a rock mass characterization system, originally calculated qualitatively based on the rock mass structure and the joint condition. The calculation procedure was developed under the premise that qualified geologists or geological engineers would carry out the observations of the rock mass characteristics; however, currently in engineering practice, it has been observed that many times the qualitative GSI calculation is carried out by inexperienced personnel or engineers who do not feel comfortable using descriptive methodologies (Hoek et al., 2013), resulting in an index with a high subjective component. Subsequently, to reduce the subjectivity in the GSI calculation, various researchers (e.g. Ulusay, 1999, 2002;Cai et al., 2004;Russo, 2009) proposed quantitative formulations based on specific rock mass parameters, such as RQD (Deere, 1963), joint condition or block volume (V b ), or with approaches based on fuzzy logic (Sonmez et al., 2003), in accordance with Hoek (1999), who indicated that engineers are more comfortable using rock mass parameters that numbers can express. Despite this, it was observed that in some cases, the application of quantitative formulations could result in very dispersed values of the GSI, so it is necessary to previously evaluate the particular characteristics of the rock masses on which these formulations were defined, such as lithology, range of application, exposure conditions, structure, etc. Hoek (1998) states that it is more realistic to indicate a range of the GSI values instead of a single value, so in engineering calculations, it is suggested to consider the variability of this parameter through statistical distributions.

Review of GSI versions
Since its appearance, the GSI has undergone various modifications, so after reviewing the extensive technical information available regarding the adaptations of the GSI system and its applications, in this investigation, it has been decided to group the different versions of the GSI into three groups or development lines.

Original approach
It includes the research line carried out under the authorship, co-authorship, or supervision of the developers of the original Hoek-Brown criterion, which began with Rudarsko-geološko-naftni zbornik i autori (The Mining-Geology-Petroleum Engineering Bulletin and the authors) ©, 2022, pp. 121-138, DOI: 10.17794/rgn.2022.3.10 the investigations of Hoek and Brown (1980), incorporating the GSI in the equations of the failure criterion in Hoek et al. (1995), and whose most recent chart version is presented by Hoek et al. (2013). It also includes the development of qualitative charts for application in complex rock masses such as Flysch (Marinos and Hoek, 2001;Marinos, 2017) and molasses . The basic chart for calculating the GSI in jointed rock masses is presented in Figure 1, published by Hoek and Marinos (2000).

Complementary approach
It includes the development of new charts for the GSI calculation carried out by independent researchers. In some of these publications, the scores associated with the rock mass structure and the joint condition have been quantified, indicating them directly on the chart axes

Visual Visual
Hoek and Brown (1997) The chart for calculating the GSI is presented for the first time as a series of continuous lines that define the GSI values based on the rock mass structure and the condition of the discontinuities, so it is possible to define a range of values of GSI instead of a single value (5<GSI<85).

Visual Visual
Hoek et al. (1998) To include rock masses of very poor quality, such as the schists found in the excavation of the Athens Metro, the range of GSI values is extended below 5.

Marinos and
Hoek (2001) A new chart is presented to calculate GSI in weak and tectonically disturbed rock masses, such as Flysch.

Visual Visual
Hoek et al. (2002) A new set of relationships between GSI, m b , s, and a is introduced to give a smoother transition between very poor quality rock masses (GSI < 25) and stronger rocks.

Visual Visual
Hoek et al. (2005) A new GSI chart is presented for the calculation in tectonically undisturbed sedimentary rocks deposited in a shallow marine environment (molasses).

Visual Visual
Marinos (2017) An update of the chart published by Marinos and Hoek (2001)

Specific application approach
Although the GSI system was developed for its exclusive application with the Hoek-Brown failure criterion, it has been found in engineering practice that it can be used successfully in other areas of rock mechanics. The specific application approach includes the development of GSI charts for the study of any particular problem; for example, qualitative charts have been developed to design support systems for tunnels in Peruvian mines (Mejía and Chacón, 2009), the study of rock mass excavability (Tsiambaos and Saraglou, 2009), the study of rock mass permeability (Kayabasi, 2017) or the evaluation of vibrations in the rock mass (Mesec et al., 2016).  Hoek et al. (2013), to verify its quantitative formulation, compared the values estimated visually and with the proposed formula in 75 rock outcrops, concluding that there is an acceptable correspondence between the observed and calculated GSI and that the majority of values do not exceed 5 points of difference concerning the GSI evaluated in-situ.

Review of previous researches
Winn and Wong (2018) carried out a comparison study on sedimentary rocks located in Singapore, using the formulations of Russo Bertuzzi et al. (2016) compared the GSI values obtained visually in four different rock masses, observing that many of the calculated values differ by more than 10 points for the values estimated visually, which indicates that there is a considerable uncertainty; however, in approximately half of the data there is a reasonable correlation. For this reason, Bertuzzi et al. (2016) recommend that the quantified GSI should be used as a complementary calculation and that the visual analysis must necessarily be carried out in-situ.
Win (2019) compared the in-situ GSI and those calculated quantitatively using the formulations of Hoek et al. (1995( ), Hoek et al. (2013, and a formulation depending on the parameters of the Q index (Barton et al., 1974) in stratified sedimentary rocks in Singapore, observed differences in the order of 10 to 30 points. It was indicated that the main reason for these differences is the variability of the parameters referring to the discontinuities, such as the RQD, J n , J r , and J a in the sedimentary rocks, for this reason, it was proposed to modify the coefficients of the indicated formulations to adjust the values calculated with the values explicitly observed in the rock masses.
Vásárhelyi et al. (2016) evaluated the GSI with various formulations along 70 m of a tunnel in Hungary, indicating that the GSI values calculated with the different methodologies are between 15 and 38 points, which suggests a considerable dispersion between the results when applying the different calculation methodologies. It is also observed that the results of Russo (2009) present a more significant difference concerning the rest.
Application of fuzzy set theory to the GSI system. The original and modified GSI charts published by Sonmez and Ulusay (1999,2002) were defined by fuzzy sets, using 22 "if-then" rules.      (2019) Various GSI = 1.265RMR ' -21.49 Some formulations that relate the GSI with the RMR (Bieniawski, 1976(Bieniawski, , 1989 can also be found in the technical literature; the most widespread was published by Hoek et al. (1995), presented in Equation 2.
(2) Where: RMR' -RMR in dry conditions and without correction for fracture orientation, GSI -Geological Strength Index.  Table 5, valid for their application in rock masses belonging to the Andes Mountains.
Similarly, several authors have recommended estimating the GSI value from the RMR', using a linear equation of the form: Where: RMR' -RMR in dry conditions and without correction for fracture orientation, a, b -constants that depend on the lithology. Somodi et al. (2021) compiled quantitative formulations in the form described previously, these are summarized in Table 6.
Most of the data considered in the formulations presented in Table 6 have been defined in rock masses with GSI values between 30 and 80 points, so its application in rock masses of poor and very poor quality must be carried out with care. In this sense, for poor and very poor rock masses (RMR'<30), Osgoui and Ünal (2005) suggest an exponential relationship, which is presented in Equation 4.

(4)
Where: RMR' -RMR in dry conditions and without correction for fracture orientation, GSI -Geological Strength Index. In addition, it is observed that the formulation of Cosar (2004), defined for its specific application in schists and sedimentary rocks, provides more conservative GSI values for high RMR', and very optimistic values for low RMR', so its application must be made with care in these ranges. However, this correlation is within the shaded area for RMR's values between 30 and 70, which corresponds to most of the rock masses found.
The Singh and Tamrakar (2013) correlation, defined for metamorphic rocks in Nepal, provides more conserv- ative GSI values for RMR' values greater than 40. For lower values, it is within the expected range; however, this correlation has been defined with a base of rock masses with RMR' between 36 and 82, so its application must be done with care or avoided since it corresponds to a correlation for particular rock masses. It is also observed that the nonlinear relationship of Osgoui and Ünal (2005) is within the shaded area. Hence, its application is valid for poor and very poor rock masses.

Methods
First, the dispersion of the GSI values calculated through qualitative methodologies has been studied. A virtual survey has been conducted on a group of geological engineers, mining engineers, and civil engineers from Peru, Spain, and Chile (40 participants). The purpose of this survey is to define the mean values, the standard deviation, and the coefficients of variation (COVs) of the GSI values and to verify if these values are similar to the reference values suggested by Hoek (1998) and Harr (1987). The survey presents a general photograph with the basic description of the four rock masses evaluated in this investigation and was conducted during August 2021 on the Google Surveys platform.  Subsequently, the values of the GSI have been obtained through the quantitative formulations proposed by Somnez and Ulusay (2002) the results have been compared with the qualitative GSI, allowing to identify which is the formulation that best adjusts to the value estimated visually. Finally, the data reported in previous studies and the data obtained in this research have been integrated into a unique graph, concluding that the new data show the same trend observed in previous studies.  blocky structure, three main fracture systems, with spacing between 0.60 and 2.00 m, rough, planar, clean discontinuities, with some clayey fill, slightly altered and dry, RMR' = 71. (see Figure 3).

Rock mass 3
Pseudo-metamorphized rock mass (slate shale) located on the campus of the National University of Engineering (Lima -Peru), intensely fractured, with an average spacing between fractures of 0.05 m, low resistance to simple compression (UCS< 5MPa), the discontinuities are persistent and have an opening of up to 5 mm, partially with hard fill, the rock mass is wet and altered, RMR' = 31. (see Figure 5).

Rock mass 4
Unlike the three previous cases found on slopes, in case 4, there is a rock mass corresponding to an underground excavation (see Figure 6). The rock mass corre-sponds to a gold-bearing quartz vein gallery embedded in sandstones, lutites, and folded schists, with an approximate width of 7 m and RMR' = 55. This rock mass is available in the sketchfab repository, where the rock mass can be viewed in 3D at the link https://sketchfab. com/3d-models/underground-blast-face-3659ecc6b-d684ea2ad45bdd561f2ac64.
The summary of the RMR' values defined for each rock mass is presented in Table 7.

Qualitative analysis
The survey results are presented graphically in Figure  7, considering GSI intervals every five points; these data have been statistically processed, adjusting to a normal distribution curve, defined by the mean value (μ) and by the deviation standard (σ). Figure 8 shows the normal distribution curves of the four evaluated rock masses. Figure 8 shows that rock masses 1, 2, and 3 have standard deviation values close to 10 points, which indicates that 68.2% of the data is in the confidence interval defined by μ ± σ or GSI ± 10, which is consistent with the studies by Hoek et al. (2013) and Winn and Wong (2018). It is also observed that unlike rock masses 1, 2, and 3, the normal density function of rock mass 4 presents a more flattened and elongated shape, due to the pronounced dispersion of the GSI values reported in the survey, this is reflected in a higher value of the standard deviation (σ=16.64), greater than 10 points. The expla- nation for this behaviour in rock mass 4 is attributable to the presence of quartz veinlets; according to what has been observed in engineering practice, this tends to confuse many field evaluators, as they erroneously consider that the presence of any discontinuity is necessarily equivalent to a decrease in the rock mass quality. Therefore, although the discontinuities present a gold-bearing quartz fill, with resistance even greater than the host rock, it is common for this type of rock mass to be reported with low GSI values.
The average qualitative GSI values of the four rock masses analyzed have been plotted on a Hoek and Marinos (2000) chart, presented in Figure 9. Hoek (1998) suggests reference values of the coefficients of variation (COV) of the parameters involved in the Hoek-Brown criterion. It is indicated that the values of UCS, m i , and GSI fit a normal distribution with coefficients of variation of 0.25, 0.125, and 0.10, respectively. Harr (1987) classifies the coefficients of variation as low (COV<0.10), moderate (0.15<COV<0.30), and high (COV>0.30), indicating that the values suggested by Hoek (1998) are among the low ranges. and moderate. However, the values presented in Table 8, calculat-  To evaluate the influence of COV on the behaviour of the rock mass, a probabilistic stability analysis of a 15 m-high slope excavated in rock mass 3 has been carried out. For the UCS and m i parameters, the suggested COV values by Hoek (1998) have been considered, and for the GSI, both, the COV values proposed by Hoek (1998) and the one obtained in the survey were considered.
The probabilistic analysis results presented in Figure  10 indicate that although the average safety factors do not vary since the mean value of GSI in both cases is the same, the value of the probability of failure (Pf) of the slope has practically doubled, increasing from 15.6% to 30.6%.

Quantitative analysis
The GSI value of the four rock masses studied has been estimated using the quantitative formulations of  Notes: (*) GSI calculated from the RMR' (with the general correlation) (**) GSI calculated from RMR' (depending on lithology) (***) GSI calculated based on RQD and JCond 89 NA: This does not apply to the lithology or the rock mass quality tionships. The calculation parameters involved in these formulations are presented in Table 9, obtained from the field characterization, and Table 10 shows the summary of the GSI values calculated quantitatively. The Sing and Tamrakar (2013) relationship has been developed for metamorphic rocks, so it has only been applied to rock masses 2 and 3. The Cosar (2004) relationship has only been used to rock mass 4, corresponding to schists and sedimentary rocks. The nonlinear relationship of Osgoui and Ünal (2005) is only applicable to rock mass 3 corresponds to a poor-quality rock mass (RMR'=23).

Discussion
In the calculation of the quantitative GSI, no significant variation is observed using the general formulation In terms of the statistical parameters, in the quantitative approach, the dispersion of the results is lower compared to the qualitative approach; the results are presented in Table 11, where the average values do not suffer a significant variation; however, the standard deviation   Figure 11 graphically presents the dispersion of the results, where a tendency of the GSI values to be within the range that defines a variation of GSI±10 points is observed.  On the other hand, if the comparison graph between the RMR' and the GSI (see Figure 13) is considered, similarly to what was reported by Ceballos et al. (2014) and Sánchez et al. (2016), it is observed that most of the data is within the range suggested by Ceballos et al. (2014), defined between the lines GSI = RMR' + 5 and GSI = RMR' -15. Therefore, it is verified that this is the confidence interval of the GSI calculated from the RMR'. This graph also shows that the values of Russo et al. (2009) are outside the indicated confidence interval, providing very conservative values. Figure 13 also shows that the average GSI values estimated visually (qualitative approach) fit the line defined by the relationship GSI = RMR' -5, so that, in general, and because of the results, they constitute a reasonably reliable and straightforward approximation for the calculation of the GSI from the RMR'.
Finally, by combining the data from Ceballos et al.  Figure 14), it is observed that the trend of all the data is similar. Therefore, it is confirmed that the confidence range to calculate the GSI from the RMR' is between GSI = RMR' + 5 and GSI = RMR' -15, as indicated by Ceballos et al. (2014).

Conclusions
Although the GSI system was developed to be used exclusively with the Hoek-Brown criterion in rock mass strength and deformability estimation, this system continues to evolve. It has been successfully applied for other purposes, such as rock mass evaluation, excavability, karsticity, permeability, residual resistance parameters, and support in tunnels.
The qualitative or visual methodologies provide a high subjective component, which has been observed even considering that in the survey conducted in this research, all the participants had experience in characterizing rock masses. The high variability of the GSI values obtained with the qualitative approach is reflected in the coefficients of variation (COVs) of the rock masses evaluated in this investigation, which are classified as low to moderate in the case of rock masses 1 and 2, and high in the case of rock masses 3 and 4, exceeding the values suggested by Hoek (1998) and Harr (1987). They assign low COV values for the GSI.
In the particular case of rock mass 4, there is a more elongated and flattened normal distribution curve than the other three cases, with variable GSI values between 15 and 80 points. The explanation given for this behaviour is attributable to the presence of quartz veinlets, which tends to confuse some of the rock mass quality evaluators because, in general, the presence of filler in the joints reduces the quality of the rock mass; however, in this case, the fill material has a higher resistance to the encasing rock. The GSI values calculated with the quantitative formulations present a lower dispersion concerning those evaluated qualitatively, which is reflected in the values of the COVs observed in the four rock masses evaluated. With the qualitative approach, the COV values are between 0.14 and 0.35 (moderate to high); however, with the quantitative approach, the COV values are between 0.06 and 0.15 (low to moderate), getting closer to the value of 0.10 suggested by Hoek (1998) andHarr (1987). Of all the quantitative formulations analyzed, Russo (2009) is the one that provides the most conservative values, especially for GSI values less than 50; this trend has already been observed in previous studies. The average values obtained with both methodologies show differences of less than 5 points.
Quantitative formulations should be used with caution, taking into account the characteristics of the rock masses on which the relationships have been defined.
Although the database of Ceballos et al. (2014) and Sánchez et al. (2016) correspond to rock masses located in Spain and the Andes Mountains respectively, it is observed that in both cases, the majority of data are within the range bound by the relationships GSI=RMR'+5 and GSI=RMR'-15, this trend is also observed in the four rock masses evaluated in this investigation, so this range can be considered as the confidence interval to obtain the GSI value from RMR'. However, GSI -RMR' relationships have generally been defined in rock masses with GSI values between 30 and 80 points, so their application in rock masses of poor and very poor quality must be made with care.

Recommendations
It is recommended to expand the number of rock masses evaluated and include complex rock masses, such as volcanic rocks.
In the survey conducted to estimate the qualitative GSI, 90% of the participants assigned a single GSI value to each rock mass evaluated, only 10% indicated a range of values. However, the majority of the GSI charts suggest that one should not try to be too precise in its determination and that it is more realistic to establish a range of values. For this reason, it is necessary to emphasize this point prior to the execution and supervision of the field data collection.
In the geotechnical analyzes of slopes or other structures that involve rock masses, it is advisable to consider the variability of the GSI through probabilistic analyses to define the safety factor and the probability of failure, which according to the results obtained in this investigation can be increased by up to 50%.
The development of virtual reality is a tool that has begun to be used successfully for training in mining and civil geomechanics. It could be incorporated into the study of rock masses and the estimation of the GSI.