## 6 A Discussion of the Qualitative Aspects (Joint Conditions) of Existing GSI Charts
### Consideration of a Persistence Correction Factor (Pi) in P32 Estimates for V-GSI
This is a difference of 0.24 between the adjusted P32\({}^{\prime}\) using the Pi persistence correction factor and [PERSON]'s method, which is in virtually the same order of the difference between [PERSON]'s P32 using his new method and his P32 DFN estimate of around 0.19. Assuming this level of accuracy is required for P32 when using the V-GSI chart, the Pi method outlined above can be adopted and included on the vertical axis of the V-GSI chart. Pi therefore provides a method to include joint persistence. Figure 11 also shows the adjusted mean P32\({}^{\prime}\) plotted on the vertical axis of the V-GSI chart (just below the upper green arrow plot of P32\({}_{\text{UNIT VOLUME}}\)). The inference from the result above is that if, in general, the mean trace length or persistence is observed or assessed to be less that the diameter of the unit sphere then Pi and the resulting Sti\({}^{\prime}\) correction suggested above provide a reasonable adjustment to both the P30 and P32 estimates--which in turn adjusts the VFC plot on the vertical axis of the V-GSI chart (in this case the adjusted VFC\({}^{\prime}\) is 1.84 fracture/m\({}^{3}\) compared to 2.61 fractures/m\({}^{3}\) without the adjustment (cf. Tables 3 and 4). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Set** & **Mean Sti** & **Pi** & \(\sqrt[3]{P_{i}}\) & \(St_{i}^{\prime}\) & **VFC\({}^{\prime}\)** & **Pi \(\times\) r2** & **P32\({}^{\prime}\)** \\ \hline S1 & 0.93 & 0.49 & 0.79 & 1.18 & 0.85 & 1.21 & 1.03 \\ S2 & 2.71 & 0.34 & 0.70 & 3.87 & 0.26 & 1.21 & 0.31 \\ S3 & 1.03 & 0.27 & 0.65 & 1.59 & 0.63 & 1.21 & 0.76 \\ S4 & 4.98 & 0.13 & 0.51 & 9.76 & 0.10 & 1.21 & 0.12 \\  & & & & **\(\Sigma\)VFC\({}^{\prime}\)** & **1.84** & **\(\Sigma\)P32\({}^{\prime}\)** & **2.22** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Pi persistence correction method results providing adjusted VFC\({}^{\prime}\) and adjusted P23\({}^{\prime}\) estimates. Kates Arch Dam, Lesotho: Comparisons of P32 Estimates Using REV Approach, V-GSI Method, and DFN Modelling  
The VFC parameter, which is equivalent to P30, forms the core of the quantified vertical axis of the V-GSI chart [3], and was pioneered during the excavation of the foundation keyway for the 185 m high Kates Arch Dam (Figure 12--photo). The dam is a key structure in a series of dams and tunnels that make up the Lesotho Highlands Water project in southern Africa. Quantitative data (e.g., joint spacing, trace length or continuity measurements, and joint sidewall separation) and qualitative or descriptive data (e.g., rock mass and joint degree of weathering, joint sidewall separation, joint infill, and roughness and waviness) were systematically mapped on the exposed foundation surface during excavation [3]. Structural Domains and VFC and P32\({}_{\text{\sc{limit}}\text{\sc{volume}}}\) Estimates for the Foundation Keyway of the Kates Dam  
The foundation rock mass at the Kates Dam was shown to be divisible into twelve structural domains--six domains per flank (Figure 13). Individual domains essentially correlate across both flanks of the dam foundation, thus translating into six spatially distinct, near-horizontal domains across both flanks. Individual domains are relatively homogenous within the domain but are statistically different to the rock mass of the domains above and below. Figure 12: The 185 m high Kates Arch Dam, Lesotho Highlands, completed in 1998. The photograph was taken in 2023. In general, the Katse foundation rock mass, which comprises basalt flows, shows a fairly low degree of fracturing. Two relatively more fractured Domains (than elsewhere in the foundation) are recognizable on both abutments between EL 1900 and EL 1980 approximately (Figure 13). These two Domains are referenced as left flank (LF) Domain III and right flank (RF) Domain III, respectively. RF Domain III is present between EL 1974 and EL 1894 (Figure 13) but has a minimum dimension that is limited to approximately 15 m horizontally into the foundation rock mass and discussed in more detail in Section 8.2.