## 2 A Hypothetical Example Application of the V-GSI Chart in a Tunnel
### Correction Factors (CFs) and Calculation of True Spacings (St)
This means that there may be a joint from a particular set at a greater distance than measured, but it is better to include the \(Sa\) 'edge effect' measurement than ignore it, since ignoring it may skew the spacing statistics. For example, if there were a swarm of J1 joints in the middle of the face and none near the sidewalls, then the average spacing based on measurement in the swarm only would provide the average spacing within the swarm and not the rock mass as a whole. As can be seen in Figure 3, there are joints for set J1 mapped in the sidewalls. The orientation of the sidewalls is, however, perpendicular to the tunnel heading, and, as such, the CF for set J1 would be different, i.e., the vector measurement directions are perpendicular to the tunnel heading face. While not shown in the example, it becomes possible, however, to sum the corrected 'edge effect' (true) \"greater-than\" spacings measured on the tunnel heading and the sidewalls for two adjacent joints of a particular set to provide a single corrected 'edge effect' spacing value for the right-hand and left-hand sides of the tunnel, respectively. In the example shown in Figure 3, all the discontinuities are shown as having full persistence, i.e., they have continuity longer than the dimensions of the tunnel. This is not always the case, and this effect can be partially overcome by adding more scanlines or by adding \(Sa\) measurement vectors that sample areas with less joints due to terminations. It should be noted that it is also possible to simplify the process by estimating the joint linear density per set (\(\lambda\)) by counting the number of joints per metre of sampling line, e.g., the sampling line can be the width of the tunnel heading or the span of the tunnel. In the case of the tunnel, if \(x\) is the total number of discontinuities intersected by a line of total length (\(L\)) or the tunnel width, the linear density of discontinuities is approximated by \(x/L\) and the reciprocal is average spacing, i.e. :  
\[S_{average}\approx 1/\lambda\approx L/x \tag{4}\]  
In the example provided in Figure 2, the width of the tunnel heading is approximately 12.5 m and the tunnel height is approximately 8 m. L for set J1 is 6/12.5 or 0.48 joints/m, and the reciprocal is 2.08 m, which is the average apparent spacing for this set. The true mean \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Joint Set**} & **Mean True Spacing (m)** & **True Spacing** \\  & \(St_{i}\) & **95\% Confidence Limit (m)** \\ \hline J1 & 1.54 (1.73) & 0.955 \\ B & 2.26 (2.40) & 2.31 \\ \hline \hline \end{tabular} Note: Parentheses provide the apparent spacing means before the correction factors are applied.  
\end{table} 
Table 2: Summary of true spacing (\(St\)) statistics for joint sets J1 and B.  
spacing for J1 can then be approximated as 2.08 \(\times\) 0.89 (CF for set J1), and the corrected true spacing is around 1.85 m for this method (c.f. Table 2). This value is similar enough to the result obtained when actual spacings are measured and corrected individually (1.54 m), but where possible, individual measurement should be used to improve the reliability of the assessment. 1 for set B is 3/8 or 0.38 joints / m, and the reciprocal is 2.67 m, which is the average apparent spacing for this set. The true mean spacing for B is approximately 2.67 \(\times\) 0.94 (CF for set B), and the corrected true spacing is around 2.51 m. The disadvantage of this method is that it is not possible to calculate confidence limits on the average spacings. The average is also skewed somewhat because it does not consider the smaller spacing values noted in Figure 3. For the reader interested in estimating RQD, [PERSON] [18] provides a rough correlation using joint volumetric counts (Jv). It should be noted that [PERSON]'s Jv is not calculated in exactly the same way as the bias-free _VFC_; nevertheless, once _VFC_ is calculated with Equations (5) and (6) below, it can be used to estimate RQD.