## 2 A Hypothetical Example Application of the V-GSI Chart in a Tunnel
### Correction Factors (CFs) and Calculation of True Spacings (St)  
For both sets, J1 and B, the discontinuity traces exposed in the tunnel face are not oriented at exactly \(90^{\circ}\) to the Sa spacing measurements vectors, which are horizontal and vertical, respectively. The apparent spacing (Sa) measurements per set therefore require correction to 'true' spacings measurements (St) to obtain spacings that are orthogonal to the mean dip and dip direction of each set. This is simply achieved by using vector dot product analysis [1; 2; 15] to calculate correction factors for both sets J1 and B, respectively, given that each Sa measurement is a vector. This calculation is possible because the respective angular relationships between the tunnel face, measurement vectors and discontinuities are recorded by the mapping process. Spacing correction factor (CF) per discontinuity set (J1 and B in this case) can be calculated with the use of Equation (1) [1; 2; 15]:  
\[CF=(\sin\alpha_{n}\times\cos\beta_{n})\times(\sin\alpha_{s}\times\cos\beta_{s}) +(\cos\alpha_{n}\times\cos\beta_{n})\times(\cos\alpha_{s}\times\cos\beta_{s})+ (\sin\beta_{n}\times\sin\beta_{s}) \tag{1}\]  
\(CF\) is always the absolute value, and if <0.1, then 0.1 is set as the default value, where:  
\(\alpha_{d}=\) discontinuity set mean dip direction (used to convert to pole trend);  
\(\beta_{d}=\) discontinuity set mean dip angle (used to convert to pole plunge);  
\(\alpha_{n}=\) trend of pole or normal to discontinuity set mean dip direction;  
\(\beta_{n}=\) plunge of pole or normal to discontinuity set mean dip angle;  
\(\alpha_{s}=\) strike or trend of scanline or measurement vector (azimuth 0-360\({}^{\circ}\)) on tunnel heading, sidewall or crown;  
\(\beta_{s}=\) dip or plunge of scanline or measurement vector (can be anywhere between \(0^{\circ}\), horizontal, and \(90^{\circ}\), vertical) on tunnel heading, sidewall or crown. Table 1 shows the inputs and the resulting CFs using Equation (1) for each discontinuity set shown in Figures 2 and 3. For set J1, each _Sa_ measurement needs to be multiplied by the correction factor (CF), 0.89, to convert _Sa_ measurements to _St_ values. Likewise, for set B, each _Sa_ measurement \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Discontinuity** & **Mean Vector** & **Mean Vector Orientation,** & **Mean Vector Orientation,** & **Trend and Plunge** & **Correction** & **Trend and Plunge** \\
**Set** & **Orientation** & **Pole or Normal** & & **Measurement Vectors/Scanline** & **Factor** \\ \hline  & \(\alpha_{d}\) & \(\beta_{d}\) & \(\alpha_{n}\) & \(\beta_{n}\) & \(\alpha_{s}\) & \(\beta_{s}\) & \(CF\) \\ \hline J1 & 312 & 70 & 132 & 20 & 151 & 0 & 0.89 \\ \hline B & 195 & 20 & 15 & 70 & 61 & 90 & 0.94 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Input data for the calculation of correction factors (CF) for sets J1 and B shown in Figure 3.  
needs to be multiplied by 0.94 to convert Sa measurements to \(St\) values, as shown in Equation (2):  
\[St_{i}=CF_{i}(Sa_{i}) \tag{2}\]  
Figure 3 shows the \(Sa\) measurements scaled off the map for sets J1 and B, respectively, which are then converted to \(St\) values using Equation (2). Table 2 provides a summary of the mean true spacing and 95 % confidence limits on the spacings for each set, respectively. The confidence limits on the mean true spacings per set shown in Table 2 are calculated at the 95% level, as follows in Equation (3):  
\[95\%\text{ Confidence Limits}=1.96\frac{\partial St_{i}}{\sqrt{n}} \tag{3}\]  
where \(\frac{\partial St_{i}}{\sqrt{n}}\) is the standard error of the true spacing values, \(\partial St_{i}\) is the standard deviation for the \(i\)th discontinuity set, and \(n\) = number of spacing measurements, respectively, for that set [1,2]. As can be seen in Figure 3, there are edge effects at the intersection of the tunnel heading and the sidewalls (J1) and at the crown and invert (B), and the apparent spacings are measured as \"greater-than\" (>) values.