## 4 Proposed Geometric Similarity Evaluation Methodology
### Scenario Representativeness Evaluation  
Building on the process described in Section 4.1, we extended our methodology to assess whether certain entire scenarios can represent other geometrically similar scenarios. Accordingly, based on the premise that the credibility of an XILS platform for different scenarios can be evaluated based on the results from specific representative scenarios for which the platform has already been confirmed to be credible, we propose a scenario representativeness evaluation framework involving geometric similarity calculations between different scenarios. The evaluation metrics used for the scenario representativeness assessment include parameter-based applicability (\(A_{m}\)) and correlation (\(C_{m}\)), scenario-based applicability (\(A_{k}\)) and correlation (\(C_{k}\)), and dynamics-based correlation (\(D_{k}\)). The two parameter-based metrics (Am and Cm) are averaged across the different speed conditions (low, medium, and high) within each scenario to obtain a single value that represents the overall parameter applicability and correlation for each scenario. For each \(k\)-th scenario, the parameter evaluation scores for each speed condition are calculated as follows: \(P_{m}\big{(}v_{k}^{L}\big{)}\) represents the applicability and correlation evaluation score for XILS data under the low-speed condition in the \(k\)-th scenario. \(P_{m}\big{(}r_{k}^{L}\big{)}\) represents the applicability and correlation evaluation score for real data under the low-speed condition in the \(k\)-th scenario. \(P_{m}\big{(}v_{k}^{L},r_{k}^{L}\big{)}\) represents the applicability and correlation evaluation score between XILS data and real data under the low-speed condition in the \(k\)-th scenario. The evaluation scores for the medium-speed (\(M\)) and high-speed (\(H\)) conditions are derived in the same manner. Accordingly, the average parameter evaluation scores across speed conditions in the \(k\)-th scenario are calculated as shown in Equations (51)-(53):  
\[\overline{P_{m}}(v_{k})=\frac{1}{3}(P_{m}(v_{k}^{L})+P_{m}(v_{k}^{M})+P_{m}(v_ {k}^{H})), \tag{51}\]  
\[\overline{P_{m}}(r_{k})=\frac{1}{3}(P_{m}(r_{k}^{L})+P_{m}(r_{k}^{M})+P_{m}(r _{k}^{H})), \tag{52}\]  
\[\overline{P_{m}}(v_{k},r_{k})=\frac{1}{3}(P_{m}(v_{k}^{L},r_{k}^{L})+P_{m}(v_ {k}^{M},r_{k}^{M})+P_{m}(v_{k}^{H},r_{k}^{H})). \tag{53}\]  
Through this process, a single average value for the parameter-based reliability evaluation indices can be derived for each scenario. Based on these average values, the correlation index \(C_{m}\) and applicability index \(A_{m}\) are calculated using the parameter-based reliability evaluation formulas presented earlier. This yields new \(C_{m}\) and \(A_{m}\) values for each scenario, which are used to evaluate the representativeness of different scenarios. For each \(k\)-th scenario, a scenario representativeness vector is constructed by integrating the previously calculated indicators corresponding to each speed condition, as shown in Equations (54)-(56):  
\[SR_{k}^{L}=\Big{[}D_{k}^{L},A_{k}^{L},C_{k}^{L},A_{m}^{L},C_{m}^{L}\Big{]}, \tag{54}\]  
\[SR_{k}^{M}=\Big{[}D_{k}^{M},A_{k}^{M},C_{k}^{M},A_{m}^{M},C_{m}^{M}\Big{]}, \tag{55}\]\[SR_{k}^{H}=\Big{[}D_{k}^{H},A_{k}^{H},C_{k}^{H},A_{m}^{H},C_{m}^{H}\Big{]}. \tag{56}\]  
Using \(SR_{k}^{L}\), \(SR_{k}^{M}\), and \(SR_{k}^{H}\), we quantitatively evaluate the representativeness of a scenario based on its geometric similarity to other scenarios. For this purpose, two comparison methods are proposed: comparison across the same speed condition and comparison across different speed conditions. The same-speed comparison involves calculating the geometric similarity between different scenarios by comparing the representativeness vectors of each scenario under identical speed conditions. When the number of scenarios is \(k\), the number of possible scenario pairs is \(\binom{K}{2}=\frac{K(K-1)}{2}\), and the comparisons are performed across all three speed conditions \(v\in\{L,M,H\}\) for each scenario pair.