## 3 Methods
### Propensity Matching Between Factual and Counterfactual  
To accurately quantify the causal effects of operational incidents on passenger origin-destination (OD) flows, it is essential to establish a match between factual scenarios (normal operating conditions) and counterfactual scenarios (conditions under operational disruptions) since we cannot observe the OD passenger flows under factual and counterfactual conditions simultaneously. Specifically, assume a tensor \(\mathcal{F}\in\mathbb{R}^{S\times S\times T}\) to represent the dynamic evolution of OD passenger flows under factual conditions (normal condition), where \(S\) denotes the total number of stations, and \(T\) represents the number of discrete prediction intervals within the study period. Each element \(\mathcal{F}_{ij}^{t}\) in this tensor reflects the number of passengers traveling from station \(i\) to station \(j\) during interval \(t\) under normal operation.  
Similarly, the OD flow dynamics under counterfactual conditions (incident and emergency conditions) are captured by another tensor \(\mathcal{C}\in\mathbb{R}^{S\times S\times T}\), where each element \(\mathcal{C}_{ij}^{t}\) denotes the number of passengers traveling from station \(i\) to station \(j\) during interval \(t\) when an operational incident occurs.  
To ensure matched scenarios represent equivalent underlying conditions with minimal confounding differences apart from the operational incident, we adopt propensity matching based on tensor Kullback-Leibler (KL) divergence. KL divergence is an appropriate measure to quantify the dissimilarity between two probability distributions represented by tensors. Formally, the KL divergence between factual tensor \(\mathcal{F}\) and counterfactual tensor \(\mathcal{C}\) is defined as Equation (4):  
\[\text{KL}(\mathcal{F}\|\mathcal{C})=\sum_{i=1}^{T}\sum_{i=1}^{N}\sum_{j=1}^{N} \mathcal{F}_{ij}^{t}\ln\frac{\mathcal{F}_{ij}^{t}+\varepsilon}{C_{ij}^{t}+\varepsilon} \tag{4}\]  
where \(\varepsilon\) is a small positive constant introduced to avoid division by zero, and \(N\) represents the number of stations. A smaller KL divergence indicates a higher similarity between the two tensors, suggesting these scenarios are suitable for pairing.  
Paired datasets are constructed by selecting factual scenarios from historical OD flow data that most closely match each counterfactual incident scenario based on the minimum tensor KL divergence criterion. Formally, given an incident scenario tensor \(\mathcal{C}^{(m)}\), the matched factual tensor \(\mathcal{F}^{(m)}\) is selected from a candidate set \(\left\{\mathcal{F}^{(1)},\mathcal{F}^{(2)},\dots,\mathcal{F}^{(N)}\right\}\) as:  
\[\mathcal{F}^{(m)}=\underset{\mathcal{F}^{(n)}}{\text{argminKL}}\Big{(} \mathcal{F}^{(n)}\big{\|}\mathcal{C}^{(m)}\Big{)} \tag{5}\]  
Therefore, each incident scenario \(\mathcal{C}^{(m)}\) is paired with the most similar factual scenario \(\mathcal{F}^{(m)}\) with the sample index \(m\) and \(n\). Matched datasets allow deep learning models to distinguish between normal fluctuations and incident-induced deviations in passenger flows. These paired data then serve as input for training the dual-channel deep counterfactual prediction model, effectively capturing and quantifying the causal impact of incidents on passenger OD flows. It should be noted that the matching between factual and counterfactual scenarios does not imply a temporal \"before-and-after\" implementation; rather, it involves identifying operationally comparable but separate historical conditions. Thus, our methodological design differs from conventional before-and-after comparative analyses.