## 4 Methodology
### CF Model Calibration and Validation Based on Driving Modes
#### 4.2.1 Investigated CF Models  
1. Intelligent Driver Model (IDM)  
We have thoroughly investigated and compared the CF model performances in our previous study [1]; the IDM was validated as the best in several commonly used models for its superiority in CF behavioral reproduction, and therefore, IDM was also chosen to be the main CF model in this study. The IDM will be calibrated both from DBM and DMBM perspectives.  
The IDM is a widely recognized CF model that utilizes the concept of desired measures. Unlike the Gipps model, the IDM possesses the capability to intelligently select CF modes, facilitating a smooth transition between free-flow and CF conditions. Additionally, each parameter in the IDM represents a distinct aspect of driving behavior, rendering it more parsimonious and easier to calibrate. The IDM can be defined as:  
\[a_{n}(t)=a_{max}^{(n)}\left[1-\left(\frac{V_{n}(t)}{\widetilde{V}_{n}(t)} \right)^{4}-\left(\frac{\widetilde{S}_{n}(t)}{S_{n}(t)}\right)^{2}\right] \tag{3}\]\[\widetilde{S}_{n}(t)=s_{0}+max\left(0,V_{n}(t)T-\frac{V_{n}(t)\Delta V_{n}(t)}{2 \sqrt{a_{max}^{(n)}b_{conf}^{(n)}}}\right) \tag{4}\]  
where \(a_{max}^{(n)}\) is the maximum acceleration, \(b_{conf}^{(n)}\) is the comfort deceleration, \(\widetilde{V}_{n}\) is the desired velocity, \(S_{n}\) is the relative distance, \(\widetilde{S}_{n}(t)\) is the desired spacing, and \(s_{0}\) is the minimum distance when at a standstill.  
2. Wiedemann model  
The logic of CF that the CF interaction can be divided by perceptual thresholds was proposed by [PERSON] of Karlsruhe University. The Wiedemann model is recognized as a quintessential psycho-physical CF model that incorporates human factors to more accurately replicate realistic driving behavior, of which the CF logic is shown in Figure 4. The CF process is divided by different thresholds with a certain shape, and the subject vehicle takes different acceleration maneuvers. In this study, we utilized the Wiedemann 99 model, an updated version of the original Wiedemann model that accounts for both driver characteristics and driving modes. This model has recently gained popularity in both academic research and practical applications for its ability to accurately simulate car-following behavior. The model was calibrated individually for each driver in our study. For a detailed formulation, please refer to the literature [45; 46].  
3. Hybrid Wiedemann-IDM model  
Some researchers have argued that the sub-models in the Wiedemann model negatively affect the performance rather than the driving mode division part and have thus proposed a hybrid Wiedemann-IDM model [21]. That is, the original Wiedemann model is altered by replacing the acceleration equations in different CF modes (free-flow mode, closing-in mode, following mode, and emergency braking mode, as shown in Figure 4) with the most adaptable CF model so far, the IDM. These IDMs of each CF regime of the Wiedemann will be calibrated and given a set of parameters for each of the modes in this hybrid model. The reason why we introduce this hybrid model is to figure out (1) whether it is the acceleration  
Figure 4: The CF logic of the Wiedemann model.  
equations or the driving modes that weaken the performance of the Wiedemann model, and (2) whether the objective rule-based division of driving modes in the Wiedemann can reasonably identify different modes in human CF process.