## 3 Materials and Methods  
### Design of Multi-Mode Gravity-Based Two-Step Floating Catchment Area Method  
In the traditional 2 SFCA method, the treatment of distance affecting the supply-demand ratio and accessibility calculations is dichotomy: there is no decay within the search radius, and the intensity is zero outside the search radius, as shown in Figure 1. This means that the intensity at any point within the search radius is considered to be the same. This characteristic is also present in the MM-2 SFCA method. For instance, as shown in Figure 2, assuming that the demand points have the same population, the four demand points will have equal weights in the calculation of the supply-demand ratio in car mode, and \(P_{1}\), \(P_{2}\), and \(P_{3}\) will have equal weights in the calculation of the supply-demand ratio in bicycle mode. The roles of \(P_{3}\) and \(P_{4}\) in the final supply-demand ratio calculation will be identical, even though \(P_{3}\) is closer to the supply point \(S_{j}\) than \(P_{4}\). This does not align with the objective principle, which suggests that \(P_{1}\) should be given more weight than \(P_{4}\) in the supply-demand ratio calculation in car mode, and, similarly, \(P_{3}\) should be given more weight than \(P_{4}\) in the final supply-demand ratio calculation. The same issue arises in the accessibility calculation. To address this issue, the gravity model is introduced to add support for distance decay to the MM-2 SFCA method, and a multi-mode gravity-based two-step floating catchment area method (MM-G2 SFCA) is proposed. The gravity model is not only well-developed in the field of spatial interactions [6] but is also widely used in the improvement of the 2 SFCA method [9]. It is, therefore, well suited for the improvement of the MM-2 SFCA  
Figure 1: Schematic of the distance decay function: (**a**) traditional 2 SFCA; (**b**) gravity model; (**c**) standardized gravity model. Figure 2: A sketch map for supply–demand ratio. (**a**) Multiple catchment areas by mode around a supply point using MM-2 SFCA. (**b**) Multiple catchment areas by mode around a supply point using MM-G2 SFCA. (**c**) A more intuitive schematic of Figure (**b**). method. The calculation methods for the supply-demand ratio \(R_{j}\) of supply point \(j\) and the accessibility \(A_{i}\) of demand point \(i\) are as follows:  
\[R_{j}=\frac{S_{j}}{\sum_{\sigma=1}^{n}\sum_{k\in\{d_{kj}(M_{n})\leq d_{0}(M_{n}) \}}t_{n,k}D_{k}f(d_{kj})}, \tag{5}\]  
\[A_{i}=\sum_{v=1}^{n}\sum_{j\in\{d_{ij}(M_{n})\leq d_{0}(M_{v})\}}t_{n,k}R_{j} \big{[}f(d_{ij})\big{]}^{2}, \tag{6}\]  
where \(t_{n,k}\) represents the travel proportion of residents at demand point \(k\) under transportation mode \(M_{n}\). When \(t_{n,k}\) data are lacking, \(t_{n,k}=\frac{M_{k,n}}{M_{k}}\), where \(M\) is the transportation mode, \(M_{k}\) is the total travel volume at demand point \(k\), and \(M_{k,n}\) is the travel volume under the \(n\)-th transportation mode; the meanings of \(d_{kj}(M_{n})\) and \(d_{0}(M_{n})\) are the same as in Equation (3). \(f\big{(}d_{kj}\big{)}\) is the distance decay function, which increases the supply-demand ratio within the search radius as the distance shortens, aligning more closely with the First Law of Geography [26]. [PERSON] [25] found that the power-law form of the distance decay function better characterizes the intrinsic nature of the distance effects compared to exponential and other forms, and its corresponding interaction model is the widely used gravity model in geography and regional economics. Therefore, this work adopts the gravity model as the distance decay function:  
\[f(d)=\left\{ \begin{array}{c}d^{-\beta},d\leq d_{0}\\ 0,d>d_{0}\end{array} \right., \tag{7}\]  
where \(d\) represents the cost of movement, calculated as the time under multiple transportation modes; \(\beta\) (\(\beta>0\)) represents the distance decay coefficient, also referred to as the impedance coefficient. An increase in \(\beta\) indicates a more pronounced effect of distance on the supply-demand ratio and accessibility.