## 2 Methodology
### Loss Function  
The analysis of this section includes direct and indirect economic loss and casualty estimation, excluding the impact of casualty losses. The direct economic loss of the earthquake is the sum of the loss caused by the waste of natural resources, asset damage, and the loss caused by the collapse of the buildings themselves. After thorough analysis and careful consideration, the estimation method for the economic loss ratio is defined as shown in Equation (4):  
\[L_{L}(I)_{k}=\frac{\text{Direct economic loss caused by type K structure under earthquake  
greater than the limit state of the structural carrying capacity under the premise of a certain strength index. The seismic vulnerability function can be expressed as:  
\[P_{f}=P(DI\geq LS|IM)=1-\Phi\left(\frac{ln(LS)-ln\big{(}\alpha IM^{\beta}\big{)} }{\beta_{EDP|IM}}\right) \tag{7}\]  
where \(DI\) is the damage index of the structure, \(LS\) is the limit state of the maximum bearing capacity of the structure defined for the structure, \(\Phi(\cdot)\) is the standard normal distribution function, \(\beta_{EDP|IM}\) represents the standard deviation of the logarithmic regression analysis between the seismic demand parameters and the ground motion intensity, that can be expressed as:  
\[\beta_{EDP|IM}=\sqrt{\beta_{d}^{2}+\beta_{c}^{2}} \tag{8}\]  
The values of \(\beta_{EDP|IM}\) are dependent on a lot of different factors that include characteristics of the input, the structures, the calculation method, etc. [18,19]. When the limit state \(LS\) index is not specified, \(Sc\) is used to represent the structure resistance, which is the value corresponding to the structure in the limit state \(LS\), and \(S_{d}\) represents the structure demand, which is the value corresponding to the structure failure index \(DI\). Assuming that the probability function of structural resistance also follows lognormal distribution, according to Equation (7), the failure probability of the limit state corresponding to structural resistance is  
\[P_{f}=\Phi\left[\frac{ln(S_{d}/S_{c})}{\sqrt{\beta_{d}^{2}+\beta_{c}^{2}}}\right] \tag{9}\]  
where \(\beta_{d}\) is the logarithmic standard deviation of seismic demand, \(\beta_{c}\) is the logarithmic standard deviation of structure resistance. According to the suggestion in the American Disaster Assessment Management Manual [20], when the geo-seismic strength index \(IM\) takes spectral acceleration \(S_{a}\) as the independent variable, \(\sqrt{\beta_{d}^{2}+\beta_{c}^{2}}\) can be 0. 4. In order to represent the vulnerability function more intuitively, it is assumed that event \(E_{k}(k=0,1,2,\ldots,4)\) is used to represent the probability that structure \(i\) randomly selected from the sample is in the failure state \(E_{k}\) when spectral acceleration \(S_{a}\) is \(a_{i}\). Seismic vulnerability curves of different failure states can be expressed by the lognormal distribution function:  
\[F_{j}\big{(}a_{i},c_{j},\zeta_{j}\big{)}=\Phi\left[\frac{ln\big{(}a_{i}/c_{j} \big{)}}{\zeta_{j}}\right] \tag{10}\]  
where \(c_{j}\) is the median value corresponding to the vulnerability curves of each failure state. \(\zeta_{j}\) is the logarithmic standard deviation corresponding to the vulnerability curves of each failure state. \(j=0,1,2,3,4\) represents intact, minor, moderate, severe, and complete ultimate failure states, respectively. Based on the assumption that the vulnerability curve follows the lognormal distribution, the logarithmic standard deviation is taken as the constant \(\zeta\). According to Equations (9) and (10), the earthquake vulnerability can be defined as:  
\[\left\{ \begin{array}{l}P_{i0}=P(a_{i},E_{0})=1-F_{0}(a_{i},c_{0},\zeta)\\ P_{i1}=P(a_{i},E_{1})=F_{0}(a_{i},c_{0},\zeta)-F_{1}(a_{i},c_{1},\zeta)\\ P_{i2}=P(a_{i},E_{2})=F_{1}(a_{i},c_{1},\zeta)-F_{2}(a_{i},c_{2},\zeta)\\ P_{i3}=P(a_{i},E_{3})=F_{2}(a_{i},c_{2},\zeta)-F_{3}(a_{i},c_{3},\zeta)\\ P_{i4}=P(a_{i},E_{4})=F_{3}(a_{i},c_{3},\zeta)\end{array} \right.