## 2 Methodology
### Step 2: Modal Parameters Identification  
Once divided the time signal into its fundamental modes or mono-components using SWT, a mathematical expression, which describes the behavior of each obtained mode is necessary. The obtained mono-components present a free-decay response; hence, the following equation can be used:  
\[x_{i}(t)=A_{i}e^{-2\pi\zeta_{i}f_{i}t}\cos\left(2\pi f_{i}\sqrt{1-\zeta_{i}^{2} }t\right) \tag{7}\]  
where \(A_{i},f_{i}\), and \(\zeta_{i}\) are the amplitude, frequency and damping ratio values of the \(i\)-th mode, respectively. In general, the search of these parameters for modeling a real signal can be solved by proposing an optimization scheme. GA is one of the most employed methods to solve optimization problems, since global optimal solutions can be found [30]. A flow chart of the stages required for implementation of the GA is shown in Figure 2. In Step 1, the initial population is generated, usually randomly, to ensure a diversity of potential solutions. In Step 2, the population is evaluated in one or more objective functions in order to find the best individuals and their fitness values, which are used in the next Step 3. In this step, it is checked if the population fulfills the criterion established to end the algorithm execution. If the population cannot be considered as the solution, an additional step (Step 4) must be executed, where the genetic operators (reproduction, crossover, and mutation) are applied to obtain a new population. The reproduction employs the best individuals, which have the highest fitness value, to be part of the new population. On the other hand, the crossover operation uses the current best individuals to create new ones employing several crossover points, at the same position in the used individuals. The delimited parts are swapped to create the new individuals. A crossover probability value is used to select the order of swap. Finally, the mutation modifies specific position values according to a mutation probability value. It should be noticed that the abovementioned operators manipulate the individuals in bit manner, i.e., the operations are performed in a bit-to-bit basis. Once the new population is created, they are evaluated and tested until the stop criterion is reached. When the algorithm finalizes its execution, the final population is designated as the problem solutions (Step 5). Figure 2: Flow chart of the GA execution. GA uses many potential solutions or individuals, usually about 100, to estimate the solution [43]. This can produce a low convergence speed in the best scenario, or can cause a premature convergence or stagnation, which means that the estimated value is close enough to the solution (not an optimal solution); but its value does not change through the generations [33]. In order to lessen the abovementioned limitations, the concept of micro-GA has been presented by [44], which consists of utilizing a very small population, usually four or five individuals per generation [45]. This allows ensuring that the used time to evaluate the population is less than the required by the classical GA; further, the convergence and the diversity required are guaranteed by using the same operators than the classical GA. For this reason, micro-GA is used in this work. Figure 3 illustrates the graphical description of the micro GA-based approach proposed for performing the MPI. Employing the advantages of micro-GA, in this work, four individuals are used since they provide an adequate compromise between diversity and computational time required to find the solution. On the other hand, the design space, i.e., the values that the natural frequencies and damping ratios can take, is also set. Table 1 summarizes the range of possible values for both the natural frequency and damping ratios, which will be estimated by the proposed micro-GA scheme. It should be noticed that in these ranges the natural frequencies and damping ratios values of civil structures can be found [46]. Thus, this range will allow obtaining the optimal solutions.