## 2 Methodology of SSI-COV
### Implementation of SSI-COV  
In the implementation of the Stochastic Subspace with Covariance (SSI-COV) method, the \"shmtoolbox\" was used, a specialized MATLAB toolbox designed for structural analysis and system identification. This tool provides specific functions for applying SSI-COV, facilitating the configuration of essential parameters, such as the number of blocks and the model order, which are critical for capturing the system's dynamics. The toolbox needs the data and the sampling frequency as input information and parameters. To implement the SSI-COV method, a Power Spectral Density (PSD) matrix needs to be created. The signal processing functions in the toolbox facilitate this process automatically. Specifically, the SSI-COV method requires providing the relationship in terms of spectral energy distribution to analyze how the different acceleration signals are interrelated at the frequency component level. Additionally, SSI-COV requires the number of blocks and the model order as input parameters. In this case, 200 blocks and an order of 25 were selected to optimize the analysis in terms of resolution and precision. The choice of 200 block rows and a model order of 25 was based on the authors' experience and the specific requirements of the analyzed structure. Selecting 200 blocks ensures a higher data density in the frequency domain, which enhances the method's ability to differentiate closely spaced vibration modes. Likewise, a model order of 25 was considered sufficient to accurately capture the system's essential dynamics while avoiding overfitting or introducing unnecessary complexity. It is acknowledged that the selection of these parameters can significantly influence the accuracy and precision of modal parameter estimates. The chosen values were deemed appropriate for balancing computational efficiency and the reliability of the results, particularly given the structure's dynamic characteristics and the dataset under analysis. The importance of selecting appropriate parameters has been highlighted in the literature, as [PERSON] and [PERSON] demonstrated the influence of these parameters on the accuracy and reliability of stochastic subspace identification methods [34]. With the specified parameters for analyzing the captured vibration data, the SSI-COV method calculates covariance matrices from the data blocks and then extracts the system's dynamic characteristics using singular value decomposition techniques and clustering algorithms to identify and confirm the consistency of the estimated modes across different model order configurations [35]. Subsequently, to identify the models for all the desired orders, the process begins by iterating over a predefined range of model orders up to the maximum value. For each order, it uses subspace decomposition algorithms to process the covariance matrices derived from the signal data. These algorithms help to extract the vector spaces that represent the dominant dynamics of the system. To determine the stability of the results, a stabilization diagram was used, which facilitates the selection of the appropriate model order by comparing the dynamic properties across consecutive orders. In this process, the frequencies, relative damping, and Modal Assurance Criterion (MAC) were evaluated. The Modal Assurance Criterion (MAC) [36] is generally used as a measure of the correlation between two modal shapes. The MAC value corresponding to the \(ith\) modal shape, \(\phi_{ij}\) and \(\phi_{ij}^{*}\), is defined as follows:  
\[\text{MAC}_{i}=\frac{\left[\sum_{j=1}^{n}\phi_{ij}\phi_{ij}^{*}\right]^{2}}{\sum _{j=1}^{n}\phi_{ij}^{*}\sum_{j=1}^{n}\phi_{ij}^{*2}} \tag{13}\]  
where n is the number of elements in the modal shape vectors. A MAC value close to one indicates a perfect correlation between the two modal shapes, while a value close to zero indicates modal shapes that are orthogonal. The decision on whether a mode is stable is made by comparing the frequencies, relative damping, and Modal Assurance Criterion (MAC) of neighboring orders.