# Review of Structural Modal Tracking in Operational Modal Analysis: Methods and Applications
## 3 Method of Automatic Modal Tracking  
### Development of Modal Tracking Methods  
Despite substantial advancements in AMPE, it remains challenging to ensure that each modal estimation in long-term monitoring can exclude all spurious modes without losing any physical modes. Compounded factors, such as fluctuating data acquisition quality [36], ambient condition variations (particularly in the temperature [34, 91] and wind load [92, 93, 94]), and inherent uncertainty in modal estimation [95, 96, 97, 98], impede reliable discrimination between physical and spurious modes, potentially leading to erroneous conclusions in modal analysis. Furthermore, the absence of physical modes may lead the AMPE method to misidentify the next-order mode as the current one, thereby causing confusion in the order of modal estimation results. In the context of continuous dynamic monitoring, modal tracking serves as an essential post-processing tool to refine the temporal evolution of modal parameters [37]. Several dedicated modal tracking methods have been developed to complement AMPE in long-term monitoring scenarios, which typically utilize previously estimated modes as references to assess the validity of modes obtained from the current dataset. Consequently, modal tracking methods consist of two stages: reference mode determination and current mode linking (or new mode matching). The former is predominantly achieved through clustering algorithms, while the latter is generally characterized by a similarity criterion. Table 1 presents the criteria utilized in most studies to link current modes to reference modes. For instance, as a combined dimensionless measure of the modal distance, the following constraint form is widely used [17, 61]:  
\[d_{r,c}=d(f_{r},f_{c})+d_{r,c}^{MAC}=\frac{|f_{r}-f_{c}|}{f_{c}}+\left(1- \frac{|\phi_{r}^{*}\phi_{c}|^{2}}{(\phi_{r}^{*}\phi_{r})(\phi_{c}^{*}\phi_{c}) }\right) \tag{1}\]  
where \(d_{r,c}\) represents the distance between the reference mode and the current mode; \(d(f_{r},f_{c})\) is the relative difference in the frequency; \(d_{r,c}^{MAC}=1-\text{MAC}\); \(f_{r}\) and \(\phi_{r}\) are the frequency and mode shape of the reference mode, respectively; \(f_{c}\) and \(\phi_{c}\) are the frequency and mode shape of the current mode, respectively; and superscript \({}^{*}\) represents the conjugate transpose. The purpose of this constraint is to determine whether to link the current modes according to a predefined threshold. Nonetheless, the choice of threshold directly affects tracking performance and may not be suitable for different structures. Table 2 synthesizes the prevailing modal tracking methods, explicitly dividing the two stages: reference mode determination and current mode linking. It is noteworthy that the AMPE methods listed together in the table are presented in abbreviated form, and their specific implementation procedures may not be entirely congruent. For instance, the automatic stochastic subspace identification driven by covariance (SSI-COV) method reported by [PERSON] et al. [16] utilizes hierarchical clustering to automatically interpret stabilization diagrams, whereas the automatic SSI-COV approach adopted by [PERSON] et al. [99] relies on modified FCM clustering. Further details are available in the corresponding references. The symbol--in Table 2 indicates that the item is either not explicitly mentioned or not involved in the literature. Reference mode determination remains pivotal for robust modal tracking, with traditional strategies mainly categorized into two types based on the scope of the datasets involved. The first strategy clusters modal parameters from multiple previous response sets to establish reference modes, such as in hierarchical clustering [16; 100]. By integrating multiple sets of modes, this strategy demonstrates high robustness and has a minimal  
impact on subsequent operations when an erroneous tracking event occurs.