# Review of Structural Modal Tracking in Operational Modal Analysis: Methods and Applications
## 3 Method of Automatic Modal Tracking
### Development of Modal Tracking Methods
However, when structures undergo rapid dynamic characteristic changes induced by typhoons, temperature, or damage incidents [120], historical references become obsolete, which may overshadow current modes, leading to delays in updating critical reference modes or even tracking failure. Delayed tracking can compromise the reliability of subsequent structural damage identification and condition assessment, thereby affecting maintenance and management decisions. Successful implementation of this strategy hinges critically on determining the monitoring duration of the reference modes (i.e., the number of datasets). However, defining an optimal dataset quantity remains challenging in practical applications, and a well-established theoretical framework for this purpose is currently lacking. [PERSON] et al. [37] emphasized that a one-day monitoring test is essential for more reliable determination of reference modes and for characterizing their behavior under varying environmental conditions. Meanwhile, [PERSON] et al. [61] employed the Bayesian information criterion (BIC) to analyze the clustering results of the GMM with different operational durations, and the optimal monitoring duration for reference modes for the Sutong Bridge was determined to be two days. Specifically, the posterior probability estimation of a given observation dataset (\(D_{n}\)) for a given model (\(M_{j}\)) can be derived as follows:  
\[p\big{(}D_{n}\big{|}M_{j}\big{)}\approx L_{n}(\hat{\theta}_{j})p_{j}\big{(} \theta_{j}\big{|}M_{j}\big{)}\frac{(2\pi)^{d_{j}/2}}{n^{d_{j}/2}\big{|}I_{1} \big{(}\hat{\theta}_{j}\big{)}\big{|}^{1/2}} \tag{2}\]  
where \(L_{n}\big{(}\hat{\theta}_{j}\big{)}\) denotes the likelihood function, \(j\) represents the number of components in model \(M_{j}\), \(\theta_{j}\) denotes the model parameters, \(\hat{\theta}_{j}\) corresponds to the model parameters under the maximum a posteriori estimation condition, \(d_{j}\) is the dimensionality of \(M_{j}\), and \(I_{1}\big{(}\hat{\theta}_{j}\big{)}\) is the empirical Fisher information matrix of \(D_{n}\). The BIC is then used to estimate \(p\big{(}D_{n}\big{|}M_{j}\big{)}\), as shown in Equation (3):  
\[\text{BIC}\big{(}M_{j}\big{)}\approx\text{log}L_{n}\big{(}\hat{\theta}_{j} \big{)}-\frac{1}{2}d_{j}\cdot\text{log}\ n \tag{3}\]  
The model with the minimum BIC value is generally considered to correspond to the optimal number of reference mode datasets. The second strategy dynamically updates reference modes using the most recent modal estimation results (a single mode set) [43; 111], facilitating adaptation to abrupt mode changes. Despite improved sensitivity, its vulnerability to residual spurious modes or absent physical modes may create confusion in modal tracking. Mode-linking measures predominantly fall into two categories. The first pattern accepts current modes only when similarity criteria (e.g., the MAC and relative frequency difference) satisfy predefined thresholds [16; 121]. The tracking results of this pattern are acutely dependent on the selected threshold. A stringent threshold may preclude detection of significantly changed modes, whereas a more tolerant one permits spurious mode infiltration. Additionally, uniform thresholds across all modal orders are suboptimal due to discrepancies in frequency dispersion among different orders. The second pattern directly matches the current mode with the reference mode that demonstrates the closest similarity [61]. For example, [PERSON] et al. [111] established the similarity matrix between the current set of modal parameters with \(k\) orders and the last identified mode set with \(n\) orders as  
\[\text{S}= \begin{bmatrix}s_{11}&\dots&s_{1j}&\dots&s_{1n}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ s_{i1}&\dots&s_{ij}&\dots&s_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ s_{k1}&\dots&s_{kj}&\dots&s_{kn}\end{bmatrix}  \tag{4}\]where \(s_{ij}\) denotes the Modified Modal Observability Correlation (MMOC) between the \(i\)th mode in the current identified set and the \(j\)th mode in the last mode set (the reference mode).