## 1 Introduction  
Structural health monitoring (SHM) is a multidisciplinary field combining sensor technology, signal processing, and statistical analysis. SHM involves continuously monitoring structural health, identifying structural damage at an early stage, assessing structural performance in real time, preventing catastrophic failure through safety warnings, and providing critical information for planning and maintenance. The SHM technique has been used in civil engineering and aviation for many years. In the late 1970s, the aerospace community began to investigate vibration-based damage identification methods [1]. The numerical analysis model was updated using modal parameters measured by comparing damaged and intact structures. The method quantified damage in terms of the change in stiffness [2]; however, applying the method in practice is inconvenient because of the considerable computation involved. Since the 1980s, the civil engineering community has continued to study vibration-based damage identification in bridges and buildings [3, 4] through changes in modal parameters such as the natural frequency, damping ratio, and vibration mode. During the same period, the oil and gas industry developed vibration-based damage identification technology and applied it to offshore oil industries [5]. Modal parameter monitoring is a widely used health monitoring method [6, 7, 8, 9, 10]. However, severe damage induced in the structure alters the global mode shapes,which may result in inaccurate damage prediction. The higher-order derivatives of the mode shapes of the structure can provide crucial information and are sensitive to local damage [11]. However, a structure requires high energy to become into its higher modes; it cannot be excited under ambient conditions. Modal parameter monitoring is a critical method to monitoring support and rotor structures in offshore wind turbine (OWT) structures [10; 12]. Modal parameter monitoring reveals the overall condition of supporting structures; therefore, local damage in support structures cannot be identified accurately. However, damage typically starts from the joints in the support structure, and the local damage expands gradually to affect the entire structure. Several reduction-expansion approaches for damage detection have been proposed. Expansion techniques have been commonly used for modal vectors to provide augmented data for correlation studies, model updating, and other applications. Conventionally, researchers have used Guyan condensation [13], dynamic condensation [14], the system equivalent reduction expansion process (SEREP) [15], and the improved reduced system [16]. These approaches have been commonly used for reduction-expansion purposes. Researchers have used these approaches to expand real-time operating data. [PERSON] and [PERSON] [17] used the SEREP to expand real-time operating data measured using accelerometers to predict the response of structures. [PERSON] et al. [18] used the SEREP to expand the displacement response on a rotating structure and then calculated the strain response of the structure. [PERSON] and [PERSON] [19] developed a novel strain expansion-reduction (SER) approach based on the strain mode shapes of a structure to accurately predict the dynamic strain. To evaluate the feasibility of damage identification approaches, frame structures were widely employed for investigation using experimental testing and Finite Element (FE) model validation. [PERSON] et al. [20] adopted three kinds of frame structures to evaluate the efficiency of their FE model-updating technique. [PERSON] et al. [21] employed a genetic algorithm to express identified stiffness reduction as the damage parameters, and experimental results were compared with the results of the optimization algorithm to verify the ability of their approach. In addition, jacket structure or substructures were adopted to evaluate the practical applications of damage identification approaches. [PERSON] et al.