Coherence Analysis for Vibration Monitoring Under High Variability Conditions: Constraints for Cultural Heritage Preventive Conservation

Abstract

The development of reliable sensor networks for vibration monitoring is essential for the preventive conservation of buildings and structures. The identification of natural frequencies is crucial both for sensor network planning, to ensure optimal placement, and for operation, to detect frequency shifts that may indicate structural damage. However, traditional frequency detection methods, such as peak picking of the Spectrum or Power Spectral Density (PSD), are highly dependent on structural and environmental conditions. In highly variable vibrational environments, such as cultural heritage sites, stadiums, and transportation hubs, these methods often prove inadequate, leading to false modal identification. This study applies coherence analysis to vibrational measurements as a more reliable alternative that overcomes the limitations of traditional frequency extraction techniques. To evaluate its effectiveness, Magnitude-Squared Coherence (MSC), Squared Cross-Spectrum (SCS), and Wavelet Coherence (WC) were tested and compared with PSD analysis. Vibrational data were collected from a sensor network deployed at the Civil Museum of Castello Ursino (Catania, Italy), a site characterized by high structural complexity and variable visitor-induced vibrations. Results demonstrate that coherence analysis surpasses the limitations of traditional frequency identification techniques, with SCS and WC outperforming MSC in distinguishing resonance frequencies and providing a more stable and reliable frequency estimation. This approach enhances sensor network design by improving frequency detection, ensuring data reliability, and supporting long-term monitoring through instrumental drift detection, thus strengthening structural health monitoring in heritage sites.

1. Introduction

Vibration monitoring consists of tracking the dynamic behavior of structures under low-amplitude environmental or anthropic excitations and is widely applied across engineering, industrial, construction, transportation, and energy sectors [1,2,3,4,5,6,7,8].

Additionally, this analysis forms the foundation of Structural Health Monitoring (SHM), which has emerged as a consolidated framework for assessing the structural integrity, performance, and long-term safety of buildings and infrastructures. Among SHM techniques, Operational Modal Analysis (OMA) has gained prominence due to its ability to extract dynamic properties—such as natural frequencies, mode shapes, and damping—through ambient vibration analysis, without requiring artificial excitation [9,10,11,12,13,14]. This approach is particularly effective for in-service monitoring, where accessibility is limited or the use of active testing methods is not feasible.

In the context of Cultural Heritage, ambient vibration monitoring has proven to be a crucial tool for preventive conservation, as it enables the non-invasive observation of dynamic responses in historic buildings, artworks, and their structural supports. As emphasized by recent studies, memorial museums are not only spaces for conveying historical narratives but also places of emotional engagement and cultural responsibility [15]. In memorial museums, preventive conservation is essential to safeguard both the physical integrity of the structure and its function in transmitting collective memory. Due to their intrinsic characteristics—such as architectural complexity, material heterogeneity, and exposure to irregular dynamic loads—historical structures present specific challenges that differ significantly from modern engineered buildings. Consequently, their monitoring requires passive and continuous strategies specifically adapted to these constraints, where vibration-based methods offer an effective and minimally intrusive means to support the management and mitigation of long-term degradation processes [16,17,18,19,20,21].

Vibration monitoring involves the use of sensors able to convert physical parameters such as acceleration, velocity, and displacement into electrical signals, which are processed to extract modal parameters, including natural frequencies, mode shapes, and damping coefficients useful for the monitoring of the dynamic behavior of a structure [22,23,24]. Studies have demonstrated the effectiveness of advanced accelerometers, including piezoelectric [25], PCB [26], MEMS [27,28,29,30,31], and triaxial accelerometers [32] in capturing high-fidelity vibration data essential for accurate condition monitoring. These sensors need an adequate setting, correct installation, and strategic placement on the structure [33,34]. In particular, the frequency response and sensitivity of the sensors, along with the performance characteristics, must be compatible with the vibration frequency range of the structures being monitored to avoid distortions and signal loss. In a monitoring system, a further critical issue arises from the varying sensitivity of sensors and their potential instrumental drift, which can complicate the identification of the natural frequencies.

Thereby, the accurate assessment of natural frequencies of a structure becomes essential not only for interpreting structural behavior but also for configuring and validating the monitoring system itself, ensuring reliable long-term data acquisition in complex and variable environments.

The identification of natural frequencies in complex systems such as historical buildings is inherently challenging due to the combined effects of material heterogeneity, irregular geometries, mixed construction phases, and dynamic loads of variable amplitude induced by visitors. These factors often lead to non-stationary vibrational responses and significant temporal and spatial variability both in the amplitude of the spectral response—in acceleration (mm/s²) and velocity (mm/s)—and in frequency values (Hz). In such conditions, PSD analysis—traditionally used to extract the natural frequencies of a structure—may prove inadequate and insufficiently accurate.

This work aims to provide a methodological advancement in the field of vibration-based monitoring for cultural heritage by proposing and validating the application of coherence functions, such as Magnitude-Squared Coherence, Squared Cross-Spectrum, and Wavelet Coherence, as a more stable and accurate alternative to conventional frequency identification techniques in highly variable environments (Figure 1).

This study is guided by three core research directions. First, it evaluates whether coherence-based vibration analysis ensures more accurate and stable frequency identification than conventional Power Spectral Density (PSD) methods when applied to historical structures subject to non-stationary and heterogeneous vibrational energy. Second, it investigates the comparative performance of different coherence metrics, with particular emphasis on the ability of Squared Cross-Spectrum (SCS) and Wavelet Coherence (WC) to detect and track natural frequencies more effectively than the commonly adopted Magnitude-Squared Coherence (MSC). Finally, it discusses how the stability and reliability of coherence analysis—especially in variable structural conditions—can provide a methodological basis for future applications in the configuration and validation of sensor networks for structural health monitoring in museum environments.

Figure 1 presents a schematic conceptual flowchart illustrating the methodological shift from PSD-based to coherence-based analysis for vibration monitoring in cultural heritage. The diagram highlights the main challenges: structural complexity, non-stationary excitations, and sensor-related issues such as positioning constraints and instrumental drift. These factors reduce the reliability of traditional PSD approaches. Coherence-based methods (MSC, SCS, WC) are introduced as more robust alternatives, offering improved stability, noise resilience, and better support for sensor configuration and long-term monitoring.

2. A Review of Methods for Modal Frequency Identification in Heritage Vibration Monitoring: From PSD to Coherence

This section critically reviews the main vibration-based techniques used for identifying natural frequencies in heritage buildings, highlighting their limitations under non-stationary conditions. In doing so, it outlines the methodological gaps that this study aims to address through the application and comparative analysis of coherence-based methods.

Vibration-based analysis techniques for assessing the natural frequencies of a structure have been developed in both the frequency and time–frequency domains [35,36,37,38,39]. The traditional frequency-domain approach, commonly known as the Basic Frequency Domain (BFD) method or peak picking technique, relies on straightforward signal processing using the Discrete Fourier Transform (DFT) and estimates the fundamental modes by identifying peaks in the Power Spectral Density (PSD) matrix [40,41]. Conventional modal identification methods have been extensively applied to historical buildings such as the Cathedral of Mallorca [42], the Monastery of Jerónimos [43], the Milan Cathedral [44,45], and the Cathedral of Orvieto [46] (Table 1).

PSD-based method is influenced by the environmental factors, primarily the energy level of vibrations, which is tied to the vibrational source, the modification that the signal undergoes along the way, and the structural characteristics of the site. Consequently, while the method relying on PSD is robust under low variability conditions of vibrational response, in conditions of structural complexity and variable energetic content, peaks can reflect the characteristics of the excitation or the response of specific sectors of a complex structure rather than the intrinsic modes. Such a non-stationary behavior is particularly common in heritage buildings, which are characterized by material and architectural heterogeneity and are continuously exposed to dynamic forces such as visitor movement, as well as nearby traffic. These excitations vary across time (e.g., between weekdays and weekends) and space (e.g., between highly frequented and isolated areas), often leading to frequency shifts that compromise the reliability of traditional frequency-domain identification methods.

Due to these limitations, it becomes essential to explore alternative approaches capable of improving frequency identification in variable environments. One such approach is the Coherence Analysis, which quantifies the correlation between vibration signals, provides a more reliable and constrained frequency estimation, enabling the identification of vibration modes and other dynamic characteristics of complex systems [47,48,49].

Although coherence-based methods have demonstrated diagnostic value in controlled laboratory settings and have been applied to large-scale civil infrastructure [50,51,52]—such as bridges, towers, and steel structures—their use in cultural heritage buildings, particularly museums, remains unexplored. Existing studies have focused on systems with relatively homogeneous geometry and materials, often under controlled or repeatable loading conditions, making their findings difficult to transfer to structurally complex, heterogeneous, and dynamically variable heritage sites (Table 1). To date, no research has systematically tested or compared coherence-based approaches in such environments, where irregular layouts, material heterogeneity, and non-stationary anthropic loads present significant challenges for vibration analysis and modal identification.

Table 1. Comparison of modal identification methods in literature case studies, summarizing structure type, analysis method, signal domain, ability to address non-stationary conditions, and contextualizing the methodological positioning of the present study.

Study/Case	Structure Type	Method Used	Domain	Non-Stationarity Addressed	Structural Complexity	Coherence Analysis
Cathedral of Mallorca [42]	Historic cathedral	PSD, peak picking	Frequency	No	Yes	No
Monastery of Jerónimos [43]	Historic monastery	PSD	Frequency	No	Yes	No
Milan Cathedral [44,45]	Historic cathedral	PSD	Frequency	No	Yes	No
Orvieto Cathedral [46]	Historic cathedral	PSD	Frequency	No	Yes	No
Bridges, towers, steel [50,51,52]	Civil infrastructure	MSC, SCS	Frequency	Yes	No	Yes
Castello Ursino [this study]	Medieval castle	PSD + MSC, SCS, WC	Frequency + Time–Frequency	Yes	Yes	Yes

Table 1. Comparison of modal identification methods in literature case studies, summarizing structure type, analysis method, signal domain, ability to address non-stationary conditions, and contextualizing the methodological positioning of the present study.

Study/Case	Structure Type	Method Used	Domain	Non-Stationarity Addressed	Structural Complexity	Coherence Analysis
Cathedral of Mallorca [42]	Historic cathedral	PSD, peak picking	Frequency	No	Yes	No
Monastery of Jerónimos [43]	Historic monastery	PSD	Frequency	No	Yes	No
Milan Cathedral [44,45]	Historic cathedral	PSD	Frequency	No	Yes	No
Orvieto Cathedral [46]	Historic cathedral	PSD	Frequency	No	Yes	No
Bridges, towers, steel [50,51,52]	Civil infrastructure	MSC, SCS	Frequency	Yes	No	Yes
Castello Ursino [this study]	Medieval castle	PSD + MSC, SCS, WC	Frequency + Time–Frequency	Yes	Yes	Yes

This study addresses this gap by applying multiple coherence functions, Magnitude-Squared Coherence (MSC), Squared Cross-Spectrum (SCS), and Wavelet Coherence (WC), to vibrational waves and comparing them with traditional PSD analysis, in the real-world context of a historic museum characterized by structural complexity and high variability in anthropic-induced vibrations. The selected site is the Castello Ursino Museum in Catania (Italy)—a historical structure characterized by architectural complexity, stratified construction phases, and material heterogeneity. Evidence from previous vibration monitoring campaigns at the Castello Ursino Museum has shown substantial variability in the vibrational response of the structure, in terms of acceleration amplitudes, spectral peaks, PSD peaks, and frequency [31]. Significant variability was observed in the Power Spectral Density (PSD) peak values, spanning over two orders of magnitude—from values around 10⁻¹ to over 10¹ (mm/s²)²/Hz. The corresponding peak frequencies varied by more than two orders of magnitude as well, ranging approximately from 10¹ Hz to slightly above 10² Hz. These dynamic conditions make the Castello Ursino a particularly challenging context in which to apply and evaluate coherence-based functions.

3. Materials and Methods

3.1. Castello Ursino Site and Survey

Castello Ursino, also known as Castello Svevo of Catania, is a historic fortress constructed between 1239 and 1250 during the reign of Frederick II. Originally built as a royal castle for the Kingdom of Sicily, its functions evolved over time from a royal residence to a barracks and even a prison. Today, the castle is used as the Civic Museum of Catania, showcasing a collection of artifacts from various historical periods and reflecting the region’s rich cultural heritage.

Architecturally, Castello Ursino is distinguished by its square plan and massive stone construction. Throughout its history, the castle underwent several phases of restructuring, during which it was divided into four levels by a secondary structural element—a mezzanine supported by a steel framework attached to the castle walls. This mezzanine is predominantly covered with wooden planks, with some areas on certain levels also featuring tiles.

Unfortunately, no additional technical details are available regarding the various phases of the castle restoration, nor are there any technical reports describing the structure from an engineering perspective. In the case study, the presence of a steel-framed mezzanine anchored to load-bearing stone masonry walls, with flooring made of wooden planks and partial tile covering, introduces a high level of complexity. The mixed nature of the system—combining modern materials with historic substrates and unknown connection details—makes it impractical to derive reliable quantitative estimates of dynamic stiffness or modal behavior based on analogy alone. However, a qualitative assessment of the castle’s structural configuration and expected dynamic behavior can be attempted, based on typological analogies with similar historic masonry buildings and case studies on comparable heritage constructions. Based on typical configurations observed in heritage buildings with flexible intermediate floors, an overall elastic response can be expected, with non-rigid in-plane behavior and locally increased stiffness in areas with composite tile coverings. The floor system is thus assumed to act as a non-rigid diaphragm, partially transmitting dynamic actions between walls [53]. With regard to modal behavior, previous studies on large masonry buildings (e.g., cathedrals and historic churches) suggest that fundamental global modes in such structures typically occur between 1 and 5 Hz, depending on geometry, boundary conditions, and mass distribution [44]. In buildings with flexible or mixed floor systems, local floor modes may emerge in the range of 10–25 Hz, with higher-frequency resonances (above 50 Hz) often attributed to stiff subcomponents or secondary metallic elements.

Vibrational measurements for Coherence Analysis were acquired in the accessible rooms on levels 2, 3, and 4 (Figure 2c–e).

In this study, the analyzed frequency range spans from 0.1 Hz to 256 Hz. The lower bound (0.1 Hz) is dictated by the instrumental characteristics. The upper bound (256 Hz) is chosen to include both frequencies relevant to engineering applications and natural vibration sources, providing a comprehensive view of the structural behavior under different conditions. In structural and civil engineering, frequencies of interest often fall below 20–30 Hz, as they are associated with the fundamental modes of buildings. However, extending the analysis up to 256 Hz allows for the detection of higher-frequency components arising from anthropic factors such as traffic, which contribute to a broader understanding of the vibrational response of the monitored site. Since the measurements were analyzed in the frequency domain, by applying the Fast Fourier Transform (FFT) as the first processing step, the instrumental acquisition frequency must be at least twice the highest frequency of interest to prevent aliasing, in accordance with the Nyquist theorem. Thus, the sampling frequency was set at 512 Hz, considering that the acquisition system operates with sampling frequencies following a binary progression (128, 256, 512, 1024 Hz).

Each acquisition lasted for 30 min, enabling a comprehensive understanding of the temporal behavior of the study site, taking into account both the presence of a predominantly stationary vibrational component (e.g., wind, distant traffic) and a more periodic or occasionally transient vibrational component (e.g., the passage of persons).

Table 2. List and specifications of measurements taken at Castello Ursino, divided by floor and rooms.

Level	Room	Measurement	Position
2nd	1	CU 1 M	On wooden mezzanine
		CU 2	On wooden mezzanine
		CU 3	On wooden mezzanine
		CU 4	On wooden mezzanine
		CU 2 M	On wooden mezzanine
		CU 3 M	On wooden mezzanine
		CU 4 M	On wooden mezzanine
	3	CU 7	Central position, on wooden mezzanine
		CU 8	Side position, on wooden mezzanine
		CU 9	External side position, on tiled mezzanine
	4	CU 10	Central position, on wooden mezzanine
		CU 11	Side position, on wooden mezzanine
3rd	1	CU 12	Side position, on wooden mezzanine
		CU 13	Central position, on tiled mezzanine
		CU 14	Side position, on wooden mezzanine
		CU 16	Side position, on tiled mezzanine
		CU 17	Side position, on tiled mezzanine
4th	2	CU 20	Central/marginal position, on wooden mezzanine
		CU 21	Central/marginal position, on wooden mezzanine

provides the list of measurements taken at Castello Ursino, divided by floor and room. The “Position” column includes the measurement location whether on the wooden mezzanine or on the mezzanine with tiles.

Raw data supporting the findings of this study are available from the corresponding author upon reasonable request.

3.2. Instruments

The acquisition of vibration data was carried out utilizing the Tromino^® sensing system (Figure 2a,b), developed by MoHo s.r.l. (Marghera, Venice–Italy) [54]. This device is a compact, portable 3D seismometer specifically designed for dynamic soil characterization, structural analysis, and ambient vibration monitoring.

The instrument supports two acquisition configurations. In its basic setup, Tromino^® includes three orthogonally oriented velocimeters. In the advanced configuration, it integrates both a tri-axial accelerometer and the velocimeters, significantly expanding its dynamic measurement range. In both setups, the gain settings of the velocimeters can be customized to adjust the device sensitivity depending on the vibration intensity. High-gain sensors are employed to capture fine microtremor signals, while low-gain settings allow the acquisition of stronger vibrations without signal saturation.

The Tromino^® sensor is critically damped and capable of acquiring ambient vibrations along the three spatial components, North–South (N-S), East–West (E-W), and vertical (Z) up to approximately ±1.5 mm/s. The instrument transmits the collected data to a low-noise digital acquisition system optimized to reduce vibration interference. This acquisition system offers a resolution exceeding 24 bits and an accuracy in velocity and acceleration surpassing 10⁻⁴ mm/s and 10⁻⁴ mm/s², respectively.

From a technical perspective, the sensor is set with a 51 mV full scale, ensuring that both ambient and moderate-intensity vibrations fall within the optimal acquisition range. The analog signals acquired by the sensors are digitized and subsequently converted into physical units through calibration factors provided by the manufacturer. Specifically, velocimeter data are converted into velocity values (mm/s) using the factor counts × 0.00002984663, while accelerometer data are translated into acceleration values (mm/s²) using the factor counts × 0.15259020000. During extraction, both amplitude and phase equalization can be applied for frequency ranges above 0.8 Hz or consistent with the natural frequencies of the instrument. The operational frequency range of the Tromino^® spans from 0.1 Hz to 300 Hz, with peak sensitivity around the eigenfrequency of the internal geophones (4.5 Hz), making it suitable for a wide variety of seismic applications.

The Tromino^® offers practical advantages in field deployment. The instrument is lightweight (1.1 kg), with compact dimensions (10 × 14 × 8 cm) and requires no external cabling. It runs on standard batteries with very low power consumption and stores the data internally, eliminating the need for a dedicated acquisition laptop or external modules. The setup consists of simply placing the instrument on the measurement surface without the need for anchoring or complex installation procedures. Levelling is achieved using integrated micrometric screws. In building environments, the sensor is equipped with short feet, which are specifically suited for hard surfaces such as concrete floors or other rigid indoor substrates, ensuring stable coupling between the instrument and the structure.

3.3. Coherence Analyses

Coherence analysis between two signals assesses their relationship and degree of correlation over time or frequency, offering insights into their synchronization or coupling. Frequencies at which the signals simultaneously exhibit high energy levels—indicating high coherence and spectral power—can reveal the natural frequencies of the system or structure.

Although the Tromino^® sensor records velocity and acceleration along all three spatial components, only the vertical acceleration component was considered for this analysis. This choice is justified by the fact that, given the sensor placement on a mezzanine, the maximum amplitude of movement predominantly occurs in this direction, exhibiting greater variability. Consequently, the data acquired along this axis contain the highest energy content under varying conditions, enabling a more comprehensive and detailed coherence analysis.

The results of coherence analysis have been compared with PSD results for each pair of analyzed signals to enhance the accuracy of significant frequency identification. By visualizing the PSD of individual signals, it is possible to compare coherence peaks with amplitude peaks shown in the PSD graph, thereby providing a more comprehensive understanding of the vibrational characteristics.

For the analysis to be valid, the two signals, for which the coherence is computed, must be acquired simultaneously to accurately measure the phase and amplitude relationships that coherence analysis aims to capture. However, signals that were not exactly aligned at the start and end of the acquisition were time-aligned and trimmed to match the overlapping time before proceeding with the analysis.

Furthermore, as a pre-processing treatment, “detrending” and “demeaning” are performed to remove trends and subtract the mean value, respectively. These operations allow the elimination of very low-frequency components and constant biases, thereby enhancing the trueness, precision, and reliability of signal processing.

3.3.1. PSD Analysis

The Power Spectral Density (PSD) quantifies how power is distributed across different frequency components of a signal by computing power per unit frequency as a function of frequency. PSD is commonly used to identify dominant frequency components and understand the energy distribution of the signal in the frequency domain. Higher PSD values indicate stronger vibrations at specific frequencies, which are often linked to resonant modes or dynamic forces influencing the system or structure.

The analysis was performed using MATLAB^® R2023a (MathWorks, Inc., Natick, MA, USA). The custom scripts used for PSD and coherence analysis are available at the website https://it.mathworks.com. The considered script in MATLAB employs the “pwelch” function, based on Welch’s overlapped segment averaging estimator [55,56].

This method reduces variance by dividing the time series into overlapping segments and applying a window function to each segment (Hamming window). Each segment of the signal is then transformed to the frequency domain using the Fast Fourier Transform (FFT). The periodogram of each segment is calculated by taking the magnitude squared of the FFT results (1).

$$P_x\left(f\right)={\displaystyle \frac{1}{N}}{\left|X\left(f\right)\right|}^2$$

(1)

In this formula, P_x(f) is the periodogram, X(f) is the FFT of the segment, and N is the number of points in the FFT.

The magnitude squared of the FFT gives the power at each frequency. The periodograms of the overlapping segments are averaged to produce the final PSD estimate. Since each segmentation provides approximately uncorrelated estimates of the PSD, averaging these estimates reduces the overall variability. Additionally, the use of overlapping segments and a window function mitigates the effects at the segment edges, further enhancing the stability and reliability of the estimates.

The choice of window length must meet the requirements for both good temporal and frequency resolution, balancing the trade-off between these two needs. To achieve high performance of detection when variations in the signal occur, a narrow window is necessary. On the other hand, to achieve good frequency resolution, a large analysis window in the time domain is required. The opposite need for time and frequency resolution is a direct consequence of the Heisenberg uncertainty principle adapted to signal analysis, often expressed through the Fourier transform. Finding an appropriate compromise depends on the specific context of the signal analysis. For this study, the window length was set to 4096 to achieve a good compromise between spectral and temporal resolution. Indeed, to calculate the frequency resolution, the sampling frequency is divided by the window length. In this case, 512/4096 results in a value of 0.125 Hz. This is a good resolution for the analyses being conducted with the Tromino^® system, since it allows investigation of frequencies beyond 0.2 Hz.

The output of the function, including the PSD estimate and the corresponding frequency vector, is plotted in a Frequency vs. PSD graph.

3.3.2. Magnitude-Squared Coherence Analysis

The Magnitude-Squared Coherence (MSC) is a measure of the correlation between two signals, x and y. In MATLAB^®, the MSC is obtained by employing the “mscohere” algorithm that calculates the magnitude-squared coherence function [57] between two signals using Welch’s overlapped averaged periodogram method [58,59]. In detail, the correlation between signals is obtained by computing the PSD, P_xx(f) and P_yy(f), and the cross-power spectral density, P_xy(f), of the signals as a function of frequency (2).

$${\mathrm{M}\mathrm{S}\mathrm{C}=\mathrm{C}}_{\mathrm{x}\mathrm{y}}\left(\mathrm{f}\right)={\displaystyle \frac{{\left|{\mathrm{P}}_{\mathrm{x}\mathrm{y}}\left(\mathrm{f}\right)\right|}^2}{{\mathrm{P}}_{\mathrm{x}\mathrm{x}}\left(\mathrm{f}\right){\mathrm{P}}_{\mathrm{y}\mathrm{y}}\left(\mathrm{f}\right)}}$$

(2)

This formula normalizes the cross-power spectral density P_xy(f) using the product of the power spectral densities P_xx(f) and P_yy(f) of the two signals. Coherence ranges from 0 to 1, where 1 indicates perfect correlation at that frequency, while 0 indicates no correlation.

The results are shown in a graph where the coherence (C_xy) is represented as a function of frequency (f).

3.3.3. Squared Cross-Spectrum (Cross-Power Spectral Density)

Squared Cross-Spectrum provides information about the common frequency components and the phase relationship between two signals. It helps identify frequencies at which the signals are correlated or synchronized.

In MATLAB^®, the Squared Cross-Spectrum (SCS) was obtained by performing the Cross-Power Spectral Density P_xy(ω) of two discrete-time signals (x and y), which quantifies the distribution of the power cross-correlation per unit frequency [58,59,60].

The Cross-Power Spectral Density P_xy(ω) is defined as follows (3):

$$\mathrm{S}\mathrm{C}\mathrm{S}=\mathrm{P}\mathrm{x}\mathrm{y}\left(\mathsf{\omega }\right)=\sum _{\mathrm{m}=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{R}}_{\mathrm{x}\mathrm{y}}{\left(\mathrm{m}\right)\mathrm{e}}^{-\mathrm{J}\mathsf{\omega }\mathrm{m}}$$

(3)

This formula expresses the discrete-time Fourier transform (DTFT) of the cross-correlation sequence R_xy(m) between the two signals x(t) and y(t), defined as follows:

$${\mathrm{R}}_{\mathrm{x}\mathrm{y}\left(\mathrm{m}\right)}=\mathrm{E}\left\{{\mathrm{x}}_{\mathrm{n}+\mathrm{m}}{\mathrm{y}}_{\mathrm{n}}^{\mathrm{*}}\right\}=\mathrm{E}\left\{{\mathrm{x}}_{\mathrm{n}}{\mathrm{y}}_{\mathrm{n}-\mathrm{m}}^{\mathrm{*}}\right\}$$

In this formula, x_n and y_n are zero-mean jointly stationary random processes, −∞ < n < ∞, −∞< m < ∞, and E {· } is the expected value operator.

In the Cross-Power Spectral Density P_xy(ω) formula, the term e^−jωm is the complex exponential term, where ω is the angular frequency and m is the time lag. However, the analysis was set to obtain the output in frequency expressed in Hz. Therefore, ω is converted to 2πf, where f is the frequency in Hz.

To observe the correlation of energy (or power) between two signals and how this correlation varies across different frequencies, the results of this analysis are plotted on a graph with frequency versus SCS.

3.3.4. Magnitude-Squared Wavelet Coherence Analysis

The Magnitude-Squared Wavelet Coherence (WC) is a measure of the correlation between signals x and y in the time–frequency plane. Wavelet coherence is particularly useful for analyzing nonstationary signals. For the Magnitude-Squared Wavelet Coherence analysis, the “wcoherence” function in MATLAB^® was employed. The coherence is computed using the analytic Morlet wavelet, a complex sinusoidal function modulated by a Gaussian envelope [61,62].

$$\mathrm{W}\mathrm{C}={\displaystyle \frac{{\left|\mathrm{S}\mathrm{}\right(\mathrm{C}\mathrm{x}\mathrm{}\left(\mathrm{a},\mathrm{}\mathrm{b}\right)\mathrm{C}\ast \mathrm{y}\mathrm{}(\mathrm{a},\mathrm{}\mathrm{b})\left)\right|}^2}{{\mathrm{S}\mathrm{}\left(\right|\mathrm{C}\mathrm{x}\mathrm{}(\mathrm{a},\mathrm{}\mathrm{b})|}^2{\left)\mathrm{}\mathrm{S}\mathrm{}\right(\left|\mathrm{C}\mathrm{y}\mathrm{}\right(\mathrm{a},\mathrm{}\mathrm{b}\left)\right|}^2)}}$$

(4)

where C_x(a,b) and C_y(a,b) denote the continuous wavelet transforms of x and y at scales a and positions b. The superscript * is the complex conjugate, and S is a smoothing operator in time and scale.

The result is a value between 0 and 1, which measures the cross-correlation between two time series as a function of frequency.

Such results are shown in a Time vs. Frequency graph where the x-axis shows time in minutes, while the y-axis displays frequency in Hertz on a logarithmic scale from 0.25 Hz to 256 Hz. The magnitude-squared coherence values across various times and frequencies are displayed in a color-coded matrix. The color bar on the right, ranging from blue to yellow, represents coherence levels from 0 to 1. Blue areas indicate low coherence (weak correlation), while yellow areas denote high coherence (strong correlation).

4. Results

The results of coherence analysis and PSD were evaluated across different measurement points at Castello Ursino, considering three levels of data acquisition. At Level 2, coherence peaks emerged at 100 Hz and 200 Hz, accompanied by notable PSD peaks at the same frequencies. The MSC and SCS results confirmed strong correlations, demonstrating that coherence analysis effectively captured the structural vibrational modes. At Level 3, the most significant coherence was detected around 15 Hz and 21 Hz, with MSC and SCS indicating stable vibrational responses. PSD peaks appeared more dispersed, further highlighting the advantage of coherence analysis in refining frequency identification. At Level 4, coherence analysis consistently revealed correlations at 14 Hz and 17 Hz, with SCS indicating strong spectral power at these frequencies. The results suggest that the mezzanine structure exhibits distinct vibrational characteristics at these frequencies.

Overall, these findings support the hypothesis that coherence-based analysis enables a more accurate and stable identification of natural frequencies compared to traditional PSD methods, particularly in a complex historical structure like Castello Ursino.

The pairs of measurements acquired simultaneously, for which coherence was computed, are listed in

Table 3. List of pairs of measurements, along with associated peak values of Power Spectral Density (PSD).

Level	Pairs of Measurements	Frequency (Hz) of the 1st PSD Peak	1st PSD Peak (mm/s2)2/Hz	Frequency (Hz) of the 2nd PSD Peak	2nd PSD Peak (mm/s2)2/Hz
2nd	CU1 CU2	13	0.70 (CU2)	100	0.23 (CU2)
2nd	CU3 CU4	100	1.16 (CU4)	11	0.34 (CU4)
2nd	CU1 CU4	100	1.20 (CU4)
2nd	CU1 CU3	100	0.10 (CU3) 0.02 (CU1)
2nd	CU2 CU3	13	0.46 (CU2)	100	0.21 (CU2) 0.09 (CU3)
2nd	CU2 CU4	100	1.20 (CU4) 0.21 (CU2)	13	0.41 (CU2)
2nd	CU7 CU 8	20	1.26 (CU7)
2nd	CU7 CU9	20	1.36 (CU7)	200	0.28 (CU9)
2nd	CU8 CU9	200	0.27 (CU9)	100	0.05 (CU8)
2nd	CU10 CU11	200	40.45 (CU11)
3rd	CU12 CU13	15	0.62 (CU13) 0.12 (CU12)
3rd	CU12 CU14	15	0.24 (CU12)	38	0.09 (CU14)
3rd	CU13 CU14	15	0.52 (CU13)	200	0.09 (CU14)
3rd	CU16 CU17	48	0.50 (CU17)	16	0.11 (CU16)
4th	CU20 CU21	17	55.10 (CU20)	15	46.87 (CU21)

and

Table 4. List of pairs of measurements, along with associated peak values of Magnitude-Squared Coherence (MSC) and Squared Cross-Spectrum (SCS). Values related to Wavelet Coherence (WC) refer to the central values of the frequency band for which maximum coherence is estimated.

Level	Pairs of Measurements	Frequency (Hz) of the 1st MSC Peak	1st MSC Peak	Frequency (Hz)of the 2nd MSC Peak	2nd MSC Peak	Frequency (Hz) of the 1st SCS Peak	1st SCS peak (mm/s2)2/Hz	Frequency (Hz) of the 2nd SCS Peak	2nd SCS Peak (mm/s2)2/Hz	WC Mean Value of the Frequency Band (Hz)
2nd	CU1 CU2	74	0.64	61	0.57	100	8.48	11	7.18	10	100
2nd	CU3 CU4	2	0.27	200	0.18	100	5 × 10−4			100
2nd	CU1 CU4	99	0.06	200	0.06	100	3 × 10−4			100
2nd	CU1 CU3	69	0.059	35	0.043	100	2.2 × 10−5			100
2nd	CU2 CU3	5	0.06	61	0.03	100	1.5 × 10−5			100
2nd	CU2 CU4	100	0.09			100	5 × 10−3			100	13
2nd	CU7 CU8	224	0.017	104	0.013	225	7.6 × 10−5	200	5.4 × 10−5	100
2nd	CU7 CU9	138	0.03	19	0.027	200	8.5 × 10−5			100
2nd	CU8 CU9	100	0.14			200	2 × 10−4			100	200
2nd	CU10 CU11	10	0.08	191	0.07	200	1 × 10−3			200
3rd	CU12 CU13	8	0.09			15	1.2 × 10−3			14	200
3rd	CU12 CU14	21	0.29			15	6.8 × 10−4	21	2.1 × 10−4	14	200
3rd	CU13 CU14	120	0.43			15	8.3 × 10−6	200	2.4 × 10−6	14	200
3rd	CU16 CU17	65	0.09	120	0.05	16	3.4 × 10−5			15
4th	CU20 CU21	14	0.20			17	51.87	14	30.23	16

, and their locations are shown in Figure 2c–e.

4.1. Coherence Analysis Results of Measurement Acquired on Level 2

At Level 2, coherence analysis highlighted significant coherence at 100 Hz and 200 Hz, with notable PSD peaks at 100 Hz for CU4 and CU9. MSC and SCS results confirmed a strong correlation at these frequencies, reinforcing the reliability of coherence-based methods in detecting vibrational modes in structurally complex settings.

In this section, among the overall measurements, we present the two most representative pairs of data: CU3 and CU4, for room 1, and CU8 and CU9, for room 3 (Figure 2a). The peak values related to these pairs of measurements are listed in Table 3 and Table 4.

Results of the Coherence Analysis of the pair CU3 and CU4 are graphically shown in Figure 3. In the PSD graph, two main peaks of 1.16 (mm/s²)²/Hz and 0.34 (mm/s²)²/Hz are observed for CU4 at 100 Hz and 11 Hz, respectively (Figure 3a). The MSC graph shows numerous peaks, with two prominent ones of 0.27 and 0.18 at 2 Hz and 200 Hz, respectively (Figure 3b). These results evidence that the coherence of the two signals occurs for frequencies that are not characterized by high spectral power. A single significant peak appears in the SCS plot at a frequency of 100 Hz, although it indicates a coherence value of 5 × 10^–4 (mm/s²)²/Hz, which is lower than that observed in the MSC plot (about 0.03 in Figure 3c). In the WC graph, high stationary coherence is observed in a band centered at 100 Hz (Figure 3d), confirming the results obtained by the PSD and the SCS analysis.

The results of the Coherence Analysis of the measurements CU8 and CU9 are graphically shown in Figure 4. In the PSD graph, peaks of 0.27 (mm/s²)²/Hz for CU9 and 0.051 (mm/s²)²/Hz for CU8 are observed at 200 Hz and 100 Hz, respectively (Figure 4a). The MSC graph shows a peak of 0.14 at 100 Hz (Figure 4b). A significant peak in the SCS graph of 2 × 10⁻⁴ (mm/s²)²/Hz is observed at a frequency of 200 Hz (Figure 4c). The WC graph confirms high coherence with significant temporal continuity in a band centered at 200 Hz and less continuity in a band centered at 100 Hz (Figure 4d).

4.2. Coherence Analysis Results of Measurement Acquired on Level 3

At Level 3, the most significant coherence was observed around 15 Hz and 21 Hz, with MSC and SCS values confirming stable vibrational responses. PSD peaks were more dispersed, particularly at 15 Hz (CU12) and 38 Hz (CU14), demonstrating how coherence analysis provides a clearer identification of structural frequencies under variable conditions.

The pair of measurements CU12 and CU14 was selected as the most representative of the data acquired in level 3, room 1. The peak values related to this pair of measurements are listed in Table 3 and Table 4.

The results of this analysis are depicted in the graphs of Figure 5. In the PSD graph, peaks of 0.24 (mm/s²)²/Hz for CU12 at 15 Hz and of 0.09 (mm/s²)²/Hz at 38 Hz for CU14 are observed (Figure 5a). The MSC graph shows a peak of 0.29 at 21 Hz (Figure 5b). The SCS graph shows a significant peak of 6.8 × 10⁻⁴ (mm/s²)²/Hz at 15 Hz and a minor peak of 2.1 × 10⁻⁴ (mm/s²)²/Hz at 21 Hz (Figure 5c). The WC graph indicates discontinuous coherence at 14 Hz and 200 Hz (Figure 5d).

4.3. Coherence Analysis Results of Measurement Acquired on Level 4

At Level 4, coherence analysis revealed consistent correlation at 14 Hz and 17 Hz, with SCS results indicating strong spectral power at these frequencies. These findings suggest that the mezzanine structure at this level exhibits distinct vibrational characteristics, which are better captured using coherence-based methods than traditional PSD analysis.

In level 4, Coherence Analysis was performed for the pair CU 20 and CU 21. The peak values related to these pairs of measurements are listed in Table 3 and Table 4. Figure 6 displays the graphs related to the coherence analyses. In the PSD graph, peaks of 55.10 (mm/s²)²/Hz for CU20 at 17 Hz and 46.87 (mm/s²)²/Hz for CU21 at 15 Hz are observed (Figure 6a). The MSC graph shows a peak of 0.20 at 14 Hz (Figure 6b). The SCS graph shows significant peaks of 51.87 (mm/s²)²/Hz at 17 Hz and 30.23 (mm/s²)²/Hz at 14 Hz (Figure 6c). The WC graph indicates weak but persistent coherence over time at 16 Hz (Figure 6d).

5. Discussion

Throughout the Coherence Analysis conducted at Castello Ursino, both the normalized coherence from the MSC analysis and the cross-correlation values from the SCS analysis show very low levels (Figure 3, Figure 4, Figure 5 and Figure 6 and Table 4). This low correlation between measurements may be attributed to the significant structural variability across different levels and rooms of the site. However, the goal of this investigation is not to assess the levels of coherence of the signals, which were anticipated to be highly variable. Instead, the study aims to use coherence analysis to more reliably determine the fundamental frequencies of the structure. This approach is intended to provide a more accurate evaluation of natural frequencies compared to conventional methods, especially in conditions of high variability where traditional techniques might yield incorrect results.

Across all measurement levels, coherence analysis provided a more consistent identification of resonance frequencies compared to PSD alone. Indeed, as expected in a site with high structural complexity and vibrational variability, the conventional application of the PSD analysis alone returns different and uncorrelated frequencies between signals (Figure 3a, Figure 4a, Figure 5a, and Figure 6a and Table 3), even when the acquisition occurs simultaneously and in the same room. Coherence Analysis, combined with PSD analysis, provides more stable and uniform frequency values, yielding consistent results within the same room characterized by similar structural features. The frequencies at which major PSD is reached, along with high coherence observed in various graphs (MSC, SCS, and WC), stand out as the system’s natural frequencies in these rooms. The results indicate that MSC and SCS successfully identified coherent vibrational modes, particularly at 100 Hz and 200 Hz (Level 2), 15 Hz and 21 Hz (Level 3), and 14 Hz and 17 Hz (Level 4) (Table 4). These findings confirm that coherence-based methods can mitigate the limitations of PSD in highly variable environments, supporting the hypothesis that coherence analysis enhances structural monitoring in complex settings.

MSC and SCS are two different methods to define the coherence between two signals. The MSC normalizes the Cross-Power Spectral Density by the PSD of the individual signals. This normalization can amplify correlation values, making peaks more pronounced. Additionally, since strong correlation can occur even for low spectral power, the MSC can show peaks at frequencies that are not significant. Consequently, the MSC graph can show numerous peaks, with similar values, across the various frequencies, making it difficult to observe the system’s natural frequencies (Figure 3b, Figure 4b, Figure 5b, and Figure 6b and Table 4). Conversely, the SCS directly represents the shared power between the two signals without normalization. This results in fewer and more distinct peaks, corresponding to frequencies where the signals have significant shared power. This method allows high coherence peaks to stand out more clearly from lower peaks, highlighting the characteristic frequencies (Figure 3c, Figure 4c, Figure 5c, and Figure 6c and Table 4). Since the objective of applying Coherence Analysis is to reliably obtain natural frequencies under conditions of high signal variability, the SCS analysis proves to be more effective in defining the resonance frequencies than the MSC one. The limited effectiveness of the MSC analysis in conditions of high variability is evidenced by the poor correspondence between the peaks observed in the MSC graph and that observed in the graphs of the other parameters (PSD, SCS, and WC). Another highly stable and effective parameter is the WC, which allows for the visualization of coherence over time, confirming that the coherence between the two signals is maintained for a given frequency (Figure 3d, Figure 4d, Figure 5d, and Figure 6d and Table 4).

The combined approach of PSD analysis with SCS and WC in coherence analysis not only provided a deeper understanding of the natural frequencies of the structure at Castello Ursino but also established a reliable method for monitoring and assessing vibrational dynamics in complex and variable environments. This study demonstrates that using coherence analysis can overcome the limitations of conventional peak detection techniques, ensuring accurate identification of natural frequencies even under high variability conditions. To better highlight the distinctive characteristics of the spectral and coherence-based methods applied in this study,

Table 5. Comparative performance of spectral and coherence-based methods.

Method	Sensitivity to Noise	Frequency Resolution	Time–Frequency Localization	Interpretability	Stability in High Variability	Main Limitations
PSD	High—strongly affected by environmental and instrumental variability	High (FFT-based)	None	Low—peaks are ambiguous in complex environments	Low—frequency peaks can vary significantly between acquisitions	Peaks may reflect excitation rather than structural modes
MSC	Medium—normalization can amplify spurious correlations	High— same resolution as PSD	None	Low—many similar peaks, difficult to interpret	Medium—not always consistent with PSD, SCS or WC	Coherence peaks may not match energy distribution
SCS	Low—filters out incoherent power naturally	High—same resolution as PSD	None	High—few clear peaks, directly related to shared power	High—matches PSD and WC at consistent frequencies	No time localization, interpretation may depend on power contrast
WC	Low—robust to signal variability	Medium—depends on wavelet scale	Yes—reveals transient coherence over time	High—easy interpretation of color maps	High—coherence is stable over time and space	Lower frequency resolution; need for appropriate use of comparative complementary metrics

provides a comparative summary of their performance across key analytical dimensions. These include sensitivity to noise, frequency resolution, time–frequency localization capability, interpretability of results, stability under high variability conditions, and known limitations. The comparison underscores the trade-offs inherent to each method, helping to clarify their respective roles in operational modal analysis, especially in structurally complex and dynamically active environments such as the Castello Ursino Museum.

Although not explored in the present study, coherence analysis offers promising potential for future applications in the design, implementation, and management of monitoring sensor networks in SHM contexts. This methodological synergy could enhance the reliability and effectiveness of structural monitoring, enabling more timely and accurate detection and prevention of structural issues and thus representing an advanced approach to preventive conservation. In the preliminary phase of setting up the sensor network, coherence analysis can support simulations and tests on various sensor configurations to ensure the optimal setup. This guarantees that the final system is optimized for the specific needs of the structure. During the design phase of the monitoring network, coherence analysis between the data acquired by calibrated commercial accelerometers and monitoring sensors can be useful for establishing the optimal placement of monitoring sensors. This analysis can help identify areas of the structure that respond consistently to vibrations, with higher coherence corresponding to maximum spectral density, ensuring that sensors are placed at the most significant points. Furthermore, to ensure optimal monitoring coverage and resolution, coherence analysis can identify areas of the structure with low coherence, indicating where a higher density of sensors is needed for reliable monitoring. During the initial installation of the monitoring network, coherence analysis can also verify that the sensors are properly calibrated. Additionally, throughout the monitoring phase, this analysis can be used to assess sensor health. Regular comparisons through coherence analysis can help detect discrepancies that may indicate anomalies or sensor malfunctions. For instance, a decrease in coherence over time could suggest sensor degradation, potentially indicating the need for maintenance or replacement.

The results of this study demonstrate the potential of coherence-based methods for improving the structural health monitoring of complex historical sites. Future research should explore the integration of coherence analysis with other SHM techniques, such as machine learning-based pattern recognition or multi-sensor fusion approaches, to further enhance the accuracy and reliability of frequency identification in cultural heritage conservation.

6. Conclusions

In addressing the outlined objectives, the study demonstrates that coherence-based vibration analysis offers a more consistent and reliable identification of natural frequencies in heritage structures compared to PSD-based methods, especially under fluctuating and complex loading conditions. By applying and comparing Magnitude-Squared Coherence (MSC), Squared Cross-Spectrum (SCS), and Wavelet Coherence (WC) in the vibrational analysis of the Castello Ursino Museum, we showed that SCS and WC outperform MSC in terms of resonance frequency discrimination and stability of results. This not only confirms the suitability of coherence functions for operational modal analysis in challenging environments but also highlights their value as a diagnostic tool for validating sensor placements and monitoring network performance.

From a theoretical perspective, this work extends the applicability of coherence analysis to the domain of cultural heritage buildings, suggesting its relevance in contexts where structural complexity and non-stationary vibrational loads limit the effectiveness of traditional frequency-domain techniques. On a practical level, the integration of SCS and WC into vibration monitoring workflows enhances the identification of stable resonance peaks across spatially distributed acquisition points, even under suboptimal signal-to-noise conditions. The combined use of these methods facilitates both the localization and temporal tracking of vibrational modes, making them well-suited for long-term preventive conservation strategies. Although not the primary focus of this study, the findings indicate that coherence-based techniques can support the assessment and optimization of sensor placement and monitoring network configuration in heritage environments, especially where conventional approaches may fall short.

Although the study highlights several methodological strengths of the adopted approach, some limitations must also be considered. The methodology was tested in a single site-specific context—the Castello Ursino Museum—characterized by unique environmental conditions and structural complexity. Data were acquired using a single type of sensor (Tromino^®), and the results may be influenced by factors such as sensor positioning, instrumental calibration, and local environmental noise. Additionally, coherence analysis depends on signal properties such as duration, spectral resolution, and synchronization across acquisition points, which may vary in other settings.

To extend the applicability of the method to a broader range of cultural heritage contexts, future research should test coherence-based techniques across buildings with diverse geometries, construction materials, and dynamic behaviors. This includes incorporating different sensor types—such as high-sensitivity accelerometers, MEMS devices, or fiber-optic systems—to evaluate their comparative performance in coherence analysis. Upon consistent validation, the proposed framework may serve as a transferable reference for vibration-based diagnostics in heritage environments worldwide, particularly those characterized by architectural complexity and high anthropic activity.