Dynamic Identification of the Sarcophagus of the Spouses by Means of Digital Video Analysis

Abstract

Artistic masterpieces are mostly collected in museums located in the center of urban areas, which are prone to heavy traffic. Traffic-induced vibrations can represent a significant hazard for museum objects, due to the repeated nature of the excitation and the brittle, pre-damaged condition of the artifacts. This is the case of the Sarcophagus of the Spouses, displayed at the National Etruscan Museum of Villa Giulia in Rome. Vibrations on the floor of the room are measured by means of velocimeters, highlighting substantial vertical amplitudes and recommending the design of an isolation system. For its design, the dynamic identification of the statue is essential, but the use of contact or laser sensors is ruled out. Therefore, a recent technique that magnifies the micromovements present in digital videos is used and the procedure is validated with respect to constructions where the dynamic identification was available in the literature. In the case of the Sarcophagus, identified frequencies are satisfactorily compared with those of a finite element model. The recognition of the dynamic characteristics shows the method’s potential while using inexpensive devices. Because costs for cultural heritage protection are usually very high, this simple and contactless dynamic identification technique represents an important step forward.

1. Introduction

Means of transportation induce vibrations that are transmitted through the relevant volume of soil of the infrastructure to adjacent buildings and their contents, including museum objects. Although the amplitude of these vibrations is much lower than in the case of violent earthquakes, traffic determines a repeated stress that can induce fatigue phenomena and cumulative damage, particularly in objects made of fragile materials that may have already deteriorated over time. From this perspective, the hazard posed by traffic-induced vibrations is expected to increase in the coming years. For instance, the “Strategies for a New Mobility Policy in Italy” [1] identify, as a national priority—especially for metropolitan areas—the implementation, enhancement, and development of an integrated rapid mass-transit system. The need to reduce pollutant emissions and optimize space usage is leading to an increasingly intensive reliance on rail-based public transport, with a corresponding increase in vibrations. These issues are, of course, shared by many other nations. In fact, the United Nations Global Sustainable Transport Conference [2] has acknowledged that the transport sector is one of the main contributors to air pollution. At the same time, local public transport in many countries remains unsustainable, unsafe, inefficient, and inaccessible due to architectural and cost-related barriers, making the transition to rapid rail-based mass transit increasingly urgent. Moreover, physical distancing measures introduced during the COVID-19 pandemic—and potentially during future pandemics—may lead to an increased frequency of trips to reduce passenger density.

Vibrations induced by railways and tram lines have been studied since the 1970s (e.g., [3,4]) within the broader framework of vehicle-induced vibrations (e.g., [5,6]). Vibrations induced by traffic in historical constructions can be substantial and are usually investigated by means of contact sensors, such as velocimeters [7,8]. In the case of statues, accelerometers may be used when contact is allowed [9]. However, in some cases, the fragile nature of the statue material and its surface forbid any contact. This is the case with the Sarcophagus of the Spouses (Figure 1A), held at the National Etruscan Museum of Villa Giulia in Rome (Figure 1B). The statue is probably a cinerary urn [10] manufactured in the 6th century BC. It depicts a married couple during a banquet, in which the woman is apparently showing a higher status than what was usual in the Greek and Roman societies of the time. The stylistic features of the faces, eyes, and lips pertain to the Ionic culture, while the proportions of the body are elongated, following the style of the archaic period. The Sarcophagus was discovered in Cerveteri in 1881 and is similar to the one preserved at the Louvre Museum in Paris.

The hollow statue is made of non-uniform thickness terracotta and is divided into four parts (busts-top, busts-base, legs-top, and legs-base), with the top portions lying on the base ones along irregular narrow contact surfaces. What is visible today is the product of a restoration intervention that reassembled about 400 fragments by means of both glue and metal bands. These bands, together with metal struts, create frame-like structures inside the statue (Figure 2). The statue is exhibited in a glass and steel case, placed in the middle of the room, whose floor is laid directly on the soil, as shown by ground-penetrating radar surveys (Figure 3).

The room is very close to a tramway line and above an underground railway (Figure 1B). In addition, the road just outside the museum is subjected to heavy car traffic all day long. These vehicles, with their engines, transmission components, and their interaction with the tracks, have specific dynamic characteristics and transmit to the ground time-varying excitations, to which archaeological structures could be sensitive [11]. Therefore, it became clear that a protection intervention was mandatory. Because an isolation of the whole building, for instance, as proposed against earthquakes [12], is not possible, an intervention limited to the case has been envisioned by the museum management. To design properly the vibration isolation system of the Sarcophagus, it is necessary to evaluate preliminarily the input induced by the traffic.

This article is a revised and expanded version of a paper entitled “Advanced Digital Video Analyses to Estimate the Dynamic Behavior for Proper Design of a Base-Isolation System of the Sarcophagus of the Spouses at the National Etruscan Museum in Rome: Preliminary Results”, which was presented at “17th World conference on seismic isolation, energy dissipation and active vibration control of structures”, Turin, Italy, 11–15 September 2022 [13]. Compared to this conference paper, the following additional investigations were performed: ground-penetrating radar testing of the museum room floor; an ultrasonic thickness gauge investigation of the steel frame of the window case; the weighing of the case and of a glass pane; the validation of the identification technique with respect to construction dynamic identification available in the literature and to a finite element model. Moreover, a second campaign of measurements was performed, and all data were reprocessed, considering different instruments and time windows, estimating also the horizontal components of floor motion. Finally, different portions of the statue were investigated here, focusing on the busts of the two figures, accounting for most of the mass.

2. Traffic Vibrations

Ambient vibrations are often used to characterize the dynamics of a structure. This free source of vibrations, thanks to its wide frequency content, allows one to extract eigen-frequencies, modal shapes, and damping. This information is needed to study the vulnerability of a structure to any kind of dynamic action. Ambient vibrations are usually low amplitude and thus excite the structure in its elastic range. However, when substantial traffic is present, larger stresses may be induced in the structure, especially in the case of heavy vehicles such as trams and/or trains.

Traffic vibrations were recorded in the museum in the recent past [14], but not in the room of the Sarcophagus. To fill this gap, on the first day of tests, four velocimeters were placed at the corners of the glass case of the statue (Figure 4A). The four instruments were triaxial electrodynamic velocimeters, each with a GPS time synchronization antenna, and were used with a sampling frequency of 200 Hz. During an acquisition time of about two hours, 27 large events were detected, among which were 12 tramway passages and 15 train passages. As many as 10 events of remarkable intensity occurred in just 17 min (Figure 4B).

To assess the intensity of traffic vibrations, international standards typically consider the following parameters: the duration of the vibration, being either short-term (occasional or transient events) or long-term (permanent or continuous events); the peak velocity, calculated as peak particle velocity or peak component particle velocity (PCPV); and the main vibration frequencies. The PCPV distribution showed comparatively small amplitudes along the horizontal directions (Figure 5A,B) and larger amplitudes in the vertical, z, direction (Figure 5C). In this case, two classes of vibration intensity were recorded: smaller than 0.4 mm/s and in the range of 0.6–1.6 mm/s. The first class is essentially related to the railway, whereas the second class is induced mainly by the tramway. In order to evaluate the consistency of peaks across different days, measurements were repeated on a second day a few months later. The results in Figure 6 were obtained, showing even more passages in an approximately two-hour-long time window and even larger vertical peaks. These peak values can be compared with the few threshold values available in the cultural heritage literature [15].

Johnson and Hannen [16] reported a 1.5 mm/s limit velocity in the building for extremely fragile and valuable objects made of a brittle and pre-cracked material. Rudenko et al. [17] suggested a 6.4 mm/s limit velocity for the floor of a museum. Wei and Dondorp [18] observed object wandering on shelves for a floor vibrating with a velocity amplitude of 1.5 or 2.0 mm/s. As for the recommended values for historical monuments, the German standard [19] suggests a 2.5 mm/s limit velocity, while the Swiss standard [20] recommends a more restrictive 1.5 mm/s limit. Thus, it is evident that some of the PCPVs recorded in the museum in the vertical direction were close or exceeded the literature and code thresholds and require some mitigation intervention on the Sarcophagus window case.

3. Finite Element Model

For a later designof a vibration isolation system, a finite element model (FEM), was implemented. Beam elements were used for the steel frames of the exhibition case, surveyed using a caliper and an ultrasonic thickness gauge (Figure 7). Resorting to a technique used also for heritage architecture [21,22,23], a structure from motion survey was performed without contact to define a detailed 3D mesh model of the external surface of the artwork [24]. Aiming to perform numerical FEM analyses, the original mesh size was reduced from 1,733,261 triangular faces to 13,310 quadrangular faces, while controlling the alteration of detailing as much as possible (Figure 8). The numerical model considered also the discontinuity between legs and busts parts of the Sarcophagus. This model, with 90,162 degrees of freedom, was imported in the FEM software LUSAS 21.1, wherein three-node linear shell elements were used. Moreover, the entire case was modelled with frame and four-node linear shell elements aiming to simulate the modal behavior of the entire system (Figure 9). Sonic tests might have estimated the Young’s modulus of the terracotta, but such contact investigations were deemed not compatible by the Museum, considering the already cracked condition of the statue. Disassembling the statue, similar to what is visible in Figure 2, might have allowed a survey of the internal surface of the Sarcophagus and hence of its thickness, but such an operation did require specialized personnel not available to the Museum at the time of this research. For the same reasons, a direct weighing of the Sarcophagus was not possible. Therefore, the terracotta thickness, Young’s modulus, and density were estimated as follows.

This last parameter was assessed by measuring the weight of the total exhibition case plus statue (Figure 10A) and that of a single glass pane (Figure 10B). Then, an average 32 mm thickness and a 2 GPa Young’s modulus for the terracotta skin of the statue were estimated by comparing the results of numerical modal analyses with the experimental dynamic identification, as further discussed in Section 6. The thickness value seems, on average, compatible with what is visible in Figure 2. The Young’s modulus assessment appears reasonable based on the literature data (

Table 1. Literature values of Young’s modulus, E, for uncracked fired clay brick.

Reference	E (GPa)
[25]	4.9
[26]	1.8
[27]	5.2
[28]	4.0
[29]	3.2

) and accounting for the cracked condition of the Sarcophagus. Finally, parametric analyses showed little sensitivity to Young’s modulus for terracotta, because vibration modes were heavily influenced by the Sarcophagus steel support, whose characteristics are reasonably known.

4. The Motion Magnification Method

The dynamic identification of monuments is an expensive activity, requiring well-trained personnel, modern equipment, and, in some cases, computationally demanding post-processing. Under special circumstances, like in the case of the Sarcophagus of the Spouses, the object of investigation is inhomogeneous, pre-damaged, fragile, and invaluable; therefore, a non-contact approach is absolutely recommended. However, the case at hand is yet more complex because even laser vibrometry was not approved by the Museum, to avoid any possible deterioration of the remaining color on the surface of the artifact.

Consequently, motion magnification (MM) was adopted, which is a video-based method. MM acts like a microscope for micro-motions present in digital videos, unveiling tiny visual patterns hardly visible to the naked eye, while saving the topology of the images [30]. MM can transform pixels directly into “virtual” sensors covering the whole object of investigation, while each pixel works as if it were a velocimeter. A key strength of this method is the simplicity of the technological approach, which only requires the recording of a (digital) video made with ordinary tools, without the use of special equipment, and therefore, it is an intrinsically contactless methodology. The 3D points of the recorded object are projected onto a 2D plane as a sequence of video frames, just like time series appropriately processed to expand them locally, while preserving the topology of the images themselves. Basically, it is an image-processing procedure that enlarges or expands certain areas of the frame related to the previous one, without altering the general structure or arrangement of its elements. In other words, it focuses on local modifications of the pixel time series to improve the visualization of specific details, while keeping intact the image as a whole. All this is performed with a reasonably short computation time, a crucial point, as it is well-known that video processing may take a lot of time and resources. MM offers many advantages: ease of use, a contactless approach, virtual sensors, reusability of the videos, handiness, intuitive results, and quantitative analysis capability. MM is today widely used in a disparate number of scientific and technological fields: dynamic identification, material properties, cultural heritage, damage detection, machine fault analysis, physiological parameter extraction, temperature signal recovery from thermal images, healthcare, visual microscopy of cells, fluid dynamics, education, sport, landslide detection, ground motion, and rock mechanics [31]. For an extensive review of the video-based methods applied to civil engineering, see [32,33].

Clearly, video-based methods are extremely practical and safe for the object under examination, but they are not as precise or accurate in measurements as the traditional methods based on acceleration or strain transducers. In the case at hand, however, the choice of MM was dictated by the conservation constraints of the artwork. Moreover, the costs and time compared to classic dynamic identification methods are significantly reduced, simplifying the identification stage. In fact, the video camera allows the analysis of any element of a structure, even the large or not easily accessible ones. At later times, it will always be possible to analyze specific points of the object by simply choosing new regions of interest (ROIs) and subjecting them to magnification. In practice, each point of the object can be used for monitoring, depending on current but also future needs. In structures, more or less significant displacements occur due to anthropic or natural vibrations; normally, the intensities of these vibrations are low, and the displacements produced are not detectable to the naked eye, which is not able to resolve small physical or color oscillations, but nevertheless, they are recorded on the video frames. Therefore, the magnification may act on them. Generally, the sources of environmental stress that affect works of art are road and rail traffic, wind, construction or maintenance work, and of course, earthquakes. The low amplitude of the stress that produces the displacement might seem to be the major problem to be faced during the video magnification process. In fact, as will be shown later in the case study of the Tower of Pisa, the real critical factor is the signal-to-noise ratio (SNR) between the signal extracted from the magnified video and the noise. While on the one hand, it is impossible to avoid the introduction of various types of noise in the process, on the other hand, one is forced to amplify the noise itself together with the useful signal. Therefore, in the absence of significant excitations, experience shows that it is necessary to reduce noise and disturbances to obtain good-quality magnification.

Although the basic idea is not completely new and other algorithms have been developed, the phase-based MM used herein was implemented at the MIT Computer Science and Artificial Intelligence Laboratory (CSAIL). Their algorithms have gained wide popularity and impressive examples, produced using high-performing cameras, and are downloadable from the CSAIL website [34]. Therein is also available the MM code, while some hardware implementations are today on the market. The CSAIL algorithm used in the following decomposes the image into phase-amplitude signals, similar to wavelets, then amplifies their phases and translates them into displacements [30,35]. An arbitrary amplification factor acts on the resulting phase and produces the magnification as well as artifacts due to noise. Even if some reducing techniques are available also for the MM elaborations, currently, noise remains a major issue. Unfortunately, the phase-based algorithm suffers from poor or artificial lighting, shadows, smoke, fog, spurious vibrations, unwanted camera vibrations, reflections, large motions, a line of sight not perpendicular to the object, poor pixel resolution, and a low frame per second (FPS) rate. In particular, the scene illumination should remain constant, as the background light could produce apparent motions. Furthermore, it was observed [36,37] that high-frequency wavelets have a narrow envelope, so their amplitude may be distorted during the magnification process, affecting the result. Nevertheless, MM was applied to large ancient constructions and to laboratory mockups [11,38], proving its effectiveness. However, given the abovementioned pitfalls, the best practices suggest making some practicability evaluations, for example, about the best ROI selection. To simplify the general procedure as much as possible, these steps were skipped, but to justify the adopted approach, validation was mandatory.

5. Validation of the Procedure

The accuracy of the displacement reconstructions that can be achieved from video recorded on the field by means of the MM procedure, as applied in the following, was validated by a comparison with the case studies of motion characteristics investigation available in the literature. (Such validation was not directly possible for the Sarcophagus due to the Museum request of avoiding any contact and laser instruments). First, following Buyukozturk et al. [39], a video of the tuned mass damper of the Taipei building 101 during a strong earthquake was processed. Only coarse data and poor resolution videos were intentionally used to test the accuracy of the procedure because the MM was already trialed by many researchers (see [33] for an example), but these benchmarks were mostly addressed using sophisticated equipment, while here a simpler setup was assumed. In Figure 11A, the red continuous curve represents the movement of the damper according to Buyukozturk et al. [39], while the superimposed multicolor dotted curve is the MM reconstructed displacement time series. Bearing in mind the low quality of starting data, the results may be considered reasonably good, at least from a qualitative point of view. To also obtain a quantitative evaluation, particularly in the frequency domain, the first three vibration modes of the Leaning Tower of Pisa, over the ROI area identified by the dotted rectangle of Figure 11B, are considered.

In this case the structure vibrations are generated by the wind and, to a lesser degree, by the pedestrian traffic, so the amplitude of the vibrations is small, and consequently, the SNR is very low, producing some spurious noisy peaks in the MM reconstructed frequency spectrum. Therefore, this is a very demanding validation test. On the other hand, contrary to the Sarcophagus case, the literature about the Leaning Tower is abundant enough to provide numerical benchmarks. After the magnification, the time series of the ROI pixels were averaged as indicated in Figure 11C, producing a single signal (last subplot of Figure 11C); then, its power spectral density (PSD) was calculated. For reasons discussed in the following, the identification was limited to the modes having a frequency approximately within one-third of the camera FPS rate. PSD graphs allowed an easy identification of the three modes (bending, vertical, and torsional), except in the 4–5 Hz range, where the second mode (not reported here) was erroneously identified at 4.14 Hz.

The numeric MM values (

Table 2. Modal frequencies of the Leaning Tower of Pisa. Comparison between the modal frequencies of the first three modes according to the literature, as in [41], and the motion magnification (MM) analysis.

Mode	Frequency (Hz)		MM Error (Hz)	MM Error (%)
	Literature	MM
1	0.96	0.84	−0.12	−12.50%
2	2.97	2.81	−0.16	−5.39%
3	6.29	6.25	−0.04	−0.64%

) were rather close to the literature measurements made during the 2012 Emilia earthquake event [41]. It is worth emphasizing that several other operational criteria could be implemented to improve the outcomes, but at the cost of greater computational complexity. The original and magnified videos of the Leaning Tower are available for download [40].

6. Application of the Motion Magnification to the Sarcophagus and Results

The case of the Sarcophagus of the Spouses is more difficult compared to previous studies. In addition to the abovementioned pitfalls, during the measurement campaign, it was not possible to remove the glass-case that produces very disturbing reflections (Figure 1A and Figure 4A). Moreover, it was the objective of this research to resort to low-cost cameras, thus limiting the resolution and FPS rate. On the other hand, the railway and the tramway, although dangerous to the statue, provided strong excitations, rebalancing the SNR and thus enabling the whole MM analysis.

The applied procedure consisted of the following main steps. First, a specific vantage point was chosen to examine a detail or a larger part of the statue. Then, the video was analyzed starting just a few seconds after the passage of a tram, and it was phase-based magnified. The magnification step was basically required to adjust the desired amplification level and the frequency range. The trade-off between a large amplification and a small image distortion is not immediate, especially if it is necessary to study many details and many frequency ranges. After the video was magnified, an ROI was selected, and the time series of pixel values were averaged, delivering a single signal, regarded just as a conventional displacement signal (as done in the example of Figure 11C), then processed as usual in vibration engineering. Note that a further advantage of averaging is the low-pass filtering effect that reduces the noise. The fourth step consists of analyzing the signal in the frequency domain, avoiding the aliasing phenomenon. Therefore, the Shannon–Nyquist condition f_s ≥ 2 f_max must be respected, where f_max is the maximum frequency of the original signal, and f_s ≡ FPS.

Caution is needed, because in the MM reconstruction, the Shannon–Nyquist limit does not always guarantee an accurate frequency identification of the magnified signal [37,38], especially in the case of too-low pixel resolution. Furthermore, the actual level of amplification may result in being lower than expected. In such circumstances, it is suggested to increase the pixel resolution, which often implies a reduction of FPS rate and a growth of the processing time [37]. As a more viable workaround, the highest acceptable frequency, with respect to the Shannon–Nyquist limit, should be reduced. This rough but simple approach facilitates the frequency domain analysis, increases the SNR and, as a direct consequence, improves the identification of the frequency peaks. In the case of the Sarcophagus, given the 30 FPS rate of the camera, it was possible to take into consideration only the low-frequency dynamics of the statue (

Table 3. Modal frequencies of the Sarcophagus. Comparison between the modal frequencies of the first four modes of the Sarcophagus of the Spouses according to the finite element model simulation (FEM) and to the motion magnification (MM).

Mode	Frequency (Hz)		MM Error (Hz)	MM Error (%)
	FEM	MM
1	6.40	6.72	+0.32	+5.00%
2	8.00	7.60	−0.40	−5.00%
3	9.36	9.91	+0.55	+5.88%
4	12.34	11.01	−1.33	−10.78%

) in the range of 0–11 Hz, which, however, was sufficient for the dynamic characterization of the statue and for the identification of the points undergoing the largest displacements.

The use of multiple synchronized cameras or high-frame-rate devices would have produced better results, but with a significant increase in costs and processing times. Currently, international research is trying to overcome disturbances due to spurious vibrations of the video camera, the presence of unwanted movements in the shooting area, and other issues, by acting directly on the MM algorithms rather than on the quality of the device electronics for various reasons, not least of which is cost-effectiveness. However, the fact remains that noise will always be present, due to characteristics of charge-coupled devices in cameras and signal attenuation during transmission [42]. In the case at hand, the working environment was indoors, but with lighting designed to achieve an artistic and evocative effect that could not be eliminated. Even replacing the museum lighting sources with special lamps and adding a black cloth at the back of the protective case, the glass still introduced reflections. It should also be considered that the 50 Hz national power grid frequency introduces noise perceived by MM processing. Even the use of polarizing filters is detrimental because it affects the pixel color interpretation of the MM algorithm.

To implement the MM analysis of the Sarcophagus, a video shooting was performed using a commercial tablet produced in 2010, recording at 30 FPS, and having a resolution of 720 × 1280 pixels. The tablet was supported by a tripod resting directly on the floor in front of the statue (Figure 12A). Videos were elaborated by a PC having a 16 GB RAM and a 2.20 GHz processor. Each magnification calculation required about 30 min for a 20 s long video. This test setup turned out to be the minimal one for the MM analysis of a vibrating structure. The original and the magnified videos of the Sarcophagus are available for download [40].

With respect to the ROI portrayed in Figure 12B, the frequencies recovered according to the MM procedure described before are presented in Figure 12C. The major peaks (those above the dotted line) in Figure 12C were considered to identify the first four modes of the statue. Other relevant peaks, such as those at 3.8 and 8.9 Hz, could have been due to the dynamic behavior of the glass case. Knowledge of the modal frequencies is very important for any object: in fact, if the object is excited by an input vibration with frequency content with a significant amount of energy near these modal frequencies, significant amplifications may occur, which might cause crack propagations and stresses exceeding the material strength. The MM modal frequencies were compared with the ones calculated with the FEM. As mentioned in the previous sections, the Sarcophagus thickness and elastic modulus were unknown and not uniformly distributed. Due to that, an average thickness and Young’s modulus were calculated by updating the model and reducing the error between the frequencies identified by means of the MM method and those numerically obtained. The MM identification reduced significantly the uncertainty about the terracotta properties and helped to calibrate the model. By considering the first four modes with a deformation mainly involving the xz plane, the frequencies in Table 3 were obtained, and, once the mentioned calibrations were performed, a reasonable agreement between FEM and MM was achieved. It is worth pointing out that only two parameters, terracotta thickness and Young’s modulus, were varied, while four frequencies were reasonably matched, thus validating the MM dynamic identification. The calibrated model may be used later within the design process of the vibration-isolation intervention.

Such an isolation intervention will need to protect the statue from everyday vertical vibrations, much larger than horizontal ones, as documented in Figure 5 and Figure 6. To be effective, the intervention will need to deliver a natural frequency of the system sufficiently shorter than that of the input measured with the velocimeters. However, the increased flexibility will need to be compatible with user experience, both in terms of device height and of the displacement of the statue. Additionally, implementing an intervention under the case shall be the occasion for improving earthquake safety by introducing a horizontal isolation. The selected solution will need to account for the small mass of the system, which will make rubber technology inapplicable for the case at hand. The horizontal isolation will also contribute to limit rolling and pitching that could occur due to vertical isolation.

An additional yet qualitative usage of MM is possible in terms of displacement due to the vibrations carried out from the pixel time series by means of standard statistic tools. Once the time history of each ROI pixel is made available, the root mean square or similar statistics may be used to evaluate the relevance of the displacements produced by the vibrations, on a point-by-point basis. In Figure 13A are shown images of the frontal view of the Sarcophagus of the Spouses magnified at different frequency ranges, color mapping the displacements identifying the most excited points. These images are compared with modal shapes delivered by FEM analysis for corresponding frequencies (Figure 13B). As expected, the lower the frequency, the larger the displacements, mostly on the busts; in particular, the cushion edges, the heads, and the hands showed largest values. FEM modal analysis provided similar results, although less apparent at the cushion edges.

Finally, large displacements of the pottery in the background of the Sarcophagus are clearly visible, especially at high frequencies (Figure 13C,D). These items are representative of the huge variety of sculptures, bronze votive offerings, and everyday objects displayed at the National Etruscan Museum and exposed to vibration excitation levels depending on their position with respect to the transportation lines and the floor within the building. In the future, it will be advisable to resort to special protective arrangements for these archaeological relics as well.

7. Conclusions

The performed velocimeter measurements in the National Etruscan Museum of Villa Giulia in Rome confirmed the direct human impression that the Sarcophagus of the Spouses, a masterwork sculpture, is exposed to by large-amplitude vertical velocities. To design an effective isolation intervention, the dynamic behavior of the statue was identified without contact, resorting to the magnification of the motion captured in digital videos. The technique was validated by comparison with the construction dynamic identifications available in the literature. Then, an estimate of the main modal frequencies under ambient vibration conditions was delivered for the Sarcophagus. These results were satisfactorily compared with those of a specifically developed finite element model of which two characteristics (terracotta thickness and Young’s modulus) were calibrated based on motion magnification, while density was assessed by weighing the whole case and a single glass pane. The updated finite element model was able to reasonably match four motion–magnification-defined frequencies, thus validating the proposed dynamic identification technique.

Nonetheless, this work presents some limitations. Some depend directly on the critical issues of the employed magnified motion algorithm. Illumination, shadows, reflections, and spurious vibrations are all elements that negatively influence the magnification process and, consequently, the accuracy of frequency estimates. Hardware devices and new magnified motion algorithms capable of counteracting, at least in part, these problems are currently available, but at a high economic cost, which is in contrast with the aim of adopting a widely accessible dynamic identification methodology. High costs are also necessary to use a high-frame-rate camera to increase the maximum identified frequency. However, this condition may change in the future, making such devices more affordable and allowing a wider investigation of frequencies. Additionally, the proposed procedure can be adopted only if the input vibrations are sufficiently intense to deliver an adequate signal-to-noise ratio. Additionally, it is worth noting that, for computational reasons, motion magnification is based on the analyses of approximately 20 s videos, thus precluding permanent monitoring, the prediction of multistep time histories, and the accounting for missing record sections, as well as heteroscedasticity in data that might be particularly relevant in extreme events [43,44,45].

Nevertheless, the motion magnification approach demonstrated to be an interesting technique to allow the dynamic identification of vulnerable cultural heritage assets, in which it is preferable to avoid the use of contact and laser sensors. The approach exposed in this work is general, flexible, and simple to be used in a wide variety of settings, including both museum artifacts and monumental constructions. The equipment, technical requirements, and procedures are affordable and reliable, although an in-depth analysis is time-intensive.