From monitoring to forecasting: predictive modelling of thermal-induced displacements in heritage masonry – the Batalha Monastery case study

Conserving historic masonry structures is becoming increasingly challenging due to intensifying environmental cycles driven by climate change. Although valuable, traditional monitoring approaches are often reactive, only detecting damage after it has occurred and therefore offering limited support for preventive conservation strategies.

Temperature and humidity are increasingly recognised as active agents of structural change rather than merely background conditions. Several studies have demonstrated the importance of characterising environmental effects. For instance, Zonno et al. modelled time-delayed thermal responses in thick masonry and revealed how heat propagation affects strain over time. This characterisation of the effects of temperature enabled filtering and detrending techniques to be performed, thereby improving correlation accuracy (Zonno et al., 2019). Similarly, Bajno et al.’s research involved simulating moisture and thermal diffusion to highlight the complex interplay between environmental exposure and material degradation (Bajno et al., 2020). In the context of tropical forecasting, the Corvo et al.’s study linked pollution, humidity and temperature to limestone decay, thereby reinforcing the urgent need for climate-responsive conservation strategies (Corvo et al., 2010). Pinheiro et al.’s study further advanced this narrative by simulating future degradation under projected climate scenarios at urban and building scales (Pinheiro et al., 2024). Together, these studies emphasise that environmental actions must be explicitly considered in conservation strategies and that future climate variability may further amplify thermally induced structural effects.

In parallel, long-term monitoring has become fundamental to heritage diagnostics. This is evident in the studies of Saisi et al. and Azzara et al., who used regression and principal component analysis to correlate modal frequencies with temperature (Azzara et al., 2018; Saisi et al., 2018). These approaches have paved the way for understanding the impact of environmental factors on structural dynamics. Similarly, Masciotta et al. and Gentile et al. used tiltmeters, crack metres and temperature sensors to track seasonal variations in structural behaviour, thereby emphasising the importance of incorporating environmental variables into monitoring frameworks (Masciotta et al., 2016; Gentile et al., 2019).

More advanced approaches aim to transition from monitoring to forecasting. Studies such as that of Ni et al. demonstrate the transition from monitoring to forecasting, in which neural networks normalise modal frequencies to isolate damage-related anomalies (Ni et al., 2009). Additionally, the studies of Kita et al. and Saisi et al. introduced control charts and residual analysis as passive alert mechanisms, enabling stakeholders to respond to unexpected behaviour (Kita et al., 2019; Saisi et al., 2015). These methodologies enhance diagnostic capability; however, many remain focused on global dynamic behaviour and often assume an immediate relationship between environmental excitation and structural response. Furthermore, purely data-driven models may lack physical constraints, increasing the risk of capturing spurious correlations rather than causally admissible thermo-mechanical interactions.

Environmental sensitivity is particularly critical in masonry structures, where thermal expansion may be partially restrained by boundary conditions, producing stress redistribution rather than free deformation. In this sense, the need for climate resilience in heritage conservation emerges as a recurring theme across multiple studies. For instance, Blasi and Coisson and Gentile et al. investigated cyclic fatigue and non-linear behaviour under thermal stress, emphasising the necessity for adaptive monitoring strategies (Blasi and Coïsson, 2008; Gentile et al., 2016). Meanwhile, Masciotta et al. and Ramos et al. validated structural interventions through long-term modal tracking, demonstrating how data can inform conservation decisions (Masciotta et al., 2016; Ramos et al., 2010).

Despite these advances, several limitations persist in conventional monitoring practice. Threshold-based systems relying on predefined displacement or crack-width limits remain common. While straightforward to implement, such approaches do not explicitly account for environmental variability and may generate false alarms during seasonal thermal cycles or, conversely, mask progressive deterioration when environmental compensation is uncounted. Linear regression and principal component-based filtering techniques partially address this limitation by removing temperature-driven components from structural response signals. Nevertheless, these methods typically assume an instantaneous temperature–displacement relationship, neglecting the time-dependent heat propagation within massive masonry. As a result, delayed thermo-mechanical effects may be underestimated or misinterpreted.

Dynamic-based monitoring approaches, which correlate modal parameters with temperature variations, have proven effective for global structural assessment. However, they are less suited to capturing localised phenomena such as crack opening or joint displacement, where the behaviour is influenced by complex boundary conditions and material heterogeneity.

Despite substantial progress in environmental filtering and predictive monitoring, the explicit incorporation of physically admissible thermal lags into displacement forecasting models remains limited. Ignoring delayed thermo-mechanical response may underestimate the environmental contribution to measured displacements, distort regression parameters and reduce predictive reliability.

To address these limitations, this study proposes a predictive monitoring framework that explicitly incorporates sensor-specific and interval-specific time lags between temperature excitation and structural response, constrained by the thermal diffusivity and heat-transfer characteristics of limestone masonry. This intends to provide a more accurate separation between reversible environmental effects and potentially damage-related movements, supporting the development of robust and interpretable passive alert systems. Building on these insights, this study proposes a predictive monitoring system for the Monastery of Batalha. Using fibre optic sensing and data-driven modelling, it forecasts how the masonry will behave in response to temperature fluctuations, moving from descriptive monitoring to anticipating future structural responses.

The Monastery of Batalha constitutes one of the most important heritage buildings in Portugal, due to its historical importance and architectural value. For this reason, it has been the subject of several research studies to increase information about its structure. The more recent implementation was the introduction of a monitoring system in one of its cloisters – the King Afonso V Cloister – which comprises two identical fibre optic sensor systems to monitor temperature and displacement (Bourgeois et al., 2024). Regarding the monitored elements, FBG_1 system is monitoring the displacement of a crack (see Figure 1 – red area) and FBG_2 system is monitoring the displacement of a column joint (see Figure 1 – green area). Each fibre optic system is composed of four displacement sensors (E1, E2, E3 and E4) and one temperature sensor.

The installed fibre optic sensors are based on fibre Bragg grating technology, enabling different parameters to be monitored with high accuracy and minimal physical and visual intrusion in a single optical cable, since the fibre used is transparent silica fibre (Pereira et al., 2024). The data were monitored at approximately 30-s intervals in eight separate weeks over a period of two years. This approach enables the temperature level and displacement behaviour to be observed with high precision, which can be used for a correlation study to assess the influence of temperature on the movements of the monitored elements. Figure 2 presents the time series plot of the 2-years monitoring data for FBG_1 and FBG_2 systems. All temperature values are expressed in °C and all displacement values in mm unless otherwise stated.

Regarding the data acquisition, it is important to state that temperature and displacement measurements were recorded simultaneously within the same acquisition system, ensuring intrinsic time synchronisation between variables. Consequently, no resampling or temporal alignment procedures were required prior to analysis.

The data were used in their raw form, and no smoothing, filtering or detrending procedures were applied. This decision was intentional, as the primary objective of the study was to characterise the direct influence of measured temperature variations on structural displacements. Applying smoothing techniques could attenuate short-term thermal fluctuations and potentially obscure relevant thermo-mechanical effects, particularly in the context of lag estimation.

In cases where displacement data were unavailable due to temporary sensor malfunction, the corresponding time intervals were excluded from the regression and lag optimisation procedures. No interpolation or artificial reconstruction of missing values was performed. Only time windows containing simultaneously recorded temperature and displacement data were considered in the weekly analyses.

The data analysis aims to understand the relationship between temperature and displacement measured by four sensors (E1, E2, E3 and E4) on both the monitored elements. To perform this analysis, a three-step methodology was applied for both elements, which contemplates (1) initial correlation values – enable to see the relationship between temperature and displacement; (2) isolating the displacement – allows to visualise the behaviour of the element that is not explained by the temperature; and (3) applying optimal lag – allows to understand the influence of the material thermal inertia.

The first step consisted of applying a linear regression model to each sensor’s displacement data against the corresponding temperature measurements. This analysis produced an R² value for each sensor, quantifying the strength of the linear relationship between temperature and displacement. To aid interpretation, scatter plots were generated for both systems, displaying the raw displacement data versus temperature together with the fitted regression line (see Figure 3).

With regard to the FBG_1 system (crack behaviour), the analysis of Figure 3a reveals that the initial scatter plots indicate a general negative linear relationship between raw displacement and temperature for all four sensors, with R² values ranging from approximately 0.47 to 0.51. Examining the FBG_2 system data (joint behaviour) in Figure 3b, the initial linear regression analysis revealed varying degrees of correlation between raw displacement and temperature across the sensors. E1, E2 and E4 show a weak positive linear relationship with temperature, with R² values ranging from 0.04 to 0.28, while E3 exhibits a negative linear trend with an R² of 0.35. Based on the raw data of both systems, there is a limited to moderate linear relationship between temperature and displacement in both monitored elements.

In this step, temperature corrections were applied to the displacements by subtracting predicted displacement values from the original displacement values, based on an initial linear regression model. The aim was to isolate the displacement not explained by the immediate temperature relationship.

Plots were generated to visualise this influence, comparing the original and temperature-corrected displacement over time. Scatter plots confirming the non-influence of temperature are also presented. In these plots, values of R² = 0.000 indicate that the temperature-corrected displacement signal shows no remaining linear dependence on temperature (to three-decimal precision). This confirms that the applied correction successfully removed the dominant temperature-driven component of the displacement and that the residual motion is governed by other mechanisms.

It is important to note that the temperature correction aimed to isolate the displacement components, rather than being directly attributable to the immediate temperature effect. This could potentially reveal other drivers of displacement.

Analysing the isolated displacements on FBG_1 system (see Figure 4) is possible to observe that, in all of the sensors, the temperature has a moderate influence in the crack movements, visible by the decrease in the displacement in the plotted isolated data.

For its turn, in the FBG_2 system it is possible to see that the temperature influence is slightly higher in sensors E1, E2 and E4 and lower in sensor E4 (see Figure 5). This is observable by the plotted data oscillating near the initial measured point (y = 0).

Although the temperature correction allows the displacement component directly associated with the immediate temperature relationship to be reduced, Figures 4 and 5 show that non-negligible movements remain after this isolation process. These residual displacements suggest that additional factors may contribute to the observed structural behaviour. In historic masonry structures, movements can also be influenced by variations in humidity, differential solar exposure, mechanical interaction between structural elements and local boundary condition constraints. Furthermore, material-related phenomena such as creep or micro-adjustments within cracks and joints may affect the displacement response over time. Despite the presence of additional influencing factors, focusing on the temperature-driven component remains justified, as it accounts for a substantial portion of the short-term displacement variability observed in the monitored elements.

The thermal inertia of a material is defined as its resistance to temperature variations, that is the time required for it to heat up or cool down when subjected to external thermal loading (Longo et al., 2021). Analysis of the results presented in the previous sections revealed a moderate relationship between ambient temperature and joint displacement. Furthermore, a consistent temporal offset was observed between the temperature signal and the corresponding displacement response. This delay reflects the time required for heat to propagate within the stone and induce thermal expansion or contraction. From a physical standpoint, this delay represents the combined effect of heat transfer within the material and its subsequent mechanical response. To quantify this delay, a shift-based lag analysis was performed on the temperature and displacement time series. Similar methods have been used in recent structural monitoring studies to detect and compensate for thermal delays (Zhou et al., 2024; Ju et al., 2023).

Before performing the lag optimisation, a physically admissible lag window was defined a priori. This window was constrained by three physical and experimental considerations:

1. Causality: Temperature variations act as thermal excitation, while joint displacement acts as the mechanical response. Therefore, only non-negative lags were considered to ensure that temperature changes precede the observed displacement.

2. Thermal inertia timescale of stone masonry: Due to the finite thermal diffusivity and high thermal mass of limestone masonry (Stéphan et al., 2014), the characteristic time required for heat to propagate and induce measurable expansion or contraction is on the order of minutes to hours rather than days. Based on this physical behaviour, the lag search was restricted to sub-daily delays.

3. Measurement resolution: The monitoring system acquires data at 30-s intervals. Consequently, the lag resolution is limited to discrete multiples of the sampling interval and only integer numbers of samples can be tested.

Considering these constraints, the lag search window was defined as

which corresponds to delays between 0 and approximately 8.3 h, with a temporal resolution of 30 s.

This window was derived from the thermal diffusivity and heat-transfer timescale of limestone masonry. This ensures that the optimisation does not exploit random fluctuations in the data to artificially increase R², thus remaining consistent with realistic heat-transfer behaviour.

The lag analysis was performed independently for each sensor and for each monitoring week, to account for temporal variability in environmental conditions such as humidity, solar exposure and precipitation, which influence the stone’s effective thermal inertia and heat transfer mechanisms. The following steps were applied for a given sensor and weekly monitoring interval:

1. The ambient temperature time series, T(t), and the corresponding joint displacement series, D(t), were extracted for the selected week.

2. The temperature series was systematically shifted forward in time by a candidate lag, $τ$⁠, within the predefined physically admissible window, producing a lagged temperature series, T(t – $τ$⁠).

3. For each tested lag, $τ$⁠, a linear regression was performed between T(t – $τ$⁠) and D(t), and the coefficient of determination (R²) was computed as a measure of the strength of the temperature–displacement relationship.

4. The optimal lag $τ$_opt was defined as the lag value that maximised R² within the admissible lag window:

This enables having a sufficiently large sample size per interval and prevents the algorithm from adapting to short-term noise at the hourly level. A stability check was also performed by comparing the order of magnitude of the optimal lag across consecutive weeks and sensors. The resulting $τopt$ values remained within the same physically plausible range, confirming that the procedure identified stable and repeatable thermal response delays rather than noise-driven artefacts.

Following the lag analysis, the displacement data were plotted against the temperature time series, which had been shifted by the optimal lag value for each week. Linear regression models were then fitted to these lag-adjusted datasets to assess the temperature–displacement relationship, accounting explicitly for the thermal inertia of the material (see Figure 6).

The observed delay, which reflects the limestone thermal inertia, is not constant over time but varies between weeks. The variability observed in the lags can be attributed to environmental factors that affect the thermal and mechanical response of the stone. For example, weeks with higher humidity in the stone pores increase the effective thermal capacity, delaying expansion or contraction and resulting in longer lags. Furthermore, the variability in optimal lags was different for each sensor, what reinforces the idea that the thermal response characteristics are not uniform across all sensors. This can be associated with direct solar incidence or shade on the surface that influences the rate of heating, while precipitation or sudden changes in temperature can rapidly alter the surface response, modifying the observed delays. Thus, the optimal lags measured represent the combined effect of these factors, reinforcing the need to analyse each monitoring interval separately rather than assuming a single delay for the entire period. These findings are shown in Figure 7, where the time series plot of FBG_2 system presents data for two weeks – week 1 (February) and week 2 (July), with dates shown in (month-day) format – and the optimal lag for each sensor. Importantly, the weekly values of $τopt$ do not fluctuate arbitrarily but remain within a consistent and physically plausible range across the monitoring period, indicating that the optimisation captures a stable thermally driven delay rather than noise-dependent fitting.

Considering this in the analysis, it is possible to observe a general slight improvement of the linear relationship between the temperature and the displacement across the entire dataset for both systems (see Table 1).

In addition to the temporal variability of the optimal lags, the temperature–displacement relationships also reflect distinct mechanical behaviours associated with the type of discontinuity and the structural role of the monitored element. In this case, the monitored crack is in a wall and the monitored joints are in a column supporting an arch. This implies markedly different boundary conditions and load-transfer mechanisms. Construction joints in the column are expected to accommodate a certain degree of reversible movement induced by thermal expansion and contraction; however, their response is constrained by axial loading and interaction with the arch. Consequently, temperature-induced deformation in these joints may indicate stress redistribution rather than free thermal expansion.

By contrast, the crack observed in the wall shows a more complex response because thermal strains interact with existing damage, material heterogeneity and moisture-related effects. This difference is reflected in the variable improvement of the temperature–displacement correlation after lag correction, as visible in Table 1. While most sensors show a positive $Δ$R² after applying the optimal lag – indicating a clearer thermally driven response – some particular cases exhibit a reduction in R², as the sensor E3 in the FBG_2 system. This behaviour suggests that temperature alone is insufficient to explain the observed displacements for certain sensors, and that additional mechanisms, such as humidity-induced swelling, local restraint conditions or crack-specific kinematics, may play a dominant role.

Furthermore, the variation in $Δ$R² values across sensors on the same element reflects local differences. Even within a single wall or column, boundary constraints, material heterogeneity, microclimatic effects (e.g. solar exposure, moisture) and proximity to existing damage influence how each location responds to temperature changes. Consequently, some sensors capture primarily thermally driven displacement, while others are dominated by non-thermal effects or mechanical interactions, leading to weaker or even negative correlations. This highlights the heterogeneous nature of thermal responses in historic masonry.

From a structural assessment perspective, the predominance of modest $Δ$R² improvements and the absence of systematic divergence between lagged and raw responses suggest that the measured displacements are primarily reversible and thermally driven rather than indicative of progressive damage. Furthermore, the monitoring system’s ability to capture sensor-specific lagged responses across different structural elements enhances its reliability and sensitivity. This enables the distinction between expected environmental effects and potentially anomalous behaviour, which is relevant for long-term structural risk evaluation.

Although the initial linear analysis highlighted a moderate relationship between displacement and temperature, the variability in optimal time lags across different monitoring periods and sensors demonstrated that the temperature–displacement interaction is not instantaneous and cannot be fully captured by a simple linear model. This behaviour reflects the effect of thermal inertia, whereby material response to temperature fluctuations is delayed and modulated by external factors such as solar radiation and ambient humidity. Similar lagged effects between temperature and structural deformation have been reported in bridge and dam monitoring studies (Ju et al., 2023; Cao et al., 2025). Consequently, a more complex predictive approach was required. For that purpose, the analysis was performed over the lagged data, in order to evaluate the efficiency of the predictions in both systems (FBG_1 and FBG_2).

In terms of model training and testing, the same method was used for both systems. The dataset, comprising 906,720 data points, was divided into training and testing sets – 70% and 30%, respectively – to evaluate the models’ generalised performance. A model was trained using the training data for each sensor (E1, E2, E3 and E4), and predictions were then made using the test data. However, unlike a single global model, this lagged-data strategy consisted of training individual linear regression models for each sensor and for each week of monitoring. For every weekly window, a specific time lag was applied to the temperature data based on the previously determined optimal lag values, and the corresponding weekly model was then fitted to the time-shifted temperature and displacement data. To further characterise the behaviour of the weekly time-lagged models, the distribution of the optimal lag parameters (expressed in timesteps) identified for each sensor across all monitored weeks was analysed. For this purpose, histograms with overlaid Kernel Density Estimates were generated – represented in Figure 8 – and a Shapiro–Wilk test was applied to assess the normality of the lag distributions. The results were as follows: E1: W = 0.858, p = 0.115; E2: W = 0.832, p = 0.063; E3: W = 0.904, p = 0.314; E4: W = 0.859, p = 0.116.

For all sensors, the Shapiro–Wilk test (at a 0.05 significance level) indicated that the distributions of optimal lag values do not significantly deviate from normality. This result suggests the presence of a consistent underlying pattern in the delayed response of the monitored cracks to temperature variations, even though the specific optimal lag varies from week to week. The approximately Gaussian behaviour of the lag distributions may be of practical relevance for future model generalisation or the implementation of adaptive lag-selection strategies.

The results of the predictive model are shown in Figures 9 and 10, which represent the FBG_1 and FBG_2 systems, respectively. These figures show (1) a time series plot of predicted and raw displacements over one week and (2) scatter plots showing the relationship between raw displacement values on the y-axis and predicted displacement values on the x-axis. The R² score was calculated for the predictions on the test set to evaluate the model’s performance.

Analysing the FBG_1 system plot, the R² values (above 0.93) for all sensors demonstrate that the lagged temperature is a strong predictor of smoothed displacement, explaining over 93% of the variance. Regarding the FBG_2 system, the R² values vary for each sensor: sensor E1 has the highest R² value (0.95); sensors E2 (0.86) and E4 (0.84) also show strong R² values; and sensor E3 presents a lower but still substantial R² value (0.75), suggesting that there is more unexplained variation compared to the other sensors. This indicates that accounting for individual week and sensor lag time significantly strengthens the correlation between temperature and displacement.

Evaluating the predictive performance of the weekly time-lagged models required an approach different from conventional train-test assessment applied to a single global model. Although cross-validation is commonly applied in conventional machine learning frameworks, its implementation is not directly compatible with the present methodology. The predictive process does not rely on a single global model trained on a fixed dataset but rather on localised weekly regressions with sensor-specific lag optimisation. Random or k-fold cross-validation would disrupt the temporal continuity and physically meaningful lag structure embedded in the models.

Instead, robustness is assessed through the aggregation of predictions over the entire monitoring period. Root mean squared error (RMSE), mean absolute error (MAE) and R-squared (R²) are computed for each sensor by comparing the concatenated weekly predictions with the corresponding measured displacement data – see Table 2. This evaluation reflects model performance across multiple independent weekly calibrations and varying environmental conditions, thereby providing a realistic assessment of predictive stability under operational monitoring scenarios.

The differences in R², RMSE and MAE across sensors highlight the spatial variability of structural response. In FBG_1 (regarding the crack displacements), the high and consistent R² values suggest that displacement is largely thermally driven and uniformly captured across sensors. By contrast, in FBG_2 (regarding the displacement of the joints), R² and error metrics vary significantly, indicating that mechanical constraints, local environmental conditions and non-thermal effects influence displacement.

In general, the R² values obtained suggest the capability of the models to highly accurately predict displacements caused by temperature cycles. This predictive capability has proven particularly useful in situations where displacement data are missing. For instance, the FBG_1 system had weeks during which some of the displacement sensors did not acquire any data due to sensor breakage and ongoing replacement procedures. Nevertheless, temperature values were still acquired. The FBG_1 predictive model was then used to estimate the displacements of the crack at points where no displacement data existed, based on the available temperature dataset. The results of the predictive models are shown in Figure 11, where the predicted data are plotted against the monitored data for the entire temperature dataset. Analysing it, it is possible to observe the predicted displacements in the weeks where the sensors failed to acquire data. Moreover, these models can be applied in scenarios where future temperature cycles are known or forecasted, allowing the prediction of corresponding displacements in advance.

The intended outcome of the work was to create predictive models of temperature-driven displacements for a crack in a limestone wall and the joints of a limestone column at the Monastery of Batalha. Using two years’ worth of data from fibre optic monitoring systems implemented on site, the models were trained with the aim of better understanding and characterising the temperature–displacement relationship, with a view to supporting the future development of an alert system capable of anticipating temperature-driven movements as part of a passive monitoring strategy.

Initial analysis of the raw data revealed a moderate linear relationship between displacement and temperature. To reinforce this finding, the influence of temperature was isolated, and both the raw and isolated displacement values were plotted. This approach made it possible to visually assess the temperature influence, and by comparing the R² values from the scatter plots, it was possible to quantify this influence. Then, a time lag was introduced between temperature and displacement, and the results showed slight improvements in the linear relationship between displacement and temperature in both systems. A study of optimal lags revealed that different monitoring periods and sensors within the same system produce different optimal lags. This variability is associated with material thermal inertia, which is influenced by environmental conditions such as solar incidence and humidity.

The data analysis suggested that a simple linear model had limitations when used to predict displacement caused by temperature. Alternatively, a more complex predictive model was developed that considered the thermal inertia of the materials for each sensor and week. The lagged model produced results with high predictive power, reaching R² values of 0.99 and 0.95 for the FBG_1 and FBG_2 systems, respectively. The lag estimation approach adopted here relies on the simultaneous observation of temperature and displacement, ensuring that the identified lag reflects the effective thermo-mechanical response of the masonry in each monitoring interval. Accordingly, the resulting predictive relationships apply to periods where both datasets are available, while the extension of the model to intervals with missing displacement data will depend on the development of independent lag-inference procedures in future work. As the methodology evolves towards independent lag estimation, it has the potential to form the basis of a fully autonomous and robust passive warning system for historic masonry structures.

Despite the promising results, the limitations of this study should be acknowledged. The lag optimisation was performed on a weekly basis using monitored data. While this approach is physically motivated, it may only partially capture short-term environmental correlations rather than long-term causal mechanisms. Consequently, the performance of the prediction models should be assessed in future monitoring campaigns to ensure their practical reliability, and monitoring data should be regularly updated to keep the models current. This ongoing evaluation and data refresh is how the reliability and accuracy of the developed models are maintained over time.

Future work should focus on developing adaptive methodologies that can dynamically estimate thermal lag in response to evolving environmental and material conditions. Machine learning approaches offer significant potential in this context, as they can implicitly capture non-linear relationships and time-dependent effects without requiring pre-defined lag values. Data-driven models, such as recurrent neural networks, tree-based ensembles or hybrid physics-informed learning frameworks, could be used to create a unified predictive system that incorporates temperature, humidity, solar radiation and historical displacement patterns. These approaches could reduce the need for manual lag calibration while preserving physical interpretability. Nevertheless, incorporating additional environmental or structural factors beyond temperature often requires extra measurements and more sophisticated modelling to capture complex interactions, which can be logistically challenging and may reduce interpretability – particularly in historic masonry structures.

Although the models were specifically designed to assess the behaviour of two specific concerns within the Monastery of Batalha, this site-specific focus is a strength, enabling highly accurate monitoring that captures the unique materials, geometry and environmental conditions of these structures. While the models are not directly transferable to other buildings, the methodology itself is fully replicable and can be adapted to other historic sites by implementing personalised monitoring systems and calibrations tailored to each structure’s context.

From a practical standpoint, the proposed methodology has direct implications for heritage conservation professionals, monitoring system designers and asset managers. By explicitly accounting for thermal inertia and time-delayed responses, the approach allows environmentally driven displacements to be distinguished from potentially damage-related movements, reducing the risk of false alarms in long-term monitoring campaigns. For heritage institutions, this distinction is critical for informed decision-making, as it enables maintenance and intervention strategies to be based on structural behaviour rather than short-term environmental fluctuations. Furthermore, the sensor-specific and interval-based analysis highlights the importance of tailored monitoring system design, in which sensor placement, acquisition frequency and data interpretation strategies are adapted to the structural role and exposure conditions of each element. As such, the methodology supports the development of more reliable, interpretable and sustainable monitoring frameworks for historic masonry structures.

This work was funded by the Foundation for Science and Technology (FCT) under the PhD grant PRT/BD/152876/2021 awarded to the first author. The authors acknowledge the financial support of the Foundation for Science and Technology (FCT) through the project UIDB/04625/2025 of the research unit CERIS and through the research grant under the research project ’3D Printing for the Conservation and Recovery of Built Heritage Elements’ (2024.14274.PEX) https://doi.org/10.54499/2024.14274.PEX.