Concrete dam deformation prediction model considering the time delay of monitoring variables

Abstract

Concrete dam structures respond to various influencing factors with complex nonlinear characteristics and notable time lags. Deformation serves as a crucial monitoring metric, providing a direct indication of the structural response of these dams. An effective deformation analysis and prediction model is essential for accurately assessing the health of concrete dam structures. Current deformation prediction models have limitations in simulating time-delay effects. This study introduces time-shifted correlation coefficients and time-delayed transfer entropy to analyze the direction of information transmission and the time delays among environmental temperature, dam body temperature, and deformation monitoring variables. A methodology is proposed to determine the dimensions of temperature factors and their respective time delays. Utilizing a long short-term memory (LSTM) neural network integrated with Dropout regularization, a concrete dam deformation prediction model that accounts for the time delay effect of environmental temperature is developed. The results demonstrate that the proposed deformation prediction model offers superior fitting accuracy and predictive capability, effectively elucidating how environmental and dam body temperatures influence dam deformation.

Introduction

Concrete dams face long-term cyclic static and dynamic loads during operation, as well as threats from extreme weather and geological hazards^1–3. Accidents can result in severe impacts on downstream communities, including loss of life, property damage, and disruption to social and economic progress^4–6. Structural health monitoring is essential for managing dam safety, and developing effective methods for structural diagnosis and safety control is a key research focus in dam engineering^7,8. Deformation, a critical structural response, serves as a primary monitoring variable for dams. Establishing robust monitoring models that assess the influence of environmental factors on structural behavior and comparing predicted deformations with actual measurements are crucial for diagnosing structural health of a dam and for ensuring the dam’s safe operation^9,10.

Two widely used models for dam monitoring, based on temperature factors, are the hydrostatic-season-time (HST) model and the hydrostatic-thermal-time (HTT) model^11,12. The HST model describes the temperature component using harmonic factors, while the HTT model employs measured temperature data. When adequate temperature monitoring data are available, the HTT model can more precisely predict dam deformation by selecting suitable measured temperature factors^13,14. Mata et al.¹⁵ utilized short-time Fourier transform analysis to identify the effects of daily temperature variations on concrete dam deformation. Tatin et al.¹⁶ developed a hydrostatic-thermal-season-time (HSTT) model incorporating both air and water temperatures, enhancing thermal deformation predictions for concrete dams. Kang et al.¹⁷ proposed long-term air temperature data to simulate temperature-induced deformation and built an HTT prediction model. Yuan et al.¹⁸ established an HTT deformation model for concrete dams in cold regions by considering both air and reservoir water temperatures.

Temperature conduction in large-scale concrete structures is a slow process^19,20, creating a complex, nonlinear relationship between environmental temperature, dam body temperature, and dam deformation. This relationship involves a time lag in the response of deformation to temperature changes. Zhang et al.²¹ derived the analytical solution for the lag time of arch dam displacement and examined the factors influencing this lag time. Ren et al.²² applied a lag quantification algorithm to determine the number of lag days between environmental factors and concrete dam deformation. Wang et al.²³ introduced the hydrostatic-hysteretic-season-time (HHST) model, which accounts for hysteretic hydraulic deformation and environmental temperature effects on dam displacement. Wei et al.²⁴ developed a predictive model incorporating lag and time-varying effects of water level on dam seepage. Wang et al.²⁵ analyzed the correlation and time-lag effects between environmental factors and measured dam displacements. Kang et al.^17,26,27 considered the lag effect of internal and external concrete temperatures, utilizing long-term temperature data to build a dam deformation monitoring model.

The deformation of concrete dams exhibits a nonlinear response to various influencing factors, which traditional statistical models such as multiple linear regression (MLR) cannot adequately capture. With advances in artificial intelligence, machine learning algorithms have become increasingly popular in dam deformation analysis and prediction, enhancing the field of dam health monitoring. Notable algorithms include artificial neural networks (ANNs)²⁸, support vector machines (SVMs)²⁹, random forest regression (RFR)³⁰, radial basis function networks (RBFNs)¹⁷, and extreme learning machines (ELMs)³¹. In recent years, the LSTM model^32,33has proven effective in capturing long-term dependencies and complex nonlinear dynamics in time series, making it particularly suitable for handling data with time delays and temporal dependencies. It has been widely applied in dam deformation prediction. Machine learning models are prone to overfitting, and current methods to prevent overfitting include L2 regularization³⁴, cross-validation³⁵, and Dropout technique³⁶. Among these, Dropout reduces the model’s reliance on specific neurons by randomly ignoring some neurons during training, thereby improving the model’s generalization ability and robustness. These models have substantially improved the accuracy of predictive models. However, existing intelligent algorithm prediction models for dam deformation are unable to accurately capture the nonlinear lag relationship between temperature factors and dam deformation when considering the time-delay effect. As a result, the model’s generalization ability is limited. To address the nonlinear response and long-term dependencies between temperature and deformation sequences, we employ the long short-term memory (LSTM) model, incorporating Dropout regularization to mitigate overfitting and enhance the model’s generalization capabilities.

This paper addresses the time delay effects of environmental temperature, dam body temperature, and deformation response in concrete dams by employing time-shifted correlation coefficients and time-delayed transfer entropy to analyze the information transfer process among variables. This approach aims to thoroughly explore both linear and nonlinear causal relationships between monitoring variables. A time-delayed HTT prediction model for concrete dam deformation is developed using long-term measured environmental and dam body temperatures as key temperature factors. A regularized LSTM is utilized to establish the mathematical model. Taking a high arch dam as an example, the study analyzes the information transfer, importance ranking, and time delay among multiple measured temperature and deformation variables. The resulting LSTM-CE-TE-HTT model effectively captures both linear and nonlinear relationships between dam deformation and its influencing factors, showcasing superior deformation fitting and predictive accuracy, and addressing the current model’s shortcomings in modeling time-delay effects.

The remainder of the article is organized as follows. In Sect. Analysis of information transfer in monitoring sequences, the process of information transfer and time-delay analysis methods for monitoring variables is described. In Sect. Deformation prediction time delay HTT model for concrete dams, a deformation prediction model that accounts for the time delay effect is proposed. Numerical and experimental results are presented in Sect. Engineering case studies, and the conclusion is provided in Sect. Conclusion.

Analysis of information transfer in monitoring sequences

The temperature field of a concrete dam during operation is influenced by boundary temperatures, with environmental temperature changes altering the dam’s internal temperature field and causing thermal deformation. This dynamic can be viewed as the transfer of information between variables. Time-shifted correlation coefficients and time-delayed transfer entropy are used to quantify the impact of environmental temperature on the dam’s temperature field at various measurement points and the subsequent effect of these temperature changes on deformation monitoring variables. By analyzing the correlation and transfer entropy between time series with different time delays, the approach identifies the direction and strength of information transfer, key influencing factors, and optimal time delays. This analysis elucidates the dynamic pathway through which environmental temperature variations lead to dam deformation via shifts in the temperature field. Figure 1 shows a schematic diagram illustrating the information transfer between temperature and deformation monitoring variables.

Fig. 1.

Schematic diagram of information transfer between temperature and deformation monitoring variables.

Time-shifted correlation coefficient

Pearson’s product-moment correlation coefficient (Eq. (1)) quantifies the linear relationship between two variables x and y, with its value ranging from − 1 to 1, and it is given by.

1

It follows directly that

2

where denotes the correlation coefficient; , represents the covariance matrix, and det(·) indicates the determinant.

For two time series x(t) and y(x + τ) divided by a time delay of τ, the time-shifted correlation coefficient is defined as the Pearson’s correlation coefficient between these series, and it is given by.

3

where represents the time-shift correlation coefficient.

The time-shifted correlation coefficient assesses the linear relationship between two variables, but it is insufficient for capturing complex nonlinear causal relationships.

Time-delayed transfer entropy

Transfer entropy theory is a statistical approach designed to quantify the flow of information between two time series^37–39. For any given time delay, the general definition of transfer entropy is as follows:

4

 

where ; k + 1 represents the length of the historical time series; Δ denotes the smallest sampling interval; and τ indicates the time delay. Transfer entropy is a specific form of conditional mutual information, defined as follows:

5

 

Transfer entropy quantifies the information flow from the input sequence X to the next state of the target sequence Y, conditioned on the known historical sequence y_p of the target. Unlike the correlation coefficient, transfer entropy reflects directionality, i.e.  .

The transfer entropy algorithm is sensitive to sample size and numerical limitations, requiring a threshold to determine the significance of transfer entropy results. To evaluate statistical significance, a random shuffling resampling method generates permutation sequences, disrupting the nonlinear correlations between the original time series while preserving the mean, variance, and autocorrelation function of the original time series. This process eliminates potential associations between sequences and . Ideally, the transfer entropy of permutation sequences should be zero, but due to finite sample effects, it seldom reaches exactly zero. By conducting 100 resampling iterations to form a transfer entropy distribution , the original transfer entropy is considered significant if it falls within the extreme tails (e.g., the top 5% or bottom 95%) of this distribution.

By computing the transfer entropy T_e→T_i from the environmental temperature series T_e to the dam’s temperature field series T_i at each measurement point, and the transfer entropy T_i→δ_i to the corresponding deformation monitoring variable δ_i, the information flow between each stage can be quantitatively captured. This method identifies the causal relationships and temporal dynamics between changes in environmental temperature, temperature field response, and deformation.

The dam body temperature serves as an intermediary variable between the dam’s ambient temperature and deformation. For a Gaussian joint distribution, the multivariate transfer entropy can be expressed as follows³⁷:

6

where ⊕ denotes vector concatenation. For two row vectors and , is a 1×(n + m) vector. The symbol represents along with its p−1 delayed vectors. For a specific time delay p, the shorthand indicates the time-delayed variables. The term  signifies the covariance matrix:

7

The matrix representation of the time delays between multiple environmental temperature monitoring sequences (air and water temperatures) and the dam body temperature sequences, as well as the time delays between various environmental factors and dam body temperature sequences, is given by

8

where e denotes environmental factors, a denotes air temperature, and w_k indicates the water temperature at the k-th layer. The index i = 1,…,m corresponds to the m dam body temperature monitoring points.

The time delay matrix representing the relationship between dam body temperature monitoring and deformation monitoring is given by

9

where j = 1,…,n denotes the n dam body deformation monitoring points.

Deformation prediction time delay HTT model for concrete dams

HTT model expression

Concrete dam’s deformation comprises recoverable components resulting from hydrostatic pressure and thermal loads, alongside irreversible deformation linked to factors such as hydration heat, creep, and crack formation^40,41. Over years of operation, the temperature of dam concrete exhibits seasonal cyclic variations. The HST model typically employs harmonic functions to represent thermal deformation; however, these functions are limited to capturing thermal deformation’s long-term trends and often fail to address short-term environmental fluctuations. Conversely, the HTT model utilizes measured temperature data from the dam body and underlying bedrock to simulate thermal deformation. Nonetheless, selecting suitable temperature measurements to accurately represent the temperature field poses a challenge, as incorporating excessive temperature factors can complicate the model and result in the curse of dimensionality^42,43.

The deformation model for a concrete dam based on the HTT framework can be represented as the sum of three components: hydrostatic pressure, temperature effects, and time-dependent factors. The expression is given by

10

where  represents the deformation due to the hydrostatic pressure, denotes the deformation due to the temperature effect, indicates the deformation due to the time effect and is a constant.

This study examines the time-dependent deformation of a concrete dam, which encompasses long-term deformation trends, transient hydrostatic deformation under pressure, and short-term dynamic thermal deformation influenced by environmental and dam body temperatures. The time delay HTT model for concrete dam deformation is given by.

11

where is a constant; denotes the measured deformation at the initial modeling date; and indicate the water depths on the monitoring day and initial modeling date, respectively; m₁ denotes the fitting order, set to 3 for gravity dams and 4 for arch dams; and indicate the air temperature measurements on the monitoring day and the initial modeling date, respectively; and are readings from the i-th water temperature sensor; and and are measurements from the i-th dam body temperature sensor on the monitoring day and initial modeling date, respectively. The time delays , , and represent the delays for air, water, and dam body temperature sequences relative to the dam body deformation sequence, respectively. The expressions for and are and , where t and indicate the cumulative days from the monitoring start date to the current monitoring date and the initial modeling date, respectively. The statistical parameters are a_i, b_1i, b_2i, c₁, and c₂; s indicates water temperature sensors, and m denotes dam body temperature sensors.

Construction of the regularized LSTM model

To examine the long-term interactions between concrete dam deformation and its influencing variables, accounting for time delays in monitoring, an LSTM network is employed to develop the analysis and prediction model. The LSTM model effectively addresses the vanishing gradient issue commonly encountered during backpropagation by enhancing gradient propagation, ensuring accurate and stable predictions over long sequences. The LSTM model is primarily composed of three gate structures and memory cells^44,45 and is given by.

Forget Gate f_t:

12

where σ represents the sigmoid function; denotes the weight matrix for the forget gate; h_t−1 indicates the output of the unit at time t − 1; x_t signifies the input at time t; and b_f represents the bias term for the forget gate.

Input Gate i_t:

13

14

15

where and represent the weight coefficients for the input gate and memory cell, respectively; b_i and b_c denote the bias values for the input gate and memory cell, respectively; indicates the memory cell at the previous time step; and c_t signifies the memory cell at the current time step.

Output Gate o_t:

16

17

where represents the weight coefficient, b_o denotes the bias value for the output gate, and h_t indicates the output of the unit at time t.

Excessive temperature factors can cause overfitting in the prediction model, reducing its generalization ability. To address this, the Dropout regularization algorithm⁴⁶ is applied to prevent overfitting and to enhance the model’s generalization performance. The forward propagation of the i-th neuron between layers n and n + 1 with Dropout is given by.

18

19

where represents the random mask vector associated with the neurons in the n-th layer, where each element is either 0 or 1, following a Bernoulli distribution with a retention probability of p. The matrix indicates the weight matrix between layers n and n + 1, and b denotes the bias from the n-th layer to the n + 1-th layer.

To preserve the LSTM’s capacity to learn time dependencies, a time-consistent Dropout is employed, which applies the same Dropout mask across all time steps in the sequence. This approach ensures that the same subnetwork is utilized at each time step, thereby preventing the instability caused by random drops in neuron connections. The flowchart of the model establishment process is shown in Fig. 2.

Fig. 2.

Flowchart of the modeling process.

Engineering case studies

Project background

A high arch dam in southwestern China stands at a maximum height of 240 m. Temperature monitoring for the project includes air, reservoir water, and 42 dam body temperature monitoring points. The reservoir water temperature monitoring points offer continuous and reliable data at elevations between 1100 and 1160 m. Specifically, points TW1, TW2, and TW3 are located at 1100, 1130, and 1160 m, respectively. Of the 42 dam body temperature monitoring points, 40 are functioning normally to generate reliable measurements, while two have failed and are sealed.

The dam body has 10 vertical monitoring points, from which five representative points were selected for analysis: arch crown beam points TCN8, TCN9, and TCN10, along with the left bank point TCN3 and the right bank point TCN15. Daily monitoring data from February 1, 2011 to December 31, 2019 were used to examine the correlation between monitoring variables and the information transmission process, leading to the creation of a deformation prediction model. Figures 3 and 4, and 5 show the layout of the dam body temperature and vertical monitoring points, air/water/dam body temperature measurements, and the deformation and reservoir water level measurements, respectively.

Fig. 3.

Layout of dam body deformation monitoring points.

Fig. 4.

Temperature measurement process lines: (a) air temperature, (b) water temperature, and (c) dam body temperature.

Fig. 5.

Dam deformation and reservoir water level measurement process lines.

Analysis of monitoring variable correlation and information transfer

To mitigate the impact of hydrostatic deformation, time delay calculations concentrate on the period each year when the reservoir reaches its maximum water level, typically from late August to early January. During this time, the water level is stable or undergoes minimal fluctuations, allowing for significant variations in environmental temperature. Since dam body deformation is primarily influenced by these temperature changes, the calculation results effectively capture the characteristics of temperature variable information transfer.

The time-shift correlation coefficients and transfer entropies among environmental temperature, dam body temperature, and dam body deformation monitoring variables have been computed, along with their respective time delays. The correlation analysis and transfer entropy time delay statistics for each temperature monitoring point in relation to dam body deformation are summarized in Tables 1 and 2. To facilitate analysis, the results are organized in descending order based on the average values from the five deformation monitoring points. Furthermore, temperature monitoring points are classified into surface and internal temperature points according to their locations. Figure 6 shows the resulting heatmap, with blue points indicating surface monitoring points (including air, shallow water, and dam surface temperatures monitoring points), and red points representing internal monitoring points (including deep water and dam internal temperatures monitoring points).

Table 1.

Time delay statistics of correlation coefficients for temperature monitoring points (average correlation coefficient greater than 0.8).

Monitoring point	TCN3		TCN8		TCN9		TCN10		TCN15
	CE	TD	CE	TD	CE	TD	CE	TD	CE	TD
Te17	0.86	20	0.87	20	0.87	20	0.86	21	0.89	16
Te01	0.85	56	0.85	56	0.84	56	0.84	58	0.86	51
Te37	0.83	20	0.85	20	0.85	19	0.84	20	0.87	16
Te34	0.85	0	0.85	1	0.84	1	0.83	4	0.87	0
Te36	0.84	47	0.83	46	0.83	46	0.83	49	0.85	38
Te28A	0.83	47	0.83	45	0.83	44	0.82	48	0.86	37
Te49	0.84	42	0.83	39	0.83	38	0.82	41	0.86	30
Te16	0.84	49	0.83	49	0.83	49	0.82	52	0.85	40
Te15	0.83	42	0.83	41	0.83	40	0.81	43	0.85	35
Te02	0.80	33	0.81	29	0.81	28	0.81	31	0.83	24
Te48	0.80	30	0.81	22	0.80	22	0.79	23	0.83	18

Note CE and TD represent the correlation coefficient and time delay (in days), respectively.

Table 2.

Time delay statistics of transfer entropy for temperature monitoring points (average transfer entropy above 0.3).

Monitoring point	TCN3		TCN8		TCN9		TCN10		TCN15
	CE	TD	CE	TD	CE	TD	CE	TD	CE	TD
Te14	0.94	36	1.16	40	1.04	58	0.61	6	0.78	45
Te13	0.83	47	1.09	35	1.03	35	0.63	24	0.82	53
Ta	0.79	40	1.10	38	0.92	59	0.51	4	0.71	43
Te33	0.76	33	0.86	34	0.82	34	0.48	36	0.62	34
Te47	0.65	58	0.80	56	0.71	26	0.45	0	0.65	53
Te48	0.53	31	0.74	52	0.62	53	0.32	2	0.47	43
Te16	0.60	38	0.51	37	0.39	17	0.49	53	0.48	57
Te34	0.47	34	0.53	48	0.47	47	0.32	39	0.37	42
Tw1	0.40	48	0.27	58	0.33	39	0.36	30	0.55	60
Te36	0.28	41	0.42	29	0.36	31	0.27	29	0.43	39
Te19	0.36	38	0.29	34	0.26	54	0.37	58	0.42	52
Te01	0.27	1	0.36	37	0.32	17	0.27	33	0.41	59
Te49	0.26	4	0.29	50	0.25	52	0.28	56	0.46	50

Note CE and TD represent the correlation coefficient and time delay (in days), respectively.

Fig. 6.

Heatmap analysis of information transfer between temperature monitoring points and deformation monitoring points: (a) correlation coefficient heatmap and (b) transfer entropy heatmap.

Tables 1 and 2 show that the average time delays between temperature and deformation monitoring data range from 20 to 50 days. Moreover, the correlation coefficients and transfer entropy time delays for reservoir water level and deformation monitoring points are both zero, indicating an immediate response of deformation to water level changes. In contrast, the effect of temperature on deformation shows a prolonged time lag.

The generated heatmap clearly illustrates the distribution of transfer entropy and correlation coefficients among the 40 temperature variables and five dam body deformation variables. Darker colors in the heatmap represent higher values, signifying stronger relationships. Analysis of the heatmap reveals that surface monitoring points exhibit deeper colors for transfer entropy, indicating a greater influence on the information flow affecting dam body deformation compared to internal temperature points. However, the correlation coefficients between surface, internal, and dam body deformation points show no significant differences.

Prediction model validation

The predictive model is constructed based on the model expression in Eq. (11). It utilizes daily displacement, temperature, and water level monitoring data from March 1, 2011, to December 31, 2019. Records with missing values or outliers exceeding three standard deviations are excluded, leaving 3,002 records as the basis for modeling. The data is then divided, with 80% allocated to the training set, 10% to the validation set, and 10% to the prediction set.

To enable comparison and validation, model regression coefficients are determined using both MLR and LSTM methods. Three models are established based on different selections of the temperature factor: (1) without time delays (no time delay model, ND model), (2) considering highly correlated temperature sequences and time delays (time-shifted correlation coefficient model, CE model), and (3) considering highly correlated temperature sequences, sequences with high transfer entropy, and corresponding time delays (time-shifted correlation coefficient and time-delayed transfer entropy model, CE-TE model). The Adam algorithm is used to optimize the model’s hyperparameters. The LSTM hyperparameters to be optimized, their value ranges, and the optimization results are shown in Table 3. By inputting monitoring data and model hyperparameters, a deformation prediction model is established, outputting the regression coefficients for each deformation component and generating the dam deformation fitting and prediction results.

Table 3.

The LSTM hyperparameters to be optimized, their value ranges, and optimization results.

No.	Hyperparameters	Value range	Optimization results
1	Number of hidden layers	[1, 3]	2
2	Number of hidden units	[0,100]	98
3	Initial learning rate	[0.0001, 0.01]	0.002
4	Dropout rate	[0.1, 1]	0.3
5	Epoch	[10, 100]	100

The modeling results are shown in Figs. 7, 8 and 9. Figs (a) present the measured and fitted values for the training sets. Figs (b) show the measured and predicted values for the prediction sets. Figs (c) and (d) display the residuals for the training and prediction sets, respectively. The figures show that models incorporating both correlation coefficient and transfer entropy time delays for temperature factors align closely with the observed deformation patterns. Additionally, models accounting for time delays demonstrate a better fit to the measured deformation, with smaller residuals compared to models without time delays. The boundary conditions such as water level and external loads are primarily incorporated into the model through the deformation component due to the hydrostatic pressure and the time effect. In the specific modeling calculations, since different models use the same expressions for the hydrostatic pressure and the time effect components, the influence of boundary conditions like water level and external loads on deformation predictions is consistent. The accuracy of deformation predictions mainly depends on the combination of time-delay factors in the temperature component and the modeling methods. This suggests that temperature factors with time delays can effectively capture the complex causal relationships between temperature and deformation monitoring values.

Fig. 7.

Performance of the MLR-ND model for monitoring point TCN9: (a) fitting value accuracy, (b) prediction results, (c) fitting segment residuals, and (d) prediction segment residuals.

Fig. 8.

Performance of the MLR-CE-TE model for monitoring point TCN9: (a) fitting value accuracy, (b) prediction results, (c) fitting segment residuals, and (d) prediction segment residuals.

Fig. 9.

Performance of the LSTM-CE-TE model for monitoring point TCN9: (a) fitting value accuracy, (b) prediction results, (c) fitting segment residuals, and (d) prediction segment residuals.

To assess the fitting and prediction accuracy of the models, several statistical metrics are employed: the coefficient of determination R², mean absolute error (MAE), mean square error (MSE), and mean absolute percentage error (MAPE). The equations for these evaluation metrics are as follows:

20

21

22

23

where denotes the observed value, represents the estimated value, and indicates the mean of the observed deformations.

The statistical results presented in Table 4 indicate that the LSTM-CE-TE model significantly outperforms the LSTM-ND model in terms of prediction accuracy at monitoring point TCN8. The MAE for both fitting and prediction values decreased by 21.34% and 44.16%, respectively. Additionally, the MSE showed even greater reductions, with a decrease of 36.53% for fitting values and 70.48% for prediction values.

Table 4.

Statistical comparison of evaluation metrics for monitoring point TCN8.

Item	Training				Test
	R 2	MAE (mm)	MSE (mm2)	MAPE (%)	R 2	MAE (mm)	MSE (mm2)	MAPE (%)
LSTM-ND	0.9946	1.1938	2.2075	1.1070	0.9788	2.6198	9.5833	2.4387
LSTM-CE-TE	0.9966	0.9390	1.4011	0.8949	0.9937	1.4629	2.8294	1.4001
Performance improvement (%)	0.20	21.34	36.53	19.16	1.53	44.16	70.48	42.59

Table 5 presents the statistical results for the evaluation metrics of the MLR-CE-TE model compared to the MLR-ND model at monitoring point TCN9. The MLR-CE-TE model demonstrates significantly higher prediction accuracy than the MLR-ND model, with an MAE reduction of 77.39% for fitting values and 40.44% for prediction values. The MSE decreased by 94.03% for fitting values and by 66.01% for prediction values.

Table 5.

Statistical comparison of evaluation metrics for monitoring point TCN9.

Item	Training				Test
	R 2	MAE (mm)	MSE (mm2)	MAPE (%)	R 2	MAE (mm)	MSE (mm2)	MAPE (%)
MLR-ND	0.9891	1.4681	3.2197	1.4993	0.9738	2.5025	8.4103	2.4714
MLR-CE-TE	0.9994	0.3319	0.1921	0.3413	0.9911	1.4905	2.8589	1.6588
Performance improvement (%)	1.03	77.39	94.03	77.23	1.78	40.44	66.01	32.88

Figure 10 presents the monitoring model metrics for each monitoring point. The CE and CE-TE models exhibit coefficients of determination closer to 1 compared to the ND model, which omits time delays. Additionally, the MAE, MSE, and MAPE values for both fitting and prediction segments are lower, indicating that models incorporating time delays outperform those without. While there are no significant differences in fitting metrics between the CE and CE-TE models, the prediction metrics for the CE-TE model align more closely with those of the fitting segment, indicating the CE-TE model’s enhanced prediction performance.

Fig. 10.

Evaluation metrics for modeling performance.

Furthermore, the evaluation metrics for the fitting segment show no significant differences between the MLR-CE-TE and LSTM-CE-TE models. Except for TCN15, the other four measure point show that the LSTM-CE-TE models exhibit coefficients of determination R² closer to 1 compared to the MLR-CE-TE model. Additionally, the MAE, MSE, and MAPE values for the prediction segments are lower, indicating that the LSTM-CE-TE models outperform the MLR-CE-TE models in the prediction segment. Additionally, the evaluation metrics for both fitting and prediction segments of the LSTM-CE-TE model exhibit minimal variation, suggesting that there are no concerns regarding overfitting or underfitting.

In summary, the LSTM-CE-TE model demonstrates the highest fitting and prediction performance, making it an effective tool for achieving accurate predictions of dam deformation. The improvements in evaluation metrics achieved by the LSTM-CE-TE model demonstrates significant improvements in prediction accuracy, which directly enhances the reliability of practical applications such as early warning systems and predictive maintenance, enabling more timely and accurate risk assessments and intervention strategies.

Conclusion

This paper presents a time-delay HTT model for predicting dam deformation, incorporating the lag effects of temperature factors through time-shift correlation coefficients and transfer entropy. Utilizing measured data from a high arch dam, the study examines the time-delay relationships among deformation, environmental temperature, and dam body temperature. The proposed model effectively analyzes dam deformation behavior, showing enhanced fitting and prediction performance, and addressing the current model’s shortcomings in modeling time-delay effects. The findings offer substantial theoretical insights and practical applications for accurately predicting concrete dam deformation. The key conclusions are as follows:

1) Concrete dam deformation exhibits a notable lag effect in response to environmental and dam body temperatures. Surface temperature measurements, such as air temperature, shallow water temperature, and surface dam temperature, impact deformation more significantly than internal temperature measurements, including deep water and internal dam temperature, demonstrating stronger correlations and information transfer.

2) Incorporating the lag effect of temperature factors through time-shift correlation coefficients and time-delay transfer entropy enhances the deformation analysis and prediction models. This approach improves fitting accuracy and uncovers both linear and nonlinear relationships between dam deformation and explanatory variables, thereby clarifying how environmental and dam body temperatures affect dam deformation.

3) The regularized LSTM-CE-TE model shows the greatest prediction accuracy, demonstrating superior predictive performance while effectively addressing overfitting and enhancing the model’s generalization capabilities. This improvement in accuracy is crucial for practical applications, particularly in early warning systems and predictive maintenance.

Author contributions

Cao and Jiang drafted the core content of the article, Sheng reviewed and refined the manuscript, Yuan handled data processing and engineering case calculations, while Zhang contributed to example calculations and proofreading.

Data availability

The data used in this study are derived from monitoring records of a high arch dam, including water level, deformation, environmental temperature, and dam body temperature measurements. Due to confidentiality agreements with the reservoir management authority, the raw data cannot be shared publicly. However, processed data and analysis results that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.