High-altitude area arch dam deformation safety early warning index zoning drawing-up method

Abstract

The invention discloses a high-altitude area arch dam deformation safety early warning index zoning drawing-up method, which comprises the steps of collecting high-altitude area arch dam deformation safety multi-source monitoring data, and determining a measured value sequence composite similarity measurement index of the high-altitude area arch dam deformation safety multi-source monitoring data; performing arch dam deformation behavior clustering analysis under the driving of multi-monitoring-point monitoring data, completing arch dam deformation homogeneous feature recognition, and realizing objective partitioning of the arch dam deformation behavior in the high-altitude area; determining marginal probability distribution of measuring points in each partition, establishing a joint probability distribution function of a strong correlation measuring point group in the partition, and estimating a multi-measuring-point joint probability density function parameter; and carrying out graded early warning index drawing-up on the deformation safety of the arch dam in the high-altitude area in the multi-dimensional probability space. According to the method, the situation that a traditional single-measuring-point deformation early-warning mode and method do not fully consider the behavior relevance between the measuring points is broken through, the high-altitude area arch dam deformation safety partition multi-point combined early-warning index is drafted under the driving of multi-measuring-point monitoring data, and the deformation behavior of the high-altitude area arch dam can be judged more comprehensively; and higher robustness and higher accuracy are shown.

Description

Technical Field

[0001] This invention relates to the field of reservoir dam safety monitoring technology, and in particular to a method for determining zoning safety early warning indicators for arch dam deformation in high-altitude areas. Background Technology

[0002] A number of arch dam projects have been constructed or are planned for construction in high-altitude areas with extremely abundant hydropower resources. The long-term safe operation of these projects is not only crucial to the effectiveness of the projects but also to the sustainable economic and social development of the regions. Deformation, as the most comprehensive reflection of the structural behavior of arch dams, is of great significance for scientific monitoring in order to promptly detect potential safety hazards and signs of impending failure. Among these, deformation safety early warning indicators are one of the most important and intuitive bases for monitoring the safety status of dams and making emergency decisions.

[0003] The main task of formulating dam safety early warning indicators is to evaluate and predict the future working condition of the dam based on its historical working status, and to determine the warning values ​​reflecting the dam's working condition under unfavorable conditions. While using single-point deformation early warning indicators for arch dam deformation behavior is relatively convenient, it has limitations in predicting the overall condition of the dam, especially when the information from the measuring point is affected by internal or external uncertainties, often hindering the reasonableness and reliability of the early warning results. As a spatial shell structure, arch dams exhibit stronger correlations in the deformation behavior of their various points compared to gravity dams. Therefore, where conditions permit, multi-point joint early warning indicators should be established for strongly correlated measuring points to monitor the arch dam's deformation behavior from a spatial, overall perspective. Summary of the Invention

[0004] Purpose of the invention: In order to overcome the shortcomings of the existing technology, the present invention provides a method for zoning early warning indicators for the deformation safety of arch dams in high-altitude areas. It fully considers the spatial shell characteristics of the arch dam structure and the strong correlation characteristics of the deformation behavior of each measuring point. Based on the objective zoning of the deformation behavior of the arch dam according to multi-source monitoring data, it formulates joint early warning indicators for strongly correlated measuring point groups within the zoning, thereby supporting the monitoring of the deformation behavior of arch dams in high-altitude areas from a spatial overall perspective.

[0005] Technical Solution: To solve the above-mentioned technical problems, the present invention provides a method for determining the zoning of early warning indicators for deformation safety of arch dams in high-altitude areas, comprising the following steps:

[0006] Step 1: Collect multi-source monitoring data on the deformation safety of arch dams in high-altitude areas and determine the composite similarity index of their measurement sequences;

[0007] Step 2: Perform cluster analysis of arch dam deformation behavior driven by multi-point monitoring data to identify homogeneous characteristics of arch dam deformation and achieve objective zoning of arch dam deformation behavior in high-altitude areas.

[0008] Step 3: Determine the marginal probability distribution of the measurement points within each partition, establish the joint probability distribution function of the strongly correlated measurement point group within the partition, and estimate the parameters of the joint probability density function of multiple measurement points.

[0009] Step 4: Formulate graded early warning indicators for the deformation safety of arch dams in high-altitude areas under a multi-dimensional probability space.

[0010] Specifically, in step 1 above, a composite similarity metric consisting of multiple indicators such as "absolute distance", "incremental distance", and "growth rate distance" is used to mine the similarity and difference characteristics between the monitoring data sequences of various measuring points of arch dam deformation in high-altitude areas.

[0011] d ij (CD)=α1d ij (AD)+α2d ij (ID)+α3d ij (RGD) (1)

[0012] In the formula: d ij (CD) is a composite similarity metric between spatial measurement point i and measurement point j; d ij (AD) represents the absolute distance, indicating the difference in the magnitude of the measured values ​​between measuring point i and measuring point j; d ij (ID) represents the incremental distance, indicating the difference in the magnitude of change between the measured values ​​at measuring point i and measuring point j; d ij (RGD) represents the growth rate distance, indicating the difference in the rate of change of the measured values ​​between measuring point i and measuring point j; α1, α2, and α3 are the weight coefficients of the three similarity distance indicators, and satisfy α1+α2+α3=1.

[0013] The absolute distance d mentioned above ij (AD), Incremental Distance d ij (ID) and relative growth rate distance d ij The specific formula for calculating (RGD) is as follows:

[0014]

[0015] In the formula: x it x is the deformation amount. it =δ it ;y it For deformation amplification, y it =δ it -δ i,t-1 ;z it For the deformation rate, δ it and δ i,t-1 Let x and y represent the deformation values ​​of the i-th measuring point at time t and time t-1, respectively; Z() represents the original index value x after Z-score standardization.it y it z it Functions that undergo standardization, such as x it For example, perform standardized operations as follows:

[0016]

[0017] Where: μ x With σ x For x it The mean and variance of.

[0018] The CRITIC method is used, and the following steps are followed to determine the weight coefficients α1, α2, and α3 of the three similarity distance indicators in the above composite similarity measurement index, in order to overcome the limitations of subjective determination.

[0019] ① Constructing the decision matrix:

[0020]

[0021] In the formula: This represents the "absolute distance", "incremental distance", and "incremental distance" between the measurement point pair (a1, b1), and so on.

[0022] ②Standardized matrix:

[0023]

[0024] In the formula: d ab (i,j) represents the value in the i-th row and j-th column of the original decision matrix; min(d(j)) represents the minimum value in the j-th column; and max(d(j)) represents the maximum value in the j-th column.

[0025] ③ Calculate the correlation coefficient matrix:

[0026]

[0027] In the formula: ρ jk r is the Pearson correlation coefficient between standard j and standard k. ij ,r ik Let be the values ​​of the i-th row, j-th column, and k-th column in the standardized matrix; is the mean of the j-th and k-th columns; m is the number of measurement point pairs.

[0028] ④ Calculate the information content of each criterion:

[0029]

[0030] In the formula: C j σ is the weight of the j-th criterion; j ρ is the standard deviation of the j-th standard;jk It is the correlation coefficient between standard j and standard k.

[0031] ⑤ Calculate the normalized weights:

[0032]

[0033] In step 2 above, the composite similarity metric index d is used based on the arch dam deformation measurement sequence. ij (CD) integrates Density Peak Clustering (DPC), Fuzzy C-Means Clustering (FCM), and Pelican Optimization (POA) algorithms to perform hybrid clustering analysis on arch dam deformation monitoring data in high-altitude areas, thereby determining the zoning of arch dam deformation monitoring points in high-altitude areas. Details are as follows:

[0034] First, the density peak clustering algorithm is applied to determine the initial cluster centers. For each data point, two key parameters are defined: local density ρ and relative distance δ. i The mathematical equation for local density is as follows:

[0035]

[0036] In the formula: d ij For x i With x j The distance between them, i.e., the aforementioned composite similarity index d ij (CD); d c To cut off the distance.

[0037] relative distance δ i Represents data point x i The minimum distance between x and data points with higher local density. i When the local density is at its maximum, δ i The value is the data point x. i The maximum distance to other data points is expressed as follows:

[0038]

[0039] Cutoff distance d c The choice of local density ρ i The calculation has a significant impact, and unreasonable d c This can lead to poor clustering results. To avoid this, due to d... c The adverse effects of improper selection can be addressed using the following information entropy H. p Measure each cutoff distance d c The degree of disorder corresponding to the local density estimate:

[0040]

[0041] From the above equation, we can see that the function Hp It is d c The convex function has a maximum density entropy when the local density ρ values ​​of all sample points are approximately equal. This indicates that the uncertainty of the local density estimate is relatively large at this point, which is detrimental to the stability of the clustering results. Conversely, the greater the difference in the local density ρ values ​​among the sample points, the smaller the density entropy, and the more favorable the results are for identifying the clustering pattern. Therefore, the cutoff distance d c The selection of H can be viewed as the density entropy function. p The optimization problem of finding the minimum value is as follows:

[0042]

[0043] In the formula: For the cutoff distance d c The potential solution set; for The corresponding fitness function.

[0044] Using the Pelican optimization algorithm, the following basic operations are performed to search for the optimal cutoff distance d. c This is to achieve the rational selection of cluster centers.

[0045] ① Population initialization

[0046] Suppose there are n pelicans in an m-dimensional space, and the position of the i-th pelican in the m-dimensional space is X. i =[X i1 ,X i2 ,…,X im ], which is the initial cutoff distance, then the positions X of n pelicans in m-dimensional space are:

[0047]

[0048] The initialization of a pelican population is random, and its mathematical expression can be described as follows:

[0049] x ij =l j +rand(u j -l j (16)

[0050] In the formula: x ij Let be the value of the j-th variable specified by the i-th candidate solution; rand is a random number in [0,1]; u j l j These represent the upper and lower boundaries of the problem in dimension j, respectively.

[0051] ② Moving towards prey stage

[0052] In each iteration, the Pelicans' new positions are:

[0053]

[0054] In the formula: Let x be the position of the i-th pelican in the j-th dimension after the first phase update; ij This represents the position of the i-th pelican in the j-th dimension before the first-stage update, where δ is a random number in [0,1]; I is a random integer of 1 or 2; P j f represents the position of the prey in the j-th dimension; P f is the fitness function value of the prey; f is the fitness function value of the pelican, which is the fitness function of formula (14).

[0055] After the update, the position of the i-th pelican in the j-th dimension is... Meets the requirements of a normal distribution:

[0056]

[0057] Where: σ is the variance of the normal distribution.

[0058] If the objective function value improves at this position, then update the position as follows:

[0059]

[0060] In the formula: The new position for the i-th Pelican in the first phase; This represents the fitness value of the i-th pelican in its new position after the first phase update.

[0061] ③ Skimming the water surface stage

[0062] In each iteration, the Pelicans' new positions are:

[0063]

[0064] In the formula: Let x be the position of the i-th pelican in the j-th dimension after the second-stage update; ij Y represents the position of the i-th pelican in the j-th dimension before the second-stage update; Y is a non-zero number between 0 and 1; t is the current iteration number; T is the maximum iteration number; β is an integer of 1 or 2.

[0065] After the second phase update, the i-th pelican is positioned in the j-th dimension. Meets the requirements of a normal distribution:

[0066]

[0067] If the objective function value improves at that position, then update the position to the newly discovered position:

[0068]

[0069] In the formula: The new position for the i-th Pelican in the second phase; This represents the fitness value of the i-th pelican in its new position after the second phase update.

[0070] Repeat the above process in the Pelican optimization algorithm until the minimum density entropy is obtained or the maximum number of iterations is reached to complete the cutoff distance d. c The optimal selection is then made by choosing points with high density and large distance as cluster centers for the arch dam deformation partition. The remaining non-center points are assigned to the corresponding cluster centers along the nearest neighbor direction with increasing ρ, thereby automatically determining the number of clusters k, which realizes the initial spatial partitioning of the arch dam deformation.

[0071] After the above operations are completed, the obtained cluster centers are used as the initial cluster centers for the fuzzy C-means clustering algorithm. The membership degree u is continuously updated by iteratively optimizing the following objective function J. ij and cluster center v j The process continues until convergence, thereby decomposing the spatiotemporal differentiation characteristics of arch dam deformation and aggregating similar characteristics, thus obtaining the objective zoning of arch dam deformation measurement points in high-altitude areas.

[0072]

[0073] In the formula: n is the number of data points; c is the number of clusters; u ij This represents the membership degree of the i-th data point to the j-th cluster, satisfying 0 ≤ u ij ≤1 and m is the fuzziness index, which controls the degree of fuzziness in clustering. It is typically set to 1.5 ≤ m ≤ 3.0. The larger m is, the more fuzzy the clustering; the smaller m is, the closer the clustering is to a hard partition. d ij It is the i-th data point x i With the j-th cluster center v j The distance between them, i.e., the aforementioned composite similarity index d ij (CD).

[0074] The specific steps of fuzzy C-means clustering are as follows:

[0075] ① Initialization. Randomly initialize cluster centers and initialize the membership matrix U = [u ij ].

[0076] ② Update membership degrees. Based on the current cluster centers, update the membership degrees:

[0077]

[0078] In the formula: d ik For data point x i Similarity to other clusters k.

[0079] ③ Update cluster centers. Update cluster centers based on current membership degrees:

[0080]

[0081] In the formula: For data point x i The weighting coefficients for the weighted summation.

[0082] ④ Calculate the change in the objective function J. If the change is less than the preset threshold, the algorithm ends; otherwise, return to step ② to continue iterating.

[0083] In step 3 above, based on the previously obtained zoning of deformation measuring points for arch dams in high-altitude areas, to fully consider the strong correlation characteristics of deformation behavior among measuring points, the kernel density estimation method is first used to determine the marginal probability distribution of each measuring point for deformation measuring points located in the same zoning area, as follows:

[0084] ① Kernel density estimation. Given a sample X1, X2, ..., X... n Its nuclear density is estimated as follows:

[0085]

[0086] In the formula: K(·) is the kernel function, usually a Gaussian kernel is chosen. h represents the bandwidth, controlling the smoothness; the Silverman criterion is commonly used. This represents the sample standard deviation.

[0087] ② Calculation of the cumulative distribution function (CDF). The CDF is obtained through numerical integration:

[0088]

[0089] After determining the marginal distribution of the random variable, considering that the determination of the arch dam deformation early warning index needs to focus on the characteristics of the distribution tail, a suitable type is selected from the following multivariate Copula functions with tail-related characteristics to establish a joint probability distribution model of deformation monitoring points.

[0090] ①Gaussian Copula function

[0091] Let X1, X2, ..., X d It is a d-dimensional random variable with a marginal distribution F. i (x), then the joint distribution function of the Gaussian Copula can be expressed as:

[0092] C Σ (u1,u2,…,u d )=Φ Σ (Φ -1 (u1),Φ-1 (u2),…,Φ -n (u d (28)

[0093] In the formula: u i =F i (x i ) is the variable X i The marginal probability integral transformation is used to make it follow a U(0,1) distribution; Φ Σ It is the joint CDF of a multivariate standard normal distribution with a mean of 0 and a covariance matrix of Σ; Φ -1 It is the inverse CDF function of the standard normal distribution.

[0094] For two-dimensional random variables u1 and u2, let the linear correlation coefficient between the variables be ρ, then the corresponding bivariate normal Copula distribution function is:

[0095]

[0096] Where s and t are symbols within the integral.

[0097] The Gaussian Copula function has symmetric tail dependency, meaning that the correlation between variables at extreme high and low values ​​is the same; its dependency structure is completely determined by the covariance matrix Σ, and it is suitable for linear or approximately linearly correlated variables and for cases where the tail dependency at extreme values ​​is weak, i.e., the probability of extreme events occurring simultaneously is low.

[0098] ②t-Copula function

[0099] t-Copula is a Copula function based on a multivariate t-distribution. Let X1, X2, ..., X... d It is a d-dimensional random variable with a marginal distribution F. i (x), then the joint distribution function of t-Copula can be expressed as:

[0100]

[0101] In the formula: u i =F i (x i ) is the variable X i The marginal probability integral transform is used to make it follow a uniform distribution U(0,1); t Σ,ν It is a joint CDF of a multivariate t-distribution with ν degrees of freedom and ∑ covariance matrix; It is the inverse CDF function of a standard univariate t-distribution with ν degrees of freedom.

[0102] For two-dimensional random variables u1 and u2, let the linear correlation coefficient between the variables be ρ. Then the bivariate t-Copula distribution function with k degrees of freedom is:

[0103]

[0104] t-Copula exhibits symmetrical tail dependence. The smaller the degree of freedom ν, the stronger the tail dependence, and the higher the probability of simultaneous occurrence of extreme events. When ν→∞, t-Copula degenerates into Gaussian Copula.

[0105] ③Archimedean Copula function

[0106] Archimedean Copulas are a class of copula functions constructed based on generating functions, characterized by their simple form and convenient computation. They can flexibly characterize asymmetric tail dependencies and are suitable for situations with complex dependency structures between variables. Let X1, X2, ..., X... d It is a d-dimensional random variable with a marginal distribution F. i (x), then the joint distribution function of Archimedes Copula can be expressed as:

[0107] In the formula: It is a generating function that satisfies It is monotonically decreasing and convex; yes The inverse function. Based on the generating function. Depending on the type, it can be further divided into Clayton Copula, Gumbel Copula, Frank Copula, etc.

[0108] The appropriate Copula function should be selected based on its statistical characteristics and distribution pattern. Specifically, observe the overall distribution pattern and tail characteristics of the variable frequency distribution histogram, measure the asymmetry of the distribution, and assess the sharpness of the distribution. If the histogram shows symmetrical tails for the two variables or an elliptical shape with a "high in the middle and low around the edges," prioritize Gaussian Copula or t-Copula. If the histogram shows asymmetrical tails or complex dependency structures, choose Archimedesian Copula. For upper-tail dependencies (such as extremely high-value correlations), choose Gumbel Copula; for lower-tail dependencies (such as extremely low-value correlations), choose Clayton Copula; and for no significant tail dependencies, choose Frank Copula.

[0109] The marginal distribution function of a random variable may contain unknown parameters, and all chosen Copula functions contain unknown parameters. Therefore, semi-parametric estimation is used to estimate the parameters based on the sample data of the random variable. Semi-parametric estimation is an estimation method that combines parametric and non-parametric ideas. It typically consists of two steps: first, non-parametric estimation of the marginal distribution, and then parametric estimation of the parameters of the Copula function, as detailed below:

[0110] ① Use the following empirical distribution function to estimate the marginal distribution:

[0111]

[0112] In the formula: I(·) is the indicator function; n is the sample size; d is the dimension; i is the summation sign variable, representing discrete summation; x j Let X be the j-th dimension empirical distribution function variable. ij This represents the measured value of the i-th sample in the j-th dimension.

[0113] ② Use pseudo-maximum likelihood estimation to estimate the unknown parameters in the Copula function. First, transform the original data using the empirical distribution function: Then, based on the transformed data, the likelihood function of the Copula is maximized to obtain the estimated values ​​of the unknown parameters in the Copula function.

[0114]

[0115] Where c() is the Copula density function, and θ represents the unknown parameter in the Copula function.

[0116] In step 4 above, the deformation values ​​of the measuring points within the same partition follow their joint distribution function and joint probability density function. Given a significance level of α, for each measuring point within the same partition, there exists a corresponding quantile α in the probability density function, denoted as y. i,1-α (i = 1, 2, ..., n). The region formed by the quantiles is the normal deformation region of the measurement point at a given significance level α, denoted as D. 1-α The typical low probability method was used to formulate early warning indicators for deformation at measuring points. Based on the significance level, the early warning indicators for arch dam deformation were divided into first-level early warning indicators (D). 99.00 -D 95.00 ) and Level II early warning indicators (DD) 99.00 ).

[0117] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0118] The method for formulating early warning indicators for arch dam deformation safety in high-altitude areas in this invention fully relies on multi-source monitoring data of arch dam deformation in high-altitude areas. By rationally partitioning the deformation behavior of arch dams in high-altitude areas, it formulates joint early warning indicators for arch dam deformation safety at multiple measurement points in a multi-dimensional probability space. This effectively overcomes the problem that the traditional single-measurement-point formulation mode is easily affected by various unknown factors, and strongly supports the monitoring of the deformation behavior of arch dams in high-altitude areas from a spatial overall perspective. Attached Figure Description

[0119] Figure 1 This is a flowchart illustrating the implementation of a method for determining zoning safety early warning indicators for arch dam deformation in high-altitude areas, as proposed in this invention.

[0120] Figure 2 A flowchart illustrating the zoning of the deformation behavior of arch dams;

[0121] Figure 3 This is a diagram showing the layout of vertical measuring points for the arch dam in the embodiment (elevation unit: m);

[0122] Figure 4 This is the sequence curve of the measured values ​​at each measuring point along the vertical line of the arch dam in the embodiment;

[0123] Figure 5 This is the iteration process line of the POA algorithm in the embodiment;

[0124] Figure 6 This is the initial cluster center decision graph in the embodiment;

[0125] Figure 7 This is a schematic diagram of the arch dam deformation behavior partitioning results based on the inventive method in the embodiment.

[0126] Figure 8 The results of the arch dam deformation behavior partitioning based on the method of the present invention are shown in the embodiments.

[0127] Figure 9 The results of the arch dam deformation behavior partitioning based on the POA-DPC method in the embodiments are shown below;

[0128] Figure 10 The results of the arch dam deformation behavior partitioning based on the FCM method in the embodiments are shown below;

[0129] Figure 11 The cumulative probability density curves of the measured sequences of the arch dam PL3-1 and PL3-2 in the embodiment;

[0130] Figure 12 This is a binary frequency distribution histogram of the measured sequences of the arch dam PL3-1 and PL3-2 in the embodiment.

[0131] Figure 13 This is a joint probability distribution diagram of the measured sequences of the arch dam PL3-1 and PL3-2 in the embodiment;

[0132] Figure 14 The image shows the contour map of the joint probability distribution of the measured values ​​of the arch dam PL3-1 and PL3-2 in the embodiment. Detailed Implementation

[0133] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0134] Taking a certain arch dam in a high-altitude area as an example, we selected the attached... Figure 3 The 15 vertical measuring points shown are PL1-1~PL1-2, PL2-1~PL-2-2, PL3-1~PL-3-5, PL4-1~PL4-3, and PL5-1~PL5-1. These measurements are taken according to the attached diagram. Figure 1 and attached Figure 2 The process shown utilizes the attached... Figure 4 The monitoring data from the 15 vertical measuring points shown are used to conduct a spatial deformation behavior zoning analysis of the arch dam. Furthermore, joint early warning indicators are proposed for measuring points PL3-1 and PL3-2 within zoning 2, and the results of zoning and early warning indicator proposals are compared with those obtained using other methods. The specific steps are as follows:

[0135] (1) Spatial Deformation Behavior Zoning Analysis of Arch Dams

[0136] Select Appendix Figure 3 The data shown are the displacement data along the river at 15 vertical measuring points of the dam from March 1, 2018 to May 30, 2020 (as attached). Figure 4 As shown in the figure, the proposed method is used to perform spatial deformation behavior zoning analysis of the arch dam, and the zoning results are compared with those of other methods.

[0137] ① Calculation of composite similarity index for dam deformation

[0138] Based on the appendix Figure 4 The deformation monitoring data of the arch dam shown were used to determine the weighting coefficients of the composite similarity metric index for arch dam deformation using the CRITIC method: α1 = 0.3278, α2 = 0.3654, α3 = 0.3068. It can be seen that the absolute distance d... ij (AD), Incremental Distance d ij (ID), relative incremental distance d ij The weight coefficients of the three indicators (RGD) are relatively evenly distributed, and the incremental distance has the greatest impact on the overall distance.

[0139] ② Cluster center search for dam deformation behavior based on a combination of Pelican Optimization Algorithm (POA) and Density Peak Clustering Algorithm (DPC).

[0140] The Pelican Optimization Algorithm (POA) parameters are set as follows: maximum number of iterations T = 30, population size N = 30; and the Density Entropy (DPC) algorithm parameter cutoff distance d. c The value is set to [0, 1, 10], and the POA algorithm is used to optimize it. The iterative process line and decision graph of the POA algorithm are attached. Figure 5 and attached Figure 6 As shown. (From the appendix) Figure 5 It can be seen that after 4 iterations, the density entropy reaches its minimum value, and the optimal cutoff distance is 8.93; from the attached... Figure 6 It can be seen that five cluster centers were calculated using the density entropy (DPC) algorithm, namely PL2-1, PL4-2, PL3-1, PL3-4, and PL1-2.

[0141] ③ Dam deformation behavior zoning based on fuzzy C-means clustering (FCM)

[0142] The ambiguity of the FCM algorithm was set to 1.5. Then, using the five cluster centers obtained above, the cluster centers of the FCM algorithm were initialized, identifying measurement points on the dam with similar spatial deformation characteristics. Related results are attached. Figure 7 and attached Figure 8 As shown. (From the appendix) Figure 7 It can be seen that the deformation distribution of the arch dam is roughly symmetrical about the transverse axis of the dam body, exhibiting a symmetrical, upward-facing semi-circular distribution with openings on both banks, and the overall deformation is coordinated. Based on the spatial characteristics of the deformation behavior, the dam body can be divided into 5 regions, each corresponding to a cluster. (See attached...) Figure 8 It can be seen that: partition 1 contains 3 measurement points, namely PL2-1, PL2-2, and PL2-3; partition 2 contains 2 measurement points, namely PL3-1 and PL3-2; partition 3 contains 3 measurement points, namely PL3-3, PL3-4, and PL3-5; partition 4 contains 5 measurement points, namely PL1-1, PL1-2, PL4-1, PL4-2, and PL4-3; and partition 5 contains 2 measurement points, namely PL5-1 and PL5-2. The deformation time series curves of the measurement points within each partition have similar fluctuation characteristics, indicating the rationality of the clustering results.

[0143] ④ Comparison with POA-DPC and FCM cluster analysis methods

[0144] To verify the superiority of the arch dam deformation behavior zoning method proposed in this invention, it is compared with the POA-DPC method and the single FCM method. The time series diagrams of the measurement points for each zone obtained from the cluster analysis of the POA-DPC and FCM methods are attached. Figure 9 and attached Figure 10As shown in the figure, the proposed method has similar clustering results to the POA-DPC and FCM methods. However, the POA-DPC and FCM methods have significant flaws, namely grouping deformed sequences with low similarity into the same cluster, as shown in the attached figure. Figure 9 Partitions 1, 2, and 4 in the image are attached. Figure 10 Partitions 3 and 4 in the diagram show that using the POA-DPC method and the FCM method alone cannot comprehensively and reasonably obtain all measurement points with similar deformation time series.

[0145] The above analysis shows that the POA algorithm optimizes the cutoff distance d of DPC. c This approach allows for more reasonable selection of cluster centers, avoiding the poor clustering results caused by improper selection of cutoff distances in the traditional DPC algorithm. Using the optimized cluster centers as the initial input to the FCM algorithm improves its clustering accuracy and avoids the problem of the FCM algorithm getting trapped in local optima due to random selection of initial cluster centers. The combination of DPC, FCM, and POA algorithms not only solves the limitations of using POA-DPC and DPC alone, but also leverages the advantages of DPC in handling datasets with significant density differences and robustness to noisy data, as well as the relatively simple implementation and applicability of FCM to complex nonlinear data structures with ambiguous boundaries.

[0146] (2) Formulation of early warning indicators for arch dam deformation safety

[0147] Historical monitoring data of PL3-1 and PL3-2 measuring points within deformation zone 2 of the dam obtained from the aforementioned cluster analysis were selected as samples for the proposed joint early warning index of multi-measuring point deformation.

[0148] ① Determination of the joint probability density function of dam deformation at multiple measuring points

[0149] The cumulative probability curve of the empirical distribution is a strictly discrete description of the sample data, and its shape depends entirely on a finite number of observation points. If the cumulative probability curve estimated using the kernel distribution fits the empirical distribution function well, it indicates that the cumulative probability curve estimated using the kernel distribution has a good representation of the statistical characteristics of the sample, and its smoothing result can represent the population distribution behind the sample. The kernel density estimation method is used to fit the cumulative probability curve of the empirical distribution, with K(·) taking a Gaussian kernel function. Figure 11 The cumulative probability density curves are derived from the empirical distribution function and kernel density estimation of the measurement sequences for measurement points PL3-1 and PL3-2. (See attached...) Figure 11 It can be seen that the cumulative probability curves of the measurement sequence of PL3-1 and PL3-2 obtained by kernel density estimation fit well with the cumulative probability curve of the empirical distribution. Therefore, the cumulative probability density curve obtained by kernel density estimation can be used to characterize the distribution of the overall sample.

[0150] The Copula function is a connection function between the marginal distributions of each random variable and the joint distribution of the variables. The selection of the Copula function mainly depends on the distribution type of each random variable. Therefore, the most suitable Copula function can be determined by analyzing the joint distribution characteristics of the random variables. (Appendix) Figure 12 This section presents the bivariate frequency distribution histograms of the measurement sequences from measuring points PL3-1 and PL3-2. When formulating a joint spatiotemporal early warning index using arch dam deformation monitoring data, the tail characteristics of the measurement sequence distribution are extremely important. Therefore, the Gaussian Copula function, which has a thicker tail and better reflects the tail correlation characteristics between variables, is selected to construct the joint distribution function of the displacements at measuring points PL3-1 and PL3-2, respectively. Based on the deformation measurement sample space of measuring points PL3-1 and PL3-2, the linear correlation parameters in the bivariate Gaussian Copula function are obtained through parameter estimation. The estimated value is:

[0151]

[0152] Based on this, the estimated bivariate Gaussian Copula function can be obtained as follows:

[0153]

[0154] The above formula describes the deformation measurement distributions of PL3-1 and PL3-2 measuring points, and the relationship between their joint distribution. To intuitively reflect the correlation among the three, an appendix is ​​attached. Figure 13 A joint probability distribution diagram of the deformation measurements at measuring points PL3-1 and PL3-2 was plotted, with attached... Figure 14 A contour map of the joint probability distribution of deformation measurements at measuring points PL3-1 and PL3-2 was drawn.

[0155] ② Drafting of Spatiotemporal Joint Early Warning Indicators for Dam Deformation at Multiple Measurement Points

[0156] After calculating the joint distribution function of the measurement sequences of PL3-1 and PL3-2 measuring points using the bivariate Copula theory, based on the small probability theory, the deformation critical values ​​corresponding to the distribution functions at significance levels of 5% and 1% (corresponding to guarantee rates of 95.00% and 99.00%) were taken as the primary and secondary early warning indicators, respectively. Since the range of the bivariate distribution function is a surface domain, the proposed spatiotemporal joint early warning index for arch dam deformation is the critical value of the surface domain. The primary early warning index is defined as the deformation range (i.e., D) when the guarantee rate is 95.00%–99.00%. 99.00 -D 95.00 The critical value of ); the secondary early warning index is defined as the deformation range (i.e., DD) when the guarantee rate is 99.00% to 100.00%. 99.00 The critical value of ) is shown in the appendix. Figure 14 When the deformation of measuring points PL3-1 and PL3-2 (y′1, y′2) ∈ D 95.00 When the deformation of measuring points PL3-1 and PL3-2 is normal, it indicates that the deformation behavior of measuring points PL3-1 and PL3-2 is normal; when the deformation of measuring points PL3-1 and PL3-2 is (y′1, y′2)∈D 99.00 -D 95.00 When deformation of measuring points PL3-1 and PL3-2 is detected, a first-level early warning is issued, the alarm situation is analyzed, the source of the alarm is investigated, and potential safety hazards are eliminated; when the deformation of measuring points PL3-1 and PL3-2 (y′1, y′2) ∈ DD 99.00 At that time, a level II early warning was issued for the deformation of measuring points PL3-1 and PL3-2, and an immediate investigation of the source of the warning was carried out. Based on the investigation results, a control decision was made.

[0157] Table 1 shows the early warning indicators at different significance levels. It can be seen that when the deformation value of PL3-1 exceeds the early warning indicator due to a large error, while the deformation value of PL3-2 does not exceed the indicator, the single-point early warning method will result in false alarms. However, the spatiotemporal joint early warning indicator formulation method combines the distribution types and correlations of multiple measurement points, and will not issue early warnings for such measurement points. This proves that the method proposed in this paper has stronger robustness and higher accuracy. Compared with the typical low probability method, at significance levels of 1% and 5%, the early warning indicators proposed by the method of this invention are more conservative, and their values ​​are slightly smaller than those of the typical low probability method.

[0158] Table 1. Early warning indicators at different significance levels (unit: mm)

[0159]

[0160] Obviously, the above embodiments are merely examples for clear illustration and are not intended to limit the implementation. Those skilled in the art can make other variations or modifications based on the above description. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of this invention should be included within the scope of protection of this invention.

Claims

1. A method for determining zoning early warning indicators for deformation safety of arch dams in high-altitude areas, characterized in that, Includes the following steps: Step 1: Collect multi-source monitoring data on the deformation safety of arch dams in high-altitude areas and determine the composite similarity index of their measurement sequences; Step 2: Based on the composite similarity metric index of the measured value sequence, perform cluster analysis of the arch dam deformation behavior driven by multi-point monitoring data to complete the identification of homogeneous features of arch dam deformation and realize the partitioning of arch dam deformation behavior in high-altitude areas; Step 3: Based on the partitioning in Step 2, determine the marginal probability distribution of the measurement points within each partition, establish the joint probability distribution function of the strongly correlated measurement point group within the partition, and estimate the parameters of the joint probability density function of multiple measurement points; Step 4: Based on the joint probability density function determined in Step 3, formulate graded early warning indicators for the deformation safety of arch dams in high-altitude areas under multidimensional probability space.

2. The method for determining the zoning of early warning indicators for deformation safety of arch dams in high-altitude areas according to claim 1, characterized in that, A composite similarity metric consisting of "absolute distance," "incremental distance," and "growth rate distance" is used to mine the similarity and difference characteristics among monitoring data sequences of arch dam deformation at various monitoring points in high-altitude areas. d ij (CD)=α1d ij (AD)+α2d ij (ID)+α3d ij (RGD) (1) In the formula: d ij (CD) is a composite similarity metric between spatial measurement point i and measurement point j; d ij (AD) represents the absolute distance, indicating the difference in the magnitude of the measured values ​​between measuring point i and measuring point j; d ij (ID) represents the incremental distance, indicating the difference in the magnitude of change between the measured values ​​at measuring point i and measuring point j; d ij (RGD) represents the relative growth rate distance, indicating the difference in the rate of change of the measured values ​​at measuring point i and measuring point j; α1, α2, and α3 are the weighting coefficients of the three similarity distance indicators, and satisfy α1 + α2 + α3 = 1, x it x is the deformation amount. it =δ it ;y it For deformation amplification, y it =δ it -δ i,t-1 ;z it For the deformation rate, δ it and δ i,t-1 Let x and y represent the deformation values ​​of the i-th measuring point at time t and time t-1, respectively; Z() represents the original index value x after Z-score standardization. it y it z it A function for standardization.

3. The method for determining the zoning of early warning indicators for deformation safety of arch dams in high-altitude areas according to claim 2, characterized in that, Using the multi-criteria decision analysis method CRITIC, the weight coefficients of the three similarity distance indicators in the composite similarity measurement index are determined by constructing a decision matrix, standardizing the decision matrix, calculating the correlation coefficient matrix, calculating the information content of each criterion, and calculating the normalized weights.

4. The method for determining the zoning of early warning indicators for deformation safety of arch dams in high-altitude areas according to claim 1, characterized in that, In step 2, the composite similarity metric of the arch dam deformation measurement sequence is used as a clustering index. Density peak clustering algorithm and fuzzy C-means clustering algorithm are integrated and applied to perform cluster analysis on the multi-point monitoring data of arch dam deformation, as detailed below: First, the density peak clustering algorithm is applied to determine the initial cluster centers and the number of clusters k, thus realizing the initial spatial partitioning of the arch dam deformation; The obtained cluster centers are used as the initial cluster centers for the fuzzy C-means clustering algorithm. The objective function J is iteratively optimized, and the membership degree u is continuously updated. ij and cluster center v j The process continues until convergence, thereby decomposing the spatiotemporal differentiation characteristics of arch dam deformation and aggregating similar features, thus obtaining the partitioning of arch dam deformation measurement points in high-altitude areas. In the formula: n is the number of data points; c is the number of clusters; u ij This represents the membership degree of the i-th data point to the j-th cluster, satisfying 0 ≤ u ij ≤1 and m is the fuzziness index, which controls the degree of fuzziness in clustering. It is set to 1.5 ≤ m ≤ 3.

0. The larger m is, the more fuzzy the clustering; the smaller m is, the closer the clustering is to a hard partition. d ij It is the i-th data point x i With the j-th cluster center v j The distance between them, i.e., the composite similarity index d ij (CD).

5. The method for determining the zoning of early warning indicators for deformation safety of arch dams in high-altitude areas according to claim 4, characterized in that, When applying the density peak clustering algorithm to determine the initial cluster centers, two key parameters are defined for each data point: local density ρ. i The mathematical equation for the local density, relative to the relative distance δ, is as follows: In the formula: d ij For x i With x j The distance between them, i.e., the aforementioned composite similarity index d ij (CD); d c The cutoff distance; relative distance δ i Represents data point x i The minimum distance between x and data points with higher local density. i When the local density is at its maximum, δ i The value of x i The maximum distance from other data points; The cutoff distance d in the density peak clustering algorithm c The problem is transformed into an optimization problem of finding the minimum value of the density entropy function. Using the pelican optimization algorithm, the optimal cutoff distance is searched by simulating the pelican's foraging process, thus achieving the rational selection of cluster centers. In the formula: For the cutoff distance d c The potential solution set; for The corresponding fitness function; H p This is the density entropy function, used to measure the distance d at each cutoff point. c The degree of disorder in the local density estimate corresponds to the density entropy. When the local density values ​​of all sample points are approximately equal, there is a maximum density entropy, indicating that the uncertainty of the local density estimate is relatively large, which is not conducive to the stability of the clustering results. Conversely, the greater the difference in the local density values ​​of each sample point, the smaller the density entropy, and the more conducive the results are to the identification of the clustering pattern.

6. The method for determining the zoning of early warning indicators for deformation safety of arch dams in high-altitude areas according to claim 1, characterized in that, In step 3, for deformation measurement points located in the same partition, the kernel density estimation method is first used to determine the marginal probability distribution of each measurement point, the marginal probability distribution of each measurement point is analyzed, and a multivariate Copula function with tail correlation characteristics is selected to establish a joint probability distribution model of deformation monitoring points based on the binary frequency distribution histogram of the measurement value sequence of the measurement points. Finally, the semi-parametric estimation method is used to estimate the parameters of the joint probability density function.

7. The method for determining the zoning of early warning indicators for deformation safety of arch dams in high-altitude areas according to claim 6, characterized in that, A joint probability distribution model for deformation monitoring points is established using a multivariate Copula function with tail correlation characteristics, as detailed below: The appropriate Copula function is selected using the following method: observe the overall distribution shape and tail characteristics of the frequency distribution histogram of the variables, measure the asymmetry of the distribution, and assess the sharpness of the distribution. If the histogram shows that the tails of the two variables are symmetrical or have an elliptical shape with "high in the middle and low around the edges", the Gaussian Copula function or the t-Copula function is preferred. If the histogram shows that the tails are asymmetrical or the dependency structure is complex, the Archimedesian Copula function is selected, with Gumbel Copula for upper tail dependencies, Clayton Copula for lower tail dependencies, and Frank Copula for no significant tail dependencies.

8. The method for determining the zoning of early warning indicators for deformation safety of arch dams in high-altitude areas according to claim 7, characterized in that, The parameters of the joint probability density function are estimated using a semi-parametric estimation method, specifically as follows: ① Use the following empirical distribution function to estimate the marginal distribution: In the formula: I(·) is the indicator function; n is the sample size; d is the dimension; ② To estimate the unknown parameters in the Copula function using pseudo-maximum likelihood estimation, first transform the original data using the empirical distribution function: Then, based on the transformed data, the likelihood function of the Copula is maximized to obtain estimates of the unknown parameters in the Copula function.

9. The method for determining the zoning of early warning indicators for deformation safety of arch dams in high-altitude areas according to claim 1, characterized in that, The deformation values ​​of measuring points within the same zone follow their joint distribution function and joint probability density function. Given a significance level, the critical deformation value corresponding to the joint probability density function at the significance level is taken as the early warning indicator.

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