Eutrophication Risk Assessment Method for Black and Odorous Water Bodies Based on KDE-Vine Copula

Abstract

This invention discloses a KDE-Vine Copula-based method for assessing the eutrophication risk of black and odorous water bodies. The method uses a black and odorous water body identification model for initial identification, employs the SEaTH algorithm to identify feature importance, constructs a multi-dimensional feature space, optimizes the initial identification results based on the optimal support vector principle, estimates the marginal distribution using radial basis functions, selects the Copula function with the best fitting effect, and constructs a Vine Copula tree structure to assess the eutrophication risk of black and odorous water bodies. Compared to the original parametric method for constructing marginal distributions, KDE is not limited to any distribution assumptions and can maximize the satisfaction of the random variable distribution form, i.e., a better fitting effect. Overall, the method of this invention integrates multiple water quality parameters to construct an eutrophication risk assessment model, providing a new approach for water quality assessment.

Description

Technical Field

[0001] This invention belongs to the field of water environment remote sensing and relates to a method for assessing the risk of eutrophication of water bodies, specifically a method for assessing the risk of eutrophication of black and odorous water bodies based on KDE-Vine Copula. Background Technology

[0002] Water quality assessment is a fundamental task for the rational development, utilization, and protection of water resources. It is also the most important means for the government to manage water ecology and the water environment. Furthermore, it is a comprehensive task involving multiple factors such as pollution sources and water quality parameters. In recent years, the China National Environmental Monitoring Centre has assessed the eutrophication level of water bodies using 3,641 surface water monitoring sections and the Total Tropical Liability Index (TLI), providing a basis for relevant government departments to prevent or rectify the water ecological environment.

[0003] With the acceleration of urban industrialization, the discharge of wastewater rich in organic matter will inevitably pollute urban rivers, leading to secondary disasters such as eutrophication. This causes rivers to exceed their self-purification capacity, further accelerating water quality deterioration and becoming one of the factors hindering the construction of a water ecological civilization. Therefore, controlling the degree of eutrophication is an essential stage in urban river management, and water quality monitoring is a crucial part of urban water environment governance.

[0004] Traditional monitoring methods are limited in scope and efficiency, while satellite remote sensing water quality monitoring offers advantages such as low cost and high efficiency, providing a new technological approach for urban water quality monitoring. Domestic and international research has utilized remote sensing technology to study the formation, hazards, assessment, and remediation of eutrophication in water bodies. Li Yunmei established a water reflectance simulation model using analytical modeling, and calculated the trophic status index of sampling points by solving optimization functions to assess eutrophication. Wang Shengrui used mathematical statistics, quantitative inversion and interpretation methods from remote sensing, and the analytic hierarchy process (AHP) to determine the ecological risks of Dongting Lake and proposed a technical route for its prevention and control. Jing Xia used a semi-empirical regression model and the modified Carlson index method to evaluate the eutrophication level of Miyun Reservoir. Azevedo used a cumulative probability density model to predict eutrophication risk. Biggs proposed a method based on a maximum chlorophyll a regression model, using a function established by concentration and cumulative days to predict eutrophication levels. Currently, large-scale water quality monitoring is mainly carried out using remote sensing technology, while localized small water bodies can be assessed through on-site sampling and testing. However, current research largely relies on single indicators to evaluate eutrophication levels, with limited use of multi-parameter risk assessment models and a lack of comprehensive analysis of water quality. The Vine Copula function, as a mathematical model connecting multiple variable distributions, can effectively combine multiple indicators for risk probability assessment, aiding in the analysis of correlation characteristics within high-dimensional variables. It has been applied in small-scale water trophic status assessments; for example, Zhang Yan used the Copula function to assess the eutrophication status of Meihu Lake, an artificial lake at Zhengzhou University. However, the application of remote sensing technology combined with the Vine Copula function in large-scale eutrophication risk assessments of urban water bodies is lacking. Summary of the Invention

[0005] Addressing the limitations of previous studies, such as limited use of overall indicators and lack of large-scale remote sensing monitoring, this invention provides a novel KDE-Vine Copula-based method for eutrophication risk assessment of black and odorous water bodies. This method utilizes KDE (Kernel Density Function Estimation) and Copula theory to reconstruct the marginal distribution. Compared to traditional parametric methods for constructing marginal distributions, KDE is not limited by any distribution assumptions and can maximize the fulfillment of the random variable distribution form, resulting in a better fit. Overall, this invention integrates multiple water quality parameters to construct a eutrophication risk assessment model, providing a new approach to water quality assessment.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] A method for eutrophication risk assessment of black and odorous water bodies based on KDE-Vine Copula includes the following steps:

[0008] Step 1: Remote sensing image feature optimization and black and odorous water body identification:

[0009] The black and odorous water body identification model is used to make a preliminary judgment on black and odorous water bodies. The importance of features is identified by the SEaTH (SEparability and THresholds) algorithm, a multi-dimensional feature space is constructed, and the preliminary identification results are optimized by combining it with the optimal support vector theory.

[0010] Step 2, Construction of the edge distribution:

[0011] The inversion results of water quality parameters Chl-a (chlorophyll a), TSS (suspended solids), and SD (transparency) were used as one-dimensional index sequences. The KDE method was selected based on the kernel function to perform smoothing estimation, i.e., inference of the overall one-dimensional distribution based on existing data samples.

[0012]

[0013] In the formula, Represents any sample value, For a certain water quality monitoring indicator, if the simulated sequence medium to small The number is , For sample size, express The corresponding cumulative frequency;

[0014] Step 3, Choosing the Vine Copula function:

[0015] Based on the marginal distribution constructed in step 2, the fitting effect and applicability of the Copula function are judged according to the squared euclidean metric in the state of two-dimensional random variables. In the high-dimensional random variables, a tree structure is selected to simplify the calculation. The optimal Copula function is determined according to the AIC (Akaike information criterio) of each edge, and then the optimal Copula joint probability distribution model is established.

[0016] Step 4: Solve for the joint probability distribution:

[0017] The joint probability distribution is solved based on the optimal Copula joint probability distribution model established in step 3, and the eutrophication risk of water bodies is analyzed in a comprehensive manner.

[0018] Compared with the prior art, the present invention has the following advantages:

[0019] (1) This invention addresses the risk assessment of eutrophication in large-scale water bodies by combining the Vine Copula function with remote sensing technology to study and analyze the risk of eutrophication. Compared with traditional water quality monitoring, this method improves monitoring efficiency and expands the monitoring scope to a certain extent. Based on the original evaluation of a single indicator, it comprehensively assesses the probability of eutrophication risk from multiple factors, which helps to further evaluate the aquatic ecological environment and more accurately reflect the water quality status.

[0020] (2) This invention improves the original marginal distribution structure of the Vine Copula function using KDE and combines it with the two-dimensional model of the Vine Copula function. This method does not require prior knowledge of the original data, is not limited to any distribution assumptions, and can maximize the satisfaction of the distribution form of the random variable. A kernel density function is established based on the radial basis function (RBF), and the kernel density function value of each sample point is weighted and averaged to obtain the probability density estimate of the sample point. When the number of samples approaches infinity, the kernel density estimate probability density function is infinitely close to the true probability density distribution. Attached Figure Description

[0021] Figure 1 The flowchart shows the eutrophication risk assessment method for black and odorous water bodies based on KDE-Vine Copula.

[0022] Figure 2 Let (Chl-a,SD), (Chl-a,TSS), and (TSS,SD) be the two-dimensional joint risk probabilities.

[0023] Figure 3 The three-dimensional joint risk probability is given by the (Chl-a, TSS, SD) combination. Detailed Implementation

[0024] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0025] This invention provides a method for assessing the eutrophication risk of black and odorous water bodies based on KDE-Vine Copula. It uses a black and odorous water body identification model for initial identification, employs the SEaTH algorithm to identify feature importance, constructs a multi-dimensional feature space, optimizes the initial identification results based on the optimal support vector principle, estimates the marginal distribution using radial basis functions, selects the Copula function with the best fitting effect, and constructs a Vine Copula tree structure to assess the eutrophication risk of black and odorous water bodies. Figure 1 As shown, the specific steps include:

[0026] Step 1: Remote sensing image feature optimization and black and odorous water body identification:

[0027] A black and odorous water body identification model is used to initially identify black and odorous water bodies. The SEaTH algorithm is used to identify the importance of features, a multi-dimensional feature space is constructed, and the preliminary identification results are optimized by combining the optimal support vector theory. The specific steps are as follows:

[0028] Step 1.1: Due to the high concentration of suspended solids in black and odorous water bodies, their reflectance changes gradually across the green and red light bands, whereas the changes are more pronounced in general water bodies. Therefore, this invention uses the normalized ratio model BOI (Black and Odorous Water Index) to make a preliminary judgment on black and odorous water bodies.

[0029]

[0030] In the formula, For remote sensing reflectance in the blue light band, The remote sensing reflectance is in the green light band. This represents the remote sensing reflectance in the red light band.

[0031] Step 1.2: Extract the spectral mean, variance, contrast, correlation, entropy, homogeneity, dissimilarity, and second moment features from the remote sensing image. 1 eigenvalue, Represents the characteristic number, Indicates the number of bands The SEaTH algorithm is used to identify feature importance, typically employing the Jeffries-Matusita (JM) distance to measure sample separation and ranking feature importance. The feature corresponding to the largest JM distance between two classes is selected as the optimal feature, and further selection is based on feature importance. Construct a multi-dimensional feature space from the best features. :

[0032]

[0033] In the formula, Indicates the Bach distance. and This represents the mean of a feature between two categories. and The standard deviation of the features of the two categories, Indicates the distance between J and M;

[0034] Step 1.3: In the multidimensional feature space, the preliminary identification results are used as training samples, and support vector machines are used to identify black and odorous water bodies, thereby obtaining the final optimized results.

[0035] Step 2, Construction of the edge distribution:

[0036] The inversion results of water quality parameters Chl-a (chlorophyll a), TSS (suspended solids), and SD (transparency) were used as one-dimensional index sequences. The KDE method was selected based on the kernel function to perform smoothing estimation, i.e., inference of the overall one-dimensional distribution based on existing data samples.

[0037]

[0038] In the formula, Represents any sample value, For a certain water quality monitoring indicator, if the simulated sequence medium to small The number is , For sample size, express The corresponding cumulative frequency.

[0039] The expression for the kernel density estimation function is:

[0040]

[0041] In the formula, In order to be in The estimated probability density function at that location. It is the sample size. Indicates bandwidth , This represents the Gaussian kernel function.

[0042] Bandwidth is an important parameter that affects the smoothness and accuracy of the estimation. A larger bandwidth results in a smoother function but weakens the detailed features of some data points; a smaller bandwidth improves the function fitting accuracy but introduces noise. Therefore, the optimal bandwidth must be selected during kernel density estimation, expressed as:

[0043]

[0044] In the formula, , Indicates the sample standard deviation. This represents the interquartile range of the sample.

[0045] The Gaussian kernel function expression is:

[0046]

[0047] In the formula, Indicates the input value. This represents the Gaussian kernel function value.

[0048] Step 3, Choosing the Vine Copula function:

[0049] Based on the marginal distribution constructed in step 2, a multivariate function model needs to be constructed for analysis of the joint risk probability of eutrophication indicators. That is, in the case of two-dimensional random variables, the fit and applicability of the Copula function are determined based on the squared Euclidean distance. However, in high-dimensional random variables, the "curse of dimensionality" increases the complexity of the model as the dimensionality increases. A tree structure is usually chosen to simplify the calculation. The optimal Copula function is determined based on the AIC (Akaike information criterio) of each edge, and then the optimal Copula joint probability distribution model is established. The specific steps are as follows:

[0050] Step 3.1: Under the condition of two-dimensional random variables, determine the optimal Copula function according to the principle of minimizing the squared Euclidean distance. The expression for the squared Euclidean distance is:

[0051]

[0052] In the formula, This represents the Copula sample frequency, i.e., the empirical frequency. This represents the Copula calculation frequency, i.e., the calculation frequency. It refers to the sample size.

[0053] Step 3.2: Under the state of multidimensional random variables, determine the optimal Copula function based on the AIC of each edge. The AIC expression is:

[0054]

[0055] In the formula, Indicates the number of model parameters. This represents the log-likelihood function of the model.

[0056] Step 3.3: Solve for the Copula model parameters using the maximum likelihood estimation method. The joint probability density function can be expressed as:

[0057]

[0058] In the formula, This indicates a certain water quality monitoring indicator. Let represent the parameter to be determined, and let Seeking .

[0059] Step 3.4: Sum the AIC values ​​of all edges in the tree structure and select the tree structure with the minimum total AIC as the optimal function model.

[0060] Step 4: Solve for the joint probability distribution:

[0061] Copula functions are random variables. joint distribution With their respective marginal distribution functions If a function is connected to another function, then there exists a function...

[0062] satisfy The joint probability distribution is solved based on the optimal Copula joint probability distribution model, and this is used to comprehensively analyze the risk of eutrophication of water bodies. The specific steps for solving the joint probability distribution are as follows:

[0063] Step 4.1: To solve for the joint probability distribution, the joint probability density function is expressed as:

[0064]

[0065] In the formula, For joint probability density, For Copula functions, For the marginal distribution function, This is the edge density function.

[0066] Step 4.2: According to conditional probability, we can obtain:

[0067]

[0068] The three-dimensional joint probability density function is decomposed into:

[0069]

[0070] Inferred:

[0071]

[0072] Step 4.3: Combining Steps 4.1 and 4.2, we obtain the joint probability density function:

[0073]

[0074] Figure 2 and Figure 3 This is a diagram showing the eutrophication risk assessment results of black and odorous water bodies based on the KDE-Vine Copula method of this invention. Figure 2In the table, (a), (b), and (c) represent the two-dimensional joint risk probabilities under the combinations (Chl-a,SD), (Chl-a,TSS), and (TSS,SD), respectively. Figure 3 This represents the three-dimensional combined risk probability under the (Chl-a, TSS, SD) combination. As the concentration of each parameter increases, the overall eutrophication risk probability gradually increases. When the concentrations of Chl-a, TSS, and SD are all low, the eutrophication risk probability is low. When the Chl-a or SD concentrations remain constant, the impact of TSS concentration changes on the eutrophication risk probability is more significant, indicating that TSS has a substantial impact on water quality. This is because TSS comprises many components, and different components and concentrations all affect water quality. Eutrophication is not limited to a single water quality parameter but is caused by multiple factors. To maintain a low eutrophication risk probability and good water quality, it is insufficient to rely solely on a single indicator for evaluation. Precise control of the concentrations of each water quality parameter is necessary, with particular attention to monitoring the TSS concentration and its components. A comprehensive analysis considering the influence of multiple parameters is crucial for accurate water quality assessments.

Claims

1. A black and odorous water eutrophication risk assessment method based on KDE-Vine Copula, characterized by The method comprises the following steps: Step 1, remote sensing image feature optimization and black and odorous water body identification: The black and odorous water body is preliminarily discriminated by using a black and odorous water body identification model, the feature importance is identified by using a SEaTH algorithm, a multi-dimensional feature space is constructed, and the preliminary identification result is optimized in combination with optimal support vector theory; Step 2, construction of edge distribution: The Chl-a, TSS and SD water quality parameter inversion results are respectively taken as one-dimensional index sequences, the KDE method is selected based on a kernel function to perform smooth estimation, that is, the one-dimensional distribution of the population is inferred according to the existing data samples: wherein denotes an arbitrary sample value, is a water quality monitoring index, if the simulated sequence has less than the number of , is the sample size, denotes the corresponding cumulative frequency; Step 3, selection of Vine Copula function: On the basis of the edge distribution constructed in step 2, the fitting effect and applicability of the Copula function are discriminated according to the square Euclidean distance in the two-dimensional random variable state, the calculation is simplified by using a tree structure in the high-dimensional random variable, the optimal Copula function is determined according to the AIC of each edge, and then the optimal Copula joint probability distribution model is established; Step 4, solving the joint probability distribution: The joint probability distribution is solved according to the optimal Copula joint probability distribution model established in step 3, so as to comprehensively analyze the eutrophication risk of the water body. 2.The black and odorous water eutrophication risk assessment method based on KDE-Vine Copula according to claim 1, characterized in that The specific steps of step 1 are as follows: Step 1.1, a normalized ratio model BOI is selected to preliminarily discriminate the black and odorous water body: wherein is the remote sensing reflectance for the blue wavelength band, is the remote sensing reflectance for the green wavelength band, is the remote sensing reflectance for the red wavelength band; Step 1.2, extract the spectral mean, variance, contrast, correlation, information entropy, homogeneity, heterogeneity and second moment features of the remote sensing image, a total of feature values, representing the number of features, representing the number of bands, , identify the feature importance using the SEaTH algorithm, measure the sample separation degree using the J-M distance, sort the feature importance, take the feature corresponding to the maximum J-M distance between the two classes as the best feature, and select best features to construct a multi-dimensional feature space: wherein denotes the Bhattacharyya distance, and denotes the mean of a certain feature of the two classes, and denotes the standard deviation of a feature of the two classes, denotes the J-M distance, ; Step 1.3, in the multi-dimensional feature space, the preliminary identification result is taken as a training sample, the black and odorous water body is identified by using a support vector machine, and then the final optimization result is obtained. 3.The black and odorous water eutrophication risk assessment method based on KDE-Vine Copula according to claim 1, characterized in that In step 2, the kernel density estimation function expression is: wherein is the probability density function estimated at , is the sample size, represents the bandwidth , represents the Gaussian kernel function. 4.The black and odorous water eutrophication risk assessment method based on KDE-Vine Copula according to claim 1, characterized in that The specific steps of step 3 are as follows: Step 3.1, the optimal Copula function is determined according to the minimum principle of square Euclidean distance in the two-dimensional random variable state, and the square Euclidean distance expression is: where denotes the Copula sample frequency, i.e. the empirical frequency, denotes the Copula computed frequency, i.e. the calculated frequency, is the sample size; Step 3.2, in the multi-dimensional random variable state, the optimal Copula function is determined according to the AIC of each edge, and the AIC expression is: wherein denotes the number of model parameters, denotes the log-likelihood function of the model; Step 3.3, the Copula model parameters are solved by using a maximum likelihood estimation method, and the joint probability density function is represented as: In the formula, represents a certain water quality monitoring index, represents a parameter to be solved, let , and ; Step 3.4, the AIC values of all edges of the tree structure are summed, and the tree structure with the minimum total AIC is selected as the optimal function model. 5.The black and odorous water eutrophication risk assessment method based on KDE-Vine Copula according to claim 1, characterized in that The specific steps of step 4 are as follows: Step 4.1, for solving the joint probability distribution, the joint probability density function is represented as: wherein is the joint probability density, is the Copula function, is the marginal distribution function, is the marginal density function; Step 4.2, according to the conditional probability, it is obtained that: The three-dimensional joint probability density function is decomposed as: It is derived that: Step 4.3, steps 4.1 and 4.2 are combined to obtain the joint probability density function: 。

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