METHOD FOR DETERMINING A THRESHOLD OF VEGETATION LOSSES UNDER COMPOSITE DROUGHT-HEAT-STRESS CONDITIONS

Description

2. The effects of individual stress factors on vegetation were addressed. For example, a probability distribution model of vegetation loss was created using the Vine Copula model in combination with environmental factors, or the recovery capacity of vegetation was assessed using resilience analyses. However, these methods have the following weaknesses: first, they do not fully consider the non-linear effects and interaction effects of the composite stress factors on vegetation loss; second, they do not quantify the specific relationship between the probability of loss and the resilience of vegetation, in particular, clear criteria for threshold determination are lacking; and third, the applicability and promotion of the models are limited, making it difficult to meet the application requirements of multi-regions and multi-ecosystems. Currently, there is no technical solution for the compound drought-heat stress that describes the relationship between the probability of vegetation loss andthe resilience to be comprehensive shows, in which a composite probability model, an autoregressive model and a generalized additive model are created,15 and which furthermore determines the threshold of vegetation resilience to reduce the probability of loss. In summary, current research on the effects of composite drought-heat stress on vegetation often focuses solely on predicting the probability of vegetation loss or the effect of a single stress factor. The non-linear synergistic effects of the composite stress factors on vegetation are not comprehensively analyzed. Furthermore, existing studies lack clear criteria for determining thresholds between vegetation resilience and the probability of loss, which complicates application in ecosystem management and conservation. Therefore, quantifying the effect of composite drought-heat stress on vegetation remains a challenge.Determining the dynamic relationship between the probability of loss and the resilience of vegetation, as well as developing a resilience threshold to reduce the probability of loss, presents pressing technical challenges. BE2026 / 7026 3 CONTENT OF THE PRESENT INVENTION The objective of the present invention is to provide a method for determining a threshold of vegetation loss under combined drought-heat-stress conditions in order to solve the existing problems in the prior art. By constructing a composite probability model, an autoregressive model, and a generalized additive model, the method can predict the probability of vegetation loss, quantify the resilience of the vegetation, and determine the threshold. To achieve the above-mentioned goal, the present invention offers the following solution: 10 a method for determining a threshold value of vegetation loss undercomposite drought-heat stress conditions, comprising: collecting ecological response data of vegetation; where the ecological response data include drought data, high-temperature data, and vegetation data; obtaining a probability of vegetation loss and quantifying vegetation resilience based on the ecological response data of vegetation; and analyzing a non-linear relationship between the probability of vegetation loss and the resilience of vegetation under drought-heat stress conditions to determine a resilience threshold with the probability of vegetation loss. Optionally, the drought data use a standardized precipitation evapotranspiration index (SPEI). The high-temperature data use a standardized temperature index (STI); and the vegetation data use a kernel-normalized vegetation index (kNDVI).25 Optionally, determining the probability of vegetation loss consequences of:Fitting a marginal distribution and estimating parameters of the vine copula based on ecological response data of the vegetation to construct a probability distribution model of vegetation loss under composite drought-heat-30 stress conditions; and quantifying an effect of different stress intensity combinations on vegetation loss based on the probability distribution model of vegetation loss to obtain the probability of vegetation loss. Optionally, the fitting of marginal distributions and the estimation of parameter 35 of the vine copula based on the ecological response data of the vegetation includes the following: BE2026 / 7026 4 Fitting a kNDVI sequence using five distribution fitting procedures, namely normal, gamma, log-normal, logistic and Weibull distributions, determining an optimal distribution using an Akaike information criterion and the Kolmogorov-Smirnov test, and fitting an SPEI sequence and an STI-Sequence using the normal distribution during the growing season; where kNDVI represents the kernel-normalized vegetation index, SPEI represents the standardized rainfall-evapotranspiration index, and STI represents the standardized temperature index; combining marginal variables using a CDVineCondFit function and by selecting multiple copula functions to construct an optimal vine-copula structure for kNDVI, SPEI, and STI; and creating a probability distribution model of vegetation losses under composite drought-heat-stress conditions based on the calculated optimal vine-copula structure and in combination with kNDVI, SPEI, and STI data. Optionally, the probability of vegetation loss is: 15 , where F(ndvi|spei,sti) denotes the probability of vegetation loss, ∂ is a symbol of the partial derivative, CNDVI,SPEI|STI is a common copula function of NDVI and SPEI under STI conditions and represents a common dependency structurederbeidendar, u(ndvi|sti)isteineRandwahrscheinlichdesNDVIunterSTIbedingungen,20 und u(spei|sti)isteineRandwahrscheinlichdesSPEIunterSTIbedingungen. Optionally, maintaining quantification of vegetation resilience includes: creating an autoregressive model based on the ecological response data of the vegetation, analyzing a vegetation response to drought and heat anomalies using long-term time series to quantitatively describe vegetation resilience under drought, heat, and stress conditions. Optionally, the autoregressive model is: , where Yt represents a kNDVI at time t, SPEIt represents a SPEI at time 30 t, STIt represents a STI at time t, εt represents an intersection point, α represents the recovery capacity of the vegetation, β represents the resistance of the vegetation to precipitation anomalies, and γ represents the resistance of the vegetation to temperature anomalies. BE2026 / 7026 5 Optionally, the determining threshold of resilience with the probability-The following: Analyzing the non-linear relationship between the probability of vegetation loss and the resilience of vegetation using the generalized additive model; characterizing a complex response mechanism by introducing a non-parametric smoothing function; and identifying the driving factors influencing the probability of vegetation loss by combining the F-test statistic in the generalized additive model with the weighting of the variables in the random forest model; and subsequently determining the resilience threshold with the probability of vegetation loss. The present invention has the following advantageous effects: 1. Precise quantification of synergistic effects By using the vine copula model, the synergistic effects of drought and high-temperature factors are included in the analysis, thereby enabling a precise quantification of the probability of vegetation loss under composite conditions.Stress conditions are realized, thus overcoming the limitations of single-factor analysis. 2. Determination of the dynamic threshold: By calculating vegetation resilience using an autoregressive model and combining it with the GAM model to reveal the dynamic non-linear relationship between the probability of vegetation loss and resilience, a clear criterion for determining resilience thresholds is provided, which has important application guidance. 3. Scientific accuracy in data processing: Replacing the traditional normalized vegetation index (NDVI) with kNDVI, in conjunction with its subsequent trend adjustment and seasonal adaptation, significantly improves the accuracy of the vegetation state quantification and the applicability of the model. 4. Breadth of Applicability and Practical Value 30 The present invention is characterized by easily accessible data sources, asimple modeling process and highly reproducible results. It can be used on a large scale for the assessment of stress risks and management practices in different regions and ecosystems and offers a scientific and implementable technical tool for vegetation protection and ecosystem management. BE2026 / 7026 6 DESCRIPTION OF THE DRAWING In order to more clearly illustrate the embodiments of the present invention or the prior art technical solutions, the accompanying drawings, which are to be used in the exemplary embodiments, are briefly described below, and it is evident that the drawings accompanying the following descriptions are merely some exemplary embodiments of the present invention, and persons skilled in the art can obtain further drawings in accordance with these drawings without inventive effort. Fig. 1 is a schematic diagram of a probability of the global terrestrialVegetation losses according to an embodiment of the present invention; Fig. 2 is a schematic diagram of the resilience of global terrestrial vegetation according to an embodiment of the present invention; Fig. 3 is a schematic diagram of the drought resistance of global terrestrial vegetation according to an embodiment of the present invention; Fig. 4 is a schematic diagram of the heat resistance of global terrestrial vegetation according to an embodiment of the present invention; Fig. 5 is a schematic diagram of results of the generalized additive model of an embodiment of the present invention; and Fig. 6 is a schematic flow diagram of a method for determining the threshold for vegetation losses under composite drought-heat-stress conditions according to an embodiment of the present invention. DETAILED DESCRIPTION The technical solutions of the embodiments of the present invention areThe invention is described in detail and completely below with reference to the attached drawings.25 Of course, the described embodiments represent only some, but not all, embodiments of the present invention. All further embodiments that become known to a person skilled in the art without inventive step on the basis of the embodiments of the present invention fall within the scope of protection of the present invention.30 In order to make the above-mentioned objectives, features and advantages of the present invention clearer and more understandable, the present invention is described in more detail below with reference to the attached drawings and specific embodiments. As shown in Fig. 6, this implementation proposes a method for determining a threshold of vegetation loss under composite drought-heat stress conditions. By constructing a composite probability model, an autoregressive model, and a generalized additive modelThe method enables the prediction of the probability of vegetation loss, the quantitative calculation of vegetation resilience, and the determination of the threshold. The method comprises: 5. Acquiring ecological response data of the vegetation, where the ecological response data include drought data, high-temperature data, and vegetation data; obtaining a probability of vegetation loss and a quantification of vegetation resilience based on the ecological response data of the vegetation; and 10. Analyzing a non-linear relationship between the probability of vegetation loss and the resilience of the vegetation under drought-heat stress conditions to determine a resilience threshold with the probability of vegetation loss. Specifically, in this implementation, the probability distribution of vegetation loss under the composite drought-heat-stress is first created using the Vine Copula model. Subsequently, the resilience of the vegetation is determined using an auto-regressive model. Finally, the nonlinear relationship between the probability of vegetation loss and resilience is investigated using a generalized additive model to derive the resilience threshold for reducing the probability of loss. This method provides a scientific and quantifiable solution for investigating the effects of composite drought-heat stress on vegetation and simultaneously provides a theoretical basis as well as practical tools for formulating strategies for ecosystem management and vegetation protection.25 Furthermore, the drought data use a standardized precipitation evapotranspiration index (SPEI); the high-temperature data use a standardized temperature index (STI); and the vegetation data use a kernel-normalized vegetation index (kNDVI). Specifically, in this implementation form, the recording and processing of ecological reaction data of the vegetation is carried out as follows:Drought data: To quantify drought stress intensity, this format uses the Standardized Rainfall-Evapotranspiration Index (SPEI). The SPEI is a drought index that comprehensively considers the water balance between precipitation and potential evapotranspiration (PET) and can more accurately reflect the water deficit caused by climate change. First, the water surplus or deficit (D) is calculated for each timescale, as shown below: . Second, the calculated D is aggregated across different timescales: 5 . Here, P and PET denote precipitation and potential evapotranspiration, respectively. j and k stand for the analyzed month and the aggregation timescale, respectively. Subsequently, a three-parameter logistic distribution is used to fit the total moisture balance for each calendar month, and the data sequence 10 is created by fitting the logistic probability density function: .where a is a scale coefficient, β is a shape coefficient, and γ is an original parameter value that can be calculated using an L-moment parameter estimation method. Therefore, the cumulative probability on a given time scale is obtained as follows: 15: . Finally, the cumulative probability is transformed using a standard normal distribution to obtain the corresponding time scale SPEI. 20. In this process, p is determined by fitting the log-logistics to derive the cumulative probability distribution function. The values ​​of the coefficients C0, C1, C2, d1, d2 and d3 are as follows: C0 = 2.515517, C₁ = 0.802853, C₂ = 0.010328, d₁ = 1.432788, d₂ = 0.189269, d₃ = 0.001308. BE2026 / 7026 9 High-temperature data: To comprehensively assess the impact of combined drought and heat events on vegetation, this version uses the Standardized Temperature Index (STI) based on temperature data to quantify heat stress intensity. The purpose of quantifying the heat-The purpose of the STI is to illustrate the independent contribution of temperature anomalies to vegetation growth and, in combination with the SPEI, to enable the identification of composite drought-heat events. The STI calculation process can be expressed as follows: where denotes the rank, n the number of observations, φ the standard normal distribution, and p the cumulative probability. Vegetation data: To quantify the response characteristics of the vegetation ecosystem under composite drought and heat stress, this implementation uses the normalized difference vegetation index (NDVI) as a fundamental indicator to characterize vegetation status. The NDVI is one of the most frequently used vegetation indices in remote sensing and effectively reflects changes in vegetation greenness, canopy density, and biomass. However, in areas with high vegetation cover (such as forests or monsoon grasslands), the NDVI shows a significantThis results in a saturation effect, which makes it insensitive to subtle changes under these conditions and thus impairs its ability to identify stress reactions. To address this problem, this implementation additionally calculates the kernel-normalized vegetation index (kNDVI), which amplifies subtle changes in the high NDVI range and allows for a more accurate representation of vegetation greenness. The kNDVI calculation process can be expressed as follows: 25 . Here, tanh is the hyperbolic tangent function. The use of kNDVI instead of the traditional NDVI, followed by trend adjustment and seasonal adaptation, significantly improves the accuracy of vegetation state quantification and the applicability of the model. 30 BE2026 / 7026 10 The calculated kNDVI value can be affected by long-term trends and seasonality, which is why Z-score standardization for the kNDVI value is necessary to eliminate disturbances caused by long-term trends and seasonality. The Z-score-The standardization process proceeds as follows: .5 Here, i and j represent month and year, X̅ and σj denote the mean and standard deviation of the time series of the variable X in the j-th month, respectively. The determination of the probability for vegetation loss also includes: Based on the data on the ecological response of the vegetation, the marginal distribution is adjusted and the parameter of the vine copulage is estimated in order to create a probability distribution model for vegetation loss under combined drought and heat stress conditions. Furthermore, determining the probability of vegetation loss includes the following: fitting a marginal distribution and estimating parameters of the vine copula based on ecological response data of the vegetation to construct a probability distribution model of vegetation loss under composite drought-heat stress conditions; and quantifying the effect of different stress intensity combinations on vegetation loss based on the probability distribution model of vegetation loss.tion losses to obtain the probability of vegetation losses.20 Furthermore, fitting the marginal distribution and estimating the parameters of the vine copula based on the ecological response data of the vegetation includes the following: fitting a kNDVI sequence using five distribution fitting procedures, namely normal, gamma, log-normal, logistic and Weibull distributions; determining an optimal distribution using an Akaike information criterion25 and the Kolmogorov-Smirnov test; and fitting an SPEI sequence and an STI sequence using the normal distribution during the growing season; where kNDVI stands for the kernel-normalized vegetation index, SPEI stands for the standardized rainfall-evapotranspiration index, and STI stands for the standardized Temperature index; 30 Combining marginal variables using a CDVineCondFit function and by selecting multiple copula functions to construct an optimal vine copula structure for kNDVI, SPEI and STI; and BE2026 / 7026 11Creating a probability distribution model of vegetation losses under combined drought-heat-stress conditions based on the calculated optimal vine copula structure and in combination with kNDVI, SPEI and STI data. Specifically, in this implementation, the probability of vegetation loss is calculated as follows. 5. Creation of the Vine Copula Model: A two-stage approach is used to create the vine copula model, which includes fitting the marginal distributions and estimating the vine copula parameters. During the vegetation period, five distribution fitting methods—normal, gamma, log-normal, logistic, and Weibull distributions—are selected to fit the kNDVI sequence. The Akaike Information Criterion (AIC) and the Kolmogorov-Smirnov test (KS test) were used to determine the optimal distribution. The SPEI and STI sequences during the vegetation period are fitted with the normal distribution. The CDVineCondFit function from The R package CDVineCopulaConditional is used, and six copula functions are included.Gaussian, Student's, Clayton, Frank, Gumbel, and Joe copulas are selected to link the 15 marginal variables and thus construct the optimal vine copula structure for kNDVI, SPEI, and STI (using AIC). Subsequently, based on the calculated optimal vine copula structure and using kNDVI, SPEI, and STI data, a probability distribution model for vegetation loss under composite drought and heat stress conditions is created to quantify the influence of different stress intensities on vegetation loss. The probability of vegetation loss is calculated as follows: Here, du(ndvi|spei) = h(ndvi,spei,θns), u(spei|sti) = h(spei,sti,θss), θns and θss are the parameters of the joint distribution between NDVI, SPEI and STI.25 This implementation uses the vine copula model to incorporate the synergistic effects of drought and high-temperature factors into the analysis, thereby enabling a precise quantification of the probability of vegetation loss under combined-The imposed stress conditions are realized, thus overcoming the limitations of single-factor analysis.30 Furthermore, maintaining the quantification of vegetation resilience includes the following: creating an autoregressive model based on the ecological response data of the vegetation, analyzing the vegetation's response to drought and heat anomalies under BE2026 / 7026 12, and using long-term time series to quantitatively describe vegetation resilience under drought and heat stress conditions. Specifically, in this implementation, vegetation resilience is quantified as follows. To more precisely characterize the response and recovery capabilities of vegetation under drought-heat stress, an autoregressive model (AR model) based on kNDVI, SPEI, and STI data is created in this implementation. Using a long-term time series, the vegetation's response to drought-heat anomalies is analyzed, thus quantitatively assessing its resilience under drought-heat stress conditions.(Table 1 explains the coefficients of the AR model). The calculations of the 10 autoregressive model are as follows: . Here, Yt represents a kNDVI at time t, SPEIt represents a SPEI at time t, STIt represents a STI at time t, εt represents an intersection point, α represents the recovery capacity of the vegetation, β represents the resistance of the vegetation to precipitation anomalies, and γ represents the resistance of the vegetation to temperature anomalies. Table 1: Explanation of the coefficients in the autoregressive model. Coefficient: Explanation of the absolute value: Explanation of positive and negative: Larger absolute values ​​indicate low recovery capacity of the ecosystem and low ability of the vegetation to recover from previous anomalies. Positive values ​​indicate that the kNDVI anomaly is similar to previous anomalies. Negative values ​​indicate that the kNDVI anomaly is similar to previous anomalies, but with a reversed trend. Larger absolute values ​​indicate lower drought resistance / hit-Resistance to. Positive values: Precipitation / temperature above average trigger a positive kNDVI reaction (higher kNDVI value). Negative values: Below-average precipitation / temperature lead to a negative kNDVI reaction (lower kNDVI value). 20 This implementation calculates the resilience of the vegetation using an autoregressive model and, in combination with the GAM model for revealing the dynamic nonlinear relationship between the probability of vegetation loss and resilience, it first provides a clear criterion for determining resilience thresholds, which has an important application. Furthermore, the determining threshold of resilience with the probability of vegetation loss includes the following: Analyzing the non-linear relationship between the probability of vegetation loss and the resilience of vegetation using the generalized additive model, characterizing a complex response mechanism by introducing a-non-parametric smoothing function; and identified the driving factors influencing the probability of vegetation loss by combining the F-test statistic in the generalized ad-10 divisive model with the weight rating of the variables in the random forest model, and subsequently determining the resilience threshold with the probability of vegetation loss. Specifically, in this implementation, the relationship between the probability of vegetation loss and resilience is analyzed as follows.15 For the systematic analysis of the nonlinear relationship between the probability of vegetation loss and the resilience of vegetation, a generalized additive model (GAM) is used. A nonparametric smoothing function is introduced to characterize the complex response mechanism. By combining the F-test statistic of the model with the weighting of the variables in the Random-Forest model (the Random-Forest weight is mainly determined by the weighting of theBy improving node purity or increasing the prediction error of each variable in the decision tree, the main factors influencing the probability of vegetation loss are identified, and the resilience threshold corresponding to the probability of vegetation loss is then determined.25 The generalized additive model is calculated as follows: . Here, S(·) represents the smoothing function; resilience, SPEI resistance, and STI resistance represent vegetation resilience, drought resistance, and heat resistance, respectively. The F-value is calculated as follows:30 . Here, SSR represents the sum of squares explained by the model, SSE represents the sum of squares of the residuals, and df1 and df2 represent the corresponding degrees of freedom. BE2026 / 7026 14 The larger the F-value, the greater the explanatory contribution of the variable to the response variable and the higher the significance. Through model fitting and parameter optimization, the resilience threshold is identified that can significantly reduce the probability of vegetation loss.and the applicability of the threshold is verified regionally. Step 5) Model Validation and Application Using real monitoring data from typical regions and ecosystems, the applicability of the created model and the thresholds is verified; and a scientific basis for the formulation of targeted strategies to reduce stress and strengthen resilience in ecological management is provided on the basis of research findings. This implementation is characterized by easily accessible data sources, a simple modeling process, and highly reproducible results. It can be used on a large scale for the assessment of stress risks and management practices in different regions and ecosystems and offers a scientific and implementable technical tool for vegetation protection and ecosystem management. The data processing and analysis in this implementation form are carried out entirely using the R programming language. To ensure the reproducibility of the research and theTo ensure methodological rigor, the following R packages are mainly used: SPEI (for calculating the standardized precipitation-evapotranspiration index SPEI), STI (for calculating the standardized temperature index STI), fitdistrplus (for fitting the probability distribution), goftest (tests for the goodness of fit of the distribution), CDVineCopulaConditional (joint modeling and conditional fitting of vine copula), gets (creation of an autoregressive model), mgcv (generalized additive model), and randomForest (assessment of variable importance). The specific implementation steps are as follows: Step 1 (standardization of the meteorological and vegetation indices): first, the standardized precipitation- The evapotranspiration index (SPEI) is calculated using monthly precipitation and evapotranspiration data. Then, the standardized temperature index (STI) is calculated using the "sti" function from the STI package, based on monthly temperature data.To eliminate the saturation effect of the NDVI, a desaturation process is then applied to the NDVI using formula (8) to obtain the kernel NDVI (kNDVI). 35 BE2026 / 7026 15 Step 2 (Calculating the Susceptibility of Vegetation): This step comprises two main parts: (1) Fitting the Marginal Distribution: The function "fitdist" from the package "fitdistrplus" is used to fit a normal distribution to the SPEI and STI data. Subsequently, the marginal probabilities (uSPEI and uSTI) are calculated using the function "pnorm". In parallel, five common distribution functions (normal 5 times, gamma, log-normal, logistic, and Weibull distributions) are selected to rethink the NDVI to adapt. The optimal fitting distribution is determined by combining the function “gofstat” from the package “goftest” with the AIC criterion and the Kolmogorov-Smirnov tester. The corresponding marginal probability (ukNDVI) is calculated. (2) Joint distribution modeling and calculation of the probability of vegetation-10Vegetation losses: The function CDVineCondFit from the package CDVineCopulaConditional is used to model uSPEI, uSTI, and uNDVI together using six Vine copula structures: Gaussian, Student's, Clayton, Frank, Gumbel, and Joe copula. The optimal model is then selected based on the AIC criterion. Subsequently, based on the selected copula structure, the probability of vegetation losses under composite drought-heat-stress is calculated using the function BiCopHfunc1. Step 3 (Calculation of vegetation resilience): First, the kNDVI value is standardized using the Z-score according to formula (9). Then, an autoregressive model is created based on the function "arx" from the package "gets", including the standardized kNDVI value, SPEI, and STI as explanatory variables. The regression coefficients of the model fit serve to measure the resilience of the vegetation to drought, heat, and stress, and their significance is detailed in Table 1.Step 4 (Calculating the relationship between the probability of vegetation loss and resilience): In this step, a generalized additive model (GAM) is created based on the GAM function of the mgcv package. The probability of vegetation loss serves as the dependent variable and the resilience of the vegetation as the independent variable. After the model is created, the F-value is calculated according to formula (13) to evaluate the explanatory power of the variables in the model. At the same time, the importance of the variables is analyzed using the randomForest package to identify the variables with the greatest influence on the explanatory power of the model. The critical threshold of the resilience response is determined accordingly. Fig. 1 shows the probability of global terrestrial vegetation loss (SPEI <-1.3 & STI >1.3, kNDVI <50% quantile), Fig. 2 shows the resilience of global terrestrial vegetation, and Fig. 3 illustrates the drought resistance of global terrestrial vegetation. Fig. 4 shows the heat resistance of global terrestrial vegetation.

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