Method for evaluating propagation probability from non-stationary meteorological drought to hydrological drought in changing environment

Abstract

The invention discloses a method for evaluating the propagation probability from non-stationary meteorological drought to hydrological drought in a changing environment, and the method comprises the steps: S1, constructing a non-stationary standardized rainfall index NSPI and a non-stationary standardized runoff index NSRI based on a GAMLSS model; s2, identifying drought events by adopting a three-threshold run-length theory, and identifying and extracting duration and intensity characteristics of the drought events from NSPI and NSRI sequences; s3, correlation coefficients between the NSPI sequences and the NSRI sequences of different time scales are calculated, and the propagation time from meteorological drought to hydrological drought is determined according to the maximum correlation coefficient; s4, according to the propagation time, calculating a drought propagation rate and a sensitivity index; s5, establishing time-varying marginal distribution of meteorological and hydrological drought characteristic variables by using a GAMLSS model, and constructing joint distribution of the meteorological and hydrological drought characteristic variables by using a time-varying Copula function so as to calculate a dynamic propagation probability threshold for triggering different levels of hydrological drought; according to the invention, the accuracy and early warning capability of drought propagation assessment in a changing environment are effectively improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of hydro-meteorological monitoring and risk assessment, and particularly relates to a method for evaluating the propagation probability of non-stationary meteorological drought to hydrological drought under changing environment. BACKGROUND

[0002] Drought is a natural disaster with long duration, wide influence and high frequency. According to statistics, 70% of natural disasters are meteorological disasters, and drought disasters account for about 50% of meteorological disasters. In recent years, with the significant increase in the frequency, intensity and influence range of drought in China, the risk of drought disaster is also increasing.

[0003] Hydrological drought is one of the important types of natural drought, and is closely related to the hydrological cycle and water balance in nature. In order to quantitatively evaluate the characteristics of hydrological drought, domestic and foreign scholars have proposed a variety of hydrological drought indices, such as the surface water supply index (SWSI), the Palmer drought index (PHDI), the standardized runoff index (SRI), the runoff drought index (SDI), etc., which have been widely applied.

[0004] However, in recent years, due to the strong interference of climate change and human activities, the hydro-meteorological extreme process presents obvious non-stationary characteristics, that is, the statistical parameters (mean, variance, etc.) of the hydro-meteorological sequence have time-varying characteristics, and the traditional drought evaluation method based on the assumption of stationarity is facing severe challenges. Under this background, it is urgent to build a non-stationary drought detection index considering the double influence of climate change and human activities, and to explore the evolution characteristics of non-stationary hydrological drought under changing environment, which has important practical significance for regional drought resistance and disaster reduction and water resources management.

[0005] In the field of non-stationary drought research, domestic and foreign scholars have made certain progress. Rigby and Stasinopoulos proposed the generalized additive model (GAMLSS), which can flexibly describe the linear or nonlinear relationship between the statistical parameters of the random variable sequence and the explanatory variables, providing important methodological support for subsequent non-stationary drought research. Russo et al. proposed a non-stationary standardized precipitation index, which significantly improves the stability of drought monitoring by considering the time-varying distribution parameters. Li et al. established a non-stationary standard precipitation index with climate factors as covariates, which proves to be more accurate in reflecting the drought characteristics of the basin than the traditional SPI. Bazrafshan et al. constructed a non-stationary comprehensive drought index, which shows obvious advantages in characterizing the time-varying trend of drought. Wang further proposed a non-stationary standardized runoff index with climate change factors and human impact factors as covariates, which shows better evaluation effect than the traditional method in the application of Luanhe River Basin.

[0006] However, the existing research still has obvious deficiencies: on the one hand, most of the non-stationary drought index researches mainly aim at meteorological drought, and mostly take time or climate factors as explanatory variables, and less systematically consider the influence of human activities. The fundamental reason is that there is a lack of dynamic indicators that can effectively quantify the influence of human activities (such as reservoir regulation) on hydrological processes, and couple them into the non-stationary drought assessment framework. On the other hand, current researches mainly focus on the construction of single drought type index, and the research on the propagation mechanism of meteorological drought to hydrological drought, especially the dynamic propagation law under non-stationary conditions, is still insufficient. In the basin with strong human activities, such as Dongting Lake Basin, the regulation effect of reservoir group has significantly changed the natural runoff process, and it is difficult to accurately assess the probability characteristics of drought propagation by only considering climate factors.

[0007] Therefore, establishing a non-stationary assessment framework that can simultaneously couple climate variability and human activities as double driving factors, and systematically reveal the dynamic influence of them on the propagation process of meteorological-hydrological drought, has become a key scientific problem to be solved in current drought research. SUMMARY

[0008] The purpose of the present application is to overcome the above-mentioned deficiencies, and provide a method for evaluating the propagation probability of non-stationary meteorological drought to hydrological drought under changing environment, which can simultaneously consider the double influence of climate change and human activities, and effectively improve the accuracy and early warning ability of drought propagation evaluation under changing environment.

[0009] To solve the above technical problems, the technical scheme adopted by the present application is as follows: a method for evaluating the propagation probability of non-stationary meteorological drought to hydrological drought under changing environment, which comprises the following steps: S1, based on the GAMLSS model, calculating the non-stationary standardized precipitation index NSPI and the non-stationary standardized runoff index NSRI, which are respectively used to represent non-stationary meteorological drought and non-stationary hydrological drought; S2, identifying drought events by using three-threshold run theory, and extracting the duration and intensity characteristics of drought events from the NSPI and NSRI sequences; S3, calculating the correlation coefficient between the NSPI sequence of different time scales and the NSRI sequence of 1 month time scale, and determining the propagation time from meteorological drought to hydrological drought according to the maximum correlation coefficient; S4, according to the propagation time, matching the associated meteorological-hydrological drought event pairs, and calculating the drought propagation rate and sensitivity index; S5, based on the matched drought event pairs, using the GAMLSS model to establish the time-varying marginal distribution of meteorological and hydrological drought characteristic variables, and using the time-varying Copula function to construct the joint distribution, and then calculating the dynamic propagation probability threshold of triggering different levels of hydrological drought.

[0010] Preferably, the method for constructing a model to calculate the non-stationary standardized precipitation index (NSPI) in step S1 includes: selecting multiple low-frequency climate indices related to the precipitation sequence of the study area; performing moving average processing on the climate indices and conducting lag correlation analysis to screen climate indices that are significantly related to the cumulative precipitation sequence as climate change factors (CI); using the climate change factors (CI) as explanatory variables of the GAMLSS model, and establishing an optimal non-stationary model through distribution function optimization and parameter estimation; and performing standard normalization processing on the cumulative precipitation probability obtained by fitting the optimal non-stationary model to obtain the NSPI.

[0011] Preferably, the method for constructing a model to calculate the non-stationary standardized runoff index (NSRI) in step S1 includes: calculating the reservoir index MRI, which reflects the impact of human activities; selecting multiple low-frequency climate indices related to the runoff sequence of the study area, and screening out the optimal climate factor CI through lag correlation analysis; using the climate factor CI and the reservoir index MRI together as explanatory variables of the GAMLSS model, and establishing an optimal non-stationary model through distribution function optimization and parameter estimation; and performing standard normalization on the runoff cumulative probability obtained by fitting the optimal non-stationary model to obtain the NSRI.

[0012] Preferably, the formula for calculating the reservoir index MRI is as follows: ; In the formula, N represents the number of reservoirs upstream of the hydrological station; A i The controlled catchment area (km²) above the i-th reservoir dam site 2 A T The controlled catchment area (km²) of the hydrological station 2 ); Let the beneficial storage capacity of the i-th reservoir be (in billions of m³). 3 );C t Annual runoff (100 million m³) at hydrological stations 3 ).

[0013] Preferably, in step S1, the GAMLSS model is used to establish the relationship between probability distribution parameters and explanatory variables, and its general expression is: ; In the formula, k = 1, 2, ..., p; η k and θ k Let β be a vector of length n. k For a length of J k The regression parameter vector, X k For n×J k The explanatory variable matrix, Z jk Given a known n×q jk Fixed design matrix, γjk It is a q jk A vector of normally distributed random variables of dimension Z jk jk Let q represent the j-th random effect. jk This represents the dimension of the random influence factor in the j-th random effect; If we assume that the random variable follows a two-parameter probability distribution and neglect the influence of random effects on the distribution parameters, then the general expression for the GAMLSS model is: ; ; In the formula, , is the connection function used to establish the mapping relationship between the distribution parameters and the explanatory variables; μ and σ are the location parameter and scale parameter of the selected probability distribution, respectively.

[0014] Preferably, in step S2, the three-threshold run theory is used to identify drought events, specifically including: when the drought index sequence first falls below the threshold Y1, a drought event is initiated; minor drought events with a duration of only 1 month and a drought index higher than the threshold Y2 are eliminated; and adjacent drought events with an interval of 1 month and a drought index lower than the threshold Y0 within that interval month are merged.

[0015] Preferably, step S3 specifically involves calculating the Pearson correlation coefficient between NSPI sequences on a timescale of 1 to 12 months and NSRI sequences on a timescale of 1 month, and selecting the NSPI timescale corresponding to the point where the correlation coefficient reaches its maximum value as the drought propagation time.

[0016] Preferably, in step S4, when matching meteorological-hydrological drought events, the propagation time length preceding the start time of the hydrological drought is used as the starting point of the matching window, and the end time of the hydrological drought is used as the ending point of the matching window. Meteorological drought events existing within this time window are considered as events that may trigger the hydrological drought. When there are multiple meteorological drought events within the matching window, the meteorological drought event with the greatest drought intensity is selected as the primary triggering event. When the matching windows of adjacent hydrological drought events overlap, the starting point of the matching window for subsequent hydrological drought events is adjusted to the end time of the previous hydrological drought event. The drought propagation rate T1 is the proportion of meteorological drought events that trigger hydrological drought to the total number of meteorological drought events; the drought sensitivity T2 is the proportion of hydrological drought events caused by meteorological drought to the total number of hydrological drought events.

[0017] Preferably, in step S5: a time-varying Clayton Copula function is used to establish the time-varying joint distribution of meteorological and hydrological drought characteristic variables, and the conditional probability P(NSRI ≤ hydrological drought threshold | NSPI ≤ meteorological drought threshold) ≥ 0.95 is used as the criterion to calculate the dynamic thresholds for the duration and intensity of meteorological drought corresponding to different levels of hydrological drought; the formula for the time-varying Clayton Copula function is:

[0018] Where C is the Clayton Copula function. These are the time-varying parameters of the Clayton copula. This is the constant term of the time-varying parameter model, i.e., the intercept. For the i-th external covariate The corresponding regression coefficients, where n is the total number of selected covariates. and These represent the marginal probabilities of drought duration and drought severity, respectively. It is a potential covariate.

[0019] Preferably, the specific process of establishing the time-varying marginal distribution of meteorological and hydrological drought characteristic variables using the GAMLSS model in step S5, and then deriving the propagation probability assessment method, is as follows: S5.1 Extract the feature variables of all successfully matched meteorological-hydrological drought event pairs from step S4, including the duration of meteorological drought. Meteorological drought intensity D Hydrological drought duration Hydrological drought intensity Using the GAMLSS model, time-varying marginal distributions are established for each feature variable: For meteorological drought characteristic variables and D The relationship between the link function of the distribution parameters of its GAMLSS model and the climate index CI is established, namely: , ; The distribution parameters of the GAMLSS model include the mean μ and standard deviation σ, for hydrological and drought characteristic variables. and The link function of the distribution parameters of its GAMLSS model is used to establish a relationship with the climate index CI and the reservoir index MRI, namely: , ; The marginal distribution parameters obtained through the above method are no longer fixed values, but dynamically change with the external covariates CI and MRI, thus accurately depicting the time-varying pattern of drought characteristics under changing environments. S5.2. Using a time-varying Copula function, the time-varying marginal distributions of the meteorological drought characteristic variables and the hydrological drought characteristic variables obtained in step S5.1 are coupled to construct a time-varying joint distribution function for the two: ; ; The parameters of the time-varying Copula function It is also related to covariates such as climate index and reservoir index, i.e. This allows us to capture the dynamic evolution of the dependency structure between meteorological drought and hydrological drought; S5.3. Based on the constructed time-varying joint distribution, conditional probability is used as the criterion for determining whether meteorological drought triggers hydrological drought; assuming X and Y are two random variables, the conditional probability of Y ≥ v given X ≥ u can be expressed as:

[0020] In the formula, x and y are the marginal distributions of random variables X and Y, respectively; x(u) is the cumulative probability when X ≤ u, y(v) is the cumulative probability when Y ≤ v; and C(x(u), y(v)) is the joint probability when X ≤ u and Y ≤ v.

[0021] Beneficial effects of this invention: 1. This invention can simultaneously consider the dual impacts of climate change and human activities, effectively improving the accuracy of drought propagation assessment and early warning capabilities under changing environments.

[0022] 2. This invention constructs a non-stationary drought index based on the GAMLSS model, breaking through the traditional stationarity assumption. By introducing climate and reservoir indices as time-varying covariates, it dynamically fits the distribution parameters of hydrological and meteorological sequences, accurately characterizing the non-stationary characteristics of drought under changing environments. It also establishes a dynamic joint distribution by combining a time-varying Copula function, enabling accurate assessment of the probability of meteorological-hydrological drought propagation. The obtained dynamic propagation threshold can significantly improve the accuracy of drought early warning, providing reliable technical support for watershed water resource management under changing environments. Attached Figure Description

[0023] Figure 1 : Overall flowchart of the method of this invention; Figure 2 Unit root test results of precipitation series from meteorological stations in the four river basins of Dongting Lake; Figure 3MRI Changes in the Corrected Reservoir Index of the Four Rivers in the Dongting Lake Basin Figure 4 The preferred method is to construct an AIC plot based on the probability distribution function (Gamma, Lognormal2, Weibull, Gumbel, Logistic, Normal) from the precipitation sequence. Figure 5 The preferred method is to construct an AIC plot based on the probability distribution function (Gamma, Lognormal2, Weibull, Gumbel, Logistic, Normal) constructed from the runoff sequence. Figure 6 For precipitation sequences, diagnostic plots of GAMLSS model fit (QQ plot, residual worm plot). Figure 7 For runoff sequences, diagnostic plots of GAMLSS model fit (QQ plot, residual worm plot). Figure 8 Comparison chart of SPI and NSPI sequences; Figure 9 Comparison of SRI and NSRI sequences; Figure 10 : Schematic diagram of the three-threshold run theory for identifying drought events; Figure 11 : Graph showing the time of drought spread and its seasonal variation; Figure 12 Comparison of AIC values ​​of drought event characteristics (drought duration and drought intensity sequences) on the preferred marginal distribution function. Figure 13 : Dynamic propagation threshold curve of drought duration.

[0024] Figure 14 : Dynamic propagation threshold curve of drought intensity. Detailed Implementation

[0025] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0026] Example 1: A method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, comprising the following steps: S1. Based on the GAMLSS model, construct the non-stationary standardized precipitation index (NSPI) and the non-stationary standardized runoff index (NSRI): ① For NSPI, several key low-frequency climate indices related to the precipitation sequence in the study area were selected. The climate indices were averaged over a 12-month period to reduce randomness. The Kendall rank correlation test was used to perform lag correlation analysis for 0-12 months. Climate indices that were significantly correlated with the cumulative precipitation sequence were selected as climate change factors. The selected climate indices were used as explanatory variables in the GAMLSS model. The optimal non-stationary model was established through distribution function optimization and parameter estimation. The cumulative precipitation probability obtained from the model fitting was normalized to obtain NSPI. ② For NSRI, the improved reservoir index MRI was calculated, and several key low-frequency climate indices related to the runoff sequence in the study area were selected. The same lag correlation analysis method was used to obtain the optimal climate factors. The climate indices and MRI were used together as explanatory variables of the GAMLSS model. The optimal non-stationary model was established through distribution function optimization and parameter estimation. The runoff cumulative probability obtained by model fitting was processed by standard normalization to obtain NSRI.

[0027] S2. For the NSPI and NSRI sequences, the three-threshold run theory is used to identify meteorological and hydrological drought events, and the drought duration and intensity characteristic variables are extracted: when the drought index sequence is lower than the threshold Y1, a drought event is initiated, and minor events with a duration of only 1 month and a drought index higher than the threshold Y2 are removed, and adjacent drought events with an interval of 1 month and a drought index lower than the threshold Y0 in that month are merged. S3. Calculate the correlation coefficient between NSPI and NSRI sequences at different time scales, and determine the propagation time from meteorological drought to hydrological drought based on the maximum correlation coefficient. S4. Based on the propagation time, match the associated meteorological-hydrological drought event pairs and calculate the drought propagation rate and sensitivity index: the drought propagation rate T1 is defined as the proportion of the number of meteorological drought events that trigger hydrological drought to the total number of meteorological drought events; the drought sensitivity T2 is defined as the proportion of the number of hydrological drought events caused by meteorological drought to the total number of hydrological drought events. S5. Based on the matched drought event pairs, the time-varying marginal distribution of meteorological and hydrological drought characteristic variables is established using the GAMLSS model, and its joint distribution is constructed using the time-varying Copula function, thereby calculating the dynamic propagation probability threshold for triggering different levels of hydrological drought. In step S1, a 12-month time scale was used to calculate the cumulative precipitation and runoff series for the 12 months, and the unit root test method was used to analyze their stationarity characteristics.

[0028] Preferably, the definition of the GAMLSS model in step S1 is as follows: ; In the formula, k = 1, 2, ..., p; ηk and θ k Let β be a vector of length n. k For a length of J k The regression parameter vector, X k For n×J k The explanatory variable matrix, Z jk Given a known n×q jk Fixed design matrix, γ jk It is a q jk A vector of normally distributed random variables of dimension Z jk γ jk Let q represent the j-th random effect. jk This represents the dimension of the random influence factor in the j-th random effect; If we assume that the random variable follows a two-parameter probability distribution and neglect the influence of random effects on the distribution parameters, then the general expression for the GAMLSS model is: .

[0029] In step S1, eight low-frequency climate indices with physical origins related to the runoff sequence in the study area—Niño3.4, DMI, SOI, PDO, AO, NAO, AMO, and NPO—are selected. A 12-month moving average is applied to the climate indices to reduce their randomness. The Kendall rank correlation test is used in the lag correlation analysis to screen climate indices that are significantly correlated with the cumulative runoff sequence (significance level α=5%) as climate change factors.

[0030] The improved reservoir index MRI definition in step S1 is as follows: ; In the formula, N represents the number of reservoirs upstream of the hydrological station; A i The controlled catchment area (km²) above the i-th reservoir dam site 2 A T The controlled catchment area (km²) of the hydrological station 2 ); Let the beneficial storage capacity of the i-th reservoir be (in billions of m³). 3 );C t Annual runoff (100 million m³) at hydrological stations 3 ).

[0031] In step S1, six commonly used two-parameter probability distribution functions (Gamma, Lognormal2, Weibull, Gumbel, Logistic, Normal) are used to fit a stationary model (with fixed distribution parameters), and the optimal distribution function is selected according to the principle of minimizing the AIC value.

[0032] In step S1, the parameters of the GAMLSS model are estimated using the maximum likelihood method. Then, two non-stationary models based on the optimal distribution function are constructed. One is a non-stationary model with time as the explanatory variable, and the other is a non-stationary model with CI or CI and MRI as explanatory variables. The cubic spline function is used as the connection function between the distribution parameters and the explanatory variables. The model with the smallest AIC value is selected as the optimal non-stationary model.

[0033] In step S1, the fitted values ​​of precipitation or runoff sequences are obtained based on the optimal non-stationary model, the cumulative probability of the fitted sequence is calculated, and the sequence is normalized to obtain the NSPI and NSRI index values.

[0034] In step S2, the threshold Y1 is set to -0.3, the threshold Y2 is set to -0.5, and the threshold Y0 is set to 0. By combining these three thresholds, we can effectively identify drought events with practical significance, while eliminating noise interference and reasonably merging adjacent drought processes.

[0035] In step S2, drought duration is defined as the number of months that a drought event lasts from start to finish; drought intensity is defined as the cumulative absolute value of the difference between the drought index value and the threshold Y1 for each month during the drought event.

[0036] In step S3, the Pearson correlation coefficient between the NSPI sequence with a time scale of 1 to 12 months and the NSRI sequence with a time scale of 1 month is calculated, and the NSPI time scale corresponding to the maximum value of the correlation coefficient is selected as the drought propagation time.

[0037] In step S3, the propagation time of drought has obvious seasonal characteristics. The propagation time is shorter in summer, ranging from 0 to 2 months, while the propagation time is longer in spring and autumn. There are obvious spatial differences in the propagation time of each sub-basin.

[0038] In step S4, when matching meteorological-hydrological drought events, the propagation time length preceding the start time of the hydrological drought is used as the starting point of the matching window, and the end time of the hydrological drought is used as the ending point of the matching window. Meteorological drought events existing within this time window are considered as events that may trigger the hydrological drought.

[0039] In step S4, when there are multiple meteorological drought events within the matching window, the meteorological drought event with the greatest drought intensity is selected as the main triggering event; when the matching windows of adjacent hydrological drought events overlap, the starting point of the matching window for subsequent hydrological drought events is adjusted to the end time of the previous hydrological drought event.

[0040] In step S5, the meteorological and hydrological drought characteristic variables include drought duration and drought intensity, and their time-varying marginal distributions are established respectively. Similarly, six commonly used two-parameter probability distribution functions (Gamma, Lognormal2, Weibull, Gumbel, Logistic, Normal) are preferred. The marginal distribution of the meteorological drought characteristic variable uses the climate index as the explanatory variable, and the marginal distribution of the hydrological drought characteristic variable uses the climate index and MRI as explanatory variables.

[0041] In step S5, a time-varying Clayton Copula function is used to construct a joint distribution of meteorological drought characteristics and hydrological drought characteristics. The parameters of the Copula function change over time and establish a linear functional relationship with external covariates such as climate indices and MRI. The formula for the time-varying Clayton Copula function is:

[0042] in These are the time-varying parameters of the Clayton copula. and These represent the marginal probabilities of drought duration and drought severity, respectively. It is a potential covariate.

[0043] In step S5, the time-varying marginal distribution of meteorological and hydrological drought characteristic variables is established using the GAMLSS model, and the specific implementation process of the propagation probability assessment method is as follows: ① Extract the feature variables of all successfully matched meteorological-hydrological drought event pairs from step S4, including the duration of meteorological drought. Meteorological drought intensity D Hydrological drought duration Hydrological drought intensity Using the GAMLSS model, time-varying marginal distributions are established for each feature variable: For meteorological drought characteristic variables ( ,and D The relationship between the GAMLSS model distribution parameters (such as mean μ and standard deviation σ) and the climate index CI is established, i.e.: , .

[0044] For hydrological drought characteristic variables ( and ), and establish a relationship between the link function of its GAMLSS model distribution parameters (such as mean μ, standard deviation σ) and the climate index CI and reservoir index MRI, that is: , .

[0045] The marginal distribution parameters obtained through the above method are no longer fixed values, but dynamically change with the external covariates CI and MRI, thus accurately depicting the time-varying pattern of drought characteristics under changing environments.

[0046] ② Using a time-varying Copula function (preferably the time-varying Clayton Copula), couple the time-varying marginal distributions of the meteorological drought characteristic variables obtained in step ① with the hydrological drought characteristic variables to construct a time-varying joint distribution function for the two:

[0047]

[0048] The parameters of the time-varying Copula function It is also related to covariates such as climate index and reservoir index, i.e. This allows us to capture the dynamic evolution of the dependency structure between meteorological drought and hydrological drought.

[0049] ③ Based on the constructed time-varying joint distribution, conditional probability is used as the criterion for determining whether meteorological drought triggers hydrological drought. Assuming X and Y are two random variables, the conditional probability of Y ≥ v given X ≥ u can be expressed as:

[0050] In the formula, x and y are the marginal distributions of random variables X and Y, respectively; x(u) is the cumulative probability when X ≤ u, y(v) is the cumulative probability when Y ≤ v; and C(x(u), y(v)) is the joint probability when X ≤ u and Y ≤ v.

[0051] In this embodiment, X represents meteorological drought characteristics, and Y represents hydrological drought characteristics of different levels. The cumulative probability intervals for drought events of mild, moderate, severe, and extreme drought levels are (0, 0.5), [0.5, 0.75), [0.75, 0.9), and [0.9, 1). After constructing a joint distribution model of meteorological and hydrological drought characteristics, the conditional probability that the hydrological drought level exceeds moderate, severe, or extreme drought when the meteorological drought characteristic is greater than a certain value is calculated. As the meteorological drought condition increases, the conditional probability gradually approaches 1. This paper selects a confidence level of 0.95 for the drought propagation threshold, that is, when the conditional probability is equal to 0.95, the corresponding meteorological drought characteristic value is regarded as the drought characteristic propagation threshold of that level of hydrological drought.

[0052] Example 2: As Figure 1As shown, the method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments includes the following specific steps: S1. Based on the GAMLSS model, construct the non-stationary standardized precipitation index NSPI and the non-stationary standardized runoff index NSRI to characterize non-stationary meteorological drought and non-stationary hydrological drought, respectively. S2. For the NSPI and NSRI sequences, run theory is used to identify meteorological and hydrological drought events, and drought duration and intensity characteristic variables are extracted; S3. Calculate the correlation coefficient between NSPI and NSRI sequences at different time scales, and determine the propagation time from meteorological drought to hydrological drought based on the maximum correlation coefficient. S4. Based on the propagation time, match the associated meteorological-hydrological drought event pairs and calculate the drought propagation rate and sensitivity index; S5. Based on the matched drought event pairs, the time-varying marginal distribution of meteorological and hydrological drought characteristic variables is established using the GAMLSS model, and its joint distribution is constructed using the time-varying Copula function, thereby calculating the dynamic propagation probability threshold for triggering different levels of hydrological drought. In step S1, a 12-month time scale was used to calculate the cumulative runoff series over 12 months, and the unit root test method was used to analyze its stationarity characteristics.

[0053] In step S1, the parameters of the GAMLSS model are estimated using the maximum likelihood method. Then, two non-stationary models based on the optimal distribution function are constructed. One is a non-stationary model with time as the explanatory variable, and the other is a non-stationary model with CI or CI and MRI as explanatory variables. The cubic spline function is used as the connection function between the distribution parameters and the explanatory variables. The model with the smallest AIC value is selected as the optimal non-stationary model.

[0054] In step S1, the fitted values ​​of precipitation or runoff sequences are obtained based on the optimal non-stationary model, the cumulative probability of the fitted sequence is calculated, and the sequence is normalized to obtain the NSPI and NSRI index values.

[0055] In step S2, the threshold Y1 is set to -0.3, the threshold Y2 is set to -0.5, and the threshold Y0 is set to 0. By combining these three thresholds, we can effectively identify drought events with practical significance, while eliminating noise interference and reasonably merging adjacent drought processes.

[0056] In step S2, drought duration is defined as the number of months that a drought event lasts from start to finish; drought intensity is defined as the cumulative absolute value of the difference between the drought index value and the threshold Y1 for each month during the drought event.

[0057] In step S3, the Pearson correlation coefficient between the NSPI sequence with a time scale of 1 to 12 months and the NSRI sequence with a time scale of 1 month is calculated, and the NSPI time scale corresponding to the maximum value of the correlation coefficient is selected as the drought propagation time.

[0058] In step S3, the propagation time of drought has obvious seasonal characteristics. The propagation time is shorter in summer, ranging from 0 to 2 months, while the propagation time is longer in spring and autumn. There are obvious spatial differences in the propagation time of each sub-basin.

[0059] In step S4, when matching meteorological-hydrological drought events, the propagation time length preceding the start time of the hydrological drought is used as the starting point of the matching window, and the end time of the hydrological drought is used as the ending point of the matching window. Meteorological drought events existing within this time window are considered as events that may trigger the hydrological drought.

[0060] In step S4, when there are multiple meteorological drought events within the matching window, the meteorological drought event with the greatest drought intensity is selected as the main triggering event; when the matching windows of adjacent hydrological drought events overlap, the starting point of the matching window for subsequent hydrological drought events is adjusted to the end time of the previous hydrological drought event.

[0061] In step S5, the meteorological and hydrological drought characteristic variables include drought duration and drought intensity, and their time-varying marginal distributions are established respectively. Similarly, six commonly used two-parameter probability distribution functions (Gamma, Lognormal2, Weibull, Gumbel, Logistic, Normal) are preferred. The marginal distribution of the meteorological drought characteristic variable uses the climate index as the explanatory variable, and the marginal distribution of the hydrological drought characteristic variable uses the climate index and MRI as explanatory variables.

[0062] In step S5, a joint distribution of meteorological drought characteristics and hydrological drought characteristics is constructed using a time-varying Clayton Copula function. The parameters of the Copula function change over time and establish a linear functional relationship with external covariates such as climate indices and MRI.

[0063] Taking the four tributaries of Dongting Lake as an example, the unit root test method was used to analyze the stationarity characteristics of the runoff series from 1956 to 2020 over a 12-month timescale of 76 rain gauge stations and 4 hydrological stations in the four tributaries of Dongting Lake. Table 1 shows that the p-values ​​of the runoff series from the four hydrological stations in the four tributaries of Dongting Lake are all greater than 0.05, indicating that the runoff series exhibit non-stationarity.

[0064]

[0065] Using the Kendall rank correlation test at a 5% significance level, key climate indices influencing monthly precipitation and runoff sequences in the four river basins were analyzed. As shown in Table 2, the climate indices significantly correlated with precipitation and runoff sequences varied across different months, with different lag times. Notably, NAO, WP, and DMI exhibited the strongest and most persistent correlations with precipitation and runoff in most months. For example, NAO showed a significant influence from February to June, while WP had a particularly strong influence in February, April, and November. These identified climate indices were subsequently incorporated as explanatory variables into the GAMLSS framework to simulate time-varying parameters.

[0066]

[0067] like Figure 3 This study presents a modified reservoir index (MRI) to represent human impacts, particularly reservoir regulation. Each reservoir is assigned to its respective sub-basin based on geographical location, and MRI values ​​are calculated annually, taking into account the operational schedule of each reservoir, thus capturing the dynamic impact of human activities on hydrological drought.

[0068] Six probability distribution functions were used to fit the stationary model, and the best-fit distribution of precipitation and runoff sequences was selected according to the principle of minimizing the AIC value.

[0069] like Figure 4 and Figure 5 For precipitation series, the Weibull distribution showed the best fit, followed by the Gamma distribution, while the Lognormal2, Gumbel, and Logistic distributions showed poor fit. For runoff series, the Lognormal2 distribution showed the best fit, followed by the Gamma distribution, while the Weibull, Gumbel, and Logistic distributions showed poor fit.

[0070] The GAMLSS model is constructed based on the optimized probability distribution function (as shown in Table 3), and the best model is selected according to the AIC criterion and the model parameter test results.

[0071]

[0072] Table 4 shows the stationary models (Model 0) for fitting precipitation and runoff data in the Xiangshui River basin, one of the four river basins of Dongting Lake; the non-stationary models (Model 1) with time as the explanatory variable; and the non-stationary models (Model 2) with CI or MRI as explanatory variables. For the precipitation series, Model 3 has the lowest AIC value compared to the other models, indicating that Model 3 has the best fit to the precipitation series in the study area. For the runoff series, Model 5 has the lowest AIC value compared to the other models, indicating that Model 5 has the best fit to the runoff series in the study area. It is worth noting that, by incorporating climate and anthropogenic factors as covariates, the non-stationary models exhibit a more robust fit than the stationary models.

[0073]

[0074] like Figure 6 and Figure 7 As shown, to further verify the reliability of the optimal nonstationary model, the normal QQ plot and residual worm plot of the June precipitation and runoff series in the Xiangjiang River Basin were analyzed. The figures show that the results of the normal QQ plot and worm plot do not deviate significantly from normality, indicating that the optimized model effectively fits the precipitation and runoff series.

[0075] like Figure 8 , Figure 9 As shown, the temporal evolution of SPI and NSPI, and SRI and NSRI at a 12-month timescale is illustrated in the four sub-basins. Overall, the two sets of indices are highly consistent in characterizing the temporal dynamics of drought evolution and can effectively identify major drought events.

[0076] like Figure 10 As shown, the three-threshold runs theory method is used to identify drought events in the NSPI and NSRI index sequences. The method consists of three steps: (1) initial identification is performed when the drought index first falls below the threshold Y1 (−0.3); (2) events that last only one month and have an index value higher than Y2 (−0.5) are removed; (3) if two drought events are only one month apart and the index for that month is lower than Y0 (0), they are merged, and the duration and severity are calculated by arithmetic sum. Therefore, there are three drought events in Figure 9, namely b, c + d and e.

[0077] like Figure 11As shown, the propagation time pattern of the four rivers in the Dongting Lake basin is illustrated. Summer has the shortest propagation time (0-2 months) across all basins, likely due to increased evaporation and accelerated hydrological response. In contrast, spring and autumn show longer propagation lags, with the Xiangjiang River basin experiencing particularly long propagation periods during these seasons. The other three basins exhibit a more consistent seasonal pattern. Spatial variability is also significant: in the Zijiang River basin, drought propagation occurs year-round except from July to September; while the Yuanjiang and Lishui River basins have a higher probability of propagation from April to May, and the peak period in the Xiangjiang River basin is concentrated in June and July.

[0078] Table 5 shows that further analysis of drought propagation efficiency and sensitivity in the four river basins of Dongting Lake reveals important spatial distribution patterns. In terms of propagation rate (T1, the proportion of meteorological drought triggering hydrological drought), the Zijiang and Lishui basins have higher rates (both 0.57), indicating a greater likelihood of drought propagation; while the Xiangjiang and Yuanjiang basins have relatively lower propagation rates (0.49 and 0.45, respectively). Hydrological drought sensitivity (T2, the proportion of hydrological drought caused by meteorological drought) is highest in the Lishui basin (0.70), followed by the Xiangjiang basin (0.67), with the Yuanjiang and Zijiang basins at 0.60 and 0.57, respectively. These results indicate that the Lishui basin is most sensitive to meteorological drought input, while the Zijiang basin, although having a high propagation rate, has relatively low sensitivity, possibly related to the regulating effect of reservoir or groundwater interaction.

[0079]

[0080] As shown in Table 6, the Weibull marginal distribution function was used to fit the univariate marginal distribution functions of meteorological (hydrological) drought duration and meteorological (hydrological) drought intensity, respectively. For successfully matched drought events, for meteorological drought characteristics (drought duration / intensity), the climate index was introduced as an explanatory variable into the generalized additive model (GAMLSS) for time-varying marginal distribution fitting; for hydrological drought characteristics (drought duration / intensity), both the climate index and the modified reservoir index were used as explanatory variables, and the time-varying marginal distribution fitting was completed through the GAMLSS model.

[0081]

[0082] like Figure 12As shown, the AIC values ​​of well-matched drought event features (drought duration and drought intensity sequences) on the preferred marginal distribution function are compared. Compared to the static model (SM), all five time-varying models (NSM1-NSM5) achieved lower AIC values, confirming their superior fitting performance. Specifically, the location parameter (μ) and scale parameter (σ) of drought features showed significant correlations with specific climate indices and modified reservoir index (MRI), revealing the combined role of natural climate variability and human activities in regulating drought characteristics.

[0083] Table 7 shows the AIC values ​​of each Copula model. In all sub-basins and for both drought characteristics (duration and severity), the time-varying parameter Clayton Copula achieved lower (more negative) AIC values ​​than the constant parameter models. Taking the Xiangjiang River basin as an example, after introducing the time-varying parameter, the AIC value of the drought duration model improved from -7.06 to -8.07, and the drought severity model improved from -4.53 to -5.74. Similar improvements were observed in other basins, indicating that the dynamic Copula model can more accurately capture the dynamic evolution of the dependency structure among drought variables under non-stationary conditions.

[0084]

[0085] Table 8 shows the classification of different drought levels based on the meteorological drought index and the hydrological drought index.

[0086]

[0087] like Figure 13 , Figure 14 As shown, the propagation thresholds of drought duration and intensity at different drought levels are presented in the four river basins of Dongting Lake. Average values ​​are taken across the four sub-basins. The duration thresholds for moderate, severe, and extremely severe hydrological drought are 7.6 months, 8.6 months, and 10.4 months, respectively, while the corresponding severity thresholds are 7.5, 8.3, and 10.5. Spatially, the Lishui River basin has the lowest thresholds (e.g., moderate drought duration of 4.46 months), indicating its high vulnerability to drought propagation; while the Xiangjiang River basin has the highest thresholds (e.g., moderate drought duration of 8.01 months), showing stronger resilience or buffering capacity. These thresholds serve as key benchmarks for drought early warning, indicating the minimum meteorological drought conditions required to trigger different levels of hydrological drought.

[0088] The above embodiments are merely preferred technical solutions of the present invention and should not be considered as limitations on the present invention. The scope of protection of the present invention should be limited to the technical solutions described in the claims, including equivalent substitutions of the technical features described in the claims. That is, equivalent substitutions and improvements within this scope are also within the scope of protection of the present invention.

Claims

1. A method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, characterized in that: It includes the following steps: S1. Based on the GAMLSS model, calculate the non-stationary standardized precipitation index (NSPI) and the non-stationary standardized runoff index (NSRI) to characterize non-stationary meteorological drought and non-stationary hydrological drought, respectively. S2. Drought events are identified using the three-threshold run theory. The duration and intensity characteristics of drought events are identified and extracted from NSPI and NSRI sequences. S3. Calculate the correlation coefficient between NSPI sequences at different time scales and NSRI sequences at a 1-month time scale, and determine the propagation time from meteorological drought to hydrological drought based on the maximum correlation coefficient. S4. Based on the propagation time, match the associated meteorological-hydrological drought event pairs and calculate the drought propagation rate and sensitivity index; S5. Based on the matched drought event pairs, the time-varying marginal distribution of meteorological and hydrological drought characteristic variables is established using the GAMLSS model, and its joint distribution is constructed using the time-varying Copula function, thereby calculating the dynamic propagation probability threshold for triggering different levels of hydrological drought.

2. The method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, as described in claim 1, is characterized in that... The method for constructing a model to calculate the non-stationary standardized precipitation index (NSPI) in step S1 includes: selecting multiple low-frequency climate indices related to the precipitation sequence of the study area; performing moving average processing on the climate indices and conducting lag correlation analysis to select climate indices that are significantly related to the cumulative precipitation sequence as climate change factors (CI); using the climate change factors (CI) as explanatory variables of the GAMLSS model, and establishing an optimal non-stationary model through distribution function optimization and parameter estimation; and performing standard normalization processing on the cumulative precipitation probability obtained by fitting the optimal non-stationary model to obtain the NSPI.

3. The method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, as described in claim 1, is characterized in that... The method for constructing a model to calculate the non-stationary standardized runoff index (NSRI) in step S1 includes: calculating the reservoir index MRI, which reflects the impact of human activities; selecting multiple low-frequency climate indices related to the runoff sequence in the study area, and screening out the optimal climate factor CI through lag correlation analysis; using the climate factor CI and the reservoir index MRI together as explanatory variables of the GAMLSS model, and establishing the optimal non-stationary model through distribution function optimization and parameter estimation; and performing standard normalization on the runoff cumulative probability obtained by fitting the optimal non-stationary model to obtain the NSRI.

4. The method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, as described in claim 3, is characterized in that... The formula for calculating the reservoir index MRI is as follows: ； In the formula, N represents the number of reservoirs upstream of the hydrological station; A i The controlled catchment area (km²) above the i-th reservoir dam site 2 A T The controlled catchment area (km²) of the hydrological station 2 ); Let the beneficial storage capacity of the i-th reservoir be (in billions of m³). 3 ); C t Annual runoff (100 million m³) at hydrological stations 3 ).

5. The method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, as described in claim 1, is characterized in that... In step S1, the GAMLSS model is used to establish the relationship between probability distribution parameters and explanatory variables, and its general expression is: ； In the formula, k = 1, 2, ..., p; η k and θ k Let β be a vector of length n. k For a length of J k The regression parameter vector, X k For n×J k The explanatory variable matrix, Z jk Given a known n×q jk Fixed design matrix, γ jk It is a q jk A vector of normally distributed random variables of dimension Z jk jk Let q represent the j-th random effect. jk This represents the dimension of the random influence factor in the j-th random effect; If we assume that the random variable follows a two-parameter probability distribution and neglect the influence of random effects on the distribution parameters, then the general expression for the GAMLSS model is: ； ； In the formula, , is the connection function used to establish the mapping relationship between the distribution parameters and the explanatory variables; μ and σ are the location parameter and scale parameter of the selected probability distribution, respectively.

6. The method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, as described in claim 1, is characterized in that... In step S2, drought events are identified using the three-threshold run theory, specifically including: initiating a drought event when the drought index sequence first falls below the threshold Y1; eliminating minor drought events with a duration of only 1 month and a drought index higher than the threshold Y2; and merging adjacent drought events with an interval of 1 month and a drought index lower than the threshold Y0 within that interval month.

7. The method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, as described in claim 1, is characterized in that... Specifically, step S3 involves calculating the Pearson correlation coefficient between NSPI sequences with a timescale of 1 to 12 months and NSRI sequences with a timescale of 1 month, and selecting the NSPI timescale that results in the maximum correlation coefficient as the drought propagation time.

8. The method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, as described in claim 1, is characterized in that... In step S4, when matching meteorological and hydrological drought events, the propagation time length preceding the start time of the hydrological drought is used as the starting point of the matching window, and the end time of the hydrological drought is used as the ending point of the matching window. Meteorological drought events existing within this time window are considered as events that may trigger the hydrological drought. When there are multiple meteorological drought events within the matching window, the meteorological drought event with the strongest drought intensity is selected as the main triggering event. When the matching windows of adjacent hydrological drought events overlap, the starting point of the matching window for subsequent hydrological drought events is adjusted to the end time of the previous hydrological drought event. The drought propagation rate T1 is the proportion of meteorological drought events that trigger hydrological drought to the total number of meteorological drought events; the drought sensitivity T2 is the proportion of hydrological drought events caused by meteorological drought to the total number of hydrological drought events.

9. The method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, as described in claim 1, is characterized in that... In step S5: a time-varying Clayton Copula function is used to establish the time-varying joint distribution of meteorological and hydrological drought characteristic variables, and the conditional probability P(NSRI ≤ hydrological drought threshold | NSPI ≤ meteorological drought threshold) ≥ 0.95 is used as the criterion to calculate the dynamic thresholds for the duration and intensity of meteorological drought corresponding to different levels of hydrological drought; the formula for the time-varying Clayton Copula function is: Where C is the Clayton Copula function. These are the time-varying parameters of the Clayton copula. This is the constant term of the time-varying parameter model, i.e., the intercept. For the i-th external covariate The corresponding regression coefficients, where n is the total number of selected covariates. and These represent the marginal probabilities of drought duration and drought severity, respectively. It is a potential covariate.

10. The method for assessing the probability of non-stationary meteorological drought spreading to hydrological drought under changing environments, as described in claim 1, is characterized in that... In step S5, the time-varying marginal distribution of meteorological and hydrological drought characteristic variables is established using the GAMLSS model, and the specific process of deriving the propagation probability assessment method is as follows: S5.1 Extract the feature variables of all successfully matched meteorological-hydrological drought event pairs from step S4, including the duration of meteorological drought. Meteorological drought intensity D Hydrological drought duration Hydrological drought intensity Using the GAMLSS model, time-varying marginal distributions are established for each feature variable: For meteorological drought characteristic variables and D The relationship between the link function of the distribution parameters of its GAMLSS model and the climate index CI is established, namely: , ； The distribution parameters of the GAMLSS model include the mean μ and standard deviation σ, for hydrological and drought characteristic variables. and The link function of the distribution parameters of its GAMLSS model is used to establish a relationship with the climate index CI and the reservoir index MRI, namely: , ； The marginal distribution parameters obtained through the above method are no longer fixed values, but dynamically change with the external covariates CI and MRI, thus accurately depicting the time-varying pattern of drought characteristics under changing environments. S5.

2. Using a time-varying Copula function, the time-varying marginal distributions of the meteorological drought characteristic variables and the hydrological drought characteristic variables obtained in step S5.1 are coupled to construct a time-varying joint distribution function for the two: ； ； The parameters of the time-varying Copula function It is also related to covariates such as climate index and reservoir index, i.e. This allows us to capture the dynamic evolution of the dependency structure between meteorological drought and hydrological drought; S5.

3. Based on the constructed time-varying joint distribution, conditional probability is used as the criterion for determining whether meteorological drought triggers hydrological drought; assuming X and Y are two random variables, the conditional probability of Y ≥ v given X ≥ u can be expressed as: In the formula, x and y are the marginal distributions of random variables X and Y, respectively; x(u) is the cumulative probability when X ≤ u, y(v) is the cumulative probability when Y ≤ v; and C(x(u), y(v)) is the joint probability when X ≤ u and Y ≤ v.

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